Sanet - ST 3642642640

Hydraulic Parameter Identification
Generalized Interpretation Method for Single and Multiple Pumping Tests

Springer
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Luc C. Lebbe
Hydraulic Parameter
Identification
Generalized Interpretation Method
for Single and Multiple Pumping Tests
With 112 Figures and 34 Tables
, Springer
Author
Prof. Dr. Luc C. Lebbe
Research Associate
Fund ofSdentiflc Research-Flanders
University of Gent
Geological Institute
Krijgslaan 2.81 (58)
B'9000 Gent
Belgium
B·rnait: Luc.Lebbe@rug.ac.be
ISBN-13: 978'3.642.64264-7 Springer. Verlag Berlin Heidelberg New York

Library of Congress Cataloging-in-Publication Data
Lebbe. Lue. 19~0-
Hydraulic parameter identification: generalited interpretation method for single
and multiple pumping tests I Luc Lebbe.
p. em. Includes bibliograpbical references.
ISBN'13: 9'78-3-642-64264"7 e-ISBN"31 978-3-642-60"7-0
001 : ,0,' 007/978-3-642-60117-<)
I. Groundwater tlow-- Mathem. tical moods. 1. Hydrog«llogy-- M" hematical models. I. Ti de.
TC176.L43 1999
55'·49'OI·~1111--dc21 99-191140
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C Springer.Veriag Berlin Heidelberg 1999

Sofl<:over reprint of th e hardcove r lst edition 1999
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To Bernadette,
Joris, Komeel and Adriaan
Preface
This book is intended for all hydrogeologists or geohydrologists, environmental and

mining engineers who want to tackle a hydrogeological problem in general. The identifi-
cation of the hydraulic parameters is a crucial step in any hydrogeological study and is the
fIrst aim of the book. This identifIcation process is demonstrated by using a unique and
generalized interpretation method for single and multiple pumping tests. The identified
parameters are the water conducting and water storing parameters of the groundwater
reservoir. Therefore, the work starts with an overview of the different ways to define and
determine these hydraulic parameters. The related parameters that are frequently used in
this work are also described. This part is followed by a review of the successive "classi-
cal" analytical models that were developed to derive the frequently applied interpretation
methods. The most important drawbacks of these methods are their large variety, the
fragmented analyses of the observed drawdowns and the different derived values for one
identifIed hydraulic parameter. In this book a unique and generalized interpretation
method is presented which overcomes all these drawbacks.
This generalized interpretation method is obtained by a combination of a numerical
method and a non-linear analysis. The numerical model is an axi-symmetric two-dimen-
sional model. This method is preferred because it requires much less computer time than
the three-dimensional model. This is an important characteristic because during sensitivity
analyses and the non-linear regression a large number of runs are made. The applied
numerical model also allows the drawdown simulation with similar accuracies at distances
close to the pumped well, in the order of centimeters, and at larger distances from the
pumped well, in the order of meters to hectometers. By the non-linear regression method
one set of optimal values for the hydraulic parameters is obtained along with the joint
confidence region and a set of corresponding residuals, i.e. the differences between the
logarithms of the observed and the calculated drawdowns. If the spatial and the temporal
trends of the residuals are suffIciently small the applied schematization of the flow can be
accepted and the optimal values of the parameters are then representative for the actual
physical parameters. The joint confIdence region gives us an insight of the reliability with
which the hydraulic parameters can be derived from the observed drawdowns. This region
and a number of statistical parameters that characterize this region allow the evaluation of
the degree of mutual dependency of the derived parameters or in other words to make a
collinear diagnostic. If the interdependency of a number of the required parameters is too
high then the generalized interpretation method can help to design an additional test. This
could result in the performance of a double or even a multiple pumping test. The general-
ized interpretation method also allows the interpretation of a multiple pumping test.
The applicability of the axi-symmetric numerical model is augmented by the
possibility to simulate the drawdown in a laterally anisotropic aquifer. In addition the
drawdown in a pumped well can be calculated as the sum of the well loss and the aquifer
loss. With a supplementary computer program the drawdown due to a mUltiple well fIeld
can be calculated in a multilayered isotropic or anisotropic groundwater reservoir without
or with lateral constant hydraulic head and/or impervious boundaries, or with a vertical
plane of discontinuous lateral conductivity change. In all these cases the subsidence due to
VIII Preface
pumping can also be estimated assuming that the changes of the specific elastic storage,
the vertical and horizontal conductivity are constant in each layer during the compaction
process and are independent of the drawdown. The latter possibility allows the planning
of the development and the management of the groundwater resources.
This book can, however, also be considered as an introduction to inverse modeling
and its continuous interaction with the consecutive steps in hydrogeological investigations.
The design, the performance and the interpretation of a pumping test can be regarded as
an example of these consecutive steps. Pumping tests are simple and well-defined
hydrogeological studies that can be performed under favourable conditions. They are
simple because in most cases their lateral influence is rather limited so that the lateral
heterogeneity can be ignored. In hydrogeological studies on a larger scale this is not the
case so that the number of parameters can grow to an unacceptable large number. In the
pumping tests, the discharge rates are generally accurately measured. In hydrogeological
investigations on a larger scale and over a longer period the discharge rates of the
pumping wells and of the rivers are less well known. The installation of the pumping and
observation wells for a pumping test allows the collection of detailed information on the
lateral and layered heterogeneity of the groundwater reservoir which can be used during
the design of the test.
The effect of the layered heterogeneity on the water conducting properties of the
groundwater reservoir can be studied by the execution of multiple pumping tests. By the
design of these pumping tests one can create and observe large horizontal gradients in the
pervious layers to, deduce their horizontal conductivities and large vertical gradient over
semi-pervious layers to deduce their vertical conductivities. In hydrogeological studies on
the other hand, it is much more difficult to observe such large gradients. In most cases
the hydraulic gradients are smaller in regional groundwater flow problems. Here, the
location of the large gradients can be seasonally dependent.
It is easier to remedy some measurement errors during pumping tests then during
most of the hydrogeological studies on a larger scale. For example, the well storage can
be eliminated by the installation of packers above the pressure transducers. This is
necessary in wells with a rather large diameter, installed in rocks with a small permeabil-
ity with a relatively fast growing drawdown. So, the interpretation of the pumping test is
an inverse problem of the first (simplest) type where the hydraulic parameters are the
only unknowns. Experiences with this simplest type of inverse modeling is preferred to
start with. These experiences can help to tackle more complicated inverse problems as for
example hydrogeological investigations on a large scale where sink and/or source term,
initial and/or boundary conditions must be identified besides a rather high number of
physical parameters.
Until now several FORTRAN programs are available on the world wide web.
They allow: the simulation of pumping tests without or with an artificial created error
(forward problem), the optimisation of a given number of hydraulic parameters (inverse
problem of first type), the information of the required input data and the creation of input
files needed in the forward and the inverse problem and the drawing of graphs with
results of the forward or the inverse problem. These programs are described in the book.
Although sufficient examples of simulations are given in the book, these programs allow
the reader to develop his/her own exercises, problems and pumping tests.
Preface IX
Acknowledgments
I would like to thank the Fund of Scientific Research - Flanders, under the auspices of
which the scientific work and the development of the computer programs were carried
out. I also 'wish to thank Prof. Dr. W. De Breuck for the opportunities he gave me to
design, to perform and to interpret a considerable amount of pumping tests at the
Laboratory of Applied Geology and Hydrogeology, Ghent University. Some of these
pumping tests are selected and represented in the last chapter. The double pumping test in
the quaternary sediments was performed in the framework of a hydrogeological study
which was financed by the Ministry of the Flemish Community and the Water Company
(T.M.V.W). The double pumping test in a laterally anisotropic aquifer formed by
fractured rocks of Palaeozoic and Mesozoic age (Sect. 7.2) was made with the financial
support of the Belgian Geological Survey. The triple pumping test in the layered ground-
water reservoir (Sect. 7.3) on one of the campuses of Ghent University was made as part
of my research program. The single pumping test to determine the conductivity of the
tertiary, silty clay (Sect. 7.4) was made on order of SIDMAR, N.V. Finally, the artificial
recharge test in the natural bare dune valley (Sect. 7.5) was performed in collaboration
with the Intermunicipal Water Company of the Veurne Region (I.W.V.A.). I am greatly
indebted to all these organizations and companies.
I also thank the colleagues and students of the Laboratory of Applied Geology and
Hydrogeology, with whom I had numerous enlightened discussion about the design, the
performance and the interpretations of the pumping tests. The help of Ing. Eric Beeuw-
saert and Ing. Jacques Vandenheede is also acknowledged for their work in performing
the pumping tests, for the development of a data logger and the numerous discussions that
lead to the installation of packers above the pressure transducers to eliminate the well
bore storage. I am also indebted to Freddy De Leeuw, Marleen Lacroix and Martine
Bogaert for the meticulous drawing of some of the figures. I thank Marisa Boesman who
helped me overcome my computer problems. Last, but not least I thank Drs. M.
Mahauden, Drs. Y. Vermoortel, Dr. I Gaus and Drs. N. Van Meir for the discussions
and correction of some parts of the work. Any comments anyone would care to make will
be received with great interest.
Computer codes
The executable codes are available on http://allserv.rug.ac.be/-luclebbe. The FORTRAN

codes were compiled by Lahey Fortran 90 v4.5 (Lahey Computer Syst., Inc). With the
exception of inrmp code the other codes can be subdivided in three groups. The inrmp
code allows to make the input files interactively for the forward and the inverse problems.
The first group of codes treat the forward problems. They allow the calculation and the
representation of the drawdown or the subsidence due to pumping on a single well or due
to pumping on a multiple well field: This group consists of the codes sipurS, outpuS,
sidap7 and multpl. They are described in Sect. 4.7 and further developments are given
in Chapter 5. The code sipurS simulates the drawdown of a pumping tests and compares
the calculated with the observed drawdown. The code sidap7 allows to make observed
drawdown files of hypothetical pumping test of which the drawdowns are simulated and
x Preface
whether or not increased with an artificial error. The code outpu5 plots the simulated
drawdown in time-drawdown and distance-drawdown graphs either with or without the
observed data. This last code allows also to plot the drawdown contour lines in a cross
section. With the code outpu5 the evolution of the subsidence can also be plotted in time-
subsidence and distance-subsidence graphs and by means of subsidence contour lines in
cross sections (see Sect. 5.7). Additional input data to simulate the drawdown for
pumping test with variable discharge rates, in laterally anisotropic aquifers and in
pumping wells are described in respectively the Sects. 5.1, 5.2 and 5.3. With an
additional code multpl the drawdowns can be calculated due to a multiple well field. The
results can be plotted by the code mulpl2. With this code it is also possible to simulate
the drawdown in groundwater reservoirs with lateral bounds and with lateral discontinu-
ous conductivity change. These possibilities are given in Sect. 5.6 and 5.7.
The second group of codes allows to run and control the inverse numerical model
for the interpretation of single and multiple pumping tests. These codes are described in
Sect. 6.6. The code inpur5 allows to run the successive iterations of the inverse model
for the interpretation of a pumping test. The code solpu5 is mostly used for the step by
step interpretation of a multiple pumping test. Based on residuals and sensitivities of
several pumping tests, which are joined in one data file, the adjustments of the considered
hydraulic parameters are calculated. The code etabdi plots the residuals versus their
probability along with the best fitted normal distribution. The codes plprcr, susqln,
susql3, sumsqr and sumsq2 plot two- or three-dimensional cross-sections through the
approximate or exact joint confidence region. The last group of codes allows the approxi-
mation of the drawdown confidence intervals based on the optimal values and some
statistical parameters characterizing the joint confidence region. Four different computer
program packages are described in Sect. 6.7.3.
The author has written, researched and tested the source codes to ensure their
accuracy and effectiveness. Neither the author nor the publisher give any warranty of any
kind, expressed or implied, with regard to the performance of the codes. No warranties,
expressed or implied, are given by the author or the publisher that the codes are free of
errors, or are consistent with any particular standard of merchantability, or that they meet the
reader's requirements for any particular application. In no event shall the author or the
publisher be liable for incidental or consequential damages in connection with or use of the
codes. Support for the codes on the Internet is available in this book. All graphs are in HP-
GU2 codes (trademark of Hewlett-Packard Company).
Although this book and its programs are copyrighted, the reader is authorized to make
one machine-readable copy of each program for personal use. Distribution of the machine-
readable programs (either as copied by the reader or as available on the Internet) is not
authorized. Programs will only give meaningful results for reasonable problems. The
programs represent a basic and general model that can be modified for efficient application to
specific field problems. The user is cautioned that in some cases the accuracy and the
efficiency of the models can be affected significantly by the selection of values for certain
user-defined options, for example, the use of a large initial time and a reduced number of
rings per layer as demonstrated in Sect. 4.6.1. Another example is a too simplified
conceptualization of the flow as shown in Sect. 6.5.
Contents
Chapter 1 / Introduction .................................................................................... .

1.1 Previous literature on pumping test interpretation ............................................. .
1.2 Proposed generalized interpretation method ..................................................... 2
1. 3 Additional aims of the book ........................................................................ 5
1.4 Arrangement of subject matter............................... ...... ............................ .... 6
Chapter 2 / Hydraulic Parameters .............. ......................................................... 9

2.1 Hydraulic parameters describing water conducting properties ................................ 9
2.1.1 Darcy's law .............................................................................. 9
2.1.2 Hydraulic head and fresh water head................................................ 10
2.1.3 Intrinsic permeability................... ................................................ 12
2. 1.4 Heterogeneity and anisotropy of geological formations
with respect to hydraulic conductivity ...................................................... 13
2.1.4.1 Heterogeneity ............................................................... 13
2.1.4.2 Anisotropy....... ............................................................ 14
2.1.5 Generalized law of Darcy..................................................... ......... 19
2.1.6 Hydraulic conductivity ellipsoid...................................................... 21
2.1.7 Classification of layers according to their water conductive properties ....... 24
2.1.8 Hydraulic parameters derived from hydraulic conductivity ...................... 25
2.1.8.1 Transmissivity .............................................................. 25
2.1.8.2 Hydraulic resistance ....................... ................................ 26
2.1.8.3 Leakage factor.......... .................................................... 26
2.1. 9 Methods to derive the hydraulic conductivity ..................................... 27
2.1.9.1 Direct methods .................... ......................................... 27
2.1. 9.2 Indirect methods ............................................................ 30
2.2 Hydraulic parameters describing water storing properties ............... ..................... 36
2.2.1 Conservation of mass in a completely saturated volume of material
fixed in space ................................................................................... 36
2.2.2 Specific elastic storage................................................................. 39
2.2.2.1 Compressibility of water......................... ......................... 40
2.2.2.2 Effective stress concept .............................................. ..... 41
2.2.2.3 Compressibility of matrix ................................................. 42
2.2.2.4 Movement of solids in deforming medium............................ 43
2.2.3 Hydraulic parameters derived from specific elastic storage... .................. 44
2.2.3.1 Elastic storage coefficient... ............................................. 44
2.2.3.2 Diffusivity ................................................................... 45
2.2.4 Methods to derive specific elastic storage................................ .......... 45
2.2.4.1 Direct methods. ............................................................ 45
2.2.4.2 Indirect methods............................................................ 46
2.2.5 Storage coefficient near the water table... .............. ............................ 46
2.2.6 Hydraulic parameters derived from storage coefficient near watertable ....... 49
2.2.7 Methods to derive storage coefficient near water table ........................... 49
2.2.7.1 Determination by pF-curves .............................................. 49
2.2.7.2 Determination by pumping tests ......................................... 50
2.2.7.3 Determination by inverse models of unsteady state flow............ 51
XII Contents
Chapter 3 / Evolution of analytical models of pumping tests

and their interpretation methods ......................................................................... 55
3.1 Model of Thiem ...................................................................................... 55
3.1.1 Introduction.............................................................................. 55
3.1.2 Derivation of basic differential equation ......... ................. ......... ......... 57
3.1. 3 Solution of basic differential equation............................................... 58
3.1.4 Application of Thiem method .................... ............... ................... ... 59
3.2 Model of Theis ....................................................................................... 59
3.2.1 Introduction. .......... .......... ............ ........ ............... ...................... 59
3.2.2 Derivation of basic differential equation ......... .............. ............... ...... 60
3.2.3 Solution of basic differential equation for constant discharge rate ............ 62
3.2.4 Theis and Cooper-Jacob interpretation methods ... ................. ............... 65
3.2.5 Comments concerning Theis and Cooper-Jacob interpretation methods ....... 67
3.3 Model of Jacob-Hantush .......... ........... .......... .......... ............... ................... 67
3.3.1 Introduction .............................................................................. 68
3.3.2 Derivation of basic differential equation ....... ........ .... ......................... 69
3.3.3 Solution of basic differential equation for a constant discharge rate ........... 71
3.3.4 Interpretation methods of De Glee, Hantush-Jacob,
Hantush I-II and Walton....................................................................... 75
3.3.4.1 Interpretation of distance-drawdown curve at steady flow .......... 75
3.3.4.2 Interpretation of time-drawdown curve ................................ 78
3.3.4.3 Comments concerning interpretation methods of
semi -confined aquifer ................................................................ 82
3.4 Model of Hantush ................. ............ ......... ......................... .................... 83
3.4.1 Introduction. ................. ............ ..................... .................. ........ 83
3.4.2 Derivation of basic differential equations. ......... ........................ ......... 83
3.4.3 Initial and boundary conditions .... ........... ......... .............. .......... ....... 83
3.4.4 Solution of basic differential equation............................................... 90
3.4.5 Interpretation methods derived from Hantush model............................. 91
3.4.5.1 Interpretation of first part of time-drawdown curves. ... ............ 91
3.4.5.2 Interpretation of last part of time-drawdown curves ................. 92
3.4.5.3 Interpretation of maximum drawdowns of cases 1 and 3 .. ......... 93
3.4.6 Concluding considerations about interpretation methods derived from
Hantush model .................................................................................. 94
3.5 Model of Hantush-Weeks ............ ............ ........ .......... ............... ... ............... 95
3.6 Model of Boulton-Cooley ........................................................................... 99
3.7 Model of Neuman and Witherspoon .............................................................. 104
3.8 Retrospective view on analytical models and their derived interpretation methods........ 110
Chapter 4 / Numerical model of pumping tests in a layered groundwater reservoir 117
4.1 Finite-difference grid ................................................................................ 117
4.2 Mean drawdowns ..................................................................................... 119
4.2.1 Mean drawdown over horizontal plane through nodal circle in ring ........... 119
4.2.2 Mean drawdown over cylindrical surface through nodal circle in ring...... ... 122
4.2.3 Mean drawdown over entire ring volume ........................................... 124
4.2.4 Mean drawdown during a time step.................................................. 124
Contents XIII
4.3 Continuity equation in numerical model.. ........... ........... ...................... ............ 126
4.3.1 Mean velocities through boundary surfaces of rings ............ ..... ............. 126
4.3.2 Storage change in rings ................................................................ 126
4.3.3 In- and outflow difference of rings. ........... ............. ................. ......... 127
4.3.4 Continuity equation for rings ...... ......... ............... ............. ............... 127
4.3.5 Storage change for rings bounded by water table ................................. 128
4.3.6 In- and outflow difference of rings bounded by water table.............. ....... 128
4.3.7 Continuity equation for rings bounded by water table ......................... ... 129
4.4 Initial and boundary conditions ........ ............ ........... ................ ..................... 129
4.5 Solution of the numerical equations............................................................... 131
4.5.1 Alternating direction implicit method ................................................ 131
4.5.2 Verification of iteration process and number of iteration per time step....... 133
4.6 Verification of numerical model........ ......... .... .......... .................. .................. 133
4.6.1 Verification of numerical model with Theis model.... ........................... 134
4.6.2 Influence of grid parameters on results of numerical Theis model.. ........... 135
4.6.3 Verification of numerical model with Jacob-Hantush model........... ......... 141
4.6.4 Influence of grid parameters on resnlts of Jacob-Hantush model............... 145
4.6.5 Verification of numerical model with Hantush model............................ 145
4.6.6 Examination of validity limits of the Hantush asymptotic solution ............. 150
4.6.7 Verification of numerical model with Hantush-Weeks model................... 154
4.6.8 Verification of numerical model with Boulton-Cooley model................... 157
4.6.9 Consequences of numerical model verification .................................... 161
4.7 Program package for numerical simulation of pumping tests. ................................ 162
4.7.1 Program infinp .......................................................................... 162
4.7.1.1 Input space-time grid parameters and hydraulic parameters ........ 163
4.7.1.2 Input of observed drawdowns ............................................ 163
4.7.3 Program sipur5 .......................................................................... 170
4.7.4 Program outpuS ......................................................................... 172
4.7.5 Program sidap7 ......................................................................... 173
Chapter 5 / Further developments of pumping test model .................................................. 175
5.1 Drawdown of pumping tests with variable discharge rate..................................... 175
5.1. 1 Theoretical considerations ............................................................. 175
5.1.2 Additional input data .................................................................... 178
5.1.3 Example of a pumping test with variable discharge rate ......................... 178
5.2 Drawdown in a laterally anisotropic aquifer ..................................................... 184
5.2.1 Theoretical considerations.... .......... .......... ................ ..................... 185
5.2.2 Additional input data ...... ............ ........ ............. ........................ ..... 187
5.2.3 Example of a pumping test in a laterally anisotropic aquifer .................... 187
5.3 Drawdown in pumping wells ....................................................................... 190
5.3.1 Theoretical considerations ............................................................. 190
5.3.2 Additional input data ..... .......... .......... ............. ....................... ....... 192
5.3.3 Example of drawdowns in a pumped well
during a step drawdown test .................................................................. 192
5.4 Drawdown due to a multiple well field ........................................................... 194
5.4.2 Additional input data .................................................................... 195
5.4.3 Drawdown due to pumping on wells in laterally isotropic layers ............... 197
5.4.4 Drawdown due to pumping on wells in laterally anisotropic layers ............ 198
XIV Contents
5.5 Drawdown in groundwater reservoir with lateral bounds ..................................... 203

5.5.1 Drawdown in groundwater reservoir
with lateral impervious boundary ........................................................... 203
with lateral constant head boundary ........................................................ 205
bounded by several straight boundaries .................................................... 207
5.6 Drawdown in groundwater reservoir with
lateral discontinuous conductivity change............................................................ 211
5.6.2 Additional input data ........ ............. .......................... .................... 213
5.6.3 Example of drawdown approximation ............................................... 213
5.7 Land subsidence due to groundwater withdrawal ............................................... 218
5.7.2 Additional input data and representation of results................................ 219
5.7.3 Example of subsidence calculations .................................................. 219
Chapter 6 I Inverse model as tool for pumping test interpretation ....................... 229
6.1 Residual vector ...................................................................................... 230
6.1.1 Definition................................................................................. 230
6.1.2 Contributing factors of residuals...................................................... 231
6.2 Sensitivity matrix..................................................................................... 232
6.2.1 Definition................................................ ................................. 232
6.2.2 Program package to calculate sensitivity matrix ....... ................ ..... ....... 233
6.2.3 Example of a sensitivity matrix................................ ....................... 236
6.2.4 Graphical representation of sensitivities............................................. 241
6.3 Numerical nonlinear regression.................................................................... 245
6.3. I Optimal values of hydraulic parameters ............................................. 245
6.3.2 Uniqueness, identifiability and stability....... ............. ......................... 246
6.3.3 Analysis of residuals ...... .............. ........ ............... ................... ...... 248
6.3.4 Joint confidence region of hydraulic parameters ............. ...................... 249
6.3.4.1 Joint confidence region approximated by
p-dimensional ellipsoid............................................................. 249
6.3.4.2 Cross sections through joint confidence region... .................... 253
6.3.5 Condition indexes and matrix of
marginal variance-decomposition proportions......... . ............ . . .... ................. 255
6.3.6 Practical steps in interpretation by means of inverse model... ....... ........... 256
6.4 Validation of inverse numerical model ........................................................... 257
6.5 Factors influencing accuracy of results ......... ......... .............. ........................... 261
6.5.1 Influence of flow conceptualization .... .......... .................. .................. 261
6.5.2 Influence of observed drawdown accuracy and discharge magnitude .......... 264
6.5.3 Influence of observation time ........... ........... ......... ...... .................... 264
6.5.4 Influence of observation distance..................................................... 264
6.5.5 Conclusions .............................................................................. 266
6.6 Program package for the nonlinear regression .................................................. 267
6.6.1 Program solpuS ............ ........... .......... .............. ....................... ... 268
6.6.2 Program inpurS .............. .......... ........... ............... ....................... 269
6.6.3 Program etabdi .......................................................................... 270
6.6.4 Program plprcr .. .............. ..................... .......... ........ ....... ............ 272
6.6.5 Program susqln ............... .......... .......... .............. ..................... .... 273
6.6.6 Program susq13 ........... ............ ......... ............ ....................... ....... 274
6.6.7 Program sumsqr ........................................................................ 276
6.6.8 Program sumsq2 ..................... ............ ............. .......................... 276
Contents XV
6.7 Confidence interval for optimal estimated drawdown .......................................... 277

6.7.1 Drawdown confidence intervals based on two- and three-dimensional cross
sections through joint confidence interval of hydraulic parameters .................... 278
6.7.2 Drawdown confidence intervals derived by optimization
of constrained problem ........................................................................ 279
6.7.3 Program packages to approximate drawdown confidence intervals ............. 281
6.7.3.1 Program package contil .................................................. 282
6.7.3.2 Program package contill ................................................ 282
6.7.3.3 Program package conf"t3 .................................................. 283
6.7.3.4 Program package conti4 .................................................. 284
6.8 Hypothetical example to demonstrate noulinear regression and approximation of
drawdown confidence intervals ......................................................................... 284
6.8.1 Conceptual model of 'actual' groundwater flow................................... 284
6.8.2 Creation of 'observed' drawdowns ................................................... 286
6.8.3 Conceptual model of groundwater flow used during interpretation ............. 286
6.8.4 Interpretation by means of ordinary least square method ........................ 286
6.8.5 Joint confidence area of hydraulic parameters .......... ........................... 290
6.8.6 Combined influence of hydraulic parameters
at some observation points..... . . .. ........ ........ .. ......... . ... . ................ . . . .. .. .. .. 293
6.8.7 Approximation of drawdown confidence intervals ................ ................ 297
Chapter 7 / Example of performance and interpretation of pumping tests........... 303

7.1 Double pumping test in layered groundwater reservoir formed by
Quaternary sediments .................................................................................... 303
7.1. 1 Lithostratigraphical cross section..................................................... 305
7.1.2 Location of pumping and observation wells........................................ 305
7.1. 3 Performance of double pumping test ................................................ 305
7.1.4 Discretization of groundwater reservoir in numerical model.................... 306
7.1.5 Hydraulic parameters derived from observed drawdown ........................ 306
7.1. 6 Interpretation with ordinary least square method .................................. 306
7.2 Double pumping test in a laterally anisotropic aquifer formed by fractured rocks
of Palaeozoic and Mesozoic age ........................................................................ 310
7.2.1 Lithostratigraphical cross section..................................................... 310
7.2.2 Location of pumping and observation wells ........................................ 312
7.2.3 Discretization of groundwater reservoir in numerical model.................... 312
7.2.4 Drawdowns used as input data ........................................................ 312
7.2.5 Hydraulic parameters derived from observed drawdowns ....................... 312
7.2.6 First interpretation results ............................................................. 314
7.2.7 Two-dimensional cross sections through exact joint confidence region........ 316
7.2.8 Three-dimensional cross sections through
approximate joint confidence region ........................................................ 318
7.2.9 Condition indexes and matrix of marginal variance-decomposition ............ 319
7.2.10 Observational constraints for unique solution..................................... 322
7.2.11 Second interpretation phase and post-optimization............................... 324
7.2.12 Conclusion .............................................................................. 326
XVI Contents
7.3 Triple pumping test in layered groundwater reservoir formed by

Tertiary sediments ......................................................................................... 327
7.3.1 Lithostratigraphical cross section ..................................................... 327
7.3.2 Location of pumping and observation wells ........................................ 329
7.3.3 Performance of triple pumping test .................................................. 329
7.304 Discretization of groundwater reservoir in numerical model .................... 329
7.3.5 Hydraulic parameters derived from observed drawdowns ....................... 330
7.3.6 Interpretation of results ................................................................ 331
7.3.7 Outliers ............ .................................... .................... ............... 334
7.3.8 Conclusions .............................................................................. 337
7.4 Single pumping test to determine the conductivity of Tertiary silty clay................... 338
704.1 Lithostratigraphical cross-section..................................................... 338
704.2 Location of pumping and observation wells ........................................ 339
704.3 Performance of single pumping test ................................................. 339
7 A A Discretization of groundwater reservoir in numerical model .................... 341
704.5 Hydraulic parameters derived from observed drawdown ........................ 341
704.6 Interpretation results .................................................................... 343
7.5 Artificial recharge test in a natural bare dune valley ........................................... 348
7.5. 1 Lithological cross section and discretization
of groundwater reservoir ..................................................................... 348
7.5.2 Location of observation wells and
performance of artificial recharge test ...................................................... 348
7.5.3 Hydraulic parameters derived from observed rises of hydraulic head ......... 349
Index ................................................................................................................. 355
Chapter 1 / Introduction
Nowadays, it is becoming common practice to apply mathematical models to simulate
groundwater flow, solute and heat transport and subsidence. The present-day problem is
not gaining access to computers or obtaining computer codes but the acquisition of
numerical values for the different hydraulic parameters to be used in these mathematical
models. Other problems on the order of the day are: how to calibrate the mathematical
model; what is the reliability of the results of a calibrated mathematical model; and which
data should be gathered to have a sufficiently precise simulation of the problem under
consideration? It is one of the additional aims of this book to contribute to the solution of
all these problems.
The principal goal of this work is the presentation of one generalized interpreta-
tion method for a wide range of pumping test designs. In this way all possible instances
of single or multiple pumping tests in groundwater reservoir with an arbitrary alternation
of pervious and semi-pervious layers can be dealt with. The layers can be transversely
and/or laterally anisotropic and the pumping wells are fully or partially penetrating. The
interpretation method not only results in the optimal values of hydraulic parameters but
also in their joint confidence region. This statistical region in the parameter space gives
us an idea of the accuracy or the reliability with which the hydraulic parameters have
been deduced. Also, observations made in the pumped well can be included in the
interpretation process. In the last case parameters that define the well loss can be
considered along with the hydraulic parameters of the groundwater reservoir. Although
this interpretation method has primarily been developed for pumping tests, injection and
artifical recharge tests can also be interpreted with it. By means of a supplementary
computer program the drawdowns and/or the subsidence can be calculated due to
pumping on a single well or on several wells simultaneously, in laterally isotropic or
anisotropic, continuous or discontinuous, multilayered groundwater reservoirs.
The present work has several additional aims. The first is the presentation of
nonlinear regression as a calibration method and the elucidation of characteristics of the
inverse problem such as: uniqueness versus non-uniqueness, identifiability versus
unidentifiability and stability versus instability. The second additional aim is the presenta-
tion of a method to obtain information about the reliability of the results of the calibrated
model. The third additional aim is the design of a pumping test that enables the deduction
of the most important hydraulic parameters for the study of a certain problem with the
greatest possible precision.
1.1 Previous literature on pumping test interpretation
The interpretation of pumping tests has already been extensively treated by Walton (1970,
1987), Kruseman and De Ridder (1970, 1990) and Reed (1980). In Chapter 4 of his book
'Ground water resources evaluation' Walton (1970) treats the interpretation of different
2 Chapter 1 / Introduction
aquifers with fully and sometimes partially penetrating wells and a constant discharge
rate. Kruseman and De Ridder (1970) describe the formulas and methods to evaluate
pumping tests data in single and laterally extended aquifers. They conclude their work
with formulas and methods to evaluate data of pumping tests made under special
conditions. In the second edition (Kruseman and De Ridder, 1990) their objectives remain
the same. They complete the formulas and methods, more particularly these for the
pumping test data obtained under special conditions. In both books, the interpretation
methods are deduced from analytical models and analyses requiring computers are
excluded. Reed (1980) presents type curves and related material for eleven conditions of
flow with regard to wells in confined aquifers. Solutions are presented for constant
discharge, constant drawdown and variable discharge of fully or partially penetrating
pumping wells in leaky and nonleaky aquifers. Each problem includes the partial
differential equation, the boundary and initial conditions and the solution(s). For most of
the solutions FORTRAN computer algorithms are used to calculate the different type
curves. Walton (1987) presents the nonlinear regression method as a tool to match the
different type curves related with the analytical models which treat single, horizontal,
extended aquifers (Kruseman and De Ridder, 1970, 1990 and Reed, 1980).
All these 'classical' interpretation methods are based on more or less simple
analytical models. Different interpretation methods can be derived from one solution of an
analytical model. In each method the interpreted drawdowns must fulfil different
additional constraints. These constraints mostly result in fragmented analyses of the
observed drawdowns, e.g. the match of time-drawdown curves of an observation well
with one or a series of type curves. The fragmented analyses of the data by an interpreta-
tion method that is derived from a model where the assumed flow differs from the actual
flow, results in a set of different values for each parameter. These different values are
traditionally ascribed to the lateral heterogeneity of the investigated layer (e.g. p. 71 of
Kruseman and De Ridder (1990». Frequently, deduced values of the hydraulic conducti-
vity increase according to the augmenting observation distance of the interpreted time-
drawdown curve. As shown in the present work (see Sect. 6.5), this result is mostly due
to an underestimation of the leakage from layers adjacent to the pumped layer or as a
consequence of ignoring the storage in these adjacent layers.
1.2 Proposed generalized interpretation method
In the present work, which is a sequel to Lebbe (1988), a two-dimensional axi-symmetric

numerical model coupled to a nonlinear regression analysis is proposed as a generalized
interpretation method for single as well as multiple pumping tests. These tests can be
made in a wide range of horizontally or quasi-horizontally stratified groundwater
reservoirs which can be transversely and/or laterally anisotropic. The results of the
inverse model are the optimal values of the hydraulic parameters along with the joint
confidence region in the parameter space. With these results it becomes possible to
calculate the drawdowns with their confidence intervals.
Chapter 1 / Introduction 3
The proposed method differs in several aspects from existing interpretation
methods. First, it is a generalized interpretation method which is based on a single
numerical model and not on different analytical models. Second, the drawdowns and the
parameters are both considered in their logarithmic spaces. Third, the proposed interpre-
tation method not only allows the interpretation of the drawdowns in the pumped layer but
also of drawdowns observed in the adjacent layers and in the pumped well. Finally, the
method allows the simultaneous interpretation not only of single pumping tests but also of
multiple pumping tests. In the following paragraphs the reason for the introduction of
these differences will be explained briefly.
The coupling of a single numerical model to a nonlinear regression analysis has
different advantages with respect to the coupling of different analytical solutions to the
nonlinear regression as proposed by Walton (1987). The different analytical solutions are
derived from a rather simple schematization of the groundwater reservoir, e.g. a confined
aquifer, a semi-confined aquifer or a phreatic aquifer. In addition, the conceptualization
of the flow in some layers can also be simplified, e.g. only vertical inelastic flow in the
adjacent semi-pervious layers. As the result of the application of the numerical model
such simplifying assumptions are no longer necessary. Using the numerical model it is
possible to simulate the drawdown at any level in groundwater reservoirs with an
arbitrary alternation of pervious and semi-pervious layers. During these simulations the
horizontal as well as the vertical flow and the storage decrease are considered in each
layer. Consequently, fully or partially penetrating pumping wells can also be simulated.
In the nonlinear regression the observed and calculated drawdowns are compared
in their logarithmic spaces. Consequently, it becomes possible to consider drawdowns of
different orders of magnitude simultaneously. This allows the unfragmented interpretation
of a pumping test. In this way, it becomes possible to interpret rather small drawdowns,
e.g. as observed in the pumped layer just after the start of a pumping test or in the
adjacent semi-pervious layer after a longer time of pumping, simultaneously with rather
large drawdowns, e.g. as observed in the pumped layer after a longer time of pumping at
a rather small distance from the pumped well or in the pumped well itself.
In the generalized interpretation method, an axi-symmetric two-dimensional model
(AS2D model) is preferred to a three-dimensional model. The first reason for this
preference is that the axi-symmetric model requires much less computer time than the
three-dimensional model. The second reason is that it allows the calculation of the draw-
downs at distances from the pumped well which differ largely. In the axi-symmetric
model the layers are subdivided in rings of which the inner and outer radii form a
logarithmic series. This allows the calculation of the drawdowns with an equal order of
accuracy at distances from the pumped well which are in the order of centimeters,
decimeters, meters, decameters and hectometers. In a three-dimensional finite-difference
model the drawdowns in the pumped cells and in their adjacent cells are rough approxi-
mates due to the strongly converging flow near the pumped well. The drawdowns
obtained in this way can hardly be used for comparison with observations.
By means of the proposed model it is not only possible to interpret single pumping
tests but also multiple pumping tests. In a multiple pumping test, all pervious layers of a
groundwater reservoir are pumped separately. During each pumping the drawdown is
measured in a set of observation wells which are placed in the pervious as well as in the
semi-pervious layers. The observation wells in the pervious layers are situated at differing
distances from the pumped well and those in the semi-pervious layers are placed at
relatively small distances from the pumped well on one or several levels. Through the
simultaneous interpretation of all drawdowns observed in the different layers during the
different pumpings, it is possible to obtain a unique solution. By interpreting these tests,
it is possible to obtain the horizontal conductivity values of the pervious layers and the
vertical conductivity values of the semi-pervious layers with an accuracy that is compara-
bly high. It is possible to deduce the specific elastic storages of the pervious and the
semi-pervious layers and the specific yield with a lower but sufficient accuracy. Using
this unique set of hydraulic parameter values a good agreement between the observed and
the calculated drawdowns is obtained.
The use of the AS2D numerical model allows the introduction of information
collected during the drilling of the pumping and observation wells. During the drilling of
the first test hole, a careful record, or log, is kept of the various geological formations
and the depths at which they are encountered. Samples can be collected for further study
and analysis of the grain size distribution. By means of geophysical borehole measure-
ments a detailed picture of the vertical variation of some geophysical parameters can be
obtained of the different geological formations. This picture can easily be introduced in a
numerical model in which a large number of layers can be considered. By carrying out
geophysical borehole measurements in the different boreholes used for the installation of
pumped and/or observation wells, knowledge is acquired about the lateral extension of
these formations. By means of empirical relations between geophysical and hydraulic
parameters some prior estimates of the magnitude of the hydraulic parameters can be
made.
In most pumping tests the initial and boundary conditions are very simple. The
locations of the different screens of the pumped well(s) are measured with large precision.
The discharge rate of the water withdrawn from the known depth intervals can be
determined very accurately by periodical or continuous measurements. In the case of a
considerable drawdown in a pumped well with a rather large diameter, the drawdown in
the pumped well must be measured so that the water delivered from the pumped well
storage can be determined and subtracted from the pumped water to obtain the discharge
rate delivered by the groundwater reservoir. Also, the boundary conditions at the outer
boundary can be held simple in most pumping tests. Usually the location of a pumping
test is chosen in such a way that the groundwater flow is not influenced by the lateral
boundaries during the pumping test. If lateral boundaries such as constant hydraulic heads
occur, the locations of these boundaries are well-known in most cases.
1.3 Additional aims of the book
The interpretation of a properly planned pumping test can be considered as a rather

simple inverse groundwater flow problem. Therefore, the generalized interpretation
method of a pumping test also allows to demonstrate the application of an inverse model
under favorable conditions. Other inverse problems such as regional groundwater flow
problems are not so simple. In such problems the lateral change of the hydraulic
parameters influences the observations to a larger degree than in most pumping tests. In
the inverse regional groundwater flow problem these hydraulic parameters must be
identified along with their lateral change. In most cases, however, the discharge-recharge
rates and the initial and/or boundary conditions must be simultaneously identified. During
the interpretation of a well-designed and performed pumping test the hydraulic parameters
are the only which should be inferred. In pumping tests the succession of layers in the
groundwater reservoir is well-known and also the evolution of the discharge rate. The
influence of the lateral change of the hydraulic parameters on the induced flow pattern is
rather limited. So, the initial and boundary conditions are well-known in most pumping
tests. In comparison to inverse regional groundwater flow problems, large amounts of
observation data are available during a pumping test interpretation.
This generalized interpretation method is also suited to demonstrate some problems
that can arise during the solution of an inverse problem, i.e. non-uniqueness, unidentifia-
blity and instability. Some of these problems can be detected by a number of collinear
diagnostic statistical tools. Because the number of estimated parameters is rather limited,
the treated problems allow to explain the relations between the different collinear diagnos-
tic tools such as the eigenvalues and eigenvectors of the variance-covariance matrix, the
partial correlation coefficients, the marginal and conditional standard deviations, the
condition indexes and the matrix of marginal variance-decomposition proportions. The
relation between some of these numerical collinear diagnostic tools and the shape of the
joint confidence region in the p-dimensional parameter space can also be shown. The
letter p stands here for the number of estimated parameters. A visual insight into the
location of the boundaries of this region is obtained by the two- and three-dimensional
cross-sections through it. These cross-sections can then be considered as graphical
collinear diagnostic tools. All these tools help us to determine whether the inverse model
is well-posed. An inverse problem is well posed if there is a solution and if the solution is
unique and stable (Carrera and Neuman, 1986).
Once the optimal values of the hydraulic parameters have been deduced along with
their joint confidence area, the drawdown can be calculated according to the optimal
values of the hydraulic parameters. With the aid of the joint confidence region of the
hydraulic parameters, the confidence intervals of the calculated drawdowns can be
approximated. These confidence intervals can be used to evaluate the relative importance
of the model inputs and the anticipated effect of new data and analyses on the reliability
of the model results (Hill, 1989). At the end of a study the confidence intervals of the
drawdown can be used to indicate the likely errors in the simulated results clearly.
This simple inverse groundwater problems also allow us to show how the observa-
tions should be collected in an optimal way for a hydrogeological problem. So, the first
step is the identification of the hydraulic parameters that have a considerable influence on
the results of the studied problem. In the case of a groundwater reservoir which is an
arbitrary alternation of pervious and semi-pervious layers, the knowledge of the layering
is very important along with a good estimation of the horizontal conductivity of the
pervious layers and of the vertical conductivity of the semi-pervious layers. In the case of
rather thick homogeneous layers the knowledge of the horizontal as well as the vertical
conductivity of these layers is important. By means of a sensitivity analysis or by the
application of the inverse model in which the schematization of the groundwater reservoir
and the conceptualization of the flow is _already made, using rough estimates of the
hydraulic parameters, an insight can be obtained into the location of the screens of the
pumped and the observation wells if one wants to determine these important hydraulic
parameters with sufficient accuracy. The inverse model makes the design and the
interpretation of these pumping tests possible.
1.4 Arrangement of subject matter
In Chapter 2 two groups of hydraulic parameters are considered which are important
within the framework of pumping test analyses. First, all hydraulic parameters related to
the water conducting properties are treated. Then the hydraulic parameters which are
related to the water storing properties of the porous media are dealt with.
In Chapter 3 the most important steps in the evolution of the analytical models of
pumping tests are treated. For each model the derivation of the basic differential equation
is given along with the solution. For each solution one or more derived interpretation
methods are given.
In Chapter 4 the two-dimensional axi-symmetric numerical model which allows the
simulation of a pumping test in a layered groundwater reservoir is described. The
numerical model is verified with the help of several analytical models: the model of
Theis, the model of Jacob-Hantush, the model of Hantush, the model of Hantush-Weeks
and the model of Boulton as explained by Cooley. At the end of this chapter the program
package for the numerical simulation of the pumping test is proposed and explained.
In Chapter 5 further developments are treated which enlarge the applicability of
the numerical model. It concerns the treatment of pumping tests with variable discharge
rates, the ability to simulate the drawdown in a pumping test in a laterally anisotropic
aquifer, the calculation of the drawdown in a pumped well. With a supplementary
computer program the drawdown as a result of several wells can be calculated in a
laterally continuous homogeneous multilayered groundwater reservoir. With the same
program it is possible to simulate the drawdown in groundwater reservoirs which are
laterally bounded by an impervious or a constant head boundary. Finally, the drawdown
in a multilayered laterally isotropic or anisotropic groundwater reservoir with a vertical
plane of discontinuous lateral conductivity change can be approximated. In all these cases
the subsidence caused by the pumping process can also be estimated assuming that the
changes in the specific elastic storage and the vertical and horizontal conductivity values
are constant in each layer during compaction.
In Chapter 6 the development of an inverse model as a tool for pumping test
interpretation is treated. First, the used residuals and the sensitivity coefficients are
defined along with the possibility of graphical representation. This is followed by the
nonlinear regression in which the optimal values are searched by minimizing an object
function. This function is either the sum of the squared residuals or the sum of the
weighted squared residuals. In addition to the optimal values, the joint confidence region
of the hydraulic parameters is treated along with the statistical parameters and graphs
which can help in the collinear diagnostic. The inverse model is validated with a synthetic
problem. Also, the influence of the different factors applied during the interpretation
process are studied with the aid of the same synthetic problem. It is followed by a
representation of the computer package which helps us to use the nonlinear regression or
the interpretation process. Finally, some possibilities are treated to approximate the
confidence intervals for the drawdown at a certain distance from the pumped well, in a
certain layer and after a certain time of pumping. The latter calculations require the
results of the inverse model, the optimal values of the hydraulic parameters and some
statistical parameters which characterize the joint confidence region.
In Chapter 7 some examples are given of the interpretation of pumping tests with
the aid of the inverse model. In the given examples the observed drawdowns in the
pumped layers are simultaneously interpreted with drawdowns observed in the adjacent
layers. Not only the interpretation of single pumping tests is demonstrated but also the
interpretation of multiple pumping tests. As an example, the performance and interpretati-
on of a double pumping test and of a triple pumping test are shown. By observing these
multiple pumping tests it was shown that the vertical conductivities of the semi-pervious
layers can be derived with a similar level of accuracy than the horizontal conductivities of
the pervious layers. The elastic storages of each layer are also derivable. An example of
the interpretation of a double pumping test made in fractured rocks enables us to deduce
the hydraulic parameters which characterize the flow in a laterally anisotropic aquifer. In
the last example it is shown that artificial recharge tests can also be interpreted by use of
the inverse numerical model. All these examples help to demonstrate that the inverse
model can be of help in the optimal design of pumping tests, injection tests or artificial
recharge tests.
REFERENCES
Carrera, J., and Neuman, S.P., 1986, Estimation of aquifer parameters under transient
and steady state conditions, 2. Uniqueness, stability and solution algorithms: Water
Resources Research, v. 22, no. 2, p. 211-227.
Hill, M.C., 1989, Analysis of the accuracy of approximate, simultaneous, nonlinear

confidence intervals on hydraulic heads in analytical and numerical test cases.
Water Resources Research, v. 25, no. 2, p. 177-190.
Kruseman, G.P., and De Ridder, N.A., 1970, Analysis and evaluation of pumping test
data: Wageningen, Int. Inst. for Land Reclamation and Improvement(ILRI), Bull.
11, 200 p.
data. zrd edition: ILRI publ. 47, Wageningen, 377 p.
Lebbe, L., 1988, Execution of pumping tests and interpretation by an inverse numerical
model (in Dutch): Gent, Thesis Geagg. Hoger Onderw., Geologisch Instituut,
Rijksuniv. Gent, 563 p.
Reed, J.E., 1980, Type curves of selected problems of flow to wells in confined aquifers:
Tech. of Water Resources Inv., U.S.GeoI.Surv., Book 3, Chap. Bl. Washington,
DC., US Government Printing Office, 106 p.
Walton, W.C., 1970, Groundwater resources evaluation: New York, Mc Graw-Hill,
664 p.
Walton, W.C., 1987, Groundwater pumping tests: Michigan, Lewis Publishers, Inc.
201 p.
Chapter 2 / Hydraulic Parameters
In the framework of pumping test analysis two groups of hydraulic parameters are
important. The first group of hydraulic parameters describes the water conducting
properties of the porous medium and the second group describes the water storing
properties of the porous medium. The first group is described in the first section of this
part, the second in the second section.
2.1 Hydraulic parameters describing water conducting properties
In this study it is assumed that the groundwater flow is fully described by Darcy's law.
So coupled flow phenomena like groundwater flow induced by temperature, voltage or
concentration gradient as in respectively thermo-, electro- and chemical osmosis, are not
further considered.
2.1.1 Darcy's law
The hydraulic conductivity, K, is now the coefficient in the empirical law of Darcy.
Darcy (1853) states that the velocity of flow is proportional to the hydraulic gradient (Fig.
2.1):
Q -K(h -h )
_=q 2 1 K ah (2.1)
A 1 al
where Q is the volumetric flow rate (U/T),
A is the cross-sectional area (U),
I is the length of the sand column (L),
h2-h. is the difference in hydraulic head (L).
The dimensionless quantity (hz-h.)/l represents the change of the hydraulic head
over the distance through which the change takes place. This is the hydraulic gradient
often expressed as ahlaJ. q is the specific discharge through the cylindric sand column
with the units of velocity. Because the gradient is dimensionless, the coefficient in the law
of Darcy also has the units of velocity.
The specific discharge is also called the Darcy velocity. It is the velocity that a
water particle would have if the porous material was absent in the cylindric cross-section.
The Darcy velocity is a macroscopic concept. This velocity is different from the micro-
scopic velocities of the actual paths of individual water particles moving through the
porous medium. The mean actual velocities of these water particles are always higher
10 Chapter 2 / Hydraulic parameters
than the Darcy velocities and can be defined as the volumetric flow rate per unit area of
connected pore space.
(2.2)
where n., is the effective porosity (UL-3).
h,
z,
A. Cross secllon ~-:-'-_-'--'-L.:::.Z=O

reference level
Fig. 2.1. Flow through an inclined sand filter to illustrate Darcy's law
The quantities q and v have both a magnitude and a direction and are consequently
vector quantities. Darcy's law is valid as long as the flow is laminar. For materials with
very low permeability a minimum threshold gradient may be required before the flow
takes place.
2.1.2 Hydraulic head andfresh water head
On the field the hydraulic heads are measured by means of piezometers or by means of
observation wells. Piezometers are tubes which are placed into a borehole to a certain
depth. The bottom of the tube is open to facilitate the entering of the water. The angular
space between the borehole and the tube is filled with material which has a conductivity
which is smaller or at least equal to the hydraulic conductivity of the surrounding porous
medium. The tube is open at the top to measure the depth of the water.
Chapter 2 / Hydraulic parameters 11
In most cases however observation wells were installed. In these cases the tubes
are closed at the bottom but have a filter screen over a limited depth interval. The angular
space is sealed and the top is open to measure the depth of the water. The advantage of
observation wells is that they facilitate the collection of water samples to study the water
quality occurring around the filter screen.
ob •• rveUon
piezometer weU
hp
It h
h
reference level
® ®
Fig. 2.2. Hydraulic head, h, as the sum of the elevation head, h., and the pressure head,
hp. Field measurement procedure as difference between the level of the top of the
observation well, I" and the depth of the water below this top. Fresh water head, bt, as a
function of the elevation head, h., the length of the water column in the observation well
Ocd-hJ and the ratio of the specific weight of the water column in the observation well
(Pg)ob, on the specific weight of the fresh water (Pg)f
A piezometer is shown in Fig. 2.2a. The hydraulic head, h, is equal to the sum of
the elevation head, h., and the pressure head, hv. The hydraulic head is the energy per
unit weight. The total head, hT , is the total energy of a fluid per unit weight. It is the sum
of the elevation head, h., the pressure head, hv, and the velocity head, hv. Because the
velocity of the groundwater is very small the velocity head can be ignored. So the total
head is almost equal to the hydraulic head, h. The dimension of the heads are those of
lengths. It can be expressed as meters of water above a chosen datum level. The
hydraulic heads are mostly obtained by measuring the depths of the water below the
topsides of the observation wells and by leveling these topsides (Fig. 2.2b). In this case
one must be sure that all observation wells are filled with water of the same quality over
their entire length. Its specific weight must approximate the unity.
In groundwater reservoirs filled by salt, brackish and fresh water it is not enough
to measure the depth of the water and to level the topside of the observation well but the
specific weight of the water column in the observation well must be determined. This is
the case when the observation well is filled with the same water over its entire length. If
however the observation wells are filled with different water types then the different
specific weights and their representative depth intervals must be known. Most of the
observation wells are cleaned by pumpage. So the different observation wells are
normally filled with waters of different quality. Usually they are filled over their entire
length with water of the same quality (Fig. 2.2c). In groundwater reservoirs with waters
of different specific weight one option is to work with freshwater heads. Another option
is to work with pressures but is often less convenient than the first option. To determine
the fresh water head one needs a reference level and a reference liquid. This liquid has a
reference specific weight equal to the specific weight of fresh water. The fresh water head
can then be deduced by the following equation:
hf=h z+ (1 t-d-hz) (pg) obs (2.3)

(pg) f
where hf is the fresh water head (L),

hz is the elevation head (L),
It is the level of the topside of the observation well (L),
d is the depth of the water below the topside of the observation well (L),
(Pg)obs is the specific weight of the water column in the observation well (ML-2T 2),
and (Pg)f is the specific weight of fresh water (ML-2T-Z).
2.1.3 Intrinsic permeability
The hydraulic conductivity, the constant of proportionality in Darcy's law, is a function

of the characteristics of the porous medium and of the fluid. The parameter which
characterizes only the conductive property of the medium is called the intrinsic permeabi-
lity. The relation between the hydraulic conductivity, K, and the intrinsic permeability, k,
is given in Eq. (2.4):
K=kJ!Jl. (2.4)
I-L
where K is the hydraulic conductivity (LTl),

k is the intrinsic permeability (U),
pg is the specific weight of the water (ML-~2),
and p. is the dynamic viscosity (ML-1T 1).
In most hydrogeological studies the variation of the dynamic viscosity is more
important than the variation of the specific weight. The mass density of fresh water varies
around 1000 kg/m3 while the mass density of salt water varies around 1025 kg/m3.
Therefore the hydraulic conductivity of a porous medium increase with a factor 1,025
when the fresh water is replaced by salt water. The viscosity of the water at 10° C is
about twice the viscosity of water at 40° C. The viscosity of water around 10° C
decreases with a factor of about 1,029 when the temperature increases about 1° C and
consequently the hydraulic conductivity of the porous medium will increase with the same
factor. The intrinsic permeability is a function which is widely used in the petroleum
industry, where the existence of gas, oil and water in multiphase flow systems, makes the
use of a fluid-free conductance parameter attractive (Freeze and Cherry, 1979).
2.1.4 Heterogeneity and anisotropy of geological formations with respect to hydraulic

conductivity
The hydraulic conductivities of a geological . formation usually show varIation through

space. At a given point the hydraulic conductivity depends on the direction of measure-
ment. The first property is termed heterogeneity and the second property anisotropy.
2.1.4.1 Heterogeneity
A geological formation is heterogeneous if the hydraulic conductivity depends on the

pOSItion in space. The hydraulic conductivity depends on the coordinates x,y,z or
K(x,y,z)=f(x,y,z). The formation is called homogeneous if the hydraulic conductivity is
independant of the coordinates x,y,z or K=C, where C is a constant value. Homogeneous
geological formations are rather exceptional, although the concept of a homogeneous
formation is frequently used in theoretical considerations because of its simplicity.
There are probably as many types of heterogeneous configuration as there are

geological environments. Three broad classes of heterogeneity are here considered. In
sedimentary rocks there is usually a layered heterogeneity. The individual beds of a
geological formation can be considered as being homogeneous but each bed has a
different hydraulic conductivity value. Also typical for sedimentary rocks is the trending
heterogeneity which is related to sedimentary processes. Discontinuous heterogeneity can
be caused by the presence of faults, large-scale stratigraphic features and the overburden-
bedrock contact. These discontinuities are characterized by a large contrast in hydraulic
conductivity.
A quantitative description of the degree of heterogeneity in geological formation is

obtained by the study of the probability density function. It is now generally accepted that
the probability density distribution for the hydraulic conductivity is log-normal. According
to Freeze and Cherry (1979), Warren and Price (1967) and Bennion and Griffiths (1966)
this is the case in oilfield reservoir rocks, and Willardson and Hurst (1965) and Davis
(1969) support the conclusion for unconsolidated water-bearing formations. A log-normal
distribution for K is one for which a parameter Y, defined as Y = 10gK, shows a normal
distribution. Following Freeze and Cherry (1979) the standard deviation on Y (which is
independent of the units of measurement) is usually in the range 0.5-1.5. This means that
the K-values in most geological formations show internal heterogeneous variation of one
to two orders of magnitude. In contrast to hydraulic conductivity, porosity values from a
single formation have normal rather than log-normal distributions. A quantitative
description of the trending heterogeneity can be obtained by giving the trend of the mean
value of the probability distribution.
2.1.4.2 Anisotropy
A geological formation is isotropic at a point if the hydraulic conductivity K is indepen-

dent of the direction of measurement at that point. The formation is anisotropic at a point
if the hydraulic conductivity varies with the direction of measurement at that point.
Consider first a vertical plane through a horizontally bedded sedimentary forma-

tion. The principal directions in that vertical plane are the two directions corresponding
with the maximum and the minimum conductivity. In the chosen example the direction of
maximum hydraulic conductivity will be the horizontal axis and the direction of minimum
hydraulic conductivity will be the vertical axis. The principal directions are always
perpendicular to each other.
In a three-dimensional space there are three principal directions which are also
perpendicular to each other. If a plane is chosen perpendicular to one of the principal
directions the two other principal directions correspond with the directions of maximum
and minimum hydraulic conductivity in that plane. In a sedimentary formation which
consists of an alternation of sand and silty and/or clayey sand layers which are horizon-
tally bedded the smallest hydraulic conductivity will always be measured in the vertical
direction. The hydraulic conductivity measured in all horizontal directions can be the
same. This formation is then isotropic in the horizontal plane or is transversely anisotro-
pic and laterally isotropic.
With the aid of the two adjectives, "homogeneous" or "heterogeneous" on one

hand and "isotropy" or "anisotropy" on the other hand, one can distinguish four different
types of geological formation with respect to their hydraulic conductivity. In Table 2.1
the characteristics of those four types of geological formations are further exposed.
Table 2.1. The characteristics of the four different types of geological formations with
respect to their hydraulic conductivities.
Homogeneous, isotropic Homogeneous, anisotropic
Kh(x,y ,z) = Kv(x,y ,z) =C Kh(x,y,z)=Cl
for all x y z coordinates in the formation. Kv(x,y,z)=C2
Cl;t: C2, for all x y z
Heterogeneous, isotropic Heterogeneous, anisotropic
Kh(x,y,z)=Kv(x,y,z)=f(x,y,z) Kh(x,y ,z) =f1(x,y ,z)

Kv(x,y ,z) =f2(x,y ,z)
f1(x,y,z) ;t: f2(x,y ,z)
Geological formations which have a layered heterogeneity behave on a large scale

as a single, homogeneous anisotropic layer. Let us consider a sedimentary formation
which is horizontally bedded and where each layer can be considered as a homogeneous,
isotropic pervious medium. The horizontal flow through a cross-section with a unit width
through the whole formation is the sum of the horizontal flow through the individual
layers or Qt= ED;=IQ;.
Let Llh be the head difference over a horizontal distance I (Fig. 2.3), then:
Qh=-K.D. Llh (2.5)

, "1
where D; is the thickness of the individual layers, then:
Qc=LJ=lQi=-LJ=l (KiD) At (2.6)
The total flow through a cross-section with a unit width can now also be written
by the help of the equivalent horizontal hydraulic conductivity of the whole formation and
the equation of the total horizontal flow through the formation can also be written as:
(2.7)
where D is the total considered thickness of the formation.

I ..
Fig. 2.3. Horizontal flow through a layered heterogeneous formation
From the Eqs. (2.6) and (2.7) one can deduce that
(2.8)
The vertical flow through the whole formation QV is equal to the vertical flow
through each individual layer or QV=QI =Q2= ... = Qn.1 =Qn (Fig. 2.4)
According to the law of Darcy one can write the vertical flow, Q, through a unit
cross-section of the individual layer i
(2.9)
where K;,Ah j and d j are respectively the hydraulic conductivity of, the hydraulic differen-
ce over and the thickness of the individual layer i.
Chapter 2 I Hydraulic parameters 17
-~
- -
... f--i Ll.h n
JM - - - - I""
n dn I a Yn Kn
,
-
n-1 d n-1
f a~-1 Kn-1
-
-
I--
1a y I--
4 d4 1 a~ K4 I--
3 d3 ~ a 3
Y K3
I--
2 d2 ~ a~ K2
I--
1 K1
d1
,l a~
~I--
Fig. 2.4. Vertical flow through a layered heterogeneous formation
The vertical flow through the whole formation can also be written by means of the
equivalent vertical conductivity of the whole formation, K V •
(2.10)
where D is the sum of the thickness of the individual layers Ei.1 d i .
The sum of these hydraulic head differences of all the individual layers is equal to
the total head difference over the whole formation Ll.h, or
or from Eqs. (2.9) and (2.10) one can derive that
(2.11)
and because QV=Q. =Q2= ... =Q.
if fj is again dJD.
As already mentioned, an unconsolidated water-bearing formation shows a log-
normal distribution for the hydraulic conductivity. If we assume now that the hydraulic
conductivities of the individual layers of a layered formation also show a log-normal
distribution and that each layer behaves as an isotropic unit one can calculate the
equivalent horizontal and vertical conductivity of this formation. The probability density
function of the log-normal distribution can be formulated as follows:
(2 .12)
where I-'logK and CTlogK are the mean and the standard deviation of the log-normal distributi-
on of the hydraulic conductivity. With the help of Eqs. (2.8), (2.11) and putting f;, equal
to the relative frequency density (Wonnacott and Wonnacott, 1985) the equivalent
horizontal and vertical conductivity can be deduced. If the whole distribution is conside-
red then the result is that the equivalent horizontal conductivity is always infinitely large
and the equivalent vertical conductivity is always infinitely small. Therefore, only
meaningful values for the equivalent horizontal and vertical conductivity are obtained if
only the central part of the log-normal distribution is considered while the tails are
ignored. One could argue that the probability in the tails is so small that the layers can
not be laterally continueous on the large scale of the formation so that these tails of the
log-normal distribution can be ignored.
In Fig. 2.5 the relation is shown between the equivalent anisotropy of the layered
formation VKb/Kv and the standard deviation of the log-normal distribution for different
central parts of the log-normal distribution taken into consideration. The first result of
such a calculation is that the mean value of the log-normal distribution, I-'logK, i~ ~ways
equal to the equivalent effective conductivity of the layered formation or VKbKv. A
second result is that when the heterogeneity increased or the standard deviation of the log-
normal distribution then the equivalent anisotropy of the layered formation increases. A
third result is that the equivalent anisotropy is strongly dependent of the considered
central part of the log-normal distribution. This is a simple example for the upscaling of
the hydraulic conductivity. Extend studies of upscaling of the hydraulic conductivity are
made by Zijl and Starn (1992), Starn and Zijl (1992), Bierkens (1994) and Bierkens (1996).
99,5%
99%
98%
'I'"
I
95%
90%
II,
II,
III I I 80%
1 1//// 68%
-
//
'J. :/
~ ~ :-:H'
STANDARD DEVIATION 100
Fig. 2.5. Relation between the equivalent anisotropy of a layered formation v'Kb/Kv and
the standard deviation of the log-normal distribution of the conductivities of the microsco-
pie isotropic layers when different percentages of central parts of the distribution is taken
into consideration
2.1.5 Generalized law of Darcy
The one-dimensional form of Darcy's law (Eq. 2.1) can now be generalized for a
three-dimensional flow. In three dimensions the Darcy velocity is a vector with three
components <Ix, qy and 'lz. The simplest generalization of Darcy's law is then:
(2.13)
where K" Ky and Kz are the hydraulic conductivity values in the x, y and z direction.
This simplified generalized form of Darcy's law can only be applied if the principal
directions of the anisotropy coincide with the x, y and z axes. This form of Darcy law
can be applied on most sediments which are horizontally stratified. This is even the case
when these sediments show a small inclination without causing a large difference between
the approximated and the real flow.
20 Chapter 2 I Hydraulic parameters
If however the principal directions of the anisotropy do not coincide with the x, y
and z coordinate axes the generalized form of Darcy's law must be applied.
(2.l4)
In the most general case the hydraulic conductivity has nine components. These compo-
nents placed in a matrix form give the hydraulic conductivity tensor
Kxx Kxy Kxz

KyX Kyy KyZ (2.15 )
Kzx KZy K zz
This is a second order symmetric tensor that has the property Kxy = Kyxo Kxz = Kzx and
Kzy=Kyz . This means that actually only six distinct components in three-dimensional flow
are needed for fully defining the hydraulic conductivity. For the special case where the
principal directions coincide with the x, y and z axes then the hydraulic conductivity
tensor becomes:
Kxx 0 0
0 Kyy 0
0 0 K zz
The components of the second order symmetric tensor for a two-dimensional case can be
deduced with the help of Mohr's circle (Fig. 2.6). Two cases are considered. In the first
case the coordinate directions x, y are given along with the corresponding components of
the hydraulic conductivity tensor Kxx, Kyy and Kxy. The problem is the derivation of the
principal directions x' and y' and the corresponding diagonal components of the hydraulic
conductivity tensor (Bear, 1972). The non-diagonal components are then equal to zero.
From Mohr's circle (Fig. 2.6a) one can deduce the equation for the hydraulic conductivi-
ty Kx'x' in the principal direction x'
(2 .16)
and the equation for the hydraulic conductivity Ky'Y' in the principal direction y'
(2.17)
The angle (J between the direction x and the principal direction x' can also be deduced
with the aid of Mohr's circle
2K
tan (26) = xy (2.18)
KXX-Kyy
In the second case the coordinate principal directions x and y are given together with the
hydraulic conductivities in these principal directions K"" and Kyy • The problem is now the
derivation of the components of the hydraulic conductivity tensor corresponding with the
directions x' and y' which are rotated an angle (J with respect to the principal directions x
and y (Fig. 2.6b). According to the circle of Mohr (Fig. 2.6c) the equation for these
components can be written as follows (Bear, 1972):
K +K K-K
K,
xx
,= xx
2
yy + xx
2
YYcos (28) (2.19)
K -K
xx YY sin (28) (2.20)
2
(2.21)
Long et al. (1982) deduced that the fractured rocks do not always behave as
homogeneous, laterally anisotropic porous medium with a symmetric conductivity tensor.
Fractured rocks behave more like an equivalent porous medium which is laterally
anisotropic and which has a symmetric hydraulic conductivity tensor if the rocks show
following four characteristics: the fractured density is rather large, the fracture aperture is
constant rather than distributed, the fracture orientation is rather large. Increased fracture
interconnectedness enhances the validity of the equivalent continuum (Smith and
Schwartz, 1984).
2.1.6 Hydraulic conductivity ellipsoid
Consider an arbitrary flow q, in a homogeneous, anisotropic medium with principal

hydraulic conductivities Kx and Kz (Fig. 2.7a). Along the flowline
(2.22)
where Ks is unknown, although it presumably lies in the range Kx-K..

Given x, y, Kxx, K~,Kxy.What are

the principal directions x'y', derive
e, Kx'x', and Kyy.
Kx'x'
Kxx
Kx'x'
y'
Ky
x
®
Given x, y, K x , Ky , are the
principal directions, derive Kx'x:'
Kyy and Kx'y'.
Fig. 2.6. The use of Mohr's circle to deduce the components of the hydraulic
conductivity tensor in two dimensions (after Bear, 1972)
q, can be separated in the components qx and <Iz, where
(2.23)
Now, since the hydraulic head h is a continuous function of x and z
(2.24)
Geometrically, ax/as=coso and azlas=sinO. Substituting these relationships together with

Eqs. (2.22), (2.23) and (2.24) and simplifying yields:
1 cos 2 8 sin 2 8
-=---+--- (2.25)
Ks Kx Kz
® @
I
,/
.- --\lKz "- , \
, Vi<; /
I
"-
" '-
--- ----
/
®
Fig. 2.7. Hydraulic conductivity ellipse and ellipsoid
This equation relates the principal conductivity components Kx and K. to the resultant K,
in an angular direction O. If we put Eq. (2.25) into rectangular coordinates by setting
x=r.cosO and z=r.sinO then:
(2.26)
which is the equation of an ellipse with the major axes V"Kx and ..JKz. In Fig. 2.7b, the
conductivity value K, for any direction of flow in an anisotropic medium can be determi-
ned graphically if Kx and Kz are known. In three dimensions, it becomes an ellipsoid with
major axes ..JK., .../Ky and ..JKz and it is known as the hydraulic conductivity ellipsoid
(Fig. 2.7c). In contrast to isotropic media, flowlines are not perpendicular to the
equipotential lines in anisotropic media.
2.1.7 Classification of layers according to their water conductive properties.
This classification is only comparative with respect to the other layers of a groundwater
reservoir. The pervious layers are those layers with the largest hydraulic conductivity.
The semi-pervious layers have a hydraulic conductivity which is significantly smaller.
Their horizontal conductivity is mostly one to one and a half order smaller than the
horizontal conductivity of the adjacent pervious layers. The vertical conductivity contrast
between the pervious and the semi-pervious layers is mostly even larger, from one and a
half to two orders of ten. With this classification one can state that the natural groundwa-
ter flow in pervious layers has a horizontal component which is much larger than its
vertical component. In the semi-pervious layer the natural groundwater flow velocity is
much smaller than in the pervious layers. When the global flow pattern is considered then
the vertical component of the flow velocity in a semi-pervious layer is much more
important than its horizontal component. This is also the case for artificially influenced
flow as for example the flow in a groundwater reservoir where the water is abstracted
from the pervious layers. Only in special cases of artificial flow the above given rules
about the flow do not longer hold.
An impervious layer is a simple theoretical concept. An impervious layer has a
hydraulic conductivity equal to zero or its logarithmic value is equal to minus infinity. In
nature such rocks with such extreme characteristics about it conductive properties for
water seldom exist. It is therefore preferable to use the term "impervious" only for the
schematization of the groundwater reservoir. A layer is then considered "impervious" if
the flow in it is so small that it can be ignored. This ignorance may not have an influence
on the studied characteristics of the flow, e.g., the hydraulic head, the concentration, a
balance component, etc. . In the classification of layers according to their conductive
properties it is better to speak of a semi-pervious layer with very low hydraulic conducti-
vity than to speak about an "impervious" layer.
The occurrence of the groundwater can also be considered on a larger scale in
geological formations. In this context a similar classifications can be made for geological
formation as for layers although other terms are used, such as, aquifers, aquitards,
aquicludes and aquifuges.
An aquifer is a geological formation, or group of formations, that contains
sufficient saturated permeable material to yield significant quantities of water to wells and
springs. This implies an ability to store and to transmit water. Synonyms frequently
employed include groundwater reservoir and water bearing formations. Aquifers are
generally areally extensive (Todd, 1980). On the other hand the terms pervious, semi-
pervious and impervious layers are used on a smaller scale, in the context of a relatively
small study area. In such studies, it is possible that the groundwater reservoir or the
aquifer is subdivided in pervious and semi-pervious layers.
An aquiclude is a geological formation which may contain water but is incapable

of transmitting significant quantities under field conditions. An aquiclude does not yield
appreciable quantities to wells. A clay is an example of it.
An aquitard is a geological formation or a group of formations which is saturated
but is constituted of poorly permeable stratum. An aquitard impedes groundwater
movement and does not yield water freely to wells. If aquitards are sufficiently thick,
they may constitute an important groundwater storage zone. Sandy clay is a typical
example.
According to the aquifer boundaries one can distinguish phreatic or unconfined,
confined and semi-confined aquifers. The upper boundary of a phreatic or unconfined
aquifer is a free water table which is the top of the saturated zone and is defined as the
surface of atmospheric pressure. A confined aquifer is completely filled with water and is
bounded above and below by "impervious" material or an aquiclude. Because "impervi-
ous" material is a theoretical concept so will be the concept "confined aquifer". Semi-
confined aquifers are also completely saturated with water. They are bounded above and
below by a semi-pervious material or aquitards. One of the bounded layers can also be
"impervious" or an aquiclude whereas the other bounded layer is a semi-pervious material
or an aquitard.
2.1.8 Hydraulic parameters derived from hydraulic conductivity
A number of hydraulic parameters are derived from the hydraulic conductivity. In this
way the transmissivity helps to define the total horizontal flow through a pervious layer.
The hydraulic resistance is used for the calculation of the vertical flow through the semi-
pervious layers. The distribution of the leakage into a semi-confined aquifer is determined
by the leakage factor.
2.1.8.1 Transmissivity
The transmissivity is the product of the average hydraulic conductivity and the thickness
of a pervious layer or an aquifer. Transmissivity is consequently the amount of water
which flows through a vertical cross-section of an aquifer of unit width and under a
hydraulic gradient equal to unity. It is designated by the symbol T. It has the dimension
Length 2 Time- 1 (Uti) and can for example be expressed in m2 /day. From the above
definition one can state that
T=KrP (2.27)
where D is the thickness of the pervious layer or the aquifer (L) and ~ is the horizontal
conductivity (Lt l ).
2.1. 8.2 Hydraulic resistance
The hydraulic resistance is a property of semi-pervious layers or an aquitard. It is the

ratio of the saturated thickness of the semi-pervious layer 0' and the vertical hydraulic
conductivity of the semi-pervious pervious layer Kv. It characterizes the resistance of the
semi-pervious layer to upward or downward vertical flow through the semi-pervious
material. It is designated by the symbol c and has the dimension of Time (T). For
example, it can be expressed in days. From the above definition one can say that
c=.K
Kv
(2.28)
The reciprocal value of the hydraulic resistance is called the leakance as it is used by
Konikow and Bredehoeft (1978).
2.1.8.3 ~age factor
The leakage factor is a hydraulic parameter which characterizes a semi-confined aquifer.

It can be defined as follows
L=..jTCres (2.29)
where T is the transmissivity of the pervious layer

and Cres is the resulting hydraulic resistance of the bounding semi-pervious layers.
The resulting hydraulic resistance is derived from the hydraulic resistances of the semi-
pervious layers above and below the pervious layer or
1 1 1
--=-+- (2.30)
c res c1 c2
where C 1 and C2 is the hydraulic resistance of respectively the semi-pervious layer below
and above the considered pervious layer. If one of the bounding layers is "impervious"
then C 1 or c2 is equal to infinity and c r • s is equal to C 2 or c 1 • If both bounding layers are
impervious then the leakage factor is equal to infinity. This is the leakage factor of a
"confined" aquifer. In this case there is only horizontal flow possible in the pervious
layer. From the Eqs (2.29) and (2.30) one can derive that the smaller the leakage factor
the more important can be the flow through the bounding semi-pervious layers in
comparison with the flow in the pervious layer and vice versa. The leakage factor L has
the dimension of Length (L) and can, for example, be expressed in meters.
2.1.9 Methods to derive the hydraulic conductivity
Two kinds of methods can be distinguished. The first kind is called the direct methods. In
these methods the hydraulic conductivity is measured by means of an experiment where a
flow is induced through a material and the hydraulic head or the change of it is measured.
In the indirect methods some physical properties of the material are measured and the
hydraulic conductivity is derived indirectly through an imperical relation between the
measured physical property and the hydraulic conductivity.
2.1.9.1 Direct methods
In the direct methods one can make a distinction between laboratory methods and methods
for the determination of hydraulic conductivity in the field. Laboratory determination of
hydraulic conductivity is made by a permeameter. Two types of permeameters are simple
to operate and are widely employed. They are the constant head and the falling head type
(Todd, 1980).
Permeameters
In the constant head type there is a constant flow through a cylindric sample of the
investigated material under a constant head difference or under a constant hydraulic gra-
dient (Fig. 2.8a). The volume of water which flows through the sample during a chosen
time is measured. Knowing the surface of the cross-section perpendicular on the flow and
the length of the sample along with the head difference one can derive the hydraulic
conductivity with Darcy's law.
In the falling head permeameter there is a decreasing flow through the cylindric
sample of the investigated material due to a falling head difference or a falling hydraulic
gradient (Fig. 2.8b). The sample is no longer connected with a reservoir which is held at
a constant hydraulic head and delivers the water for the flow through the sample as in the
constant head type. In the falling head type water is delivered by a tall tube. The rate of
fall of the water in this tall tube is measured. This rate defines the rate of flow through
the tube and the water sample.
(2.31)
where r, is the radius of the tall tube

and -dh/dt is the rate of fall of the head in the tall tube.
28 ChapuT 2 I HydrQufie jHmuntttrs
(.J ~J
Fig. 2.S. Constant head (a) and falling head (b) permeameters (afer Domenico and
Schwartz)
Applying Darcy's law the flow through the sample is
O= 1f. I~Kh/L (2 . 32)
where L is the length of the sample.

After equating and integrating,
(2 . 33)
where .61 is the time interval for the fall of the water level from h, to h1 •
Permeameler results may bear little relation to actual field hydraulic conductivity.
The tests are carried out on small samples. If the samples are undistu rbed core samples
then the measured values should represent the in situ point sample . Undisturbed samples
are however not easy to obtain. Disturbed samples experience changes in porosity,
packing and grain orientation which modify the hydraulic conductivities. Variation of
several orders of magnitude frequently occur for different depths and locations in an
aquifer.
WeU tests
Besides the laboratory test one has a al rge variety of field test to determine the
hydraulic conductivity. The two methods which are now frequently used [0 measure [he
hydraulic conductivity in situ are the well tests and the pumping tests. Two well tests are
considered. One is suitable for wells that are open over a short interval at their base and a
second for wells that are open over the entire thickness of a confined aquifer.
The well tests are initiated by causing an instantaneous change of the water level
in the well through a sudden introduction or removal of a known volume of water. The
recovery of the water level with time is then observed. When water is removed, the tests
are called bail tests; when it is added, they are known as slug tests. These tests can also
be performed by the sudden removal or introduction of a solid cylinder of known volume.
The interpretation methods of well tests depend on the construction of the tested wells.
The first is applied on wells that are open over a short interval. The second concerns
wells that are open over the entire thickness of the confined aquifer. In the first case the
method of Hvorslev (1951) is applied, in the second case the interpretation procedure
evolved by Cooper et al. (1967) and Papadopoulos et al. (1973) is used.
A disadvantage of the well test methods is that they depend on the well characte-
ristics. Erroneous values may be deduced from wells with corroded or clogged screens.
Also developed wells by surging or backwashing before testing may result in increased
hydraulic conductivity values. They may be a complex function of the conductivity of the
installed gravel pack, of the artificial developed gravel pack and of the disturbed natural
material around the used borehole.
Step-drawdown pumping tests
By the performance of step-drawdown pumping tests the drawdown at the inner limit of
the aquifer at the well and the well loss can be found. The well loss is here considered to
be the head loss caused by the flow through the disturbed zone at the borehole wall,
through the placed gravel pack and through the well screen and due to the flow inside the
well to the pump intake. The well loss is assumed to be proportional to the n-th power of
the discharge rate. The well loss is then equal to CQ', where n is a constant larger than
one and C is a constant governed by the radius, construction and condition of the well.
Jacob (1947) suggested that a value n=2 might be reasonable. Rorabough (1953) pointed
out that n can significantly deviate from two. An exact value for n cannot be stated
because of differences of individual wells. Detailed investigation of flows inside and
outside the wells shows that considerable variation occurs from the assumed flow
distributions. From the drawdown measurements in a pumped well during a step-draw-
down test one can deduce the constant C and the power n of the well loss along with the
transmissivity of the aquifer situated around the well. In a following part it is explained
how these parameters describing the well loss can also be deduced from pumping tests
where not only the drawdown is measured in the pumped well but also in observation
wells placed around the pumped well.
Pumping tests
Laboratory tests provide point values of the hydraulic conductivity. Well tests provide in
situ values more or less representative of a small volume of porous media in the imme-
diate vicinity of the well screen. The pumping tests provide in situ measurements that are
averaged over the whole aquifer volume. Initially pumping tests were performed with one
pumped well with its well screen in a pervious layer and a number of observation wells
with their screens in the same layer. With such tests the transmissivity and the elastic
storage coefficient are determined with a rather high degree of accuracy. The resulting
hydraulic resistance of the bounding layers is far less accurate.
Through observation of rising drawdowns in the bounding layer one can determine
the average vertical conductivity of the bounding layer along with the parameters that
describe their elastic storage characteristics. Through execution of multiple pumping tests
the horizontal conductivity of the pervious layers and the vertical conductivity of the
semi-pervious layers can accurately be derived along with their elastic characteristics. In
this multiple pumping test every permeable layer is separately pumped while the draw-
down is measured in the pumped as well as in the bounding layer. The water conducting
and storing characteristics of all these layers are derived by the simultaneous interpre-
tation of all the measured drawdown. It is self-evident that a multiple pumping test is
expensive. The major part of this work concerns the performance and interpretation of
pumping tests.
2.1. 9.2 Indirect methods
Four types of empirical correlation are presented here. In the first type empirical relations
are given between the hydraulic conductivity and the grain size distribution of different
sediments. In the second type the hydraulic conductivity is estimated from natural gamma
radiation and in the third kind from electrical resistivity measurements. In the last
considered type of methods the hydraulic conductivity is estimated from the joint
characteristics of fractured rocks.
Grain size distribution
Numereous researchers try to find a relation between the hydraulic conductivity and one
or more characreristics of the grain size distribution. Principally, two types of relation
can be distinguished. The first type of relation is based on a grain size diameter. The
second type is based on the specific surface of sand and a part of the silt fraction.
For each type an example is given.
An example of the first type of relation is the well-known equation of Hazen
(1911).
(2.34)
where K is the hydraulic conductivity in centimeter per second, dlO is the effective grain
size in centimeter and C is a constant with the unit cm-1sec-1. The largest range for the
values of this constant is given by Taylor (1948). The effective grain size is defined as
the value where 10% of the particles are finer and 90% of the grains are coarser. Later
the same relation was adjusted so that one can take into account the temperature of the
water
K=Cdto (0.7 +0.03 t) (2.35)
where t is the temperature in °c.

An example of the second type of relations is the formula of Ernst as given by De
Ridder and Wit (1965).
K=54, 000 u/;;"ABC (2.36)
where K is the hydraulic conductivity in meter per day, U is the specific surface of all
grains between 2000 and 20 ILm fraction and A, B, and C are correction factors for
respectively the sorting of the sand, the percentage of the grains smaller than 16ILm and
the gravel content (particles larger than 2000lLm). The specific surface is the ratio
between the total surface of all particles between 2000 and 20 ILm and the surface of a
same weight of spheroidal particles of the same material with a diameter of one centime-
ter. The specific surface of grains comprised between the diameters d, and d i + 1 is given by
the following equation
(2.37)
where di is the largest and di + 1 is the smallest sieve fraction expressed in centimeter. The
specific surface of all the grains between 2000 ILm and 20 ILm is obtained by the multipli-
cation of the corresponding weight fraction or
Eu.w.
u =--~-~ (2.38)
t:at: wt:at:
where Wi is the weight of the sieve fraction between the diameters d, and d,+1 and Wtot is
the total weight of the fraction between 2000 and 20 ILm.
The correction factors A, Band C can be deduced from Fig. 2.9. The parameter
that characterizes the sorting of the sand fraction is given here by the weight percentages
of the three largest sieve fractions. If the sorting is 70%, the factor A is equal to one,
larger sortings result in values of A larger than one and sortings smaller than 70 % result
in values of A smaller than one. The correction factor B depends of the percentage of
particles smaller than 161Lm. If the sample has more than 6% of these particles the factor
B becomes very small and the method cannot be used to deduce the value of the hydraulic
conductivity. The factor B is equal to or smaller than one. The correction factor C is
defined by the gravel content. If gravel occurs between the smaller particles, it decreases
the hydraulic conductivity. Usually gravel occurs as small separate layers even it seems as
mixed in disturbed samples. In these cases the hydraulic conductivity increases with the
gravel content. The correction factor C takes only these cases into account. The factor C
is larger or equal to one.
To calculate the hydraulic conductivity by computer it was necessary to replace the
curves, shown in Fig. 2.9, by equations. By means of simulated grain size distribution
Lebbe (1978) shows that the correction factor A can be expressed in function of the
inclusive standard deviation CTj • This parameter is defined by Folk and Ward (1957) by
means of four diameters f/J5, f/J16' f/J84 and f/J95 where respectively 5, 16, 84 and 95% of the
grains are larger than the given values
(J . = /j)84 -/j)16 + /j)9S -/j)s (2.39)

~ 4 6.6
The diameters are expressed in f/J-values. These values correspond with the negative
logarithm in base two of the diameters expressed in millimeters. When CTj is larger than
0.63 then the correction factor A is equal to the reciprocal value of CTj and when CTj is
smaller than 0.63 the correction factor is equal to 1.6.
The correction factor B is calculated with the fraction of the sample which is
smaller than 16~m, fs
B=O. 998-25. 4fs+231f~-994f~ (2.40)
The correction factor C obtains a value one if the gravel fraction of the sample is smaller
than 0.1. If the gravel fraction is larger than 0.1 but smaller than 0.47 then the correction
factor C can be calculated by the following relation:
C=O. 944+0. 926fG -2. 49f~+6. 26f~ (2.41)
where fo is the gravel fraction. If the gravel fraction is larger than 0.47 then the correcti-
on factor is calculated with :
C=-0.385+3.958fG (2.42)
Gamma ray
The natural gamma radiation of sediments depends on the clay content. Because the clay
content of the sediments defines strongly their hydraulic conductivities, a relation is
searched between the natural gamma radiation and the hydraulic conductivity. Laboratory
investigation of Repsold (1989) on hose cores of unconsolidated sediments of the
Gorleben investigation area in Northern Germany results in a relation of the form
(2.43)
sorting percentage in 3 tcp-fractions
~-
'll.4 0.6
7%
6
0.8
particles <O.016mm
,~
1'.
1.0 1.2 1.4 1.6
corractlon factor A
5
4
......
3 r-...
....... "'-.
2
......
r--. ....
1
o 0.2 0.4 O. 6 0.8 1.0 1. 2

correction factor B
gravlll>2mm
60" 0
I- ....
50
j...-- po
40
30
V-
ii"
20
10
17
01.0 1.2 1.4 1.6 1.8 2.0 2.2
correction factor c
Fig. 2.9. Correction factors for estimating hydraulic conductivity of sands from grain
size distribution (U-figure) (Data from Ernst, 1955, unpublished, from Kessler and
Oosterbaan, 1974)
where K is the hydraulic conductivity in meter per second and GR is the gamma radiation
in API units. The relation is however not ap licable if the sediments contain important
changing amounts of glouconite, micas or other radioactive minerals belonging to the
sand fraction.
On three parallel samples of the same material, porosity and hydraulic conductivity
were determined along with the determination of the natural gamma activity. A linear
regression analysis was made on the logarithmic values of the hydraulic conductivity and
the natural gamma activity. The residuals of the twenty-five data points show a rather
large variation. Because of two reasons Repsold (1989) asserts that this was not unexpec-
ted. The first is that the determination of the hydraulic conductivity is necessarily
subjected to large variations. The second is that it was not possible to consider the poro-
sity as a separate parameter because of the small number of samples.
The given relation can only be applied to estimate the hydraulic conductivity if the
gamma ray measurements are calibrated to API units and the measurements are adjusted
to standard conditions in hydrogeological studies: a borehole diameter of five inches, a
specific weight of the borehole mud of one gram per cubic centimeter and a probe
diameter of one inch and a half. The method is limited to about 30 API, corresponding
with a K-value of lxlO-4 mis, where the sand contains no longer radiation activity. The
upper limit of application is 100 API and corresponds with a K-value of lx1Q-4 m/s.
Repsold (1989) states that the application of the given relation is restricted to material
comparable to that of the surveyed area. On the other hand, it should be possible to esta-
blish similar relations for other investigation areas.
Electrical resistivity
Based on the Kozeny-Carman relation which was also used as base for the method of
Ernst, the relation between the electrical resistivity and the hydraulic conductivity can be
derived. Initially this relation was established by Kozeny (1927) in the scope of field
irrigation, thus for unconsolidated sediments. This relation was later generalized and
theoretically supported by Carman (1939). It was established that the range of validity is
not confined to unconsolidated sediments. The Kozeny-Carman relation relates the
hydraulic conductivity with the specific surface and the porosity:
K~C n 3
(2 .44)
Ko (l-n) 2
where CKo is the Kozeny constant, n is the porosity, U is the specific surface and Ub is
the specific bulk surface. The specific bulk surface is the internal surface related to the
volume or Ub = (l-n) U. According to Repsold (1989) the relation between the electrical
resistivity of the matrix PItJA' and the specific bulk surface is of the form uba - 11PItJA'.
Introducing this last relation in the Kozeny-Carman relation one becomes
(2.45)
where C and the power b are material dependent.

The model of the parallel conductivities states that the electrical conductivities of
the sediments are equal to the sum of the electrical conductivity due to the solid matrix
and the electrical conductivity due to the pore water or
1 1 1
-~--+-- (2.46)
Pt Pmat Fpw
If one can dispose over a number of sediment resistivities p, along with their correspon-
ding pore water resistivities Pw in about the same material, one can deduce the matrix
resistivity PmJj' and the formation factor F of this material using the inverse model of the
parallel conductivities (Lebbe, 1994).
Archie (1942) formulated the relation between the formation factor F and the
porosity n as
F=~ (2.47)
nm
According to the Humble formula, the proportionality factor a is equal to 0.62 and the
power m, also known as sedimentation factor, is equal to 2.15. This formula is most
frequently used for slightly consolidated and unconsolidated sediments. Also customary is
the very similar formula with the value 0.81 for a and 2 for the power m. From the
chosen relation one can calculate the porosity n from the formation factor F which was
deduced with the inverse model of the parallel conductivities
(2.48)
Introducing the calculated values for the matrix resistivity PJDA' and the porosity, one can
estimate the hydraulic conductivity of the sediments from the electrical resistivity using
the relation (2.45) derived from the Kozeny-Carman relation.
Characteristics of joints
Relation between the hydraulic conductivity and the characteristics of joints has been
studied by Snow (1968) by means of hydraulic arguments. For a parallel array of planar
joints of aperture width b, with N joints per unit distance across the rock face, the
permeability may be described by
k=Cnb 2 =CNbb 2 K= Pwg Nb 3 (2.49)

1..1 12
where k is the permeability with dimensions of lengths squared, C is a dimensionless

constant, in this case (1/12) and the porosity n is taken to be a planar type of porosity
equal to Nb. Here, N has the units L· t . As an example, for a spacing of 1 joint per meter
and an opening of 0.1 cm, the permeability is 8.3x10"7 cm2 and according to Hoek and
Bray (1981) the equivalent hydraulic conductivity for pure water at 20°C is 8.lxlO·z
cm/s. This mean that one joint per meter with an opening of 0.1 cm will conduct as much
water as will 1 mZ of porous material with a hydraulic conductivity of 8.lx10"z cm/s under
identical hydraulic gradients.
2.2 Hydraulic parameters describing water storing properties
Water can be stored in a groundwater reservoir by two different phenomena. In a

completely saturated material, water can be stored due to the elastic properties of the
water and of the material. The hydraulic parameter that describes this elastic storing
property of a completely saturated material is called the specific elastic storage. Water
can also be delivered or stored by the lowering or rising of the water table. Here water is
delivered by the replacement of water by air in a part of the pores situated around the
lowering water table or the inverse around the rising water table. The hydraulic parameter
that describes the storing property of a material where the water table is fluctuating, is
called here the storage coefficient near the water table. In hydrogeological literature this
parameter is, however, more frequently indicated by the specific yield.
2.2.J Conservation of mass in a completely saturated volume of material fixed in space
The conservation of fluid mass can be formulated as follows: the sum of the mass inflow
rates minus the mass outflow rates is equal to the change in mass storage with time.
AZ
B~______ -+____~
F
AX X
AZ 0 H
L::.y Ay
AX
A E
Fig. 2.10. Small cube of saturated porous material fixed in space
Consider a small cube of saturated porous material with the sides .:lx, .:ly and .:lz
equal to unity (Fig. 2.10). The mass inflow through the face ABCD expressed in MIT is
equal to
(2.50)
with Pw the density (MIU) and q. is the Darcy velocity in the x-direction (LIT). The mass
outflow through the face EFGH is
[P",qx+ a(P.Rx)
ax
Ax 1 A A
Y Z
(2.51)
The mass inflow through the face CDHG is
Pwq~xAz (2.52)
with Pw the density (MIU) and qy is the Darcy velocity in the y-direction (LIT). The mass
outflow through the face BAEF is
[Pwqy+ a(Pwqy)
ay Ay 1 ..... x ..... z (2.53 )
The mass inflow through the face ADHE is

(2.54)
with Pw the density (MIU) and <iz is the Darcy velocity in the z-direction (Lin. The mass
outflow through the face BCGF is
[ a(pwqz)AzlA A (2.55)
Pwqz+ az x y
The net mass outflow rate is the sum of the mass outflow rates given in the Eqs.
(2.50), (2.52) and (2.54) minus the sum of the inflow rates given by the Eqs. (2.51),
(2.53) and (2.55)
(2.56)
Dividing the net mass outflow rate by the volume AxAyAz results in the net mass outflow
rate per unit volume
(2.57)
The abbreviation div(Pwq) or V·(Pwq),which is read "divergence of Pwq", describes the

space rate of change of the mass flux per unit volume (or of a vector).
The net mass outflow rate per unit volume is due to a mass storage change with respect to
time. The product Pwn is the mass of fluid per unit volume where Pw is the mass per unit
volume of material and n is the porosity of the material. So, the equation expressed in
words given at the start of this section becomes
(2.58)
Assuming that the density of the fluid does not vary spatially, the density term of the left-
hand is taken as a constant so that the Eq. 2.58 becomes
(2.59 )
The net fluid outflow rate for the unit volume .1.x.1.y.1.z equals the time rate of fluid
volume change within the unit volume.
Assuming that the principal directions coincide with the x, y and z axes, the Darcy
velocity components q., qy and q. may directly be substituted in Eq. 2.59
~ (K ah) +~ (K ah) +~ (K ah) =~ a(p..,n) (2.60)

ax x ax ay Y ay az z az Pw at
This expression now has a positive sign because the q's are negative. In the case of
isotropic and homogeneous material the right-hand side of Eq. 2.60 becomes
(2.61)
or
(2.62 )
The term (alax)(ah/ax) represents a space rate of change in the gradient across the unit
volume. Accordingly, there must follow velocity variations in the three component
directions. If the fluid mass per unit volume is not changing with time the velocity
variations cancel each other. For this condition, the Laplace's equation is obtained
(2.63 )
The solution to this equation describes the value of the hydraulic head at any point of the
three-dimensional flow field.
If, however, the hydraulic head changes with time there is a storage change of
fluid. In most hydrogeological problems, one can assume that the gains or losses in fluid
volume are proportional to the changes in hydraulic head. Water is going into storage
when the hydraulic head increases and is removed from storage if the hydraulic head
decreases. The right-hand side of Eq. 2.60 can also be formulated as follows
(2.64)
where S, is the proportionality constant that corresponds with the amount of water stored
in a unit volume of material for a unit increase of the hydraulic head. From the definition
of the proportionality constant S, one can deduce its dimensions is equal to Length-lor
L- 1 • This proportionality constant is here called the specific elastic storage although it is
called the specific storage in most hydrogeological studies.
Assuming a homogeneous and anisotropic medium for the hydraulic conductivity
and a homogeneous medium for the specific elastic storage the equation of mass conserva-
tion becomes
(2.65)
which is called the diffusion equation. The quantity KIS, is called the diffusivity and has
the unit UIT. This equation describes unsteady state, or transient, flow problems. The
solution results into values of the hydraulic head at any point of the three-dimensional
saturated flow field at any time or how the head is changing at any point of the three-
dimensional saturated flow field.
2.2.2 Specific elastic storage
When the hydraulic head decreases, water is removed from the storage of a completely
saturated material and when the hydraulic head increases water is taken into storage. A
question that arises immediately is where the water is coming from if a fully saturated
portion of a confined aquifer releases water during a drop of the hydraulic head while it
remains completely saturated. The answer to this question is that water is delivered partly
by the compaction of the aquifer and partly by the expansion of the water itself when the
hydraulic head or the pressure head decreases. When the hydraulic head increases or the
water pressure increases water is taken into storage because of the expansion of the
aquifer and because of the compression of the water itself.
From Eq. 2.64 which defines the specific elastic storage, one can deduce that the
change in water content is partly due to the change in the water density and partly due to
the change in the porosity of the water content or
(2.66)
In the first term in the right hand side of Eq. 2.66, nOPw/ot, the change of the water
density is included. This change will occur because of the expansion or the contraction of
the water. In the second term, p~nlot, the change of the porosity is included. This
change is due to the compaction or the expansion of the aquifer material.
Although the expansion or the compression of the water and the aquifer material is
very small, the total amounts cannot be ignored in the water balances of confined aquifers
with a large lateral extension and a significant drop of the head. The contribution of the
water compressibility in the specific elastic storage will first be considered. It will be
followed by the considerations on the contribution of the rock matrix compressibility in
the specific elastic storage.
2.2.2.1 Compressibility of water
Domenico and Schwartz (1990) define the isothermal compressibility of water as
(2.67)
where {jw is the fluid compressibility, M·1L'Tl,

Ew is the bulk modulus of fluid compression, M1L·1T·2 ,
Vw is the bulk volume, U,
and P is the pressure, M1L· 1T-2 •
Here, (jw reflects the bulk fluid volume response to changes in fluid pressure at a constant
temperature T and mass M. The minus sign is used because the fluid volume decreases as
the pressure increases.
Mass conservation requires that the product of fluid density and fluid volume
remain constant. A decrease in fluid density in response to a decrease in pressure is
accompanied by an increase in fluid volume. Because PwVw is a constant
(2.68)
From Eqs. 2.67 and 2.68 one can derive that
ap .. ap ah
at =.....P..Tt =.....Pwg at
A A 2
(2.69)
The last term of Eq. 2.69 is obtained if one assumes that the elevation head is invariant in
time and that the variation of the hydraulic head, h=z+P/Pwg, is only due to the variation
of the pressure head or ohlot=(1IPwg)oP/ot.
The first term of Eq. 2.66 which includes the change of the water density now
becomes
--L [n ap ",] = [p gnR ] ah

p", at '" 1'", at (2.70)
The quantity [Pwgn/3w] is the specific elastic storage with an incompressible matrix. For a
porosity of 20% and a compressibility of water at 25 °e, an amount of 9x1o-7 m3 of water
is released from one cubic meter of saturated material when the pressure head declines
1m.
2.2.2.2 Effective stress concept
The total vertical stress acting on a horizontal plane at any depth is resolved into the
intergranular pressure and the pore water pressure or
(2.71)
where is the total vertical pressure or stress,

CTt
CTjis the intergranular pressure or the effective stress,
and P is the pore water pressure or the neutral stress.
The pore water pressure acts on all sides of the grains but does not cause the
grains to press against each other. The total weight of a sediment column is partly beared
by the grains and partly by the pore water. If, for example, the pore water pressure
decreases due to a pumpage, the intergranular pressure increases because the total vertical
pressure stays constant. The part of the weight carried by the grain structure must
increase in proportion to the decrease in fluid pressure. If one assumes that the total
weight stays constant then one can derive from Eq. 2.71 that
(2.72 )
The negative sign indicates that a decrease in fluid pressure is accompanied by an

increase in intergranular pressure. This formula explains the manner by which water is
removed from storage of a completely saturated material in the vicinity of wells. Because
of the pore water pressure reduction, the water expands to the extent permitted by its
elasticity. At the same time, compression of the saturated material results in a porosity
decline, which releases water from the material. When pumping stops and the pore water
pressure restored, the grains return to their original position and the water is compressed.
If the pore water pressure does not recover, for example due to a continuous pumpage,
but instead declines steadily, there is a continuous increase of intergranular pressure,
causing the compression of the saturated material. This compression causes land subsiden-
ce.
2.2.2.3 Compressibility of matrix
An increase in intergranular pressure as defined so far merely means that the individual
grains are pushed more closely together. If the individual grains are assumed to be incom-
pressible then the grain rearrangement causes the pore volume reduction. During the
compression, it is assumed that water can drain freely out of the deforming material. So
the compressibility is described at a constant pore water pressure. The compression is
assumed to take place only in the vertical direction. Within these constrains of a constant
pore water pressure, incompressible grains and one-dimensional (vertical) compression,
the vertical reduction of pore volume is exactly equal to the volume of pore fluid
expelled. The compressibility of the matrix can then be defined as (Dominico and
Schwartz, 1990)
(2.73)
where .:iVb =.:i V p for the special case of incompressible grains. In this formulation, {3b is
the bulk (total) compressibility in units of pressure-I, Eb is the bulk modulus of compres-
sion, {3p is the vertical compressibility, Ep is the modulus of vertical compression referring
to the pores only, Vb is the bulk volume, Vp is the pore volume and CTj is the intergranular
pressure. The negative sign is again used to indicate that the volume decreases as the
intergranular pressure increases. The compressibility {3p is perfectly equivalent to the bulk
(total) compressibility (3b, that is av/aCTj = aVblaCTj • When the individual grains are
compressible, the pore volume change is no longer equal to the bulk volume change.
The change in porosity is partly due to a change in porosity and partly due to a
change of the total volume of the deforming material as can be shown by the following
equations
(2.74)
This results in
(2.75)
where Vp is the pore volume and V T is the total volume. Because the total volume
consists of the sum of the pore volume and the solid volume the change in the total
volume equals the change in the pore volume in the case where the volume of solids is
not changing with time, that is, dVT=dVp , so that
dV
dn= (l-n) --p (2.76)
VT
It is now clear that the change pore volume per unit volume is simply dn/(l-n). Substitu-
ting for dVplVT from the expression for compressibility (Eq. 2.73) gives
(2.77)
Assuming that the total vertical pressure remains constant

(2.78)
where dh=(lIp~)dP for an invariant elevation head z.

The second term of Eq. 2.66 which includes the change of the porosity
(2.79 )
Substituting the results of Eqs. 2.70 and 2.79 describing respectively the change of the
water density and the change of the porosity in Eq. 2.66, gives
~[nap ... +p
p... at at
. . an1=p g[(l-n)A +nA
W ... p......1 ah
at (2.80)
The expression Pwg[ (1-n){jp + n.Bw] corresponds with the specific elastic storage if,
however, we assume that the movement of the soil grains is negligibly small.
2.2.2.4 Movement of solids in deforming medium
A complication arises due to the fact, that the resistance of flow and the intergranular
pressure are related to the moving soil grains rather than to a fixed coordinate system.
In this respect the basic fluid mass conservation is reconsidered according to Barends
(1978)
-V.(q +nv ) =~ D(p".n) (2.81)

11' • P... Dt
where qw denotes the Darcian velocity of the pore water with respect to the solid particles
and Vs is the velocity of the solid particles with respect to a fixed coordinate system and
D(Pwn)lDt is the material derivative that follows the motion of the solid phase. This
material derivative is equal to the expression of the specific elastic storage for nonmoving
solids as derived in Eq. 2.80. The new conservation equation for moving solids becomes
now
(2.82 )
The divergence of the solid velocity field, V·v" may be interpreted as the relative rate of
growth of the bulk (total) volume or
(2.83)
From the definition of the matrix compressibility (Eq. 2.73) we can derive that
(2 .84)
Assuming again that the total vertical pressure remains constant
(2.85)
Substitution of Eq. 2.85 in Eq. 2.82 gives us the specific elastic storage considering the
movement of the grain skeleton
(2.86)
2.2.3 Hydraulic parameters derived from specific elastic storage
Several hydraulic parameters are derived from the specific elastic storage. The elastic
storage coefficient defines the storage of water in a layer of a certain thickness. The
diffusivity is a hydraulic parameters which combines storing and conducting properties of
a material. In ground mechanics this parameter is also known as coefficient of consolida-
tion.
2.2.3.1 Elastic storage coefficient
The elastic storage coefficient is the volume of water released of or taken into storage by
a layer over a certain thickness per unit of surface and per unit decline or rise of the
hydraulic head. From the definition one can derive that the dimension is m3 m·2 m- 1 or
dimensionless. It is equal to the specific elastic storage multiplied by the considered
thickness of the layer or
s=s,p (2.87)
where S is the elastic storage coefficient, S, is the specific elastic storage (L-1) and D is
the thickness (U) of the considered layer. The elastic storage coefficient is just like the
transmissivity a characteristic of a pervious layer or an aquifer where the flow is
essentially horizontal.
2.2.3.2 Diffusivity
The diffusivity is the ratio of the hydraulic conductivity and the specific elastic storage or
(2.88)
where Di is the diffusivity (UT- 1), K is the hydraulic conductivity (LTI) and S, is the
specific elastic storage (L- 1).
This hydraulic parameter represents the ratio of the 'ease of flow' to the 'ease of
volume change' or Barends (1978) defines it as the ratio between the rate of change of
hydraulic head with respect to time to the amount of water squeezed out of the pores
during the considered interval. The coefficient of consolidation used in soil mechanics
literature, corresponds merely with the hydraulic diffusivity which is here defined as the
ratio between the vertical hydraulic conductivity Kv and the specific elastic storage if the
fluid expansion is ignored.
2.2.4 Methods to derive specific elastic storage
As in the case of the hydraulic conductivities two kinds of methods are distinguished, the
direct and indirect methods. In the direct methods the specific elastic storage is measured
by an experiment. In the indirect methods the specific elastic storage is derived from
some physical properties.
2.2.4.1 Direct methods
The specific elastic storage can be determined by laboratory tests on one hand and by
field tests on the other hand. In the laboratory the specific elastic storage can be derived
by the performance of a permeameter and a compression test. As explained in Sect.
2.1.9.1 the vertical hydraulic conductivity is derived by the permeameter test. By a
compression test with confined lateral expansion the coefficient of compressibility is
measured (Barends, 1978). An undisturbed (clayey) sample is carefully fitted into a
massif ring (diameter 65 mm and height 20 mm). Pore pressures are relaxed through
porous closure stones at both sides. Compression of the sample by coaxial loading is
registered and form this measurement the soil compressibility can be obtained. According
to Eq. 2.88 the specific elastic storage can be found from the vertical hydraulic conducti-
vity and from the coefficient of consolidation.
In the field the specific elastic storage of pervious layers as well as semi-pervious
layers can be determined by single or multiple pumping tests. The specific elastic storage
of a pervious layer can be derived from the rise of the drawdown in this layer which is
pumped. The specific elastic storages of semi-pervious layers are deduced from rises of
drawdowns in these layers when one of the adjacent pervious layer is pumped.
2.2.4.2 Indirect methods
In the Netherlands the specific elastic storage can be successfully estimated according to
the depth of the sediments (Van der Gun, 1979). If the specific elastic storage is only due
the compressibility of the porous medium and the water then:
Ss 1.8x10-6 + 2. 59x10-4 d-Q-7 (2.89)
where d is the depth below the ground surface. This formula cannot be applied if there
are dissolved and undissolved gases in the water or if there are suspended solids in the
water. The formula is not applicable on sediments which underwent a large geological
preloading condition neither is it applicable on sediments with a large vertical upward
flow.
2.2.5 Storage coefficient near the water table
The storage coefficient near the water table, So, is defined as the volume of water per unit
area of soil, drained from a soil column extending from the water table to the ground
surface, per unit decline of the water table. From the definition, it can be derived that the
dimension of the storage coefficient near the water table is m3 m-2 m- 1 or dimensionless just
as the elastic storage coefficient. In hydrogeological literature this hydraulic parameter is
most frequently indicated by the term specific yield, Sy. Because this hydraulic parameter
is defined by the water holding characteristics of the material which is situated in the zone
where the water table fluctuates and also above it, the term storage coefficient near the
water table is here preferred.
The storage coefficient near the water table is determined by the interaction
between the saturated and unsaturated zone. Therefore some characteristics of the
unsaturated zone as the moisture content and the suction head, is first discussed as well as
the water table. The water table is defined as the surface on which the pressure in the
pore water of the medium is equal to the atmospheric pressure. The location of this
surface corresponds with the level at which stands the water in a shallow well. This well
has to be open along its length and is just deep enough to encounter standing water at the
bottom. At the water table the pressure head is equal to zero and the hydraulic head
equals the elevation head.
The volumetric moisture content () is defined as the ratio of the volume of water
Vw to the total volume of the soil or the material VT or () = V ';V T. The total volume VT
consists of a volume of solids V" a volume of water Vwand a volume of air Va. In the
saturated zone the volumetric water content is equal to the porosity n. The volumetric
water content is smaller than the porosity in the larger part of the unsaturated zone. In the
capillary fringe which often exists above the water table, the volumetric water content is
equal to the porosity.
In the saturated zone the pressure head hp is positive, in the unsaturated zone the
pressure head is negative. The water is held in the unsaturated zone by surface-tension
forces. In soil the negative pressure head is called the tension head or the suction head.
The negative pressure head in the unsaturated wne is measured by means of a tensiome-
ter. A tensiometer consists of a porous cup attached to an airtight, water-filled tube. The
porous cup is inserted into the soil at the desired depth, where it comes into contact with
the soil water and reaches hydraulic equilibrium. The equilibrium process involves the
passage of water through the porous cup from the tube into the soil. The vacuum created
at the top of the airtight tube is a measure of the pressure head in the soil. It is usually
measured by a vacuum gauge attached to the tube above the soil which acts as an inverted
manometer. The hydraulic head h is now equal to the algebraic sum of the elevation head
hz with the pressure head hp. Last mentioned head has a negative value.
Both the moisture content () and the hydraulic conductivity K are functions of the
pressure head hp. In Fig. 2.11 the characteristic curves are given which represent the
relation between the moisture content and the hydraulic conductivity are given for a
natural occurring sand soil (after Liakopoulos, 1965). Both relations are hysteretic; it has
a different shape when soils are wetting than when they are drying. In a sample of this
soil which is saturated, the pressure head is lowered step by step until it reached a large
negative value. The moisture content at each step would follow the drying curve (or the
drainage curve). If water is added to the dry soil in small steps, the pressure head will
take the return route along the wetting curve (or the imbibition curve). The internal lines
are called scanning curves. They show the course that the moisture content and the
pressure head will follow if the soil is only partially wetted, then dried, or vice versa
(Freeze and Cherry, 1979).
The storage coefficient near the water table So is best visualized in Fig. 2.12 . It
shows the water table position and the vertical profile of the moisture content versus the
depth in the unsaturated zone at two times, t1 and ~. The crosshatched area represents the
volume of water released from the storage from a column of unit cross section. If the
water table drop represents a unit decline, the crosshatched area represents the storage
coefficient near the water table So.
- Unsaturated ---t--Saturated
I
Tension-saturated - t
I Saturated
I moisture
~--:..-r-eontent 30
• porosity
of soil.
n-30"lo 20
I
10
!
-400 -300 -200 -100 o 100
(a)
:s
I Saturat~ 0.03 ~
_-.,.'I"-t,.h~ydrauhc 5
I nductiv~ ~
Ko-0.026
em/min 0.02.?i'
.~
0.01 1
.2
-400 -300 -200 -100 o 100

I
:t:
Pressure head. ,,(em of water)
(b)
Fig. 2.11. Characteristic curves relating hydraulic conductivity and moisture content to
pressure head for a naturally occuring sand soil (after Liakopoulos, 1965)
Source: Freeze, R.A. and Cherry, I.A., 1979, Groundwater, p.42, Copyright © 1979
Prentice-Hall, Inc. reprinted by permission of Prentice-Hall Inc.
The usual range of the storage coefficient near the water table is 0.01 to 0.30.
This means that 10 to 300 liters of water are drained from a soil column with a base of
unit surface and extending from the watertable to the ground surface when the watertable
is lowered with one unit of length. This volume of water released in the vicinity of the
water table is mostly larger than the volume of water which is released from storage due
to the elasticity of the unconfined aquifer. An unconfined aquifer of 20 m thick and a
mean specific elastic storage of 6xlO·5 m· l will release only 1.2 liters of water from
storage if the hydraulic head decreases with one unit over the whole thickness of the aqui-
fer. The storage coefficient near the water table range between 0.3 and 0.15 for gravels
and sands, between 0.1 and 0.05 for silts or loams and 0.03 to 0.01 for clays.
Moisture content, a
,, o n
'II,
,
,,
.c
g.
~
~o~__~~~~__ t,
I
I
I ---+------+---"-- t2
I
I
I
,
Fig. 2.l2.Concept of specific yield viewed in terms of the unsaturated moisture profile
above the water table (after Freeze and Cherry, 1979)
2.2.6 Hydraulic parameters derived from storage coefficient near the water table
The 'Boulton delay index' was first an empirical constant used in the interpretation of
pumping tests in unconfined aquifer with delayed yield or in semi-unconfined aquifer. It
is expressed in days and is used with the 'Boulton delay index curves' (see Sect. 3.6) to
determine the time that the delayed yield ceases to affect the drawdown. Cooley and Case
(1973) show that the 'Boulton delay index' lIa can be expressed as a function of the
storage coefficient near the watertable So and also of the thickness D2 and the vertical
conductivity KV2 of the covering semi-pervious layer or
2= SOD2 =s c (2.80)
ex KV2 0 2
where C2 is the hydraulic resistance of the covering semi-pervious layer. The dimension of
the 'Boulton delay index' is the same as the dimension of the hydraulic resistance.
2.2.7 Methods to derive storage coefficient near the water table
Three methods to determine the storage coefficient near the watertable will be treated.
The first method is derived from the knowledge of soil physics, by means of pF-curves.
The second method is a pumping test where the watertable is sufficiently influenced. In
the last sited method the storage coefficient is deduced by an inverse process based on
data about water table fluctuations.
2.2.7.1 Determination by pF-curves
The pF-curves shows the relation between the soil suction and the volumetric moisture
content. The soil suction is here defined as the logarithm in base 10 of the soil suction ex-
pressed in centimeters water column. As was already explained, these relations are not
simple but are hysteretic. From these pF-curves one can deduce the field capacity and the
drainable porosity. The field capacity is defined as the water content that remains in a
unit volume of soil after gravity drainage has ceased. By convention the field capacity
corresponds with the volumetric moisture content at a pF-value of 2, i.e., a soil suction of
100 centimeters water column or pressure head of -1 m. The complement of the field
capacity (JpF=2 is called the drainable porosity nd • It is defined as the volume of water
drained by gravity from a unit volume of saturated soil. The sum of the field capacity and
the drainable porosity is equal to the porosity n.
The storage coefficient near the water table is indicated by the crosshatched area
of Fig. 2.12 . For a homogeneous isotropic soil, the two curves which limit the crosshat-
ched area and represents the vertical profile of the moisture content at two different times
t1 and tz, are identical in shape. If, in both cases the water table is sufficiently deep below
the ground surface, the two curves will merge at (JPF=2. For a homogeneous isotropic soil
and a very deep water table, the storage coefficient near the water table or the specific
yield is identical to the drainable porosity. In heterogeneous and anisotropic soils with
eventually a relatively undeep water table, the storage coefficient near the water table can
only be deduced by the simulation of the flow in the unsaturated zone due to a lowering
of the water table with one unit. For this simulation, the relations between the hydraulic
conductivity and the soil suction and the moisture content for every layer has to be known
and taken into account.
2.2.7.2 Determination by pumping tests
The storage coefficient near the water table can be determined by a pumping test in an
unconfined aquifer. The screens of the observation wells can be placed on two different
levels. They can be located either in the directly pumped part of the aquifer or in the
upper part of the unconfined layer at a small distance below the water table.
The storage coefficient near the water table can be deduced from the last segment
of the time-drawdown curves measured in observation wells which are situated not too
close to the pumped well. These time-drawdown curves plotted on bi-logarithmic paper
have three distinct segments. The first segment, covering a short period after pumping has
begun shows that an unconfined aquifer reacts initially in the same way as a confined
aquifer. Water is released instantaneously from the elasticity of the porous medium and
the water. Gravity drainage is not started. The second segment of the time-drawdown
curve shows a decrease in slope because of the vertical flow of the water of the zone just
below and above the water table. The third segment, which may start from several
minutes to several days after pumping has begun, shows again an increase in slope. This
is due to the drop of the water table. With the Boulton interpretation method, the storage
coefficient near the water table can be deduced from this last segment (Kruseman and De
Ridder, 1970).
By placing a short well screen of an observation well just underneath the water
table the drop of the water table can be followed. Usually the water table will show only
a relatively small drop after a relatively long period of pumpage. This drop is very
sensitive to the storage coefficient near the water table. Using an inverse numerical model
this drawdown at the top of the unconfined layer can be interpreted (Lebbe, 1988). It
results in a value of the storage coefficient with a relatively good accuracy.
2.2.7.3 Determination by inverse models of unsteady state flow
By studying the long term fluctuations of the water table and the hydraulic head the
storage coefficient can be deduced. A typical long term fluctuation is the seasonal
fluctuation of the water table and the hydraulic head of unconfined and semi-confined
aquifers. Long term variations of the hydraulic head can also occur due to seasonal
change of discharge rates or to seasonal variation of canals, rivers or lake levels. By the
interpretation of this long term fluctuation using an inverse model the storage coefficient
near the water table can be deduced with good accuracy. The deduced values are in most
cases an average value for a relative large representative area. Along with the other
parameters and their joint confidence region they can be inserted in forecasting models
which will yield probability distributions of the responses. The other parameters are
derived from the inverse model such as horizontal conductivity of the pervious layers and
vertical conductivities of the semi-pervious layers.
Tarhouni and Lebbe (1993) have derived the storage coefficient near the water
table by means of an inverse three-dimensional numerical model for the dune area of De
Panne (Belgium). In some parts of the dune area the water table fluctuates principally due
to the seasonal fluctuation of the recharge rates. Around and in the water catchment area
the water table fluctuations are principally caused by the seasonal fluctuations of the
pumping rates. The input data of the inverse model were merely the monthly or half
monthly observations in 106 different wells during the period June 1975 to March 1978.
Most of the observation wells have screens a few meters under the water table. These
wells were well distributed over the area. A few wells have screens which are located in
deeper layers. The optimized value for the storage coefficient near the water table was
0.19. The water table was principally situated in the very well sorted medium to fine
dune sands.
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delingen, part XIX, Delft, Delft Soil Mechanics Laboratory, 150 p.
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Cooley, R.L., and Case, R.M., 1973, Effect of a watertable aquitard on drawdown in an
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Cooper, H.H., Jr., Bredehoeft, J.D., and Papadopoulos, I.S., 1967, Response of a finite-
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Davis, S.N., 1969, Porosity and permeability of natural materials: Flow Through Porous
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De Ridder N.A., and Wit, K.E., 1965, A comparative study on the hydraulic conductivi
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Domenico,P.A., and Schwartz, F.W., 1990, Physical and Chemical Hydrogeology: New
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Freeze, R.A. and Cherry, J.A., 1979, Groundwater: Englewood Cliffs, New Jersey,
Prentice-Hall, 604 p.
Folk, R.L. and Ward, W.C., 1957, Brasos river bar: a study in the significance of grain
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Hazen, A., 1911, Discussion of "Dams on Soil Foundations": Trans. Amer. Soc. Civil.
Engrs., v. 73, p. 199.
Hoek, E. and Bray, J.W., 1981. Rock Slope Engeneering: Londen, Inst. of Min. and
Met.
Hvorslev, M.J., 1951, Time lag and soil permeability in groundwater observations: U.S.
Army Corps Engrs. WatelWays Exp. Sta. Bull. 36, Vicksburg, Miss.
Jacob, C.E., 1947, Drawdown test to determine effective radius of artesian well: Trans.
Amer. Soc. Civil Engrs., v. 112, p. 1047-1070.
Kessler, J., and Oosterbaan, R.J., 1974, Determining hydraulic conductivity of soils:
Wageningen ILRl, Publ. 16, v. III, p. 253-296.
Konikow, L.F., and Bredehoeft, J.D., 1978, Computer model of two-dimensional solute
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Kozeny, J., 1927, Uber die kapillare Leitung des Wassers im Boden: Sitz. b. Akad. Wiss.
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563 p.
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Hydrogeology. Vol. 9. Hannover, Heise. 136 p.
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artesian well. 23p. Proc. Soc. Civil. Engrs., v. 79.
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Starn, 1.M.T., and Zijl, W., 1992, Modelling permeability in perfectly layered porous
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Chapter 3 / Evolution of analytical models of
pumping tests and their interpretation methods
A large number of analytical models was developed. Here only a few steps in the
evolution of these models are treated. The steps are first selected to give the reader an
idea about the historical evolution of the analytical models so that it becomes clear that
the first models deal exclusively with flow in the directly pumped pervious layer. In the
following developed models the flow from the adjacent layers to directly pumped layer is
incorporated with increasing accuracy. The application frequency of the interpretation
methods that are derived from these models was another criterion for the selection of the
treated models.
3.1 Model of Thiem
In the model of Thiem (1906) a steady state flow to a pumped well is considered in a
confined aquifer. The screen of the pumped well is situated over the full thickness of the
pumped pervious layer. The flow from the bounding layers to the pumped layer does not
exist or does not affect the drawdowns in the pumped layer. Consequently, it is assumed
that the bounding layers are impervious.
3.1.1 Introduction
The transmissivity of a confined aquifer is derived by the performance of a pumping test

with a constant discharge rate. The solution requires the knowledge of the discharge rate
and the drawdown in at least two observation wells which are placed at different distances
from the pumped well. The required drawdowns are those reached after a long period of
pumping when the equilibrium is approached.
The assumptions and conditions of the model are as follows (Fig. 3.1):
- the pumped layer is bounded above and below by an impervious layer and forms a
confined layer with a seemingly infinite lateral extent,
- the pervious layer is homogeneous and has a constant thickness,
- the discharge rate is constant,
- the pumped well has a screen over the entire thickness of the layer so that there is only
horizontal radial flow in the pumped layer,
- the flow approaches the steady state and the hydraulic head is almost constant during the
last part of the test.
56 Chapter 3 / Evolution of analytical models of pumping tests
MODEL OF THIEM
"IMPERVIOUS" LAYER
PERVIOUS LAYER
REFERENCE LEVEL
OF
HYDRAUUC HEAD
"IMPERVIOUS" LAYER
Fig. 3.1. Schematization of the groundwater reservoir, the flow and the drawdown in
the model of Thiem (1906)
01 __ II 01 = 02
--_ . . . . ,1
-.......... "-
'J ,/
..... .,.,-
Fig. 3.2. The discharge rates through successive coaxial cylinders are the same as the
pumped discharge rate when the steady flow in a confined aquifer is approached
Chapter 3 / Evolution of analytical models of pumping tests 57
3.1.2 Derivation of the basic differential equation
Because the flow reaches a steady state, one can say that the discharge rate is the same
through all the successive coaxial cylinders around the pumped well (Fig. 3.2). This
discharge rate equals the discharge wells through the pumped well and so:
Q = 21trD V{ (3.1)
where Q is the discharge rate (UT"l),

r is the distance from the pumped well (L),
D is the thickness of the confined aquifer (L),
Vb' is the horizontal Darcian flow at a distance r from the pumped well (LT"l).
The constant discharge rate through the successive coaxial cylinders is symbolically
represented in Fig. 3.1 by an arrow of constant thickness.
The horizontal Darcian flow velocity at a distance r from the pumped well can be
derived by the law of Darcy
-K ah (3.2)
h ar
where Kb is the horizontal hydraulic conductivity of the pervious layer (LT"l) and h is the
hydraulic head.
The drawdown s is defined as the difference between the hydraulic head before the start
of the pumping test, the initial hydraulic head H, and the hydraulic head at the end of the
pumping test, h, or
s = H-h (3.3)
and consequently,
as (3.4)
ar
By substitution of the equation (3.2) in equation (3.1) and replacing of -oh/or by os/or
one obtains the basic differential equation.
Q = 21trD Kh ar
as (3.5)
or
Q ar (3.6)
r
3.1.3 Solution of basic differential equation
By the integration of both sides of the equation, the relation is obtained between the
discharge rate, Q, the drawdown, s, on a distance, r, from the pumped well and the
transmissivity KbD of the pervious layer.
Q lnr = 21t KhD s + Cst (3.7)
where Cst is the integration constant.

This relation was already deduced earlier by different researchers as for example Dupuit
(1863), Slichter (1899), Turneaure and Russell (1901). For the determination of the
integration constant they assume the unpractical boundary condition of a very small
drawdown (or s=O) at a distance R from the pumped well. So, the integration constant
becomes QlnR. By introducing this constant in Eq. 3.7 the Dupuit equation is obtained.
(3.8)
According to Wenzel (1942), Thiem (1906) was the first who derived an interpretation
method based on Eq. 3.7. The required data are the observed drawdowns in two
observation wells at two different distances from the pumped well: the drawdown s) at a
distance r) and the drawdown S2 at a distance r 2 • The introduction of these values in Eq.
3.7 result in:
Q In (rl) =21t KhD Sl + Cst
(3.9)
Q In (r2) =21t KhD 3 2 + Cst
By means of these two equations the integration constant can be eliminated to arrive at the
Thiem equation
(3 .10)
With a similar derivation the relation between the horizontal conductivity, the discharge
rate and the drawdown at two different distances from the pumped well in a phreatic
aquifer is found:
K- Q In(r 1 /r)
(3.11)
h- 1t (Dl+D2) (Sl- S 2)
In this equation Dl and D2 are the thicknesses of the phreatic aquifer at the distances r)
and r2 when steady state flow is almost reached (Dl=H-s) and D2=H-s 2). The Eq. 3.11
differs only from Eq. 3.10 by the replacement of 2D by (Dl+D2).
3.1.4 Application of Thiem method
Wenzel (1942) describes the application of this method using a pumping test where a
large number of observation wells were placed at different distances from the pumped
well in different directions. The pumping test was performed in 1937 in the Valley of the
Platte in Nebraska. The hydraulic head was measured in eighty-four observation wells.
The distances from the pumped well varied between 1 and 360 m. The surveyed aquifer
consists of sands and gravel and has a thickness of 30 m. During a period of 48 hours
water was pumped with a constant discharge rate of 2943 m3 /d.
Based on this large amount of field observations, Wenzel (1942) concluded that
there were large differences between the real flow conditions and those assumed in the
model of Thiem. The difference is so important that the results obtained by the unconditi-
onal application of the Thiem interpretation method were completely wrong. Therefore,
Wenzel (1942) developed a method to apply the equilibrium equation empirically on the
field data. The proposed method required, however, many observation wells so that the
deduction of the horizontal conductivity with this method is not practical.
3.2 Model of Theis
In the model of Theis (1935) the unsteady state flow is treated in a confined aquifer. The
well screen is situated over the full thickness of the pumped pervious layer. The pumped
discharge rate is constant during the pumping test.
3.2.1 Introduction
The transmissivity and the elastic storage coefficient of a confined layer can be derived
from a time-drawdown curve which is observed in the pumped layer at a known distance
from the pumped well. From this well a constant discharge is withdrawn. For the Thiem
interpretation method it is necessary to observe the drawdown evolution for the selection
of the appropriate drawdowns at the end of the test. By the observation of this evolution,
there was soon a need for a method for the interpretation of the time-drawdown curves
which are measured in different observation wells. Theis (1935) was the first who
proposed such an interpretation method. He derived the equation that describes the
evolution in time of the drawdown after starting the pump. This equation was derived by
analogy to the theory of head conduction in solid media.
The assumption and the boundary conditions of the model of Theis are:
- the pumped pervious layer is bounded above and below by impervious layers and forms
a confined aquifer with a seemingly infinite lateral extent,
- the pervious layer is homogeneous and has the same thickness over the entire influence
area of the pumping test,
- the discharge rate is constant during the pumping test,

- the pumped well has a screen over the full thickness of the pumped pervious layer so
that there is only horizontal flow,
- the flow to the pumped well has not reached a steady state so that the hydraulic head
and the drawdown in the pervious layer change in time,
- the diameter of the pumped well is very small so that the well storage can be ignored.
When the flow has not reached equilibrium, the discharge rate through the
successive coaxial cylinders decreases with increasing distance from the pumped well.
The decrease of the discharge between two successive coaxial cylinders is equal to the
amount of water that is released per unit of time by storage decrease. This decrease in
storage is released by the volume of confined aquifer that is situated between the two
coaxial cylinders by the decrease in hydraulic head. The discharge rate through the
successive coaxial cylinders is symbolically represented by arrows in Fig. 3.3. Because
these discharge rates decrease with increasing distances from the pumped well the arrows
are plotted with decreasing thicknesses. The storage decreases are symbolically represen-
ted by bold circles between the arrows.
3.2.2 Derivation of basic differential equation
The flow through a coaxial cylinder of the pumped well can be derived in a similar
manner as Eq. 3.5.
as
Q = 21t KiP r ar (3.12 )
When water is withdrawn from the pumped well, the discharge rate Q has a negative rate.
This value is negative because the flow is then in the opposite direction of the r-axis.
The drawdown depends not only on the distance from the pumped well r like in the
steady state flow but also on the time. The discharge rate through the successive coaxial
cylinders is not the same. Here, this rate depends on the time since starting the pump and
is function of the radius of the coaxial cylinder. The change of the discharge rate between
coaxial cylinders around a pumped well with radii equal to rand r+..:lr (Fig. 3.4) is
(3.13)
From Eq. 3.12 one derive that
aQ
ar (3.14)
MODEL OF THEIS
"IMPERVIOUS" LAYER
PERVIOUS LAYER
REFERENCE LEVEL
OF
HYDRAULIC HEAD
"IMPERVIOUS" LAYER
Fig. 3.3. Schematization of the groundwater reservoir, the flow and the drawdown in
the model of Theis (1935)
01(t) < 02(t)
--------
Fig. 3.4. Flow through two successive cylinders around the pumped well at a conside-
red t after the start of the pump Q\(t) and Q2(t)
Combining Eqs. 3.12 and 3.14 one obtains:
Q Q 2 K n (r iPs + as) t.r (3.15)

I+~I- I = 1t lr' ar2 ar
In a confined aquifer the discharge rate between two successive cylinders changes
because water is released from storage by the hydraulic head drop in the aquifer volume
situated between the two coaxial cylinders. The water released by this volume of confined
aquifer comes from the compression of the pervious layer and in most cases to a lesser
extent from the expansion of the water when the hydraulic head decreases. The change in
discharge rate between two coaxial cylinders is proportional to the volume of the
concerned confined aquifer (21!Tt.rD), the specific elastic storage of the layer (S,) and the
rate of drawdown (as/at) or
(3.16)
From Eqs. 3.15 and 3.16 it follows that:
21t KhD (r iPs + as) t.r as (3.17)

21trt.rD Ss at
ar2 ar
Dividing both parts of Eq. 3.17 by the volume which is comprised between the two
coaxial cylinders and by the transmissivity KbD results in the basic differential equation of
the unsteady-state flow in a confined aquifer:
a2 s 1 as
--+-- (3.18 )
ar2 r ar
3.2.3 Solution of basic differential equation for constant discharge rate
Following expression satisfies the basic differential equation of the unsteady-state flow in
the confined aquifer:
s =
v (3.19)
41tKj,Dt
This is a special solution of this basic differential equation. This expression describes the
drawdown in the confined aquifer when a volume of water V is abruptly withdrawn at the
time t=O in a well with a screen over the full thickness of the confined aquifer. It is
assumed that the well has a very small diameter which coincides with r=O. The draw-
down caused by the withdrawal of a volume of water VI from the same well at a time
t=tl is then given by:
v _ Spr
s= I e~ (3.20)
41TKh D( t - t l )
Because the differential equation (Eq. 3.18) is linear in the drawdown s, one can apply
the method of superposition. By this superposition, the sum of the effect of several
separate actions is equal to the effect of all those actions together.
During a pumping test with a constant discharge rate Q, a volume of water is
withdrawn on each moment equal to Qd1'. The drawdown due to this withdrawal on the
time t=1' is:
s Q dt' (3.21)
41tKJP( t-t')
The drawdown created by all withdrawals between the time l' =0 and l' =t is the sum of
all the drawdowns caused by all those discrete volume withdrawals or:
S"D r'
S = r t'=t Q e 4KJP( t-t') dt' (3.22)
Jt,=o 41tKhD( t-t')
Because Q/(4'lIKh D) is constant one can put this constant outside the integration sign.
S"D r'
S = __Q_ _ rt'=t 1 e 4KJP(t-t') dt' (3.23 )
41tKJPJ t ,=o (t-t')
Replacing the variable l' by the variable y which is defined as:
r2 SJ]
Y= (2.24)
4KJP( t-t')
then for l' =0
(3.25)
for l' =t
r 2 sJ]
Y= 00 (3.26)
4KJP( t-t')
and
dy (3.27)
dt l 4KJP( t - t / ) 2
or
dt l = r 2 S,pdy (3.28)
4KJP y2
Substitution of rS,D/(4KbD(t-t') by y and dt' by r2S,Ddy/(4KbDyZ) in the integral of Eq.

3.23 and also its limits results in:
S - Q f~ 1 (3.29)
41tKJP r 2S"D/ (4KJP) (t- t/)
From the equation above and Eq. 3.22 one derives that
S = --Q-f~ e -Y dy (3.30)
41tKJP r 2S"D/4KJPt y
The value r2S,D/(4KbDt) is indicated by the letter u in the hydrogeological literature, so:
S = -_Q- f~ e-Y dy (3.31)

41tKJP u y
The integral in the right part of the above given equation is equal to an infinite series in
which the lower limit is concerned:
fu~e-Y dy
y
= -0.5772 -lnu +u _ u2
2.2!
+_u_3 _
3.3!
_ u4
4.4!
+ (3.32)
In the hydrogeological literature the above given integral is called the well function of
Theis and is indicated by the symbol W(u). The relation of the drawdown s can now be
given by the very simple relation:
S _ Q - W(u) (3.33)
41tKJP
This relation satisfies the following initial and boundary conditions:

s(r,O) 0, r ~ °
s(oo,t) 0, t ~ ° (3.34)
meaning respectively that the initial drawdown is zero at the time t=O and at every
distance from the pumped well and that the drawdown stays zero at an infinite distance
from the pumped well. The initial and the boundary condition at the pumped well are:
Q = 0, t < 0
Q=cst, t ~ 0
(3.35)
lim ras=~, t 0
x-o ar 21CKhD
~
signifying that there was no pumping before the start of the pumping test; the pumping
rate is constant from the start of the test, t=O; and that the discharge rate of the pumped
well with a very small diameter is the same as the pumped discharge rate.
3.2.4 Theis and Cooper-Jacob interpretation methods
The Theis type curve is first plotted on double logarithmic paper with the function l/u on
the abscissa and the Theis well function W(u) on the ordinate. The observed drawdowns
of one observation well are also plotted on double logarithmic paper with the same
modulus length. The drawdown is put in the ordinate and the time is put on the abscissa.
In the interpretation method of Theis the best match between the plotted observed
drawdown data and the Theis type curve is searched. The coordinate axes of the type
curve and of the date plot must be held parallel. Once the best position is found, the
position of the best match is fixed. An arbitrary match point is selected on the overlap-
ping portion of the two data sheets. The coordinates W(u), 1/u, sand t of the match point
are determined. For all points of the overlapping sheets the ratios W(u)/s and t/(l/u) are
the same. Usually, a point is chosen corresponding to one or a power of ten for W(u) and
u. With the four coordinates values of the matching point the transmissivity ~D and SsD
are derived. By means of the ratio W(u)/s, the transmissivity is first derived.
K D = ~ W(u) (3.36)
h 41CS
where Q is the discharge rate (UT· 1),

s the drawdown of the arbitrary chosen point (L),
W(u) the value of the Theis well function of the chosen point (dimensionless).
The ratio t/(l/u) and the obtained value of the transmissivity KhD enable us to calculate
the elastic storage coefficient S:
(3.37)
where t is the time of the arbitrary chosen point (T),

u is the reciprocal value of l/u of the point (dimensionless),
and r is the distance from the pumped well (L).
Cooper and Jacob (1946) pointed out that for small values of u, smaller than 0.01, the
terms following the term In u in Eq. 3.32 can be ignored. The observations that satisfy
the condition u<O.Ol, show a flat course when they are plotted in time-drawdown graphs
on double logarithmic paper. This course can be compared with the course of the last part
of the Theis type curve (curve corresponding with r/L=O in Fig.3.8). Because of this flat
course, the comparison of the Theis type curve with the observed data is very difficult. In
those cases the transmissivity and the elastic storage coefficient can best be found with the
interpretation method of Cooper-Jacob.
In the interpretation method of Cooper-Jacob the Theis well function W(u) is
simplified to the following relation:
W(u) ~ -0.5772 -In u (3.38)
The Eulerian constant -0.5772 corresponds with the Neperian logarithm of 2.25/4 and so
one can give another approximate expression for the Theis well function:
(3.39)
Because 4u r2S,D/(KbDt) the approximate expression for the Theis well function is
also:
W(u) '" In( 2.25KJJt) 2.25KJJt

2.301og lO ( ) (3.40)
r 2 S,D r 2 S,D
When u is smaller than 0.01 the drawdown in a confined aquifer due to a pumping test
with a constant discharge rate can be calculated by the following approximate expression:
s = 2.30 log (2. 25KhDt) (3.41)

10 r2 S,jJ
The Cooper-Jacob interpretation method is based on this equation.

In this interpretation method the observed drawdowns are plotted against the time
on semi-logarithmic paper. The drawdown is put on the arithmetic ordinate and the time
on the logarithmic abscissa. A tangent is drawn through the points corresponding with an
u-value smaller than 0.01. The slope of the tangent line ~s is deduced. This is the
drawdown difference per log cycle of time. Sometimes, the slope ~s is also called the
drawdown difference over one modulus length. Besides the slope of the tangent, the
intercept of the tangent with the time axis (s =0) is deduced and is here indicated by the
symbol ~.
The transmissivity KbD is defined by the slope ~s with the relation:
KhD = 2. 30Q (3.42)

41t~S
The specific elastic storage coefficient S is derived by means of the intercept of the
tangent with the time axis ~:
S = SJJ = 2.25 KtP~

r2
(3.43)
with r the distance frqm the pumped well.
3.2.5 Comments concerning Theis and Cooper-Jacob interpretation methods
Both interpretation methods are the first valuable steps in the development of pumping
test interpretation methods. During the application of these methods one must keep in
mind that both interpretation methods are derived from the model of Theis. The boundary
conditions of this model are, however, seldom met in a practical problem.
Wenzel (1942) was one of the first to apply the Theis interpretation method on a
large amount of field data. He found that the derived hydraulic conductivities depend
strongly on the observation well of which the data were used. These differences were
ascribed by Wenzel to the slow drainage of water from the material. Wenzel pointed out
that there were some differences between the theoretically assumed and the actual flow
condition of the pumping test. Some of these differences are worth mentioning. They are
the heterogeneity of the aquifer, the partially penetrating well and the ignorance of the
vertical flow. Although Theis (1935) had set an important step in the interpretation of
pumping tests, it was soon clear that the theoretical model of unsteady-state flow to a
pumped well in a confined aquifer is only applicable in a few cases. Therefore, this
theoretical approximation of the unsteady-state flow was adjusted very soon. The first
important adjustments of the Theis model were related to the boundary conditions above
and below the pumped layer. One of these boundaries or both are no longer considered
impervious. In the model, which will be treated in the next part, a vertical flow is
assumed through at least one of these boundaries.
3.3 Model of Jacob-Hantush
The radial flow in a semi-confined aquifer to a pumping well with a screen over the full
thickness of the aquifer was treated in a large number of works. First, De Glee (1930)
treats the steady-state flow in a semi-confined aquifer and presented his interpretation
method. The unsteady-state flow in the semi-confined aquifer was first treated by Jacob
(1946). This theory was further adjusted in Hantush and Jacob (1955). According to this
model a new interpretation method was presented by Walton (1962). This method is
similar to the Theis interpretation method. Similar to the Cooper-Jacob interpretation
method corresponding with the model of Theis three different interpretation methods were
developed: the Hantush-Jacob, the Hantush I and Hantush II interpretation method.
3.3.1 Introduction
The transmissivity and the elastic storage coefficient of the semi-confined aquifer can be
derived from a time-drawdown curve. This curve was measured in an observation well at
a known distance from the pumped well. During the test water was pumped with a
constant discharge rate. The hydraulic resistance of one of the bounding semi-pervious
layers or the resulting hydraulic resistance of both layers can be derived from the way the
time-drawdown curve reaches a constant drawdown after a certain time of pumping.
The differential equation of the unsteady-state radial flow to the pumped well with
a linear leakage was first treated by Jacob (1946). He proposed a solution for the equation
with the following boundary conditions: a constant discharge rate to the pumped well and
a constant hydraulic head in the pumped layer at a well-defined distance from the pumped
well. In Hantush and Jacob (1955) a solution is proposed for the same differential
equation with a constant hydraulic head boundary in the pumped layer that is at an infinite
distance from the pumped well. Based on the preceding work, Walton (1962) developed
an interpretation method similar to the Theis method. Instead of one type curve as in the
interpretation method of Theis, the interpretation method of Walton is characterized by a
series of type curves.
The assumptions and the boundary conditions of the model can be formulated as
follows (Fig. 3.5):
- the pervious layer is bounded above and below by semi-pervious layers or by one
impervious layer and a semi-pervious layer,
- the pervious layer is homogeneous, isotropic and has the same thickness over the
influence area of the pumping test,
- the discharge rate is constant,
- the pumped well has a screen over the entire thickness of the pumped layer,
- the flow to the pumped well is in an unsteady-state, the hydraulic head or the drawdown
in the impervious layer varies with time,
- the diameter of the pumped well is very small so that the storage in it can be ignored,
- the leakage from the bounding semi-pervious layer(s) is proportional to the drawdown in
the pervious layer at every distance from the pumped well so that the storage decrease in
the semi-pervious layers is ignored.
When the flow has not reached equilibrium, the discharge rate through the
successive coaxial cylinders decreases with increasing distance from the pumped well.
This decrease of the discharge between two successive coaxial cylinders is not only due to
the storage decrease. It is also a consequence of the amount of water delivered by the
leakage from the adjacent semi-pervious layer(s). In Fig. 3.5 the increasing discharge rate
through the successive coaxial cylinders with decreasing radius is symbolically represen-
ted by arrows with increasing thickness. The storage decrease is represented by bold
circles between the arrows. The vertical arrows with a constant thickness represent the
exclusive vertical flow through the semi-pervious layer(s) without storage decrease.
MODEL OF JACOB - HANTUSH
PERVIOUS LAYER AT
CONSTANT HYDRAULIC HEAD
SEMI-PERVIOUS
LAYER
PERVIOUS LAYER
SEMI-PERVIOUS
LAYER
PERVIOUS LAYER AT CONSTANT HYDRAULIC HEAD
Fig_3.5. Schematization of the groundwater reservoir, the flow, and the drawdown in
the model of lacob-Hantush (1946, 1955)
3.3.2 Derivation of basic differential equation
Based on the law of Darcy the change in discharge rate between two successive
coaxial cylinders around a pumped well can be formulated as in Eq. 3.15:
(3.44)
The change in discharge rate between two successive coaxial cylinders due to the storage
decrease can be formulated as in Eq. 3.18:
(3.45 )
The change in discharge rate between two successive coaxial cylinders due to leakage
from the adjacent semi-pervious layer(s) is proportional to the surface between the two
concentric circles with radius rand r+.ilr (21r1".ilr) and is assumed to be proportional to
the drawdown in the pervious layer at a distance r from the pumped well and the sum of
the reciprocal values of the hydraulic resistances of the bounding semi-pervious layers:
21tr.1r s (~+~) (3.46)

c1 c3
where C1 is the hydraulic resistance of the subjacent semi-pervious layer

and ~ is the hydraulic resistance of the superjacent semi-pervious layer.
When the reciprocal value of the resulting hydraulic resistance Cr is the sum of the
reciprocal values of the hydraulic resistance of the adjacent semi-pervious layers then the
change of the discharge rate between two successive adjacent cylinders due to leakage has
the simple expression of
21tr.1r (3.47)
The value of the resulting hydraulic resistance is equal to the hydraulic resistance of the
semi-pervious layer with the smallest hydraulic resistance when the hydraulic resistance of
the other semi-pervious layer is much higher or when the other layer can be considered as
impervious.
The total change of the discharge rate between two successive coaxial cylinders is
due to storage decrease in the pumped pervious layer and due to leakage from the
adjacent semi-pervious layers:
(3.48)
From Eqs. 3.44 and 3.48 it follows that:
21t KIP (r a 2 s + as) .1r (3.49)

ar2 ar
Dividing both parts of Eq. 3.49 by the surface which is comprised between the two
concentric circles with radius rand r+.ilr and by the transmissivity KbD results in the
basic differential equation of the unsteady-state flow of water in a semi-confined aquifer:
~s 1 as s S"D as (3.50)
--+----=----
ar2 r ar L2 KIP at
where L is the leakage factor which is defined as:

L=';KhfJ C r (3.51)
The leakage factor is a measure for the ratio between the horiwntal flow in the pumped
layer to the vertical flow in the adjacent layers. The larger the leakage factor, the larger
the discharge is in the pervious layer and/or the smaller the leakage is from the adjacent
layers.
In the basic differential equation (Eq. 3.50) one assumes that the leakage from the
adjacent pervious layer(s) is proportional to the drawdown in the pumped layer. As a
consequence of this assumption the storage decreases in the semi-pervious layers are
ignored. The flow in the adjacent semi-pervious layer due to pumping in the pervious
layer is thus strongly simplified. In reality, an elastic flow which is principally vertical
arises in the semi-pervious layer. Therefore, this flow depends strongly on the vertical
conductivity and specific elastic storage of the semi-pervious layer or its diffusivity. The
horiwntal component of the flow in the semi-pervious layer is usually very small so that
the horizontal conductivity is less important.
3.3.3 Solution of basic differential equation for a constant discharge rate
Particular solutions of the basic differential equation (Eq. 3.50) are the expressions:
(,,'+1) KIP t
J. ..!!.:£ e (3.52)
o L
S"D L' and K (..£)
o L
where 10 is the Bessel function of the first kind and of zero order
and Ko is the modified Bessel function of the second kind and of zero order.
Because the basic differential equation is linear in s and homogeneous both equations in
Eq. 3.52 can be combined in one solution:
(,,'+1) KIP t
s = Cst [rmA(a)Jo ( aIle S"DL2 da+Ko(..£)] (3.53 )
Jo L L
where Cst is an arbitrary constant and A(a) is a function which depends only on a.
Consequently, A(a) does not depend on the variables of the basic differential equation.
By means of the initial and boundary conditions one can derive the constant Cst and the
function A(a).
The initial and boundary conditions are:
- s(r,O)=O, r~O or the initial drawdown is zero at every distance from the pumped well,
- s(oo,t)=O, t~O or the drawdown is zero at every time after starting the pump and at an
infinite distance from the pumped well,
- lim.-.o ras/ar= - Q/(2'lrKh D), t~O or the discharge rate is equal to the pumped discharge
rate in the vicinity of the pumped well with a very small diameter and at every time after
the start of the pump.
The solution in Eq. 3.53 satisfies the condition that the drawdown is zero at every
time after starting the pump and at an infinite distance from the pumped well. The value
of the constant Cst is found by the condition of the discharge rate to and near a small
diameter pumped well.
Because limr-oO Ko(r/L) = -0.5772 -In(r12L) one can derive that Cst= Q/(2'lrKbD). The
function A(a) can be derived from the condition that the initial drawdown is zero at every
distance from the pumped well. So:
o (3.54)
Hantush (1955) used the integral relation given by Watson (1944):
K (..:£) = r~ ( _a_) J. (~) da (3.55)

o L )0 a2+1 0 L
to rewrite Eq. 3.54 as:
(3.56)
From this last equation one can see that:
A (a) (3.57)
When the constant Cst and the function A(a) are substituted in Eq. 3.53 then:
s Q [K ( r) -J:~_a_J.o (~) e -P("2+ 1 ) da] (3.58 )

n
2 1tK""n- 0 -L
0 a 2 +1 L
where p = KbDt/(S,D U).

When pumping continues, the integral in Eq. 3.58 becomes zero at t~oo and one
obtains the solution of the steady-state flow which was already obtained by De Glee
(1930). The maximal drawdown sm is than:
s m = -_Q-
21tKtp
K (..:£)
0 L
(3.59)
For the evaluation of the integral in Eq. 3.58, which is further indicated by the symbol I,
one proceeds this way. The integral representation of 1/(if+ 1) is
fo~ e -x(,,'+l) dx. (3.60)
When one substitutes 1/(~+ 1) by its integral form and one changes the integration
sequence then one obtains:
I = r~e-(P+x)dx. r mJ (~)e-(P+x)a.'a.dx. (3.61)

Jo Jo 0 L
The change in integration sequence is justified because the integral I converges. Integra-
tion with respect to ()l results in:
r mJ (~) e-(p+x)a.'a.da. 1 e 4 (p+x)L' (3.62 )

Jo 0 L 2 (p+x)
Consequently, Eq. 3.61 results in:
I=..!.lm_1_e -p-x 4(p+x)L' dx. (3.63 )

2 0 p+x
Substitution of (p+x) by y results in:

r'
I = ..!.Jm..!. e -Y- 4yL' dy (3.64)
2 p Y
The solution of the basic differential equation that satisfies the initial and boundary
condition can now be written as:
r2
S = -_Q- [2K (..:£) _Jm..!. e -y- 4yL 2 dy] (3.65)
4-n KaD 0 L p Y
Using the integral relation:
(3.66)
one obtains:
r2 x2
s -_Q_- [l~ ..!. e -y- 4yL 2 dy _Jm..!. e -y- 4yL 2 dy]
41tKaD 0 y p y
r2
(3.67)
or s --Q--l P..!. e -Y- 4yL 2 dy

4-nKaD 0 y
Substitution of the variable y by z where z = r2/4yU then
z
(3.68)
Then Eq. 3.66 becomes:
s __Q_
41tKIP
[1 ~
2
u 4L Z
r2
r2
e ----z
4L 2 z
r2
(3.69)
or s = --Q-J~ 1:. e -z- 4L 2 z dz
41tKIP u z
This is the solution appropriate to calculations for small times after starting the pump
when u~ lor if t~r2S.D/(4KhD) (Cooper, 1963). Eq. 3.69 can be written as:
(3.70)
when the leakage factor is large (large values of hydraulic resistance Cres and/or the
transmissivity KhD), or for rather small leakages during pumping, or for very small
pumping times when the leakage does not yet contribute to the flow to the pumped well.
This is the solution of the problem for small leakages or for flow in a confined aquifer
(Eq. 3.31 of Theis (1935».
The solution of the treated problem can be written in general as:
s=--Q- w(u ~) (3.71)

4 rr; KIP ' L
where W(u,r/L) is the well function of a semi-confined aquifer with a full screen and a
constant discharge rate and with ignorance of the water delivered by storage of the
adjacent semi-pervious layers. This well function W(u,r/L) is calculated in a different way
according to different values of u and r/L (Hantush and Jacob (1955), Cooper (1963) and
Reed (1980». A FORTRAN program to calculate the well function is given by Reed
(1980). The well function is in this paper represented by another notation L(u,v)
according to Cooper (1963), where v = r!2L. The well function is calculated on three
different manners according to the value of u and v2 (=r 2/(4U».
For u~ 1:
L(U,v) = (~ e-Y-~ dy= (f(Y)dY (3.72)
as given in Eq. 3.69. This integral is transformed in the form where y=x+u
- u -v -
ia e-
2
w e X'U (3.73)
X [ ] dx
x+u
which is evaluated by a Gaussian-Laguerre quadrature formula.

For v2 <u< 1:
(3.74)
The first integral is calculated with a Gaussian-Laguerre quadrature formula. The second
integral is calculated by means of a series expansion, when
(f(Y) dy = s(l) -stu) (3.75)
where
w v2
stu) = logU[Ln=o---]
(n! ) 2
(3.76)
~ {V2
+L-m=l [U m_ ( _ ) m] [L~=o (
v 2n
)
(-l)m}
u m+n In! m
For u< 1 and usv2 :
L(u, v) = 2Ko (2v) - r

Jv 2 /u
w
fey} dy (3.77)
The integral in Eq. 3.77 can also be evaluated by the Gaussian-Laguerre quatradure
procedure as described previous Iy .
3.3.4 Interpretation methods of De Glee, Hantush-lacob, Hantush I-II and Walton
3.3.4.1 Interpretation of distance-drawdown curve at steady flow
The transmissivity of the pumped layer and the resulting hydraulic resistance can be
derived by the formula of De Glee (1930) (Eq. 3.59).
76 Chapter 3 I Evolution of analytical models of pumping tests
Q K ( r) (3.78)
2rr;KJ!) 0 L
where Sm is the maximum drawdown (at steady-state flow) in the pumped layer at a
distance r from the pumped well.
This method requires sufficient observed drawdowns in the pumped layer at
different distances from the pumped well during steady-state flow. For the interpretation
of these drawdowns, the De Glee-type curve is plotted on double logarithmic paper. The
values of the modified Bessel function of the second kind and zero order K,,(r/L) is
plotted in the ordinate and the r/L in the abscissa. The drawdown in the pumped layer at
steady-state flow sm is plotted against the distance r from the pumped well on double loga-
rithmic paper with the same modulus length as the type curve: the drawdown in the
ordinate and the distance in the abscissa. In the De Glee interpretation method the best
match between the plotted observed and the De Glee type curve is searched. The
coordinate axis of the type curve and of the data plot must be held parallel. Once the best
position is found, an arbitrary match point is selected. The coordinates Ko(r/L), r/L, Sm
and r of this arbitrary point are determined on both sheets. Usually the point is chosen so
that the values of Ko(r/L) and r/L are equal to one or a power of ten.
The transmissivity KbD is calculated using the ratio Ko(r/L)/sm with the formula:
(3.79)
where Q is the discharge rate,

sm is the drawdown of the arbitrary chosen point,
Ko(r/L) is the determined value of the arbitrary chosen point.
The resulting hydraulic resistance c, is calculated with the ratio r/(r/L).
cr = (3.80 )
where KbD is the value of the transmissivity derived with Eq. 3.79
r is the distance from the pumped well of the arbitrary chosen point
(r/L) is the determined value of the arbitrary chosen point.
The reciprocal value of the resulting hydraulic resistance c, is equal to the sum of the
reciprocal values of the hydraulic resistances of the adjacent semi-pervious layers or
1
(3.81)
When one adjacent semi-pervious layer has a hydraulic resistance that is much larger than
the other then the derived value of the resulting hydraulic resistance approximates the
value of the hydraulic resistance of the most pervious adjacent semi-pervious layer. When
both adjacent layers have the same texture or do not differ considerably then the exact
values of the hydraulic resistances C 1 and c3 remain unknown. Here, it can be assumed
that both values are the same. Consequently, these are equal to twice the value of the
resulting hydraulic resistance cr'
The relation between the drawdown at steady-state flow and the distance from the
pumped well in a fully screened semi-confined aquifer was also derived by Jacob (1946).
This was deduced independently from the work of De Glee (1930). Based on the
characteristic that lim.....o Ko (r/L) = -0.5772 -In (r/2L), Hantush and Jacob (1955) develop
an interpretation method for small value of r IL « 0.05). When the Euler constant
(-0.5772) is expressed as the Neperian logarithm of 2.25/4, one can approximate the
modified Bessel function of the second kind and zero order as follows:
Ko( Lr) '" In(2.25 2L) or K (..£) '" In(1.125..f.) (3.82)

4r 0 L r
Because Sm = (Q/2'lIKbD) Ko(r/L) the maximum drawdown can be approximated by:
S '" -_Q_ 1n ( 1.125 L) = 2.30Qlog( 1.125 L) (3 .83)

m 21tKaD r 21tKhD r
The Hantush-lacob interpretation method is based on this last equation. For this
method the maximum drawdown is plotted against the distance from the pumped well on
a single logarithmic paper. The drawdown is put on the arithmetic ordinate and the
distance on the logarithmic abscissa. A straight line is drawn through the points for which
the value rlL is smaller than 0.05. The slope of the line ~sm is derived. This slope is
equal to the drawdown difference with respect to one log cycle (base 10) of the distance.
The intercept of the straight line with the distance axis ro is also determined.
The transmissivity KbD can then be calculated as follows:
(3.84)
The resulting hydraulic resistance Cr can be derived with:
(3.85)
With the obtained values of the transmissivity and the resulting resistance one can
calculate the leakage factor L. The value of rlL can then be calculated for the different
observation wells. One can now check if all the points used to draw the straight line,
satisfy the condition rlL ~0.05.
3.3.4.2 Interpretation of time-drawdown curve
Hantush (1956) proposed three different interpretation methods of the observed drawdown
of unsteady state flow. They are known as the interpretation methods of Hantush-l, -II, -
III (Kruseman and De Ridder, 1970). The first two methods, which are frequently used,
are given here. These methods are given along with the Walton interpretation method.
Hantush-l interpretation method
The first method of Hantush (1956) is based on the characteristics of the inflection point
of the time-drawdown curve plotted on single logarithmic paper (Fig. 3.6). To find the
inflection point the steady-state drawdown should be known from direct observation or
from extrapolation. The drawdown of the inflection point sp is equal to half the maximum
drawdown Sm or
8m
8 =- (3.86)
p 2
The slope of the time-drawdown curve on single logarithmic paper is given by:
(3.87)
The relation between the drawdown and the slope of the curve at the inflection point is:
(3.88)
The time at the inflection point can be expressed in function of L, r, S,D and ~D:
(3.89)
For the interpretation method of Hantush-I, the maximum drawdown must first be
determined directly or estimated by extrapolation. The drawdown at the inflection point is
half the maximum drawdown. With the aid of this drawdown the inflection point is
determined as well as the time of the inflection point tp. Finally, the slope of the curve is
approximated .
By means of Eq. 3.88 the value of ef/LKo(r/L) is derived with the values of the
inflection point drawdown sp and the slope of the curve at the inflection point .asp. With
the aid of tables giving the relation between x and e'Ko(x) (Kruseman and De Ridder,
1970, p. 192 or Ambramowitz and Stegun, 1965, p. 417-421) the value rlL can be
derived.
The values rlL and r enable us to calculate the leakage factor L:

I
(3.90)
L = IlL
The transmIssIvIty is determined by the slope of the time-drawdown curve at the

inflection point and the value rlL according to Eq. 3.87:
r
K D = 2. 30Q e-r; (3.91)
h 41t.6.sp
From the value of the leakage factor L and the value of the transmissivity KbD one can
calculate the resulting hydraulic resistance:
L2
C =-- (3.92 )
r K~
Finally, with the help of Eq. 3.89 and the time of the inflection point ~ the elastic storage
coefficient S can be derived:
S S,)J (3.93)
s in metre
,
,, --
0.15
---5;'=0 147m
0.10
./
/.
'/
/'
6 • • 0.072
0.05
r...J!l!I~tLll.n~r..('
sp.00735m
,
,,
,.. ,
I
--t--- - -
I
---
log eye Ie
---- L
i
j
2 1_.2 6 a 10-1 4 6 e 100
tp'" 2.8 x1ud
tin d
Fig. 3.6. Analysis of the pumping test "Dalem" according to the Hantush-I interpreta-
tion method (after Kruseman and De Ridder, 1970)
Hantush-II interpretation method
The second interpretation method of Hantush (1956) requires at least the time-drawdown
curves in two observation wells at two different distances from the pumped well in the
pumped aquifer. The slopes of the straight parts of the time-drawdown curves on single
logarithmic paper are first determined. These slopes are plotted against the distance from
the pumped well on an arithmetic axis. The best-fitted straight line is drawn through the
points (Fig 3.7). This line is the graphic representation of Eq. 3.87 which can also be
written as:
r 2.30L (log 2. 30Q -log.1.s ) (3.94)

41CKJJ P
This equation allows us to find the slope of the straight line ..:ir and the interception of this
straight line with the abscissa (r=O), ..:is,~o:
.1.r=2.30L
(3.95 )
The leakage factor L, the transmissivity KbD and the resulting hydraulic resistance c, can
be calculated according to the following set of equations:
l:l.r 2.30Q
L = 2. ]0' KhD = .1. and c r (3.96)
41C sr=o
s in metre r m metre
O.2!:! '00
,('
"
./
0.20
/ ,~o
I
V-f-' 65(30)·
. I
O.072m ~
,,~
,.,,0 V
/
~
./-(
0.15 100
I
../'
/
",-
/",£_- -- -- - - -
~-/~r r./ I
.,I
./ vf
,'. 1~ ~60)~i
~
J
0"- , I
.069
(,f. ,~
0.10 ~o
'0(60'-' ~
~ ~.,<
6.(120)·
-"
O.066m
~.
/'
'-- --6"5(90)"
inflection
point f;;7"=
12:' ,-- f---~q=::~:: j:6S)O
k
,--
-. -..,-1"H
/ ~,,/ log cycle
o
• •
0.05
'0- 2 2 r • 6 8 10- 6 e , cP '0 4 6 e
10-'
tmd 6S in metre
Fig. 3.7. Analysis of the pumping test "Dalem" according to the Hantush-II interpreta-
tion method (after Kruseman and De Ridder, 1970)
Once the values are known for Land KbD, the inflection point drawdowns of the
observed time-drawdown curves are found with Eq. 3.86. With the drawdowns at the
inflection points one can find the corresponding times of these points. The elastic storage
coefficient can then be calculated with the aid of Eq. 3.93.
Walton interpretation method
The given interpretation methods of Hantush are based on the characteristics of the time,
the drawdown and the slope of the inflection point. According to Kruseman and De
Ridder (1970), Walton (1962) develops an interpretation method similar to the Theis
interpretation method. For the Walton interpretation method, the Walton type curves must
be plotted on double logarithmic paper for different values of r/L. The values of the weJl
function of a semi-confined layer W(u,r/L) and 1/u can be read from tables (e.g.,
Kruseman and De Ridder (1970) or (1990». These curves can also be calculated by the
FORTRAN-program given in Reed (1980).
s in metre
10 0
8
6
W (U,I"'/L) 4
101
--
rlL
0.00
2 0.05
0.10
~ 0.20
10-1
8 ~-
--
6 ~ 0.50
4 ~ ---- )(A90
~
1.0
4
10-1 o data from ciezom. at 90 m
2 I I
1cr 3
'0- 2 2 4 6 810-' 2 4 6 8100
I
2 4 6 810'
tin d
Fig. 3.8. Analysis of pumping test "Dalem" (r= 90 m) with the Walton interpretation
method (after Kruseman and De Ridder, 1970)
The observed drawdowns of the different observation weJls are plotted on double
logarithmic paper with the same modulus length as that of the type-curves. The best
match between the plotted observed time-drawdown curves and the family of type-curves
is searched (Fig 3.8). The coordinate axis of the type curves and of the data plot must be
held parallel. Once the best position is found for a chosen type curve and one observed
time-drawdown curve, the position is fixed. First, the r/L value of the best fitted Walton-
type curve is noted. Then, an arbitrary match point is selected on the overlapping portion
of the two data sheets. The coordinates W(u,r/L), l/u, sand t of the match points are
determined. For all points of the overlapping sheets the ratios W(u,r/L)/s and t/(1/u) are
the same. Usually the point is chosen so that to the values of W(u,r/L) and l/u are one or
a power of ten.
The transmissivity is calculated by means of the ratio W(u,r/L)/s:
(3.97)
The elastic storage coefficient can be infered with the values of KhD and the ratio t/(l/u):
(3.98)
The leakage factor is derived from the noted value of r/L and the distance r from the
pumped well. With the last equation of the set given in Eq. 3.96 one can find the
resulting hydraulic resistance cr.
3.3.4.3 Comments concerning interpretation methods of semi-confined aquifer
These methods were and are used frequently because of the fact that the observed
drawdown curves resemble the course of the Walton-type curves. By the application of
one of the interpretation methods of the time-drawdown curves, one mostly ascertains that
the derived values of the transmissivity, the elastic storage coefficient and the resulting
hydraulic resistance vary strongly according to the distance from the pumped well of the
interpreted time-drawdown curve. The resulting hydraulic resistance is best estimated
from the distance-drawdown curve of steady-state flow with the De Glee method.
Hantush (1960) posed that only consistent values can be obtained with these
interpretation methods when the following three suppositions approximate the reality.
First, the hydraulic heads above and below the adjacent layers are constant. The contrast
between the hydraulic conductivity between the pumped pervious layers and the adjacent
semi-pervious layers must be so large that the flow is exclusively vertical in the semi-
pervious layers and horizontal in the pervious layer. Finally, the storage decrease in the
semi-pervious layer can be ignored.
Because of these last two conditions, the applicability of the method is strongly
reduced. Sedimentary rocks are mostly characterized by a trending heterogeneity which
results in a gradual change of hydraulic conductivity (see Sect. 2.1.4 and Sect. 7.4).
Sometimes, the amount of water delivered by storage of the adjacent semi-pervious layers
can be more important that the amount of water that is delivered from the storage of the
unpumped layer. In most cases the semi-pervious layers has a larger specific elastic
storage than adjacent pervious layers. This is specially true if one deals with recent
sediments. To cope with the neglect of the storage decrease in the semi-pervious layer,
Hantush (1960) adjusted the model of lacob-Hantush. In the adjusted model, the model of
Hantush, the storage decreases in the adjacent semi-pervious layers are considered.
3.4 Model of Hantush
3.4.1 Introduction
This model considers the radial flow in a semi-confined aquifer with storage decrease in
the adjacent semi-pervious layers (Hantush, 1960). The model is an extension of the
theory of the radial flow toward a pumped well with a complete filter screen in a semi-
confined aquifer and with a constant discharge rate (see Sect. 3.3). Here, one considers a
pumped pervious layer, which is bounded above and below by semi-pervious layers (Fig.
3.9). The water is delivered by horizontal flow and storage decrease in the pumped
pervious layers and by vertical flow and storage decrease of the adjacent semi-pervious
layers. One supposes that the hydraulic conductivities of the semi-pervious layer are very
small with respect to the conductivity of the pumped pervious layer.
Depending on the boundaries of the semi-pervious layers, Hantush distinguishes
three different cases. In the first case a constant hydraulic head boundary is assumed at
the top of the superjacent semi-pervious layer and at the base of the subjacent semi-
pervious layer. In the second case, the forementioned boundaries are both impervious. In
the third case, the base of the subjacent semi-pervious layer is impervious and the top of
the superjacent layer has a constant hydraulic head.
3.4.2 Derivation of basic differential equations
The total difference of the discharge rate through two successive coaxial cylinders around
the pumped well is due to storage decrease in the volume of the pumped layer located
between these two cylinders and is due to leakage from the super- and subjacent semi-
pervious layers. This leakage is not supposed to be proportional to the drawdown in the
pumped layer but is derived from the vertical gradient of the hydraulic head in the semi-
pervious layer at their boundaries with the pumped layer.
The change of the discharge rate in layer 2, the pumped layer, between two
successive coaxial cylinders can again be formulated as in Sect. 3.2.2 (Eq. 3.15):
(3.99)
The change of the discharge rate due to storage decrease in the pumped layer 2 is given
as in Eq. 3.16:
(3.100)
The leakage of the semi-pervious layer is proportional to the surface located

between the two concentric circles rand r + ~r, to the vertical conductivity of this layer
and to the vertical gradient in this layer at its boundary with the pumped aquifer. The
change of the discharge rate between two successive coaxial cylinders in the pumped layer
is due to leakage from the lower semi-pervious layer 1 which can be formulated as:
(3.101)
where Kvt is the vertical hydraulic conductivity of the semi-pervious layer 1

and ast(r,D1,t)/az is the vertical hydraulic gradient in the semi-pervious layer 1 at a
distance r from the pumped well at the boundary of the pumped layer 2, z=D1 (Fig. 3.9),
and after a time t of pumping.
The discharge rate change between two successive coaxial cylinders in the pumped
layer is also partially due to leakage from the upper semi-pervious layer 3 which can be
formulated as:
(3.102)
where KV3 is the vertical hydraulic conductivity of the semi-pervious layer 3

and asir,Dl+D2,t)/aZ is the vertical hydraulic gradient in the semi-pervious layer 3 at a
distance r from the pumped well at the boundary of the pumped layer 2, z=D t +D 2 (Fig.
3.9), and after a time t of pumping.
The total change of the discharge rate between two successive coaxial cylinders is
the sum of storage changes of the pumped layer 2 and the leakage from the adjacent
layers 1 and 3. From Eqs. 3.99, 3.100, 3.101, 3.102 one derives that:
(3.103)
Dividing both parts of Eq. 3.103 by the surface which is comprised between the two
concentric circles with radius r and r+~r results in the basic differential equation of the
unsteady-state flow in the pumped layer 2:
(3.104)
Beside the basic differential equation of the flow in the pumped layer 2, one also needs
the basic differential equations for the flow in the adjacent semi-pervious layers 1 and 3.
These equations are obtained by the application of the law of continuity. If one assumes
that there is a sufficient contrast between the pumped layer and its adjacent semi-pervious
layers then the flow in the semi-pervious layer is nearly exclusively vertical. The vertical
upward flows in the adjacent semi-pervious layers are described by following equations:
aS1 aS3
VVl = -KV1 az and VV3 = -Kv3 az (3.105)
The flow velocity at a certain distance from the pumped well and after a certain time of
pumping is no longer taken constant over the different levels of the semi-pervious layers.
It is not assumed that they are proportional to the drawdown in the pumped layer as in the
model of lacob-Hantush (Sect. 3.2.3). Because the storage decrease in the semi-pervious
layers is considered, the vertical flow in these layers depends on the distance from the
boundary of these layers with the pumped layer. The vertical flow is thus level depen-
dent. In the beginning of the pumping test, there is a large vertical gradient in vertical
flow in the semi-pervious layer near the pumped layer. In Fig. 3.9 the increasing vertical
flow through the semi-pervious layer is symbolically represented by vertical arrows which
increase in thickness to the pumped layer. The storage decrease is symbolically represen-
ted by circles between the arrows.
The discharge rate Qv;(r ,z, t) which flows upward through a certain horizontal
surface in the semi-pervious layer i after a certain time t after the start of the test is:
as.
Qvi (r, z, t) = -21tro6.r KVi ( - a
Z
1) I Z •
",
(3.106)
where 2'lITo6.r is the surface limited by two concentric circles with radius rand r+o6.r at
the level z.
The change of the discharge rate between two successive surfaces at the level z
and z+o6.z in the semi-pervious layer i is then equal to:
Qvi (r, z+o6.z, t) -Qvi (r, z, t) = (aQ vi (3.107)

az ) I,Z,e o6.z
MODEL OF HANTUSH
CASE 1
PERVIOUS LAYER AT
SEMI-PERVIOUS
LAYER
PERVIOUS LAYER
SEMI- PERVIOUS
LAYER
PERVIOUS LAYER AT
CASE 2. CASE J
IMPERVIOUS BOUNDARY
Fig. 3.9. Schematization of the groundwater reservoir in the three cases of the model
of Hantush (1960)
Because
( aQVi ) -21tr.6.r K v ' (--~

CPs·
) I Z t (3.108)
aZ I,Z,t ~ aZ 2 "
then:
Qvi(r,z+.6.z, t) -Qvi(r,z, t) =
a2 S· .6.z
(3.109)
-21tr.6.r KVi ( aZ2~) I,Z, t
In a ring-shaped volume bounded by two successive surfaces on a distance AZ of each

other and by two coaxial cylinders with radii rand r+ Ar (Fig. 3.10), the storage
decrease is proportional to the considered volume 21ITArAZ, the specific elastic storage SSI
or S'3 and to the drawdown increase rate as/at or as 3/at. In the subjacent semi-pervious
layer 1, the discharge rate increases toward a higher level Z or toward the pumped layer:
aS1
Qv1 (r, z+.6.z, t) = QV1 (r, z, t) + 21tr.6.r.6.z SS1(ft (3.110)
.I
.I
r
L
Q~(r.z.t)
I_ Ar
Fig. 3.10. Vertical discharge Qvj(r,z,t) and QVj(r,z+Az,t) through two successive
surfaces (21ITAr) at the levels z and Z+AZ in the semi-pervious layer i
In the superjacent semi-pervious layer 3, the discharge rate also increases toward the
pumped layer 2 or toward a lower level z:
(3.111)
From Eqs. 3.109 and 3.110, the basic differential equation for the flow in the subjacent
semi-pervious layer 1 can be derived:
21tr ~ r ~Z SS1
(as
at1 )
I,Z, t
(3.112)
Dividing both parts of Eq. 3.112 by the volume 27rl"~r~z results in:
a2s 1)
-K
v1 az
( __
2 I,Z,t
(3.113)
From Eqs. 3.109 and 3.111, the basic differential equation for the flow in the superjacent
semi-pervious layer 3 can be derived:
Dividing both parts of Eq. 3.114 by -27rr.6.r.6.z results in:
(3.115)
3.4.3 Initial and boundary conditions
Following initial and boundary condirions can be formulated for the pumped pervious
layer 2:
S2 (r, 0) = 0, r :? 0
S2 (00 0)
I = 0, t :? 0
Q = 0, t <a (3.116)
Q = cst, t :? 0
1.lm r -
aS2
-
-Q
t :? 0
I-O ar 21tKh2D2 '
This means that the initial drawdown at every distance from the pumped well r is equal to
zero; that the drawdown is zero at an infinite distance from the pumped well; that there
was no pumping any time before the pump started (t < 0); that the discharge rate is
constant after the pump started and that the discharge rate through the cylinder with an
infinite small radius is equal to the pumped discharge rate after any time after the pump
started.
For the subjacent semi-pervious layer 1, following initial and boundary conditions
are valid:
8 1 (r , z, 0) = 0, r > 0, 0 < Z < D1
8 1 (r , D 1 , t:) = 82 (r, t:) , t: ;;, 0 (3.117)
8 1 (r, 0, t:) = 0, t: ;;, °
Verbally expressed these conditions mean that the initial drawdown at every level and
distance from the pumped well in the subjacent semi-pervious layer equals zero; that the
drawdown at the top of the subjacent semi-pervious layer 1 is equal to the drawdown in
the pumped pervious layer 2 at every moment after the start of the pumping test; and that
the lower boundary of the subjacent layer (z=O) is a constant hydraulic head boundary as
assumed in the first case of the Hantush model. In the second and third case, one
supposes that the semi-pervious layer 1 is bounded by an impervious boundary or
81 (r,0,t)
t ;;, 0 (3.118)
az = 0,
which means that the vertical gradient or the vertical flow is zero at the lower boundary
of layer 1.
For the superjacent semi-pervious layer 3, following initial and boundary conditi-
ons are valid:
83 (r , z, 0) = 0, r > 0, Dl +D2 < z < Dl +D2+D3
83 (r , D1 + D 2 , t:) = 8 2 (r, t:) , t: ;;, ° (3.119)
8 3 (r, Dl+D2+D3 , t:) = 0, t: ;;, °
In words these conditions mean that the initial drawdown at every level and distance from
the pumped well in the superjacent semi-pervious layer 3 equals zero; that the drawdown
at the base of the superjacent semi-pervious layer 3 is equal to the drawdown in the
pumped pervious layer 2 at every moment after the pumping start; and that the upper
boundary of the subjacent layer (z = 0 1 + O 2 + 0 3) is a constant hydraulic head boundary as
assumed in the first and third case of the Hantush model. In the second case, one
supposes that the semi-pervious layer 3 is bounded above by an impervious boundary or:
8 3 (r, D1 +D2+D3' t)
az = 0, t;;, ° (3.120)
which means that the vertical gradient or the vertical flow is zero at the upper boundary
of layer 3.
3.4.4 Solution of basic differential equation
The detailed derivation of the solution of the first case of the Hantush model is not given
here. For this derivation, the reader is referred to the appendix of the original work of
Hantush (1960). He obtains an asymptotic solution enabling only the calculation of the
drawdowns for the pumped layer.
For small values of time, t<SslD/I(10K.I) and t<S,3D/1(10Kv3), or
t < Slc/lO and t < S3cilO, the solution of the different cases of the Hantush model is
the same:
(3.121)
H(u,{j) is the well function of the first case of the Hantush model and can be formulated
as:
H(u, 11) =J~ e-Y erfc IlIU dy

u y .jy(y u} (3.122)
erfc(x) = 2... J~ e-Y'dy
.(it x
For large values of time, the solution depends on the case of the Hantush model.
In case 1 for values of time larger than both 5SsI D/IK.I and 5S s3 D/IKv3
(3.123)
where u is as defined previously and
a: = r (3.124)
In case 2, the drawdown can be calculated for time values larger than both lOSsID/IKvl
and lOSs3D/IKv3
(3.125)
In case 3 for values of time larger than both 5S'ID/IK,,1 and lOS'3D/IKv3
(3.126)
where u is as defined previously and
(3.127)
3.4.5 Interpretation methods derived from Hantush model
3.4.5.1 Interpretation of first part of time-drawdown curves
The first part of the time-drawdown curve shows the same course in the three case of the
Hantush model. Before the given time limits (see Sect. 3.4.4), the different boundary
conditions assumed in the three cases have no influence on the drawdown in the pumped
layer. The first part of the time-drawdown curve can be interpreted as in the interpretati-
on method of Theis or Walton. Therefore, the well function H(u,{j) must first be plotted
on double logarithmic paper. The Hantush well function H(u,{j) is put on the ordinate and
1/u on the abscissa. The values of H(u,{j) and lIu can be read from tables given in
Hantush (1960) or can be calculated by a FORTRAN program developed by Papadopolus
and given in Reed (1980). The shape of the type curves changes gradually with increasing
values of {j. When there are observation in one well with a screen in the pumped layer,
the observed drawdowns are plotted versus time t on double logarithmic paper of the
same modulus length as the type-curves. The best match between the plotted observed
time-drawdown curves and the series of Hantush type curves is searched holding the
coordinate axises of the two graphs parallel to each other. The {j-value of the best
matched type curve is read as well as the coordinates of one arbitrary chosen point on the
overlapping part of the two graphs. The transmissivity of the pumped layer ~D2 is
calculated with a formula which is derived from Eq. 3.121.
(3.128)
The elastic storage coefficient of the pumped layer is derived according to the formula
(3.129)
From the J3-value one can derived the following value
(3.130)
When one of the adjacent semi-pervious layers is less pervious than the other then the
product of the vertical hydraulic conductivity and the specific elastic storage of the most
pervious of the adjacent semi-pervious layer can be derived
(3.131)
The index i indicates the most pervious layer of the two adjacent semi-pervious layers.
Because of the gradual change of the Hantush-type curves, there will be uncertainty about
the chosen type-curve that matches best with the observed drawdown curve.
When more than one observation well is available then the observed drawdowns
are better plotted in one graph with respect to t/r2 where t is observation time and r is the
well distance from the pumped well. Each time-drawdown curve must now fit with a
different Hantush type curves of which the J3-value is proportional to the distance r. This
can reduce the uncertainty of the chosen type curves. Instead of the time t the value of
t/r2 of the arbitrary point of the overlapping portions of both graphs is read.
3.4.5.2 Interpretation of last part of time-drawdown curves
The last part of the time-drawdown curve obtained from case 1 and case 3 of the model
of Jacob can be interpreted by type-curves which show a same course as the Walton type
curves. The solution of case 1, for large times, can also be written as in Eq. 3.132.
From these formulas one can derive that the last part of the time-drawdown curve can be
interpreted with the Walton type curves. The determined value of the elastic storage
coefficient corresponds with Sri given in third part of Eq. 3.132. Usually, the last parts of
the time-drawdown curves show a flat course. This results in a difficult determination of
the values of 1/ul and t and consequently of the value of Sri. For the same reason, the
determination of Lz13 is also problematic.
(3.132)
The solution for large time values of case 3 can also be written as
(3.133)
When the last part of the time-drawdown curves, which satisfy the time constraint, is
interpreted with the Walton type-curves then one can find the value of Sr3. Again here,
the interpretation of the last part of the time-drawdown curves is difficult because of their
flat course.
The solution for large time values of case 2 of the Hantush model can also be written as:
(3.134)
The last part of the time-drawdown curves can thus be interpreted by the Theis type
curve. The values which can be determined are the transmissivity of the pumped pervious
layer Kb2D2 and the resulting elastic storage coefficient Sa. This last parameter is the sum
of the elastic storage coefficients of the pumped layer and of the two adjacent semi-
pervious layers.
3.4.5.3 Interpretation of the maximum drawdowns of cases 1 and 3
The flow and the drawdown reach an equilibrium in the cases 1 and 3 of the Hantush
model. The maximum drawdown sm! of case 1 is given by the Eq. 3.135 in which ~!3 is
defined as in Eq. 3.132. The maximum drawdown sm3 of case 3 is then:
(3.135)
(3.136)
where ~3 is defined in Eq. 3.133.

From these formulas, it follows that maximum drawdowns plotted versus the observation
distance on bilogarithmic paper can be interpreted with the method of De Glee. In the
first case, the resulting hydraulic resistance is found. The reciprocal value of this
hydraulic resistance is equal to the sum of the reciprocal values of the hydraulic resistan-
ces of the adjacent semi-pervious layers. In the third case, the hydraulic resistance of the
semi-pervious layer that is bounded by the constant hydraulic head boundary is derived.
3.4.6 Concluding considerations about interpretation methods derived from Hantush

model
The first parts of the time-drawdown curves are only determined by the transmissivity and
the specific elastic storage coefficient of the pumped layer and by the hydraulic resistan-
ces and the specific elastic storages of the adjacent semi-pervious layers. These parts are
not influenced by the boundary conditions which prevail at the upper boundary of the
superjacent layer and the lower boundary of the subjacent layer. The time-drawdown
curve of the Hantush model starts to rise later than the curve of the model of Jacob-
Hantush. The larger the j3-value, the later the rise of the time-drawdown curve in the
pumped pervious layer. Consequently, the larger the distance from the pumped well, the
larger the possible error in the obtained values when the time-drawdown curves corres-
ponding with the Hantush model are interpreted with a method derived from the Ja-
cob-Hantush model. The possible errors are also large when the specific elastic storages
and vertical conductivity of the adjacent semi-pervious layers are relatively large. The
thus obtained values of transmissivity, the elastic storage coefficient and the hydraulic
resistance are larger than the real values. This error is also influenced by the transmissivi-
ty and the elastic storage coefficient of the pumped pervious layer. The larger their real
values are, the larger the overestimates will be.
The influence of the boundary conditions on the drawdowns of the pumped layer
becomes more pronounced as the pumping test prolongs. In the first and third case of the
Hantush model the drawdown reaches a maximum. Here, the last part of the time-
drawdown curves has the same course as the last part of a Walton type curve. In the first
case the location of the Walton type curve is defined by the transmissivity of the pumped
pervious layer and by its elastic storage coefficient enlarged by one third of the sum of
the elastic storage coefficients of the adjacent semi-pervious layers. In the third case, the
location of the Walton type curves is defined by the transmissivity of the pumped
pervious layer and the elastic storage coefficients of the pumped pervious layer enlarged
with the elastic storage coefficient of the semi-pervious layer which is bounded by the
impervious boundary and by one third of the storage coefficient of the semi-pervious
layer which is bounded by the constant hydraulic head boundary.
The choice of the Walton-type curve becomes very difficult if one has only one
observed time-drawdown graph. This is mostly due to the flat course of this curve.
Different time-drawdown graphs observed at different distances from the pumped well can
be interpreted simultaneously. Therefore, the drawdowns are plotted versus the ratio t/r2
on double logarithmic paper. By the joint interpretation of all the drawdown curves by
means of the Walton-type curves the transmissivity of the pumped layer can be found.
Because the chosen r/L-values must be proportional with the distance from the pumped
well, the uncertainty in the choice of the type curves is reduced. There will still be an
uncertainty in the choice of the ratio of t/r2 value on the llu value so that the derivation
of the resulting elastic storage coefficient is still difficult and uncertain.
The maximum observed drawdown at different distances from the pumped well in
the pumped layer in case 1 and 3 can be interpreted with the De Glee interpretation
method (Sect. 3.3.4). In case 1, the transmissivity of the pumped pervious layer is
determined as well as the resulting hydraulic resistance of the adjacent semi-pervious
layers. In case 3, the hydraulic resistance of the semi-pervious layer that is bounded by
the constant hydraulic head boundary is found.
The drawdown in the pumped layer continues to rise in case 2 of the Hantush
model because of the infinite lateral extension of the impervious boundaries. This is
similar to the model of Theis so that the last part of the time-drawdown curves shows a
similar course as the Theis type curve. The location of this curve is found by the
transmissivity of the pumped layer and the sum of the elastic storage coefficient of the
pumped pervious and of the adjacent semi-pervious layers.
In practice, however, the interpretation of the last part of the time-drawdown
curve can only be made if the shape of these time-drawdown curves shows the same
course as the type curves. This is only the case if the flow is in reality influenced by the
boundary conditions as assumed in the three cases of the Hantush model. However, long
term variations of the hydraulic head very often cause deformation of the last part of the
time-drawdown curve. The correction of these curves by means of observation in a
reference well is theoretically possible but is hardly applicable.
3.5 Model of Hantush-Weeks
This model is named after Hantush and Weeks. The first author derived the basic
differential equation and proposed a solution for this equation (Hantush, 1961 and
Hantush, 1964). The second author derived an interpretation method from this solution
(Weeks, 1969). In this model a pervious layer is assumed that is bounded above and
below by an impervious boundary over an infinite lateral extension. The pervious layer is
assumed to be homogeneous. However, this layer can be transversely anisotropic. The
pumped well has a filter screen over a limited interval of the pumped layer, not over the
full thickness of the pumped layer. Because of this, the flow is not completely horizontal
but has an important vertical component near the pumped well. The diameter of the well
screen is assumed to be very small. Consequently, the storage in the well must be
negligibly small. The discharge rate is assumed to be constant over the whole period of
the test. In such a pumping test, the observations are made in wells which are at different
distances from the pumped well but are also at different levels. Wells at different levels
near the pumped well are very important to measure the vertical gradient.
The basic differential equation of the radial and vertical unsteady state flow is
s as (3.137)
Sat
The initial and boundary conditions can be summarized as follows:

s(r,z,O) = 0, r :<: 0 and 0 ~ z ~ D
s(oo,z,t) =0, t :<: 0 and 0 ~ z ~D
as(r, 0, t) 0
az =, r o and t :<: 0
as(r,b,t) =0, r :<: 0 and t :<: 0 (3.138)
az
limr as = -Q , at U<z<L
x-o ar 2rrKh (L-U)
limr aas = 0 at 0 <z <U and L < z <D
x-o r
The symbols U, Land D are given in the Fig. 3.11.
Which means that:

- the drawdown is zero before the start in all points of the pervious layer,
- the drawdown is zero at an infinite distance from the pumped well and at every level in
the pervious layer during the whole pumping time,
- the vertical gradient of the drawdown at the upper and lower boundary of the pervious
layer is zero or there is no vertical flow during the whole pumping period,
- the horizontal flow around the pumped well is radial and is the same at every level of
the well screen,
-the horizontal flow is zero at every level above and below the filter screen of the pumped
well over the whole pumping time.
MODEL OF HANTUSH - WEEKS
IMPERVIOUS LAYER
ANISOTROPIC
PERVIOUS LAYER D-
- PO
Kh
o
UP
--+
LP
0 .....
S.
O_~....:._+--_t:::t::::I
1-___
r
IMPERVIOUS LAYER
Fig. 3.11. Schematic representation of the groundwater reservoir and the flow around a
partially penetrating well. The pervious layer is anisotropic.
The solution of the basic differential equation according to the posed initial and
boundary conditions is given by Hantush (1961). The drawdown in a piezometer is
s = __0 - [W(u) +f1 (u, ,ar ~ !:! p) 1

41tKr/J p' D' D' D
w her e f (u ar L UP) _ 2 D
1 'V' D' D' D - 1t(L-U)
(3.139)
Ln=1 ~ (sinn~L-sin n~u)COS n~pW(U, n~ar)
a=~ and
~~
W(u) and W(u,x) are respectively the well function of the confined (Theis) and semi-
confined aquifer. The symbols rand P are represented in Fig 3.11.
The drawdown in an observation well is
Q [() f ( ar L U LP UP ) 1
s = 4rcKJP W u + 2 U, D' D' D' D' D
wher e f (u ar L U LP UP) =
2 'D' D' D' D' D (3.140)
2 D2 ~ 1 ( . nrcL . nrcu)
.£...n-1- Sln-- -Sln-- .
rc 2 (L-U) (LP-LU) - n 2 D D
(sin nrc:p -sin nrc)/p) w( u, nrc;r)
W(u), W(u,x), a and u are defined in Eq 3.139. The symbols LP and UP are also
represented in Fig 3.11.
For large values of the time, that is, for t > rJ2S,J(2a2KJ or t > D 2S,J(2K.), the
effect of the partially penetrating well in the nonleaky aquifer is constant in time. For
large values of time and small values of u, the function W(u,ll1rar/D) in Eqs. 3.139 and
3.140 can be replaced by the function 2Ko(ll1rar/D). Weeks (1969) defined the drawdown
correction factor 5s as the difference between the drawdown with a partially penetrating
well and the drawdown with a complete well after a sufficient time of pumping. The
drawdown correction factor 5s 1 for a piezometer is derived from Eq. 3.139.
(3.141)
The drawdown correction factor 5s 2 for an observation well is derived from Eq. 3.140.
as = __ Q_. 4 D2 L:-1 ~ .
2 4rcKh D rc2 (L-U) (LP-LU) n- n2 (3.142)
(sin nrcL -sin nrcu) (sin nrcLP -sin nrcup) K ( nrcar)
D D D DOD
In Weeks (1969), a dimensionless drawdown correction factor, 5s 1 4'lIKhD/Q, is given in

tables for different values of ariD, LID, UID and P/D. Reed (1980) constructed a large
number of graphs where the dimensionless drawdown correction factor is plotted against
ariD for different values of LID, UID and PID.
Weeks (1969) proposed three different interpretation methods to derive the
horizontal and vertical hydraulic conductivity and the specific elastic storage from the
observed drawdowns. The different interpretation methods depend on the amount of
observations and also on the location and the time of the observations. Each interpretation
method is rather complicated and has eight or nine steps.
From these interpretation methods one can derive that a sufficient number of
observed drawdowns must be available after a sufficient long time of pumping. The
necessary piezometers or observation wells must be located inside and outside the
influence zone of the partially penetrating well. Inside the influence zone, piezometers or
observation wells must be available at different levels.
Reed (1980) proposed to plot all the observed drawdowns in one graph versus the
t/r2-values. The drawdowns of the different observation wells must be compared with
different series of type curves. Each group of standard curves correspond with one value
of anisotropy. A criterion for determining the match between observed and type curves is
that the value of ariD for different observation wells should all indicate the same
anisotropy. These different type curves can be calculated by the FORTRAN-program
given in Reed (1980).
3.6 Model of Boulton-Cooley
In the original model of Boulton the flow to a pumped well is treated in a phreatic or
unconfined aquifer. Boulton (1955, 1963) assumed that at a distance r from the pumped
well after a certain time t of pumping, the part of "delayed yield" caused by a drawdown
increase in the pumped layer os at a time t1 (tl < t) can be expressed by an empirical
relation:
(3.143)
where OV1 is the velocity of the delayed yield at every time t (t > tl) caused by a
drawdown increase os in the pumped layer during an interval ot1 (L.'), So is the storage
coefficient near the water table (UL-3) and ex is an empirical constant (T-') of which the
reciprocal value is sometimes called the Boulton delay index. The minus sign results from
the choice of the direction of the z-axis (positive in the upward direction) whereas OV 1 is
directed downward.
From this empirical relation follows that after a long time the amount of water
released by the delayed yield for a drawdown unit equals the storage coefficient near the
water table or:
(3.144)
Boulton (1955) assumed further that the total velocity of the "delayed yield" Vd at a
distance r from the pumped well and a time t after starting the pump can be obtained by
the integration of all the "delayed yields" caused by all the drawdown increases which
took place in the pumped layer since the start of the pump until the considered time t
Vd = - IX S
o
r~
atl
Jo
t e-«(t-t1) dtl (3.145)
Boulton (1955) did not give a theoretical explanation for the above given formula which is
called the Boulton integral.
The basic differential equation of the flow to the pumped well in the phreatic
aquifer can be written as (Boulton, 1955):
S n
s-
as +s
at 0
a,rJo atl
t as e-«(t-U) dtl (3.146)
In Fig. 3.12, the structure of the unconfined aquifer in the model of Boulton is represen-
ted schematically. The increasing discharge rate through the successive coaxial cylinders
with decreasing radii is symbolically represented by some horizontal arrows that increase
in thickness to the pumped well. This increase is on one hand due to storage decrease of
the pumped layer (symbolically represented by the circles between the arrows) and on the
other hand due to the "delayed yield" of water in the vicinity of the water table (symboli-
cally represented by a circle with a downward directed arrow near the water table).
Cooley (1971), Cooley (1972) and Cooley and Case (1973) demonstrated that the
Boulton integral describes the vertical flow in a semi-pervious layer which overlays the
pumped layer. The water table is located in this semi-pervious layer (Fig. 3.13). It is
assumed that the compressibility of the semi-pervious layer is very small so that it can be
ignored. Finally, it is assumed that the drawdown of the water table is small in respect of
the thickness of the semi-pervious layer. In contradiction to the concept of the "delayed
yield" in the theory of Boulton, Cooley assumes an immediate yield of water when the
water table lowers.
Fig. 3.13 gives a schematic representation of the groundwater reservoir and the
flow to the pumped well according to Cooley. With a numerical calculation Cooley shows
that his model regenerates the same drawdowns as the model of Boulton with its empirical
relation about the "delayed yield". The scheme represented in Fig. 3.13 will further be
indicated as the model of Boulton-Cooley. In this model the groundwater reservoir is
formed by two layers. A pervious layer 1 which is bounded below by an impervious
boundary and above by a semi-pervious layer 2. One assumes that the flow is exclusively
horizontal and radial toward the pumped well with a complete well screen in layer 1. The
discharge rate through the successive cylinders with decreasing radii increases. This is
symbolically represented by the horizontal arrows with increasing thickness toward the
pumped well. The increases in the discharge rate are due to two causes. The first cause is
storage decrease due to compression of the aquifer and expansion of the water in the
aquifer (symbolically represented by circles between the arrows). The second cause is the
"immediate yield" of water due to a dropping water table (symbolically represented by a
circle near the water table). The flow through the semi-pervious layer is assumed to be
exclusively vertical and inelastic. An inelastic flow means that the storage increase due to
the elasticity of this layer and the water is completely ignored in this layer. This last flow
is symbolically represented by one vertical arrow of the same thickness.
MODEL OF BOULTON "delayed yield"
GROUND SURFACE-
So
WATER TABLE
PERVIOUS LAYER
IMPERVIOUS LAYER
Fig. 3.12. Schematization of the phreatic aquifer in the model of Boulton (1955) with
"delayed yield" during a drop of the watertable
MODEL OF BOULTON - COOLEY "immediate yield"
GROUND SURFACE
WATER TABLE
SEMI·PERVIOUS LAYER
PERVIOUS LAYER
SEMI- PERVIOUS LAYER
Fig. 3.13. Schematization of the model of Boulton-Cooley with a hydraulic resistance

between the pumped layer and the watertable and an "immediate yield" of water due to
a watertable drop
The flow in the semi-pervious layer 2 (Fig. 3.13) can be represented by the
following basic differential equation and boundary conditions:
a 2S 2
= 0 D1 :s; z < D1 +D2 t > t1
aZ 2
a 2s as"
K -- =
az 2 -so 7ft Z = D1 +D2 t > t1
v2 (3.147)
S2 s" Z = D1 +D2 t > t1
S2 S1 Z = D1 t > t1
S2 = 0 D1 :s; Z :s; D1 +D2 t = t1
where S2 is the drawdown in the semi-pervious layer 2 which depends on the distance
from the pumped well r, the level z and the time t,
s\ is the drawdown in the pumped pervious layer which is a function of rand t,
Sw is the drawdown near the water table,
Kv2 is the vertical conductivity,
D\ is the level of the base of the semi-pervious layer and
D\ + D2 is the level of the top of the semi-pervious layer, the water table.
The solution of this basic differential equation is:
(3.148)
where u(t-tl) is the unit step function which is equal to zero for t s tl and is equal to 1
for t > t1. Because the compressibility of the semi-pervious layer is ignored, the vertical
Darcian velocity is the same on the different levels in this semi-pervious layer or
(3.149)
The Eqs. 3.148 and 3.149 give the evolution of respectively the drawdown and the
vertical Darcian velocity due to a drawdown unit in the pumped pervious layer at time tl.
The drawdown and the vertical Darcian velocity in the semi-pervious layer caused by the
drawdown increase in the pumped pervious layer since starting the pumped at the consi-
dered time t is obtained by integration. According to Cooley and Case (1973), this
integration can be made by adaption of the theorem of Duhamel as proposed by Venetis
(1970). If the solution of the linear partial differential equation has the form
s=s1(tl).F(z,t-tl) with the initial condition s=O and the boundary condition s=s\(tl) then
the solution at every time t for the boundary condition s=s\(t) is:
s = i asa
o
t (tl)
tl
F(z, t - t l ) dtl (3.150)
where s, is the drawdown in layer 1 and consequently the drawdown at the base of the
semi-pervious layer 2 and F(z,t-tl) is the unit step solution.
To derive the vertical Darcian velocity, the law of Darcy must be applied to Eq. 3.150
v v2
_ Jr
-
t aS 1 (tl)
atl Kv2
aF(z, t - t l )
az
d
tl
(3.151)
o
Applying the Eqs. 3.150 and 3.151 to the Eqs. 3.148 and 3.149 one obtains
s
2
= s
1
_z-D
__ 1
D2
it as
0
__
1
atl
e-a;(t-tl) dtl
i as
(3.152)
v = -a S t
__1 e-u(t-tl) dtl
v2 0 0 atl
The equation for the vertical flow in the semi-pervious layer is identical to the empirical
relation of Boulton which describes the velocity of the "delayed yield" Vd (see Eq.
3.145). The only difference between the model of Boulton-Cooley and the model of
lacob-Hantush is the boundary condition at the top of the semi-pervious layer. This
boundary is a constant hydraulic head boundary in the model of lacob-Hantush. In the
model of Boulton-Cooley, this boundary is a constant flux boundary with an immediate
water release due to a water table drop. In both models, the storage decrease due to the
elasticity of the semi-pervious layer is ignored.
The delay index of Boulton can be defined in function of the storage coefficient
near the water table So, the vertical conductivity KV2 and the thickness of the semi-
pervious layer 2 D2 or the hydraulic resistance of layer 2 Cz
or l:.=SC
a 0 2 (3.153)
Hence, the drainage factor of an unconfined aquifer as defined by Kruseman and De

Ridder (1970), B=«Kh,D,)/aSo)'I2, can be replaced by the leakage factor of the semi-
confined aquifer, L12 = (Kh' D,CJ'I2, formed by a pervious layer 1 which is overlaid by
the semi-pervious layer 2.
Boulton (1955) supposes initially that the "delayed yield" appears in two cases: the
first case with the water table in a fine-grained material which overlays the pumped layer
and in a second case where compressible layers of fine-grained material occurs in the
pumped pervious layer which is bounded above and below by impervious layers. Boulton
does not describe the precise character of the "delayed yield". He assumes only a simple
empirical relation. Later, Boulton (1963) specifies that the "delayed yield" results from
the flow above the water table. Therefore, Cooley and Case (1973) study the time-
drawdown curves of three different cases and compare the results mutually.
The first studied case of Cooley and Case is just extensively described. It is the
case of exclusively vertical flow in the semi-pervious layer where the elasticity is ignored
and with an immediate water release. In the second case, the specific elastic storage of
the semi-pervious layer is considered. In the third case, the flow in the unsaturated zone
during the lowering of the water table was considered. Cooley and Case (1973) carne to
the conclusion that the flow in the unsaturated zone has only a small influence on the
drawdown in the pumped layer. This last result was obtained in two different ways: by
analytical derivation of the solution and by numerical calculations.
The flow after a relative long time of pumping, t > lOS,2D2c2' is approximated by
the assumption of an immediate drop of the water table. The water is then principally
released by the water table drop while the water released from the elastic compression of
the layers is negligibly small.
In their conclusions, Cooley and Case (1973) pointed out that the typical S-shaped
time-drawdown curves are not only related to unconfined aquifers with lowering water
tables but can also occur in semi-confined aquifers with a dropping water table in or
above the covering semi-pervious layer. A typical S-shaped time-drawdown curve can
also be obtained in cases where the drawdown is first dominated by a rather small elastic
storage coefficient of the pumped layer and is dominated after a longer time by the flow
in the adjacent semi-pervious layers with larger elastic storage coefficients. In Kruseman
and De Ridder (1970 and 1990) an extensive description is given of an interpretation
method which is derived from the model of Boulton.
3.7 Model of Neuman and Witherspoon
Confined aquifers of which the pervious layers are isotropic or anisotropic as in the
models of Theis and Hantush-Weeks, are not so frequent in nature. Generally, groundwa-
ter reservoirs consist of an alternation of pervious and semi-pervious layers. To treat the
flow to a pumped well in multilayered aquifers, one must follow the storage change and
the drawdown in as many points as possible. A first step of the flow analysis in a
multilayered aquifer is the model of Neuman and Witherspoon (1969a and 1969b). In this
model the unsteady state flow of water is treated in a groundwater reservoir that consists
of two pervious layers which are separated by a semi-pervious layer. The upper pervious
layer is bounded above by an impervious layer and the lower pervious layer is assumed to
be bounded below by an impervious layer (Fig. 3.14). One assumes that the lower
pervious layer, layer 1, is pumped and that the flow is exclusively horizontal in the
pervious layers and exclusively vertical in the semi-pervious layers. The thickness of the
pumped pervious layer must be such that the horizontal flow is considerably larger than
the vertical flow in it. It is also assumed that the thickness of the unpumped pervious
layer is not so large. Consequently, the drawdowns at different levels but at the same
distance from the pumped well do not differ much. So, the flow in this pervious layer can
be characterized by one gradient of the drawdown.
MODEL OF NEUMAN -WITHERSPOON
IMPERVIOUS LAYER
PERVIOUS LAYER
SEMI- PERVIOUS LAYER
PERVIOUS LAYER
IMPERVIOUS LAYER
Fig. 3.14. Schematization of the multilayered aquifer in the model of Neuman and
Witherspoon
The basic differential equation of horizontal flow in the pumped pervious layer is:
(3.154)
The initial and boundary conditions are:

8 1 (r, 0) 0, at:r'2:0
8 1 (00,t:) 0, at:t:'2:0
(3.155)
lim r aa81
r-O r
The basic differential equation of the vertical elastic flow in the semi-pervious layer can
be written as:
(3.156)
with the initial and boundary conditions:

82 (r , z, 0) ~ 0, at: r '2: 0 and O,,;,z,,;,Dl
82 (r, 0, t:) ~ 81 (r, t:), at: r '2: 0 and t: '2: 0 (3.157)
S2(r,Dl,t:) ~s3(r,t:), at:r'2:0andt:'2:0
The basic differential equation of the horizontal flow in the unpumped pervious layer 3
can be formulated as follows:
azS3 1 as3 ) aS2 (r,D 2 , t:)

K D ( - - + - - - -K (3.158)
h3 3 ar2 r ar v2 az
with the initial and boundary conditions:

S3 (r, 0) 0, at:r'2:0
S3(00,t:) 0, at:t:'2:0
(3.159)
. aS3
11m r ar ~ 0, at: t: '2: 0
r-O
Neuman and Witherspoon (1969a) elaborate a complete analytical solution for the above
described problem.
The drawdown in the pumped pervious layer is a function of five dimensionless
parameters. They were selected so that these parameters can be compared with the
forgoing models. The first parameter rfLJ2 is defined by the distance r from the pumped
well and the leakage factor LJ2 of the lower semi-confined aquifer. This aquifer is formed
by the pervious layer 1 and bounded above by the semi-pervious layer 2
or (3.160)
The second parameter rfL32 is also defined by the distance r from the pumped well and
the leakage factor L32 of the upper semi-confined aquifer. This last aquifer is formed by
the upper pervious layer 3 and bounded below by the semi-pervious layer 2. The third
parameter (312 is defined by the distance r from the pumped well and the characteristics of
the lower semi-confined aquifer. In addition to the parameters that define rfL 12 , the
parameter (312 is also defined by the elastic storage coefficients of the pervious layer 1,
SsIDJ, and the storage coefficient of the semi-pervious layer 2, SS2D2:
r Ss:zD2 (3.161)
~12
4 S SlD1 KhlD1 C 2
The fourth parameter (332 is defined by the distance r from the pumped well and the
characteristics of the upper semi-confined aquifer. In addition to the parameters that
define rf~2' the parameter (332 is also defined by the elastic storage coefficients of the
pervious layer 3, SS3D3, and the elastic storage coefficient of the semi-pervious layer 2,
Ss2D2:
r Ss:zD2 (3.162)
4 SS3D3Kh3D3C2
The fifth parameter lIu I depends on the distance r from the pumped well, the time of
pumping t and the characteristics of the pumped pervious layer 1, Knl and SSI:
(3.163)
The description of the drawdown in the semi-pervious layer 2 requires an additional

dimensionless parameter, zfD2 , which is defined by the level z in the semi-pervious layer
and the thickness of layer D2. The origin of the z-axis coincides with the base of the
semi-pervious layer (Fig. 3.14). This large number of parameters makes it practically
impossible to draw a sufficient number of type curves which are necessary for the
interpretation of the observed drawdowns. For the Walton type curves there are only two
independent dimensionless parameters, lIu I and rfL l2 , which define the dependent
dimensionless variable W(uJ,rfL12).
To reduce the number of parameters Neuman and Witherspoon (1972) propose to

diminish the number of observations which will be involved in the interpretation. They
consider only the observations after a short period of pumping. It is the time before the
occurrence of a drawdown in the unpumped pervious layer or not even at the boundary
between this layer and the semi-pervious layer. The drawdown in the pumped pervious
layer depends then only on the parameters U1 and (3IZ while the drawdown in the influen-
ced part of the semi-pervious layer depends on the same parameters as the unpumped
layer along with the additional dimensionless parameter 1/uz = Kvzt/(zZS,z).
By approximating the ratio of the drawdown in the semi-pervious layer Sz to the
drawdown in the pumped layer Sl, at the same distance from the pumped well and after a
same time of pumping, one can derive the dimensionless parameter 1/Uz from a graph
(Fig. 3.15). In this graph, a series of type curves represents the relation between 1/uz and
the ratio SziSI for the different values of 1/u l . The type curves that correspond with values
for 1/u1 equal to or larger than 100, do not differ from each other. The curves show a
steep course for values of S2/S1 equal to 0.1 or smaller. As a consequence the value 1/uz
can be derived accurately from the ratio SziSI which can only be estimated for large values
of 1/u1 and small values of sisl. The critical value which must be defined is not the
accurate value of the drawdown in the semi-pervious layer but is the time t for which the
first well distinct drawdown appears in the semi-pervious layer.
The interpretation method of Neuman and Witherspoon can be summarized in
following steps. The first step consists of the interpretation of the time-drawdown curve
according to the interpretation method of Theis or Cooper-Jacob. Because the leakage
from the adjacent layers is not considered, the derived value of the transmissivity
increases with the increasing distance r of the observed time-drawdown curve. The
transmissivity is then estimated with the aid of the smallest value derived from the time-
drawdown curve of the nearest observation well from the pumped well and with the aid of
the variation of these derived values with respect to their distances from the pumped well.
The deviations of the elastic storage coefficient values, derived with the interpretation
method of Theis or Cooper-Jacob, with respect to the real values are not predictable. The
elastic storage coefficient is mostly estimated with the help of the time-drawdown curve
observed in the directly pumped layer closest to the pumped well. The next step is the
derivation of the diffusivity of the semi-pervious layer 2, Ciz (=Kvz/S,z). This value is
defined by the time t of the first clear appearance of the drawdown Sz in the semi-
pervious layer due to the pumping. At that time the drawdown in the unpumped pervious
layer is negligibly small. With the drawdown Sl, observed at a same distance from the
pumped well as the drawdown sz, the ratio sisl is obtained. Also the value z must be
deduced. This is the difference between the level of the screen of the observation well in
the semi-pervious layer and the level of the base of this layer. From the estimated values
of the transmissivity and the elastic storage coefficient of the pumped pervious layer the
value of U1 is calculated. With this value the type curve which gives the relation between
SZ/Sl and 1/uz is chosen (Fig. 3.15). The value sisl allows us to derive the value 1/uz.
Chapter 3 / Ellolution of analytical models of pumping tests 109
With this value the diffusivity of the semi-pervious layer is derived according to the
equation:
(3.164)
To derive the vertical hydraulic conductivity of the semi-pervious layer, the specific
elastic storage of the semi-pervious layer should be estimated. This parameter can be
determined in another way, e.g., by compressibility tests in the laboratory or by placing
extensometers in the borehole.
100
i ~~ :;.--
~ ~ ../ I
'.,=IO'~jII ~
,/
I~ ~~
10'
II
I let.
10
0.2
~
7111l In 1'1
BI
r7II/Tl r7 11'2.1 i
rill II / I
VI/I II I
, ,
I
, I
! , i
"
i I
I I:
/ I.
I
1/ I
,
!
rJ I
I
Jr1.11 I- I
Fig. 3.1S. Family of type curves representing the relation between s2/s1 and 1/u2 for
different values of tD (=1I(4u)) (after Neuman and Witherspoon, 1972).
3.8 Retrospective view on analytical models and

their derived interpretation methods
The first analytical model, the model of Thiem (1906) treats the steady-state flow to a
pumped well in a confined aquifer. Only the transmissivity or the horizontal hydraulic
conductivity can be derived from the maximum drawdowns at different distances from the
pumped well. With the model of Theis (1935), it becomes possible to use not only the
maximum drawdowns but also to interpret all the observed time-drawdown curves at
different distances from the pumped well. With this interpretation method, one can derive
the elastic storage coefficient and the transmissivity of the pumped layer. It soon becomes
clear that unsteady-state flow in a confined aquifer is seldom a realistic situation. The
time-drawdown curves of most pumping test reach a maximum value after a relatively
short period of pumping. In some pumping test, this drawdown increases again after a
longer period of pumping so that a S-shaped time drawdown curve is obtained. The
theoretical time-drawdown curve of the model of Theis increases continuously. Therefore,
theoretically, no steady-state flow can exist. The drawdown increase rate diminishes
continuously as the area of influence expands. The shapes of the time-drawdown curves
and the differences between the derived values of transmissivity and elastic storage coeffi-
cient from the different time-drawdown curves bring us to the conclusion that the flow
conditions as assumed in the model of Theis are rarely met in reality. The layers
bounding the pumped pervious layers seldom have such a small conductivity that they can
be considered as impervious.
In the model of lacob-Hantush (1955), a leakage from the bounding layers to the
pumped pervious layer is assumed. This happens by a vertical flow. The vertical flow
velocity is assumed to be proportional to the drawdown in the pumped layer at every time
after the start of the pumping. This means that the vertical flow is considered inelastic
and that the drawdown is zero at the outer boundary of the bounding semi-pervious
layers. Generally, the theoretical time-drawdown curve of the model of lacob-Hantush
resembles the time-drawdown curve of most pumping tests or at least the first part of an
observed S-shaped time-drawdown curve. With the interpretation method, derived from
the model of lacob-Hantush, the transmissivity and the elastic storage coefficient of the
pumped pervious layer are derived as well as the resulting hydraulic resistance of the
bounding semi-pervious layers. The reciprocal value of this resulting hydraulic resistance
is equal to the sum of the reciprocal values of the hydraulic resistances of the adjacent
semi-pervious layers. Also, here, the derived values of the transmissivity increase with
increasing distance of the observed time-drawdown curves. This can be ascribed to the
fact that the storage decrease in the bounding semi-pervious layers is ignored.
Therefore, Hantush (1960) adjusted the model of lacob-Hantush so that the storage
decreases in the bounding semi-pervious layers are considered. By applying the interpreta-
tion method derived from this last model, a (J-value is derived beside the values for the
transmissivity and the elastic storage coefficient of the pumped pervious layer. This (J-
value is a function of the transmissivity and the elastic storage coefficient of the pumped
layer and of the specific elastic storage and vertical conductivity of the bounding semi-
pervious layers. The Hantush type curves H(u,{3) show a gradual transition in shape so
that it is difficult to select the appropriate type curve.
From the three different cases of the model of Hantush, it is inferred that the last
part of the time-drawdown curves are determined by the boundary conditions of the
adjacent semi-pervious layers. Depending on these boundaries, the last part of the time
drawdown curve can be interpreted by the Theis type curves or with the Walton type
curves. The derived values are mostly a combination of parameters of different layers. In
this way the derived r/L value is defined by a distance from the pumped well, the
transmissivity of the pumped layer and the hydraulic resistances of the adjacent semi-
pervious layers. The {3-value is a combination of the elastic storage coefficients of the
adjacent layers along with the parameters which define the r/L value. All these hydraulic
parameters cannot be derived unequivocally. Frequently, a value must be assigned to
some hydraulic parameters or to a ratio between hydraulic parameters if the values of
other hydraulic parameters should be derived. Hantush proposed a number of other
analytical models which treat the unsteady-state flow to pumped wells in aquifers of
which the hydraulic parameters of the different layers have special characteristics or of
which the boundary conditions are special. One of these models treats the unsteady-state
flow to a partially penetrating well in a confined aquifer of which the pumped pervious
layer may be transversely anisotropic (Hantush, 1962). Weeks (1969) derived an interpre-
tation method based on this model.
In the same period as Jacob and Hantush, Boulton proposed an analytical model of
which the theoretically derived time-drawdown curve shows a typical S-shape. Boulton
(1955) assumes that for a water table drop not all the water is released immediately but
that there occurs a certain "delayed yield". Initially he proposed that the delayed yield can
occur in two cases: a first case where the water table is situated in fine-grained material
which overlies the pumped pervious layer and a second case with compressible fine-
grained layers in the pumped layer which is bounded above and below by two impervious
layers. Boulton proposed an empirical relation between the velocity of "delayed yield"
and the drawdown increase, the storage coefficient near the water table (in case of an
unconfined aquifer) and an empirical constant ex which is called the Boulton delay index.
The total velocity of the "delayed yield" is obtained by integration of all the "delayed
yields" caused by all the drawdown increases that occur between the start of the pumping
test and the considered time. Cooley (1971) shows that the Boulton integral describes the
vertical inelastic flow in a semi-pervious layer. This layer is bounded on top by a water
table and overlies the pumped pervious layer. The compressibility of the semi-pervious
layer is ignored. The drawdown of the water table is small in comparison with the
thickness of this semi-pervious layer. In contrast to the concept of "delayed yield" in the
model of Boulton, an immediate release of water is assumed for a drawdown of the water
table. The only difference between the model of lacob-Hantush and the model of Boulton-
Cooley is the boundary condition at the top of the semi-pervious layer. In the model of
lacob-Hantush a constant hydraulic head boundary is assumed. In the model of Boulton-
Cooley, the boundary condition at the top of the semi-pervious layer is a constant flux
with an immediate release of water when the water table drops. In both models, the
elastic storage decrease in the semi-pervious layer is ignored or a vertical inelastic flow is
assumed. The Boulton delay index in the case of a semi-confined aquifer is the product of
the storage coefficient near the water table and the hydraulic resistance of the covering
semi-pervious layer or fine-grained material. The drainage factor of the unconfined
aquifer with "delayed yield", as defined in Kruseman and De Ridder (1970) is equal to
the leakage factor of the semi-confined aquifer. Because the model of lacob-Hantush and
the model of Boulton-Cooley differ only in the boundary condition at the top of the
covering semi-pervious layer, the first part of the two theoretical derived time-drawdown
curves do not differ. The Walton type curves and the first part of the Boulton type curves
(Kruseman and De Ridder, 1970) are the same. Because Boulton (1963) ascribes the
"delayed yield" to the flow above the water table, Cooley and Case (1973) examine the
effect of this flow above the water table on the time-drawdown curve. This was done with
the help of a numerical model which simulates the flow between a large number of layers
that are located above and below the water table. They arrive to the conclusion that the
flow in the unsaturated zone has little or no influence on the time-drawdown curves of the
pumped pervious layer.
The last considered analytical model is the model of Neuman and Witherspoon
(I969a). This model treats the flow in a multilayered aquifer where two pervious layers
are separated by a semi-pervious layer. These layers are bounded above and below by
impervious layers. This model was a first step in the analysis of the flow in a groundwa-
ter reservoir which consists of an alternation of pervious and semi-pervious layers. In this
model, the drawdowns and the storage increases are followed in a large number of layers.
In this model following simplifying assumptions are made: exclusive horizontal flow in
the pervious layer and exclusive vertical flow in the semi-pervious layer. Neuman and
Witherspoon elaborate a complete analytical solution for this problem. The drawdown in
the pervious layers is a function of five dimensionless parameters. The drawdown in the
semi-pervious layer is determined by the same five dimensionless parameters and an
additional dimensionless parameter. This last parameter is the level in the semi-pervious
layer divided by its thickness. This large number of hydraulic parameters results in a
large amount of type curves which are needed for the interpretation of the drawdowns in
the pervious layers and in the semi-pervious layer.
To reduce the number of parameters involved in the interpretation, Neuman and
Witherspoon (1972) proposed to analyze only a reduced number of observations such as
the drawdown after a relatively short period of pumping until the drawdown arrives in the
unpumped pervious layer. The ratio of the drawdown in the semi-pervious layer S2 to the
drawdown in the pumped pervious layer S1 at the same distance from the pumped well
and at a same time of pumping is only defined by the dimensionless parameters l/u1 and
l/u2 • The dimensionless parameter l/u2 has only a small influence on the ratio S1/S2 when
this value is smaller than 0.1. This ratio also has a small sensitivity for the values of the
dimensionless parameter l/u1 when this value is larger than 100. If l/u 1 > 100 and S1/S2
< 0.1 then a rough estimate of S2/S1 after a certain pumping time t is sufficient to
determine the dimensionless parameter l/u2 • With the value of I/u2 and the delay t by
which the drawdown occurs in the semi-pervious layer, the diffusivity Kv2/S'2 of the semi-
pervious layer is found. The diffusivity of the semi-pervious layer is determined by this
delay. So, the term, ratio method, is misleading and would better be indicated by the term
delay method.
Based on the evolution of the analytical models and their derived interpretation
methods of pumping test, following concluding remarks can be made. The first require-
ment for the application of an interpretation method is that a thorough knowledge of the
different assumptions of the applied model is required. The "classic" interpretation
methods are very different among each others, even between the methods which are
derived from a same model. The additional conditions of each interpretation method
should be known and satisfied. Simple conditions as assumed in the model of Theis are
rarely met in nature. Moreover, the knowledge of the horizontal conductivity of one
pervious layer is generally not sufficient to resolve some hydrogeological problem.
In reality, groundwater reservoirs consists of an alternation of pervious and semi-
pervious layers with layered and/or trending heterogeneity (Sect. 2.1.4). As a conse-
quence, the knowledge of more than one hydraulic parameter is required. The larger the
number of identifiable parameters, the more complex and more specific the applied
models become. Mostly, this number of parameters is reduced by the combination of
parameters. In spite of these combinations the number of parameters that should be
identified is still large (e.g. model of Neuman and Witherspoon) so that a considerable
number of type curves are required. In some cases, a reduction of the number of
hydraulic parameters is obtained by the choice of the drawdowns that are interpreted
(e.g., the first part or the last part of the time-drawdown curve, the maximum drawdown
or a given ratio of drawdowns, etc.). This choice of some drawdowns leads to "frag-
mented" analyses of the observed drawdowns. As a consequence, the observed draw-
downs are then not interpreted simultaneously in one group. The final remark is that the
treated interpretation methods of this chapter do not allow to infer the accuracy of the
identified hydraulic parameters. One of the purposes of this book is now to present a
unique and generalized interpretation method of pumping test which overcome all the
above mentioned drawbacks. This generalized interpretation method demonstrates at the
same time the application of a simple inverse model under favorable conditions.
REFERENCES
Ambramowitz, M., and Stegun, LA., 1965, Handbook of mathematical functions with
formulas, graphs and mathematical tables: New York, Dover Publications, Inc.,
1046 p.
Boulton, N.S., 1955, Unsteady radial flow to a pumped well allowing for delayed yield
from storage: lASH AssembLee Generale de Rome, Tome II, Pub\. N°37.
Boulton, N. S., 1963, Analysis of data from non-equilibrium pumping test allowing for
delayed yield from storage, Proc. Inst. Civ. Eng., 26, p. 469-482.
Cooley, R.L., 1971, A finite difference method for unsteady flow on variably saturated
process media, application to a single pumping well: Water Resources Research,
v. 7, no. 6, p. 1607-1625.
Cooley, R.L., 1972, Numerical simulation of flow in an aquifer overlain by a water table
aquitard: Water Resources Research, v. 8, no. 4, p. 1046-1050.
Cooley, R.L. and C.M. Case, 1973. Effect of a watertable aquitard on drawdown in an
underlying pumping aquifer: Water Resources Research, v. 9, no. 2, p. 434-447.
Cooper, H.H., 1963, Type curves for nonsteady radial flow in an infinite leaky artesian
aquifer, in Bentoll, Ray, compiler. Shortcuts and special problems in aquifer tests:
U.S. Geol. Survey Water-Supply Paper, 1545-C, p. 48-55.
Cooper, H.H., and Jacob, C.E., 1946, A generalized graphical method for evaluating
formation constants and summarizing well field history: Am. Geophys. Union
Trans., 27, p. 526-534.
De Glee, G.J., 1930, Over grondwaterstromingen bi} wateronttrekldng door middel van
putten: Thesis. Delft (The Netherlands), Waltman, J., 175p.
Dupuit, J., 1963. Etudes theoriques et pratiques sur Ie mouvement des eaux dans les
canaux decouverts et ii travers les terrains permeables. deuxieme edition. Dunot,
Paris, 304 p.
Hantush, M.S., 1956, Analysis of data from pumping tests in leaky aquifer: Am.
Geophys. Union Trans., 37, p. 702-714.
Hantush, M.S., 1960, Modification of the theory of leaky aquifers: loum. Geophys.
Res., 65, p. 3713-3725.
Hantush, M.S., 1961, Drawdown around a partially penetrating well: lour. Hydraul. Div.
Proc. Amer. Soc. Civil. Engrs., 87 (HY4), p. 83-98.
Hantush, M.S., 1964, Hydraulics of wells, in Chow, V.T., ed., Advances in hydro-
science, Vol. 1: New York, Academic Press, p. 281-432.
Hantush, M.S., 1966, Analysis of data from pumping tests in anisotropic aquifers: loum.
Geophys. Res., 71, p. 421-426.
Hantush, M.S., and Jacob, C.E., 1955, Non-steady radial flow in an infinite leaky
aquifer: Trans. Amer. Geophys. Union, 36, p. 95-100.
Jacob, C.E., 1946, Radial flow in a leaky artesian aquifer: Trans. Amer. Geophys.
Union, v. 27, no. 2, p 198-205.
data: Wageningen (The Netherlands), Int. Inst. Land Recl. and Improv. (ILRI)
Bull. 11., 200p.
data. Second Edition: Wageningen (The Netherlands), Int. Inst. Land Recl. and
Improv. (ILRI), Publ. 47., 375p.
Neuman, S.P., and Witherpoon, P.A., 1969a, Theory of flow in a confined two aquifer
system: Water Resources Research, v. 5, no. 4, p. 803-816.
Neuman, S.P. and Witherpoon, P.A., 1969b, Applicability of current theory of flow in
leaky aquifers: Water Resources Research, v. 5, no. 4, p. 817-829.
Neuman, S.P. and Witherpoon, P.A., 1972, Field determination of the hydraulic proper
ties of leaky multiple aquifer systems: Water Resources Research, v. 8, no. 5,
p. 1284-1298.
Reed, J.E., 1980, Type curves for selected problems of flow to wells in confined
aquifers: U.S. Geol. Survey Techn. of Water Resources lnv., Book 3, Chap. B3,
106 p.
Slichter, C.S., 1899. Theoretical investigation of the motion of ground waters. U.S.
Geological Survey 1f)'i' Ann. Rept., part 2, 322p.
Theis, C.V., 1935. The relation between the lowering of the piezometeric surface and the
rate and the duration of discharge of a well using groundwater storage. Trans.
Amer. Geophys. Union, 16, p. 519-524.
Thiem, G., 1906. Hydrologische Methoden, Liepzig, Gebhardt, 56p.
Turneaure, F.E. and H.L. Rusell, 1901. Public water supplies: New York, John Wiley
and Sons. Inc., 766 p.
Venitis, C., 1970. Finite aquifers, Characteristics responses and applications: Joum. of
Hydrology, v. 12, no. 1, p. 53-62.
Walton, W.C., 1962. Selected analytical methods for well and aquifer evaluation: Illinois
State Water Survey Bull., 49, 81p.
Weeks, E.P. 1969. Determining the ratio of horizontal to vertical permeability by aquifer-
test analysis: Water Resources Research, v. 5, no. 1, p. 196-214.
Wenzel, L.K., 1942. Methods for determining permeability of water-boring materials,
with special reference to discharging-well methods: U.S. Geol. Survey Water
Supply Paper, 887, 192 p.
Chapter 4 / Numerical model of pumping tests
in a layered groundwater reservoir
In this chapter a two-dimensional axi-symmetric model is described which allows the
simulation of a pumping test in a layered groundwater reservoir. In the axi-symmetric
model the layers are subdivided in rings of which the inner and outer radii form a
logarithmic series. This allows the calculation of the drawdowns with a same order of
accuracy at distances from the pumped well which are in the order of centimeters,
decimeters, meters, decameters and hectometers. The points of time for which the
drawdowns are calculated also form a logarithmic series so that it is also possible to
calculate the drawdowns with the same accuracies after secondes, minutes, hours, days,
weeks and months after starting the pump.
The numerical model is verified with the help of several analytical models which
are treated in Chapter 3: the model of Theis, the model of Jacob-Hantush, the model of
Hantush, the model of Hantush-Weeks and the model of Boulton as explained by Cooley.
At the end of this chaper the program package for the numerical simulation of a pumping
test is proposed and explained.
4.1 Finite-difference grid
In the numerical model, the groundwater reservoir is schematized in an alternation of

pervious and/or semi-pervious layers which are laterally continuous and have an infinite
extension. According to the location of the screens of the pumping and observation wells
and to the expected flow, the pervious and semi-pervious layers are further discretized in
the numerical model in several layers. The layers of the numerical model are numbered
upward (Fig. 4.1). The lowest considered layer is layer 1 and is always bounded below
by an impervious boundary. The uppermost considered layer is always bounded above by
a water table. The layers are subdivided in a number of coaxial rings around the pumped
well. Each layer is characterized by one value of the horizontal conductivity and one
value of the specific elastic storage. Consequently, it is assumed that the layers are
laterally homogeneous and continuous. The horizontal flow and the storage decrease are
considered in each layer simultaneously with the vertical flow between the layers. This
vertical flow is governed by one value of the hydraulic resistance between the layers. By
the subdivision of the groundwater reservoir in a number of layers and a further subdivi-
sion of the layers in a number of coaxial rings, the three-dimensional flow around the
pumped well is simplified to an axial-symmetric two-dimensional flow problem.
In each coaxial ring a nodal circle is considered. This nodal circle is at half height
of the ring and its radius is equal to the geometric mean of the inner and the outer radius
of the ring. The inner and the outer radii of the rings form a logarithmic increasing
118 Chapter 4 / Numerical model of pumping test
I '\
I \ I \ I \
1-l.J~~ ---I.J+1-, ;:: 1+1.J+1--, 0(3+1 )
I \ " ".... 1 \ ........... \
'\.-I , I',
I , I "
1-1.3', ___ I.J_~ 1 __
__ 1+1.J " '\_ O(J)
I \ \
1 \ \ \
I -, \
" I ''1-
'"I', I'
I ",
- 1.J-1~- -1+1.J-1 -~'\ 0(J-1)
I \ \
I \ \
I \ \
R(1-1/2) = R1.A(I-1.5)
R(1) = R1.A(I-1)
R(I+1/2) = R1.A(I-O.5)
Fig. 4.1. The (I,I)th ring and the surrounding rings of the axisymmetric grid. RI is the
initial radius and A is a factor which is larger than I, mostly, 10°·1.
series. As derived from Fig. 4.1, this series is defined by the initial radius Rl, the inner
radius of the smallest considered ring, and a factor A which is mostly chosen equal to
1.259. In each nodal circle, a drawdown is considered and is indicated by the symbol
s(I,J). The index I indicates the number of the rings. The first ring is the smallest. The
rings are counted in the outward direction. The index 1 indicates the number of the layer.
The layers are counted in the upward direction.
The drawdowns are calculated at well-defined points of time. These points of time
are defined by the initial time Tl and the same factor A which defines also the radii of
the rings. The first calculated drawdown corresponds with the point of time TIA. The
calculated drawdown at the nodal circle of ring 1,1 after the 'J"h calculation is indicated by
the symbol s(l,Jh- It is the drawdown after a time TlAT of pumpage.
In an earlier version of the numerical model the finite-difference method was
applied (Lebbe, 1983a). Here, the drawdown was calculated at well-defined distances
from the pumped well, in the different layers and at well-defined points of time after the
start of the pump. In this version, the central and backward finite-difference approxima-
tions were applied to calculate the drawdown. Here, a discontinuous change of the draw-
down is assumed between the nodal circles and between the successive points of time. In
the newly developed version, the discharge rates which flow through the boundaries of
Chapter 4 / Numerical model of pumping test 119
the rings are calculated according to the drawdowns which vary continuously between the
nodal points and between the successive points of time. This is also the case when the
storage decrease is calculated within the considered rings. For all these calculations, one
considers different kinds of mean drawdowns. They will be treated in the following part.
4.2 Mean drawdowns
4.2.1 Mean drawdown over a horizontal plane going through the nodal circle in a ring
One assumes that the drawdown between two successive nodal circles of the same layer
shows a logarithmic change versus the distance from the pumped well. This assumption is
made because this change is found in some instances of some analytical models, as for
example the model of Theis when u<O.OI (Cooper and Jacob, 1946). The drawdown
s(r,J) in the ring I,J at the same level as the nodal circles of layer J can be described by
means of two equations. The first equation describes the drawdown change versus the
pumped well distance from the inner radius of the ring I,J , which is here indicated as
R(I-1I2), to the nodal circle in the ring I,J, with a radius indicated by R(I) or for
R(I-II2) ~ r ~ R(I):
s(r,J) = s(I,J) + (s(I,J) -s(I-l,J» (In r-In R(I» (4.1)

In R (I) - In R (I -1 )
where r is the distance from the axis of the pumped well and R(I-l) is the radius of the
nodal circle of the (I -l)th ring.
The second equation represents the drawdown change from the nodal circle in the ring I,J
to the outer radius of the ring I,J , indicated by R(l + 112),
or for R(l) ~ r ~ R(I + 112):
s(r,J) = s(I,J) + (s(I+l,J) -s(I,J» (In r-In R(I» (4.2)

In R(I+l) - In R(I)
where R(I + 1) is the radius of the nodal circle of the (I + l)th ring.
The mean drawdown in ring I,J on the same level of the nodal circles of the Jth
layer, sr(l,J), can be found by integration of 2u s(r,J) over r between the boundaries
R(I-1I2) and R(l + 112) divided by the integration of 2u over r between the same bounda-
ries or:
R(I) J,R(I+l/2)
J, 21trs(r,J)dr+ 21tr s(r,J) dr
R(I-l/Z) R(I)
(4.3)
R(I+l/2)
J, 21tr dr
R(I-l/2)
where R(I-1I2) = RIA<I.L5) or In R(I-II2) = In Rl + (1-1.5) In A

R(I) = RIA<I-l) or In R(l) = In Rl + (1-1.0) In A
R(l + 112) = RIA(1-0.5) or In R(I+ 112) = In Rl + (1-0.5) In A.
The denominator in Eq. 4.3 is the surface of the plane bounded by the two concentric
circles with the radii R(I-l/2) and R(I + 112) or:
J,R(I-1/2)
R(I+l/2)
2rtrdr = rt R12 A 2I - 3 (A2-1) (4.4)
The numerator of Eq. 4.3 can be written with the help of Eqs. 4.1 and 4.2 as:
(R(I) 2rtr(s(I,J) +(s(I,J) -s(I-l,J) Inr~lnR(I) dr

JR(I-l/2) n A
+(R(I+l/2)2rtr(s(I,J) +(s(I+l,J) -s(I,J» Inr-InR(I) dr
JR(I) In A
= s(I,J)
R(I+l/2)
J,R(I-l/2)
2rtrdr (4.5)
+ 2rt (s(I,J) -s(I-l,J» (R(I) (Inr-InR(I» rdr

In A JR (I-l/2)
+ 2rt (s(I+l, J) -s(I, J» (R(I+l/2) (Inr-InR(I» rdr
In A JR(I)
The first term of the right part of Eq. 4.5 is equal to the product of the drawdown in the
nodal circle in ring I,J , s(I,J), with the surface of the horizontal cross-section through the
ring I,J or:
(4.6)
The second term of the right part of Eq. 4.5 after integration is equal to:
21t (s(I,J) -s(I-l,J)
lnA
r2( )]R(I) 1 ()[r2]R(I)
{[ 2" In r -0.5 R(I-l/2) - nR I 2" R(I-l/2) } or (4.7)
1t (s ( I, J) - s ( I -1 , J) {R ( I) 2 (lnR (I) - 0 . 5 )
In A
-R(I-1/2) 2 (lnR(I-1/2) -0.5) -lnR(I) (R(I) 2-R(I-1/2) 2)}
When the distances R(I) and R(l-1I2) in Eq. 4.7 are written in function of the initial
radius Rl, the factor A and the number of the ring I then the second term of the right
part of Eq. 4.5 becomes:
1 rt R12 A 2I- 3 s(I,J) -s(I-l,J) ( -A+lnA+l) (4.8)

2 In A
ChapteT 4 / Numerical model of pumping test 121
The third term of the right part of Eq. 4.5 after integration is equal to:
2'1t (s(I+1.,J) -s(I,J»
lIlA
2 2
{[ ~ (In r -0.5) 1~~i;-1/2) -lnR (I) [ ~ 1 ~~i;-1/2)} or (4.9)
'It (s(I+1., J) -s(I, J» {R(I+1./2) 2 (lnR(I+1./2) -0.5)
ln A
-R(I) 2 (lnR(I) -0.5) -lnR(I) (R(I+1./2) 2-R(I) 2)}
When the distances R(I + 112) and R(I) in Eq. 4.9 are written in function of the initial
radius R1, the factor A and the number of the ring I then the third term of the right part
of Eq. 4.5 becomes:
(4.1.0)
The mean drawdown in ring 1,1 on the level of the nodal circles of the J'" layer, sr(l,1), in
function of the drawdowns in three nodal circles s(l-l,1), s(l) and s(l+l) (see Fig.
4.1) are derived by dividing the sum of the three terms of the right part of Eq. 4.5 as
given in Eqs. 4.6, 4.7 and 4.8 with the denominator given in Eq. 4.4:
(4.1.1.)
where hI =(A-lnA-1)/(2(N-1)lnA),
h3=(NlnA-A2+A)/(2(N-1)lnA)
and h2=1-hI-h3.
As one can derive from the above given formula the interpolation constants hI, h2 and h3
are independent of the ring number I. In Table 4.1, the interpolation constants are given
for different values of the factor A. The interpolation constants hI and h3 are strongly
dependent of the factor A. This is much less the case for the interpolation constant h2
which approximates 0.75 for the different values of the factor A (see Table 4.1). For very
small values of the factor A hI and h3 approximate 0.125. When the factor A increases hI
decreases and h3 increases. When the value of the factor A is equal to 1.6 then h3 is
nearly two times hI. Consequently, the contribution of the drawdown s(l + 1,J) is twice as
important as the contribution of the drawdown s(I) in the mean drawdown sr(I,J).
Table 4.1. Different interpolation constants h" h2 and h3 for different values of factor A
and loglo A.
A loglOA hi h2 h3
1.011579 0.005 0.1240420 0.7499972 0.1259608
1.023293 0.010 0.1230868 0.7499890 0.1269243
1.035142 0.015 0.1221344 0.7499752 0.1278904
1.047129 0.020 0.1211850 0.7499558 0.1288592
1.059254 0.025 0.1202385 0.7499310 0.1298305
1.071519 0.030 0.1192950 0.7499006 0.1308043
1.096478 0.040 0.1174173 0.7498234 0.1327592
1.122018 0.050 0.1155522 0.7497242 0.1347235
1.161449 0.065 0.1127791 0.7495344 0.1376865
1.202264 0.080 0.1100363 0.7492955 0.1406682
1.258925 0.100 0.1064286 0.7489013 0.1446701
1.333521 0.125 0.1020020 0.7482883 0.1497097
1.445440 0.160 0.0959680 0.7472102 0.1568218
1.584893 0.200 0.0893186 0.7456735 0.1650078
1.778279 0.250 0.0813979 0.7433181 0.1752840
4.2.2 Mean drawdown over cylindrical sUiface going through a nodal circle in a ring
The assumption of the way how the drawdown varies between two nodal circles of rings
of the same size but of different layers is not so obvious as the assumption of the way
how the drawdown varies between two successive nodal circles of a same layer. Initially
one can assume the drawdown shows a linear change versus the level z between two
successive nodal circles. Then the change of the drawdown s(I,z) on the cylindrical
surface running through the nodal circle of the ring 1,1 can again be described by two
equations. The first equation describes the change of the drawdown from the base plane
of the ring to the nodal circle.
_ ( )) 2(z-Z(J))
s(I, z) -s I, J) + (s(I, J) -s(I, J-1 (J) ( ) (4.12)
D +D J-1
where the level z varies between the levels indicated by the symbols Z(J-1I2) and Z(J);
Z(J) represents the level of all the nodal circles of the Jh layer and Z(J-1I2) represents the
level of the base of the Jh layer. The second equation describes the change of the
drawdown from the nodal circle to the top plane of the ring.
S(I, z) =s(I, J) + (S(I, J+1) -s(I, J» D~g:~~~~i) (4.U)
where the level z varies between the levels indicated by the symbols Z(J) and Z(J + 112),
Z(J + 112) is the level of the top of the ]'h layer.
The mean drawdown in the ring I,J over the vertical cylindrical surface through the nodal
circle, sz(I,J), can now be found by integration of s(l,z) over z between the limits
Z(J-II2) and Z(J+II2). This results in an equation where the mean drawdown, sz(l,J), is
a function of the drawdowns in three nodal circles s(l,J-l), s(l) and s(l) + 1):
(4.14)
SZ (I, J) =v1 (J) s (I, J-1) +V2 (J) s(I, J) +V3 (J) s (I, J+l)
where Vb v2 and V3 are the vertical interpolation constants.
These vertical interpolation constants are equal to

vt(J) =D(J)/(4(D(J)+ D(J-l))),
viJ) =D(J)/(4(D(J)+ D(J + 1»)
and vz<J)=I-v t(J)-v3 (J)
where D(J) is the thickness of layer J.
When the thicknesses of the layers J-l, J, J + 1 are the same then the vertical interpolation
constants vt(J), viJ) and viJ) are respectively equal to 0.125, 0.750 and 0.125. When the
thickness of the layer J is smaller than the thicknesses of the adjacent layers J-l and J + 1
then the values of vt(J) and v3 (J) are also smaller and the value of vZ<J) approximates the
value of 1. When, on the contrary, the thickness of layer J is large with respect to the
thicknesses of the adjacent layers J-l and J + 1 then the values of vt(J) and v3 (J) are larger
than 0.125 and the value of v2 (J) is smaller than 0.750. Consequently, the contribution of
the drawdown of a nodal circle of a thin layer is important in the mean drawdown over
the cylindrical surface of an adjacent thick layer. The contribution of the drawdown of a
nodal circle of a thick layer is less important than in the mean drawdown over the
cylindrical surface of a thin layer.
In reality, however, the cases where the drawdown shows a linear change versus
the level z between two nodal points during a transient flow are exceptional. Only after a
very long time of pumpage, such a change can occur in a semi-pervious layer when
steady-state flow is approximated. Such a linear change of the drawdown with the level
can certainly not be assumed between a pumped pervious layer and the adjacent semi-
pervious layer. Even when a semi-pervious layer is discretized in a few layers into the
model, then the true change of the drawdown versus the level is not linear between the
nodal circles of these layers at the beginning of the pumping test. Therefore, the influence
of the drawdown in the nodal circles of the adjacent layers on the mean drawdown of a
layer over its cylindrical surface is attenuated by dividing the interpolation constant vt(J)
and v3(J) by 10. When the hydraulic parameters of the adjacent layers are different then
the vertical interpolation constant v3(J) is further multiplied by the factor f3(J):
(4.15)
where k,,(J) is the vertical conductivity between layer J and J + 1 and is equal to
(D(J)+D(J+1»l2c(J). The vertical interpolation constant vt(J) is further multiplied by the
factor ft(J), which is given by a same equation as f3(J) but where the indices J + 1 are
replaced by J-l. vl(J) and V3(J) approximate zero and v2(J) approximates one when there
is a high contrast between the hydraulic parameters.
4.2.3 Mean drawdown over entire ring volume
The mean drawdown over the entire volume of a ring, sn(l,J), is a function of the
drawdown in the nodal circle of the ring itself s(l,J), the drawdowns of the nodal circles
of the adjacent rings of the same layer s(l-I,J) and s(I+I,J) and the drawdowns in the
nodal circles of the rings above and below the rings already mentioned. This mean
drawdown can be inferred with the help ofSz(I-I,J), sz(l,J), sz(l+I,J) as:
(4.16)
From the above given equation and the mean drawdowns Sz given in Sect. 4.2.2 one can
define the mean drawdown over the entire volume of a ring, s,,(I,1), in function of the
drawdown in the nodal circle of the ring I,J itself and of the drawdown in the nodal
circles of its adjacent rings:
Srz(I,J)
hVll (J) s(I-1, J-1) +hV21 (J) S(I,J-1) +hV31 (J) s(I+1,J-1)
(4.17)
+hV'2 (J) s (I-1, J) +hV22 (J) s (I, J) +hV32 (J) s (I+1, J)
+hV'3 (J) s (I-1, J+1) +hV23 (J) s (I, J+1) +hV33 (J) s (I +1, J+1)
where hvll(J)=h1vit(J), hvdJ)=h tvi2(J), hv13(J)=h1vi3(J);

hv21 (J) = h2vi l (J), hV2z(J) =h2vi2(J), hv23 (J) =h 2vi3(J);
hv 31 (J)=h3vi l (J), hV32(J)=h3viz(J), hV33(J)=h3vi 3(J);
and vil(J) =vl(J)fl(J)/lO, viiJ) =v3(J)f3(J)/10, vi 2(J) = I-vil(J)-viz(J).
4.2.4 Mean drawdown during time step
During the calculated time step, a linear course is assumed between the drawdown and
the logarithm of the time elapsed since starting the pump. This is also the case when
u<O.OI in the analytical model of the confined aquifer (Cooper and Jacob, 1946).
Chapter 4 I Numerical model of pumping test 125
The drawdown in a nodal circle of a ring I,J at a point of time t within the interval
T1AT-' , T1AT is then described by following equation:
S( I, J) t =s (I, J) T-l. +
In t-In T1AT-l. (4.18)
(s(I,J) ,,-S(I,J)T_l.) T Tl.
In T1A -In T1A -
where s(I,Jh_, is the drawdown in the nodal circle of ring I,J at the time point TlAT-' and
s(I,J)T is the drawdown in the same nodal circle at the time point T1AT.
The mean drawdown during the T th time step, s.(I,J)T_,.T, can be found by the integration
of s(I,J), over the time t between the limits TlAT-' and TlAT and by the division of the
result by the duration of the considered interval T1AT-'(A-1) or:
S (I J)
l TlAT
S(I,J) t dt
= -=TlA~T-_1-::--:-:----
(4.19)
t I T-l.,T T1AT-l. (A-1)
By substitution of Eq. 4.18 in Eq. 4.19 one can find the mean drawdown during the Tth
time step, St(I,Jh_'.T, as a function of the drawdowns at the start and the end of the Tth
interval T1AT-', T1AT
(4.20)
where t,=(AlnA-A+1)/(lnA(A-1» and tz=l-t,.

In Table 4.2 the interpolation constants t, and tz are given for different values of A and
10g,oA. Small values of 10g\oA result in values of t, and tz which are very close to 0.5.
Table 4.2. Different interpolation constants t\ and tz for different values of factor A and
10g\oA.
A 10g,oA t, tz
1.011579 0.005 0.4990406 0.5009594
1.023293 0.010 0.4980812 0.5019188
1.035142 0.015 0.4971219 0.5028781
1.047129 0.025 0.4961625 0.5038375
1.096478 0.040 0.4923258 0.5076742
1.148154 0.060 0.4884907 0.5115093
1.202264 0.080 0.4846581 0.5153419
1.258925 0.100 0.4808287 0.5191713
1.445440 0.160 0.4693681 0.5306319
1.584893 0.200 0.4617586 0.5382414
1.778279 0.250 0.4522924 0.5477076
4.3 Continuity equation in numerical model
4.3.1 Mean velocities through boundary sUrfaces of rings
The mean horizontal flow velocity through the common boundary plane of the rings
1+ 1,J and I,J which are not bounded by the water table can be derived by a central finite-
difference approximation.
(4.21)
where Kh(J) is the horizontal conductivity of layer J, and sz(I + 1,J) and sz(I,J) are the
mean drawdown over the cylindrical surface running through the nodal circles of the rings
l+l,J and I,J (these are negative values, positive values correspond with a rise). This
horizontal velocity is positive in the direction of the pumped well.
The mean vertical flow velocity through the common boundary plane of the rings
I,J + 1 and I,J can also be derived by a central finite-difference approximation.
sr (I, J+1) -Sr (I, J)
(4.22)
Vvr =
c(J)
where c(J) is the hydraulic resistance between the half heights of layers J + 1 and J,
sz(I,J + 1) and sz(I,J) are the mean drawdown over the horizontal plane running through
the nodal circles of the rings I,J + 1 and I,J. This vertical flow velocity is positive in the
downward direction.
4.3.2 Storage change in rings
The storage change in ring I,J that is not bounded by the watertable after time TIAT of
pumpage .6.S(I,Jh_l.T during the time step between the time points TIAT-1 and TIAT can be
obtained by a backward finite-difference approximation with respect to time:
(4.23)
where 'll"R1 2 NI-3(N_l) is the surface of the base and top plane of ring I,J,
S.(J) is the specific elastic storage of layer J, D(J) is the thickness of layer J,
srz(I,Jh-to is the mean drawdown over the entire volume of ring I,J at time point
TIAT - 1,
srz(I,Jh, is the mean drawdown over the entire volume of ring I,J at time point
TIAT.
The storage change is calculated with the mean drawdowns over the entire volume of the
ring I,J.
4.3.3 In- and outflow rate difference of rings
In the rings which are not bounded by the water table there is inflow and outflow through
the inner and outer cylindrical surface of the ring as well as through the base and top
plane of the ring. The difference between the inflow and outflow rate can be derived with
the mean horizontal and vertical flow velocities or:
Il.Q(I,J) = 21tRl AI-loS D(J) (y;"'z(I,J)A-y;"'z(I-1,J»

(4.24 )
+1tR1 2 A 2I- 3 (A 2 -1) (Vv:r(I,J) -Vv:r(I,J-1»
Substitution of Eq. 4.21 which gives the mean horizontal flow velocity and of Eq. 4.22
which gives the mean vertical flow velocity in Eq. 4.24 results in:
Il.Q(I, J) = rtR1 2A 2I - 3 (A 2 -1) .

{Sr(I,J+1) -sr(I,J) _ Sr(I,J) -Sr(I,J-1)}
c(J) c(J-1) (4.25)
5
+21t AO. Kh(J)D(J) (Sz(I+1,J) -2Sz (I,J) +sz(I-1,J»
A-1
By the introduction of S, and Sz given in Eqs. 4.11 and 4.14 in Eq. 4.25, one obtains
Il.Q(I,J) in function of the drawdown in the nodal circles:
Il.Q(I, J) = 1tR1 2A 2I- 3 (A 2 -1) .

{ h1s (I -1, J-1) +h 2s (I, J-1) +h3S (I +1, J-1)
c(J-1)
-(h1 s(I-1,J) +h2 s(I,J) +h 3s(I+1,J) (1 ) +( (1J)
J-1 c c
+ hlS (I -1, J+1) +h 2s (I, J+1) +h3S (I+1, J+1) } (4.26)
c(J)
AO. 5
+21t A-1 Kh (J) D(J) .
{v1 (J) (s (I-1, J-1) -2s(I, J-1) +s (I+1, J-1) )
+V2 (J) (s(I-1,J) -2s(I,J) +s(I+1,J)
+V3 (J) (s (I-1, J+1) -2s (I, J+1) +s (I+1, J+1) )}
4.3.4 Continuity equation of rings
The law of continuity poses that there is no gain or loss of water in any ring during a
time step elapsing from time point TlA T·! to time point TlAT. For rings not bounded by
the water table, this law can be written as:
tl Il.Q(I,J)T_l+ t 2 Il.Q(I,J)T = Il.S(I,J)T_l.T (4.27)
where Il.Q(I,Jh_\ and Il.Q(I,Jh is the difference in in- and outflow rates of ring I,J at
respectively the time points T1AT-! and T1AT. These differences in in- and outflow rate
can be obtained by applying Eq. 4.26 for the respective time points.
4.3.5 Storage change for rings bounded by water table
The storage change in the rings of the uppermost layer I,N9 which is bounded by the
water table can be described by the following equation:
(4.28)
where So is the storage coefficient near the water table or the specific yield,
(D(N9)+sr(l,N9)T_l) is the real thickness of the uppermost layer N9 at time
point T1A T-1 •
In the above mentioned equation two types of mean drawdowns are considered. For the
calculation of the storage change due to the elasticity, the mean drawdown over the entire
volume is used. Here, the mean drawdown over the horizontal plane is applied to
approximate the thickness of the uppermost layer in the ring. For the calculation of the
change in storage due to the water table drop, the mean drawdown over the horizontal
plane is considered. Here, it is assumed that the water released by the lowering of the
water table is delivered immediately by vertical flow near the water table. This assump-
tion is based on the experience of Cooley and Case (1973) (see Sect. 3.6). These authors
also state that the influence of the flow in the unsaturated zone on the drawdown in the
saturated zone is small.
Because the storage change in the ring of the uppermost layer as formulated in Eq.
4.28 enhanced the numerical process strongly, this formulation is simplified to:
(4.29)
In this formula the calculation of the change in storage due to the compressibility of the
water and the layer is simplified and consequently a little less accurate than in Eq. 4.28.
This simplification is justified. Only a small error is made in the calculation of the elastic
storage change in the uppermost layer. Mostly, this error is negligibly small with respect
to the storage change due to the water table drop.
4.3.6 In- and outflow rate difference of rings bounded by water table
The water flow through the inner and outer cylindrical surfaces and through the base
planes of the uppermost rings are considered. The heights of inner and outer cylindrical
surfaces are defined by the drawdown in these rings. Because it is assumed that the
drawdown between the nodal circles shows a linear change versus the logarithm of the
distance from the pumped well and because the radii of the inner and outer cylinders of
the rings are located in the geometric mean of the radii of the nodal circles, the heights of
the cylinders situated between the nodal circles (I-l,N9) and (I,N9) are equal to
(2D(N9)+s(l-N9,J)+s(I,N9»/2. These heights vary from ring to ring. Therefore, the
formulation of the mean horizontal velocity through these cylindrical surfaces is rather
complex. To avoid this complex formulation, only the horizontal velocities at the levels of
the nodal circles are considered which can be written as:
V h (I,N9) = K h (N9) s(I+i,N9) -s(I,N9) (4.30)
Rl A I-1 (A-i)
The difference of the inflow and the outflow rate of a ring of the uppermost layer can
then be formulated as:
Il.Q(I,N9) = -TtR12A2I-3(A2-1) Vvr(I,N9-1)
+TtR1A I -1.5.<A V h (I,N9) (2D(N9) +s(I+i,N9) +s(I,N9» (4.31)
-Vh (I-1,N9) (2D(N9) +s(I,N9) +S(I-1,N9»}
With the aid of the formulation of the mean vertical velocity of Eq. 4.22 and of the mean
drawdown over the horizontal plane given in Eq. 4.11 the differences between the inflow
and the outflow rates can be formulated in function of the drawdowns in the nodal circles:
fl.Q(I,N9) = TtR1 2A 2I - 3 (A Z -1) .

{(h 1s(I-1,N9-1)+h 2 s(I,N9-1)+h 3 s(I+1,N9-1)
-h1 s (I-1, N9) -hzs (I, N9) -h 3 s (I+1, N9)}/ c(N9-i)
AO.5 (4.32)
+Tt A-1 Kh (N9) .
{(2D(N9) +s(I+1,N9) +s(I,N9» (s(I+1,N9) -s(I,N9»
-(2D(N9)+s(I,N9)+S(I-1,N9» (s(I,N9)-S(I-1,N9»}
4.3.7 Continuity equation of rings bounded by water table
The law of continuity can be formulated analogous as for the rings which are not bounded
by the water table:
t1 Il.Q(I,N9)T_1+t2 Il.Q(I,N9)T = Il.S(I,N9)T_1,T (4.33 )
where fl.Q(l,N9h-! and fl.Q(l,N9h are the difference in in- and outflow rates of ring I,N9
at respectively the time points TIAT.! and TIAT. These in- and outflow rate differences
can be obtained by applying Eq. 4.32 for the respective time points.
4.4 Initial and boundary conditions
The initial condition for the drawdown in each nodal circle is that the drawdown at the
initial time is equal to zero. This initial time is chosen at to +T1 where to is the time of
starting the pumping test and T1 is a first small lapse of time of pumpage. During this
first lapse of time, one supposes that there is no groundwater flow to the pumped well
and that all the water which is pumped is removed from the storage of the pumped well
itself. The initial time T1 is mostly chosen equal to 0.1 minutes. Only when the pumped
aquifer has a horiwntal conductivity larger than 20.0 mid, it is necessary to choose a
smaller initial time, for example T1 = 0.01 minutes.
The inner cylindrical surface of the first or smallest ring is considered as an
impervious boundary. Corresponding with the layer or the layers where the well screen of
the pumped well is located, water is withdrawn from the smallest ring(s) of the numerical
model. The base of the lowermost layer in the numerical model is an impervious
boundary. The top of the uppermost layer is a water table which delivers water immedi-
ately by the lowering of the water table.
During a well-defined time step (TlAT-!,TlAT), a constant drawdown is assumed
in the nodal circle of the outermost or largest rings of all the layers. After each time step,
the drawdowns of the largest rings are calculated. Therefore, the increases of the
drawdowns in the nodal circles of the three rings just before the largest considered ring
during the former time step are used. These increases are indicated by the ZI(J), Z2(J)
and Z3(J) and are defined as:
Z1 (J) dS(V1-l,J)T_l,T = S(V1-l,J)T_l- S (V1-l,J)T
Z2(J) ds(V1-2,J)T_l,T = S(V1-2,J)T_l-S(V1-2,J)T (4.34)
Z3 (J) dS(V1-3,J)T_l,T = S(V1-3,J)T_l- S (V1-3,J)T
where VI is the number of rings in a layer,

The drawdown increase in the outermost ring is derived by extrapolation with the above
defined drawdown increases. In this extrapolation, it is assumed that the drawdown
increase shows a linear change versus the logarithm of pumped well distance. The radii of
the nodal circles increase according to the relation RIN-! where I varies between I and
Vi. Therefore, these radii increase with a constant step on a logarithmic scale. The
drawdown increase in the last ring is now derived by means of three characteristics of the
drawdown increases in the adjacent rings. The first is the drawdown increase in the last
ring but one V 1-1,J or Z I (J). The second is the first derivative of the drawdown increase
versus the logarithm of the pumped well distance and derived by finite-difference
approximation or (Zl(J)-Z2(J»/logA. The third and last used characteristic is the second
derivative of the drawdown increase versus the logarithm of the pumped well distance or
(ZI(J)-2Z2(J)+Z3(J)/(logA)2. If one assumes that these finite-difference approximations
can also be considered as the first and second derivatives of the drawdown increase
versus the logarithm of the pumped well distance between the last but one and the last
ring then the drawdown increase in the last ring ZO(J) can be approximated using a
Taylor series:
ZO(J) = 2.50 Z1(J) -2.0 Z2(J) +0.50 Z3(J) (4.35 )
This solution with changing drawdown at the boundaries after the different time
steps is an attempt to approximate the analytical solutions where the constant hydraulic
head boundary is at an infinite distance from the pumped well. This attempt is preferred
to the simple boundary condition of zero drawdown at finite distance from the pumped
well. Because of the artificial character of the boundary condition, one has to try to
minimize the influence of this boundary condition on the calculated drawdowns. This can
be achieved by the continuous verification of the ground water balance after each
calculated time step (see Sect. 4.5.2). The storage decreases of the considered layers must
be maximized and the horizontal flow through the outer boundary minimized. The
influence of the posed boundary condition in the last nodal circle will further be treated in
Sect. 4.6 where the results of the numerical calculation are verified with classical
analytical solutions.
4.5 Solution of the numerical equations
4.5.1 Alternating direction implicit method
By means of the alternating direction implicit method, the drawdown in each nodal circle
is calculated at different points of time after starting the pump. First the drawdowns of
the rings with the same index I are considered as unknown. The drawdowns of the
adjacent rings are considered as known. They correspond with the drawdowns calculated
in the proceeding iteration for the larger adjacent rings and with the drawdown of the
ongoing iteration for the smaller adjacent rings. For each ring, except for rings belonging
to the lowermost and uppermost layer, the numerical equation (Eq. 4.27) can be
reformulated in function of the unknowns which must be resolved:
AC(J) s (I, J-1.) T+BC(J) s (I, J) T+CC(J) s (I, J+1.) T=DC(J) (4.36)
where AC(J), BC(J), CC(J) and DC(J) can be derived from the Eqs. 4.23, 4.26 and 4.27.
The continuity equation of the lowermost ring has only two unknown drawdowns. So, this
equation can be written in the following form:
BC(2) s(I,2)T +CC(2) S(I,3)T = DC(2) (4.37)
where BC(2), CC(2) and DC(2) can be derived from the Eqs. 4.23, 4.26 and 4.27.
For the uppermost ring, the numerical continuity equation can be reformulated in function
of the two unknowns s(I,N9-1),. and s(l,N9h as:
AC(N9) s(I,N9-1.) T +BC(N9) s(I,N9) T = DC(N9) (4.38)
where AC(N9), BC(N9) and DC(N9) can be derived from the Eqs. 4.28, 4.32 and 4.33.
In this way, one obtains as many equations as there are unknowns. This equation series
can be resolved by the Thomas algorithm as described in Von Rosenberg (1969).
In this algorithm first the values of V(N) and G(N) are calculated for N which varies
between 2 and N9 according to following equations:
V(2) BC(2) and V(N) = BC(N) -AC(N) CC(N)

V(N-1. ) (4.39)
G(2) DC(2) and G(N) = DC(N) -AC(N) G(N-1.)
BC(2) V(N)
After the calculation of the vectors V and G the drawdowns s(l,N) are calculated using
these vectors and where N varies from N9-1 to 1 with step -1:
s(I,N9) = G(N9) and s(I,N) = G(N) - CC(N):<,~,N+1.) (4.40)
The iteration per column of the same rings is started with the smallest rings, 1=2, and is
continued with the larger ones, I varies from 3 to VI-I. VI is the number of rings per
layer. After all these iterations per column of the same rings, a first approximation of all
drawdowns is obtained.
In the second step, the drawdowns of the different rings of one layer are conside-
red as unknown while the drawdowns of the adjacent layer(s) are considered as known.
First, the drawdowns are calculated for the uppermost layer. Next, the drawdowns of the
successive underlying layers are calculated. So, the drawdowns of the superjacent layers,
which are considered as known, are the results of the ongoing iteration. The drawdowns
of the subjacent layers, which are also considered as known, are the results of the former
iteration. For each ring, except for the first and the last ring, the numerical continuity
equation (Eq. 4.27) can be reformulated in function of the unknowns which must be
resolved:
AR (I) s (I-1., J) T+BR (I) s (I, J) T+CR (I) s (I+1., J) T=DR (I) (4.41.)
where AR(I), BR(I), CR(I) and DR(I) can be derived from the Eqs. 4.23, 4.26 and 4.27.
For the first ring, there are only two unknowns resulting in following equation:
BR (2) s (2, J) T +CR (2) s (3, J) T = DR (2) (4.42)
where BR(2), CR(2) and DR(2) can also be derived from the Eqs. 4.23, 4.26 and 4.27.
For the last but one ring, there are also two unknowns resulting in following equation:
AR( Vl-l) s (Vl-2, J) T+ BR( Vl-l) s (Vl-l, J) T=DR(Vl-1) (4.43)
where AR(Vl-l), BR(Vl-l) and DR(Vl-l) can also be derived from the Eqs. 4.23, 4.26
and 4.27.
The drawdown in the outermost ring, I=Vl, is a constant drawdown boundary during a
considered time step. So one obtains as many equations as there are unknowns. This new
series of equations is again resolved by the Thomas algorithm (Von Rosenberg, 1969).
Chapter 4 / Numerleal model 0/ pUlll[Jing test 133
4.5.2 Verification of iteration process and number of iteration per time step
The groundwater balance is used for the verification of the accuracy of the calculated
drawdowns. This balance is the difference between the pumped discharge rate and the
sum of the storage decrease rate and the horizontal inflow rate through the inner boundary
of the last ring. The storage decrease is calculated according to the Eqs. 4.23 and 4.29,
the horizontal inflow rate according to the Eqs. 4.21 and 4.30. Initially, the ground water
balance was made after each pair of iterations, iterations in the two directions. When the
iteration process converged, the balance decreased continuously. This balance was first
used as criterion to stop the iterations of the treated time step. When this balance became
smaller than a certain percentage of the pumped discharge rate then the iteration of the
treated time step was stopped. The drawdowns obtained during the last iteration are then
considered as accurate enough and the calculation of a new time step was started.
After a certain experience with the iteration process, the balance was no longer
calculated after each iteration. So, the calculation time of the simulation is considerably
reduced. Now, for each time step a fixed number of iteration pairs are made. This
number of iteration pairs depends on the number of the treated time step T, the initial
time T1 and the factor A. When the initial time T1 is equal to 0.1 minutes and the factor
A is equal to lOO.1 then the number of iteration pairs is greater or equal to (T+5)/5. If a
different value is chosen for the factor A, then one must deduce the number of iteration
empirically. This can be checked by the water balance which is still calculated after each
time step. An increase in the number of iteration pairs can be necessary for large values
of the factor A and small values of hydraulic resistance between the layers.
4.6 Verification of numerical model
The numerical model is verified with different analytical models: the model of Theis, the
model of Jacob-Hantush, the model of Hantush, the model of Hantush-Weeks and the
model of Boulton as explained by Cooley. For each analytical model the assumed
suppositions are repeated. They are followed by the equation which gives the relation
between the drawdown, the hydraulic parameters of the ground water reservoir, the
distance from the pumped well, the time since starting the pumping test and the discharge
rate. Hereafter, the way how the analytical models can be treated by the numerical model
is described so that the same results are obtained. For each model the parameters defining
the fInite-difference grid are discussed along with the appropriate hydraulic parameters. A
comparison is made between the results of the analytical model and the results of the
numerical model. Finally, the influence of the chosen parameters on the results is studied.
4.6.1 Verification of numerical model with Theis model
In the model of Theis, a confined aquifer with an infinite lateral extension is assumed.
The aquifer is pumped by a fully penetrating well. It is assumed that the discharge rate is
constant and that the pumped layer is homogeneous and has a constant thickness over an
infinite distance from the pumped well. The sub- and superjacent layer is completely
impervious. This last supposition is also valuable over an infinite distance from the
pumped well. Because of these assumptions, the pumped water is only delivered by the
elastic storage due to the drawdown in the pumped layer. Finally, it is assumed that the
diameter of the pumped well is very small so that the well storage is negligible.
The solution of the analytical model of Theis is given in Sect. 3.2.3. and can be
written as:
(4.44)
Using Eq. 3.32 to determine W(u) and Eq. 4.44, the drawdown is calculated at 1 m, 2.5
m, 6.3 m, 12.5 m, 40 m, 100 m and 250 m from the pumped well and at the times 10°·11
minutes since the start of the pump, where I varies between zero and thirty-two with a
step of one. The transmissi-vity and the elastic storage coefficient of the pumped layer are
put equal to 100 m2 /d and lxlO~3. The discharge rate is equal to 180 m3/d.
In the numerical model, two layers and forty-three rings are considered. With a
discharge rate of 180 m3/d, water is pumped from layer 1. The initial time, T1, is set
equal to 0.1 minute and the value of log lOA is 0.1. The drawdown is calculated until 1600
minutes after starting the pump. The thickness, the horizontal conductivity and the
specific elastic storage of the pumped layer (layer 1) are 10 m, lOm/d, and lxl()4 m~l. A
very large value (1010 d) is attributed to the hydraulic resistance between layer 1 and 2.
The hydraulic parameters of layer 2 are only introduced as completion of the input data.
Because of the large hydraulic resistance between layer 1 and 2, they have a negligible
influence on the drawdown in the pumped layer. So, the same values are chosen for the
thickness, the horizontal conductivity and the specific elastic storage of layer 2 as for
layer 1. The storage coefficient near the water table is set equal to 0.2.
With the FORTRAN program sipurS, the drawdowns are calculated in all the
nodal circles of layer 1 and layer 2. For all considered times the drawdowns in layer 2
are very small. At the end of the calculation the largest drawdown of layer 2 occurs in
the smallest ring and is equal to 0.4xlO~9 m. With the output program outpuS, the draw-
downs calculated with the analytical and numerical model are plotted in time-drawdown
and distance-drawdown graphs (Fig. 4.2). All the axes of the graphs are logarithmic. The
analytically calculated drawdowns are represented by crosses, the numerically calculated
drawdown by solid lines. Looking at Fig. 4.2, one can conclude that the drawdowns
derived with both methods are almost the same.
Chapter 4 I NUlllflrical model of pumping test 135
DRAWDOWN (m)
•OO TIME (min)

LAYER'
...
'.' ...
SO-0.2OO
SA(2)-0.000100 "-I
C 11)-9999999999.9 D - - - - - - - - - -
K 11)-10.00 HID SA(I)-O.OOOtOO H-I
THEIS - Rl-0.l. - TI-O.laln - loaA-O.l - 43 Rings - a-180 .. 3/d
Fig. 4.2. Analytically (crosses) and numerically (continuous lines) calculated draw-
down corresponding with the model of Theis (1935)
4.6.2Irifluence of grid parameters on results of numerical Theis model
First, the influence of the number of rings is considered. In the fIrst calculation 43 rings
are chosen. This means that the outer boundary is situated at a diSlance of 1600 m from
the pumped well. Verifying the balance of the last calculated time step from 1258 minutes
to 1600 minutes after the start of the pump, it appears that 179.1 m3/d is delivered by the
storage decrease of the pumped layer and 0.9 m3/d flows horizontally through the outer
boundary. This small flow causes no remarkable differences between the numerically and
analytically calculated drawdown. A decrease of the number of considered rings will
decrease the distance of the outer boundary of the model from the pumped well. When
only 38 rings are considered, this distance decreases to 630 m. Here, the water which is
delivered by the storage decrease of the layer is reduced to 32.4 m3/d during the last time
step (1258, 1600 minutes) and the horizontal inflow through the outer boundary of the
layer during the same time step is now 147.6 m3 /d.
DRAWDOWN (m)
'rT-r---~--~~-----+-----+-----4
,.. nME(min)
LAYER 1
, .. ,.' ,.- DISTANCE
(m)
SO-0.200
0(2)-10.0/1 K (2)-10.00 /lID SA(2)-0.000100 11-1
C(Il-9999999999.9 D - - - - - - - - - - - - - - - - - -
0(1)-10.0/1 K(1)-IO.OO tIID SA(I)-O.OOOIOO /I-I
THEIS - Rl=O.I. - Tl-0.laln - logA-O.l - 38 Rings - Q=180.3/d
Fig. 4.3 Analytically (crosses) and numerically (continuous lines) calculated drawdown
corresponding with the model of Theis (1935) when only 38 rings per layer are
considered
In Fig. 4.3, the drawdowns corresponding with this last case are shown. The
differences between the numerically and analytically calculated drawdowns are small but
already perceptible for the last considered time steps at distances larger than 250 m. In
Fig. 4.4, the calculated drawdowns are represented for a model where only 33 rings are
considered. This means that the outer boundary is situated at only 160 m from the
pumped well. Now, the difference becomes very important after 125 minutes of pumpage.
With this particular example it is shown that from this time on nearly all pumped water is
flowing horizontally through the outer boundary of the model. Then the calculated draw-
downs are too strongly influenced by this boundary. From that moment on, there are
large differences with the results of the analytical model where an infinite lateral
extension of the pumped aquifer is assumed. For the simulation of the drawdown in an
aquifer with an infinite lateral extension, one can conclude that the storage decreases must
be maximized and the horizontal inflow through the outer boundary must be minimized.
DRAWDOWN (m)
.0-1---+-1--+---+---;
LAYER 1
... '0' ... ... DISTANCE
(m)
SO-0.200
0121-10.011 K(21-10.00 11/0 SAI21-0.000100 11-1
C (1)-9999999999.9 0 - - - - - - - - - -
0(1)-10.011 K (11-10.00 11/0 SAIII-0.000100 11-1
THEIS - Rl=O.l. - Tl=O.l.ln - logA-O.l - 33 R~gs - a-180.3/d
Fig. 4.4 Analytically (crosses) and numerically (continuous lines) calculated drawdown
considered
A second parameter that defines the finite-difference space-time grid is the factor
A. This factor defines the radii of the nodal circles and of the considered rings as well as
the points of time for which the drawdown is calculated. In Fig. 4.5, the drawdowns,
which are calculated with the numerical model with 24 rings and a value of 10glOA equal
to 0.2, are represented as solid lines. Even with such a numerical model with a coarse
grid of nodal circles and time points, the calculated drawdowns correspond quite good
with the analytically calculated drawdowns. The other possibility is a value of 0.05 for
138 Chapter 4 I Numerical model of pumping test
loglOA. In this case, a large number of values are obtained for different distances from the
pumped well and for different times after the start of the pump. Mostly, such a fine grid
is not required for a sufficient accurate calculation. One can conclude that a value of 0.1
for loglO A is a good value for most simulations.
DRAWDOWN (m)
··-,....--+-+--1H--f-H---j
•aO
'.' ... ... LAYER 1
.aO
'.' ... ... DISTANCE
(m)
50-0.200
K (2)-10.00 tvO 5A(2)-O.OOOIOO M-I
C (1l-9999999999. 9 0 - - - - - - - - - -
KIIl-IO.OO tvD 5A(I'-O.OOOIOO M-I
THEIS - Rl=O.lm - Tl=O.lmln - logA=O.2 - 24R~gs Q=180m3/d
down with the model of Theis (1935) when only 24 rings per layer are considered
(log A= 0.2)
The third parameter that defines the grid is the initial radius, Rl. Mostly, a value of 0.1
m is chosen. Only in the cases where the drawdown must be calculated in pumped wells
with a small diameter (see Sect. 5.3), one must consider an initial radius which is at least
1.6 times smaller than the radius of the well screen (if log,oA=O.I). In Fig. 4.6, the
drawdowns are represented by solid lines which are calculated by means of a numerical
model where the initial radius is put equal to 0.8 m and 34 rings are considered. The
calculated drawdowns are nearly the same as in the first calculation. They coincide also
with the drawdowns calculated with the Theis well function W(u) (crosses).
DRAWDOWN (m) DRAWDOWN (m)
'O-~......,I--+-J.--I..J----Itt---+---l ,O-"!----+_+-44~\__++_--..,
'O-~-I--4---H----,I-+---+---l
.. ..
'O-,....---I--+--I~-I--+----l
,00 '0' , •0> ,._ TIME (min) ,00

'0' , ,0> DISTANCE
LAYER 1 (m)
SO-0.200
0(2)-10.011 1«2)-10.0011/0 S~(2)-0.000100 11-1
ell) -9999999999M'9 1 - & 9 - - - - - - - - - - -
I< (1)-10.00 11/0 S~(11-0.000100 11-1
THEIS - Rl=0.8 - Tl=O.I .. ln - logA z O.l - 34 Rings - Q=180m3/d
Fig. 4.6. Analytically (crosses) and numerically (continuous lines) calculated drawdown
considered (R1 = 0.8 m)
The small difference between the two numerical calculations, with R1 = 0.1 m
and 43 rings at one side and with R1 = 0.8 m and 34 rings at the other, can easily be
understood. In the last calculation, the flow and storage decrease are not considered in a
volume of the pumped layer which has the shape of a ring with inner radius 0.1 m and
outer radius 0.8 m. This volume difference is very small with respect to the total
considered volume of the pumped layer. The differences between the two calculations are
entirely due to the neglect of the elastic storage decrease in this very small volume. As a
conclusion one can state that the value of R1 has no large influence on the results of the
calculations as long as this value remains sufficiently small. Mostly, this value is set equal
to the radius of the pumped well or to the radius of the bore hole if the drawdown in the
pumped well is not required.
DRAWDOWN (m) DRAWDOWN(m)
'O-'!!+--.~+-.I---+"f---A------1f----! ,r~--4--~~-r-~--~
'O-'-I--I--*--f+--+-+--~---I 'r'-l---~r--~-r--t--~
'0- , .. TIME (min) , .. '0' DISTANCE

(m)
LAYER 1
SO-0.200
K(2)-10.00 tvD SA(2)-0.000100 M-l
ell) -9999999999r!.9 r & 9 - - - - - - - - - - -
SA(1)-0.000100 M-l
THEIS - Rl=O.l. - Tl=O.8mln - logA-O.l - 43 R~gs - a-180m3/d
down corresponding with the model of Theis (1935) when the initial time is 0.8
minutes
The fourth and last parameter that defines the space-time grid is the initial time,
Tl. This initial time is mostly put equal to 0.1 minutes. In Fig. 4.7, the drawdowns are
represented by solid lines that are calculated with the numerical model where the initial
time is taken equal to 0.8 minutes. All other parameters are taken equal to the first
calculation (Fig. 4.2). The results show two striking differences with respect to the first
calculation. Near the pumped well the drawdown oscillates at the start of the test. This is
due to the time interpolation constants tl and ~. The largest deviation with respect to the
analytically calculated drawdowns occurs just after the initial time, Tl. This is clearly
illustrated by the distance-drawdown curve after 1 minute of pumpage, at the end of the
first time step, which runs from 0.8 until 1 minute of pumpage. The distance drawdown
curve after 10 minutes of pumpage corresponds already quite well with the analytically
calculated drawdown. As a conclusion one can say that ignoring the storage decrease
before the initial time, T1, has no influence on the results for times that are at least ten
times larger than the initial time. This cannot be said for times that are smaller than five
times the initial time.
4.6.3 Verification of numerical model with lacob-Hantush model
The following assumptions are made in the analytical model of lacob-Hantush. The semi-
confined aquifer has an infinite lateral extension. The aquifer is pumped by a completely
penetrating pumped well with a discharge rate which is constant during the pumping test.
The pumped layer is homogeneous and has a constant thickness over an infinite lateral
extension. Only one of the adjacent layers is impervious or both are semi-pervious. The
pervious characteristics of the adjacent layers do not change laterally. The leakage from
the adjacent layers is proportional to the drawdown in the pumped layer at any distance
from the pumped layer and at any time after starting the pumping test. Consequently, a
vertical inelastic flow is assumed in these semi-pervious layers. The diameter of the
pumped well is small, so that the water delivered from the storage of the pumped well
can be ignored.
The solution of the analytical model of lacob-Hantush is derived in Sect. 3.3.3 and
can be written as:
s -_Q- w(u,r/L) (4.45)

41tKaD
where s is the drawdown in meter,

Q the discharge rate in m3/d,
KbD is the transmissivity of the pumped pervious layer in m2/d
and W(u,r/L) is the well function of the semi-confined aquifer.
Using Eq. 4.45 the drawdowns are calculated for different distances from the pumped
well and for different times after starting the pump. The considered distances are 1.0 m,
2.5 m, 6.3 m, 12.5 m, 40 m, 100 m and 250 m. For each distance, the examined times
are 10°·11 minutes after starting the pump where I varies between 0 and 32 with a step of
1. Drawdowns smaller than 0.005 m are not considered. The well function W(u,r/L) is
calculated with the FORTRAN program (Reed, 1980). The trans-missivity and the elastic
storage coefficient of the pumped pervious layer are 100 m2 /d and lxlO-3. It is assumed
that the subjacent layer is impervious and that the hydraulic resistance of the superjacent
layer is 100 d. The discharge rate is 180 m3/d. These analytically calculated drawdowns
are compared with the numerically calculated ones.
In a first calculation with the numerical model, two layers of forty-three rings are
considered. Layer 1 corresponds with the pumped layer, layer 2 with the pervious layer.
This layer is located above the covering semi-pervious layer and is supposed to be under
a fixed hydraulic head. The horizon between layer 1 and layer 2 corresponds with the
semi-pervious layer where only a vertical inelastic flow is considered. The initial radius is
0.1 m, the initial time is 0.1 minutes and loglO A is 0.1. Until 1600 minutes after the start
o~ the pump the drawdowns are calculated. The thickness, the horizontal conductivity and
the specific elastic storage of the pumped pervious layer are respectively 10 meter, 10
mid and lxlO-4 m- I . The hydraulic resistance between layer 1 and 2 is 100 d. By assigning
a very large value (1010 mid) to the hydraulic conductivity of layer 2, the drawdowns in
this layer stay negligibly small. Because of this very large value, the values assigned to
the other hydraulic parameters of layer 2 have no influence on the calculations. The
thickness, the specific elastic storage and the storage coefficient near the water table are
respectively 10 m, lxlO-4 m- I and 0.2.
'~~~~~-----1------+-----~
,0> ,.. TIME (min) ,00 ,.' ,.> DISTANCE

LAYER 1 (m)
SO-0.200
D (2)-10.0 11 K(2)-9999999999.99 1110 SA(2)-0.001000 11-1
C (1) -100.0 0
0(1)-10.011 K(1)-10.00 1110 SA(I)-0.000100 11-1
JACOB - RI-O.l. - TI=O.l.'n - logA=O.l - 43 R~gs - Q=1801113/d
down corresponding with the model of lacob-Hantush (in the numerical model only two
layers are considered)
With the FORTRAN program sipur5, the drawdowns of all nodal circles of layer
and 2 are calculated. With the output program outpu5, the drawdowns calculated with
the analytical and numerical model are plotted in time-drawdown and distance-drawdown
graphs (Fig. 4.8). All the axes of these graphs are logarithmic. The analytically calculated
drawdowns are represented by crosses, the numerically calculated drawdown by continu-
ous lines. Looking to Fig. 4.8, one can conclude that the drawdown calculated with both
methods are almost the same. Small deviations between the analytically and numerically
calculated drawdowns occur just after starting the test at a relative large distance from the
pumped well. These small deviations can be explained by the fact that in the numerical
model, T1 and R1 are respectively equal to 0.1 min and 0.1 m. In the analytical model
both parameters are equal to zero. The difference between the two results is mainly due
to the different assumed values of T1 in both methods. According to the good agreement
between the analytically and numerically calculated drawdowns, it is concluded that the
inelastic vertical flow through the semi-pervious layer can be simulated by means of the
vertical flow through the horizon between the two layers in the numerical model. This
flow is only characterized by one hydraulic parameter, the hydraulic resistance.
In the second calculation, it is demonstrated how the vertical inelastic flow through
a semi-pervious layer can also be simulated through a layer of the numerical model when
a very small value is assigned to the horizontal conductivity and the specific elastic
storage of this layer. In the numerical model the same groundwater reservoir is treated as
in the first simulation of the lacob-Hantush model. Now, three layers are considered in
the numerical model. Each layer is subdivided in forty-three rings. The same values as in
the first equation are assigned to the hydraulic parameters of layer 1. Also, the discharge
rate is the same. In the lacob-Hantush model, it is assumed that leakage through the semi-
pervious layer is proportional to the drawdown in the pumped pervious layer. This
corresponds with the supposition that there is an inelastic vertical flow through the semi-
pervious layer and that the upper boundary is under a fixed hydraulic head. Layer 2 in
the numerical model replaces here the semi-pervious layer. Therefore, the hydraulic
conductivity and the specific elastic storage are put equal to a very small value. The sum
of the hydraulic resistances between layer 1 and 2 and between layer 2 and 3 is equal to
100 d. The hydraulic parameters of layer 3 are the same as the parameters of layer 2 in
the first calculation. A very large value is attributed to the horizontal conductivity of layer
3 to guarantee a fixed hydraulic head in this layer.
In Fig. 4.9, the drawdowns of layer 1, calculated with the numerical model, are
compared with the drawdowns calculated with the well function W(u,r/L). Both calcula-
tions are almost the same. The results of the first and the second numerical calculation
are identical. Because the hydraulic resistance between layer 1 and 2 is chosen equal to
the hydraulic resistance between layer 2 and 3, the drawdown of layer 2 is at each
distance from the pumped well and at each time after starting the pump the same as half
the drawdown in the layer 1 at the same distance and time. This is inherent to the
assumptions made in the numerical model. According to these assumptions, the drawdown
in the semi-pervious layer at each level can be deduced from the drawdowns in the
144 Chapter 4 I Numerical model of pumping test
,~+------+------+-----~----~
,·-'t------>,rr....".-->~I_----+----__t
,.-.t------+-I---++-',....,....---t-------!
,.-i------t--t--4tt--++-+-----i
'.' '.' '0_ TIME (min) ,..

LAYER 2
,oo~~..... rl------t------+-----i
1Q·I'+------\---\-~~----_+----__i
,.-''t-----_+-+-++-'I-\----+----__t
min
,.-''t------t--t---'i--++_+----__i
,.. '.' '0_ TIME (min)

LAYER 1
'0. '.' '.' DISTANCE
(m)
50-0.200
O(J)-IO.O M KIJ)-9999999999.99 MID SAIJ)-O.OOIOOO M-I
C(2)-SO.0 0
0(2)-10.0 M K(2)-0.00 MID SA(2)-0.000000 M-I
cm-so.o 0
0(1)-10.0 " K (tl-IO.OO "/0 5A(I)-0.000100 M-I
JACOB - Rl=O.lm - Tl=O.l"ln - logA=O.l - 43 Rings - Q=180m3/d
down corresponding with the model of Jacob and Hantush (in the numerical model
three layers are considered)
pumped pervious layer at the corresponding times and distances. These drawdowns must
be multiplied by the ratio between a level difference and the thickness of the semi-
pervious layer. This level difference corresponds with the distance between the top of the
semi-pervious layer and the considered level.
4.6.4 Influence of grid parameters on results of numerical lacob-Hantush model
The influence of the parameters which define the space-time grid is similar to the
influences shown during the verification of the Theis model. The number of rings must be
chosen so that the horizontal inflow through the outer boundary of layer 1 stays very
small. Considering 43 rings, this horizontal inflow rate is 1. 7x 10.5 m3 /d after 1600
minutes of pumpage. This rate is equal to 7.1 m3 /d when 38 rings are considered and
equal to 99.5 m3 /d when 33 rings are considered. When one compares the numerical
results with the analytical, one can conclude that the results where 33 rings are con-
sidered, have insufficient precision. The value of R1 has a small influence on the
calculations, at least if this value stays smaller than 1 m. The distance-drawdown curve at
the start of the pumping test depends strongly on the chosen value of T1 as in case of the
Theis model. Especially, the small drawdowns at a relatively large distance from the
pumped well are very sensitive for the values of the initial time, T1. Therefore, a value
smaller or equal to 0.1 minutes is advised.
4.6.5 Verification of numerical model with Hantush model
The assumptions of case 1 of the model of Hantush are the same as the assumption of the
model of Jacob-Hantush with the exception that in the last model the storage change is no
longer ignored. In the Jacob-Hantush model, the specific elastic storage of the semi-
pervious layer is equal to zero. In the Hantush model, a certain value is assumed. So, a
vertical elastic flow is assumed in the adjacent semi-pervious layers. The solution of the
model of Hantush is derived in Sect. 3.4 as:
(4.46)
where s is the drawdown in meter,

Q the discharge rate in m3 /d,
KbD is the transmissivity of the pumped pervious layer in m2/d,
and H(u,,B) is the well function of the semi-confined aquifer with vertical elastic flow
in the adjacent semi-pervious layers.
By means of Eq. 4.46 the drawdowns are calculated at different distances from the
pumping well and for different times after starting the pump. Following distances are
considered: 1.0 m, 2.5 m, 6.3 m, 12.5 m, 40 m, 100 m and 250 m. For each distance
the examined times are lOG.II minutes after starting the pump where I varies between zero
and thirty-two with a step of one. Drawdowns smaller than 0.005 m are not considered.
The well function H(u,,B) is calculated with the FORTRAN program given in Reed
(1980). The transmissivity and the elastic storage coefficient of the pumped pervious layer
are set equal to 100 m 2/d and 1. Ox 10-2. Here, a special case is considered because the
subjacent layer is impervious whereas the superjacent semi-pervious layer has a hydraulic
resistance of 100 d. This semi-pervious layer is assumed to be very elastic. The specific
elastic storage of this layer is put equal to 1.0x1Q-2 m- I. This large value is chosen,
because the formula given above is only valid for times smaller than 0.lCzS,2D2 (see Sect.
3-4-4). When Cz = 100 d and S,2D2 = 0.1 then the time limit of application is 1 d
according to Hantush (1960). The discharge rate is 180 m 3 /d.
In the first calculation with the numerical model three layers and forty-three rings
are considered. Water is pumped with a constant discharge rate of 180 m 3 /d. The
horizontal conductivity and the specific elastic storage of layer 1 are respectively 10 mid
and 1.0xlO-3 m- I. In the model of Hantush, an elastic vertical flow through the semi-
pervious layer is assumed with a fixed hydraulic head boundary at the top of this layer.
Layer 2 in the numerical model replaces the semi-pervious layer. Therefore, a very small
value is assigned to the horizontal conductivity of layer 2. The sum of the hydraulic
resistance between layer 1 and 2 and between layer 2 and 3 is equal to 100 d. The
specific elastic storage of layer 2 is now put equal to LOx1Q-2 m-I. A very large value is
assigned to the horizontal conductivity of layer 3 to guarantee the fixed hydraulic head in
this layer. In Fig. 4.10, the drawdowns of layer 1, which are calculated with the
numerical model, are compared with the drawdowns calculated with the well function
H(u,,8).
The drawdown calculated in the two different ways shows a rather large deviation,
particularly the drawdowns at large distances from the pumped well and for a relative
large time after starting the pump. These deviations are due to an inferior discretization of
the semi-pervious layer. This is specially true when this semi-pervious layer is ten times
more elastic then the pumped pervious layer. To treat the vertical flow in this very elastic
semi-pervious layer, this layer must be discretized in several layers. In Fig. 4.11, the
analytically and numerically calculated drawdowns are shown for the case where the
semi-pervious layer is discretized in seven layers of the same thickness. The layers 2 to 8
have the same values for the horizontal conductivity and for the specific elastic storage,
respectively 10-9 mid and 1_0xlO-2 m- I. The sum of all the hydraulic resistances between
the lowermost and uppermost layer is equal to 100 d. By attributing a very large value to
the horizontal conductivity of the uppermost layer, the hydraulic head is fixed in this
layer. Studying the results in Fig. 4.11, it is established that the deviations between the
numerically and the analytically calculated drawdowns are considerably smaller.
The numerically calculated drawdwowns in the middle of the semi-pervious layer
are represented in Fig 4.8 (layer 2) and in Fig. 4.11 (layer 5). Comparing these graphs,
one can see that the start and the steepness of the time-drawdown curve of the semi-
pervious layer are strongly dependent of the discretization of the semi-pervious layer. The
drawdowns toward the end of the simulation which are larger than 0.01 m, are less
dependent from this discretization. From this comparison one can conclude that a
,00 ,00
10- 1 IIr ,I--

.........
'0- '0- 2 ~\
~
'0- '0- 1\ 1\
.. ~o '~O" Vo~ .w,
, 10" TIME (min) 100 101 102 '0> DISTANCE

LAYER 2 (m)
..
, ,00~~~+---4---+---~
,0-'f-----f4--i<i-4-f-.;.r>-+---+----i ,0-'t-----=H-4-+H--+---4
n
'o-"t--f--++--If---i\---+----i 'O-'+-_ _-\!-_+--<l_-\-_-\-_ _ ~
'0' TIME (min) ,00 '0' ,0> DISTANCE

LAYER 1 (m)
50-0.200
0(3)-10.0 M K(3)-9999999999.99 MID SA(3)-O.OOIOOO M-l
C(2)-SO.0 0
0(2)-10.0 M K(2)-0.00 MID SA(2)-O.OI0000 M-l
cm-so.a 0
0(11-10.0 M K (1) -10.00 MID SA(1)-O.OOIOOO M-I
HANTUSH - Rl=O.lM - Tl=O.lmln - iogA=O.l - 43 Rings - Q=180M3/D
corresponding with the model of Hantush. The specific elastic storage of the pumped
pervious layer is ten times smaller than those of the semi-pervious layer. Layer 2 of the
numerical model corresponds with the semi-pervious.
,--
'~~----~-----4------+-----~----~ ,~
I0" ,-I-----+-----+---fl.!L,..---4----I ,.-
,.-.!I---~----+-m,r--;.+----+----I ,.- '\

,.-
\
,.-~--_l_--~f.II+~-A----~--__I
~102 1\ 103 min
,OG ,.' ,02 ,02 , .. TIME (min) ,~ ,.' '.' ,., DISTANCE
(m)
LAYER 5
DRAWDOWN (m) DRAW DOWN (m)
'oG~~~~---+---+---~
"-~---J.~--+':'-I-~.IL-I------I-----l ,.-.'f---lr+-+-.lrl4---+----'
,.-3!1--+_~+---++_--+-----+-----i ,.-3!!-___*_\--4t-\_ _+-___~
,OG ,.' '.' I.' '.' TIME (min) ,oG ,.' '.' , •• DISTANCE
(m)
LAYER 1
50-0.200
0(91-10.0/1 K(91-9999999999.99/1/0 SA(9)-0.001000 M-l
D{SI-I.~
D (71-1 • ~ M
D {61-1.1 M
M K(SI-O.OO MID
K m-o.oo MID
K(61-0.00 MID
C(SI-7.1 0
C(71-I~.3
C(61-1~.3
0
0
,.
D {Sl-I.~ /1 K(S)-O.OO MID C(SI-11.3 D
0(4).1.4 t1 K(~I-O.OO MID
C(11-11.3 D
O{31-1.1 M K (J) -0.00 MID C(31-11.3 D
P(2) - , ... t1 K(2)-O_OO MID C(21-11.3 D
C(Il-T.I 0
0(11-10.0 M K (1)-10.00 M/O SA(I)-O.001000 /1-1
HANTUSH - RI-O.IM - TI=O.I .. ln - logA-a.1 - 43 Rings - Q=180M3l0
pervious layer is ten times smaller than those of the semi-pervious layer. The semi-
pervious layer is discretized in seven layers in the numerical model.
... ••0
..-
• /,~ ..-.r---
..-• /!/;II ..-• 1\
II/I II \
..- wr / /
..- ~" \
-,103 min
... '.' '.' ••> '.' TIME (min)

LAYER 5
... '.' '.' ••> DISTANCE
(m)
DRAW DOWN (m) DRAW DOWN (m)
'''~~~~-----4------+-----~
'0-"+----~~~M+----+---1
'o-'q-.---++~\-lt+--+----I
.•->.!----+-+-~++--+------1
... '.' '.' ••> ,.' TIME (min)
LAYER 1
.0°
SO-0.200
0(9)-10.0 M K(9)-9999999999.99 MID SAI91-0.001000 M-I
0(8)-1 . ~ M K(8J-O.OO MID C(81-7.t 0

C(Tl-14.3D
D (Tl-I .4 M K(ll-O.OO MID
D (S)-I .4 M K(S)-O.ao MID C(61-H.3
C(SI-I~.3 0
D
a (5)-" 4 M K(S)-O.OO Mia C(4)-14.3
0(4)-1.4 M K(4J-O.OO MID C(3)-I~.3 DD
a (3) -I .4M K(3)-0.OO MID C(2)-H.3D
D (2)-1 .4 M K(2)-O.OO MID
CI1l-7.1 0
0(1)-10.0 M K (1) -I 0.00 MID SAO)-O.OOOIOO M-I
HAN TUSH - Rl=Q.lm - Tl=Q.lmln - logA=O.l - 43 Rings - Q=180m3/d
pervious layer is hundred times smaller than those of the semi-pervious layer. The semi-
pervious layer is discretized in seven layers in the numerical model.
rather elastic semi-pervious layer should be discretized in a number of layers if one wants
to calculate the drawdowns accurately at a required level in this semi-pervious layer.
To illustrate that these differences are typical for large differences between the
elasticity of the pumped and the semi-pervious layer, the following two simulations are
made. In these simulations, the specific elastic storage of the pumped pervious layer is the
only difference with the above given verification. The extremely large value for the
specific elastic storage of the semi-pervious layer is still used so that the asymptotic
solution of Hantush (1960) is valid for the treated times « 1 d). In Fig. 4.12, the
analytically obtained results are represented along with the numerically obtained results
for the case where the specific elastic storage of the pumped layer is a hundred times
smaller that the specific elastic storage of the semi-pervious layer. The semi-pervious
layer is also discretized in seven layers as in the former calculation. In this case with an
extremely large elasticity contrast, the discretization in seven layers is even not enough to
obtain a sufficient close approximation of the analytically calculated drawdown. In Fig.
4.13, the analytical results are compared with the numerically obtained results for the
case where the specific elastic storage of the pumped layer is the same as the specific
elastic storage of the pumped layer. Here, the analytically and numerically calculated
drawdowns are almost identical for the pumped layer as well as the semi-pervious layer.
4.6.6 Examination of validity limits of Hantush asymptotic solution
In the last series of calculation, one of the validity limits of the asymptotic solution of
Hantush (1960) is examined. Therefore, the drawdowns are calculated analytically and
numerically for the case where the specific elastic storage of the pumped pervious layer
and of the semi-pervious layer are both equal to 1. Ox 10-3 m· l (Fig. 4.14). This is ten times
smaller than the last calculation represented in Fig. 4.13. The other hydraulic parameters
and grid parameters are unchanged. Comparing Fig. 4.14 with Fig. 4.13, one can derive
that the same drawdown values arrive ten times earlier at the same distances from the
pumped well. As an example, the drawdown after ten minutes of the calculation represen-
ted in Fig. 4.13 arrives in the last calculation already after 1 minute of pumping (Fig.
4.14). According to Hantush (1960), the time limit of the validity of the asymptotic
solution is now 0.1 d or 144 minutes. From Fig. 4.14, one can derive that the analytically
and the numerically calculated drawdown are still identical until thousand minutes of
pumpage. In the last simulation of this paragraph, the specific elastic storages of the
pumped pervious and of the semi-pervious layer are set equal to 1.0xlQ4 m· l . This is
again ten times smaller than the last but one calculation. The time limit of the validity of
the asymptotic solution is now reduced to 0.01 d or 14.4 minutes. From Fig. 4.15, one
can see that the analytically and the numerically calculated drawdowns are identical until
hundred minutes after starting the calculation. From this time on, the deviation becomes
considerable. From these three last calculations one can conclude that the time limit as
proposed by Hantush (1960) is seven times too severe. This corresponds with the results
of Neuman and Witherspoon (1969) which state that the criterion for the validity of the
Hantush solution is on the conservative side and could be relaxed somewhat.
.. ..
, ,
'0- , I? '0- ,I"----
'0- II II
,0-,
~
1\103 mi
'0-
J,~/Il
VII /
..- \1
,02 ,0> .., TIME (min) ,00 ,.' '0' , .. DISTANCE
LAYERS (m)
DRAWDOWN (m) DRAWDOWN(m)
'''~~--~-----4------+-----~
min
'0' TIME (min)

LAYER 1
,00 ,.' ,02 '0' DISTA
(m)
CE
50-0.200
D(9)-10.0 /1 K (9) -9999999999.99 /1/D 5A(9)-0.001000 /1-1
D(8) -I.~ /1 K (8) -0.00 MID C (8) -7.1 0

DI71-1.~ M K (7) -0.00 MID C 171-H.3 D
D(6)-I.~ M K (6) -0.00 MID C (6) -14.3 0
P IS) -I. ~ M K(5)-0.OO MID C (S) -14.3 0
D(~)-I.~ M K I~) -0.00 MID C(4)-14.30
D(3)-I.~ M K el) -0.00 MID C (3) -14.3 D
0(2) -I. 4 M K (2) -0.00 MID C (2) -14.3 D
CII) -7.1 0
K (1)-10.00 MID SAU)-O.OIOOOO M-I
HANTUSH - Rl=O.IM - Tl=O.lmln - logA=O.1 - 43 Rings - Q=180M3/D
pervious layer and of the covering semi-pervious layer are the same (lx1o-2 m- I ). The
semi-pervious layer is discretized in seven layers in the numerical model.
DRAW DOWN (m) DRAWDOWN (m)
r---.. ~
,~~-----+------~-----+------~----~ ,~
,.- ,t--...
,·-'t------4----IfH+.f---t4-------+------j ,.- 2
"'"1\ \
,.-••.\-----_!_-/I-,I-I--I'l--I---_!_+-----+------4 '.- • t=l~-\ \,03m:·r
100 10 1 ,0> ,._ TIME (mln),~ ,.' ,0>
\ ,.> DISTANCE
(m)
LAYERS
,.-.'-1-----\-+--\--++-1----+--------1
,.-.!I------U----1----'~-+--_!_----___l
,~
'.' ,.. '.' ,._ TIME (min)
LAYER 1
,~
,.' ,0> '.' DISTANCE
(m)
50-0-200
0(9)-10.011 ~(9)-9999999999.99 11/0 5"(9)-0.001000 11-1
C (S) -7.1 0
o (S) -1.4 11 K(S)-O.OO 11/0 C(71-14.3 0 5"181-0.001000 11-1
0171 -1.4 11 ~ 171 -0.00 MID C (6) -14.3 0 5"(7)-0.001000 11-1
0(61 -1.4 11 KIS)-O.OO MID C(S)-14.30 5" (6) -0.001 000 11-1
0(51 -1.4 M K(S)-O.OO MID $1'(5) -0.001000 11-1
C(4)-14.30
0(4) -1.4 11 ~(41-0.00 11/0 C(3)-14.3 0
0(3) -1.4 11 ~(3)-0.OO 11/0
C(2)-14.30
P (2) -1.4 11 ~(2)-0.00 11/0 CIIl-7.1 0
0(1)-10.011 ~(\)-IO.OO 11/0 5AII)-0.001000 11-1
HANTUSH - Rl=O.lM - Tl=O.lmln - logA=O.l - 43 Rings O=180M3/D
pervious layer and of the covering semi-pervious layer are the same (lxlQ-3 m-I). The
'''rt~rrl
'0··~--.1I-I-++-I---I-1----l="",---+----4
.r·~I1-1-4-~-~-HL--4----+----4
min
'0' 10' TIME (min) ... 10' .o' DISTANCE

LAYERS (m)
DRAWDOWN (m)
.rl-l-_ _..lI-~........:...Jo~_ _+-__-I
10··2J..----+-+--\-H4--+---l
.0·'+----+--\--\11-+-+-+---1
104
LAYER 1
loG '0 ' ... 10' DISTANCE
(m)
50-0.200
D(91-10.0 11 K(91-9999999999.99 11/0 SA(91-0.0DOIOD /1-1
K ) -0. 0 11/0 C (81-7.1 0

K (Tl -0.00 11/0
C(71-14.J 0
(6)-1.4 /1 K (61-0.00 11/0 C(61-14.J 0
I ) -1.4 11 K (S) - .00 /1/0 C(SI-14.J D
(4) -1.4 /1 K (41-0.00 /1/0 C(41-14.J 0
( ) -1.4 11 K (JI- .00 /1/0 C(JI-14.J 0
( ).1.. M K ( )-0. 0 /11
C(2)-14.J 0
C(1I-7.1 D
0(1)-10.0/1 K (11-10.00 /1/0 SA(I)-0.000100 /1-1
HANTUSH - Rl s 0.1M - Tl=O.laln - logA=O.1 - 43 Rings - Q=180M3/D
pervious layer and of the covering semi-pervious layer are the same (lxlO"' m· I ). The
4.6.7 Verification of numerical model with Hantush-Weeks" model
In the model of Hantush-Weeks, a confined aquifer with an infinite lateral extension is

assumed. The pumped layer is assumed to be homogeneous, transversely anisotropic and
of a constant thickness over an infinite lateral extension. The adjacent layers are both
impervious. The aquifer is pumped by a partially penetrating pumped well. Consequently,
the flow to the pumped well is not exclusively horizontal. The vertical component of the
flow is important in the vicinity of the pumped well. It is assumed that the discharge rate
is constant during the pumping test. The diameter of the pumped well is small, so that the
water delivered from the storage of the pumped well can be ignored.
The solution of the analytical Hantush-Weeks model is described in Sect. 3.5. Two
kinds of solutions are given: a first solution for piezometers (Eq. 3.139) and a second
solution for observation wells (Eq. 3.140). With the FORTRAN program given in Reed
(1980), the drawdown is calculated at different levels of the pervious layer. In these
calculations, it is posed that the pervious layer has a thickness of 10 m, a horizontal
conductivity of 10 mId, a vertical conductivity of 0.1 mId and a specific elastic storage of
1. Ox 104 m-I. Consequently, the transverse anisotropy of these layers is equal to 10. It is
assumed that the filter screen of the pumped well is located in the lowest three meters of
the pumped pervious layer. The drawdowns are calculated on three different levels,
respectively on 8.5 m, 5.0 m and 1.0 m under the top of the upper impervious boundary.
On each level the drawdowns are calculated at a distance of 1.0 m, 2.5 m, 6.3 m, 12.5
m, 40 m and 100 m from the pumped well. For each distance the drawdowns are calcula-
ted for the times 10°·11 minutes after starting the pump. The variable I varies between zero
and thirty-two with a step of one. The drawdowns smaller than 0.005 m are not con-
sidered.
In the first calculations of the numerical model, the number of layers is limited as
much as possible. Five layers are considered of which four replace the pervious layer.
The lowermost layer, layer 1, has a thickness of 3 m. This length corresponds with the
length of the filter screen of the pumped well. The second layer is 4 m thick, the third 1
m and the fourth is 2 m. The middles of the layers correspond with the levels of the
analytically calculated drawdown. The hydraulic resistance between layer 4 and 5 is set
equal to a very large value, 101o d. This large resistance is introduced in the numerical
model to replace the upper impervious boundary. Because of this large resistance, the
values of the hydraulic parameters of layer 5 have no influence on the calculations. The
horizontal conductivity of the lowest four layers is 10 mId and the specific elastic storage
is 1. Ox 104 m- I. The hydraulic resistances between these layers are derived from the
distances between the middle of the layers and the vertical conductivity which is put equal
to 0.1 mId. Layer 1 is pumped with a constant discharge rate of 180 m3 /d. The chosen
value for the initial radius is 0.1 m. For the initial time 0.1 minute and for 10gioA a value
of 0.1 is selected. The drawdowns are calculated until 1600 minutes after starting the
pump.
,,'1---1---+--+---1----1 ...
~
t_l0'
"'"1\\ m'.
x x
\ \\
... ,,' ... 104 TIME (min) .00 ; 101 102
\
DISTANCE
(m)
LAYER' DRAWDOWN (m)
~~~~~-.--~--~
...
.,f---1L-1L X~ ~
-~ ~ \.\ moo
1\ \ \ \
X
\ \
,,' ", DISTANCE
(m)
'''I-':::''::=f~,"+--l---1
E(mln) ... ,,' DISTANCE

LAYER I (m)
50-0.200
KISI-IO.OO MID SAC5J-O.0001QO 1"1-1
C (4) .. 9999999999 ".>-+,- - - - - - - - -

II: (4) -10.00 MID SA(4)-O.OOOIQO M-l
-.!D!.!.(J~J,:,·!..:''~0..c"!..-_--,KC!(",J>.:.·,-,I0",.0",0,--"",,/0 ~ ~~~ :~;: ~ ~ 5"(3)-0.00010011-1
0(2) -4.0 J1 )( 12l -10.00 1'1/0 5"(2)-0.000100 1"1-1
CIO-JS.O D
KIII·IO.OO MID 5AO)-0.000100 11-1
WEEKS - RI=Q.IM - TI=Q.I.'n - l.gA-Q.I - 43 Rings - Q-180M3/0
corresponding with the model of Hantush-Weeks. The transvers anisotropic layer is
discretized in four layers in the numerical model.
'~t-----r---~-----+~--~----1
,,.
"'" \\ ""\
\
toolO'
\
,<4 ,. .,.
'''l1ME (min) ... ,.. let "'DISTANCE
(m)
DRAWDOWN (m) LAVER • DRAWDOWN (m)
~~~~~~---r--~
'''F=*~-r--jt-----j
'''DISTANCE
(m)
,,.t-~~~~-+-----+----~
_________________
'.TlMCi~~~).
. . . . . . .00
,,. _ _ _ _ _ _ _ _'''DISTANCE
_--'(""m.)
$Altl-O .. OOOloo "..,
C (1)-•••, " . . . . . . D - - - - - - - - - - - -
5A(I)-0.000Ioo ,...,
SAI1l-0.000IDO tt-I
Cell-It.O 0
"'I
0(6)-1.6 " Ie ctl ·'0.00 ",D SA elJ -0.000100 ~1
Ot$J-O.9 " "(5)-'0.00 CCSJ-'2.S D SA (51 -0.000100 rt-1

iiiii.li i h kiii aid 88 A, ~t~t:'i~o8D Uliiad IIddidS Q4!
K(2)·'O.OO tvD 5,\(2) -0.000100 ".,
blll.O,I H $6111.0 000100 tfl
WEEKS - RI-O.III - TI-O.I.,. - I •• A-opl - 43 Rings· - g-1801lJ/D
corresponding with the model of Hantush-Weeks. The transvers anistropic layer is
discretized in eight layer in the numerical model.
In Fig. 4.16, the numerically calculated drawdowns are represented by continuous
lines along with the analytically calculated drawdowns (crosses). Looking to this figure,
one can remark that the numerically and the analytically calculated drawdowns deviate a
little from each other. In the pumped pervious layers the deviations are small. They are
the smallest at relatively large distances from the pumped well after a relatively long time
of pumpage. In the second and fourth layer, the deviations are the largest just after
starting the pumping test.
In a second numerical calculation, the homogeneous transversely anisotropic
pervious layer is replaced by eight layers. The location and the thicknesses of the layers
are selected according to the estimated vertical gradient in the layer. Several layers with
relatively small thicknesses are considered at the levels where the largest vertical gradient
is expected. This is just above the directly pumped part of the homogeneous, transversely
anisotropic, pervious layer or just above the well screen. The discretization of the
groundwater reservoir is represented in Fig. 4.17. In this figure the thicknesses and the
hydraulic parameters of the different layers are shown. The horizontal conductivities of
the eight layers of the numerical model which replace the homogeneous anisotropic layer
are all put equal to 10 mId and all the values of the specific elastic storage is 1.0xlQ-4 m- 1 •
The hydraulic resistances between the layers are derived from the distances between the
middle of the layers and the vertical conductivity which is put equal to 0.1 mId. Water is
pumped on the three lowermost layers. The total discharge rate is equal to 180 m3 /d. The
discharge rates on each of these three layers are assumed to be proportional with the
transmissivity of the layers and here also proportional with the thickness of the layers.
The other parameters that define the space-time grid are equal to the values used in the
former calculation. The results of these last calculation correspond better with the
analytical results. Now the differences between the two calculations are very small. By
this verification, it is shown that an appropriate discretization can be very important. At
the levels with the largest vertical gradient, the discretization must be sufficiently fine.
4.6.8 Verification of numerical model with Boulton-Cooley model
Boulton (1955) assumes a "delayed yield" when the water table is lowered in a fine
grained deposition. He formulated this "delayed yield" using an empirical relation. The
total vertical velocity is obtained by the integration of these "delayed" yields. The
so-called Boulton integral is given in Sect. 3.6. Cooley (1971), Cooley (1972) and Cooley
and Case (1973) proved that the Boulton integral describes the vertical inelastic flow in a
semi-pervious layer which is bounded above by the water table and below by the pumped
pervious layer. The drawdown of the water table is negligibly small with respect to the
thickness of the semi-pervious layer. However, it is here assumed that the water delive-
rance is immediate when the water table drops. This concept is further indicated by the
model of Boulton-Cooley. The Boulton delay index l/a is equal to the product of the
storage coefficient near the water table and the hydraulic resistance of the semi-pervious
layer. The drainage factor of the semi-unconfined aquifer in the Boulton model corres-
ponds with the leakage factor of the semi-confined aquifer.
The only difference between the lacob-Hantush model and the Boulton-Cooley
model is the boundary condition at the top of the semi-pervious layer. In the lacob-
Hantush model, this boundary is a fixed hydraulic head and in the Boulton-Cooley model
it is a constant flux boundary with an immediate yield of water when the water table
lowers. Consequently, the time-drawdown curves in the beginning of the pumping test are
very similar for both models.
The numerical model allows to show the relations between the model of Boulton,
the model of Boulton-Cooley and the model of lacob-Hantush. In the first calculations
with the numerical model, two layers and 43 rings are considered. Water is pumped on
layer 1 with a constant discharge rate of 180 m3 /d. The initial radius, the initial time and
log lOA are chosen respectively equal to 0.1 m, 0.1 minute and 0.1. The thickness, the
horizontal conductivity and the specific elastic storage of the pumped pervious layer are
put equal to 10 m, 10 mid and 1.0xlO-4 m- I . The hydraulic resistance between layer 1 and
2 is equal to 100 d. The thickness of the semi-pervious layer, layer 2 of the numerical
model, is set equal to 10 meter. This is ten times the maximum drawdown which will
occur in this layer.
In the Boulton-Cooley model, vertical inelastic flow is considered in this semi-
pervious layer. Therefore, the horizontal conductivity and the specific elastic storage of
layer 2 are set equal to a very small values. The storage coefficient near the water table is
equal to 0.2. This means that the total volume of "delayed yield" due to a unit drop of the
water table over a unit area is equal to 0.2, Sy of Boulton (1963). The amount of water
which is delivered immediately from storage per unit drawdown and per unit area, S., as
was formulated by Boulton (1963) corresponds with the elastic storage coefficient of layer
1 and is equal to 1. Ox 10-3 • Consequently, 'Y = 1 + SyiS. = 201. According to Boulton
(1963) the first part of the time drawdown curves will be identical to the course of the
time-drawdown curves of the lacob-Hantush model if the value 'Y is larger than hundred.
The middle part of the time-drawdown curve will then show a horizontal course.
Finally, one can remark that all parameters introduced in the numerical model to
simulate the lacob-Hantush model are identical as the parameters used to simulate the
Boulton-Cooley model except for one parameter, the horizontal conductivity of layer 2. In
the lacob-Hantush model, this value is equal to a very large value, 1010 mid, so that the
water table is fixed. In the Boulton-Cooley model this value is equal to a very small
value, so that there is a lowering of the water table due to the vertical inelastic flow in
this layer. With the numerical model the drawdowns are calculated until l(f minutes after
starting the pump.
The numerically calculated drawdowns are compared with drawdown which are
obtained by analytical calculation. These drawdowns are calculated for the same interval
for three different distances from the pumped well, 40 m, 100 m and 250 m. These
drawdowns until 1600 minutes after starting the pump are calculated with the FORTRAN
program given in Reed (1980). The other values are derived by means of the type curves
given in Boulton (1963).
DRAW DOWN (m) DRAWDOWN (m)
,00t------r------~----~----~--
IQ-Z lo-t
t=l02 min
10· ' '0-
,00 '00 '0' '0> DISTANCE

(m)
DRAWDOWN (m)
'00
'o-'t------t-t--++-t-lIri+------i
'0-r------t--+---.I!--t4-11-------i
'0> '0' TIME (min) ,00 '0' DISTANCE

LAVER 1 (m)
50-0.200
0(2)-10.0M K (2) -0.00 MID SA (2) -0.000000 M-l
C (I) -I 00.0 D
0(1)-10.0 M K (I) -I O. 00 MID SA (I) -0.0001 00 M-I
BOULTON - Rl=O.lM - Tl=O.lmln - logA=O.l - 43 Rings - Q=180M3/0
corresponding with the model of Boulton-Cooley. In the numerical model two layers are
considered.
ORAWOOWN (m) ORAWOOWN (m)
'~~---+----4----4----~ '''~:::::''4~--+----4-----l
.rl'---~-~+--~-
.,. "'OISTANCE
(m)
ORAWOOWN (m)
'''OISTANCE
(m)
.,.
TI't~ ~~~), .,.
_ _ _ _ _ _ _ _ _ _ _...so.o .• oo _ _ _ _ _ _ _ _.._
, DISTANCE
_ (m)
K(3)-O.OQ rtJD ·SA C3J -0.000000 Pt-1
- - - - - - - - - - - C(2)-SO.O 0
DtZ,-IQ.O" SA (2) -0.000000 tt-I
C{I)-SO.Q D
0(0-10.0 " KlU-l0.QO KID 5,,(1)-0.000100 """
BOULTON' RI-O.IM - TI-O.I.'n - I.,A-O.I - 43 Rings - O-180M3/0
corresponding with the model of Boulton-Cooley. In the numerical model three layers are
considered.
In Fig. 4.18, the numerically and analytically calculated drawdowns are represen-
ted. The numerical results are represented for layer 1 and layer 2 in time-drawdown
graphs as well as in distance drawdown graphs. The drawdown of layer 2 is the draw-
downs of the water table. The last parts of the time-drawdowns curves which are derived
with the type curves (the crosses) shows a small difference with the numerical model.
These differences are probably due to the small accuracy with which these curves can be
read from the Boulton (1963) paper. So, by the application of the numerical model, the
relation between the lacob-Hantush, Boulton and Boulton-Cooley model is demonstrated.
In the second calculation, it is shown how the vertical inelastic flow in the semi-
pervious layer can also be simulated by a numerical model where three layers are
considered (Fig. 4.19). This is analogous to the second calculation of the lacob-Hantush
model. The layer 2 and 3 of the numerical model replaced now the semi-pervious layer.
The vertical inelastic flow in these layers are obtained by putting their horizontal
conductivities and specific elastic storages equal to very small values. The sum of the
hydraulic resistances between layers 1 and 2 and between layers 2 and 3 is equal to 100
d. This is the hydraulic resistance between layer 1 and layer 2 during the first calcula-
tions. In these second calculation, both hydraulic resistances are equal to 50 d.
Comparing the results of the two numerical calculations, one can see that the
drawdowns are identical in the uppermost and lowermost layer. Because the hydraulic
resistance between layer 1 and 2 is equal to the hydraulic resistance between layer 2 and
3, the drawdown in the middle of layer 2 is equal to the average of the drawdown in
layers 1 and in layer 3 at the same distance from the pumped well and after the same time
after starting the pump.
4.6.9 Consequences of numerical rrwdel verification
The different models of which analytical solutions are known can exactly be simulated
with the numerical AS2D model if an appropriate choice of the space-time grid parame-
ters and of the hydraulic parameters are made. The value of the initial radius, Rl, has a
small influence on the calculated drawdowns, at least if this value stays sufficiently small,
for example smaller than 1 m. So, this value can approximately be put equal to the
diameter of the borehole of the pumped well. If, however, the drawdown in the pumped
well is required then the initial radius must be chosen at least 1.6 times smaller than the
diameter of the well screen (1.6 or 1021og10A). Mostly, the proper value of 10gloA is 0.1.
The number of rings determines the distance of the outer boundary from the
pumped well. Although, the outer boundary is not a constant hydraulic head boundary, it
is recommended to reduce as much as possible the influence of this boundary on the
calculations and consequently the flow through this boundary. This can be checked by
means of the discharge balance which is calculated after each time step. Herein, the
balance component describing the horizontal flow through the outer boundary must be
kept as small as possible until the end of the simulation.
In the beginning of the calculation the value of the initial time Tl has a large
influence on the numerically calculated drawdowns. The deviations between numerically
and analytically calculated drawdowns are usually reduced to an acceptable range after a
time which is five to ten time larger than the initial time Tl. Consequently, this initial
time must be chosen ten times smaller then the smallest time of which the drawdown must
be accurately known to compare with the observation.
During the verification of the numerical model with the Hantush model, it is
established that the undirectly pumped semi-pervious layers with an elastic storage
coefficient which is much larger than the elastic storage coefficient of the directly pumped
pervious layer should be discretized in several layers. Only with a fine vertical discreti-
zation of this semi-pervious layer, the drawdowns of the directly pumped and undirectly
pumped layers are obtained with a sufficient accuracy. During the verification of the
numerical model with the Hantush-Weeks model, it appears that a fine vertical discretiza-
tion is required at the level where a large vertical hydraulic gradient is expected during
the simulation.
4.7 Program package for numerical simulation of pumping tests
The program package consists of four programs:

- the program inrmp informs the user about the required input data and allows to create
the input files with the space-time grid parameters and the hydraulic parameters on one
hand and the observed drawdowns on the other hand,
- the program sipur5 to simulate the evolution of the drawdown in the axi-symmetric
grid and to compare the simulated with the observed drawdowns,
- the program outpuS to plot the results of the simulated pumping test in time-drawdown
graphs and distance-drawdown graphs either with or without the observed data or to plot
the simulated drawdown in a cross section.
- the program sidap7 to simulate the evolution of the drawdown in the axi-symmetric
grid and to calculate the drawdown for the given observation times and locations whether
or not increased with an artificial error. The artificially added errors approximate a log
normal distribution.
4.7.1 Program inJinp
The program inrmp allows the user to create three input files. The first two input files
are necessary to run the programs sipur5, outpu5 and sidap7. The third input file allows
the user to make sensitivity analyses and to run the inverse model which is treated in
Chapter 6. Here, only the creation of the first two input files will be treated. The creation
of the third input file will be treated in Chapter 6.
4.7.1.1 Input of space-time grid parameters and of hydraulic parameters
With the program infinp, the parameters which define the space-time grid and the
hydraulic parameters can be introduced interactively. This program is mostly used for a
new group of parameters at the start of a simulation or of an interpretation. The parame-
ters which must be introduced interactively are given in Table 4.3. This input program is
a part of one program which gives information about all the necessary input data and
allows also the input of these data interactively. The name of this global input program is
infinp. Each treated problem must be denominated. The here used example of the name
of the treated problem is Ramel. The procedure is started by typing: 'p.infinp name 1'. In
this case the parameters are stored in a file name1.papu (UNIX) or name1.pap (MS-
DOS). In this table some parameters are mentioned which are not treated until now.
These parameters are described in Chapter 5. The sequence and the format of these
parameters in the file are given in Table 4.4. When a small number of parameters should
be adapted, it is better to change this file in an editor.
4.7.1.2 Input of observed drawdowns
The comparison of the calculated drawdown with the observed drawdown is

possible. The program for the input of the observed drawdowns is also a part of the
global input program inrmp. The first input parameter is here the number of observation
wells. For each observation well, one must introduce the layer where the screen of the
observation well is located, the number of observations made in the well and the distance
from the pumped well. When a groundwater reservoir is treated with a lateral anisotropy
(see Sect. 5.2), an additional parameter is necessary. This parameter describes the
direction observation well - pumped well with respect to the north. This is the angle
between the direction pumped well - north and the direction pumped well - observation
well if one turns to the east. Finally for each observation well the observed drawdowns
with their corresponding observation times must be given.
The sequence in which these parameters must be introduced interactively is given
in Table 4.5. In the case that observations made in the pumped well are included in the
data, the information blocks, which contain these data, must be mentioned at the end of
the file. These data are stored in a file name1.dapu (UNIX) or a file name1.dap
(MSDOS). The sequence and the format of these data are given in Table 4.6. Also in this
table some parameters are mentioned which are not defined until now. These parameters
are described in Chapter 5. After the interactive input of the data, the data file should be
checked for possible errors. The corrections are best made by means of the editor.
It is advisable to classify the observed drawdowns in the input file according to the
observation time, distance and level. First, the observations made in the pumped layer are
given. These observations are followed by those made in the adjacent layers. The wells
which are located in the same layer are classified according to their distance to the
pumped well. The well which is the closest to the pumped well is first given. The
observations of a same well are arranged according to the observation times.
.....
Table 4.3. First part. Input to create the file with the space-time grid parameters and hydraulic parameters (name1.pap(u» by means of the interactive input program inlinp ~
Variable Description Limits and/or units Format

Rl initial radius or inner radius of axi-symmetric grid meter G8.0
Tl initial time which defines the treated time steps along with L9 minute G8.0
L9 10glOA, logarithm of factor A which defines the increase of the radii of the nodal circles and of the time steps dimensionless G8.0
T2 time which defines the last calculated time step, calculation ends just before the given time minutes G8.0
N9 number of layers 02,;;N9,;;18 12
VI number of rings OS,;;Vl,;;60 12
D(J) thickness of layer I, I varying from 1 to N9 with step 1 meter G8.0
Kh(J) horizontal conductivity of layer J, I varying from 1 to N9 with step 1 mid G8.0 Q
c(I) hydraulic resistance between layer J and I + I, J varying from 1 to N9-1 with step 1 d G8.0 {:
S,(I) specific elastic storage of layer J, I varying from 1 to N9 with step 1 m- I G8.0 ...~
So storage coefficient near the watertable m'/m' or dimensionless G8.0 .....
FLl flag which defines the location of wellscreen, OO,;;FLl ,;;01 12 .....
FLl =00, wellsreen located in one layer
FLl =01, wellscreen located in more than one layer
If FLl=OO
i[
HI number of layer where wellscreen is located 01 ,;;Hl ,;;N9-1 12
Q discharge rate which is pumped from layer HI m'/d G8.0 ~
if FLl=l
QLAYER(J) discharge rate of layer I, I varying from 1 to N9 with step 1 m'/d G8.0
!t
FL2 flag which defines the evolution of discharge rate 00,;;FL2,;;OI 12 ~
~
FL2=00, there is no variation in discharge rate I::
FL2=01. there is a variation in discharge rate l ~
if FL2=OO skip to FLANIS ~.
if FL2=Ol
NTIM number of changes in discharge rate 03,;;NTIM,;;20 12 a
For IT varying from 2 to NIIM +1 o';;T(I)';; T2 - minute
T(IT) time of IT" change of discharge rate' QT(I)';;Q or G8.0
QT(lT) discharge rate before the time of the IT" change of discharge rate QT(I)';; E~~IQLAYER(J) G8.0
m'/d
I A pumping test followed by a recovery test can be considered as a pumping test with changing discharge rates.
, The first discharge change is the start of the pump. Consequently T(I) and QT(l) are both equal to zero.
T(NTIM + 1) is always equal to T2. QT(NTIM + 1) is discharge rate just before and during the last observation or it is the discharge rate during the time step T(NTIM) and T2.
Table 4.3. Second part. Input to create the file with space-time grid parameters (namel. pap(u» by means of the interactive input program infmp
Variable Description Limits and/or units Ponnat

FLANIS flag which deflnes the anisotropy in the horizontal plane 00:5:FLANIS:5:01 I2
FLANIS=OO, there is no anisotropy in the horizontal plain
FLANIS=OI, there is anisotropy in the horizontal plain
if FLANlS=OO skip to OBPUWE
if FLANIS=Ol ~
ANISOT anisotropy of hydraulic conductivity in horizontal plain dimensionless G8.0
PHI angle between the north direction starting from the pumping test and the direction of maximum horizontal radianoo:5: G8.0 i...
OBPUWE conductivity turning to the eastflag which deflnes if there are observations in pumped well OBPUWE:5:01 I2
OBPUWE=OO, there are no observations in pumped well "-....
OBPUWE=OI, there are observations in pumped well
if OBPUWE=OO skip to CORREC f
if OBPUWE=Ol ~.
NPOWER Power N well loss function CQN dimensionless G8.0 It
CVALUE C value of well loss function CQN m(l-,NjdN G8.0
CORREC flag which derIDes if the observed drawdown should be corrected by means of observations in a reference well oo:5:CORREC~09 12
CORREC=oo, no correction of observed drawdowns
CORREC > 00, correction of observed drawdowns are needed ~
l
CORREC derIDes the number of ampliflcation factors and retardation factors
if CORREC=OO skip to end i'"
if CORREC >00 ~
S.,(t)= s",(t)-AMPLIP(I)*s,..(t+ RET ARO(I» I ~
where s",(t) is the corrected drawdown at time t iii.
s",(t) is the observed drawdown at time t I
s,..(t+ RET ARD(I» is the drawdown of the reference well at time t +RET ARO(I)
RET ARO(I) is the retardation
I
AMPLIP(I) is the amplifIcation factor
For Ie varying from 1 10 CORREC with step 1
I
AMPLIP(lC) is the Ie"' ampliflcation factor dimensionless G8.0
RETARD(lC) is the IC" retardation minute G8.0
-B;
Table 4.4. First part. Sequence and format of parameters in file name l.pap(u)
Line Variable Description and remarks Limits and/or units Pormat
1 N9 number of layers 02,,;N9,,;18 12

2 VI number of rings OS,,;V1,,;60 12
3 HI number of pumped layers (if PULAMI ~O) 01 ,,;H1 ,,;N9-1 12
4 PULAMI 00,,;PULAMl,,;03 12,
FLANIS flags determining kind of treated problem' OO,,;FLANIS";Ol 1x,12,
OBPUWE oo,,;OBPUWE,,;OI IX,12,
CORREC OO,,;CORREC ,,;01 IX,12
S R1 initial radius or inner radius of axi-symmetric meter P20.9
6 Q grid m'/d P20.9
7 T1 discharge rate (if PULAMI ~O) minute P20.9
8 L9 initial time dimensionless P20.9
9 T2 log lOA minute P20.9
time defining last treated time step
10 BUOY1 first buoy 0000.000000 P20.9
11 O(J) J~I thickness meter P20.9

12 J~2 of meter P20.9
... . .. layer J
1O+N9 J~N9 meter P20.9
11+N9 BUOY2 second buoy 0000.00000 P20.9
12+N9 K.(J) J~I horizontal conductivity mid P20.9

13+N9 J~2 of mid P20.9
... ... layer J .... ...
II +2N9 J~N9 mid P20.9
12+2N9 BUOY3 third buoy 0000.00000 P20.9
13+2N9 c(J) J~l hydraulic resistance d P20.9

14+2N9 J~2 between d P20.9
... ... layer J and J+1
11 +3N9 J~N9-1 d P20.9
12+3N9 S,(J) J~I specific elastic storage mol P20.9

13+3N9 J~2 of mol P20.9
... . .. layer J ... . ..
11+4N9 J~N9 mol P20.9
12+4N9 So storage coefficient near water table m 3/m 3 P20.9
if FLANIS ~ I'
13+4N9 ANISOT anisotropy of horizontal conductivity' dimensionless P20.9

14+4N9 PHI angle of maximum conductivity' radian P20.9
, PULAMI ~PLl +2*PL2 (for definition PLl and PL2 see Table 4.3)
4 for detailled description see also Table 4.3
Table 4.4. Second part. Sequence and format of parameters in file name1.pap(u)
Line Variable Description Limits and! or Fonnat

units
ifOBPUWE=l
Ll' CVALUE C value of well function CQN m(l-3N) dN F20.9

L2' NPOWER power N of well function CQN dimensionless F20.9
if CORREC ~ I'
L3' AMPLIF(lC). IC~1 amplification factor IC dimensionless 2FI0.4

RETARD(lC) and 2FlO.4
L4' IC~2 retardation IC
... ... . ..
LS' IC~CORREC minutes 2FI0.4
if PULAMl~OI or PULAMl~031
L6' NTIM number of changes in discharge rate 03 ,; NTIM ,; 20 12

T(I)~O.O and QT(I)~O.O
L7' T(IT). IT~2 time of IT"' change of discharge rate m'/d 2FlO.4
QT(IT) and 2FI0.4
L8' IT~3 discharge rate before IT"' change ... . ..
minute 2FI0.4
L9' IT~NTIM
L1 ~ 1O+4N9+2FLANIS+3
L2 ~ 1O+4N9+2FLANIS+4
L3 ~ lO+4N9+2FLANIS+4+1
L4 ~ 1O+4N9+2FLANIS+4+2
LS ~ 10+4N9+2FLANIS+4+CORREC
L6 ~ lO+4N9+2FLANIS+4+CORREC+l
L7 ~ lO+4N9+2FLANIS+4+CORREC+2
L8 ~ 1O+4N9+2FLANIS+4+CORREC+3
L9 ~ lO+4N9+2FLANIS+4+CORREC+NTIM
.....
0'1
00
Table 4.5. Input to create the file with observed data (name1.dap(u)) by means of the interactive input program infmp
Variable Description Limits or units For-

mat
IASB number of blocks filled with observed drawdowns (maximum SO observations per block) 01 :<;;IASB :<;;20 12
For I varying from 1 to IASB
Q
LAYO(I) 01 :<;;LAYO(I):<;;N9 12 ~
layer in which observations of block I are made
DIST(I) distance from pumped well of observation in block I meter GS.O ~
IAVW(I) OI:<;;IAVW(I):<;;SO 12
.....
number of observations in block I .....
ifFLANIS=1
ALFA(I) angle measured from the direction pumping well - north turning to the east until the direction
pumped well-observation well of observation in block I
f
For J varying from 1 to IAVW(l) t
TIME(J,I) time of the JIh observation in block I minute GS.O iii
For J varying from 1 to IAVW(I) ~
DRAW(J,!) drawdown of the JIh observation in block I meter GS.O
if OBPUWE=I
...~
INFBLO number of blocks with pumped well observation 01 :<;;INFBLO:<;;lO 12 ~~.
For m varying from 1 to INFBLO
NUINFB(IB) mlh block number with pumped well observation 01 :<;;NUINFB(I):<;;IASB 1014 ~
~
ifCORREC=1
NINFRW number of blocks with observations in pumped well 01:<;;NINFRW:<;;09 12
For IN varying from 1 to NINFRW
NUBCOR(IN) block number with reference well observation 01 :<;;NIBCOR(I):<;;19 12
For IC varying from 1 to CORKEC
NIBCOM(lC,J) block numbers corrected by the Ie" group of correction factors 00 < NIBCOM(IC,J) :<;; IASB 12
Table 4.6. Sequence and format of parameters in file name1.dap(u)
Line Variable Description and remarks Limits and/or units Format

I IASB number of observation blocks (wells) 01 :;;IASB:;;20 Ix.l2
2 LAYO(I),I=I,IO layer in which observations of block 1 are made 01 :;;LAYO(I):;;N9 1014
3 LAYO(l),1=11,20 layer in which observations of block 1 are made 01 :;;LAYO(I):;;N9 1014
4 IAVW(I),I=I,IO number of observations in block 1 01 :;;IAVW(l):;;SO 1014
S IAVW(l),I=II,20 number of observations in block 1 01 :;;IAVW(l):;; SO 1014
6 DIST(I),I=I,IO distance from pumped well of observations in block 1 meter IOF6.1
7 DIST(I),1=11,20 distance from pumped well of observations in block 1 meter IOF6.1 &l
{;
8 ALFA(I), 1= 1,10 if FLANIS = II angle measured from the direction radian IOF6.1 ~
9 ALFA(I), 1= 11 ,20 pumped well-north turning to the east until the direction radian IOF6.1 ..
pumped well-observation well of observation in block 1 .....
""
L1' TIME(1,J), J=I,IO for 1=1,20,1 time of J1h observation in block 1 minute IOF6.1
L1 +1 +S(1-I) TIME(I,J), J = 11 ,20 time of Jib observation in block 1 minute IOF6.1 f
L1 +2+S(I-I) TIME(1,J), J =21,30 time of J1h observation in block 1 minute IOF6.1 ~.
L1 +3+S(I-I) TIME(1,J), J=31,40 time of Jt> observation in block 1 minute IOF6.1 ~
L1 +4+S(I-I) TIME(I,J), J=41,SO time of Jt> observation in block I minute IOF6.1 ~
L2' DRAW(1,J), J=I,IO for 1=1,20,1 drawdown of J1h observation in block 1 meter IOF6.3 ~
L2+ I +S(I-I) DRAW(1,J), J=11,20 drawdown of J1h observation in block 1 meter IOF6.3
~
L2+2+S(I-1) DRAW(I,J), J=21,30 drawdown of J1h observation in block 1 meter IOF6.3 ....
I:
L2+3+S(I-I) DRAW(1,J), J=31,40 drawdown of J1h observation in block 1 meter IOF6.3
L2+4+S(I-I) DRAW(I,J), J=41,SO drawdown of J1h observation in block 1 meter IOF6.3 ~
~.
L2+IOO INFBLO if OBPUWE = II number of blocks with observations in pumped well 01 :;;INFBLO:;;10 12
~
L2+IOI NUINFB(1B),1B = I,INFBLO block numbers with pumped well observation 01 :;;NUINFB(I):;;IASB 1014 ~
L3' NINFRW if CORREC = II number of blocks with observations in reference well 01 :;;NINFRW :;;09 12
L3+1 NIBCOR(1N),IN = I,NINFRW block numbers with reference well observations 01 :;;NIBCOR(1):;; 19 1014
L3+I+IC NIBCOM(lC,J) for IC = I,CORREC 1 block numbers corrected by 00< NIBCOM(IC,J):;;IASB ISI4
J=I,NIBCOR the ICIb group of correction factors
1 for detailed description see also Table 4.4

2 L1 = 7 + 2FLANIS
3 L2 = L1+100
4 L3 = L2+100+20BPUWE
$
-
4.7.3 Program sipur5
With the parameters which define the space-time grid and the hydraulic parameters, the
drawdowns are calculated for all nodal circles and all time steps. After these calculations,
the observed drawdowns are then compared with the calculated drawdowns. The name of
the proper simulation program is sipurS. It is a program written in FORTRAN. It can be
started with the procedure p.sipurS. The procedure can be started by typing: 'p.sipur5
name 1'. Again, namel is the name of the treated problem. The results of the calculation
are stored on two different output files. All calculated drawdowns of all the nodal circles
and all time steps are stored on the file name1.oupu (UNIX) or name1.oup (MSDOS).
This file is used for the plot program outpu5 (see Sect. 4.7.4).
Table 4.7. Echo of the used space-time grid parameters of the numerical model sipurS
read from the file name1.pap(u)
RADIUS OF WELLSCREEN, R, IN M, ------------------------------------------ 0.100
DISCHARGE OF PUMPED WELL, Q, IN M 3 /DAY----- ______________________ _ 320.100
INITIAL TIME, T1, IN MIN, ------------------------------------------------------- 0.100
LOGARITHMIC INCREASE OF TIME AND OF RADIUS OF RINGS
LOGA, ---------------------------------------------------------------------------------- 0.100
LATEST CALCULATED TIME, T2, IN MIN, ---------------------------------- 1600.
NUMBER OF LAYERS, N, -------------------------------------------------------- 3
NUMBER OF RINGS, M, ----------------------------------------------------------- 43
THE WELLSCREEN IS SITUATED IN LAYER ---------------------------------
THICKNESS OF THE SUCCESSIVE LAYERS, IN M
NUMBERED FROM LOWER TO UPPER
THICKNESS OF LAYER 1, IN M,------------------------------------------------- 7.000
THICKNESS OF LAYER 2, IN M, ------------------------------------------------ 4.500
THICKNESS OF LAYER 3, IN M, ------------------------------------------------ 3.500
HYDRAULIC CONDUCTIVITY, Kh (I), IN M/DAY, --------------------------- 18.800
HYDRAULIC CONDUCTIVITY, Kh (2), IN M/DA Y, --------------------------- 5.000
HYDRAULIC CONDUCTIVITY, Kh (3), IN M/DA Y, --------------------------- 0.100
HYDRAULIC RESISTANCE, c(I), IN DAY, ----------------------------------- 15.
HYDRAULIC RESISTANCE, c(2), IN DAY, ----------------------------------- 900.
SPECIFIC ELASTIC STORAGE, S,(1), IN M-I, ------------------------------- 0.74D-04
SPECIFIC ELASTIC STORAGE, S,(2), IN M- ' , ------------------------------- 0.63D-04
SPECIFIC ELASTIC STORAGE, S,(3), IN M- I, ------------------------------- 0.lOD-03
STORAGE COEFFICIENT AT THE WATER TABLE, So, -------------------- 0.200000
The printed results are given in the file name1.lst. The main purpose of this file is
the control of the calculation process. The first lines of this file are echoes of the used
parameters which define the space-time grid and the hydraulic parameters of the different
layers. An example of such an echo is given in Table 4.7. After this echo, the results of
each time step are given. For each time step, the corresponding time point is given along
with the calculated discharge rate balance for the whole groundwater reservoir. In the
balance, following discharge rates are considered: the pumped discharge rate, the
horizontal inflow through the outer boundary and the rate delivered by a storage decrease.
After that, for each layer the discharge rate balances are given along with the drawdowns
in all the nodal circles of this layer. The drawdowns are printed in the format IOF8.4.
This means that the drawdowns of the first ten rings are written on the first line. The
drawdowns of the second ten rings are written on the second line, and so on until the
drawdown in the last rings is given. With the number of the ring, the radius of the nodal
circle can be calculated with the formula given in Fig 4.1. In Table 4.8, an example is
given of the calculation results after a time step.
Table 4.8. Example of results printed after every time-step of the numerical model
sipur5
TIME(MIN)~100.0
IN M'/D DlSCHARGE~ 320.1 HOR. O.I644D-OS STORAGE DECREASE~ 320.1 BALANCE~ 0.ISSJD-02
INF1..0W~
LAYER = 1
HORIZONTAL IN- AND OUTFLOW IN M 3 /D OUTJ<LOW OR DISCHARGE= 320.1 INl<LOW= 0.1214D-05 STORAGE DECRHASE= 191.3
-2.7093 -2.6200 -2.5392 -2.4581 -2.3765 -2.2943 -2.2113 -2.1273 -2.0422 -1.9557
-1.8680 -1.7790 -1.6892 -1.5990 -1.5090 -1.4195 -1.3307 -1.2423 -1.1542 -1.0665
-0.9795 -0.8935 -0.8087 -0.7252 -0.6435 -0.5637 -0.4861 -0.4110 -0.3385 -0.2695
-0.2047 -0.1458 -0.0948 -0.0541 -0.0254 -0.0089 -0.0019 -0.0002 0.0000 -0.0000
-0.0000 0.0000 0.0000
LAYER~2
HORIZONTAL IN- AND OUTFLOW IN WID OUTFLOW OR DlSCHARGE~ 0.000 INF1..0W~ 0.430SD-06 STORAGE DECREASE~ 103.S
-0.7057 -0.7057 -0.7058 -0.7058 -0.7059 -0.7060 -0.7060 -0.7060 -0.7060 -0.7056
-0.7056 -0.7051 -0.7044 -0.7034 -0.7017 -0.6993 -0.6958 -0.6907 -0.6835 -0.6735
-0.6597 -0.6413 -0.6170 -0.5858 -0.5471 -0.5005 -0.4466 -0.3869 -0.3234 -0.2592
-0.1969 -0.1397 -0.0901 -0.0508 -0.0234 -0.0079 -0.0016 -0.0001 0.0000 -0.0000
0.0000 0.0000 0.0000
LAYER~3
HORIZONTAL IN· AND OUTFLOW IN M'/D OUTFLOW OR DlSCHARGE~ 0.000 INF1..0W~ 0.8119D-12 STORAGE DECREASE~ 2S.27
-0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002
-0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002
-0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001
-0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000
0.0000 -0.0000 0.0000
After the print of all the calculated drawdowns, these drawdowns are compared
with the observations. The observation distances and times should not correspond with the
radii of the nodal circle and with the time points at which the drawdowns are calculated.
The drawdowns at the observation distances and times are inferred from the numerical
calculated drawdown by bilogarithmic interpolation versus the distance and the time. For
each observation well, a table is made where the calculated drawdowns are compared
with the observed drawdowns. First, the observation number is given in the first column.
It is followed consecutively by the observation time (in minutes after starting the pump),
the logarithm in base 10 of the calculated drawdown (expressed in m), the logarithm in
base 10 of the observed drawdown (in m) and in the last column, the difference between
the two last mentioned values or the residuals. So, a numerical overview of all the
residuals is obtained. At the end of each table the mean and the standard deviations of the
residuals are given for the observation before the 31.6 minutes of pumpage and for the
observations after this time. These tables, means and standard deviations can help in the
analysis of the residuals (see Sect. 6.3.2).
4. 7.4 Program outpu5
By the plot of the calculated drawdown against the distance from the pumped well and the
time since starting the pump, a clear graphical overview can be obtained over the results
of the numerical model. This results can also be compared with the observed data. The
name of the plot program is outpu5. It is a program written in FORTRAN. The output is
a file with HP-GLl2 codes (Hewlett-Packard's Graphics Language standard). Only the
program possibilities relevant with this part will be described here shortly. After the start
of the program, some questions must be answered interactively. There are two possibil-
ities to plot the calculated drawdowns. The first possibility is the plot of time-drawdown
and distance-drawdown graphs. The second possibility is the plot of drawdown contours
in a vertical cross section after different time of pumpage.
In the first possibility, the calculated drawdown can be compared with the
observed drawdowns. In the time-drawdown graphs, the drawdowns are plotted versus the
time after starting pump. Both axes of the graph are logarithmic. Different time-draw-
downs lines are represented for different distances from the pumped well. In the distance-
drawdown graphs, the drawdowns are plotted versus the pumped well distance for
different times after starting the pump. These times correspond with the different orders
of ten which are larger or smaller than one minute, including this last time. Both axes of
the drawdown-distance graph are also logarithmic. There are now three options to plot the
calculated and/or the observed drawdown. In the first option, only the observed draw-
downs are plotted in the time-distance graphs and in the distance-drawdown graphs. In the
second option, only the calculated drawdowns are plotted. Here, in the time-drawdown
graphs, the lines correspond with different distances from the pumped well starting with
1m and increasing by a factor A2. In the last option the calculated and the observed
drawdowns are plotted simultaneously. In the time-drawdown graphs, only the calculated
drawdowns for the distances corresponding with the distances of the observations are
plotted. The calculated drawdowns are always represented by solid lines. The observed
drawdowns are always represented by crosses (see e.g. Fig. 4.18).
In the second possibility, the drawdown contours are plotted in cross-sections after
different times of pumpage. There are two options to give the required pumpage times. In
the first option, only the smallest required time is asked. The other times are then put
equal to 10, 100, 1000, ... times larger than the given time. In the second option, the
number of required cross-sections must be given followed by the times of each cross-
section. The vertical axis of the cross-section is arithmetic. It has a constant height of 10
cm. The scale of this axis is thus calculated by the program. The depth and the different
layers are shown along this axis. The horizontal axis or the distance from the pumped
well has a logarithmic scale. This origin of the horizontal axis corresponds with the
minimum radius (distance) that one wants to consider in the cross-section. This minimum
distance is mostly equal or larger than the initial radius Rl. The horizontal axis is further
defined by the number of moduli and length of these moduli. The plotted contour lines
are defined using a minimum value Spbni and a logarithmic interval LI. The plotted
contours lines correspond then with the values Spbni, 10LI spbnb 102LISpbni, 103LISpbni, ••••• until
the largest possible value. This value is smaller than the largest calculated drawdown in
the treated cross-section (see Fig. 5.25).
4.7.5 Program sidap7
With the parameters which define the space-time grid and the hydraulic parameters, given
in the file namel.pap(u), the evolution of the drawdowns in the axi-symmetric grid is
first calculated. From these results the drawdowns are obtained by a bilogarithmic
interpolation according to the observation times and distances, given in file namel.-
dap(u). It is possible to add at random errors on the calculated drawdowns. These at
random errors approximate a log normal distribution. The variance of this log normal
distribution must be given interactively. These errors are obtained by 'a random number
generator' consisting of two function ranI and gasdev given by Press et ai, 1992. When
a very small value is attributed to the variance of the log normal distribution then the log
normal distributed errors become so small that only the truncation error remains which
arises when the results are written down. The obtained drawdowns with their correspon-
ding observation times and locations are written down in a file namel.dapu.result
(UNIX) or namel.dar (MSDOS) in the same sequence and format as given in Table 4.6
for the file namel.dap(u). When one want to use this file to make an interpretation of
these artificially obtained drawdown, e.g., in the programs sipurS, outpu5 or iopurS (for
program iopu5, see Chapt. 6), one should be aware that the calculated drawdown is not
smaller than 0.0005 m. These values will be truncated to zero during the writing down in
the file. Because these values will hamper the running of the above metioned programs,
these zeros should be removed from the obtained file. A part of the results of program
sidap7 is written down in the files namel.oup(u) and namel.lst in the same sequence
and format as the program sipurS (see Sect. 4.7.3)
REFERENCES
Boulton, N.S., 1963, Analysis of data from non-equilibrium pumping test allowing for
delayed yield from storage: Proc. Insf. Civ. Eng., 26, p. 469-482.
Cooley, R.L., 1971, A finite difference method for unsteady flow on variably saturated
porous media, application to a single pumping well: Water Resources Research,
v. 7, no. 6, p. 1607-1625.
Cooley, R.L., 1972, Numerical simulation of flow in an aquifer overlain by a water table
aquitard: Water Resources Research, v. 8, no. 4, p. 1046-1050.
Cooley, R.L., and C.M. Case, 1973, Effect of a watertable aquitard on drawdown in an
underlying pumping aquifer: Water Resources Research, v. 9, no. 2, p. 434-447.
Cooper, H.H., and Jacob, C.E., 1946, A generalized graphical method for evaluating
formation constants and summarizing well field history: Am. Geophys. Union
Trans., 27, p. 526-534.
Hantush, M.S., 1960. Modification of the theory of leaky aquifers. Joum. Geophys.
Res., 65, p. 3713-3725.
Neuman, S.P. and P.A. Witherpoon, 1969. Applicability of current theory of flow in
leaky aquifers. Water Resources Research, v. 5, no. 4, p. 817-829.
Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P., 1992, Numerical
recipes in FORTRAN, The art of scientific computing, Second Edition: Cambridge:
Cambridge University Press, 963 p.
aquifers: Washington, D.C., U.S. Geol. Survey Techn. of Water Resources lnv.,
Book 3, Chap. B3, 106 p.
Von Rosenberg, D. V. (1969). Methods for the numerical solution of partial differential
equations. New York: American Elsevier, 128 p.
Chapter 5 / Further developments of
pumping test model
With the numerical model the evolution of the drawdown can be calculated in a layered
groundwater reservoir for a constant discharge rate. The layers of the groundwater
reservoir are supposed to be laterally homogeneous, isotropic and of an infinite lateral
extension. When pumping tests are performed in practice, the above mentioned con-
straints of the numerical axi-symmetric model often limit the possibility to compare the
observed and the calculated drawdown. These constrains limit the use of the numerical
model as a base for an inverse model which allows the interpretation of a wide range of
pumping tests.
To raise the applicability of the numerical model, some further developments were
made. The first development concerns the possibility to treat a pumping test with a
variable discharge rate. A second development is the ability to simulate the drawdown of
a pumping test in a laterally anisotropic aquifer. In addition the drawdown in a pumped
well can be calculated as the sum of the well loss and the aquifer loss. With a supplemen-
tary computer program the drawdown due to a multiple well field can be calculated in a
lateral continuous homogeneous multilayered groundwater reservoir. With the same
program it is possible to calculate the drawdowns in groundwater reservoirs that are
laterally bounded by impervious or constant head boundaries. Finally, the drawdown in a
multilayered laterally isotropic or anisotropic groundwater reservoirs with a vertical plane
of discontinuous lateral conductivity change can be approximated. Of all these cases the
subsidence due to pumping can also be estimated assuming that the specific elastic
storage, the vertical and horizontal conductivity are constant in each layer during the
compaction and are independent of the drawdown.
5.1 Drawdown of pumping tests with variable discharge rate
5.1.1 Theoretical considerations
A pumping test can be followed by a recovery test. After the shut down of the pump
water levels will begin to rise in pumping and observation wells. This is referred to as the
recovery of groundwater levels (Todd, 1980). During the recovery test the residual
drawdown is observed during the recovery period. The difference between the hydraulic
head just before the start of the pump and the hydraulic head at a certain time after the
stop of the pump is called the residual drawdown. Also, residual drawdowns can be used
for the derivation of the values of the hydraulic parameters. Therefore, one can compare
the simulated and the observed drawdowns and residual drawdowns of a pumping test
followed by a recovery test. A pumping test followed by a recovery test can be con-
sidered as a special case of a pumping test with a variable discharge rate. In this part the
more general case, a pumping test with variable discharge rate is treated.
176 Chapter 5 / Further developments oj pumping test model
When one treats a pumping test with a variable discharge rate, the fIrst step is the
discretization of the time discharge rate course in a number of consecutive pumping
periods with constant discharge rate (see Fig. 5.1). During this discretization, the sum of
the amounts withdrawn in the consecutive pumping periods should be equal to the total
withdrawal during the whole pumping test. This discretized time discharge rate curve can
now be described by means of a time table of discharge rates. Two variables are
introduced in this table: the times T(I) and the discharge rates QT(I). T(I) is the ph time
where the discharge rate changes from QT(I) to QT(I + 1). T(l) is always equal to zero
and corresponds with the time of the fIrst start of the pump in a test. QT(I) is the
discharge rate between the times T(I-1) and T(l). QT(l) is also always equal to zero and
corresponds with the discharge rate before the first given time T(l).
If it is assumed that the drawdown behaves always linearly with the discharge rate
then the drawdown of a pumping test with a variable discharge rate can be calculated by
applying the rule of superposition. Here, the drawdown of a pumping test with a variable
discharge rate is considered as the sum of drawdowns of a series of pumping tests. Each
test has a constant but different discharge rates and starts at different times (Fig. 5.1). So,
the time-drawdown curves are first calculated for a constant discharge rate. The draw-
down corresponding with a constant discharge rate Q, at a distance r, in a layer J and
after a time t is here indicated by sQ(r,J,t). After the simulation of drawdown with a
constant discharge rate with the numerical model, the drawdown of a pumping test with a
variable discharge rate s(r,J,t) is searched by superposition:
"NTIM QT(I) -QT(I-l) (5.1)
s(r,J,t)=£""I=2 sQ(r,J,t-T(I» Q
where NTIM is the number of considered discharge rate changes increased with one. The
pumping periods for which t-T(l) is negative are not considered.
When the simulated and observed drawdown of a pumping test with a constant
discharge rate Q should be compared along with the simulated and observed residual
drawdown of the successive recovery test then the number of considered discharge rate
changes is three or NTIM=3. The considered times and discharge rates are then:
T(l) =0, T(2) =Tp ' T(3) =Tp+Tr
(5.2)
QT(l) =0, QT(2) =Q, QT(3) =0
where Tp and Tr are respectively the duration of the pumping and the recovery test.
Eq. 5.1 is only applicable if the groundwater flow equation is linear. This is
mostly the case. This equation is not applicable in the exceptional cases where a consider-
able horizontal flow in the uppermost layer occurs and where the drawdown in this layer
is large with respect to its thickness.
Chapter 5 / Further developments of pumping test model 177
DISCHARGE RATE (m 3 /d)
OT(5) 7"-"""-- -OT(6)

OT(4)-1T"" ,....~~OT(7)
OT(3)- J p---===-- OT(8)
OT(2)-
1 3 4 5 6 7 8 9
o OT(1)..Q OT(9)=O
T(1) T(3) T(4) T(5) T(S) T(7) T(8) ~9)

T(2)
o TIME (MIN)
__T
, ol_Q_T_(_2ra _('_)LI________________________________________
T(1)..Q
2 o~-QT-(3-)-Q-T-(-:2):I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~_
T(2)
QT(4)-QT(3)1"'1------------------
3 ot~----------~--------------------------------------
T(3)
4 o!~----~Q~T~(~~Q~T~(~4)il~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
__
T(4)
5
! T(5)
0r----aQmT(~SraHnT~(~~'~==========================~
t T(6)
6 oltr-----------CQrrTQnra}QTn(;:~~I::::::::::::::::::::::::::::~
7 0 !~i ------~~~m7T~(n::::::::::::::::::::::::_
QT(8)-QT(7) I
OI~---------------------------------T~(~8)~-----------
QT(9)_QT(8) L.._ _ _ _ __
Fig. 5.1. Discretization of time-discharge rate course in a number of consecutive

pumping periods with constant discharge rate and disintegration of this pumping test in
eight pumping tests with constant discharge rate to apply the rule of superposition to
calculate the drawdown
178 Chapter 5 / Further developments of pumping test model
Pumping tests with a variable discharge rate are rather frequently encountered in
practice. Two different reasons can cause these variable discharge rate: the first is the
change of the discharge rate of the pump itself; the second is due to the change of the
amount of water that is delivered from the well storage per time unit. Centrifugal pumps,
which are located at or near the ground surface for suction lift, have a discharge rate
which is very sensitive to the hydraulic head in the pumped well. Because the hydraulic
head in the pumped well changes quickly in the beginning of the test, the discharge rate at
the start of the test can change meaningfully. After a certain time of pumping the
maximum possible drawdown is reached in the pumping well whereafter the discharge
rate gradually decline.
Also the discharge rate of submersible pumps can show meaningful variation if the
hydraulic head change in the pumped well is large. This large drawdown in the pumped
well can occur in the case that a layer with low hydraulic conductivity is pumped.
Specially here the change of the amount of water that is delivered from the storage of the
well per time unit can be considerable. The larger the diameter of the well where the
water drop takes place, the larger is the discharge rate delivered by the storage of the
well. An example of this last case is the pumping on dug wells, which have diameters
between two and five meter. Here, considerable amounts of water are delivered by the
well storage despite the rather small drawdown.
5.1.2 Additional input data
When a pumping test with a variable discharge rate is treated, the description of the
change of the discharge rate should be included in file name1.pap(u). First, the number
of considered discharge rate changes NTIM must be given followed by the vector T
which gives the times of the discharge rate changes and QT with the discharge rates
between the given times. These parameters can be included in the file name1.pap(u)
using the interactive input program inrmp (see Table 4.3). The location and the format of
these parameters in this file are given in Table 4.4.
5.1.3 Example of a pumping test with variable discharge rate
The first example of pumping test with variable discharge rate is a test with a constant
discharge rate followed by a recovery test in a transversely anisotropic aquifer as assumed
in the model of Hantush-Weeks (see Sect. 3.5). In the numerical model, the groundwater
reservoir is schematized as in the second verification of the numerical model with the
model of Hantush-Weeks (see Sect. 4.6.4). The homogeneous transversely anisotropic
pervious layer of ten meter is replaced by eight layers in the numerical model. In Fig.
4.17 the introduced thicknesses and hydraulic parameters are represented. The numerical
model simulates first the drawdown due to a constant discharge rate of 180 m3 /d until a
time that is just larger than the last observation time of the residual drawdown. These
obtained drawdowns are used to calculate the drawdowns of the pumping test with a
Chapter 5 / Further developments of pwnping test mOdel 179
.
,
DRAtIHMIOO
...
...._'"'
10-1 ..
,, ~
10-2 ,.. ~
. ~ b-L\
r-Io "IN
t-l0 "I.
, '.' 102 TltE(ftIN) loG
'.' 102 DISTANCE 00
LAYER 8
.
DftAWDOW(tD
, ••• t--...;~4"""':_:_--+--------,j
, •. 'I---"'*--.,I-i'----..,II-------~
,.. '.' 102' ntEmut)

LAYER
SO=.200
'.'
0(S)=5.0 11 K(S)=IO.OO 11/0 SA(S)-.OOOIOO 11-1
C(S)'10000000B~e~9~-----------------
D (S) -2.0 11 K(S)-IO.OO I1/D SA(S)-.OOOIOO 11-1
---------------------------- C (7) '21. 0 0
K(7)'1 0.00 11/0 SA(7)-.000100 11-1
C(6)=IS.0 0
D(6) -I .6 11 K (6) =1 0.00 MID SA(6)·.000100 M-I
--0::-:":(5"")-.-.""'S....,."I1--------.,.,K"""(S"'),....-:1"'0,...•..,.O..,.0-:11""'/0 C (5) = I 2.5 D SA(S)-.OOOIOO M-I
::R1l:B!![flI-~.SB:8==::i~:B!!£l:ffiI8~.m88Ei8~jB ~ !~! ~ ~ 0 8D =, UI!I •. 8881118 8-1
D(2)·1.8 11 K (2) -10.00 11/0 SA(2)·.000100 M-I
0(1) .... 6 H KOl.'o.OO Hlp C(1)s12 .. 0 0 SA(') •. ooOloo HEI
WEEKS -2- PUMPING TEST FOLLOWEO BY A RECOVERY TEST. O· 180 113/0
Fig. 5.2. Drawdowns calculated for a pumping test fol1owed by a recovery test in a
confined transversely anisotropic aquifer (model of Hantush-Weeks)
a constant discharge rate (180 m3 /d) which is followed by a recovery test. The pumping
test lasts only hundred minutes.
The drawdown and the residual drawdown are considered in ten observation wells.
Five observation wells have their screen in the middle of the three lowest meters of the
aquifer where the pumping well screen is situated. This corresponds with layer 2 of the
numerical model. The centers of the short screens of the other five observation wells are
situated one meter below the top of the aquifer or in layer 8 of the numerical model.
These two different series of observation wells are at the same distances from the pumped
well, namely, 6.25m, 25 m, 50 m, 100 m and 200 m.
The time discharge rate curve is here discretized in three periods with a constant
discharge rate. The first period corresponds with the time before the start of the pumping
test with discharge rate equal to zero. The second period corresponds with the pumping
test period itself between the times 0 and 100 minutes (T(l) =0 and T(2) = 100 min) and
with a constant discharge rate, QT(2) = 180 m3 /d. The third period is situated between
the times 100 and 1000 minutes (T(3) = 1000 min) and the discharge rate during the
recovery test is equal to zero (QT(3)= 0 m3 /d). After the stop of the pumping test the
residual drawdowns are calculated until 1270 minutes after the start of the pumping test.
The results of the calculations are represented in time-drawdown graphs and in
distance-drawdown graphs (Fig. 5.2). From these results one can remark that the three
closest observation wells in the middle of the dire~tIy pumped part of the layer shows a
sharp decline just after the shut down of the punip. In the other observation wells, the
residual drawdown still increases just after the sh~~ down of the pump. It is remarkable
that the residual drawdown in the two observation wells at 200 m from the pumped well
still increases until a hundred minutes after the stop of the pump.
The second example of a pumping test with a variable discharge rate is a test
where the rate changes as a block wave. This pumping test is also carried out in a
transversely anisotropic aquifer as assumed in the model of Hantush-Weeks (see Sect.
3.5). The conceptualization and the hydraulic parameters, which are used to simulate first
the drawdown with the numerical model for a constant discharge rate of 180 m3/d, are the
same as in the first example. The drawdown and the residual drawdown are considered in
six observation wells: three observation wells in layer 2 and three observation wells in
layer 8 of the numerical model. The considered distances of the two series of observation
wells are the same: viz. 25 m, 100 m and 200 m from the pumped well. The considered
block discharge rate is characterized by three pumping periods of 100 minutes with are
separated by two periods without pumping of 100 minutes. After these five periods the
pumping stopped. So, the number of considered discharge rate changes is equal to seven.
T(1)=O., T(2)=100., T(3)=200., T(4)=300., T(5)=400., T(6)=500., T(7)=1270.
(5.3)
OT(l)"O., OT(2) "180., OT(3)"O., OT(.) "180., OT(5)"0., OT(6) "180., OT(7)"0.
The residual drawdowns are calculated until 1270 minutes after starting the pump.
Chapter 5 / Funher developments of pumping test model 181
ORAWDOWCH) DRAWOOWCK)
100
10- 1 1 ~
10- 2 10' 2 \\
~ :::--l_\
ralO I1IN
r-l0 HIN
100 '0 ' 102 TII'£(I1IN) 100 10 1 t02 0tSTANCE(tn

LAYER 8
DRAWOOWN Utl
100 1.0 +--':"'-"""""'1''''''',.,.....-+---4
100 10 1 102 TJttE(HIN) 100 10 1 102 OISTAMCE tfO

LAYER 2
50=.200
0(9)=S.0 M K(9)=10.00 MID SA(9)=.000100 M-I
C(8)=1000ooooa~e~9~-----------
D(S)-2.0 M K(S)-IO.OO MID SA(S)-.OOOIOO M-I
C(7)=21.0 0
0(7)=2.2 M K (7) =10.00 MID SA(7)=.000100 M-l
C(S)=19.00
0(6) -I .6 M K(S)=IO.OO MID SA(S)=.OOOIOO M-I
o (S) =.9 M K(S)=IO.OO MID C(S)=12.5 0 SA(S)=.OOOIOO M-l
B!!l=.s 8 ~t!j;18.88 8;8 81~l~~1~080 §X!!j;.888188 8-1
D (2) -I .8 M K(2)=10.00 MID SA(2)=.000100 M-I
O(J)~.6 H KIll=la.aa H2o C(1)=12.0 0 SA(l)=.OOOlao H-I
WEEKS -3- PUMPING TEST WITH A BLOCK WAVE DISCHARGE RATE
Fig. 5.3. Drawdown of a pumping test with a block wave discharge rate change in a
confined transversely anisotropic aquifer (model of Hantush-Weeks)
The results of this second example are represented in Fig. 5.3. Studying these
results one can conclude that the fluctuation of the drawdown is very dependent on the
distance of the observation well from the pumped well. The observations in the directly
pumped part of the aquifer at twenty-five meters from the pumped well show the largest
amplitude. At the same distance from the pumped well but in the top of the aquifer the
amplitude is much smaller. At a hundred meters from the pumped well, the fluctuations
in the upper and lower part of the aquifer are not very different. The drawdown fluctu-
ations in the upper and lower part are about the same at two hundred meter from the
pumped well. From the start of the test until a hundred minutes after the last pumping
period the drawdown increases continuously. Here, only the rate of the increase fluctuates
in time.
The third example of a pumping test with a variable discharge rate is a test where
the rate changes as in a step drawdown well test. In this example the same observation
wells are considered in the same transversely anisotropic aquifer as in the first example.
Also, the same conceptualization and same hydraulic parameters are used for the calcula-
tion of the drawdown corresponding with a constant discharge rate of 180 m3 /d. Only the
evolution of the discharge rate is different. During the first hundred minutes of pumping
the discharge rate is equal to 90 m3 /d. The following 400 minutes water is pumped with a
rate of 180 m3 /d. In the last considered pumping period the discharge rate is equal to 360
m3 /d. Consequently, four different discharge rate steps are considered.
T(l)=O., T(2)=lOO., T(3)=500., T(4)=1270.
(5.4)
QT(l)=O., QT(2)=90., QT(3)=180., QT(4)=360.
Also, here the drawdowns are calculated until 1270 minutes after the start of the
pumping. The results of the third example are represented in Fig. 5.4. In the three
observation wells that are situated in the directly pumped part of the aquifer (layer 2) and
at the smallest distance from the pumped well show a step drawdown curve which is also
typical for drawdowns in a pumped well (see Sect. 5.3). This curve is characterized by a
strong increase of the drawdown each time that the discharge rate changes. The curve
shows the smallest increase at the end of each pumping period, just before the change of
the discharge rate. The time drawdown curves in the other observation wells show a
smaller variation in drawdown increase.
DRAWDOWNOU DRAWOOWNcro
.00 -L----+----~----,. .00 *----~-"""':_l---__.,j
IO-l't-_ _ _I -_ _....~~.,/£'_J
.0-2'1-_ _ _1#.1--A--l1if:..._ _'_J
.00 '0' 102 TJt£(HIN) 100

LAYER
'0-2.bl--1--I-~:"--~-----J
102 TltE (HIN) .00 '0' 102 OtSTAMCE uo

LAYER 2
SO=.200
D(9)=5.0M K(9)=10.00 MID SA(9)=.000100 M-I
C(8)=IOOOOOOoa 8 B
K (8) -I O. 00 MID SA(S)-.OOOIOO M-I
C (7)-21. 0 D
0(ll=2.2 M K (ll=1 o. 00 MID SA(7)-.000IOO M-I
CIS) =19.0 D
D (S) =I.S M K (S) -10.00 MID SA(S)=.OOOIOO M-I
0(5)=.9 M K (5) =10.00 MID C (5) =1 2.5 D SA(5)=.000100 M-I
B!3j;.s 8 ~!!j:18.88 8iB g!~l:'~~08D §g!3j;.888188 8-1
0(2) =1 • S M K (2) =10.00 MID SA(2)=.000IOO M-I
DCO-.6M K())=\O.oo Hlp C(1l=12.0 D SA(l)s.OOOlOD H-I
WEEKS -4- PUMPING TEST WITH INCREASING DISCHARGE RATE
Fig. 5.4. Drawdown of a step drawdown pumping test in a confined transversely

anisotropic aquifer (model of Hantush-Weeks)
184 Chapter 5 I Further developments of pumping test model
5.2 Drawdown in a laterally anisotropic aquifer
In laterally anisotropic aquifers the hydraulic conductivity depends on the direction in the
horizontal plane. Here, the horizontal conductivity can be described by means of three
hydraulic parameters: the effective horizontal conductivity Khe , the lateral anisotropy ."fm
and the angle (J between the north and the principal direction of maximal horizontal
conductivity turning to the east (Fig. 5.5). The effective horizontal conductivity is equal
to Khe=';K"".Khy with Kbx and Khy the horizontal conductivity in the principal directions x
(maximum value) and y (minimum value). The lateral anisotropy ."fm is the square root of
the ratio between the maximum and minimum conductivity. The angel (J is measured in
radians. In these aquifers lines of equal drawdown are concentric ellipses around the
pumped well.
Laterally anisotropic aquifers can occur in fractured rocks (Long et aI., 1982) as
well as in water-laid sedimentary deposits, i.e. fluvial, clastic lake, deltaic and glacial
outwash deposits. The horizontal conductivity in the direction of flow during sedimen-
tation tends to be greater than the conductivity in the direction perpendicular to this flow
(Kruseman and De Ridder, 1990). Based on a numerical simulation of flow in fractured
rocks Long et aI., (1982) concluded that fractured rocks do not always behave as
homogeneous porous mediums which are lateral anisotropic and have a symmetric
conductivity tensor. The fractured rocks behave more like an equivalent porous medium
which is laterally anisotropic and have a symmetric conductivity tensor if the rocks show
following four characteristics: the fractured density is rather large, the fracture aperture is
constant rather than distributed, the fracture orientation is rather distributed than constant
and the tested sample size is rather large. Increased fracture interconnectedness enhances
the validity of the equivalent continuum (Smith and Schwartz, 1984).
In this part a method to calculate the drawdown in a laterally anisotropic aquifer is
described which is based on the results of the axi-symmetric two-dimensional numerical
model. By the introduction of the apparent distance, the drawdown calculated with the
AS2D numerical model can easily be transformed to drawdown that appears in a laterally
anisotropic medium. By this transformation it becomes also possible to calculate the
drawdown in a three-dimensional anisotropic medium. In such a medium one of the
principal axes of the three-dimensional hydraulic conductivity tensor is vertical whereas
the two other principal directions are in the horizontal plane. With this method one
assumes, however, that the lateral anisotropy ."fm and the angle (J, which define the
principal direction of maximum horizontal conductivity, are the same for all the layers of
the numerical model while the vertical hydraulic conductivity can be layer dependent.
Chapter 5 I Further developments of pumping test model 185
Fig. 5.5. Representation of the angles required to characterize the horizontal conductivity
(angle 8) and the location of the observation wells (angles (Xi) in a laterally anisotropic
groundwater reservoir. All angles are measured turning from north to east. The ellips
represents a drawdown countour line around the pumped well PP.
To compare the drawdown calculated by the axi-symmetric model with the observed
drawdown, the apparent distance, aro ' is introduced. This factor can be deduced from the
equation given by Hantush (1966) for the drawdown in an anisotropic semi-confined
aquifer.
s (5.5)
where Q is the discharge rate, KheD is the effective transmissivity, W(u' ,ro/LJ is the well
function of an anisotropic semi-confined aquifer and ro is the real distance from the
pumped well to the observation well. The dimensionless parameter u' is defined as
(5.6)
U' =
where S is the elastic storage coefficient, t is the time since the start of the pumping test
and KboD is the transmissivity in a well-defined direction 0 which forms an angle 8-cx o
with the principal direction x and L.~";K..Dc where c is the hydraulic resistance of the
semi-pervious layer.
The relation between the horizontal conductivity in the direction 0, the horizontal
conductivity in the principal direction x and the angle 8-cx o between these two directions
(Fig 5.5) is:
(5.7)
The above given Eq 5.5 can now be reformulated in function of the apparent distance aro
or ar.~r.";K..IK.. ~r.JcoS'-(O-a.,)I.;m+sin2(O-a.,) . .;m
Q ar
s = --=---- W(u"-LO) (5.8)
41tK,,)) •
with ( L.~";K,..Dc ).
The dimensionless parameter u' can now also be formulated in function of the apparent
distance aro and of the effective horizontal conductivity Kbe .
U' = (5.9)
4KJJt
The two parameters KbJ) and Lo of Eqs. 5.5 and 5.6, which are both function of
the direction, are now replaced by only one parameter which is direction dependent aro
and two parameters KheD and Le, which are direction independent. Eqs 5.8 and 5.9 are
similar to the equations that describe the drawdown in an isotropic semi-confined aquifer
(Hantush, 1960). The only difference is that the distance from the pumped well r is
replaced by the apparent distance aro' From this similarity between the equations, one can
learn that the drawdown of a laterally anisotropic medium is derivable with an axi-
symmetric numerical model when one considers the apparent distance instead of the real
distance. Consequently, one assumes, however, that each layer in the numerical model
has the same lateral anisotropy and the same principal directions.
If the drawdown in a laterally anisotropic groundwater reservoir should be calculated, two

additional hydraulic parameters are required. These parameters are the lateral anisotropy
..Jm and the angle () between the north and the principal direction of maximal horizontal
conductivity. The angle should be measured in radians turning to the east. In the case that
a lateral anisotropy is treated, the different values of the horizontal conductivity corre-
spond now with the values for the effective horizontal conductivity of the different layers.
The lateral anisotropy ..Jm (ANISOT) and the angle () (FHI) should be included in the file
name1.pap(u) if one wants to deal with lateral anisotropy (see Tables 4.3 and 4.4).
When the observed drawdowns are compared with the calculated drawdowns, the
observed wells are here not only characterized by their distances from the pumped weIl
but also by their directions toward the pumping weIl. This direction 0 is here also
characterized by the angle a o between the north and the direction pumping weIl -
observation well. These angles are additional parameters which characterize each
observation weIl and must be entered the file name1.dap(u) (see Tables 4.5 and 4.6).
5.2.3 Example of a pumping test in a laterally anisotropic aquifer
The chosen example corresponds also with the model of Hantush-Weeks which treats a
transversely anisotropic aquifer. When one assumes that this aquifer is at the same time
lateraIly and transversely anisotropic then the drawdown is calculated for a confined
three-dimensionaIly anisotropic aquifer. In the treated example the aquifer is ten meters
thick. The weIl screen of the pumped well is situated in the lowest three meters of the
aquifer. The lateral anisotropy is equal to two or the ratio between the maximum and
minimum horizontal conductivity is equal to four. The transverse anisotropy is equal to
3.2 or the ratio between the effective horizontal and the vertical conductivity is equal to
ten. The direction of maximum horizontal conductivity forms an angle of 0.85 radians
with the north when one turns to the east. The effective horizontal conductivity is equal to
10 mid and the specific elastic storage is 1. Ox 10-4 m- I . The numerical model is first used
to simulate the drawdown due to a constant discharge rate of 180 m3 /d. In this model the
groundwater reservoir is conceptualized as in the second verification of the numerical
model of Hantush-Weeks (see Sect. 4.6.7 and Fig. 4.17).
With the drawdowns simulated with the numerical model, the drawdowns in six
groups of two observation wells are calculated. The locations of these six groups are
given in Fig. 5.5. The observation wells of one group are situated at a same distance
from the pumped well and in a same direction with respect to the pumped well. The first
well screen of a group is situated in the middle of the directly pumped part of the aquifer
(layer 2 of the numerical model) and the second well screen is placed at the top of the
DRAWDOWN U'D
.~~-----+------~----~----~ .~ ~---+------I----""
.0-·,.L------I---.'JI5....A~>¢...--_I_----___,I .O-·!l-----+--~I._~,...._....J
.0-2!b-__..:::::::~~-l'-iL-1-_ __I_--___,I
'0-2.~
'-10 HIN
'-10 "1111
.00 .0' .02 103 TlHE (HIN) 100 .0' 102 APPARENT
DISTANCE,")
LAYER 8
.~ .0' .02 103 TltE (HIM) 100 .0' 102 APPAREJIT

DISTANCE (1)
LAYER 2
- - - - - - - - - - - - - - - - - SO-.200
0(9)-5.0 M K (9) -1 0.00 MID SA(9)-.000100 M-l
C (8) -9999999. 01-&9--------------

K (8) -10.00 MID SA(B)-.000100 M-l
K (ll-10.00 MID SA(7)-.000100 M-l

C(S)=19.0 D
D (S) -1. S M K (S) = 10.00 MID SA(S)=.000100 M-l
D (5) -.9 M K (5) = 10.00 MID C (5) =12.5 0 SA(5)=.000100 M-l
Bm-.a 8 lI!l =1 B. B8 8iB ~ !~l :~~~OBD §tt!l·.88BI88 8-1
0(2) ·1.8 M K (2) -10.00 MID SA(2)=.000100 M-l
QO)a.6 H Kcll-Io.oo Hlp C(I)-12.0 0 SA(1)=.oo0100 A-I
HORIZONTAL ANISOTROPY=2.00 MEAN DIRECTION OF MAX.CONDUCTIVITY-N~8 42 E
WEEKS -5- DRAWOOWN IN A THREE-DIMENSIONAL ANISOTROPIC ADUIFER
Fig. 5.6. Drawdown in a three-dimensional anisotropic groundwater reservoir: the model

of Hantush-Weeks with laterally anisotropic layers. The wells are located in three
different directions (see Fig. 5.5). Drawdown in the directly pumped part of the aquifer
and drawdown in layer 8 in top of the aquifer which is pumped indirectly.
aquifer (layer 8). Three groups of observation wells are placed rather close to the pumped
well (at 10 m from the pumped well). The other three groups are situated at a distance of
100 m from the pumped well. The observation wells are placed in three different
directions with respect to the pumped well. A group at a distance of 10 m from the
observation wells (1.2 and 1.8) and an other group at 100 m (4.2 and 4.8) from the
pumped well is placed in the same direction to the pumped well. This direction coincides
with the principal direction of maximal horizontal conductivity. Two other groups of
observation wells are placed in the eastern direction with respect to the pumped well: one
group at a distance of 10 m (2.2 and 2.8) and the other group at 100 m (5.2 and 5.8).
The last two groups are placed in the direction of minimum horizontal conductivity: also
here, a distance of 10 m (3.2 and 3.8) and at a distance of 100 m (6.2 and 6.8).
LAYER BArtER 100. HINUTES OF PUHPING LAYER B AFTER 995. HINUTES OF PUHPING
~ ~
~ g
E ~
0
IS !
! ~
; ;
., .,
~
~ :::
.. ...
II II
• •• " liS ". no H. .., ••• •• " liS ". I.t 230
'0' •••
LAYER 2 AFTER 100. HINUTES OF PUHPING LAYER 2 AFTER 995. HINUTES OF PUHPING
~ ~
.. .
~
II II
@
Ie :Il
:s :s
..,
! !
.,
:!
.,
:: ~
~
III III
." liS ". III 2. . . . . .., ~. ." liS IS' 112 230 211 ..' •••
Fig. 5.7. Drawdown (m) contour lines around a pumped well in a 3D anisotropic
aquifer: in directly pumped part of the aquifer (lower two maps, layer 2) and in the
indirectly pumped top of the aquifer (upper two maps, layer 8).
190 Chapter 5 / FUl1her developments of pumping test model
In Fig. 5.6 the calculated drawdown of a confined three-dimensionally anisotropic

aquifer is represented in time-drawdown curves and in apparent distance-drawdown
curves. The apparent distances of the observation wells which are situated on the principal
direction of maximum conductivity (7.071 m and 70.711 m) are smaller than the real
distance ( 10 m and 100 m). The apparent distances of the observation wells which are
situated east of the pumped well (10.741 m and 107.409 m) are only 7.41 % larger than
the real distances (again 10 m and 100 m). The apparent distances of the observation
wells which are situated on the principal direction of minimum horizontal conductivity
(14.142 m and 141.421 m) are considerably larger than the real distances (also 10 m and
100 m). Consequently, drawdowns in observation wells situated on the principal direction
of maximum horizontal conductivity are always larger than drawdowns at the same
distance but situated on an other direction. The smallest drawdowns at a same distance
from the pumped well occur in observation wells that are situated on the principal
direction of minimum horizontal conductivity.
The drawdowns in the upper part of the three-dimensional anisotropic aquifer at a
relatively small distance from the pumped well (10 m) are not strongly dependent on the
direction of the well with respect to the pumped well. This is due to the fact that the
drawdown is not strongly apparent distance dependent close to the pumped well in this
part of the aquifer, which is not directly pumped. This is not true for the observation
wells that are situated at a relative large distance to the pumped well (e.g. 100 m).
In Fig. 5.7 the lines of equal drawdown are represented in four maps. Two maps
represent the drawdowns in the middle of the directly pumped layer (layer 2) and two
maps represent the drawdowns in the upper part of the aquifer (layer 8). The results are
shown at two different times after the start of the pump (after 100 and 995 minutes). The
lines of equal drawdown are here concentric ellipses around the pumped well.
5.3 Drawdown in pumping wells
The drawdown in a pumping well is partially due to the aqUifer loss and partially due to
the well loss. The aqUifer loss is here defined as the drawdown which would occur at the
borehole wall situated at the same depth interval of the well screen if the pervious layer
just outside the borehole wall is unaltered and has the same hydraulic parameters as the
rest of the aquifer. The well loss is here defined as the difference between the drawdown
in the pumping well and the aquifer loss. Consequently, the well loss is a result of
different flow phenomena which occur in the vicinity and inside the pumped well. These
phenomena are the flow of the water inside the well to the pump intake; the flow through
the well screen; the flow through the filter pack which is placed in the annular space
between the well screen and the borehole wall; the flow through the borehole wall itself
which can be clogged by mud cake residues; the flow in the pervious layer in the vicinity
of the well screen which can be altered in flow properties due to drilling and/or well
development. The well efficiency is the ratio of the aquifer loss to the drawdown in the
pumped well expressed as a percentage.
The aquifer loss as defined is directly calculated by the numerical model and
corresponds with the drawdown at a distance from the center of the axi-symmetric grid
equal to the borehole radius. The aquifer loss must now be increased with the well loss to
obtain the drawdown in the pumping well. The well loss is assumed to be proportional to
power N of the discharge rate or equal to C~ (Todd, 1980). The C value is the well loss
function constant. The power N is a constant greater than one. Jacob (1947) suggested
that N=2 might be reasonably assumed. Rorabaugh (1953) pointed out that the power N
can deviate significantly from 2. An exact value for N cannot be stated because of
differences of individual wells.
To evaluate the C value and the power N of the well loss function a step-draw-
down pumping test is required. This consists of pumping a well initially at a low rate
until the drawdown within the well stabilizes. The discharge is then increased through a
successive series of steps. Only, by means of the drawdown observed in the pumping well
and in some observation wells it is possible to find simultaneously the C value and the
power N of the well loss function. If, however, drawdown data in the pumping well and
in some observation well are merely available for a pumping test with only one constant
rate, it is not possible to infer these two hydraulic parameters simultaneously. This is also
the case when the drawdowns measured in the pumped well during a step-drawdown test
are the only available data.
According to Driscoll (1986) it is in this last case possible to distinguish a laminar
term and a turbulent term by the graphical interpretation method of Bierschenk (1964).
The laminar term is proportional with the discharge rate (BQ) and the turbulent term is
proportional to the discharge rate to the second (CQ). The sum of the laminar term and
the turbulent term is equal to the drawdown in the pumping well. Jacob (1947) called the
laminar term the aquifer loss and the turbulent term the well loss. Driscoll (1986) states,
however, that analyses of real wells have shown that this correlation is not correct,
because the BQ term almost always includes a major portion of the well losses and the
CQ2 term occasionally includes some aquifer loss. For this reason computing well
efficiency from a step drawdown test with only observation in the pumped well results in
an erroneous value of the well efficiency when this efficiency is defined as the percentage
of the pumping well drawdown that is attributed to the aquifer loss. For the same reason
an estimation of the transmissivity from the BQ term as proposed by Logan (1964), Hurr
(1966) and Gabrysch (1968) can also be erroneous. In these estimation methods B is put
equal to In(rolrw)I2'IIKbD for steady state conditions where KbD is the transmissivity of the
pumped part of the pervious layer, ro is the distance from the pumped well where the
drawdown is negligibly small and rw is the well radius. When unsteady state conditions
are assumed B is put equal to (2.30/41r~D)log(2.25KbDt/rw2S) where t is the time since
the start of the pump and S is the elastic storage coefficient.
When the drawdowns are only measured in the pumped well during a step draw-
down test, it is then only possible to distinguish a term BQ and a term C~ when N is
192 Chapter 5 / Funher developments of pumping test model
significantly different from one. Adding drawdowns measured in observation well at

different distances to the pumped well makes it mostly possible to distinguish the aquifer
loss, the drawdown due to flow in the aquifer and the well loss or the drawdown due to
flow phenomena in the well and in the vicinity of it. With the above given definition of
the well loss, it is possible that the well loss has a negative value. In this case the
drawdown in the pumped well is smaller than the drawdown deduced from the measure-
ments made in the observation wells with a screen in the pumped pervious layer and the
well efficiency is then larger than 100%. A negative well loss can be encountered in a
pumping well which was extensively developed. By the development of wells in deposits
the finer material is removed from the natural formation surrounding the well screen. So,
a zone around the well screen can be created with a hydraulic conductivity that is much
larger than the conductivity of the natural material. Such zones of larger hydraulic
conductivity can also be obtained by a large variety of development procedures such as
pumping, surging, use of compressed air, hydraulic fracturing and use of explosives.
These procedures are briefly described in Todd (1980).
If one wants to calculate the drawdown in a pumping well, two additional hydraulic
parameters must be known. These parameters are the C value and the power N of the
well loss function C~. The unit of the C value depends on the power N. These parame-
ters should then be included in the file name1.pap(u) (see Table 4.3 and Table 4.4).
The well loss is equal to a height expressed in meter. When one wants to compare
the observed drawdown in the pumped well then one has to indicate the observations
made in the pumped well in the file with the observations (name1.dap(u». Therefore,
first the number of blocks that contains observations made in the pumped well must be
given. This number must be followed by the numbers of the blocks that contain these data
(see Table 4.5 and 4.6).
5.3.3 Example of drawdowns in a pumped well during a step drawdown test
The step drawdown pumping test is made in the transversely anisotropic aquifer as
assumed in the model of Hantush-Weeks (see Sect. 3.5). The groundwater reservoir is
conceptualized as in the second verification of the numerical model of Hantush-Weeks
(see Sect. 4.6.7). The same hydraulic parameters are also used. During the first hundred
minutes of pumping the discharge rate is equal to 90 m3 /d. The following 400 minutes
water is pumped with a rate of 180 m3 /d and during the last considered pumping period
the discharge rate equals 360 m3 /d. For two observation wells at a distance of 8 and 40 m
from the pumped well the drawdown is calculated in the pumped layer (layer 2 in the
numerical model). The drawdowns in the pumped well are calculated assuming that the
power N of the well loss function is two and that the C value of this function is equal to
0.0002 m-sd2 • The assumed borehole radius is 0.2 m. In the directly pumped part of the
layer the drawdown is calculated at the borehole wall. This drawdown should also occur
in a pumped well where the well loss is zero or when there is 100% well efficiency.
OItAwoowcPO ORAWDDWI (1)
I~t------r-----i------t-----, 10' t------+------+----~
'0 ' t-----+-----4-~IPe~!+---_J '0 ' F'\"-"""'::I------+----~
10-1.+------+-----"?f~_"t____>n
10-'e~o
10-~*"---+------I1__--+_--__l
'-'0 "IN
,-, "I"
101 I~ 103 TlttEutlH' 10- 1 100 101 DISTMCEO'O
LAYER 2 x
- - - - - - - - - - - - - - - - - - - - - SO-. 200
0(9)-5.0 H K(9)=10.00 HID SA(9)-.000100 H-I
C (8) -I OOoooooo ... ee-eel------------

0(8)-2.0 H K(8)-10.00 HID SA(8)-.000100 H-l
K (7) =1 0.00 HID SA(7)=.000100 H-l
o (S) -I.S H K[S)-10.00 HID

C (S) ·19.0 0
SA[S)-.000100 H-l
o (S) -.9 H K(5)-10.00 HID C(S)=12.S 0 SA(S)·.000100 H-l
B!!I.J 8 ll!I.18.88 8~B ~!~!:~1~oBo n !!I-.088188 M-I
0(2)-1.8 H K(2)-10.00 HID SA(2)-.000100 H-I
0(1) •. $ H K (j) .)0.00 HID C (I) -12.0 0 SACl) •. o0010D H-I
ORAWOOWN IN PUHPEO WELL WITH AND WITHOUT WELL LOSS
Fig. 5.S. Drawdown in pumped well (1) (rw=O.2m) with a well loss (s=C(f, C=2x10-4
m-5d2) in a partially penetrated transversly anisotropic confined aquifer during a step
drawdown test along with the theoretical drawdown at the borehole well (2) and the
drawdown in the directly pumped part of the aquifer at a distance of 8 and 40 m from the
pumped well.
The results of the calculation are represented in Fig. 5.8. From the drawdown-
distance graph one can derive the well loss. This is the difference between the drawdown
in the pumped well and just outside the pumped well. The well loss is 1.62 m during the
194 Chapter 5 / FUl1her developments of pumping test model
first hundred minutes of pumping with a discharge rate of 90 m3 /d, is 6.48 m during the
second pumping period with a discharge rate of 180 m3 /d and is 25.92 m during the third
step with a discharge rate of 360 m3/d. These differences correspond with the well loss of
the pumped well. The well efficiency at the end of the first pumping period is equal to
61 % (2.564 m aquifer loss, 4.184 m drawdown inside pumped well), at the end of the
second period equal to 45% (5.375 m aquifer loss, 11.855 m drawdown inside pumped
well) and equal to 30% (10.971 m aquifer loss, 36.891 m drawdown inside pumped well)
at the end of the third pumping period.
5.4 Drawdown due to a multiple well field
In some circumstances it is necessary to have an estimate of the drawdown due to

simultaneous pumping in different wells. By application of the rule of superposition it
becomes possible to calculate this drawdown. This requires the conceptualization of the
groundwater reservoir in pervious and semi-pervious layers and the knowledge of the
horizontal and vertical conductivity and the specific elastic storage of the layers as well as
the storage coefficient near the water table. In the best case these hydraulic parameters
can be deduced by the performance of a simple or multiple pumping tests in a preliminary
investigation. Once these data are known, the drawdown due to a pumping with a
discharge rate of Om.. on one well can be calculated with the program sipurS. With this
program the evolution of the drawdown in the different layers is calculated for an axi-
symmetric grid.
With the program multpl it is now possible to calculate the drawdown in a certain
layer and after a certain time of the pumping in a mesh-centered grid with squared
meshes for nw wells which are pumped simultaneously in or in the vicinity of the
considered grid:
(5.10)
where s.(xm,yn,t) is the drawdown in layer I at the x-coordinate x", of the mth row of the
grid, the y-coordinate Yn of the nth column and at the time t after the start of the
pump,
xp,yp,Qp are respectively the x- and y-coordinates and the discharge rate of
the pth pumped well,
sAs2D(i,rp,t} is the drawdown in the layer I at a distance from the pumped
well rp at the time t after the start of the pump calculated in the axi-sym-
metric grid with the discharge rate Q".x.
Chapter 5 / Fulther developments of pumping test model 195
The discharge rate <1n.x which is used to calculate the drawdown with the program sipur5
is at least the largest value of the different discharge rates Qp of the different wells p con-
sidered in the program multpl. As with pumping tests with variable discharge rate, the
rule of superposition and consequently Eq. 5.10 is only applicable if the groundwater
flow equation is linear. This rule of superposition is, however, not applicable in a number
of cases. A first case is that the pumpings on several wells result in a drawdown which is
considerable in respect of the thickness of the uppermost layer whereas its horizontal
conductivity is sufficiently large so that the horizontal flow is no longer negligibly small
in comparison with the total flow. Therefore, the obtained drawdown in the uppermost
layer must always be compared with the total thickness of this layer. If this drawdown is
rather large as well as the horizontal conductivity the results of the calculation can only
be considered as a first approximation. The rule of superposition is also not applicable if
there is a considerable interaction between the pumped groundwater reservoir and the
surface waters. Here, the pumpings can reduce the flow toward the rivers or can even
cause an irrigating river over a certain length. In this case the calculated drawdowns
obtained by simple superposition are overestimated.
The drawdown due to the pumping on one well is first calculated for different layers in an
axi-symmetric grid with the program sipur5. The results of this calculation are stored in a
data file name1.oup(u) where name I is the name of the treated problem. With the
program multpl the drawdown is calculated for a set of chosen layers and a set of chosen
times after the start of the pump. The additional input data required for the multpl
program can be given interactively or by means of an input file name1.ml(p)2. When the
data are given interactively, a data file name1.ml(p)1 is made. This file can then be
renamed to name1.ml(p)2 and used after the necessary adjustments to treat a similar
problem. The sequence and the format of the parameters in the files name1.ml(p)1 and
name1.ml(p)2 are given in Table 5.l.
The required additional input data are shortly mentioned and described. Firstly,
the number of layers must be given for which the drawdown must be calculated followed
by their corresponding numbers. Secondly, the number of times must be given for which
the drawdown must be calculated followed by the corresponding times after the start of
the pumping. After the indication of the layer(s) and the time(s) for which the drawdown
should be calculated, three parameters must be given which define the square-meshed
grid. These parameters are the total width and total length of the grid and the side of the
squared mesh. The unit of these three last parameters is meter. The following parameter
defines the scale of the map with the lines of equal drawdown which can be plotted after
the calculations. The next input parameter is the number of pumping or injection wells.
This parameter is followed by the x- and y-coordinates (in m) and the discharge rates of
the different wells (in m3 /d). The origin of the used coordinate system is situated on the
left lower corner of the left lower mesh of the grid. The axises of the coordinate system
are parallel with the axises of the meshed grid. So, the x-coordinate x." of the mth row of
......
~
Table 5.1. Sequence and format of parameters in file namel.ml(p)l or namel.ml(p)2

Line Variable Description and remarks Limits and/or units Format
I ILAY,lTlM number of demanded layers, number of demanded times 01 ,,;ILAY,,; 10,01 ,,;lTlM,,; 10 212
2 LAYOL(JL),IL=I,ILAY demanded layer number 01 ,,;LAYOL(IL) ,,;N9' 1012
3 TlME(lT), IT= l,lTlM demanded times TI *IO"A";TlME(lT) ,,;T2' 10FB.0 g
minute ~
l?
....
4 WIDTH total width of the squared-mesh grid 3*SISQUC ,,;WIDTH,,; lOO*SISQUC FB.2
meter
...
"-
5 LENGTH total length of the squared-mesh grid 3*SISQUC ,,;LENGTH,,; lOO*SISQUC FB.2 ~
meter s.
6 SISQUC side of the squared mesh RI*IO"A";SISQUC ";RI*ION9'Al2 FB.2 ~
meter !}
~
7 SCALE scale of map dimensionless FB.2
B NUMWEL number of pumping or injection wells 01,,;NUMWEL,,;99 12 ~
9 XWELL(J), YWELL(J),QWELL(J) J=I x- and y-coordinates of well and the discharge rate m, m, m'/d 3FB.0
(positive)
i
~
10 J=2 or recharge rate (negative). 3FB.0 ~
... "" .
B+ J=NUMWEL 3FB.0 ~~.
NUMWEL
l?
9+ NBCCH flag indicating vertical plane of discontinue lateral conductivity 01 ,,;NBCCH ,,;02 12 ~
NUMWEL change, 00 if there is no vertical plane ~

or 0 I if there is a vertical plane It
10+ XVBI,YVBI,XVB2,YVB2,K2KI x- and y-coordinates of two vertical lines of vertical plane of m, m,m, m, dimensionless SFB.2
NUMWEL discontinue lateral conductivity change:
XVBI,YVBI and XVB2,YVB2
and ratio of conductivities on positive side of vertical plane
to conductivities on negative side of vertical plane K2KI
1 for detailed description of variables see Table 4.3

the grid is m-112 times the side of the square mesh and the y-coordinate Yn of the n1h
column is n-l12 times this side. It is possible to introduce wells in the problem which are
situated outside the squared meshed grid but which has an influence on the treated area
which is covered by this grid. The discharge rates of injection or deep infiltration wells
are negative.
5.4.3 Drawdown due to pumping on wells in laterally isotropic layers
In the first example the drawdown is calculated due to the pumping on three wells in a
groundwater reservoir which has the following conceptualization. The groundwater
reservoir is composed of two pervious and two semi-pervious layers. The reservoir is
bounded below by a layer with a very small conductivity and above by the water table
(Fig. 5.9). The lowest considered layer is a pervious layer with a thickness of 10 m and a
horizontal conductivity of 10 mid. This layer is covered by a semi-pervious layer. This
layer is 8 m thick and have a vertical conductivity of 0.1 mid. The second pervious layer
has a thickness of 4 m and a horizontal conductivity of 2 mid. The uppermost layer of the
groundwater reservoir, which is bounded above by the water table, has also a thickness of
four meters and a vertical conductivity of 0.02 mid. The storage coefficient near the
water table is equal to 0.16. All layers have the same specific elastic storage and the same
transverse anisotropy, respectively 5xlO·s m- 1 and l.4l.
In the numerical model the groundwater reservoir is discretized in nine layers. The
lowermost layer in the numerical model (layer 1), which is always bounded by an
impervious boundary, corresponds with the lowermost pervious layer (Fig. 5.9). The
covering semi-pervious layer is discretized in five layers, layer 2, 3, 4, 5 and 6 of the
numerical model. The uppermost pervious layer corresponds with layer 7 of the numerical
model. The uppermost semi-pervious layer is discretized in two layers: layers 8 and 9 of
the numerical model. With program sipur5 the drawdown is calculated in an axi-
symmetric grid due to pumping on one well with a discharge rate of 120 m3 /d. The
screen of the pumped well is situated in layer 1 of the numerical model. In Fig. 5.10
some of the calculated drawdowns are represented in time-drawdown and in distance-
drawdown graphs.
With program multpl the drawdown in the pumped layer and of the water table is
calculated due to the simultaneous pumping on three wells. All three wells have screens
in the lowermost pervious layer. The width and length of the considered mesh-centered
grid are both 1000 m. The sides of the squared meshes is 10 m. The x- and y-coordinates
of the pumped wells are as follows:
XWELL(1),350m. YWELL(l),400m. QWELL(l),l20m'fd

XWELL(2),650m, YWELL(2),400m. QWELL(3),120m'fd (5.11)
XWELL(3),500 m. YWELL(3),600m. QWELL(3),180m'fd
SchemaUsaUon of
groundwater reservoir Layers and hydraulic parameters In numerical model
dept.h em)
o 5.=0.16
laye,. 9 K h (9) - 0.04 mid
c(e)=t76.d
ayer 8 (8l = 0.04. mid c(7)=25.0d
layer 1 K h (7) = 2.0 mid
c(8)=lO.Od
layer 6 K h (6) .., 0.2 mid
10 Cl(5)=20.0d
layer 5 K h(6) - 0.2 mid
c(.)=18.6d
layer. Kh(4) ,.. 0.2 mid
c(3)=15.0d
layer 3 Kh(3) - 0.2 mid
15 c(2)=11.5d
layer 2 K h (2) - 0.2 mId
e(1) .. 6.0d
20
pervious layer 1 K h (l) '" 10 mid
layer
25
Fig. 5.9. Schematisation of the groundwater reservoir to calculate the drawdown due to
pumping on several wells and discritisation in layers of numerical model
The drawdowns in the directly pumped layer (layer 1 of the numerical model) and
of the water table (layer 9 of the numerical model) are calculated for two different times
after the start of the pumping, after l(f and 106 minutes after the start of pumping. This
is approximately after two months and two years after the start of the pumping. The
results are represented in maps with lines of equal drawdown plotted (Fig. 5.11) by the
plot program mulpl2.
5.4.4 Drawdown due to pumping on wells in laterally anisotropic layers
In this example the drawdown is calculated due to pumping on three wells in a ground-
water reservoir that is composed of laterally anisotropic layers. The conceptual-ization of
the groundwater reservoir is the same as in Sect. 5.4.3 as well as the discretization of the
layers in the numerical model (Fig. 5.9). The effective horizontal conductivities of all the
layers are now equal to the horizontal conductivity of the layers in the former part. The
lateral anisotropy of all the layers is the same and is equal to 1.41. So, the maximum
horizontal conductivity is equal to twice the minimum horizontal conductivity. The
direction of maximum horizontal conductivity is also the same for all the layers. This
direction forms an angle of one radian with the northern direction. The angle is measured
DRAWOQWU'O 00...,.,.,'"
.o0t-______+-______+-______+-______+-______+-____ ~
.00 t-------+-------+-----~
10-1t-------+-------+-------+-------+-7IJ~7"lT"_,,c...?""iI
,.-
'00 '0' '0'
..._'"'
TItE.mUI)
LAYER
ORAWOOlAI (H)
.00 .00
10- 1 10- 1
T-l0 HIN
Y-l0 'UN
Y-IO I1IN
'0- 10- 2
T"
"'M
... .03 .04 TUtEC"UI)
LAYER 7
...
T-lO .tlN
ORAVDOWCrD DRAWOOWJf<tO
.00 ...
Tll'tEutllO
LAYER
DRAWOQW(1)
'OO~_II
.03 .04 TUtEotllO

LAYER
Fig. 5.10. Time-drawdown and distance-drawdown curves for the layers 4, 7 and 9 of
the numerical model calculated for a pumping rate of 180 m3 /d in layer 1
LAYER. AFTER 100000. HIMUTES OF PUI1PIMG LAyER. AFIER .tI•••. "IMUIES OF PUI1PIMG
I I ~O~.
I I
.~
~ ~
! I
I I
! !
•
!
•
!
! !
100 20CI HO 400 SOCI IGO 100 100 HO 100 20G 300 400 500 100 100 100 toO
LAYER I AFTER 100000. HIMUIES OF PUI1PIMO LAYER I AFIER tttltl. HI MUlES OF PUnPIMO
I I
"[5;
I I
~ ~
! ! ,. <p.
I
I
I
I . .
•
!
§
!
~ ~
.N 200 HI 400 aGO HO 100 100 toO 100 aoo 100 400 lOG 100 100 100 100
Fig. 5.11. Drawdown (m) in directly pumped layer (1) and of water table (layer 9)
around three pumped wells in laterally isotropic groundwater reservoir (schematisation see
Fig. 5.9) after HP and 106 minutes of pumping
starting from the northern direction and turning toward the east. The transverse aniso-
tropy is chosen so that the vertical conductivity is twice smaller than the effective
horizontal conductivity. Consequently, in the given example the layers are three-
dimensionally anisotropic. The values for the specific elastic storage and the storage
coefficient near the water table are also the same as in the former part.
During the first run with the program multpl the drawdown is calculated for a
square grid with a side of 1000 m. In the grid squared meshes are considered with a side
of 10 m. Also, here the lowermost layer is pumped by means of three wells. The
coordinates and the discharge rates are the same as in Sect. 5.4.3. The drawdowns in the
directly pumped layer (layer 1 of the numerical model) and of the water table (layer 9 of
the numerical model) are calculated for two different times after the start of the pumping,
namely after HP and 1()6 minutes.
LAYER 8 AFTER 89998. IIINUTES OF PU"ING LAYER 9 AFTER 999999. IIINUTES OF PU"ING
C;O'
o 100 200 300 400 SOO eoo 100 100 '00 100 aoo ~oo 400 100 100 100 100 900
LAYER I AFTER 88899. IIINUTES OF PU"ING LAYER I AFTER 888989. IIINUTES OF PU. . ING
o 100 200 300 400 lOG 100 700 aoo 100 o 100 200 ~OO 400 lDO 100 700 100 900
Fig. 5.12. Drawdown (m) in directly pumped layer (1) and of water table (layer9)
around three pumped wells in laterally anisotropic groundwater reservoir after l(f and 106
minutes of pumping. Location of grid used in Fig. 5.13 to zoom on the area between the
wells
The results are represented in maps where the lines of equal drawdown are plotted
(Fig. 5.12). Studying these maps one remarks immediately the large mutual influence of
the pumping wells which are aligned following the direction of maximum hydraulic
conductivity. The directions of the lines going through the third well and the two other
wells with strong mutual influence deviate strongly from the direction of maximum
horizontal conductivity. Therefore, the mutual influence of this third well on the two
other wells is small. From Fig. 5.12 it is, however, difficult to deduce the exact place of
the stagnation points or the points without horizontal groundwater flow in it. To locate
them better, a second run of the problem is made with a grid which covers a smaller area
and has much smaller meshes. The sides of the squared meshes are now equal to three
meters. The rectangular grid with a length of 300 m and a width of 201 m is situated
between the three pumping wells. The lower left corner of the grid, which corresponds
with the origin of this grid, is situated in well number 1. The new x- and y-coordinates of
the pumped wells become now:
XWELL(1)=Om, YWELL(I)=Om, QWELL(I) =120m'ld
XWELL(2)= 300 m, YWELL(2)=Om, QWELL(2) = 120m'/d (5.12)
XWELL(3)= 150m, YWELL(3)=200m, QWELL(3) = 180m'ld
The drawdown for different layers is represented in Fig. 5.13 for two different times after
the start of the pumping. From these figure one can better deduce the place of the
stagnation points.
LAYER 9 AFTER 100000. MINUTES OF PUMPING LAYER 9 AFTER 999999. MINUTES OF PUMPING
N
o
o
..
~
o
~~~~~~~~~~
o 20 40 60 80 100 121141161181201221241261281 o 20 40 60 80 100121141161181201221241261281
LAYER 1 AFTER 100000. MINUTES OF PUMPING LAYER 1 AFTER 999999. MINUTES OF PUMPING
o 20 40 60 80 100 121 141 161 181 201 221 24. 261 281 o 20 40 60 80 100 121141161181201221241261281
Fig. 5.13. Drawdown (m) in an area situated between the three pumped wells in a
laterally anisotropic groundwater reservoir for the same problem as represented in Fig.
5.12. The location of the used grid is given in Fig. 5.12.
5.5 Drawdown in groundwater reservoir with lateral bounds
Until now, it was assumed that the layers of the groundwater reservoir have an infinite
lateral extent and that they are laterally homogeneous. By the introduction of imaginary
wells a groundwater reservoir of a finite lateral extent or with lateral changes in the
hydraulic conductivity can be transformed into a groundwater reservoir of infinite lateral
extent. Consequently, the drawdowns obtained with the ax i-symmetric numerical model,
where infinite lateral homogeneous layers are assumed, can be used to calculate the
drawdowns in aquifers with a finite lateral extent or with a lateral conductivity change.
First, the way is described how one can calculate the drawdown in a groundwater
reservoir with a finite lateral extent. In the first example the drawdown is calculated in a
groundwater reservoir that is laterally bounded by a vertical impervious boundary. In the
second example a groundwater reservoir is considered which is laterally bounded by a
vertical constant head boundary. Both examples are often described in the general
hydrogeological literature (Freeze and Cherry, 1979, Todd, 1980 and Domenico and
Schwartz, 1990). In the third example, problems are treated with several vertical
boundary planes. This simple method of images can only be applied on laterally isotropic
groundwater reservoirs or on laterally anisotropic groundwater reservoirs of which the
principal directions are parallel to the considered bounds.
5.5.1 Drawdown in groundwater reservoir with lateral impervious boundary
In this example the considered groundwater reservoir is composed of laterally isotropic

layers and is bounded above by a water table and below by an impervious layer.
Laterally, the groundwater reservoir is bounded by a straight impervious boundary. Such
case can occur e.g. near the impervious bounds of an alluvial flood plain, near dykes
consisting of material of very small conductivity or near artificial walls. With the method
of images it is possible to calculate the drawdown in this groundwater reservoir based on
the results of the axi-symmetric model. Each well which pumps water from this ground-
water reservoir must now be complemented with an imaginary pumped well. This
imaginary pumped well is located just opposite of the vertical impervious boundary and at
a same distance from the boundary. These image wells operate simultaneously and at the
same rate as the real wells (Todd, 1980).
In the example the drawdown is calculated due to the pumping on three wells in a
layered groundwater reservoir that is bounded by a vertical impervious boundary. The
conceptualization of the treated reservoir is the same as in Sect. 5.4.3 (see Fig. 5.9). The
coordinates and recharge rates of the three wells with screens in the lowermost layer
(layer 1 of the numerical model) are as follows:
XWELL(I)=3S0m, YWELL(1)=400m. QWELL(I)=120m'{d

XWELL(2)=6S0m, YWELL(2)= 4OOm, QWELL(2)=120m'{d (5.13)
XWELL(3)=SOOm, YWELL(3)= 6OOm. QWELL(3)=ISOm'{d
The vertical impervious boundary coincides with the y-axis of the coordinate system
(x=O). This boundary coincides with the right limit of the considered meshed grid.
Consequently, the coordinates and the discharge rates of the image wells are:
XWEUl(l) = -350m, YWELU(l)= 4OOm, QWEUl(l)= 120m'/d
XWEUl(2) = -6SOm, YWELU(2)=400m, QWEUl(2)= 120m'/d (5.14)
XWEUl(3) = -500m, YWELU(3) = 6OOm, QWEUl(3) = 180m'/d
LATER t AFTER 100000. HIMUTES OF PUlI'lMC LAYER t AFTER tltUO. "I"UTES OF PUlI'lMO
I I
I I
0
! !
I I
II II
! !
II II
! ! .tt'!l
! !
100 ~ 300 400 100 aoo 100 100 900 'DO 200 300 400 soo 100 roo 100 100
LATER I AFTER 100000. HIMUTES OF PUlI'lMO LAYER I AftER 9999tt. HIMUTES OF PIM'IMO
I I
I i
Q~
! !
I ! o ,.
II II
! ! o •
II II
! !
.ft'!l
! !
100 toO .100 400 sao 100 100 100 toO 100 200 300 400 100 100 100 too .00
Fig. 5.14. Drawdown in directly pumped layer (1) and of the water table (layer 9)
around three pumped wells in layered groundwater reservoir laterally boun ded by vertical
impervious boundary after 105 and 106 minutes of pumping
The drawdowns are calculated for a squared grid with a side of 1000 m. In this grid the
squared meshes have a side of 10 m. The x- and y-coordinates of the grid points range
between 5 and 995 m. For two different times after the start of the pumping, lOS and 106
minutes, the drawdowns in the directly pumped layer (layer 1 of the numerical model) as
well as the drawdown of the water table (layer 9 of the numerical model) are calculated.
The results of the calculations are shown in four maps where the lines of equal
drawdown are represented (Fig. 5.14). Studying these maps one remarks immediately that
the lines of equal drawdown are perpendicular to the impervious boundary (x=O). Along
the boundary the wells offset their imaginary well, causing no flow across the boundary,
which is the desired condition. The drawdown along this boundary is now equal to twice
the drawdown in Fig 5.11. In this figure the results are represented for the same pumping
in the same layered groundwater reservoir but without the vertical impervious boundary.
The drawdowns around the pumping wells with the same discharge rate are in this
example slightly different from each other. The drawdown in the well which is situated
closer to the impervious boundary is slightly larger than the drawdown in the well which
is situated farther from the impervious boundary.
5.5.2 Drawdown in groundwater reservoir with lateral constant head boundary
In this example the same pumping is considered in the same layered groundwater
reservoir as in the former part. The only difference with this example is that the vertical
boundary which is situated along the y-axis (x=O) is here not an impervious boundary but
is a constant hydraulic head boundary. The drawdown in this groundwater reservoir can
be calculated with the aid of the results of the ax i-symmetric model. Here, each pumped
well must now be completed with an imaginary well. This imaginary well is located just
opposite of the vertical constant head boundary at a same distance from the boundary.
These image wells operate simultaneously and at the same discharge rate but with an
opposite sign. Here the image of a pumped well is an injection well and the image of an
injection well is a pumped well (Todd, 1980).
The coordinates and the recharge rates of the three wells with screens in the
lowermost layer (layer 1 of the numerical model) are given in Eq. 5.13. Because the
vertical boundary with a constant hydraulic head coincides with the y-axis (x=O) the
coordinates and the discharge rates of the image wells are as follow:
XWELU(l) = -350m, YWELU(l) =4OOm. QWELU(l)= -l20m'/d
XWELU(2) =-650m. YWELU(2) =4OOm. QWELU(2)= -120m'/d (5.15)
XWELU(3) =-500m. YWELU(3) =6OOm. QWELU(3) =-180m'/d
The drawdown is calculated for a squared grid with a side of 1000 m with squared
meshes of 10 m side for two times after the start of the pumping, l(f and 1<Y' minutes and
for two different layers, the directly pumped part of the groundwater reservoir (layer 1 of
the numerical model) and of the water table (layer 9 of the numerical model).
The results are shown in four maps (Fig. 5.15). In the vicinity of the constant
hydraulic head boundary the flow is perpendicular to the boundary. There is a large
hydraulic gradient near this boundary. On the boundary itself, the absolute value of the
drawdown of the real well and of its corresponding image well are the same. Because
they have an opposite sign, they cancel each other in their summation which results in a
drawdown equal to zero. Also, here, the drawdown around the wells with a same
discharge rate is slightly different from each other. The drawdown in the well which is
situated closer to the constant head boundary is slightly smaller than the drawdown of the
well which is situated farther from this boundary.
LAYER' AfTER 100000. HIMUTES Of PUIf'IMG LAYER' AfTER "tv". HIMUTES Of PUIf'IMG
100 200 300 4DO lOG 100 100 100 toO 100 200 HO 400 soa 100 100 aoo HO
LAYER I AfTER 100000. HIMUTES OF PUIf'IMG LAYER I AfTER tv9UI. "IMUTES DF PUIf'IMO
I
§
~
I
II 0
! 0 '":
II
2
~
'00 200 300 400 500 100 100 100 Mel 100 aoo MQ 400 aoo 100 100 100 toO
Fig. 5.15. Drawdown (m) in directly pumped layer (1) and of the water table (layer 9)
around three wells in layered groundwater reservoir laterally bounded by a constant
hydraulic head boundary after 105 and 106 minutes of pumping
A straight vertical boundary of constant head occurs in nature where a relative thin
groundwater reservoir is traversed by a canal or river over the whole thickness of the
reservoir. When the river depth approximates only the thickness of the reservoir, the
assumption of a constant head boundary is still acceptable. This case is, however, not
applicable if the watercourse traverses only for a small part the groundwater reservoir.
5.5.3 Drawdown in groundwater reservoir bounded by several straight boundaries
A large number of groundwater boundary problems can be solved by the method of

images. As already demonstrated in the two former parts, the actual boundaries are
replaced by a groundwater reservoir with infinite lateral extension where a number of
imaginary wells are included. In Fig. 5.16, 5.17 and 5.18 several cases with more than
one straight boundary are represented. In Fig. 5.16 three cases of pumping on a well
between two parallel boundaries are considered: a pumped well between two parallel
impervious boundaries, a pumped well between two parallel constant head boundaries and
finally a pumped well between two parallel vertical boundaries of which one is impervi-
ous and the other is a constant head boundary. Here, an infinite number of imaginary
wells should be included into this problem. In practice, however, it is only necessary to
include the pairs of image wells closest to the real well because the others have a
negligible influence on- the drawdown.
I
15
0
14
• I1.~
• I
12
0 13
I
I
I
lmpervlou. bounda,.,.
con.tan! bead boundary
12 13 real pumped ...n

• 0 ~
0 Imace pumped ..ell
• tmalll injection ..eU
Fig. 5.16. Image well system for a pumping well between two parallel straight vertical
boundaries
In Fig. 5.17 some cases are shown where the groundwater reservoir is laterally
bounded by two vertical boundaries which form a right angle. For the case left above the
image pumped wells I1 and 12 provides the required flow but, in addition, a third image
well I3 is necessary to balance the drawdown along the extension of the impervious
boundaries. The resulting system of the four pumped wells in an extensive aquifer
represents hydraulically the flow system for the physical boundary conditions. The second
example, represented right above, treats a pumped well in the case where the two vertical
bounds are now constant head boundaries. The image wells I1 and 12 are now injection
wells. The third image well 13 is a pumped well. The flow system of the second example
can be obtained in an extensive aquifer by two pumped wells and two injection wells. The
third example, represented left below, treats a pumped well in the case where one of the
vertical boundaries is impervious and the other one is a constant head boudary. This
corresponds, for example, with a well near an impervious boundary and a perennial
stream. This flow example can also be obtained by means of two pumped wells, the real
well and the image well 11, and by two image injection wells 12 and 13.
011 .11
_ . _ . _ . _ . _ . _ . .J
o 12 013 .12 013
Impentou. boundary
oOl1lltaat. head boundary
011 real pumped ....11
o 1m... pumped .ell
. 1m... mJecUon well

• 12 .13
Fig. 5.17. Image well system for a pumping well between two vertical boundaries which
form a right angle
According to Stallman and Papadopoulos (1966) the drawdown in wedge-shaped

groundwater reservoirs, such as occurring in a valley bounded by two converging
impervious boundaries or a perennial stream which does not cross an impervious
boundary at a right angle, can also be treated by the method of images. In Fig. 5.18 a
purriping on three pumped wells is considered in a groundwater reservoir which is
bounded by two impervious barriers intersecting at an angle of 60 degrees. Each pumped
real well has now five image pumped wells. The real well and its corresponding image
wells are located on a circle with its center at the wedge apex. Consequently, the radius
of this circle is the distance from the real well to the apex. The drawdown at any point
between the two barriers can then be calculated by summing the drawdown of all the
three real wells and the fifteen imaginary wells. Each case where the wedge angle is an
aliquot part of 360 degrees can be treated in the same way. The number of the required
image wells increased by one for the real well is then equal to 360 divided by the wedge
angle in degrees.
w213
Fig. S.lS. Image well system for three wells between two vertical impervious boundaries
which intersect at an angle of 60 degrees
In Fig. 5.19 the drawdown at two different times and of two different layers are
represented when water was withdrawn in three different wells. The location of the wells
and the two boundaries is given in Fig. 5.18. The coordinates and the discharge rates of
the three real wells are the same as in Sect. 5.4.3 and Eq. 5.11. The first impervious
boundary coincides with the x-axis (y=O). The second impervious boundary forms an
angle of 60 degrees with the first boundary. The angle is measured when one turns
toward the north. The wells pump water in the lower part of the groundwater reservoir
(layer 1 of the numerical model). The drawdowns in the directly pumped layer were
calculated along with the drawdowns of the water table (layer 9 of the numerical model).
As can be seen from Fig. 5.19 the lines of equal drawdown are all perpendicular to the
impervious boundaries. The well wi with the same discharge rate as well w2, causes a
210 Chapter 5 / Further developments oj pumping test model
remarkably larger drawdown than the well w2. Well wI is located closer to the second
impervious boundaries than well w2.
LAYER 9 AFTER 100000. "INUTES Of PUI1PING LAYER 9 AFTER 999999. MINUTES DF PUI1PING
.
§
Ii:
e ;Po
)
I
!
!
g
§
,..
!!
1 00 200 JOG 400 500 eGO 700 eGO toO 100 aoo 300 400 500 100 100 100 900
LAYER I AFTER 100000. "INUTES Of PUI1PING LAYER I AFTER 11911111111. "INUrES OF PUI1PING
I DO 200 JOG .00 500 100 100 100 toO o 100 200 JOO 400 100 600 700 100 tOO
Fig. 5.19. Drawdown (m) contour lines around three wells in laterally isotropic
groundwater reservoir bounded by two vertical impervious boundaries intersecting at an
angle of 60 degrees for directly pumped layer (1) and for the water table (layer 9) after
105 and 106 minutes of pumping
5.6 Drawdown in groundwater reservoir

with lateral discontinuous conductivity change
Based on the analogy between the flow of electrical current (Jakosky, 1957) and the flow
of groundwater a method is developed to approximate the drawdown due to pumping in a
groundwater reservoir that is characterized by the occurrence of a vertical plane of
discontinuous lateral conductivity change. In such a groundwater reservoir, the horizontal
conductivities of all layers change discontinuously at the vertical plane. At both sites of
the plane the conceptualization of the groundwater reservoir remains the same. The
sequence and the thicknesses of the pervious and semi-pervious layers are unaltered at
both sides of the plane. The horizontal conductivities are, however, different at both sides
of the vertical plane but the ratio between them is the same for all layers. In other words
the horizontal conductivities of the different layers at one side of the plane are equal to
the conductivities of the corresponding layers at the other side of the plane multiplied by
a same factor. Impervious and constant head boundaries are in fact special cases of
groundwater reservoirs with a discontinuous lateral conductivity change. At an impervious
boundary, the conductivities at both sides of the plane are multiplied with a same factor
which is equal to zero. At a constant head boundary, all the conductivities are multiplied
by a factor which is a very large value. In these two special cases the location of the
pumped well is only meaningful at one side of the boundary plane as well as the calcula-
tion of the drawdown. In the more general case of a groundwater reservoir with a
discontinuous lateral change of conductivity, the drawdowns should be known at both
sides of the plane of discontinuous conductivity change. It is, however, also possible that
the pumped wells are at both sides of this plane.
Because in the more general case the drawdown should be calculated and the well
can occur at both sides of the vertical plane, one must fIrst unequivocally defIne both
sides of this plane. This is done by the help of the point-slope equation of the straight line
which represents the location of the vertical boundary on the map:
(5.16)
where m is the slope of the straight line and Xl and Yl are the x- and y-coordinates of a
fIrst point on the straight line. The slope m is obtained by the help of the x- and y-
coordinates of a second point X2 and Y2:
m (5.17)
The hydraulic conductivities at the positive side of the straight line or vertical plane
correspond with the values used in the ax i-symmetric numerical model to calculate the
drawdown in a groundwater reservoir with continue lateral extension. A point is situated
on the positive side of the straight line when the x- and y-coordinate of this point are
introduced in the left part of Eq. 5.16 and result in a positive value. The vector of
horizontal conductivities of the layers at the positive side is further indicated by the
symbol K+ •.
First, the calculation method is considered of the drawdown at the positive side of
the straight line when water is withdrawn by a well at the positive side. This drawdown
can also be derived by the method of images based on the result of the axi-symmetric
numerical model. Also, the pumped well must be completed with an imaginary well. This
imaginary well is located just opposite of the vertical plane of discontinuous lateral
conductivity change. This imaginary well operates simultaneously with the real well and
at a discharge rate which is (K(1)+.-K(l>-.)/(K(l)+.+K(l).s) larger than the discharge rate
of the real well. K. is the vector of the conductivity at the negative side of the straight
line. In the case of an impervious boundary the conductivities at the negative side of the
plane are equal to zero which means that (K(l)+.-K(l).s)/(K(l)+s +K(lL) is equal to + 1
and that the imaginary well operates at the same discharge rate as the real well. In the
case of a constant head boundary the conductivities at the negative side of the plane are
equal to infinite large values. This means that (K(1)+.-K(lU/(K(1)+. + K(1U is equal to -
1. At the imaginary well, water is now injected with the same rate as water is withdrawn
in the real well. The drawdown at the positive side of the plane is now equal to the sum
of the drawdowns of the real and the imaginary well. If a laterally anisotropic ground-
water reservoir is treated, the drawdown caused by the imaginary well must be calculated
for a lateral anisotropic medium where the direction of maximum anisotropy must be
mirrored with respect to the plane of discontinuous lateral conductivity change. In this last
case, one assumes that the principal directions in the horizontal plane are the same at both
sides of the vertical plane of discontinuous lateral conductivity change.
The drawdown at the negative side of the plane when water is withdrawn by a
well at the positive side must now be calculated with a second approach. The drawdown
at this negative side of the plane is the drawdown caused by pumping on only one
imaginary well. This well is located on the same place as the real well. This imaginary
well operates simultaneously with the real well and at a rate which is equal to the real
pumped rate multiplied by 2K(l)+/(K(l)+s+K(l).s). In the case of a laterally anisotropic
medium, the drawdown at the negative side of the plane must be calculated taking into
account the real direction of maximum conductivity. This drawdown can only be
considered as an approximation. The further from the boundary plane the larger the
approximation.
When water is pumped at the negative side of the straight line, the same theory
can now be applied. With the numerical ax i-symmetric model, the drawdown must now
first be calculated with the hydraulic conductivities for the groundwater reservoir. Based
on these data the drawdown can be approximated for the negative side with the best
accuracy according to the first approach. The symbol K(l).s must be replaced by K(l)+.
and vice versa. The drawdown at the positive side of the vertical plane must now be
approximated by the second approach. The imaginary well is situated on the same
location as the real well and operates with a discharge rate which is equal to the real
discharge rate multiplied by 2K(I)j(K(IL+K(I)+s).
If water is withdrawn at both sides of the plane. Two calculations of the draw-
down must be performed: a first time with all the wells at one side of the plane and a
second time with all wells at the other side of the plane. Both obtained drawdowns must
be accumulated to obtain the final approximation of the drawdown caused by a pumping
at both sides of the plane of discontinuous lateral conductivity change. The results of this
calculation must be considered as a first approximation because the proposed method is
deduced by analogy with a steady state electrical flow (Jakosky, 1957). The accuracy of
the result depends on the area influenced behind the vertical boundary plane. The smaller
the storage decrease in this area (relative to the total storage decrease), the smaller is the
error of this approximation.
If one wants to approximate the drawdown in a groundwater reservoir with a vertical

plane of discontinuous lateral conductivity change the x- and y-coordinates of two vertical
lines of the vertical planes are required as well as the ratio of the conductivities between
the positive side and the negative side of the vertical plane, K(1)+,1K(1L (Table 5.1). The
x- and y-coordinates of these two vertical lines x1,YI and X2 ,Y2 must be selected so that the
slope m of the straight line which represents the location of the vertical plane on the map
can easily be calculated according to Eq. 5.17. The x-coordinates of the two vertical lines
must always be different, otherwise it becomes impossible to calculate the slope m with
the last mentioned equation.
5.6.3 Examples of drawdown approximation
In a first example water is withdrawn near a plane of discontinuous lateral boundary

change. The conceptualization of the groundwater reservoir is the same as assumed in
Sect. 5.4.3 (Fig. 5.9) and the coordinates and the discharge rate of the three considered
pumping wells are given in Eq. 5.13. The wells have screens in the lowermost layer,
layer 1 of the numerical model. The x- and y-coordinates of two vertical lines which
defines the location of the vertical plane are respectively 0.0 , 0.0 and 577.35 , 1000.0.
As a consequence, the well are located at the positive side of the plane and the hydraulic
conductivities at this side correspond with the values used in the axi-symmetric model. At
the positive side of the plane the horizontal conductivities of the all layers are all three
times smaller than the horizontal conductivities at the negative side of the plane. In Fig.
5.20 the lines of equal drawdown are represented for two different layers and for two
different times for each layer. From this figure one can remark that the gradient changes
abruptly at the plane of discontinuous lateral boundary change. At the negative side of the
plane the drawdown gradient is smaller than at the positive side of the plane. The abrupt
change of drawdown gradient results in a bend in the lines of equal drawdown. This is at
the place where this line crosses the plane.
LAYER 8 AFTER 100000. "IMUTES OF PUII'IMG LAYER 9 AFTER 999999. "IMUTES OF PUII'IMG
100 20G 300 4DO SOO 100 100 100 100 100 200 SOO 400 500 100 roo 100 100
LAYER 1 AFTER 100000. "IMUTES OF PUII'IMG LAYER 1 AFTER 999989. tllMUTES OF PUII'IMG
100 200 300 400 500 600 700 100 '00 100 200 300 "00 500 100 100 laO 100
Fig. 5.20. Estimated drawdown (m) around three pumping wells in a laterally isotropic
groundwater reservoir with a vertical plane of discontinuous lateral conductivity change.
This plane goes through the origin and forms an angle of 30°. The horizontal conducti-
vities of all layers at the positive side of the plane are three times smaller than the
horizontal conductivities of the corresponding layers at the negative side of the plane
In the second example the horizontal conductivities at the positive side of the plane
are all three times larger than the horizontal conductivity at the negative side. This is the
only difference with the problem given in the first example. The results are represented in
Fig. 5.21. Studying this figure one remarks also here a discontinuous change of the draw-
down gradient around the plane of discontinuous lateral boundary change. At the negative
side of the plane the drawdown gradient is now larger than at the other side of the plane.
The drawdown is everywhere larger than in the first example.
LATER \I AFTER 100000. I1INUTES OF PUttPlNG LATER 9 AFTER 9999911. I1INUTES OF PUttPlNG
100 200 ~ 400 500 100 700 100 100 o 100 zoo )00 400 100 600 700 100 90Q
LATER 1 AFTER 100000. I1INUTES OF PUttPlNG LATER 1 AFTER 99111199. I1INUTES OF PUttPlNG
100 200 300 400 $00 600 100 100 too o 100 200 He) 400 $00 600 100 100 100
Fig. 5.21. Estimated drawdown (m) around three pumping wells in a laterally isotropic
groundwater reservoir with a vertical plane of discontinuous lateral conductivity change.
This plane goes through the origin and forms an angle of 30° with the northern direction.
The horizontal conductivities of all layers at the positive side of the plane are all three
times larger than the horizontal conductivities of the corresponding layers at the negative
side of the plane.
In the third example the same problem is treated as in the first example with the
exception that the groundwater reservoir is now laterally anisotropic. The lateral aniso-
tropy is equal to 1.41 and the angle which defines the principal direction of maximum
horizontal conductivity is equal to 1.31 radians. This angle is measured starting from the
northern direction and turning toward the east. The results of the third example are
represented in Fig. 5.22.
LAYER 9 AFTER 100000. "INUrES OF I'Utl'lNG LAYER 9 AFrER 999999. "INUrES OF I'Utl'lNG
o 100 200 ~ 400 $00 100 700 100 100 100 200 300 400 500 100 700 100 100
LAYER I AFTER 100000. "INUrES OF I'Utl'lNO LAYER I AFTER 191111119. "INUrES Of I'Utl'lNC
100 aoo lOG 400 $00 100 700 aoo 100 100 200 300 4DO $GO 100 700 100 100
Fig. 5.22. Estimated drawdown (m) around three pumping wells in a laterally anisotropic
groundwater reservoir (m = 1.41, (j= 1.31 radians) with a vertical plane of discontinuous
lateral conductivity change. This plane goes through the origin and forms an angle of 30°
(or 0.52 radians) with the northern direction. The horizontal conductivities of all layers at
the positive side of the plane are all three times smaller than the horizontal conductio vities
of the corresponding layers at the negative side of the plane.
The only difference between the fourth and the second example is the lateral
anisotropy. In the second example each layer of the groundwater reservoir is assumed to
be isotropic in the horizontal plane whereas in the fourth example all the layers have the
same lateral anisotropy (1.41) and the same angle (1.31 radians) which defines the
principal direction of maximum horizontal conductivity. The results of the fourth example
are represented in Fig. 5.23.
LAYER 9 AFTER 100000. "IHUTES OF PUIIPIHG LAYER 9 AFTER 999999. "IHUTES OF PUIIPIHG
o 100 200 300 400 SOO 100 700 800 '00 o 100 200 HO 400 SOO 100 700 100 NO
LAYER I AFTER 100000. "IHUTES OF PUIIPIHG LAYER I AFTER 999999. "IHUTES OF PUIIPIHG
I I
I I
!
!
100 ~ 300 400 $GO 100 700 100 100 100 200 :JOO 400 100 100 700 100 NO
Fig. 5.23. Estimated drawdown (m) around three pumping wells in a laterally anisotropic
groundwater reservoir (m=1.41, 8=1.31 radians) with a vertical plane of discontinuous
lateral conductivity change. This plane goes through the origin and forms an angle of 30°
(or 0.52 radians) with the northern direction. The horizontal conductivities of all layers at
the positive side of the plane are all three times larger than the horizontal conductivities
of the corresponding layers at the negative side of the plane.
218 Chapter 5 / Further developmenJs of pumping test model
5.7 Land subsidence due to groundwater withdrawal
Land subsidence due to changes in reservoir pressures and/or water table lowering has
been reported in many areas of the world (Domenico & Schwartz, 1990). According to
these authors the geological requisites and qualifying conditions for the occurrence of
subsidence ar~ so well adapted to sedimentary basins that it is likely to believe that this
phenomenon is taking place much more frequently than ever been reported. The chief
reason for this is the lack of close control of benchmarks necessary to detect small
changes in land-surface altitude.
Land subsidence can cause serious problems. The normal flow of surface water
can be affected because the gradient of canals or natural watercourses can be reduced or
even reversed by the subsidence. In subsidence areas the cracking of brick structures or
concrete is common and construction on piles, as bridges, may protrude above the
land-surface. Also, well casings can protrude in these areas or even break down.
Subsidence in coastal areas can result in the conversion of pastures to tidelands.
The real three-dimensional deformation of the groundwater reservoir is here simplified to

a one dimensional compression in vertical direction. The relative compression of a layer
J, aD(J)/D(J) can be derived from the definition of vertical compressibility of layer J,
I3p(J), as given in Eq. 2.73:
aD(J) = P (J) a a (5.18)
D(J) P I
Because the total vertical stress acting on an horizontal plane at any depth does not vary
in time, the change of the intergranular pressure AUj is equal to the pore water pressure
decrease -aP due to water withdrawal. This pore water pressure decrease can also be
expressed in function of the drawdown increase as or:
(5.19)
The vertical compressibility I3 p (1) can be derived with the help of Eq. 2.86 which give the
relation between the specific elastic storage of layer J, Ss(J), the water compressibility,
I3w, the porosity of this layer, n(1), and the specific weight of the water Pwg:
(5.20)
Substituting Eq. 5.19 and 5.20 in Eq. 5.18 the relation between the reduction of the
thickness of layer J, aD(1), and the drawdown increase as is obtained:
aD(J) = D(J) (S/J) - P.$ n(J) Pw ) as (5.21)
Applying this formula, it is assumed that the specific elastic storage in each layer is
constant during the compaction and independent on the drawdown. According to Barends
(1978) the changes of the specific elastic storages and even of the vertical and horizontal
conductivities must be considered during the subsidence process if the drawdown in the
different layers of the groundwater reservoir is meaningful.
The thickness reduction of each layer can now be found by calculating the
drawdown in each layer with the numerical model (Lebbe, 1995). The subsidence
undergone by the middle of a layer is now the sum of the thickness reductions of all
underlying layers increased with half the thickness reduction of the considered layer. For
the uppermost layer of the numerical model, the calculated subsidence corresponds with
the subsidence at the top of this layer.
5.7.2 Additional input data and representation of results
The subsidence due to pumping on one well can be calculated by the program outpu5 and
directly plotted in graphs. There are two possibilities: the subsidence-time and subsidence-
(apparent) distance graphs and the lines of equal subsidence in a vertical cross-section at
different times after the start of the pumping. To calculate the subsidence one must give
interactively the values of the porosity of the different layers of the numerical model.
With the program multpl it is also possible to calculate the subsidence due to pumping
on different wells which started simultaneously and are situated in a same layer. In this
last case the line of equal subsidence can be obtained for each layer of the numerical
model and for the different times after the start of the withdrawal. Also, the porosity of
the different layers must be given interactively.
5.7.3 Example of subsidence calculations
In the example the considered groundwater reservoir is conceptualized in three pervious

layers which are separated by two semi-pervious layers (Fig. 5.24). The base of the
groundwater reservoir consists of a thick clay layer. The top is formed by the water table.
The lowermost pervious layer consists of fine to medium sands. These sands have a mean
horizontal conductivity of 10 mid and a mean vertical conductivity of 2m/d. These eighty
meters thick sands are covered by thirty meters thick clayey fine sand which can be
considered as a first semi-pervious layer. The mean horizontal conductivity is equal to
1.00 mid and the mean vertical conductivity is equal to 0.25 mid. The second pervious
layer consists of medium sands with a thickness of thirty meters, an average horizontal
conductivity of 25 mid and an average vertical conductivity of 5m/d. The second semi-
pervious layer is formed by an alternation of silty clay and fine sands. The layers are
horizontally stratified. The average horizontal conductivity is principally due to the fine
sands and is equal to 0.5 mid. The average vertical conductivity is principally due to the
silty clay layers and is equal to 0.01 mid. This semi-pervious layer is covered by four
meters of fine saturated sands. The estimated values of the horizontal and vertical
conductivity of these sands are respectively 3m/d and 1m/d.
~
Bahem.U..Uon of around-tar r.eenolr laJU'll and hJdrauUc paramltlll'll
In numerical model
depth( .m)
..at.r tabla
o -nao...-a-
..t ......tla:a ",
.at,
....0",
lID_ .....
f.
III. . . . . nzuI. '"......
eo
r~
It-
01.,.,. na.. lUi"
~
1
100
&,,(&)- 1.18.la-&
!
~
ltaele J"lmou. la,er 110(1)- I .... ur&

medl1DIl lUlU
Is·
II'Ci
1-------f---------lIC(I)-U.•• ~
1&0 Itb (1)- 10.0 mi. &,,(1)- 1.00.. 0-&

0(1)-3•.' " I-
eJ.]r
Fig. 5.24. Schematisation of groundwater reservoir and discretisation in numerical model used for subsidence calculations
In the treated example the subsidence is calculated for a pumping in the most
pervious layer which consists of medium sands. The groundwater reservoir is discretized
in eighteen layers (Fig. 5.24). The lowermost pervious layer is discretized in four layers
of which the thickness decreases from bottom to top in the direction of the flow. The
lower semi-pervious layer is discretized in four layers in the numerical model. The
pumped pervious layer is discretized in three layers of equal thickness. The upper semi-
pervious layer is replaced by six layers in the numerical model. The thicknesses of these
layers decrease from bottom to top. The uppermost layer of the numerical model
coincides with the relatively thin uppermost pervious layer.
The horizontal conductivity of layers 1 until 4 is 10 mid or Kh(I-4)=lOm/d. The
layers 5, 6, 7 and 8 have a horizontal conductivity of Im/d or Kh(5-8) = 1m/d. The
horizontal conductivity of layers 9, 10 and 11, Kb(9-1l), is 25m/d and for layers 12, 13,
14, 15, 16 and 17, Kh(12-17), is 0.5m/d. An estimated value of 3m/d has been assigned
to the horizontal conductivity of the uppermost layer (Kh(18)=3m/d).
The hydraulic resistance between the different layers has been calculated based on
the thickness and the vertical conductivities according to:
c(J) = D(J) + D(J+l) for J = 1, 16

2Kv(J) 2KP+l)
(5.22)
and c(17) = D(17) + D(18)
2Kv(17) K/18)
In the calculations it is assumed that the elasticity of the layers increases gradually in the
upward direction. These values have been deduced from the average depths of the layer
according to the Van der Gun relation (Eq. 2.89). The storage coefficient near the water
table has been put equal to 0.16.
With the program sipur5 the drawdowns are first calculated which should occur in
each layer at different times after the start of the pump. Water is withdrawn from the
layers 9, 10 and 11 of the numerical model with a discharge rate of 1440 IIi/d. These
results are first of all represented by lines of equal drawdown in vertical cross-sections
for different times after the start of the pump (Fig. 5.25). Studying this figure one can
learn that already after one minute of pumping there is a measurable drawdown within a
distance of fifty meters from the pumped well in the pumped pervious layer while in the
other layers the drawdown is negligibly small. After ten minutes of pumping, the draw-
down increases in the directly pumped layer and also in the bounding semi-pervious
layers. In these last mentioned layers the drawdown gradient is principally vertical. After
a hundred minutes of pumping there is already a measurable drawdown in layer 1 of the
numerical model. After a thousand minutes of pumping one can remark that at larger
distances from the pumped well the small drawdowns or the rise of the drawdown is
much less level dependent. A clear insight in the evolution and the lateral variation of the
drawdown in a certain layer can be obtained by the plot of time-drawdown and distance-
drawdown graphs for this layers. In Fig. 5.26 the time-drawdown and distance-drawdown
graphs are shown for the layers 4, 10, 14 and 18.
Depth(m)
0;.,,_ _ ___
o ·1
80 t ------..r 80
160' 100 min 10 1000 160' 100000 m!on 100 1000

~----~-------r------~~~~
Depth(m) Distance (m) D~th(m) Distance (m)
°tt,I_ _ ___
·1 -3
80
160' 10 min 10 160 10000 ~~

100 1000 100 10
Depth(m) Distance (m)Depth(m) Distance (m)
~
0'
-:Jo )
\
o -I -3
80 80t-------'
160 1 min 10 160 1000 min 10 100 IO~

~__--~~----~IOO~__--~I~OOO~-----
Distance (m) Distance (m)
Fig. 5.25. Contour lines of logarithm of drawdown (m) (interval 0.5) in vertical cross
section for different times after the start of the pumping test which is used for the
demonstration of the subsidence calculation (Note that the distance-axis are logarithmic!)
After calculating the drawdown, the subsidence can be approximated with the help
of the obtained drawdown and the relation given by Eq. 5.21. For all the layers a same
porosity is assumed, i.e. 0.38. These calculations are performed by the output program
outpu5 and directly plotted in vertical cross-sections or in time-subsidence and distance-
subsidence graphs. In Fig. 5.27 the lines of equal subsidence are represented in vertical
cross-sections for one, ten, hundred, thousand, ten thousand, hundred thousand and one
million minutes after the start of the pumping. In the beginning of the pumping test the
directly pumped layer and the covering layers are the only which are subjected to
subsidence. As the pumping test progresses, the subsidence extends toward the underlying
layers and increases also steadily in the directly pumped and its covering layers. In Fig.
5.28 the time-subsidence and distance-subsidence graphs are represented for the layers 4,
10, 14 and 18. From these graphs one can deduce that the subsidence in the uppermost
layer is largest after about 2 years (or more exactly one million minutes) of pumping.
_..
This maximum value is only 2.5 mm and occurs close to the pumped well. The lateral
change in subsidence is very small and so is the differential subsidence.
,~t_----~----_+----~------~--~ ,~t_----~----_r----~
IO·''r------+------+-----f------bo!IIl~~
_lID ,.. ,..

,O-'\-----_r----_t-----I~;y,"""7"cr-"7"'_i
'0' 10' n"'O'III'

LAYER
_lID 100 UlEatlln

LAYER
,~t_----~----_+----_t------r_--~ ,~t_----~----_r----~
105 TI.OIIID 101 102

LAYER 4
Fig. 5.26. Time-drawdown and distance-drawdown graphs for the layers 4, 10, 14 and 18
for pumping test used to demonstrate the subsidence calculation
Depth(m) Depth(m)
o .! -3
0:L-~3
80~ _ _ _ _ _ _~ 80
160' 100 min 10 160' 10000 nMn 100
Depth(m) Distance(m) Depth(m)
0L---J~' 0:~~3
80 80
1000 min
160 10 min 10 100 1000 160 10 100
Depth(m) Distance(m) Depth(m)
O~ O~
.1 -3
80 8
1 min 10 1000 min

160 100 1000 160 10 100
Distance(m)
Fig. 5.27. Contour lines of logarithm of subsidence (m) (interval 0.5) in vertical cross
section for different times after the start of the pumping test
With the program multpl it is possible to calculate the drawdown and the
subsidence caused by a number of wells. In Fig. 5.29 the drawdown is represented in the
middle of the pumped layer when water is pumped from ten wells with a discharge rate of
1440 m3 /d each. The wells are located in the middle of the considered area and are
aligned in rows of five wells each. The mutual distance of the wells in the row is 100 m
and the mutual distance between the rows is also 100 m. In the well field itself the
drawdown is larger than five meters_ The lines of equal drawdown at large distance from
this well field are nearly circles. With the same program multpl it is possible to calculate
the subsidence around the same withdrawal. The results of the subsidence of the upper-
most layer of the numerical model are represented in Fig. 5.30. From this figure one can
deduce that the subsidence after about two years of pumping varies is thirteen to fifteen
millimetres close to the well field. Further from the well field the lines of equal subsid-
ence shows also a nearly circular course at about 900 m from the center of the well field
the subsidence of the uppermost layer is only 6 mm. This considerable withdrawal of
water will not affect large subsidence in the considered area nor will it cause large
differential subsidence_
Chapter 5 I Further developments oj pumping test model 225
101 TIlEatl10
LAYER
loS fUll Ofl., Itl

LAYER 14
SUlSIDE.CI (to s.lIDDlCt:oo
,.- ,.-\------1---+----1
,.-
,.-
"."Cla.)
"-
,.-
,.-
,.' loS n"OIlID
LAYER.
Fig. 5.2S. Time-subsidence and distance-subsidence graphs for the layers 4, 10, 14 and
18 for a pumping test with a discharge rate of 1440 m3/d
LATER 10 AFTER . ., _ . "INUrES OF PUlPING
• - ... - - I."
Fig. 5.29. Drawdown (m) in middle of pumped layer around ten wells with a discharge
rate of 1440 m3/d each after 106 minutes of pumping.
The differential subsidence is in reality caused by two facts. The first fact is that
the drawdown after a certain time of pumping differs from layer to layer and depends on
the distance to the pumped well or the center of the withdrawal. The second fact is that
the elasticity properties and the thickness of the layers can change laterally. It is of course
clear that it is possible to study the differential subsidence due to the first fact. With the
above given programs outpu5 and multpl it is, however, not possible to study the
differential subsidence due to the second fact.
LAYER 18 AFTER 11118888. nlNUTES OF PUWING
Fig. 5.30. Subsidence (mm) of uppermost layer arround ten pumping wel1s with a
discharge rate of 1440 m3 /d each after 106 minutes of pumping
REFERENCES
Bierschenk, W.H., 1964, Determining Well Efficiency by Multiple Step-Drawdown Tests:

Pub\. 64, International Association of Scientific Hydrology.
Domenico, P.A., and Schwartz, F.W., 1990, Physical and Chemical Hydrogeology:
New York, John Wiley and Sons, 824 p.
Driscoll, F.G., 1986, Groundwater and Wells (Second Edition): St. Paul, Minnesota,
Johnson Division, 1089 p.
Freeze, R.A., and Cherry, J.A., 1979. Groundwater: Englewood Cliffs, New Jersey,
Printice-Hal1, 604p.
Gabrysch, R.K., 1968, The relation between specific capacity and aquifer transmissibility
in the Houston area, Texas: Ground Water, vol. 6, no. 4, p. 9-14.
Hurr, R. T., 1966, A new approach for estimating transmissibility from specific capacity.
Water Resources Research, vol. 2, p. 657-664.
Jacob, C.E., 1947, Drawdown test to determine effective radius of artesian well: Trans.
Amer. Soc. Civil. Engrs. vol. 112, p. 1047-1070.
Jakosky, J.J., 1957. Exploration geophysics: Newport Beach, California, Trija Publishing
Company, 1195 p.
Lebbe, L., 1995, Land subsidence due to groundwater withdrawal from the semi-confined
aquifers of southwestern Flanders: Land Subsidence, lABS Publ., no. 234, p. 47-54.
Logan, J., 1964, Estimating transmissibility from the routine production tests of water
wells: Ground Water, vol. 2, no. I, p. 36-37.
Rorabaugh, M.I., 1953, Graphical and theoretical analysis of step drawdown test of
artesian well: Proc. Amer. Soc. Civil Engrs., vol. 79, pp. 1-23.
Stallman, R.W., and Papadopolous, I.S., 1966, Measurement of hydraulic diffusivity of

wedge-shaped aquifers drained by streams: U.S. Geol. Survey Prof Paper 514, p. 1-50.
Todd, D.K. 1980. Groundwater Hydrology (Second Edition): New York, John Wiley &
Sons, 535p.
Chapter 6 / Inverse model as tool for
pumping test interpretation
In the numerical model for the simulation of pumping tests, the groundwater reservoir is
first schematized or conceptualized in a number of pervious and semi-pervious layers.
These hydrogeological units are further discretized in a number of layers in the numerical
model. Each layer of the numerical model is assumed homogeneous and is characterized
by a single value for the horizontal conductivity and one for the specific elastic storage.
The hydraulic resistance between the layers can be deduced from the thicknesses and the
vertical conductivities of these layers. Knowing the discharge rate, one can calculate the
drawdown at any distance from the pumped well, at any level and at any time after the
start of the pump. The problem solved by the numerical model is called by Sun Ne-Zheng
(1994) a forward problem.
With the inverse model for pumping test interpretation, the optimal values of the
hydraulic parameters are determined along with their joint confidence region. This
interpretation is based on the observed drawdowns, the measured evolution of the
discharge rate and the conceptualization of the groundwater reservoir. According to Sun
Ne-Zheng (1994) the inverse model for pumping test interpretation solves a type 1 inverse
problem. This author distinguishes four different types of inverse problems. In type 1
hydraulic parameters are identified whereas in type 2 it are the discharge and/or recharge
rates. In type 3 initial and/or boundary conditions are inferred. Finally, in type 4 more
than one of the former mentioned parameters are identified simultaneously.
The algorithm of the inverse model for pumping test interpretation is here obtained
by the combination of the numerical model along with its further developments, a number
of sensitivity analyses and the calculations of adjustment factors for the derivable
hydraulic parameters. By the successive execution of these three steps the optimal values
of the hydraulic parameters are derived iteratively along with their joint confidence
region. This joint confidence region in the p-dimensional parameter space informs us
about the accuracy with which the different hydraulic parameters are derived. The letter p
stands here for the number of derived hydraulic parameters. Two- and three-dimensional
cross sections through this joint confidence region give us first an insight in the location
of the bounds of this region. These cross sections can also be considered as an additional
tool in the collinear diagnostic besides the eigenvalues and the eigenvectors of the
variance-covariance matrix, the partial correlation coefficients, the marginal and condi-
tional standard deviation, the condition indexes and the matrix of marginal variance-
decomposition proportions. All these statistical parameters help us to identify if the
inverse model is well-posed. An inverse problem is well-posed if there is a solution and if
the solution is unique and stable (Carrera and Neuman, 1986b). Once the optimal values
of the hydraulic parameters are deduced along with their joint confidence region, the
drawdown can be calculated according to the optimal values of the hydraulic parameters.
230 Chapter 6 / Inverse model
With the aid of the joint confidence region of the hydraulic parameters the confidence
intervals of the calculated drawdowns can be approximated. These confidence intervals
can be used to evaluate the relative importance of the model inputs and the anticipated
effect of new data and analyses on the reliability of the model results (Hill, 1989). At the
end of a study the confidence intervals of the drawdowns can be used to indicate clearly
the likely errors in the simulated results.
6.1 Residual vector
6.1.1 Definition
After the schematization or conceptualization of the groundwater reservoir in pervious and

semi-pervious layers and the discretization in a number of layers in the numerical model,
one has to estimate the initial values of the hydraulic parameters. Using these values, the
model calculates the drawdowns at the same places and times of the observations. The
observed and the calculated drawdowns can than be compared in two different ways. In
the first way the drawdowns can be compared in an untransformed way. Here, the
residuals are the difference between the measured and the calculated drawdowns. In the
second way the measured and calculated drawdowns are first transformed before they are
compared. For the interpretation of pumping tests the second way is preferred to define
the residuals. The measured and the calculated drawdowns are first log transformed or:
(6.1)
where r are the residuals,

s* the measured drawdowns,
and s the calculated drawdowns.
The residuals are defined in this way because drawdowns of different order of
magnitude must be considered simultaneously in an inverse model for the interpretation of
pumping tests. Sometimes, it is necessary to use drawdowns of several tens of meters,
which are measured in the pumped wells, along with drawdowns of only a few centime-
ters, which are measured in a semi-pervious layer. In field applications, it may be
necessary to consider both types of drawdowns, for example, if one wants to estimate
simultaneously the well loss characteristics and the vertical conductivity of a semi-
pervious layer. The first type of drawdown determines the well loss characteristics, the
second type the vertical conductivity of a semi-pervious layer. If in this case the arithme-
tic values of the drawdowns are considered then the very large values would have an
undesirable large weight with respect to the small values, resulting in difficulties in
estimating the vertical conductivity of the semi-pervious layer.
The small drawdowns at the start of the pumping test are very important in
estimating the specific elastic storages. Consequently, their influence on the inverse
solution may not be ignored. Small drawdowns can be observed either in wells located in
Chapter 6 / Inverse model 231
the pumped layer at relatively large distance from the pumped well or in adjacent layers
of the pumped layer at relatively small distance from the pumped well.
6.1.2 Contributing factors of residuals
In practice, the residuals are a combination of the measurement errors and the errors
arising from the mathematical model (Carrera and Neuman, 1986a). The measurement
errors are the differences between the measured and the true drawdowns. The differences
between the true and the calculated drawdowns are called the errors arising from the
mathematical model. These last mentioned errors can further be subdivided in conceptual
errors and computational errors (Carrera, 1984).
The conceptual errors arise from improper or oversimplified conceptualization of
the groundwater reservoir. This is a frequently appearing error in pumping test analyses;
e.g., the use of the Theis-type curve or the Walton-type curves for the interpretation of
time-drawdown curves measured in groundwater reservoirs that do not match with the
simple conceptual models for which the type curves are derived (see also Sect. 6.5.1).
Other examples of conceptual errors occur when the heterogeneity of the layers and/or the
lateral variation of the hydraulic parameters are not exactly represented in the mathema-
tical model.
Computational errors are primarily due to discretization of the partial differential
equation used to describe aquifer behavior. An example of such an error is the discretiza-
tion of a semi-pervious layer with a large vertical hydraulic gradient in only one layer or
in an insufficient number of layers. Other computational errors are due to the computa-
tional algorithm. They depend on the applied discretization scheme (e.g., finite-difference
or finite-element method, the assumed variation of the hydraulic head, etc.).
Measurement errors are not only associated with measurement instruments and
their reading or recording but can also be caused by the erroneous interpretation of the
quantity being measured. This last error can occur when observation wells are equipped
with rather long screens. Then the determination of the representative observation level
can be cumbersome. This representative level can then depend on the layer which is
pumped. In the worst case such observation wells can also cause supplementary flow in
the groundwater reservoir, e.g., they act as small pumping wells at the levels where the
drawdown at the specified level is smaller than the drawdown in the observation well and
acts as a small injection well where the drawdown at the specified level is larger than the
drawdown in the observation well. When this flow becomes important and is not consi-
dered in the model then it can result in errors that can be called conceptual. Sometimes,
the contrast between conceptual and measurement errors is not sharp. Another measure-
ment error is due to the effect of the wellbore storage in the observation wells. Here also
a quantity of water must flow from the storage of the observation well into the ground-
water reservoir. This flow causes a drawdown difference between the observation well
and the groundwater reservoir. This retardation of drawdown rise can be minimized by
the installation of inflatable packers above the pressure transducer with which the
drawdown is measured.
From the short discussion about the different kind of errors, it is evident that the
residuals are the result of many factors. The apparent boundedness of the errors suggests,
on the basis of the central limit theorem (Carrera and Neuman, 1986a), that the residuals
show a Gaussian or a normal distribution with zero mean. Once an optimal solution is
obtained, a thorough analysis of the residuals is necessary to examine the validity of the
assumption of the nonlinear regression analysis. The analysis of the residuals is given in
Sect. 6.3.3.
6.2 Sensitivity matrix
6.2.1 Definition
The sensitivity J ij of the drawdown Si to the hydraulic parameter or group of parameters,

Hj, is here approximated by the finite-difference method in which the drawdowns and the
hydraulic parameters are considered in their logarithmic space:
log,0§i (Hj • sf) - log,0§i log,0§i (Hj • sf) - log,0§i
(6.2)
log,o (Hj • sf) - log,oHj log,osf
where sf is the sensitivity factor; Sj is the calculated drawdown at the place and time of
the i-th observation with the estimated values of the parameters for the first iteration or
calculated values of the preceding iteration; and sj(Hj.sf) is the calculated drawdown at
the place and time of the i-th observation with the estimated values of the parameters
except for the value(s) of the j-th parameter or group of parameters of which the estima-
ted value(s) is (are) multiplied by the sensitivity factor. Another term for sensitivity,
which is sometimes used, is sensitivity coefficient. The matrix of sensitivities or the
matrix of sensitivity coefficients is indicated by the bold letter J and is also called the
Jacobian matrix.
The hydraulic parameters are considered in their logarithmic space because of the
large variation of the different possible values. The horizontal conductivity varies mostly
between 10.1 and 102 mid, the hydraulic resistance between 0.1 and 105 d, the specific
elastic storage between 104 and 10-7 m- I and the specific yield between 0.005 and 0.25.
The sensitivities defined in this way are all dimensionless. The consideration of the
hydraulic parameters within their logarithmic space results also in the compression of the
parameter space. The sensitivities defined in the logarithmic space of the parameters
result also in a well-conditioned Hessian matrix (PJ) of which the inverse can easily be
calculated. Carrera and Neuman (1986a) state that the number of iterations is reduced
when the parameters are considered within their logarithmic space.
The sensitivities can also be defined in an other way. The most simple way is the
finite-difference approximation:
J .. = §i (Hi· sf) - §i (6 .3)
"J H j • sf - H j
The sensitivities defined in this way have different units which depend on the units of the
treated parameters.
McElwee and Yulkler (1978) proposed to calculate the normalized sensitivities
which can be written in the finite-difference approximation as follows:
§i (Hi· sf) - §i §i(Hj.sf) -§i
(6.4)
H j • sf - H j sf - 1
Here the sensitivities have all the same unit, the unit of the drawdown.
Tarhouni and Lebbe (1992) proposed still another way to compute the sensitivities
where all the sensitivities have also all the same unit which is also the unit of the
drawdown. The finite-difference approximation of this last definition can be written as
follows:
§i (Hi· sf) - §i
(6.5)
loglosf
also here the hydraulic parameters are considered into their logarithmic space. In the two
last cases it is assumed that the residuals show a normal distribution rather than a log-
normal distribution as it is assumed in the first definition of the sensitivities (Eq. 6.2).
6.2.2 Program package to calculate sensitivity matrix
The program package conceived for the calculation of the sensitivity matrix according to
Eq. 6.2 consists of two programs:
- the input program for the definition of the grouped hydraulic parameters to compute the
sensitivity matrix (a part of the inrmp program) and;
- the proper calculation algorithm which is a part of the inverse process (program inpur5,
see Sect. 6.6.2). During each iteration of the inverse process the sensitivities are
calculated and written consecutively on the file namel.etadfdp (UNIX) or namel.eta
(MSDOS) in a concise way.
With the last part of the input program inrmp it is possible to define all necessary
parameters needed for the sensitivity calculations. This program creates the file namel-
.pase (UNIX) or namel.pas (MSDOS) where namel is the name of the treated program.
The sequence, the limits and the format of the written variables or parameters are given
in Table 6.1.
Table 6.1. Sequence and format of the data in the file name1.pas(e)
Line Variable Description or remark Limits For-

or units mat
1 SENFAC logarithm of .03,;;ISENFACI ;;00.5 F4.2

sensitivity factor sf
2 NUSEAN number of 00,;;NUSEAN;;o27 12
sensitivity calculations
Only if NUSEAN =00 N varying from 1 until NUSEAN
number of hydraulic parameters

3+2*N-2 NUPISA(N) which must be changed in 01,;;NUPISA(N);;o15 12
the N'" sensitivity calculation
4+2*N-2 PARAM(N,M) M varying from 1 until NUPISA(N) 01 ';;PARAM(N,M)';;3*N9 1514
The first required parameter is the logarithm of the sensitivity factor sf, SENFAC.
The recommended value for SENFAC is 0.1 or -0.1. The absolute value of the logarithm
of this factor is preferably between 0.03 - 0.5. Too small values of the sensitivity factor
can result in an erroneous numerical approximation of the sensitivity. The second
required parameter is the number of sensitivities, NUSEAN, that one wants. In other
words, it is the number of parameters and/or groups of parameters of which the sensitivi-
ties must be calculated. The maximum allowed number of sensitivities that can be
calculated with the inpurS program is 27. If one wants to perform sensitivity calculations
for more than 27 parameters, then these programs must be redimensioned. When a zero
value is attributed to the second parameter then 3N9 sensitivity calculations are made. N9
is the number of layers in the treated model. Here, the sensitivities of the horizontal
conductivities, the hydraulic resistances and the specific elastic storages of each single
layer are calculated. The parameters SENF AC and NUSEAN are followed by the
numerical description of the parameters and/or the groups of parameters of which the
sensitivities are calculated. Therefore, for each sensitivity analysis the number of
considered hydraulic parameters in a group must be given along with the numbers which
indicate these hydraulic parameters.
By the assignment of a number to each hydraulic parameter, the numerical
description of the hydraulic parameters becomes possible. So the horizontal conductivities
of the different layers are first numbered. Number 1 is assigned to the horizontal
conductivity of layer 1 of the numerical model, number 2 to the horizontal conductivity of
layer 2, etc. After the numbering of all the horizontal conductivities, the numbering is
continued with the hydraulic resistances between the layers of the numerical model. If N9
layers are considered in the numerical model, the number N9+ 1 is assigned to the
hydraulic resistance between the layers 1 and 2, the number N9+2 to the hydraulic
resistance between the layers 2 and 3, etc. After the hydraulic resistances, the specific
elastic storages are numbered. The number 2N9 is assigned to the specific elastic storage
of layer 1; the number 2N9 + 1 is assigned to the specific elastic storage of layer 2 of the
numerical model; etc. The number 3N9 is assigned to the storage coefficient near the
watertable or to the specific yield.
The following parameters, which are numbered, are the two parameters that
describe the lateral anisotropy of all the layers. The number 3N9+ 1 is assigned to the
lateral anisotropy and the number 3N9+2 to the angle 8 between the north and the
direction with maximum horizontal conductivity. The numbering is continued with the
parameters that define the well loss Crr. The number 3N9+3 is assigned to the C-value
and the number 3N9+4 to the N-power.
There is also a possibility to attach a negative sign to the number of the hydraulic
parameter. In this case the so indicated hydraulic parameters are not multiplied by the
factor lOSENFAC but are devided by the factor 1OSENFAC . By the introduction of this facility
it is possible to consider horizontal conductivities and hydraulic resistances within one
group of hydraulic parameters. For example pose that a lithological layer, which is
homogeneous and transversally anisotropic, is discretized in several layers in the
numerical model. The sensitivities of the drawdown with respect to the hydraulic
conductivity of this lithologically homogeneous layer can be determined with the
assumption that the transversal anisotropy remains constant. Here, the horizontal
conductivities and the hydraulic resistances will be considered in one group where the
corresponding numbers of the hydraulic parameters obtain opposite signs, e.g., a negative
sign is attached to the numbers of the hydraulic resistances while the numbers of the
horizontal conductivities remains positive. With the facility to attach a negative sign to the
number of the hydraulic parameters it becomes also possible to study the sensitivities of
the drawdown with respect to the diffusivity (see Sect. 2.2.3) of a homogeneous layer.
During each iteration of the inverse process the residuals and the sensitivities are
calculated. The residuals are the differences between the logarithm of the calculated
drawdowns and the logarithm of the observed drawdowns. The calculated drawdowns are
according to the hydraulic parameters given in the input file narne1.pap(u) and the
observed drawdowns are given in the input file narnel.dap(u). The calculated sensitivities
for the different hydraulic parameters defined in the input file narnel.pas(e) correspond
with the given drawdowns and hydraulic parameters. The input files narnel.pap(u),
narnel.dap(u) and narnel.pas(e) are linked with the numbers 26, 28 and 29.
The results of the calculations are written concisely in the file narne1.etadfdp
(UNIX) or narnel.eta (MSDOS). This file is linked with number 30. These residuals and
sensitivities can be used as input for the program solpuS which calculates the adjustment
factors with the linearization method (see Sect. 6.6.1). This file can be read in the
EDITOR or printed to check quickly the results of the sensitivity calculations. The order
of the lines of the residual vector r and the sensitivity matrix J correspond with the order
of the given observation. The first line corresponds with the first observed drawdown
given in the first observation block of file narne1.dap(u) (see Sect. 4.7.1). In the first
column the observation number in the respective observation well is given. The residuals
are given in the second column. In the following columns the sensitivities are given in the
order of the defined sensitivity analyses. After each iteration of the inverse process the
residuals and the sensitivities are written in file narne1.etadfdp (UNIX) or narnel.eta
(MSDOS). As a consequence it is possible to study the evolution of the residuals and the
sensitivities after each iteration.
6.2.3 Example of a sensitivity matrix
In Lebbe (1988) some examples of sensitivity calculations are shown. These examples
were based on the validation of the numerical model with the help of the different
analytical models. These were the analytical models of Theis, lacob-Hantush, Hantush,
Hantush-Weeks and Boulton (see Chapter 3). For each model the sensitivity matrix is
given in a table. The sensitivities were also graphically represented in some figures. They
visualize how the sensitivities change with place and time of observation. To obtain a
clear graphical representation of these sensitivities, the concerning hydraulic parameters
were here calculated with a sensitivity factor sf equal to ten. This value is too large for
practical calculations as for example in the nonlinear regression analyses.
Here, only one example of the fore mentioned sensitivity calculations is given; it
is demonstrated for the analytical model of Hantush. For the validation of the analytical
model of Hantush, a pervious layer of 10 m thickness is considered which is pumped with
a discharge rate of 180 m3 /d. This layer is bounded below by an impervious layer and
above by a ten meters thick semi-pervious layer. The top of these semi-pervious layer is a
constant hydraulic head boundary. This boundary condition is obtained by considering a
pervious layer with a very large horizontal conductivity above the semi-pervious layer.
In the numerical model the lower pervious layer is replaced by one layer in the
numerical model (layer 1). The covering semi-pervious layer is replaced by seven layers
in the numerical model (layer 2 until layer 8). The upper pervious layer introduced to
obtain the constant hydraulic head boundary condition is the uppermost layer in the
numerical model (layer 9). The horizontal conductivity and the specific elastic storage of
layer 1 of the numerical model are set respectively equal to 10 mid and lx1O-3 m-I. The
numbers assigned to these hydraulic parameters are respectively one and eighteen.
Because only vertical flow is assumed in the semi-pervious layer, the horizontal conduc-
tivities of the seven layers, which replace the semi-pervious layer, are equal to a very
small value, lxlO-7 mid. The hydraulic resistances between these layers are calculated
according to a vertical conductivity of 0.1 mid. A value of lxlO-3 m-I is attributed to the
specific elastic storages of these layers. Because a value of 1010 mid is given to the
horizontal conductivity of the uppermost layer of the numerical model, a constant
hydraulic head is obtained in this layer. Consequently, the other hydraulic parameters of
these uppermost layer do not influence the calculated drawdowns.
The sensitivities of the drawdown in the pumped pervious layer are calculated
with respect to four different hydraulic parameters or groups of hydraulic parameters. The
first two sensitivity calculations are calculated with respect to one hydraulic parameter of
the numerical model: to the horizontal conductivity (hydraulic parameter number 1) and
to the specific elastic storage (hydraulic parameter number 18) of the pumped pervious
layer. The last two sensitivity calculations are with respect to grouped hydraulic parame-
ters. They concern hydraulic properties of the covering semi-pervious layer. The first
group encloses the hydraulic resistances within the semi-pervious layer (or the hydraulic
parameters assigned with the numbers 10, 11, 12, 13, 14, 15, 16 and 17); the second
group encloses the specific elastic storages of the semi-pervious layer (or the hydraulic
parameter numbers 19,20,21,22,23,24 and 25).
The obtained sensitivities are represented in Table 6.2. In the first part of this
table an echo is given of the parameters which define the space-time grid along with the
hydraulic parameters. The number assigned to the hydraulic parameters are inserted in
this echo. The table is continued by the numbers of the hydraulic parameters which are
included in the different sensitivity analyses. These echoes are followed by the residuals
and the sensitivities per observation well. In the first column the observation number in
the respective well is given. The residuals are given in the second column and are
indicated by the symbol ETA. In the following columns the sensitivities are given. These
columns are indicated by the number of the sensitivity analyses.
The sensitivities of the drawdowns in the pumped pervious layer to the horizontal
conductivity, which will be considered in this paragraph, can have positive as well as
negative values. The sensitivities are all negative for the drawdown between the first and
the 1600th minute of pumping in the observation wells at 1 m and 2.5 m distance from the
pumped well. This means that for a larger value of this conductivity the drawdowns
become smaller. In the observation wells at a distance of 6.3 m from the pumped well
and farther away the considered sensitivities are positive at the start of the pumping
period. This means that the drawdowns increase for increasing values of the horizontal
conductivity. For small values of t/r the considered sensitivities are positive. They are
negative for larger values of t/r2. They are, however, not equal for the same values of tlr
as holds for the Theis model. After a sufficient long time of pumping the sensitivities of
the drawdown of one observation well reach a constant value. This is when the draw-
downs reach a steady state in both models with different values for the horizontal
conductivity.
The sensitivities of the drawdowns in the pumped pervious layer with respect to
the specific elastic storage are always negative. These sensitivities obtain a zero value
when the flow reaches a steady state in the pumped as well as in the covering semi-
pervious layer. For small values of tlr the absolute values of these sensitivities are large.
They decrease for larger values of t/r2. They are, however, not equal for the same values
of tlr as for the Theis model.
the hydraulic resistance of the covering semi-pervious layer are always positive. These
sensitivities are very small just after the start of the pump and at a small distance from
the pumped well. These sensitivities increase according to the increasing pumping time
and distance from the pumped well. They obtain a maximum value when the flow reaches
a steady state in both cases. Mostly, if the values of the hydraulic resistance of the
covering semi-pervious layer and of the horizontal conductivity of the pumped pervious
layer are not too small, these maximum values are significantly smaller than the absolute
values of the sensitivities of the drawdown with respect to the horizontal conductivity of
the pumped pervious layer and to its specific elastic storage before the flow reaches the
steady state.
Table 6.2. First part. Results of the sensitivity analyses of the numerical model validated
with the analytical model of Hantush
RADIUS OF WELLSCREEN,R,IN M,---------------------------- .0.100

DISCHARGE OF PUMPED WELL,Q,IN M'IDAY,---------------- 180.000
INTI1AL TIME, T1,IN MIN, ---------------------------------------- 0.100
LOGARfl1IMIC INCREASE OF TIME AND OF RADIUS OF RINGS
LOGA,------------------------------------------------------------------- 0.100
LATEST CALCULATED TIME, TI, IN MIN ,------------------------------- 1600.
NUMBER OF LAYERS, N,----------------------------------------- 9
NUMBER OF RINGS, M,---------------------------------------------- 43
THE WELLSCREEN IS SITUATED IN LAYER----------------------------- 1
THICKNESS OF THE SUCCESSIVE LAYERS, IN M
THICKNESS OF LAYER UN M, -------------------------------------------- 10.000
THICKNESS OF LAYER 2.IN M, -------------------------------------------- 1.429
THICKNESS OF LAYER 3,IN M, -------------------------------------------- 1.429
THICKNESS OF LAYER 4.IN M, -------------------------------------- 1.429
THICKNESS OF LAYER 5,IN M, ------------------------- --------------- 1.429
THICKNESS OF LAYER 6,IN M, --------------------------------------- 1.429
THICKNESS OF LAYER 8,IN M, ------------------------------------------- 1.429
-------------NUMBER OF HYDRAULIC PARAMETER ----INRI-------------------------
HYDRAULIC CONDUCTIVITY, K(l),IN MIDAY,-------------I 1/--- 10.000
HYDRAULIC CONDUCTIVITY, K(2),IN MIDAY,--------------I 2/---- 0.000
HYDRAULIC CONDUCTIVITY, K(3),IN MIDAY,---------------I 3/---- 0.000
HYDRAULIC CONDUCTIVITY, K(4),IN MIDAY,----------------I 4/---- 0.000
HYDRAULIC CONDUCTIVITY, K(5),IN MIDAY,--------------I 51---- 0.000
HYDRAULIC CONDUCTIVITY, K(6),IN MIDA Y ,---------------1 6/---- 0.000
HYDRAULIC CONDUCTIVITY, K(7),IN MIDAY,----------------I 7/--- 0.000
HYDRAULIC CONDUCTIVITY, K(8),IN M/DA Y ,--------------1 8/---- 0.000
HYDRAULIC CONDUCTIVITY, K(9),IN MIDA Y ,----------------1 9/---- 999999999.999
HYDRAULIC RESISTANCE,c(1 ),IN DAY ,------------------------1 10/---- 7.
HYDRAULIC RESISTANCE,c(2).IN DAY,------------------------/III---- 14.
HYDRAULIC RESISTANCE,c(3),IN DAY,------------------------I 12/---- 14.
HYDRAULIC RESISTANCE,c(4).IN DAY,------------------------I 13/---- 14.
HYDRAULIC RESISTANCE,c(5).IN DAY ,------------------------1 14/---- 14.
HYDRAULIC RESISTANCE,c(6),IN DAY,-----------------------I 15/---- 14.
HYDRAULIC RESISTANCE,c(7),IN DAY,------------------------I 16/---- 14.
HYDRAULIC RESISTANCE,c(8),IN DAY, ------------------------1 17/--- 7.
SPECIFIC ELASTIC STORAGE,SA (I),IN M-',------------------/18/----- 0.100-02
SPECIFIC ELASTIC STORAGE,SA (2),IN M-' ,--------------------/191----- 0.100-02
SPECIFIC ELASTIC STORAGE,SA (3),IN M-' ,-------------------120/----- 0.100-02
SPECIFIC ELASTIC STORAGE, SA (4),IN M-' ,--------------------1211----- 0.100-02
SPECIFIC ELASTIC STORAGE,SA (5),IN M" ,----------------/22/---- 0.100-02
SPECIFIC ELASTIC STORAGE, SA (6),IN M-',------------------/23/----- 0.100-02
SPECIFIC ELASTIC STORAGE, SA (7),IN Ml ,-------------------/24/---- 0.100-02
SPECIFIC ELASTIC STORAGE, SA (8),IN M 1 ,--------------------1251----- 0.100-02
SPECIFIC ELASTIC STORAGE,SA (9),IN M-' ,--------------------126/----- 0.100-02
STORAGE COEFFICIENT AT THE WATERTABLE,SO,--------/27/----- 0.200000
SENSITIVITY ANALYSES
NR NUMBERS OF ADJUSTED PARAMETERS
1 1
2 18
3 10 11 12 13 14 15 16 17
4 19 20 21 22 23 24 25
Chapter 6 I Inverse model 239
Table 6.2. Second part. Results of the sensitiVity analyses of the numerical model
validated with the analytical model of Hantush
OBSERVATION WELL I IN LAYER I OBSERVATION WELL 2 IN LAYER I

AT 1.0 M FROM THE PUMPED WELL AT 2.5 M FROM THE PUMPED WELL
N. ETA I 2 3 4 N. ETA 1 2 3 4
1 -0.0123 -0.6317 -0.3398 0.0029 -0.0001 I -0.0278 -0.2030 -0.7059 0.0046 -0.0001
2 -0.0074 -0.6551 -0.3186 0.0035 -0.0001 2 -0.0178 -0.2966 -0.6261 0.0055 -0.0002
3 -0.0050 -0.6832 -0.2928 0.0041 -0.0002 3 -0.0110 -0.3735 -0.5595 0.0065 -0.0003
4 -0.0024 -0.7007 -0.2766 0.0048 -0.0003 4 -0.0065 -0.4366 -0.5045 0.0076 -0.0004
5 -0.0013 -0.7219 -0.2563 0.0056 -0.0004 5 -0.0033 -0.4899 -0.4571 0.0088 -0.0006
6 0.0001 -0.7356 -0.2432 0.0065 -0.0007 6 -0.0013 -0.5346 -0.4170 0.0101 -0.0009
7 0.0005 -0.7521 -0.2268 0.0074 -0.0009 7 0.0002 -0.5730 -0.3817 0.0114 -0.0013
8 0.0012 -0.7631 -0.2157 0.0083 -0.0013 8 0.0011 -0.6059 -0.3510 0.0128 -0.0019
9 0.0013 -0.7762 -0.2019 0.0093 -0.0019 9 0.0016 -0.6345 -0.3236 0.0141 -0.0027
10 0.0016 -0.7852 -0.1922 0.0102 -0.0026 10 0.0018 -0.6593 -0.2994 0.0153 -0.0037
11 0.0015 -0.7959 -0.1804 0.0110 -0.0035 11 0.0019 -0.6812 -0.2774 0.0163 -0.0049
12 0.0016 -0.8034 -0.1718 0.0117 -0.0045 12 0.0018 -0.7005 -0.2577 0.0172 -0.0064
13 0.0014 -0.8123 -0.1614 0.0123 -0.0057 13 0.0017 -0.7177 -0.2396 0.0179 -0.0080
14 0.0014 -0.8186 -0.1537 0.0128 -0.0070 14 0.0015 -0.7329 -0.2233 0.0184 -0.0098
15 0.0012 -0.8261 -0.1446 0.0133 -0.0083 15 0.0014 -0.7467 -0.2082 0.0189 -0.0116
16 0.0012 -0.8316 -0.1376 0.0138 -0.0096 16 0.0013 -0.7591 -0.1944 0.0195 -0.0133
17 0.0011 -0.8379 -0.1295 0.0145 -0.0108 17 0.0011 -0.7704 -0.1816 0.0202 -0.0149
18 0.0011 -0.8426 -0.1230 0.0152 -0.0120 18 0.0011 -0.7806 -0.1698 0.0210 -0.0164
19 0.0010 -0.8481 -0.1157 0.0161 -0.0132 19 0.0011 -0.7899 -0.1586 0.0220 -0.0179
20 0.0010 -0.8522 -0.1096 0.0170 -0.0143 20 0.0010 -0.7984 -0.1482 0.0230 -0.0193
21 0.0010 -0.8569 -0.1029 0.0181 -0.0155 21 0.0010 -0.8062 -0.1383 0.0242 -0.0207
22 0.0009 -0.8605 -0.0972 0.0191 -0.0167 22 0.0010 -0.8133 -0.1290 0.0254 -0.0221
23 0.0010 -0.8646 -0.0908 0.0203 -0.0179 23 0.0009 -0.8199 -0.1201 0.0268 -0.0235
24 0.0010 -0.8678 -0.0854 0.0216 -0.0190 24 0.0009 -0.8259 -0.1116 0.0282 -0.0249
25 0.0008 -0.8713 -0.0794 0.0231 -0.0201 25 0.0008 -0.8315 -0.1034 0.0300 -0.0261
26 0.0009 -0.8741 -0.0742 0.0250 -0.0209 26 0.0008 -0.8366 -0.0956 0.0323 -0.0270
27 0.0006 -0.8772 -0.0684 0.0276 -0.0212 27 0.0006 -0.8414 -0.0880 0.0355 -0.0272
28 0.0004 -0.8796 -0.0631 0.0313 -0.0209 28 0.0003 -0.8457 -0.0805 0.0400 -0.0267
29 -0.0002 -0.8822 -0.0572 0.0362 -0.0199 29 -0.0002 -0.8496 -0.0728 0.0460 -0.0253
30 -0.0009 -0.8842 -0.0515 0.0424 -0.0183 30 -0.0013 -0.8531 -0.0649 0.0536 -0.0231
31 -0.0020 -0.8862 -0.0450 0.0496 -0.0163 31 -0.0030 -0.8562 -0.0567 0.0624 -0.0205
32 -0.0037 -0.8878 -0.0386 0.0575 -0.0140 32 -0.0052 -0.8588 -0.0483 0.0723 -0.0176
33 -0.0064 -0.8890 -0.0317 0.0660 -0.0117 33 -0.0080 -0.8607 -0.0397 0.0827 -0.0146
Table 6.2. Third part. Results of the sensitivity analyses of the numerical model
OBSERVATION WELL 31N LAYER I OBSERVATION WELL 41N LAYER I

AT 6.3 M FROM THE PUMPED WELL AT 15.8 M FROM THE PUMPED WELL
N. ETA I 2 3 4 N ETA I 2 3 4
I -0.1175 1.6933 -2.2331 0.0073 -0.0001 I -0.0392 2.0692 -2.4920 0.0363 -0.0025
2 -0.0730 1.2011 -1.8439 0.0087 -0.0002 2 -0.0203 1.5228 -2.0625 0.0412 -0.0039
3 -0.0449 0.8291 -1.5455 0.0104 -0.0003 3 -0.0093 1.1036 -1.7255 0.0461 -0.0059
4 -0.0271 0.5418 -1.3119 0.0123 -0.0005 4 -0.0031 0.7756 -1.4566 0.0505 -0.0087
5 -0.0154 0.3154 -1.1253 0.0144 -0.0007 5 0.0003 0.5157 -1.2396 0.0541 -0.0123
6 -0.0081 0.1349 -0.9746 0.0166 -0.0011 6 0.0018 0.3075 -1.0628 0.0566 -0.0168
7 -0.0032 -0.0111 -0.8510 0.0190 -0.0017 7 0.0022 0.1392 -0.9176 0.0580 -0.0219
8 -0.0005 -0.1304 -0.7484 0.0214 -0.0026 8 0.0022 0.0013 -0.7971 0.0585 -0.0273
9 0.0013 -0.2287 -0.6624 0.0237 -0.0037 9 0.0020 -0.1123 -0.6966 0.0584 -0.0325
10 0.0020 -0.3104 -0.5897 0.0257 -0.0052 10 0.0015 -0.2070 -0.6120 0.0581 -0.0373
II 0.0025 -0.3790 -0.5276 0.0275 -0.0071 II 0.0013 -0.2865 -0.5401 0.0582 -0.0413
12 0.0025 -0.4368 -0.4742 0.0288 -0.0094 12 0.0012 -0.3539 -0.4787 0.0585 -0.0447
13 0.0024 -0.4861 -0.4279 0.0297 -0.0120 13 0.0009 -0.4113 -0.4256 0.0593 -0.0476
14 0.0020 -0.5282 -0.3876 0.0303 -0.0148 14 0.0010 -0.4606 -0.3795 0.0603 -0.0501
IS 0.0018 -0.5647 -0.3523 0.0306 -0.0176 15 0.0008 -0.5031 -0.3391 0.0615 -0.0524
16 0.0014 -0.5964 -0.3211 0.0310 -0.0203 16 0.0008 -0.5398 -0.3035 0.0629 -0.0545
17 0.0013 -0.6242 -0.2934 0.0315 -0.0227 17 0.0007 -0.5718 -0.2719 0.0645 -0.0564
18 0.0012 -0.6486 -0.2686 0.0323 -0.0248 18 0.0006 -0.5998 -0.2437 0.0666 -0.0579
19 0.0012 -0.6703 -0.2463 0.0332 -0.0267 19 0.0005 -0.6244 -0.2184 0.0697 -0.0588
20 0.0011 -0.6894 -0.2260 0.0344 -0.0285 20 0.0002 -0.6460 -0.1953 0.0745 -0.0584
21 0.0010 -0.7065 -0.2075 0.0356 -0.0303 21 -0.0002 -0.6651 -0.1739 0.0818 -0.0565
22 0.0009 -0.7218 -0.1906 0.0370 -0.0320 22 -0.0013 -0.6818 -0.1538 0.0922 -0.0528
23 0.0009 -0.7355 -0.1749 0.0385 -0.0337 23 -0.0032 -0.6963 -0.1343 0.1056 -0.0476
24 0.0009 -0.7479 -0.1605 0.0402 -0.0353 24 -0.0061 -0.7086 -0.1153 0.1215 -0.0416
25 0.0008 -0.7590 -0.1470 0.0422 -0.0367 25 -0.0102 -0.7190 -0.0966 0.1393 -0.0352
26 0.0007 -0.7691 -0.1343 0.0449 -0.0376 26 -0.0157 -0.7272 -0.0784 0.1582 -0.0289
27 0.0005 -0.7782 -0.1224 0.0489 -0.0377
28 0.0001 -0.7864 -0.1108 0.0546 -0.0367
29 -0.0007 -0.7937 -0.0994 0.0623 -0.0345
30 -0.0020 -0.8002 -0.0880 0.0721 -0.0314
31 -0.0041 -0.8052 -0.0764 0.0835 -0.0276
32 -0.0071 -0.8105 -0.0647 0.0963 -0.0236
33 -0.0110 -0.8141 -0.0529 0.1097 -0.0195
Table 6.2 Fourth part. Results of the sensitivity analyses of the numerical model
OBSERVATION WELL 5 IN LAYER I OBSERVATION WELL 6 IN LAYER I

AT 39.8 M FROM TIlE PUMPED WELL AT 100.0 M FROM TIlE PUMPED WELL
N. ETA I 2 3 4 N. ETA I 2 3 4
I -0.0216 2.0823 -2.3723 0.1306 -0.0453 I -0.0386 2.2871 -2.2167 0.2889 -0.2181
2 -0.0124 1.5569 -1.9538 0.1302 -0.0572 2 -0.0260 I. 7491 -1.8023 0.2808 -0.2248
3 -0.0072 \.1520 -1.6246 0.1283 -0.0687 3 -0.0183 1.3330 -1.4765 0.2753 -0.2284
4 -0.0045 0.8335 -1.3619 0.1259 -0.0787 4 -0.0131 1.0040 -1.2161 0.2730 -0.2290
5 -0.0030 0.5792 -1.1499 0.1238 -0.0867 5 -0.0099 0.7402 -1.0053 0.2753 -0.2256
6 -0.0020 0.3732 -0.9766 0.1225 -0.0929 6 -0.0086 0.5260 -0.8318 0.2853 -0.2168
7 -0.0014 0.2047 -0.8338 0.1218 -0.0976 7 -0.0095 0.3504 -0.6863 0.3057 -0.2014
8 -0.0010 0.0659 -0.7150 0.1217 -0.1012 8 -0.0134 0.2057 -0.5614 0.3379 -0.1798
9 -0.0007 -0.0493 -0.6156 0.1221 -0.1041 9 -0.0210 0.0870 -0.4524 0.3813 -0.1538
10 -0.0005 -0.1458 -0.5314 0.1231 -0.1061 10 -0.0328 -0.0090 -0.3562 0.4331 -0.1260
11 -0.0006 -0.2270 -0.4600 0.1254 -0.1071
12 -0.0008 -0.2958 -0.3984 0.1300 -0.1063
\3 -0.0011 -0.3544 -0.3447 0.1384 -0.1028
14 -0.0024 -0.4045 -0.2969 0.1519 -0.0963
15 -0.0049 -0.4470 -0.2536 0.1707 -0.0869
16 -0.0090 -0.4829 -0.2135 0.1945 -0.0757
17 -0.0152 -0.5125 -0.1760 0.2220 -0.0636
18 -0.0233 -0.5362 -0.1410 0.2518 -0.0517
the specific elastic storage of the covering semi-pervious layer are always negative. At
small distances from the pumped well they show a typical course in time. At the start of
the pumping test they are very small. According to the progress of the pumping test, the
absolute values of these sensitivities reach a maximum value where after they decrease
until zero when the flow reaches a steady state. At larger distances from the pumped well
these sensitivities of the first measurable drawdowns are already significant. Their
absolute values reach a maximum after a somewhat later time then at a smaller distance
from the pumped well. These maximum values for the sensitivities increase according to
the enlarging distances from the pumped well.
6.2.4 Graphical representation of sensitivities
In Fig. 6.1 a time-drawdown course (solid line) is represented. The drawdowns arise in
an observation well at 40 m from the pumped well in a semi-confined aquifer with
hydraulic parameters as described in Sect. 6.2.3. These parameters were also used for the
verification of the numerical model with the model of Hantush (Sect. 4.6.3). The dashed
line is the time-drawdown course at the same distance from the pumped well and in the
same layer for which all the hydraulic parameters are the same except one. The value of
the hydraulic conductivity of the pumped pervious layer was changed from 10 mid to 100
mid. The sensitivities correspond now with the differences between the logarithms of the
drawdowns at the same distance from the pumped well and after the same time of
pumping. These sensitivities should be obtained if the logarithms of the sensitivity factor
is one. The positive sensitivities are indicated with an upward directed arrow, the
negative sensitivities with a downward directed arrow. In the same figure, Fig. 6.1, the
sensitivities of the drawdowns after ten minutes of pumping with respect to the horizontal
conductivity are represented in a distance-drawdown graph. These sensitivities are
negative for distances smaller than 17 m from the pumped well and are positive for
distances larger than 17 m. Around this distance and after ten minutes of pumping the
sensitivities are very small.

r=40m t .... 10min
10·
= 10 mid
10-1 1 ~
.....'.tIJj.
10-2 2
Kt, = 100 II>
\. \
\
10-3 10·J
If\
10· 1'0 2 10' 10' TIME (min) 100 10 1 10 2 ."
10 DISTANCE 1m;
Fig. 6.1. Graphical representation of sensitivities of drawdowns in the pumped pervious

layer to the horizontal conductivity of this layer according to the model of Hantush
The sensitivity of the drawdown in the pumped layer with respect to its horizontal
conductivity can also be deduced from the asymptotic solution of Hantush (see Eq. 3.121,
Sect. 3.4.4). The horizontal conductivity of the pumped layer influences the three factors
in this equation. These factors are Q/41fK.,D, u and {3. The factors Q/41fKhD and u define
the location of the Hantush-type curves and the factor {3 the shape of these curves. In Fig.
6.2 some sensitivities of the drawdown in the pumped layer with respect to the specific
elastic storage of the pumped layer are represented. The sensitivities are shown in a
time-drawdown graph for drawdowns at a distance of 40 m from the pumped well and in
a distance-drawdown graph for drawdowns after ten minutes of pumping. The specific
elastic storage of the pumped layer influences the factors u and {3. The factor u defines
the locations of the Hantush-type curves. These curve locations determine in a large
extent the drawdown sensitivity in the pumped layer with respect to the specific elastic
storage. The factor {3 determines the small shape change of the time-drawdown curves.
Consequently, the contribution of this shape change in the here treated sensitivities is limited.
I
r=-'Om t ... l0min
I
Ss (1) a 0,001 m'rfnT n1DJ JJ I-'
,~ Ss(,).o,l, m,l
1
1/' I
tl~ Ss (1) c 0,01 m,l Ss(I)= 0,01 mJ~.
I z
,,
I
~
l
,
.
10-3
10'
II 10 TIME (mon)
\ 3
10 DISTANCE (m)

layer to the specific elastic storage of this layer according to the model of Hantush
DRAWDOWN (m) DRAWDOWN (ml

r=40m l-l000mln
10· ~
.........
~ :n),
~rr"fnI , 111111
./ c . 11111 d
, c .1111 d
11,·
~
/
I
•
\
l
\
'0 • '0
j
, .
'0 TIME (mon) ,11' 10
, '0
z
It l
10 DISTANCE (m)

layer to the hydraulic resistance of the covering semi-pervious layer according to the
model of Hantush
The hydraulic resistance and the specific elastic storage of the covering semi-pervious
layer determine only the fJ factor jointly with the above treated parameters of the pumped
layer. As stated already above, the fJ factor determines only the shape of the curves.
Consequently, the sensitivities of the drawdown in the pumped layer with respect to the
hydraulic parameters of the covering layer are rather small. A graphical representation is
given for the drawdown sensitivities in the pumped layer with respect to the hydraulic
resistance of the covering layer in Fig. 6.3, These sensitivities are represented at the left
part of Fig.6.3 in a time-drawdown graph for drawdowns that occur at a distance of 40 m

from the pumped well. In the right part of this figure the sensitivities are represented in a
distance-drawdown graph for drawdowns that occur after thousand minutes of pumping
In Fig. 6.4 the sensitivities of the drawdown in the pumped layer with respect to the
specific elastic storage of the covering layer are graphically represented. At one side of
this figure these sensitivities are represented in a time-drawdown graph for drawdown that
occur at a distance of 40 m from the pumped well. At the other side of this figure these
sensitivities are shown for drawdowns that occur after a thousand minutes of pumping.
DRAWDOWN (m)
...
DRAWDOWN (m)
r",,40m t= 1000 min
~ ~=O'OOlm-l
So (2-8) )0,001 m- 1
~~
.<111f£ ,
~~(2-8)=0'01 ~
I
m- 1 So 12-8) =0,01 m- 1
I~
!I
2
.~
J
10 • 10
, 10 2 10
, •
10 TIME (min) 1ff 10' 10'
\~
10· DISTANC E (m)

layer to the specific elastic storage of the covering semi-pervious layer according to the
model of Hantush
From these sensltlvtIes of the drawdown in the pumped layer one can learn that
the drawdown in the pumped layer is very sensitive to the hydraulic parameters of the
pumped layer itself. These drawdowns are, however, considerably less sensitive to the
hydraulic parameters of the covering layer. Therefore, the hydraulic parameters of the
pumped layer can be derived from the drawdown in the pumped layer with a large
precision. The hydraulic parameters of the covering layer can only be deduced with a
considerable smaller precision. If one wants to deduce these last mentioned hydraulic
parameters with a better precision, one must try to observe drawdowns that are consider-
ably more sensitive to these parameters such as the drawdown in this covering layer.
These drawdowns must in addition be sufficiently large so that the relative observation
error is negligible.
6.3 Numerical nonlinear regression
6.3.1 Optimal values of hydraulic parameters
The optimal values of the hydraulic parameters are here defined as the values of the
hydraulic parameters for which the sum of the weighted squared residuals is minimized.
This sum S can be formulated as:
n
S = LWiiI; or S = r2'wr (6.6)
i=1
where n is the number of observations and W ii is the weight of the ith observation. In the
ordinary least square (OLS) method all the weights are equal to 1. In the matrix notation
r is the vector of the residuals and rT is the transpose of the residual vector, whereas w is
the diagonal matrix of the weights.
With the help of the estimated initial values of the hydraulic parameters the
drawdowns are calculated at the same times and places as the observations. First the
residuals are calculated followed by the sensitivities of the drawdown for the hydraulic
parameters or groups of hydraulic parameters which one wants to deduce from the
observations. By means of the linearization method (Draper and Smith, 1981) one can
compute the adjustment factors of these parameters so that the sum of the weighted square
of the residuals is minimized:
A = (J .....7)-l J" .. r (6.7)
where A is the vector of the logarithms of the adjustment factors of the different parame-
ters. In this linearization method only the finite-difference approximations of the first
order derivatives of the logarithm of the drawdown with respect to the logarithm of the
hydraulic parameters are used. The second order derivations and the mutual influence of
the hydraulic parameters are not considered. Therefore the deduced adjustment factors
will result only in an approximation of the optimal values for which the sum of the
weighted squared residuals has a minimum value.
The new estimated values of the parameters are obtained by adding the logarithms
of the adjustment factors to the corresponding logarithmic values of the J"th parameter of
the former iteration or:
(6.8)
where hPt is the logarithmic value of the jth parameter during the mth iteration of the
inverse process,
and At is the logarithm in base 10 of adjustment factor of the jth parameter calculated
after the mth iteration.
The algorithm is repeated until the adjustment factors become very small and the sum of
the weighted squares of the residuals S reaches a minimum value after which the
distribution of the residuals is analysed (see Sect. 6.3.3). If the number of outliers in the
residuals is rather high, the ordinary least square (OLS) method is followed by a weighted
least square (WLS) method. The used WLS-method is the biweighted least square
(BWLS) method as described by Wonnacott and Wonnacott (1985). In this method, a kind
of standardized residual u is calculated:
u = r / (3 IQR) (6.9)
where IQR is the interquartile range of the residuals or the distance between the lower
and upper quartiles of the residuals. The lower and upper quartiles correspond with a
cumulated frequency or probability of 25 % and 75 % (also called the 25th and the 75 th
percentile) .
The weight is now given in a diagonal matrix w where the weight of the ith
observation is given in the diagonal element wli :
Wii = (l-Ul) 2 if Iuil :;; 1

(6.10 )
Usually it is, however, impossible to include all the deducible parameters at the
start of the iteration process. When very sensitive parameters are included along with
much less sensitive parameters in the iteration process, it happens frequently that the
Hessian matrix JTwJ is not well conditioned. This results in unreliable values for the
adjustment factors. The smaller the sensitivities of the parameters the less reliable are
their adjustment factors. To overcome this problem, it is advisable to introduce first the
most sensitive parameters in the iteration process. The minimization process is then
continued until it converges to an optimum solution with these limited number of
hydraulic parameters. After the optimum is reached, the other parameters are successively
included in the iteration process in a sequence according to the order of magnitude of
their sensitivities.
6.3.2 Uniqueness, identifiability and stability
Each time that the number of parameters is increased in the inverse process one should
investigate if this process is still well posed. An inverse process is well posed, if the
process fulfils following three requirements (Carrera and Neuman, 1986b): the solution
must exist, the solution is unique and the solution is stable. If the inverse problem fails to
satisfy one or more of these three requirements then the problem suffers to as being ill-
posed.
The solution is nonunique in the case of different local minima or if a global
minimum occurs at more than one point in the parameter space. Uniqueness must be
distinguished from the notion of a parameter "identifiability". The question of identifiabi-
lity is equivalent to asking whether different parameter sets may lead to one given set of
drawdown data, if so, the parameters are unidentifiable. Unidentifiable parameters are
parameters which do not influence the calculated drawdown at the corresponding times
and distances of the observations. Their sensitivities are too small.
The question of unidentifiability refers thus to the forward problem. In contrast to
this, uniqueness refers to the inverse problem. It concerns the question whether different
parameters may originate from one given set of drawdown observations, if so, the
parameters are nonunique. In the following sections the different ways to obtain informa-
tion about the uniqueness of the problem are explained. These ways concern the study of
the marginal and conditional standard deviations, of the partial correlation coefficients, of
the two- and three-dimensional cross sections through the approximate or through the
exact joint confidence region of the hydraulic parameters and the study of the condition
indexes and marginal variance-decomposition proportions. When a solution is sought by
the minimization of only the sum of the squared drawdown residuals, identifiablility is
necessary but it is not sufficient for uniqueness.
Stability is the condition that small errors in the drawdown data must not result in
large changes in the computed parameters with the inverse numerical model. Instability
manifests itself in spatially oscillating parameters. Instability often arises from the lack or
poor degree of identifiability. It is generally associated with an estimation criterion that is
flat near the minimum. Furthermore, the solution obtained with the inverse process
depends quite heavily on the point of departure which may give the false impression that
the solution is nonunique due to local minima.
According to Carrera and Neuman (1986b), it is important to recognize the
circumstances that mayor may not allow solution of the inverse problem and, if a
solution is possible, to impose on it the proper limitations to make it mathematically well
behaved and physically meaningful. To achieve this goal, one must have a clear insight of
the terms uniqueness, identifiability and stability. These terms are extensively treated by
Carrera and Neuman (1986b). The readers which want to obtain a thorough knowledge of
these terms are therefore referred to this paper. In addition, one must understand how
each effect influences the behavior of the inverse process. Carrera and Neuman (1986b)
also treat some mitigation measures to overcome some adverse process of weakly iII-
posed problems. So, they propose to include prior information in the inverse problem.
Here identifiability is neither necessary nor sufficient for uniqueness. In this work the
inclusion of prior information in the process is not treated. It is the author's opinion that a
rough estimation of nearly unidentifiable parameters is sufficient to introduce in the
inverse process and they do not influence much the result of the inverse process. By
means of several post optimizations it might be proven that these unidentifiable or nearly
unidentifiable parameters respectively do not influence the solution or influence only
slightly the results of the inverse process. The post-optimization process is further
considered in Sect. 6.3.6 where the practical steps are explained of the interpretation of a
pumping test with the inverse model.
6.3.3 Analysis of residuals
Once a numerical model is fitted to the observations by means of regression, one should
examine the residuals if they exhibit the assumption made to apply the regression analysis
(Draper and Smith, 1981). The assumptions are that the residuals are independent, have a
zero mean, a constant variance and follow a normal distribution. If the fitted numerical
model is correct, the residuals should exhibit tendencies that tend to confirm the assump-
tion, or at least, should not exhibit a denial of the assumptions. This last mentioned
possibility does not mean that the assumptions are correct.
Draper and Smith (1981) emphasize that graphical procedures involving visual
analyses are the most valuable tools to examine the residuals. Therefore, the residuals can
be plotted versus the cumulative frequency or probability along with the best fitted normal
distribution (see Sect. 6.6.3). This plot allows us to conclude if the residuals do not
approximate a normal distribution.
The dependencies of the residuals against the independent variables such as the
time since the start of the pumping test, the distance to the pumped well and the layer (or
observation level) can be examined from the tables printed as output of the proper
simulation program (see Sect. 4.7.3). In these tables the logarithms of the calculated
drawdowns corresponding with the optimal values are printed along with the logarithms
of the observed drawdowns and their differences, the residuals. Because the observations
of one well are arranged according to their observation time, it is very easy to deduce if
these residuals are time dependent. An example of time dependency is that the residuals
of the observations made at the start of the pumping test have all the same sign.
If the observed drawdowns are given in the input file in a sequence according to
their observation distance and layer, it is also easy to detect the dependency of the
residuals to the observation distance and/or layer. Usually the observations made in the
pumped layers are given first. They are followed by the observation in the adjacent layer
according to their distance to the pumped layer. The wells of one layer are given
according to their distance to the pumped well: first the observation well the closest to the
pumped well followed by the wells with an increasing distance to the pumped well.
Studying the so classified tables, it is easy to detect the distance dependency of the
residuals. An example of distance dependency of the residuals is that the residuals are all
positive at a small distance to the pumped well and are all negative at larger distances to
the pumped well or vice versa. An example of layer or level dependency is that all
residuals of observations made in a certain layer have the same sign. The level-, distance-
and time-dependency of the residuals can best be deduced by the visual analysis of the
time-drawdown and distance-drawdown graphs in which the observed and calculated
drawdowns are represented simultaneously.
Draper and Smith (1981) and Cooley and Naff (1990) proposed still another
method to study the residuals. In this method the residuals are plotted versus the calcula-
ted drawdown according to the optimal parameter values. If the chosen numerical model
is correct, the plot should display a roughly horizontal band of the residuals. If the plot
display shows an unequal band width, the variance is not constant. This may indicate a
need for an other transformation of the observed drawdown or the need to apply other
weights in the least square analyse.
6.3.4 Joint confidence region of hydraulic parameters
6.3.4.1 Joint confidence region approximated by p-dimensional ellipsoids
When the assumption that the residuals exhibit a normal distribution is not violated as
well as the assumption that the drawdowns can be approximated as a linear function
within the considered region then the joint confidence region can be described by the
optimal values and the variance-covariance matrix of the parameters covp:
(6.11)
where U,2 can be estimated as (~n;=lw;l2J/(n-p) where n is the number of observations and
p the number of parameters.
Taking the above mentioned assumptions into consideration, the bounds of the
joint confidence regions can be approximated by concentric p-dimensional ellipsoids. Let
us first consider a rather simple case where p, the number of the estimated parameters, is
equal to two. The joint confidence region for two parameters is now approximated by an
ellipse centred around the optimal values as represented in Fig. 6.5. The lengths of the
principal axes are proportional to the square root of the eigenvalues 'Y of the variance-
covariance matrix. The equation of one of these concentric p-dimensional ellipsoids in a
coordinate system XE with its origin in the center of the ellipsoid and with the axes
parallel with the principal axes of the ellipsoid can be written as:
p
E XE1
i=1
/ Yi = 1 (6.12)
The eigenvectors {j of the variance-covariance matrix defines the direction of the axes.
The first eigenvector corresponds with the projection of a unit length of the first principal
axis of the p-dimensional ellipsoid on the respective parameter axes. With the aid of the
eigenvectors the coordinates of system XE can be converted to system lIP of which the
axes are parallel to the parameter axes but of which the origin is located in the center of
the ellipsoid or in the optimal values:
p
HPj = ~ J3 ij XE i for j 1, P (6.13)

hp,= logH,
HP,
HP'
~,--------f-----~~---,~----~
hp,= logH,
Fig. 6.5. Different coordinate systems considered in the parameter space: H, coordinate
system of parameter values of which the units correspond with those of the different
hydraulic parameters, bp, coordinate system of the logarithm of the parameter values, the
units are dimensionless, HP, coordinate system with axes which are parallel to the
coordinate system bp and with the same units but with different origins (origin of the HP
coordinate system is located in the optimal solution corresponding with the optimal
parameter values hp), XE, coordinate system with same units and origin as system HP of
which the axis are parallel to the principal axes of the p-dimensional ellipsoid which
approximate a joint confidence region
So the values HP can be calculated with the aid of the coordinates in the system XE:
P
XE i = E Pj i HPj for i = 1, P (6.14)
j=l
Substitution of Eq. 6.14 in Eq. 6.12 results in the equation of a p-dimensional ellipsoid
in function of the coordinates of system HP:
p p
E [(E Pj i HPj) 2 /y i l 1 (6.15)
i=l j=l
The relation between the logarithm of the parameter values bp and the coordinates of the
system is given by:
forj=l,p (6.16)
The marginal standard deviation slI1 of the r parameter can now be approximated
as the square root of the jth diagonal term of the variance-covariance matrix. By multiply-
ing the marginal standard deviation slI1 with the square root of p.F(p,n-p,l-ex) one obtain
the term cimj is calculated for a given significance level (1-ex). This term defines the
marginal confidence interval, hpj ±cimj' in the logarithmic parameter space. F(p,n-p, I-ex)
is the F-distribution for p and n-p degrees of freedom and a significance level (1-ex). As
Draper and Smith (1981) state (p. 95) a simultaneous interpretation according to the
marginal confidence intervals hpl±cim1 and hP2±Cim2 of for example two parameters may
lead to a wrong estimate of the extension of the joint confidence region. It may be
thought that it corresponds with the rectangular area shown in Fig. 6.6 (Draper and
Smith, 1981). This rectangular region overestimates the joint confidence region especially
when two parameters are correlated. Here the joint confidence area is bounded by a long
thin ellipse. The partial correlation coefficient between the parameters hpj and hpj+l is
covp j,j+/(smj.slI1+1)'
The construction of the joint confidence ellipse is not difficult when only two
parameters are involved. When four or more parameters are involved, the interpretation
and perception of the joint confidence region are not so simple. Draper and Smith (1981)
proposed a possible solution in finding the coordinates of the points at the end of the
major axes of the region (for example, points x, x', y and y' in Fig. 6.6). Lebbe and De
Breuck (1995) pose that this possible solution is not convenient enough, especially, when
more than three parameters are estimated (p > 3). The points at the end of the major axes
of the p-dimensional ellipsoid have p-coordinates in the p-dimensional parameter space
which differ mutually so that they are not easy to interpret. It is, however, easier to
interpret the coordinates of the intersection points of the ellipsoid with the axes parallel to
the parameter axes and which go through the optimal values. In Fig. 6.6, this points
correspond with the points a, b, c and d. In this way, the description and the interpreta-
tion of these points are considerable simplified, especially, if more than three parameters
are estimated. The coordinates of these points are equal to the optimal value of the
hydraulic parameters except one. This exception is equal to the optimal value of the
corresponding parameter plus or minus the conditional standard deviation SCj of this jth
parameter multiplied by a factor Fa. This factor Fa is equal to the square root of the
product p.F(p,n-p,l-ex).
The p-dimensional ellipsoid which approximates the joint confidence region can
now be represented by cross sections which go through the optimal values and which are
in each case parallel to another combination of two parameter axes. These cross sections
are each time ellipses (see Fig. 6.6). The equation of those ellipses can be derived from
Eq. 6.15 in which all the values of lIP are put equal to zero with the exception of two
parameters HP; and HPj or:
HPrt P1k +2HP.HP.t PikP jk +HP~t P}k = Fa 2 (6.17)

k=l yk 2 J k=l Yk J k=l Yk
From this formula one can deduce the term cicj which defines the conditional confidence
interval hpj±cicj of the jth parameter by putting the value of HP; equal to zero:
(6.18)
So the conditional standard deviations of the jth parameter can be approximated by means
of the eigenvalues and the eigenvectors of the variance-covariance matrix (Lebbe. 1988):
(6.19)
The conditional standard deviations help to locate the intersections of the parameter axes
which go through the optimal values and the bounds of the confidence region which is
approximated following the linearization method. In the case where the conditional
standard deviation is only a fraction smaller than the marginal standard deviation of a
certain parameter then this hydraulic parameter has no large correlations with the other
parameters. If. however. there is a large difference between the conditional and the
marginal standard deviations of a considered parameter then there is a correlation between
this parameter and one or several other parameters that one wants to optimize. The
comparison of both standard deviations of a certain parameter can be viewed as a help in
the regression or collinear diagnostic.
Knowing the correlations between the parameters and their standard deviations one
can now better evaluate the joint confidence region as indicated by the densely dotted area
in Fig. 6.6. The correlation between the parameters in the example shown in Fig. 6.6 is
positive. Therefore the parameter values (hpi + Fa.sffi l .hp2+ Fa.sffi2 ) approximate the upper
right point of the confidence area and (hpcFa.sffil.hp2-Fa.sffi2) approximate the lower left
point of the confidence area. When the correlation between the parameters is negative
then the joint confidence region is characterized by an upper left point and a lower right
point corresponding with the parameter combinations (llPcFa.sffil.hp2+Fa.sffi2) and
(hpI+Fa.sffil.hp2-Fa.sffi2)' The points (hp 10 hP2 + Fa.sc2). (hplohp2-Fa.sc2). (hpI+Fa.sc l .hP2)
and (hpcFa.scI.hP2) are on the limits of the joint confidence region. The p-dimensional
joint confidence region is limited by the following 2p points: (hpl±Fa.sc1.hP2 •...• hpp).
(hPlohp2±Fa.sc2 •...• hpp) •...• (hpl.hP2 •... ,hpp±Fa.scp). Evidently. all these points are
located within the logarithmic parameter space.
Because bp represents the logarithmic values of the hydraulic parameters. the
interval hprFa,scj. hpj + Fa,scj given in the logarithmic space can be given in the arithme-
tic parameter space as H/Cfl_",cj• Hj.Cfl_",cj where:
(6.20)
The factor Cfl-acj is here called the conditional confidence factor. The marginal confidence
factor Cfl_.,mj can now be defined in the same way. The interval hprFa.smj, hp!+ Fa. smj
transformed to the arithmetic parameter space corresponds with H/Cfl_.,mj, Hj.Cfl_.,mj
where:
(6.21)
r x
~12
'1----;--- 'Vfi:2
0.90 0.92 0.94 0.96 0.98 1.08 1.10
Fig. 6.6. Joint confidence region in the parameter space (hp/hp),hP2/hp2) along with the
marginal confidence intervals c~( = Fa. sill;) and the conditional confidence intervals
cicl = Fa.scj) with j = 1,2. Source: Lebbe, L. and De Breuck, W. 1995. Validalion of an
inverse numerical model for the interpretation of pumping test and a study of factors
influencing accuracy of results. Journal of Hydrology 172 (1995), p. 61-84, p.71.
Copyright <D 1995, Elsevier Science, reprinted by permission of Elsevier Science.
6.3.4.2 Cross sections through joint confidence region
For nonlinear problems the exact joint confidence region is obtained by drawing the
contour lines of the sums of the weighted squared residuals, S(bp). The different values
of these sums for the different significance levels are determined with following equation:
S(12'p) S(.qp) (l+--LF(p,n-p, i-IX)) (6.22)

n-p
where S(hp) is the sum of the weighted squared residuals corresponding to the optimal
values of the hydraulic parameters, hp, where p is the number of deduced parameters, n
is the number of observation and F(p,n-p,l-a) is the F-distribution for p and n-p degrees
of freedom and a significance level I-a. For each considered parameter combination, the
sums of the weighted squared residuals must be calculated with the numerical model.
These contours are in the linear as well as in the nonlinear case exact confidence
contours. In the nonlinear case, the level of probability is approximated (Draper and
Smith, 1981).
When only two parameters are involved, the confidence contours can be drawn in
a graph of which the two axes correspond with the two parameters. If three parameters
are considered, the joint confidence space is a three-dimensional ellipsoid. This ellipsoid
can be represented in a three-dimensional graph which is still interpretable. For p
parameters the contours can be drawn in p!/(2!(p-2)!) two-dimensional cross sections
through the p-dimensional ellipsoid. In each case p-2 parameters are set equal to their
optimal values while the two other parameters vary around their optimal values. For each
set of values of the parameters, the corresponding sums of the weighted squared residuals
are calculated each time with the numerical model for the cross sections through the exact
joint confidence region. As Banchoff (1996) states this slices will help us to vizualize
dimensions higher than our own.
Cross sections through the approximate joint confidence region are obtained when
the residuals are first calculated by means of the residuals at the optimal solution and
their sensitivities by the application of the linearization method with following equation:
r(12'p) ~ r(n.p) + J(12'p-n'p) (6.23)
where J is the sensitivity matrix corresponding with the optimal logarithmic values hp.
The so obtained residuals are squared and summed for each parameter value set. These
sums of the weighted squared residuals can again be represented by contour lines. These
contours can also be called the two-dimensional conditional confidence areas in analogous
way as that the conditional confidence interval indicates the one-dimensional section
through the p-dimensional approximate joint confidence region.
Through the p-dimensional ellipsoid, also p!/(3!(p-3)!) three-dimensional cross
sections can be drawn. These three-dimensional graphs are still interpretable. For each
three-dimensional cross section, p-3 parameters are set equal to their optimal values while
the three remaining parameters vary around their optimal values. These cross sections are
the limits of the three-dimensional conditional confidence spaces. Cross sections through
the exact joint confidence region are obtained when the residuals are calculated by the
numerical model for each parameter combination. The cross sections are through the
approximated joint confidence region if the residuals are calculated according the
linearization method (Eq. 6.23).
With the conditional confidence factors of the hydraulic parameters and their
optimal values, it was possible to determine 2p parameter combinations, or in other words
2p points in the parameter space, which are situated on the bound of the approximate
joint confidence region. The two- and three-dimensional cross sections through the exact
or approximate joint confidence region can help us to define a large number of parameter
combinations which are situated on the bounds of the joint confidence region. These
parameter combinations can be used to approximate, in a first attempt, the confidence
intervals of the drawdown. However, one has to realize that these parameter combinations
represent only a small part of all points of the bounds of the joint confidence region.
6.3.5 Condition indexes and matrix of marginal variance-decomposition proportions
Besides the partial correlation coefficients, the eigenvalues and the eigenvectors of the
variance-covariance matrix, the marginal and conditional standard deviation and the cross
sections through the joint confidence region also the condition indexes and the matrix of
marginal variance-decomposition proportions can help us in the collinear diagnostic.
According to Belsley (1990), the marginal variance of the jth hydraulic parameter varmj
can be written as:
(6.24)
where POj is the jth singular value of the sensitivity matrix w 1l2J and vjk is the jkth eigen-
vector of the Hessian matrix JTwJ.
The proportion of the marginal variance of the hydraulic parameter hpj associated
to the kth singular value is given by:
4>jk • h..... U~k d ..... ~P

..... (6.25)
1t"k=--w~t
J 4>j
"'"k=--an
J I-lt ","=L.;k-1"'"k
J - J
The proportion is associated with a condition index 11k which is the ratio of the largest
singular value (pomaJ to the singular value POt (lIk=POmax/poJ. The largest value of 11k is called
the condition number of J.
Belsley (1990) extends the Kendall-Silvey suggestion as follows: there are as many
near dependencies among the columns of a matrix w1l2J as there are high condition
indexes (singular values small relative to pomaJ. Weak dependencies are associated with
condition indexes around 5 - 10, whereas moderate to strong relations are associated with
condition indexes of 30 - 100. Small singular values of the matrix wl12J and consequently
large condition indexes correspond with large eigenvalues of the matrix (jTwJ)'1 and with
large principal axes of the p-dimensional joint confidence area.
Once the near dependencies are found, the columns of the matrix w1l2J must be
indicated which are involved in these dependencies. If only one condition index is very
large with respect to the others, the columns which are involved in the dependencies are
those which show large marginal variance-decomposition proportions. In the case of
different large condition indexes, the columns involved in one or the other group of near
dependency are determinated by addition of the proportions corresponding with these
large condition indexes. The identification of the columns that form a same group of near
dependency cannot be distinguished alone from the proportions (Tarhouni, 1994).
Therefore, Belsley (1990) proposed to use the partial correlation coefficient between the
involved columns to identify the columns that form a group of near dependency.
6.3.6 PracticaL steps in interpretation by means of inverse modeL
The first step is the selection of suitable input data. Only observed drawdowns with a
possible small relative error are selected, e.g., drawdown larger than 7 mm in the
beginning of the pumping test. The second step is the schematization of the groundwater
reservoir in pervious and semi-pervious layers based on the collected lithostratigraphical
and geophysical data. Taking into account the assumed flow around the pumped well
screen and the location of observation well screens, the pervious and semi-pervious layers
are further discretized in the layers which are introduced in the numerical model. In the
third step the initial values of the hydraulic parameters are estimated. During the parame-
terization the parameters are joined into groups which should be identified. Information
about their identifiability is deduced from the study of the sensitivity matrix along with
the variance-covariance matrix and its eigenvalues and eigenvectors. The marginal and
conditional standard deviation as well as the partial correlation coefficients are verified to
obtain information about the uniqueness of the solution. The parameterization process is
based on all available information (applied in former steps and obtained in next steps).
To restrict the number of the iterations of the inverse model only the most
sensitive parameters are introduced at the start of the process (Lebbe and De Breuck,
1995). Here, they correspond to the largest sums of the weighted squared sensitivities of
the drawdown to the concerned parameters. Each time a minimum value of the sum of the
weighted squared residuals is reached the next most sensitive parameter is added. This
process continues until all the considered parameters are added or until the ratio between
the largest and the smallest eigenvalues exceeds 20,000. Carrera (1984) showed by a
sensitivity analyses that this ratio is indicative for a very unstable problem. The hydraulic
parameters causing these large eigenvalue(s) of the variance-covariance matrix cannot be
deduced from the observed drawdowns. These large eigenvalues correspond with large
condition indexes. The matrix of marginal variance-decomposition proportions allows to
indicate the parameters which are involved in the dependency(ies) which cause(s) the
large condition index(es). This study of the condition indexes and of the matrix of the
marginal variance-decomposition proportions along with the partial correlation coefficients
can be considered as an additional tool in the collinear diagnostic (Belsley, 1990). If there
is a correlation between two parameters or groups of parameters then two possible
measures can be taken: either the two concerned parameters can be joined in one group or
the least sensitive parameter can be temporarily or permanently removed.
The solution is reached when a minimum value is obtained for the sum of the
weighted squared residuals when all derivable parameters or parameter groups are
introduced. An idea about the precision of the derived values is obtained from the exact
or approximate joint confidence region. Some bounding points of the approximate joint
confidence region can be evaluated by means of the marginal and the conditional standard
deviation. This region approximates the exact joint confidence region if two conditions
are fulfiJled. The first is that the residuals with their different weights approximate a
normal distribution and the second is that the drawdown is approximately a linear function
within the considered region. Considerable larger number of bounding points of this
region can be visualized in graphs which represent two- and three-dimensional cross
sections through this region. To assert that the unidentifiable parameters have a negligible
small influence on the optimal values of the deducible parameters, some post-optimization
analyses should be performed. Two post-optimizations are made: a first with the
maximum possible values of the unidentifiable parameters such as the horizontal and
vertical conductivities, specific elastic storage and possibly the storage coefficient near the
watertable, and a second with the minimum possible values for the unidentifiable
parameters. These practical steps are demonstrated in Sect. 6.8 and Chapt. 7.
6.4 Validation of inverse numerical model
The validation of the inverse numerical model is shown by means of a hypothetical

example. In this synthetic problem the input data of the inverse problem are first
calculated by means of the numerical model. The assumed pumping test was executed in a
groundwater reservoir composed of three layers, each with a same thickness of 10 m
(Fig. 6.7). The lower layer is pervious, homogeneous and isotropic with a hydraulic
conductivity of 10 mid and a specific elastic storage of 8xl(f5m·l. This layer is bounded
below by an impervious boundary and above by a homogeneous anisotropic semi-pervious
layer. The horizontal and vertical conductivity are respectively 0.5 mid and O.lm/d and
its specific storage is 4xlO-5m- l • The upper pervious layer is a homogeneous isotropic unit
which is bounded above by the watertable. The conductivity and the specific elastic
storage are 10 mid and 4xl04m- 1 respectively. The specific yield or the storage coefficient
near the watertable is 0.2.
In the numerical model, this groundwater reservoir is subdivided in six layers.
Layer 1 is the lower pervious layer. The semi-pervious layer is discretized into three
layers (Layers 2, 3 and 4 of the numerical model) of equal thickness. This discretization
is necessary to simulate the flow, the drawdown and the storage change accurately at the
different levels of the semi-pervious layer. The upper pervious layer is discretized in two
layers, Layer 5 and 6 in the numerical model, which represent respectively the lower 9 m
and the upper 1 m. The hydraulic parameters introduced in the numerical model corres-
pond with the parameter values of the pervious and semi-pervious layer as mentioned in
the previous paragraph. Using the numerical model, drawdowns are calculated at six
observation weJls assuming that the lower pervious layer was pumped with a constant
discharge rate (180 m3/d). Three observation weJls are located in the middle of the
pumped pervious layer and three in the middle of the overlying semi-pervious layer. The
wells in the semi-pervious layer are at the same distance from the pumped well as the
wells in the pumped pervious layer, 5.01 m, 15.85 m and 50.12 m. Only drawdowns
which are larger than 5 mm are used as input data in the inverse model. The drawdowns
are rounded off to the nearest millimeter. The times are given with an accuracy of 0.1
minute for the first hour and 1 min after the first hour of pumpage. The 151 simulated
observations for the six observation wells, which are used as input data in the inverse
model, are given in Table 6.5.
LAYERS
IN
NUMERICAL
MODEL
- t t - - t t t - - - ! H - - - - - - - -........I11-- watertable
- pervious layer LAYERS
LAYER4
- - semi-pervious layer LAYER3

~~~~~~~~~~~~
LAYER 2
_ pervious layer
LAYER'
3
L- nr. observation well
Fig. 6.7. Hydrogeological conceptualization of the groundwater reservoir, discretization

of the numerical model to generate the 'observed drawdowns' of the hypothetical problem
and location of observation wells for the validation of the numerical model. Source:
Lebbe, L. and De Breuck, W. 1995. Validation of an inverse numerical model for the
interpretation of pumping test and a study of factors influencing accuracy of results.
Journal of Hydrology 172 (1995), p. 61-84, p.72. Copyright Cl 1995, Elsevier Science,
reprinted by permission of Elsevier Science.
The initial values of the hydraulic parameters, assumed at the start of the inverse
model, differ considerably from the actual values used to calculate the 'observed'
drawdowns. The initial values are given in the first line of Table 6.6, which is indicated
by a dash in its first column, while the actual values are given in the last line of this
table, indicated by an asterisk in its first column. In the same table the changes in the
parameter values are given along with the change of the sum of the squared residuals
during each iteration. During the first four iterations, only four parameters or groups of
parameters are included in the inverse process. They correspond to the parameters with
the four largest sums of the squared sensitivities of the drawdown and are located at the
left side of the dashed line in Table 6.6. These parameters are the horizontal conductivity
Kh(l) and the specific elastic storage S,(l) of the pumped pervious layer, and the vertical
conductivity Kv(2-4) and specific elastic storage Ss(2-4) of the overlying semi-pervious
layer. A reduced number of parameters are considered to enhance the rate of conver-
gence. Cooley and Naff (1990) stated that, in general, the greater the number of parame-
ters the slower the convergence rate.
Table 6.3. "Observed" drawdowns (in m) of the hypothetical problem used as input data
in the inverse model
Observation well 1 2 3 4 5 6
Layer 1 1 1 3 3 3
Distance (m) 5.01 15.85 50.12 5.01 15.85 50.12
Time (min) 1. 0 0.286 0.042 - - -
2.0 0.381 0.100 - - -
2.5 0.411 0.122 - - -
3.2 0.440 0.145 - - - -
4.0 0.469 0.168 0.007 - - -
5.0 0.496 0.192 0.012 - - -
6.3 0.522 0.215 0.019 - - -
7.9 0.547 0.237 0.027 - - -
10.0 0.570 0.259 0.037 0.006 - -
12.6 0.593 0.279 0.048 0.010 - -
15.9 0.614 0.299 0.059 0.016 0.007 -
20.0 0.634 0.318 0.071 0.024 0.011 -
25.1 0.654 0.337 0.083 0.036 0.018 -
31.6 0.673 0.355 0.095 0.052 0.029 0.005
39.8 0.692 0.373 0.107 0.072 0.042 0.008
50.1 0.711 0.391 0.120 0.096 0.060 0.014
63.0 0.731 0.410 0.133 0.124 0.082 0.021
79.0 0.750 0.429 1.148 0.154 0.106 0.030
100.0 0.769 0.447 0.162 0.185 0.133 0.042
126.0 0.788 0.465 0.177 0.215 0.159 0.055
159.0 0.806 0.483 0.191 0.242 0.184 0.069
200.0 0.823 0.500 0.206 0.266 0.206 0.083
251.0 0.838 0.515 0.219 0.285 0.224 0.096
316.0 0.851 0.528 0.231 0.299 0.238 0.107
398.0 0.863 0.539 0.241 0.311 0.249 0.117
501.0 0.871 0.548 0.249 0.319 0.257 0.124
631.0 0.878 0.554 0.255 0.324 0.263 0.129
794.0 0.882 0.559 0.260 0.329 0.267 0.133
1000.0 0.885 0.562 0.263 0.332 0.270 0.135
1259.0 0.887 0.564 0.265 0.335 0.273 0.138
1585.0 0.889 0.565 0.266 0.338 0.276 0.140
After five iterations, the parameter estimates are within 10% of the actual values.
The sum of the squared residuals changes from 42.3 to 0.0804. The adjustment factors
after the fifth iteration is negligibly small and the estimated change of the sum of the
squared residuals is also very small. Therefore the number of included hydraulic parame-
ters is increased by the horizontal conductivity of the overlying semi-pervious layer Kh(2-
4) and the specific yield So. These are the two next most sensitive parameters. After the
introduction of these new parameters the sum of the squared residuals changes markedly
during the following three iterations from 0.0804 to 0.0173. The newly introduced
hydraulic parameters approach the actual values.
Table 6.4. Evolution of hydraulic parameter values during the iterations of the inverse
model
Iteration Hydraulic parameters E 115 \ 2

no.
Kh(l) S,(l) 1(,,(2-4) S,(2-4) Kh(2-4) So Kh(5-6) c(5) S,(5-6)
mid m· 1 mid m· 1 mid m 3/m 3 mid d m· 1
- 3 O.l00xlO·' 0.200 0.100x10-' : 0.200 0.050 5 1 0.lx10' 42.29
1 11.526 0.16lxlO·' 0.136 0.186xlO-' lO.200 0.050 5 1 O.lxlO-' 41.44
2 9.110 0.774xlO" 0.137 0.899xlO-' lO.200 0.050 5 1 0.1.10-3 4.268
3 10.806 0.823x10" 0.100 0.434x10·' lO.200 0.050 5 1 0.1.10" 0.214
4 10.841 0.823x10" 0.091 0.360xlO·' : 0.200 0.050 5 1 0.1.10" 0.0806
................ ................
5 10.848 0.824xlO" 0.090 0.357x10·' 0.200 0.050 5 1 0.lx10·3 0.0804
6 10.101 0.798x10" 0.098 0.399x10·' 0.559 0.157 5 1 O.lxlO-' 0.0324
7 10.026 0.794xlO" 0.0999 0.401.10-' 0.501 0.222 5 1 0.1.10" 0.00173
................
8 10.025 0.794x10" 0.0999 0.401.10-' 0.500 0.232 5 1 0.lx10-' 0.00173
................
9 10.025 0.794xlO" 0.0999 0.401.10" 0.498 0.219 10.872 1 ~ O.lxlO·' 0.00173
10 10.026 0.794x10" 0.1000 0.400x10-' 0.497 0.201 11.443 0.804 : 0.lxlO- 3 0.00172
Sm 0.0007 0.0009 0.0020 0.0020 0.0065 0.1016 0.5519 0.2206 -
.
Sc 0.0005 0.0006 0.0012 0.0004 0.0018 0.0238 0.2472 0.0819 -
10 0.8x10" 0.1 0.4.10-' 0.5 0.2 10 0.55 0.4x10-' -
In each of the last three iterations, the next most sensitive parameter is included in
the inverse process. The values of the horizontal conductivities Kb(5-6) and of the hydrau-
lic resistance c(5) of the upper layer roughly change to the actual values. During the three
last iterations, the change in the sum of the squared residuals is very small. The influence
of the specific elastic storage S,(5-6) of the upper pervious layer on the considered
drawdown is so small that it cannot be introduced in the iteration process. A rough
estimate, about four times smaller than the actual value, is sufficient to calculate reliable
values for the other parameters. The parameter values that are the most influenced by
these rough estimates are those with the smallest sensitivities to the considered draw-
downs. They can also be calculated if another estimated value for S,(5-6) is introduced
which differs considerably from the first estimate. In Table 6.4 the conditional and
marginal standard deviations which are located in the log space are shown. The smaller
these values, the larger the accuracy with which the hydraulic parameters can be
estimated from the observed parameters. Lebbe (1988) demonstrated that the same
optimal values are reached when another set of initial values is used to start the inverse
model.
6.5 Factors influencing accuracy of results
6.5.1Irifluence offlow conceptualization
The conceptualization of. the flow supposes a set of assumptions. The groundwater
reservoir is subdivided into pervious and semi-pervious layers. These layers are either
replaced by horizons between layers or by one or more layers in the numerical model.
When layers are replaced by horizons only a vertical inelastic flow is considered. The
discretization in more than one layer can be required to simulate elastic flow in a semi-
pervious layer or flow to a partially penetrating well. The upper and lower boundary
conditions are assumed to be impervious, constant head, or a water table boundary. In
this section, the lower pumped pervious layer, the overlying semi-pervious layer and the
upper pervious layer in the numerical model are each time represented by one layer.
At first, the flow is conceptualized according to the model of Theis (1935) (see
Sect. 4.6.1), where the pumped pervious layer is bounded above and below by an
impervious boundary. From the observed drawdowns in the pumped layer, only the
horizontal conductivity and the specific elastic storage of the pumped layer can be
estimated assuming that the vertical and horizontal conductivities and the specific elastic
storage of the overlying layers are zero. The results are given in the first line of Table
6.5. On following three lines, the results are given when the observations of the three
wells are interpreted separately. These results would also be obtained from the ordinary
least square fit of the Theis-type curve.
Table 6.5. Hydraulic parameter values estimated from the 'observed' drawdowns in the
pumped layer when the flow is conceptualized according to the Theis (1935) model
Observation Distance Kh(l) mid 5,(1) m-I ~i=1 fj2 n

well (m)
value Sm Sc value Sm Sc
1,2,3 5.0,16.,50. 14.77 0.0148 0.0147 1.03xlO4 0.0185 0.0183 0.887 89

1 5.01 16.22 0.0135 0.0047 2. 14xl0" 0.0791 0.0274 0.01459 31
2 15.9 15.92 0.0141 0.0109 7.54x10 4 0.0270 0.0208 0.05441 31
3 50.1 23.26 0.0136 0.0136 1. 21 x10-2 0.0109 0.0109 0.04789 27
From Table 6.5, one can see that the estimated values of the horizontal conducti-
vity are all larger than the actual values. The estimated values from the data of the
separate observation wells increase with distance from the pumped well. This is even
more so for the specific elastic storage which ranges from four times smaller than the
actual value to more than one thousand times larger than the actual value. It is clear that
different values for the same parameter can be obtained from separate interpretation of
different time-drawdown curves. These differences are not due to lateral heterogeneity but
to large differences between the actual flow and the assumed flow in the interpretation
model.
In the second step, the flow is conceptualized according to the model of Hantush
and Jacob (1955) where the pumped layer is bounded below by an impervious boundary
and above by an overlying semi-pervious layer. An exclusively vertical inelastic flow is
considered in the semi-pervious layer and its top is under constant hydraulic head. Now
the horizontal conductivity Kh(1) and the specific elastic storage S,(1) of the pumped layer
and the vertical conductivity K.(2) of the overlying semi-pervious layer are estimated
from the observed drawdowns in the pumped layer. It is assumed that the horizontal
conductivity and the specific elastic storage of the overlying semi-pervious layer are zero
and that the horizontal conductivity of the upper pervious layer is very large, to obtain the
same conditions as in the model of Jacob-Hantush (see Sect. 4.6.3).
In Table 6.6, the results are given for the simultaneous interpretation of the three
time-drawdown curves and also for the separate interpretation of each time-drawdown
curve. The last three lines represent the results as would be determined from the best
match of the family of Walton-type curves as described in Kruseman and De Ridder
(1990), with the different time-drawdown curves. The estimated values of Kh(1) are larger
than the actual values. The parameter values estimated from the separate time-drawdown
curves depend greatly on their observation distance from the pumped well: the larger this
distance, the larger the deviation between the actual values and the estimates. The
estimated values of S,(1) range from about two times smaller than the actual value to
about two times larger than the actual value. This range is considerably smaller than the
one obtained with the interpretation model of Theis. The estimated value of 1(,,(2) is close
to the actual value if the time-drawdown curves are interpreted simultaneously but if
interpreted separately much smaller values are found.
Table 6.6. Hydraulic parameter values estimated form the 'observed' drawdowns in the
pumped layer when the flow is conceptualized according to the Jacob-Hantush model
Obser- Dis- Kh(l) mid 8,(1) m-I Kv(2) mid Ei=l~i2 n

vation tance
well (m) value Sm Sc value Sm Sc value Sm Sc
1,2,3 - 11.049 .0106 .0077 .906xI0-4 _0095 _0089 _115 .0375 .0266 _226 89
I 5_01 13_195 .0075 .0016 .464xI0-4 .0274 _0095 .2066x I 0-2 _0656 .0140 _001512 31
2 15.85 12.955 .0106 _0049 .843xI0-4 .0097 _0082 A647x10-2 .0636 _0289 .009651 31
3 50_12 20_336 _0185 .0106 .177xI0- 3 .0092 .0080 .2099xI0-2 _1229 .0686 .02690 27
* 10 - - 0.8x10-' - - 0.1 - - - -
From the interpretation according to the two first conceptualizations of the flow
system, one can conclude that it is better to make an interpretation of all the observed
drawdown values together rather than of the drawdowns of different observation wells
separately. This is certainly the case when there is a difference between the simulated and
the actual flow regime. The closer the simulated flow approximates the actual flow; the
closer are the estimates from the observations with the inverse model to the actual values.
In the third and fourth step, the observations in the pumped pervious layer as well
as the observations in the overlying semi-pervious layer are employed in the estimation
process. In the third step, the flow is conceptualized according to the model of Hantush
(1960) (see Sect. 4.6.5) where a vertical elastic flow is considered in the overlying layer
with a constant head boundary on top of it. In the fourth step, horizontal flow in the
semi-pervious layer is also considered. In both steps, the semi-pervious layer is only
replaced by one layer. In the third step the horizontal conductivity is set equal to zero,
resulting in an exclusively vertical flow. The constant head condition is obtained by
setting the hydraulic conducti-vity of the upper pervious layer equal to a very large value,
109 mid. Because the semi-pervious layer is only discretized by one layer in the numerical
model, the level depen-dence of the drawdown and of the storage decrease is less
accurately simulated than in the case for the calculations of the 'observations' of the
hypothetical example. The horizontal conductivity and the specific elastic storage of the
pumped layer, the vertical conductivity, and the specific elastic storage of the overlying
semi-pervious layer are calculated from the considered observations. The results of this
third step are represented in Table 6.7. The calculated values of Ki2) and S,(2) are 25 %
smaller than the actual values, whereas the calculated values of Kh(l) and S,(l) deviate a
little more than 10 % from the actual value. There is a large difference between the
conditional and marginal standard deviation of Kv(2) and S,(2). This is due to the rather
high correlation between the parameters. It is better to combine these parameters into
one, the diffusivity of the semi-pervious layer K(2)/S,(2). One will notice that the
calculated value of this diffusivity very closely approximates the actual value.
Table 6.7. Hydraulic parameters estimated from "observed" drawdowns in the pumped
pervious layer and in the overlying semi-pervious layer where the flow is conceptualized
on the one hand according to Hantush (1960) model (without horizontal flow in the semi-
pervious layer) and on the other hand according to Hantush (1960) model but with the
exception that horizontal flow is considered in the semi-pervious layer
Hydraulic No horizontal flow in With horizontal flow in *

parameter semi·pervious layer semi·pervious layer
value Sm Sc value Sm Sc real value
Kh(l) mid 1\ .500 0.0120 0.0078 11.112 0.0121 0.0074 10

S,(l) m-I 0.896xHr' 0.0124 0.0104 0.890x1O-4 0.0171 0.0096 0.8x10-4
Ki2) mid 0.0755 0.0418 0.0120 0.0797 0.0382 0.01\3 0.1
S,(2) m· 1 0.295x10·' 0.0458 0.0159 0.309x10' 0.0417 0.0150 0.4x10·'
Kh(2) mid (0) - - 0.098 0.1697 0.0997 0.5
In the fourth step, Kh(2) is included as parameter to be estimated. The results of

this step are also given in Table 6.7. Here the calculated values are only slightly different
from those obtained in the previous step. The calculated values are only slightly
improved. The calculated value for Kh(2), however, is very different from the actual
value.
6.5.2 Influence of observed drawdown accuracy and discharge magnitude
Mostly, drawdown is proportional to the discharge rate. This is not true, however, in the
exceptional case of considerable drawdown of the watertable in a relatively thin pervious
layer where the horizontal flow is not negligible. Because there is an inverse relationship
between the relative error of the observed drawdown and the discharge rate when the
drawdown is measured with a same absolute error, both cases can be treated in one part.
Therefore, only the influence of the absolute error of the drawdown is studied. The 151
drawdowns given in Table 6.3 are rounded off to the nearest centimeter and are used as
input data. Through this rounding off, a maximum absolute error of 5 mm is introduced.
The inverse process was started using the same initial parameter values and the same
conceptualization of the flow as during the validation of the inverse process.
The results of this problem (Problem 1) are represented in Table 6.8. Comparing
them with those of Table 6.4 one can make the following conclusion. When the relative
error of the observed drawdowns increases either by an increase in absolute error of the
measured drawdown or by a decrease in the discharge rate or both, then the standard
deviations of the calculated parameters increase and the number of parameters that can be
reliably calculated decreases.
6.5.3 Influence of observation time
To show the influence of observation time two problems were treated, Problem 2, in
which the observation between the first minute and the first hour of pumpage are used;
and Problem 3, in which observations between the first hour and the first day of pumpage
are used. Each time the drawdowns are selected from Table 6.3 and are rounded to the
nearest centimeter. In both cases, the conceptualization of the flow in the numerical
model and the initial parameters are the same as in the validation problem. In the first
problem, five parameters were identifiable, whereas in the second case there were six.
The results are given in Table 6.8. By comparing the results of these two calcula-
tions, one will note that the accuracy by which the hydraulic parameters can be calculated
is considerably higher in Problem 3 than in Problem 2, with the exception of the specific
elastic storage of the pumped pervious layer, which is principally defined by the rise of
the drawdown in the pumped layer during the first hour of pumpage. The gain in
information is greatest for the hydraulic parameters of the overlying semi-pervious layer.
Comparing the results of Problems 1 and 2, one can conclude that not only the accuracy
of the calculated parameters but also the number of parameters that can be reliably calcu-
lated increases with the duration of the observations.
6.5.4 Influence of observation distance
The influence of observation distance is shown by means of three problems. In Problem

4, the hydraulic parameters are estimated from drawdowns "observed" in Wells 1 and 4
(Table 6.3) at a relatively short distance from the pumped well (5 m) in the middle of the
pumped pervious and overlying semi-pervious layer, respectively. All observations given
in Table 6.5 are used and rounded to the nearest centimeter. The conceptualization of the
groundwater reservoir and the initial parameters are the same as in the validation
problem.
Table 6.S. Hydraulic parameters estimated from different "observations sets"
Problem Hydraulic parameters

no.
Kh(l) S,(I) Kv(2-4) S,(2-4) K.(2-4) So Kh(5-6) c(5) S,(5-6) 1:,"1',2 n
mid m" mid m" mid m3/m 3 mid d m"
ni~9 10.051 0.759xl04 0.105 0.396x10·' 0.633 0.117 0.225 1 O.lxH)"' 0.223 151
1 Sm 0.0083 0.0100 0.0203 0.0204 0.0603 0.3926 2.9532 - - - -
sc 0.0052 0.0065 0.0147 0.0045 0.0188 0.1353 1.4278 - - - -
ni~9 10.201 0.76OxlO-4 0.111 0.403xl0·' 0.960 0.050 5 1 O.lxH)"' 0.200 61

2 Sm 0.0163 0.0202 0.0520 0.0504 0.1494 - - - - - -
sc 0.0087 0.0179 0.0178 0.0088 0.0311 - - - - - -
ni~12 10.036 0.864x104 0.0971 0.386xlO·' 0.489 0.253 5 1 O.lxlo"' 0.00545 90

3 Sm 0.0029 0.0230 0.0080 0.0112 0.0186 0.1329 - - - - -
Sc 0.0013 0.0085 0.0029 0.0025 0.0047 0.0537 - - - -
ni~14 13.055 0.445xl04 0.0161 0.575.104 0.175 0.0159 5 1 O.lxlO"' 0.0467 54

4 Sm 0.0520 0.1562 0.4660 0.4706 0.3847 0.7211 - - - - -
Sc 0.0057 0.0021 0.0053 0.0368 0.0220 0.0129 - - -
ni~8 9.857 0.804xl04 0.109 (0.4xl0·') 0.724 0.0373 5 1 O.lxlo"' 0.0520 54

5 Sm 0.0107 0.0503 0.0123 - 0.0661 0.2983 - - - - -
Sc 0.0059 0.0437 0.0071 - 0.0289 0.2386 - -
ni~1O 10.037 0.753xl04 0.105 0.385xl0·' 0.688 0.038 5 1 O.lxlO"' 0.0800 81

6 Sm 0.0085 0.0102 0.0208 0.0233 0.0616 0.2606 - - - - -
Sc 0.0055 0.0067 0.0060 0.0098 0.0217 0.1904 - - - - -
* 10 0.8x104 0.1 0.4xl0·' 0.5 0.2 10 0.55 O.4x1O"'
Problem 1, From all 'observed' drawdowns (in Table 6.3) rounded off to the nearest centimeter,
Problem 2, From drawdowns 'observed' between first minute and first hour of pumpage,
Problem 3, From drawdowns 'observed' between first hour and first day of pumpage,
Problem 4, From drawdowns 'observed' in observation wells no. 1 and 4,
Problem 5, From drawdowns 'observed' in observation wells no. 1 and 4 where S,(2-4) is known,
Problem 6, From drawdowns 'observed' in the observation wells nos. I, 3 and 4,
ni ~ number of iterations, n = number of applied observations
Six parameters were included in the inverse process. After six iterations the sum
of the squared residuals approaches a minimum value and the parameter values show a
rather high oscillation. The amplitude of the oscillation decreases only slowly during the
following eight iterations. The non-diagonal elements of the correlation matrix were all
larger than 0.91. The largest non-diagonal element is the one which describes the
correlation between the parameters K.(2-4) and Ss(2-4). The absolute value of the partial
correlation coefficient is larger than 0.99. Consequently, one can conclude that it is
impossible to reach a unique solution and that there exists an infinite number of solutions
with the same ratio between the above-mentioned two parameters. The different solutions
all have about the same sum of the squared residuals. These two hydraulic parameters are
not separately derivable as one can also deduce from the theory given by Carrera and
Neuman (1986b). In such a case the conditional standard deviation is much smaller than
the marginal standard deviation.
From the results represented in Table 6.8 (Problem 4), one can conclude that the
calculated parameter values are very different from the actual ones. The results obtained
from the inverse problem with very large non-diagonal elements in the correlation matrix
are very sensitive to the residuals. It results in a large confidence region which is defined
by large conditional and especially large marginal standard deviations. In the hypothetical
example, the residuals are due to the rounded off drawdowns. In an actual problem, the
residuals are due to measurement errors and to slight differences between the actual and
the simulated flow. The input data are here the same as the drawdowns used in the ratio-
interpretation method as derived from the model of Neuman and Witherspoon (1969).
Clearly, only the diffusivity, the ratio of the vertical conductivity to the specific elastic
storage, of the semi-pervious layer can be deduced using this method.
A possibility to obtain a unique solution is to combine these two parameters in one
group but inversely related which means that one parameter is multiplied by the sensitiv-
ity coefficient while the other is divided by the sensitivity factor. Another possibility is
that the specific elastic storage is deduced by means of a compression test and that this
parameter is then considered as known. Still another possibility to obtain a unique
solution is to add observations of which their sensitivities to Kv(2-4) and to S.(2-4) are
mutually unrelated.
In Problem 5 of Table 6.8 S,(2-4) is considered known because, for example, it
has been estimated from a compression test. In Problem 6 of Table 6.8 the drawdowns
observed at 50 m from the pumped well in the pumped layer (observations of Well 3) are
added to the observed drawdown of Problem 4. In both cases a unique solution is
obtained from which the confidence region of the deduced parameters is reduced
considerably. This can be seen from the reduction of their standard deviations, especially
the marginal standard deviations.
6.5.5 Conclusions
Following conclusions can be drawn from the study of the factors that influence the
hydraulic parameter estimates and their joint confidence region. The closer the simulated
flow approximates the actual flow the closer are the estimates to the actual ones.
Hydraulic parameters estimates obtained by fitting separate time-drawdown curves can
result in very different values for the parameters if the actual flow is different from the
model corresponding to the type-curves. Some of these estimates can be very different
from the actual values. The best estimates are obtained when a large number of observa-
tions are included in the inverse process and if the simulated flow regime closely
approximates the actual one.
The number of parameters that can be reliably estimated decreases and their joint
confidence region enlarges when the relative error of the observed drawdowns increases.
Increasing the period during which observations are collected reduces the joint confidence
region and increases the number of identifiable parameters. In case where it is not
possible to obtain a unique solution because of correlation between two parameters, one
can consider the two parameters in one group. If the two parameters are both multiplied
by the same sensitivity coefficient one calculates the best absolute values of these parame-
ters corresponding with the given ratio. If the two parameters are introduced in the same
group but inversely related then the best ratio between the parameter values is calculated
but their calculated absolute values do not make sense. In both cases, the best absolute
values of the parameters can only be deduced if one of the parameters is deduced by
another test, a compression test or another pumping test.
6.6 Program package for the nonlinear regression
This package consists of eight programs.

- The program solpu5 to calculate the adjustment factor, the marginal and conditional
standard deviation based on the given residuals and sensitivity matrix and to adjust the
concerned hydraulic parameters. This program is mostly used for the interpretation of
multiple pumping tests. The residuals and sensitivities are the results of several pumping
tests which are joined in one data file.
- The program inpur5 with the inverse model which is a tool for the interpretation of
pumping test. The results are the optimal values of the hydraulic parameters and the
parameters which are required to plot the exact or approximate joint confidence region or
cross sections through this region.
- The program etabdi to plot the residuals versus their probability along with the best
fitted normal distribution or sum of normal distributions along with the optimal values of
the mean(s), the standard deviation(s) and if necessary of the weight(s).
- The program plprcr to plot two-dimensional cross sections through the joint confidence
region which is approximated by the p-dimensional ellipsoids. These cross sections are
here obtained with the help of the eigenvalues and the eigenvectors of the variance-
covariance matrix.
- The program susqln for the calculations of the sums of the squared residuals for a set of
parameter combinations to draw two-dimensional cross sections through the approximate
joint confidence region with the plot program plssql. The used residuals are obtained by
the application of the linearization method on the residuals and the sensitivities of the
optimal solution.
- The program susql3 for the calculations of the sums of the squared residuals for a set of
parameter combinations to draw three-dimensional cross sections through the approximate
joint confidence region with the plot programs pissq3 and plelo3. The residuals are
obtained by the application of the linearization method on the residuals and sensitivities of
the optimal solution.
- The program sumsqr for the calculation of the sums of the squared residual for a set of
parameter combinations to draw two-dimensional cross sections through the exact joint
confidence region with the aid of the plot program plsusq.
- The program sumsq2 makes the same calculation as the program sumsqr but calculates
and stores additionally for the same set of parameter combinations the logarithms of the
drawdown for one required observation point decreased with the logarithm of the corre-
sponding drawdown of the optimal solution. With the program plsusd the two-dimen-
sional cross sections through the exact joint confidence region are plotted along with the
contour lines of the above mentioned differences of the logarithms of the drawdowns.
An example of the results of the last six plot programs is represented in the figures of
Sect. 6.8.
6.6.1 Program so/puS
This program is mostly used during the interpretation of a multiple pumping test. The
program calculates the adjustment factors with the ordinary least square (OLS-) method or
with the biweighted least square (BWLS-) method along with the marginal and the
conditional standard deviations. These parameters are obtained by aid of the residuals and
the sensitivities calculated and stored during the successive iterations of the inpur5
program. The residuals and the sensitivities of the separate pumping tests calculated after
one iteration of the inpur5 program are joined in one file namel.etadfdp (UNIX) or
name1.eta (MSDOS). With this data the hydraulic parameters are adjusted in the soIpu5
program and stored in the file name1.papu.resinv (UNIX) and name1.par (MSDOS).
The required files are name1.pap(u) with the hydraulic parameters and the
parameters which define the discretization of the groundwater reservoir (Tables 4.3 and
4.4), name1.dap(u) with the observations (Tables 4.5 and 4.6), uame1.pas(e) with the
definition of the parameters in the sensitivity analyse and name1.eta(dfdp) (Table 6.1)
with the residuals and the sensitivity matrix. These files are linked with the numbers 26,
28, 29 and 30.
The results of the calculation are written in a file uame1.soIpu5.lst (UNIX) or
uame1.lss (MSDOS). This file can be read in the EDITOR or printed to check quickly
the results of one iteration of the inverse model. In the first line of this file the weights
W ti are given for i varying from 1 to n. These weights are followed by an echo of the
parameters given by the file name1.pap(u), the grid parameters and the hydraulic
parameters along with their assigned numbers. These parameters are followed by an echo
of the different hydraulic parameters that are included in the different sensitivity analyses.
After these echoes the proper results of the calculations are given. The results are the
adjustment factors, the marginal and conditional standard deviations along with the sum of
the weighted squared residuals, FOLD, corresponding with the given parameters and an
estimate of this sum, FNEW, after the adjustment of the parameters. This last sum is an
estimate because it is assumed that the logarithm of the drawdown varies linearly within
the logarithm of the proposed adjustments factors. These results are followed by the
partial correlation coefficients deduced from the variance-covariance matrix and its
eigenvalues and eigenvectors. Finally the adjusted hydraulic parameters are given.
The adjusted hydraulic parameters are also given in a file namel.papu.resinv

(UNIX) or in a file namel.par (MSDOS). In this last file the parameters are written in
the same format as in file namel.pap(u). So this file can easily be used for the continu-
ation of the calculation. In the file namel.dppc (UNIX) and in a file namel.dpc
(MSDOS) the parameters are written which are needed for the calculation of the plot of
the cross section through the approximate or the exact joint confidence region. In this file
the symbols of the different derived parameters are included along with their optimal
values and their units. They are followed by the eigenvalues and eigenvectors of the
variance-covariance matrix and by the square root of the product p. F (p, n-p, I-a) for the
significance levels 90%, 99%, 99.9% and 99.99%. The sequence and the format of the
data in the file namel.dp(p)c is given in Table 6.9.
Table 6.9. Sequence and format of the data in the file namel.dp(p)c
Line Variable Description and remarks Format
1 NUMPAR number of derived hydraulic parameters 12
2 (SYMB(J), UNIT(J», symbols and units of hydraulic parameters 2A4

J~I,NUMPAR
NUMPAR+2 HYDPAR(J), optimal values of hydraulic parameters SG12.4

J~l,NUMPAR
Lll EVAL(J). eigenvalues of variance-covariance matrix SG12.4

J~l.NUMPAR
L2' «(EVEC(I,J), eigenvectors of variance-covariance matrix 10F6.4

I~l,NUMPAR),
J~l,NUMPAR»
L3' TVALUE(I), (p.F(p,n-p,l-<»)'12 4F6.3

I~1,4 for the significance levels 90,99,99.9 and 99.99%
L4' PARCAL(I) sequence of sensitivity analyses 27(lx,I2)

I~l,NUMPAR
I L1 ~NUMPAR+2+K where K IS the truncated part of (NUMPAR+S)/S, 'L2~L1 +K

'L3~L2+NUMPAR*L where L is the truncated part of (NUMPAR+IO)/IO,' L4~L3+1
6.6.2 Program inpurS
The purpose of the program is the interpretation of a pumping test by the application of
nonlinear regression following the ordinary least square method or the biweighted least
square method. The algorithm is obtained by the combination of the numerical model, the
sensitivity analysis and the linearization method to calculate the adjustment factors. The
required files are namel.pap(u) with the hydraulic parameters and the parameters which
define the discretization of the groundwater reservoir (see Tables 4.3 and 4.4), namel.-
dap(u) with the observations (see Tables 4.5 and 4.6), namel.pas(e) with the definition
of the parameters which should be optimized (see Table 6.1). These files are respectively
linked with the numbers 26, 28 and 29.
The interpretation of the pumping test is done interactively. This is by writing the
necessary information about the course of the calculations and by asking some questions
on the screen which must be answered to continue the calculation in the desired way. At
the start of the iteration process, echoes of the used grid parameters are given along with
the initial values of the hydraulic parameters. These echoes are followed by the observa-
tions which are included in the interpretation process and by the definition of the
hydraulic parameters that will be optimized. After these echoes the optimization method
must be chosen: the ordinary least square method or the biweighted least square method.
Mostly the optimization method is started with the ordinary least square method.
First, the residuals are derived along with the sensitivity matrix. From these
residuals and sensitivity matrix, the adjustment factors, the marginal and conditional
standard deviations are derived and displayed on the screen. They are followed by the
sum of the weighted squared residuals, FOLD, corresponding with the former parameter
set and by a linear estimate of this sum, FNEW, corresponding with the new parameter
set obtained according to the adjustment factors. If the ratio FOLD IF NEW is smaller than
1.001 and the adjustment factor of the hydraulic parameters are sufficiently small, one
can stop the iteration process.
To obtain an idea about the conditioning of the Hessian matrix J'fJ, the partial
correlation coefficients deduced from the variance-covariance matrix and its eigenvalues
and eigenvectors are given. Parameters causing large eigenvectors or corresponding with
large absolute values of the partial correlation coefficient cannot be deduced from the
observations. Then the iteration process must be stopped and the hydraulic parameter
which one wants to optimize must be redefined. Also the condition indexes of the
sensitivity matrix are displayed along with the matrix of the marginal variance decomposi-
tion proportions. These data can be helpful in the collinear diagnostic. Finally the
hydraulic parameters are adjusted and displayed on the screen. After each iteration of the
nonlinear regression, the process can be stopped or continued either with the ordinary
least square method or with the biweighted least square method.
If the program is stopped, the adjusted hydraulic parameters are stored in a file
name1.papu.resinv (UNIX) or in a file name1.par (MSDOS) where all the parameters
are written in the same sequence and format as in file name1.pap(u). So this file can
easily be used for the continuation of the calculation. In file name1.dppc (UNIX) or in
file name1.dpc (MSDOS) the parameters are written which are required for the plot of
the cross sections through the approximate or the exact joint confidence region. The
sequence and the format of the data in the file name1.dp(p)c are given in Table 6.9 (see
Sect. 6.3.5.1).
6.6.3 Program etabdi
The purpose of this program is the plot of the residuals versus their cumulative frequency
or probability along with the best fitted normal distribution and with the optimal values of
the mean and standard deviation. If, however, the residuals exhibit a distribution which is
the sum of several normal distributions then the optimal values of the means, the standard
deviations and the relative weights of the different normal distributions are calculated and
the results are plotted in a graph.
The probability density p(r) or frequency of a normal distribution of the residuals r
with mean J.I. and standard deviation (J is given by Wonnacott and Wonnacott (1985):
p(r) = _1_ (6.26)

21<U
Pr(r<r l ) is the probability (this is the surface under the probability density curve) for the
value r smaller than r l . The probability of r smaller than infinite is 1 or Pr(r< (0)=1.
If the residuals exhibit a distribution which is the result of m different normal
distributions then the probability density Pc of the composed distribution can be formu-
lated as follows:
m
pc(r) = LPj(r)wj (6.27)
j=l
where p/r) is the probability density of the j'h normal distribution with mean J.l.j and
standard deviation (Jj and a relative weight Wj or
(6.28)
The weight of the first considered normal distribution is always equal to 1 (WI = 1). The
other weights Wj for j varying from 2 to m should be considered relative to the weights WI
of the first normal distribution.
The probability of the composed distribution Prc(r < r I) is given by:
m m
prC(r<r, ) = LPrj(r<r,)WjILWj (6.29)
f,"l j-I
where Prj (r < r l ) is the probability for the value r smaller than r1 for the jth normal
distribution. Because Pr/r < (0) = 1 also Prc(r < (0) will also be equal to 1.
The residuals which are examined are read from the file name1.etadfdp (UNIX)
or name1.eta (MSDOS) (Sect. 6.2.2). These residuals are calculated during the success-
ive iterations in the program iopurS. At the start of the program the initial values of the
estimated mean (zero) and standard deviation must be given. These are the only required
statistical parameters in the case of a single normal distribution. In the case that m
(1 < m < 6) different normal distributions are considered, these statistical parameters must
be completed with the estimates of the mean(s), the standard deviation(s) and the
weight(s) of the m-l other normal distribution(s). After the interactive input of these
estimated statistical parameters the best fitted simple or composed distribution is calcu-
lated by a regression process. In this regression process the sum of the squared differ-
ences between the observed and calculated probability is minimized. The observed
probability corresponds with the given residuals and the calculated probability corresponds
with the normal distribution of the given mean and standard deviation or with the sum of
the normal distributions of the given means, standard deviations and weights.
After the optimization of the statistical parameters (means, standard deviations and
weights) the program etabdi creates a HPGL-file plot. pit. This file allows the plot of the
residuals versus their probability and of the best fitted simple normal distribution or sum
of normal distributions. Also, the probability density curve is plotted in case of a single
normal distribution. In case of a composed distribution the probability density of the m
different normal distribution is plotted along with the probability density of the composed
distribution (see Figs. 6.9, 7.3 and 7.7).
6.6.4 Program plprcr
The purpose of this program is the drawing of two-dimensional cross sections through the
p-dimensional ellipsoid which is an approximation of the joint confidence region. Eq.
6.15 represents the equation of this p-dimensional ellipsoid in the p-dimensional parame-
ter space in function of the eigenvalues and the eigenvectors of the variance-covariance
matrix. The required data are stored in file name1.dp(p)c (see Sects. 6.6.1 and 6.6.2)
which is linked with number 32. This p-dimensional ellipsoid is an approximation of the
joint confidence region of the optimized parameters. For each cross section another
combination of p-2 parameters is set equal to the optimal values while the remaining
parameters correspond with the axes of the considered plane in the p-dimensional
parameter space.
The axes of the graphs are all logarithmic. The modulus lengths of the logarithmic
axis are not always the same. The axes of the different graphs that correspond with a
same parameter have, however, always the same modulus length. This modulus lengths of
the jth parameters are chosen according to s~.(p.F(p,n-p,O.9999»1I2. In Table 6.10 the
five possible modulus lengths are represented along with the factors by which the optimal
values of the parameters must be multiplied and divided to obtain the represented interval.
So the middle of the represented interval corresponds with the optimal value of the
hydraulic parameter which is printed in the graph. The values on the axes correspond
with the hydraulic parameter divided by his optimal value.
In each graph four different ellipses are shown. These ellipses are the cross
sections through the approximates of four different joint-confidence regions which cOrres-
pond with three different significance levels. The four considered significance levels are
90%, 99%, 99.9% and 99.99%. Eq. 6.17 represents the equation of these ellipses. These
two-dimensional cross sections through the approximate joint confidence region can help
us in the regression diagnostics (see Fig. 6.10). Parameters with a high correlation
between their sensitivities correspond with long thin ellipses of which the principal axes
forms angles of about 45° with respect of the considered parameter axes. The results are
stored in a file plot.pit which is a file written in HPGL and can be drawn on the screen
or printed with appropriate programs.
Table 6.10. Five possibilities of modulus lengths along with the factors which define the
considered interval on the parameter axis
Possibility Modulus length Factor which defines Factor which defines

number (in cm) start of interval end of interval
1 5 0.1 10
2 16 0.5 2
3 50 0.8 1.25
4 110 0.9 1.1
5 220 0.95 1.05
6.6.5 Program susqln
The purpose of this program is to calculate the sums of the squared residuals for a set of
parameter combinations. The chosen sets of parameter combinations are located in a
number of mutually perpendicular planes which are parallel to the parameter axes and
traverse the optimal point in the parameter space. For each plane 212 calculations of the
sums of the squared residuals are made. The considered intervals of the different parame-
ters are chosen with the help of the marginal standard deviation in the same way as in the
program plprcr. The five possible logarithmic intervals are given in Table 6.10. In the
different considered planes the considered intervals of a same parameter do not change.
The first calculation of a considered plane corresponds with the optimal values of
p-2 hydraulic parameters and with the optimal values of the two remaining parameters
which are multiplied by the factor which defines the start of the intervals. For this first
parameter combination the residuals are calculated by means of the residuals and the
st:nsitivities at the optimal point according to Eq. 6.23. The used residuals and sensiti-
vities are read from the file name1.etadfdp (UNIX) or name1.eta (MSDOS) (see Sect.
6.2.2). With the calculated residuals according to Eq. 6.23 the sums of the squared
residuals are made and stored in a two-dimensional matrix. One of the two considered
parameters is then multiplied by its multiplication factor. The logarithm of this factor is
equal to one tenth of the logarithm of the factor which defines the end of the considered
interval. With this new parameter combination the sums of the squared residuals are again
calculated in the same way as previously described and stored in the matrix. After 21
times the multiplication of the first chosen parameter with its multiplication factor, this
hydraulic parameter has reached the maximum value of the selected interval. Then this
first parameter is set back to the minimum value of its treated interval and the second
parameter is now multiplied by its corresponding multiplication factor. Again 21 sums of
the squared residuals are calculated where each time the first parameter is multiplied by
its multiplication factor. These calculations are continued until the two considered
parameters reach their optimal values multiplied by the factor which determined the end
of the treated interval.
The required files are namel.pap(u), namel.dap(u), namel.pas(e), namel.eta(-

dfdp) , namel.d(p)pc. These files are linked with the numbers 26, 28, 29, 30 and 32.
The matrices of the sums of the squared residuals are stored in the file namel.ssql
(UNIX) or namel.sql (MSDOS). This file is linked with number 35.
Once the matrices of the sums of the squared residuals are obtained for different
windows around the optimal point and belonging to all the different planes which are
parallel to another combination of two parameter axes. The two logarithmic parameter
axes are marked in the same way as in the graphs which are drawn with the plprcr
program. In each graph the optimal values of the p-2 parameters are printed in the upper
right corner of the graph. In each graph four contour lines of the sums of the squared
residuals can be drawn. They correspond with the approximate bounds of the 90%-, the
99%-, the 99.9%- and the 99.99% joint confidence region if the appropriate values for
the sums of the squared residuals are given. These sums can be calculated with Eq. 6.22.
The bounds are also here only approximations because it is assumed that the sensitivities
of the drawdown to the considered drawdown are constant within the considered windows
(see Fig. 6.11).
These graphs are obtained by the plot program plsusq (UNIX) or plssql (MS-
DOS). Ignoring the errors due to interpolation during the plot of the contour lines of the
sums of the squared residuals, the results of the program plprcr and susqln should be the
same. The required input files are namel.d(p)pc and namel.s(s)ql which are linked with
the numbers 32 and 33. The results are written in plot file plot. pit and in a file namel.-
prl(n) which is linked with number 39. In the file namel.prl(n) all the parameter
combinations are stored which correspond with the bounds of the 99% approximate joint
confidence region. They are located in the planes which are parallel to two other
parameter axes and go through the optimal point. This file could be used to approximate
the confidence intervals of the calculated drawdowns (see Sect. 6.7.3.1)
6.6.6 Program susql3
The purpose of the program is to calculate the sums of the squared residuals for a
parameter combination set so that three-dimensional cross sections through the approxi-
mate joint confidence region can be depicted. Each three-dimensional cross section is
parallel to another combination of three parameter axes while the other p-3 parameters are
put equal to their optimal values. These three-dimensional cross sections can also be
called conditional confidence 3D-spaces. The bounds of these conditional confidence
spaces are represented by fifteen closed polylines which are plotted in graphs with three
orthogonal axes. These lines correspond with three times five two-dimensional cross
sections through the joint confidence space. All these cross sections are parallel to two of
the three axes considered in the graph. Three of the fifteen two-dimensional cross sections
go through the optimal values. The other cross sections do not go through this optimal
point in the parameter space. The lines of these cross sections are obtained in the same
way as explained above for the two-dimensional cross section. For each cross section 2F
calculations are made. The residuals for each parameter combination are here calculated
according to the linearization method (Eq. 6.23). The used residuals of the optimal
solution and their sensitivities are read from the file namel.etadfdp (UNIX) or namel.-
eta (MSDOS). Because of the use of Eq. 6.23, the obtained three-dimensional cross
sections are through the approximate joint confidence region. The distance between the
two outer planes and the parallel plane which go through the optimal point is chosen so
that the minimum value of the sum of the squared residuals must be smaller than the sum
of the squared residuals which correspond with the 99 % significance level. These cross
sections are centred exactly around the optimal values and are pure ellipsoids (Fig. 6.12).
The bounds of these three-dimensional cross sections are parts of the bounds of the
joint confidence regions. The study of their shapes enlarges further the collinear diagnos-
tic possibilities. The ellipsoids of which one of the principal axes is much larger than the
two others and of which the direction of the principal axes deviates strongly from the
parameter axes are the most responsible for the deterioration of the conditioning of the
Hessian matrix JTJ.
The required files for the program susql3 are namel.pap(u), namel.dap(u),
namel.pas(e), namel.eta(dfdp), namel.d(p)pc. These files are linked with the numbers
26, 28, 29, 30 en 32. The matrices of the sum of the squared residuals are stored in the
file namel.ssq3 (UNIX) or namel.sq3 (MSDOS). This file is linked with number 35.
Once the matrices of the sums of the squared residuals are obtained contour lines
of these sums of the squared residuals are drawn with the program pisus3. These contour
lines are drawn in the same planes as in the program pisusq (UNIX) or pissql (MSDOS).
In each graph four times five contour lines of the sums of the squared residuals are
drawn. They correspond with four different significance levels of 90%-, 99%-, 99.9%-
and 99.99%. The appropriate values for the sum of the squared residuals must be
calculated with the help of Eq. 6.22 and given interactively. Only four of the lines are
located in the considered plane. The other four times four lines are projected lines in the
considered plane. These lines are located in planes which are parallel to the considered
plane. The results of the program pisus3 are the files plot. pit and the file namel.pr3(n).
The HPGL file plot. pit allows the representation of the graphs on the screen or their
print-outs with the appropriate programs. In the file namel.pr3(n) all parameter
combinations are stored which correspond with the bounds of the 99 % approximate joint
confidence region. This files allow the plot of the three-dimensional cross sections with
the program plelo3 and the approximation of the confidence intervals of the calculated
drawdowns (see Sect. 6.7.3.3).
The input files of the program plelo3 are the files namel.d(p)pc, namel.s(s)q3
and namel.pr3(n) which are linked with the number 32, 33 and 39. Two parameters
must be given interactively. The first parameter is the angle between the first parameter
axis (the horizontal axis in the figure) and the third parameter axis (the axis which is
perpendicular on the two axes which are located in the drawing plane). The second
parameter is the number of the columns in which the three-dimensional graphs must be
drawn. The results are stored in a HPGL file plot. pit which allows to see the three-
dimensional cross sections on the screen or allows to make a print-out.
6.6.7 Program sumsqr
The purpose of this program is to calculate the sums of the squared residuals for a set of
parameter combinations to draw two-dimensional cross sections through the exact joint
confidence region with the aid of the program plsusq. The chosen sets of parameter
combinations are located in mutually perpendicular planes which are parallel to the
parameter axes and traverse the optimal point in the parameter space. The considered
intervals of the different parameters are chosen in the same way as in the program plprcr
and susqln. Also, here, 212 calculations are made for the sum of the squared residuals.
The residuals for each parameter combination are, however, here calculated after a
complete simulation with the numerical model. With the thus obtained matrix of the sum
of the squared residuals it is possible to draw cross sections through the exact joint
confidence region (see Fig. 6.13).
The required files are namel.pap(u), namel.dap(u), namel.pas(e), namel.-
d(p)pc. These files are linked with the numbers 26, 28, 29 and 32. The matrices of the
sums of the squared residuals are stored in the file namel.susq (UNIX) or namel.ssq
(MSDOS). This file is linked with number 35.
Once the matrices of the sum of the squared residuals are obtained for different
windows around the optimal point and belonging to all the different planes which are
parallel to two other parameter axes. The two logarithmic parameter axes are marked in
the same way as in the graphs which are drawn with the pIprcr program. In each graph
the optimal values of the p-2 parameters are printed in the upper right corner of the
graph. In each graph four contour lines of the sum of the squared residuals can be drawn.
They correspond with the bounds of the 90%-, the 99%-, the 99.9%- and the 99.99%
joint confidence region if the appropriate values for the sums of the squared residuals are
given interactively. These sums can be calculated with Eq. 6.22.
This graphs are obtained by the plot program plsusq (UNIX and MSDOS). If the
treated problem is not too nonlinear, the contour lines of the sums of the squared resid-
uals, the results of the programs susqln and sumsqr are similar. The required input files
are namel.d(p)pc and namel.s(s)ql which are linked with the numbers 32 and 33. The
result are written in plot file plot. pit and in a file namel.prl(n) which is linked with
number 39. In the file namel.prl(n) all the parameter combinations are stored which
correspond with the bounds of the 99 % approximate joint confidence region and are
situated in the planes which are parallel to another combination of two parameter axes and
go through the optimal point. This file could be used to approximate the confidence
interval of the calculated drawdowns (see Sect. 6.7.3.2).
6.6.8 Program sumsq2
This program makes the same calculations as the program sumsqr but calculates and
stores additionally for the same set of parameter combinations the logarithms of the
drawdown for one given observation time and location decreased with the logarithm of
the corresponding drawdown of the optimal solution. With the program plsusd the two-
Chapter 6/ Inverse model 277
dimensional cross sections through the exact joint confidence region are plotted along
with the contour lines of the above described differences (Figs. 6.14, 6.15 and 6.16).
The required files are name1.pap(u), name1.dap(u), name1.pas(e), name1.eta-
(dfdp), name1.d(p)pc. These files are linked with the numbers 26, 28, 29, 30 and 32.
The matrices of the sums of the squared residuals are stored in the file name1.ssql
(UNIX) or name1.sqI (MSDOS). This file is linked with number 35. The same input and
output files are used as with the program sumsqr.
The results are written in a HPGL file plot. pIt. This file allows to view the graphs
on the screen or the print-out of these graphs. The different graphs are parallel with two
logarithmic parameter axes and go through the optimal solution. They are marked in the
same way as in the graphs which are drawn with the plprcr program. In each graph the
optimal values of the p-2 parameters are printed in the upper right corner of the graph. In
each graph four contour lines of the sum of the squared residuals can be drawn. They
correspond with the bounds of the 90%-, the 99%-, the 99.9%- and the 99.99% joint
confidence region if the appropriate values for the sums of the squared residuals are given
interactively. These sums can be calculated with Eq. 6.22. In this graph the contour lines
of the logarithm of the drawdowns are plotted for one well-defined observation location
and time decrease with the logarithm of the corresponding drawdown of the optimal
solution.
6.7 Confidence interval for optimal estimated drawdown
In a nonlinear analysis, the investigator is not only interested in obtaining the confidence
region of the model parameters, but also in obtaining confidence intervals on various
functions of the model parameters (Vecchia and Cooley, 1987). An example of such a
function is the drawdown at a certain location and layer after a certain time of pumping
on one or different wells. The intervals can be used to evaluate the relative importance of
model inputs and the anticipated effect of new data and analyses on the reliability of the
model results (Hill, 1989). At the end of a study the intervals can be used to clearly
indicate the likely errors in simulated results. In this section, we show how the exact and
approximate joint confidence regions of parameters developed in the previous section may
be used to derive confidence intervals for the calculated drawdown.
The confidence limits are the upper and the lower bounds of confidence intervals.
A confidence band is formed by joining a series of confidence intervals. A simultaneous
confidence interval is defined as the width of a simultaneous confidence band at one time
and location. As quoted by Hill (1989), Miller (1981) defines a simultaneous confidence
band as formed by two functions which lie entirely above and below the unknown surface
of drawdown with a probability (1-a). Hill (1989) defines the simultaneous confidence
band equivalently as two functions which contain the entire surface for (l-a)100% of all
possible values of the model inputs used to calculate the drawdown. The fact that the
simultaneous confidence interval can only be defined in relationship to the simultaneous
confidence band is the result of its being a joint, versus an independent, confidence
interval. An individual confidence interval is defined by two values which lie above and
below, respectively, one unknown point on a surface, for example, drawdown with
probability (i-a). Equivalently, an individual confidence interval can be defined as two
values between which the point calculated at one time and location occurs for (1-a)100%
of all possible model inputs used to calculate the point. Individual confidence intervals are
independent. A joint, simultaneous confidence interval is always wider than an indepen-
dent, individual confidence interval calculated for the same time and location.
Previously developed methods of error analysis for non-linear groundwater flow
problems can be divided into two groups (Dettinger and Wilson, 1981); one group
produces the full probability density (pdf) function of hydraulic head, the other produces
only the first and second moments (the means and variances). The full pdf function of
hydraulic head or drawdown has been produced through the Monte Carlo methods (Hill,
1989) which has the disadvantage of being computational intensive. First and second
moments methods provide sufficient information to accurately determine confidence
intervals on simulated head as long as higher order moments, such as skew, are small
(Dettinger and Wilson, 1981).
6.7.1 Drawdown confidence intervals based on two- and three-dimensional cross-sections

through joint confidence interval of the hydraulic parameters
The individual confidence intervals of the drawdown can be approximated by the help of
the two-dimensional cross-sections through the joint confidence region of the hydraulic
parameters. As already stated in Sect. 6.3.4.2, the parameter combinations corresponding
with the contours of the two-dimensional cross-section through the joint confidence region
correspond with points on the bounds of the joint confidence region in the p-dimensional
parameter space. For each of these points or parameter combinations the drawdown at a
certain observation time and location can be calculated. The maximum and minimum
value of all these drawdowns can now be considered as the upper and lower bound of the
confidence interval of the drawdown. This is a first approximation because only a limited
number of bounding points of the joint confidence region are used. As in the Monte Carlo
method many simulations are performed. The method differs, however, from the Monte
Carlo method. In the proposed method the edges of the joint confidence region are
searched and in the Monte Carlo method the region in and around the joint confidence
region is studied.
The above described method can best be explained with the help of the plots made
by the programs sumsq2 and plsusd. In this plot p!/(2!(p-2)!) two-dimensional cross
sections through the exact joint confidence area are represented by closed contour lines
along with the lines of equal logarithms of the drawdown ratios. These ratios are between
the drawdown for a certain parameter combination and for a certain observation time and
location, at one hand, and the drawdown corresponding with the optimal parameter values
for the same observation time and location, at the other hand. For each cross section, the
minimum and maximum value of the logarithms of the drawdown ratios are determined
when the contour lines, which represent the bounds of the 99% confidence interval, are
followed. The lower bound of the 99% confidence interval of the drawdown is now the
smallest value of all minima, whereas the largest value of all maxima is put equal to the
upper bound of this confidence interval.
To increase the number of points on the bounds of the joint confidence region of
the hydraulic parameters one can consider the three-dimensional cross-sections instead of
the two-dimensional cross-sections. Because a larger amount of points or parameter
combinations is considered, it can be expected that the so obtained confidence intervals
for the drawdown are better approximates. However, it is still possible that the parameter
combinations which correspond with the largest and the smallest values of the drawdowns
are situated just between the considering bounding points which are situated on the three-
dimensional cross-sections. In the following section a method is treated to find the points
on the bounds of the joint confidence region which correspond respectively with the
minimum and the maximum drawdown for a certain observation time and location.
6.7.2. Drawdown confidence intervals derived by optimization of constrained problem
The bounds of the joint confidence region can be approximated by a p-dimensional

ellipsoid if two assumptions are not violated. The first assumption is that the residuals
exhibit a normal distribution. The second assumption is that the drawdown can be
approximated as a linear function within the considered joint confidence region. In Sect.
6.3.3 Eq. 6.15 describes the location of all points on the p-dimensional ellipsoid in the
p-dimensional Euclidian coordinates system lIP of which the origin is situated in the
optimal point:
p p
~ [(.k,fljlHPj)2/Yil =pF(p,n-p,l-u) (6.30)
This relation can also be written as:

a11HP~ +a2,HP2HP, +a3,HP3HP, . . . . • . . +ap,HP~P,
+a'2HP,HP2+a22HPi +a 32 HP3HP2 . . . . . . . +ap2HPpHP2
+a'3HP,HP3+a23HP2HP3+a33HP; . . . . . . . +ap3 HPpHP3
+ ...................................... .
+a,~P,HPp+a2~P2HPp+a3~P3HPp' . . . . . . +ap~p;,
(6.31)
=pF(p,n-p,l-u)
or ellip(BP)=pF(p,n-p,l-u)
where a··
l.J
= f
k .. l
flikfljk
Yk
and F(p,n-p,l-a) is the F-distribution for p and n-p degrees of freedom and a significance
level (I-a).
Then the optimization problem for the (l-a)l00% confidence of the drawdown
s(lIP) may be formulated as:
min s (HP) , max s (HP)
(6.32)
HP BP
subjected to the equality constraint

ellip(HP) = pF(p,n-p,l-l1.) (6.33)
Here, it is assumed that the gradient of s(HP) with respect to HP is nonzero within the
closed region of the ellipsoid and that the extreme values are situated on the boundary of
the ellipsoid.
As a Lagrangian problem (Haimes, 1977), Eqs. (6.32) and (6.33) are equivalent to find
the extreme value of
L(HP, ').') = s (HP) +').'[ellip(HP) -pF(p,n-p, 1-11.) 1 (6.34)
where A' is the Lagrange multiplier.

Necessary conditions for a stationary point of L are:
~=~=~=
aHP aHP aHP
=~=~=o
aHP a').' (6.35)
1 •••••••
2 3 p
or by expanding:
(6.36)
.. fJi. .... iJ;' (HPi ... ; ............................... . ~
-- - - a - - +2'). (a p1 HP1 +apflP2+ap3HP3' . . . . . +appHPp) -0

aHPp HPp
aLi = ellip(HP)-pF(p,n-p,l-l1.) =0
a').
The first p equations can be reformulated in matrix notation as

a HP = -)..Jj (6.37)
where a is a symmetric squared matrix of which the elements are given by Eq. 6.31 and
J j is a row of the sensitivity matrix corresponding to the point j or is the gradient of the
drawdown with respect to the hydraulic parameters
J j = {J .. }1 = (as(HF)}1 and ).. 1

(6.38)
~J i=l aHFi i=l 2)..1
To start the iterative solution of Eq. 6.36, the value for A can be set equal to one and the
gradient of the drawdown with respect to the hydraulic parameter J j can be supposed to
be constant within the joint confidence area and can be set equal to the gradient in the
optimal point. Estimating a value for A, the elements of the vector HP can be deduced
from Eq. 6.37 or
(6.39)
Once the values for lIP are known the value for A can be deduced with the help of the
following equation
p p p
A= (pF(p,n-p, 1-«) -I: I: aijHPi-ellip(HP» II: J ij
j=l i=l i=l.
(6.40)
With this new value for A new values for lIP can again be deduced with Eq. 6.39 after
which the A value can again be calculated with Eq. 6.40. The iteration process continues
until the values of the elements of HP and the A value do not longer change from one
iteration to another. The so obtained parameter combination will approximate the lower or
upper bound of the confidence interval of the calculated drawdown.
The so obtained interval is still an approximation because it is assumed that the
drawdown gradient with respect to the hydraulic parameters or the sensitivity is constant
within the joint confidence region of the hydraulic parameters. The difference between the
logarithm of the drawdown corresponding with the optimal parameter values and the
logarithm of the lower and upper bound of the confidence interval of this drawdown st is
. P
or sf' = I: HPiJij (6.41)
i..,l.
With this parameter one can calculate the lower and upper bound of the confidence
r
interval of the drawdown Sj' s/ and st :
and S!' (6.42)

J
6.7.3 Program packages to approximate drawdown confidence interval
Four program packages have been conceived to approximate the confidence interval of the
calculated drawdown:
- the package contil with which the confidence intervals are approximated based on the
parameter combinations corresponding with the two-dimensional cross-sections through
the approximate joint confidence region of the parameters,
- the package contill for approximating the confidence intervals with the aid of the
parameter combinations corresponding with the two-dimensional cross-section through the
exact joint confidence region of the parameters,
- the package conrt3 for approximating the confidence intervals based on the parameter
combinations corresponding with the three-dimensional cross-section through the
approximated joint confidence region of the parameters,
- the package conti4 for approximating the confidence intervals based on the optimization
of a constrained problem in which the constrain is a p-dimensional ellipsoid which
approximates the joint confidence region of the parameters.
An example of the results of these program packages is given in Sect. 6.8.
6.7.3. 1 Program package conti!
The purpose of this program package is the plot of the confidence intervals of the
calculated drawdown. These confidence intervals are approximated with the help of the
parameter combinations corresponding with the two-dimensional cross-section through the
approximate joint confidence region of the parameters.
These two-dimensional cross-sections are obtained by the help of the program
susqln and the program plsusq (see Sect. 6.6.5). The required files are name1.pap(u),
name1.dap(u), name1.pas(e), name1.eta(dfdp), name1.d(p)pc. The files name1.eta-
(dfdp) and name1.d(p)pc are the results of the last iteration of the inverse model
(program inpur5). These files can also be the results of a combination of the programs
inpur5 and soIpuS. The first calculates the sensitivities and the second the adjustment
factors and the marginal and conditional standard deviations. Both calculation methods
must be performed with the optimal values for the hydraulic parameters.
The results of the program susqln are the matrices of the sum of the squared
residuals which are stored in the file name1.ssqI (UNIX) or name1.sql (MSDOS). With
the help of these matrices the contour lines of the sum of the squared residuals are plotted
with the program plsusq (UNIX) for the different two-dimensional cross-sections through
the 90%-, 99%-, 99.9% and 99.99% joint confidence regions. For different confidence
regions the respective sums of the squared residuals which are required can be calculated
with Eq. 6.22 and stored in a file name1.susqpr.inp (UNIX). The result of this plot
program is not only a graph but also a file name.prl(n) where all the parameter combina-
tions are stored which correspond with the bounds of the 99 % approximate joint confiden-
ce region. These parameter combinations are now applied in the program conti!.
With each parameter combination a simulation is performed with the numerical
model. For the locations and the times given in the file name1.dap(u) the maximum and
minimum values of the drawdown are searched. The calculated drawdowns corresponding
with the optimal values and the upper and lower bounds of their confidence intervals are
stored in the file name1.dap(u) after the data which described the observations. With the
help of the plot program outpuS the confidence intervals of the calculated drawdown can
be represented in time-drawdown and distance-drawdown graphs (see Fig. 6.17).
6.7.3.2 Program package contiU
With this program package the confidence intervals of the calculated drawdown are
approximated with the aid of the parameter combination corresponding with the two-
dimensional cross sections through the exact joint confidence region of the parameters.
These two-dimensional cross-sections are obtained by the help of the program sumsqr
and the program plsusq (see Sect. 6.6.7). The required files are name1.pap(u), na-
me1.dap(u), name1.pas(e), name1.d(p)pc. The file name1.dppc (UNIX) or name1.dpc
(MSDOS) is the result of the inverse model, program inpur5, or of a combination of the
programs inpur5 and solpu5 with the optimal values for the hydraulic parameters (see
former section).
The results of the program sumsqr are the matrices of the sum of the squared
residuals which are stored in the file name1.susq (UNIX) or name1.ssq (MSDOS). With
with the program plsusq (UNIX) for the different two-dimensional cross-sections through
the 90%-, 99%-, 99.9%- and 99.99% joint confidence regions. For different confidence
regions the respective sums of the squared residuals which are required can be calculated
with Eq. 6.22 and stored in a file name1.susqpr.inp (UNIX). The result of this plot
program is not only a graph but also a file name.prl(n) where all the parameter combina-
tions are stored which correspond with the bounds of the 99% approximate joint confiden-
ce region. These parameter combinations are now applied in the program conti!. From
here on, the same programs and files are used as described in the previous section to
obtain the confidence intervals of the calculated drawdown (see Sect. 6.7.3.1).
6.7.3.3 Program package conf"I3
The confidence intervals of the calculated drawdown are approximated with the aid of the
parameter combination corresponding with the three-dimensional cross sections through
the approximate joint confidence region of the parameters. These three-dimensional cross-
sections are obtained by the help of the program susql3 and the plot programs plsusq,
plsus3, plelo3 (see Sect. 6.6.6). The required files are name1.pap(u), name1.dap(u),
name1.pas(e), name1.eta(dfdp), name1.d(p)pc. The files name1.eta(dfdp) and
name1.d(p)pc are the results of the last iteration of the inverse model (program inpur5)
or of a combination of the program inpur5 and solpuS with the optimal values for the
hydraulic parameters (see also Sect. 6.7.3.1).
The results of the program susql3 are the matrices of the sums of the squared
residuals which are stored in the file name1.ssq3 (UNIX) or name1.sq3 (MSDOS). With
with the program plsus3. One of the results of the program plsus3 is the file namel-
.pr3(n) where all the parameter combinations are stored which correspond with the
bounds of the 99 % approximate joint confidence region according to three-dimensional
cross-sections. These parameter combinations are now applied in the program conti3.
With each parameter combination a simulation is performed with the numerical
model. For the locations and the times given in the file name1.dap(u) the maximum and
minimum values of the drawdown are searched. The calculated drawdowns corresponding
with the optimal values and the upper and lower bounds of their confidence intervals are
stored in the file name1.dap(u) after the data which described the observations. With the
help of the plot program outpuS the confidence intervals of the calculated drawdown can
be represented in time-drawdown and distance-drawdown graphs (see Fig. 6.19).
6.7.3.4 Program package confi4
This program package is conceived for approximating the confidence intervals based on
the optimization of a constrained problem. The used constraint is the p-dimensional
ellipsoid which approximates the joint confidence region of the hydraulic parameters. For
the equation of the p-dimensional ellipsoid the required eigenvalues and eigenvectors are
read from the file name1.d(p)pc. The gradients of the drawdown with respect to the
hydraulic parameters are supposed to be constant within the joint parameter region and
are set equal to the sensitivities at the optimal solution. These sensitivities are stored in
the file name1.eta(dfdp). So, the files name1.eta(dfdp) and name1.d(p)pc must be the
result of the last iteration of the inverse model (program inpur5). It is also possible that
these files are the results of one sensitivity analyse, one iteration with program inpur5,
followed by the program solpu5 which calculates the adjustment factors and the
eigenvalues and eigenvectors of the covariance matrix. The other required files are
name1.pap(u), name1.dap(u), name1.pas(e). The results are represented in Fig. 6.20.
6.8 Hypothetical example to demonstrate nonlinear regression and

approximation of drawdown joint confidence intervals
6.8.1 Conceptual model of 'actual' groundwater flow
The approximation of the confidence interval of the drawdown is shown by the aid of a
hypothetical example. In this synthetic problem the input data of the inverse model is
calculated by means of the numerical model. The assumed pumping test is executed in a
layered groundwater reservoir. The composition of this groundwater reservoir is the same
as the example which was used to validate the inverse model. (see Sect. 6.4, Fig. 6.7).
The groundwater reservoir is composed of three layers, each with a same thickness of ten
meters. The lower layer is pervious, homogeneous and isotropic with a hydraulic
conductivity of 10 mid and a specific elastic storage of 8xlO-sm· l . This layer is bounded
below by an impervious boundary and above by a homogeneous anisotropic semi-pervious
layer. The horizontal and vertical conductivity of the semi-pervious layer are respectively
0.5 mid and O.lm/d and its specific storage is 4x10·sm- l . The upper pervious layer is a
homogeneous isotropic unit which is bounded above by the watertable. The conductivity
and the specific elastic storage are 10 mid and 4xlO-4m-1 respectively. The specific yield
or the storage coefficient near the watertable is 0.2.
In the numerical model, this groundwater reservoir is subdivided in six layers
(Fig.6.7). Layer 1 is the lower pervious layer. The semi-pervious layer is discretized into
three layers (Layers 2, 3 and 4 of the numerical model) of equal thickness. This discreti-
zation is necessary to simulate accurately the flow, the drawdown and the storage changes
at the different levels of the semi-pervious layer. The upper pervious layer is discretized
in two layers, Layer 5 and 6 in the numerical model, which represent respectively the
lower 9 m and the upper 1 m. The hydraulic parameters introduced in the numerical
model correspond with the parameter values of the pervious and semi-pervious layer as
mentioned in the previous paragraph.
ORAWDOWftG DRAWOO... OO
.00 .00*-____-+______ ~----~
..- ~
,....lS QC.
/
• ..-.~----_F~--~~----_l
00- 2
L
.00 .0' .02 1~ lItEonN)
LAYER
DRAWOOW (to DRAWDOWfOO
.00 .00
10- 1
..- ..-
.00
'.' .02 I~ T1t1E("IN)
LAYER 1
100
50=.200
0(3)=10.0 M K(3)-10000000.00 MID 5A(3)=.000400 M-l
C (2) =47.9 0
0(2)=10.0 M K(2)=.00 MID 5A(2) =.000350 M-l
C (1) -47.9 0
0(1)=10.0 M K(1)=10.52 MID 5A(I) =.000090 M-l
Fig. 6.8. Graphical representation of the "observed" drawdown (crosses) of the hypo-
thetical problem in time-drawdown and distance-drawdown graphs (program outpuS) with
the best-fitted drawdown (continuous lines) following the ordinary least square method.
The assumed conceptual model during the interpretation is the model of Hantush.
6.8.2 Creation of "observed" drawdowns
Using the numerical model, drawdowns are calculated at three observation wells assuming
that the lower pervious layer was pumped with a constant discharge rate (180 m3 /d). Two
observation wells are located in the middle of the pumped pervious layer and one in the
middle of the overlying semi-pervious layer. The wells in the pumped pervious layer are
at a distance of 15.85 m and 50.12 m from the pumped well. The single well in the semi-
pervious layer is at a distance of 15.85 m. On the numerically calculated drawdown an at
random error is added. These added errors approximate a log normal distribution with a
variance equal to 0.0625. Only drawdowns which are larger than 10 mm are used as input
data in the inverse model. The drawdowns are rounded off to the nearest millimeter. The
times are given with an accuracy of 0.1 minute for the first hour and 1 min after the first
hour of pumpage. The 79 simulated observations for the three observation wells, which
are used as input data in the inverse model, are shown in Fig. 6.8.
6.8.3 Conceptual nwdel of groundwater flow used during interpretation
The conceptual model of the groundwater flow applied in the inverse model for the
interpretation of the observations is different from the conceptual model of the "actual"
groundwater flow. The applied conceptual model corresponds with the model of Hantush.
In this conceptual model, a pumped pervious layer is considered which is bounded below
by an impervious boundary and above by semi-pervious layer in which only vertical
elastic flow is possible. The upper boundary of this semi-pervious layer is considered to
be a constant hydraulic head boundary. In the applied numerical model three layers are
considered to introduce this conceptual model. The three layer have the same thickness as
in the "actual" model, each layer has the same thickness (10m). Because only vertical
flow is considered in the conceptual model the horizontal conductivity of layer 2 of the
numerical model, which corresponds with the semi-pervious layer, is put equal to a very
small value (10-7 mId). To replace the constant hydraulic head boundary, assumed in the
applied conceptual model, the hydraulic conductivity of layer 3 is put equal to a very
large value (107 mId). According to this conceptual model only four different parameters
must be determined. These four parameters, which are included in the inverse process,
are the hydraulic conductivity kh (l) and the specific elastic storage S,(l) of the pumped
pervious layer and the hydraulic resistance c(1-2) and the specific elastic storage S,(2) of
the covering semi-pervious layer.
6.8.4 Interpretation Uy means of the ordinary least square method (OLS-method)
The ordinary least square method was used to reach the optimal solution (program
inpur5). The optimal value for the hydraulic conductivity kh(l) was 10.62 mId against 10
mId for the "actual value" of the hypothetical problem and its marginal and conditional
standard deviation is respectively equal to 0.0250 and 0.0135. For the specific elastic
storage S,(1) the best estimate or the optimal value according the ordinary least square
method is 8.955xlO-s m- l against 8xlO-s m- l for the "actual value" and 0.0208 and 0.0146
for the marginal and conditional standard deviation. The best estimate for the hydraulic
resistance of the semi-pervious layer is 95.82 d against 100 d for the "actual value". Its
marginal and conditional standard deviation is respectively 0.0580 and 0.0198. Finally,
the best estimate of the specific elastic storage is 3.597xlO-4 m- l against 4xlO-4 m- l whereas
its marginal and conditional standard deviation is respectively 0.0736 and 0.0368.
..,- +
+
eo f 10
;~ 'r' eo
.jf'
10
70 70
10 lJ
~ If
109
iii 50 50&
.,
c(
j
405!It...
ID
0
"
r:r:
~
0..
:so t'\.. :so

~ '\
"
20 20
10 / V 10
.-/. t:.r ~ t--
- ... -.3 -.2 -.1 0 •1 .2 .3 ... .5
RESIDUAL
_ • IIOIIIW. DJ81RJIUTJDI J8 .OM
81 _ _ D!VIATJOII • IIClMIM. DIITRIIUTJOII J8 0172
Fig. 6.9. Residuals versus their probability along with the best fitted log normal distribu-
tion with mean 0.006 and standard deviation 0.172 (program etabdi)
After the optimization of the hydraulic parameters it is examined if the residuals

exhibit a normal distribution and the assumption made to apply the regression analysis.
Therefore, the residuals are plotted versus their probability along with the best fitted
normal distribution (Fig. 6.9, program etabdi). Studying this figure one can conclude that
the made assumption is not violated; the residuals approximate a normal distribution with
a mean zero. The time-, distance- and layer-dependence of the hydraulic parameters can
be deduced from a visual. analysis of the time-drawdown and the distance-drawdown
graphs in which the "observed" drawdown are represented along with the drawdowns
which are calculated with the optimal values of the hydraulic parameters (Fig. 6.8,
program outpu5). From this visual analysis one can conclude that the residuals are not
layer and time-dependent. Because there is only one observation well in the semi-pervious
layer (layer 2 in Fig. 6.8) it is not possible to deduce if the residuals are distance-
I.as k 01 I1ID c ' ......'0' ..,. 1.25 K 01 H.tD _I~.o-nr' 11-1 I•• K 01 ".10 PO...... , ........
~,"IO-""'I ."....... '0-..... 1 C 11-4 ... ", IW
.... • • • • .11 .M .OS •• 0 .1$ .:ao .as
I." SAOI ~I I CII_,.,elol NO

...."'... '0..........
1.21 SAOI 1"1-1 1101_•••• ,,1_
c, .......... ' _
..,.r,;;; ......,.;;-"T.,,;;-t.-::
..:-<.o;"..,.-;;; ..:;..,;;T.''ii-<7~~D
..:;.7.,,:;.-:: .10 SA02 "-1
1.00 C 12 HID II 0,_",eIl1 ItfOD
1M, ... 000, ........ ,
SA02 "-1
• • • 17 . . . . pt; •. , .11 .1$ • • • at .7• • 11
Fig. 6.10. Six two-dimensional cross sections through the four-dimensional ellipsoids
which approximate the 90%-, 99%-, 99.9% and 99.99% joint confidence regions of the
hydraulic parameter deduced with the help of the eigenvalues and eigenvectors of the
variance-covariance matrix (program plprcr)
dependent. The residuals in the pumped pervious layer (layer 1 in Fig. 6.8) show a slight
dependency for the distance. The residuals of the observation made in the observation
well which is closest to the pumped well are all of the same sign. The residuals of the
observations made at a distance of 50 m vary around the zero value. This slight distance-
dependency is due to the differences between the "actual" model of groundwater flow
resulting in the "observed" drawdown and the conceptual model which is used to deduce
the optimal values of the hydraulic parameters. By the adaption of this conceptual model
towards the actual model of groundwater flow this distance-dependency will reduce. One
can wonder if this slight distance-dependency is sufficient to conclude that the assump-
tions made to apply the regression are violated?
1.25 K Of HID
1.20 1.20 1.20
1.15 1.15 1.15
1.10 1.10 1.10
1.05 1.05 1.05
....9' 1.00
.9•
.9'
1.1_101
.9'
.92
.SS .SS .88
.8, .S' .8,
.80 .....~~~~..;:;.~~-=;::.:...;.:...:,
.So e 12 HID .80 -<"'-r--'--r---'--=+--"'--"'~~!!:'!"
.80 .84.88 .•2.961.001 .at IJ 01 T.2t· 25 .50 .57. 66 .75. 87 l.od· 1 t3~ .5~ .71,00 .50
.51
.6&
.15
.87 1.15
1.00 1.32
t .52
1.74
2.00
1.25 SAO 1 H-'

1.20
1.15 1.15
1.10 1.10
1.05 1.05
-.
...
'.00
• 9'
9. -, 1.00
.9 •
.9'
,0-1 -s
.88 .S8
.s. .84
.SO .......-..--r--r--.--+--.---.--"c-r':.::....."';:..D;. •80 SA02 tt-I
.50. 57 ···.75 .87 ,.oJ .. t3~ .5~ .rf· OC .50.57 •••. 75. 87 ,.oJ·' ~ .3~ .5t71· OO
2.00 C 12 !'VO
1. 74
1.52
I.J2
1.15
,.OOI~,.J!;=~~-,
.S7
.75
.66
.57
• 50 .......-r---.--r---.--4---.,.~....::;:::..::....:,.
.50 .57. 66 .75 .87 toad .11.~~ .5~ .71. 00
hydraulic parameter deduced with the aid of the sum of the squared residuals. The
residuals of each parameter combination is calculated with the linearization method
(program susqlo)
6.8.5 loint confidence region of hydraulic parameters
The joint confidence region of the hydraulic parameters can be approximated by concen-
tric four-dimensional ellipsoids around the optimal values if two assumptions are not
violated. The first is that the residuals exhibit a normal distribution and the second is that
the drawdown can be approximated as a linear function within the considered region.
These concentric four-dimensional ellipsoids can be deduced from the eigenvalues and
eigenvectors of the variance-covariance matrix (Eq. 6.15). In Fig. 6.10 six two-dimen-
sional cross sections through these concentric four-dimensional ellipsoids are represented
which approximates the bounds of the 90%-, 99%-, 99.9% and 99.99% confidence
regions (program plprcr). These cross-sections can help us in the regression diagnostic.
From these graphs one can remark the correlation between the hydraulic resistance (C 12)
and the specific elastic storage of the semi-pervious layer (SA02) at a glance. From these
graphs one can also deduce that the hydraulic parameters of the semi-pervious layer are
deduced with much less accuracy that the hydraulic parameters of the pumped pervious
layer.
The same cross sections are also obtained by means of another method based on
the sums of the squared residuals for the parameter combinations which are situated in the
considered windows of the mutually perpendicular planes. The residuals for each
parameter combination are deduced from the residuals and the sensitivities at the optimal
points with the linearization method (Eq. 6.23, program susqlo). The sums of the squared
residuals which correspond with the 90%-, 99%-, 99.9%- and 99.99% joint confidence
region are calculated by means of Eq. 6.22. The required parameters are the sum of the
squared residuals at the optimal point (S(hp)= 0.4182), the number of deduced parame-
ters (p= 4), the number of observations decreased with the number of deduced parame-
ters (n-p= 75) and the values of the F-distribution for the significance level 0.1, 0.01,
0.001 and 0.0001 (F(4,75,0.9O)= 2.02152, F(4,75,0.99)= 3.58120, F(4,75,0.999)=
5.16290 and F(4,75,0.9999)= 6.81022). These last values can be read from the tables of
the F-distribution (e.g., Mardia and Zemroch, 1978). The resulting values of the sum of
the squared difference are 0.46328798, 0.498075082, 0.533353322 and 0.570095142. In
Fig. 6.11 the contour lines of these values of the squared residuals are given for the same
windows as in Fig 6.10. Comparing both figures one can conclude that both methods give
nearly identical results.
Four three-dimensional cross-sections through the four-dimensional ellipsoid which
approximates the 99 % joint confidence region are represented in Fig 6.12 (program
susqI3). Here, the required sums of the squared residuals are obtained by means of the
linearization method with the aid of the residuals and the sensitivities of the optimal
solution (Eq. 6.23). The sums of the residuals which correspond with the 90%-, 99%-,
99.9%- and 99.99% joint confidence region are calculated by means of Eq. 6.22. From
this figure one can immediately remark that the hydraulic parameters of the pumped
pervious layer are deduced with higher accuracy than the hydraulic parameters of the
covering semi-pervious layer.
Ie 01-1.1-10' NO ~<f!'"
.,~
.3'
..
• 52
.00
.15
.00
.• 7
.66
1.25 .57
.50
1.20 SAOI PH
1.15
~1Ic.I!""'+--------J 1.10
1.05
...
.9'
1.00
.9'
.9'
.0_' -,
.0. .a•
. 0. .0•
. ao ~:r-..-::~"'::-I'-,c."!"":"':"::":::.=.t'
.5'
.3'
t .1~
.00
...
.07
.70
1.25 . ~S7
1.20 I( 01 HID
1.15
1.10
1.05
1.00f'L'-J.:J.\"--+-7f
.0'
...
.0'
.0'
Fig. 6.12. Four three-dimensional cross sections through the four-dimensionale ellipsoid
which approximate the 99%-confidence region of the hydraulic parameters deduced with
the aid of the sum of the squared residuals. The residuals of each parameter combination
is calculated with the linearization method (program susql3)
In Fig. 6.13 six two-dimensional cross sections are represented through the 90%-,
99%-, 99.9% and 99.99%-joint confidence region. To draw these cross sections the sums
of the squared residuals are calculated for the same parameter combinations as considered
in Fig. 6.10. Here, for each parameter combination the residuals are now calculated after
one complete numerical simulation. The values of the contour lines corresponding with
the bounds of the different joint confidence region can again be calculated with Eq. 6.22
as demonstrated and given in the second paragraph of this section. There is a slight
difference in shape between the contour lines represented in Fig. 6.10 and those represen-
ted in Fig. 6.13. The contour lines in Fig. 6.10 are symmetric with respect to the optimal
solution. This is not the case for the contour lines in Fig. 6.13. This is because the
problem deviates a little from a linear function within the considered joint confidence
area. This can be shown by the study of the contour lines of the logarithm of the
drawdowns for some observation times and locations.
1.25
'.20
.., HID
~AM:~::::g!4~1 1.25 • DI HID
1.20
~A01.9.0.10-S H-l
A02"3.6.'O-4 1'1-1
, .25
1.20
.., HID S"0'·9.0-'0-5 H-I
C 12 ..... 8.'01 "/D
1.15 loiS 1.1$

, .10 1.10 1.10
1.05 1.05 1.0.5

.
...
1.00 1.1
...
t .00 1.,·,0'
...
1.00 1.1.'
......
•• 2
......
.92
.....
.92
... ... ... ...... • 96

1.00
1.05
SAOI ,,.,
1.15 1.25
1.10 1.20
... • 5•
.57
... .75
.87 1.15
c
1.00 1.32
12 HID
1.52 2.00
,.74
... .50
.S7
.66
.7S
.87 loJS
1.00
SA02 "-,
1.52
1.32
2.00
1.14
1.2S SAOI H-l I( 01-1.1.101 ""D

C 12· ... 8.'01 tvD
1.20
1-15 1.15
1.10 '.10
1.05 1.05
...
1.00 • _, 0'
...
1.00 • -, -,
......
• '2
.
.'2
.8•
... C 12 "/0
.
···.75 ·",.oJ·
. 80 "r::-::r-"T".,.-..,..-:+-...,-...-S,.
.50. 57 ·~.75·87 I.oJ· 't3~ .Std· OO .5•. 57
••;.;;2..,.H-~'
't3~ .5t7l- 00
2.00 C 12 M/D
1.7.
1.52
1.32
'.1$
,.OO~=""",~..--,
•• 7
...
.75
. 57
• S. "r-..,-..,-.,.-.,.-+-.,.-"'-:::';=':':"'':''
.50 .S8 .87 1.151.522.00
.57 .75 1.00 1.32 1.74
hydraulic parameter deduced with the aid of the sum of the squared residuals. Here, the
residuals are calculated after a complete run of the numerical model with each considered
parameter combination (program sumsqr)
6.8.6 Combined irifluence of hydraulic parameters at some observation points
In Fig. 6.14, 6.15 and 6.16 the contour lines of the logarithm of the drawdown decreased
with the logarithm of the corresponding drawdown at the optimal solution (program
sumsq2) are represented for three different observation points. These contours are
represented along with the contour lines which deliminate the bounds of the joint
confidence regions (same contour lines as in Fig. 6.13). The three different observation
points are all located in the same layer and at 50.1 m from the pumped well.
I .•' K., ~:!:2:18:~ R=l 1.•".'",0 lAf~:::3:18;5"",

1.20 • - - - - - 1.20t-_ _ .J>I-.....,..._____
1.15
77..::::!!~--_ 1.1. I--+f,""~~
__
....
-jrmf"-.:.~\\- ".';.j---H.w---¥.~---
...
.11 ....9. 1.00
.
• 11
.a.
... .
.aa
.~~~~~~~~~~~ .••
.84
10
-4-::,..-"';=;~~=r=~~ ...
.a •
SA02 ... 1
1.25 SAOI H-I 1~:!:1:lg~4"AA, 1.25 SAOI "·1 ---*!-l'r~}:l:~

1.20..... 1.20
I .15;.L._---:;l~....... 1.15
'·'·L--·-M'-":~~--- "'.+---J~";:~~---
1 ••' 1 •••+---u.{IIf--~;\_--
1.00 •• '"' -+I-H--- 1.00 • -I ..
•M
...
.•• L_---\WI--r7"'r! ::: l--_'~l'<-+~-I#---
.11
• 8' .a• .j----~::>_t.
.eo C I. ",0 • 8. -l;=o;==;=;;==;~...,....."..~1;!!i1l-,.,

••• ••7···.7.··7 1 .ocl· , .,l·'f.rl· OO
2.00 C 12 PVD
1.74
1.52
1.32
1 ...l -........'t7·__~!lot-----
I.OO~......~~...,
.87
.75
.11
••7 I-----=t--....- - -
.•• "'r::r-..-=:...-r:::'f--o:--:-r-=SAOF.:,.r....;,'
.50. 57 oH.75 .17 I.oad .Itd "'-,1,00
Fig. 6.14. Contour lines for the logarithm of the ratio of the drawdown to the drawdown
at the optimal point along with the contours which delimitates the joint confidence region.
The considered observation point is located in the pumped pervious layer at 50.1 m from
the pumped well and after 4.5 minutes of pumping (sumsq2)
The first observation is made at 4.5 minutes after the start of the pumping test
(Fig. 6.14). From the graph in the upper left corner of the figure one can immediately
deduce that the horizontal conductivity and the specific elastic storage of the pumped
pervious layer are the parameters which influence this observation point for the largest
part. The gradient of the represented surface is the largest for these two parameters.
From the graphs in the right column of the figure one can deduce that the specific elastic
storage S,(2) has practically no influence on this observation. The contour lines in these
graphs are all very close to parallel to the axis of these hydraulic parameters .
....1' ....1'
...
. a. ......
.9'
....
.a •
.
. 8. .
~h~
12· ... 8. D
................
'OJ ....
~
...... .....
...
.
....•.
• 88
...
--
~A81:~:~:18!stv~,
.87
....7'
• 57
.•• ;""""'-~~-:3"""'f:::r..::;=~.:.:
.7'
Fig. 6.1S. Contour lines for the logarithm of the ratio of the drawdown to the drawdown
at the optimal point along with the contours which del imitates the joint confidence region.
the pumped well and after 89 minutes of pumping (sumsq2)
The graphs for the second treated observation point made after 89 minutes of
pumpage are represented in Fig. 6.15. From these graphs one can deduce that the four
treated hydraulic parameters influence the drawdown at this observation point. It is also
clear that the drawdown in this observation point is non linear with respect to these
hydraulic parameters because the contour lines deviate clearly from strait lines. The
second order derivatives are here more or less important for this observation point.
1.25 K 01 I'VO
1.20 1.20
'.'5+-___ ,...,,:;;;;:::;;;;;;;;....,,.-- 1.15
1., .+--74c...F'7.~~~:"-- '.'.+----f1~~r-~~--_

1.0S 1.05
1.00~~i>!+-4-~ ,.OO.J.l.'.!.:.LI4m-, """"Tttt"-_ _

.96 .96
::: t---'*\~+--+t\\--_
...... 1 - - - - 4 -..... _-- .. _.t,.-+-.,L-.-Ll-;h...L,~~ ......
::: 1---..l,~~~=f:;.4<;L--- :::
~ -
SAO! M-I .80
.•• -I,....-.---.--~.,....::+-...-~~~
.ao .84
.as'.92 .96 1.001.051.151.25
1.10 1.20 .ori 1.::d .st d·
.50 .57.66.75.81 I .1 QO .50. 57 .66. 75 .87 I.OJ .11.3~ ,St,l'OO
1.25 SAOI M-'

1.20
'.15
1.10
., ..
..
1.05 1.05
...
1.00 9. 1.00
.
9.0·'0-s
... ...
• 92 .92
... ...
12 MID .80 """'-.---,--~"""::+-...-,....:Irl~
.50 .57.66.75 87 l.oJ·1 ~ .3~ .5i .71,00 .50.57.66.75.87,.06.1 ~ .JJ .5~ .71,00
2.00 C 12 HID
~Agl :~:b:: g~5H(/!.1
1.7"'i---"...,.._ _ _ _ -
1.52
'.J21-....\-j~..;;::I. .-
1.15
,.OO+,,-~~~-, ~~ _ __
•• 7
.75
.•• 1----~-;;:::
.57
. 50 -I.-:-::r--,--,---~1--..._~SA~.2:..;:1t-.!,.'
.50 .51. 6 &.75 .87 I.OJ oIt3J .5~ .71,00
Fig. 6.16. Contour lines for the logarithm of the ratio of the drawdown to the drawdown
at the optimal point along with the contours which delimitates the joint confidence region.
the pumped well and after 1685 minutes of pumping (sumsq2)
In Fig. 6.16 the graphs for the third treated observation point are represented.
This third observation is made after 1685 minutes of pumpage. From this graphs it is
clear that this drawdown is the most influenced by the horizontal conductivity of the
pumped pervious layer and by the hydraulic resistance of the semi-pervious layer. These
are the hydraulic parameters which influence only the steady-state flow which is almost
reached after 1685 minutes of pumpage. The inclination of the surface in the graph of
which the axices correspond with the specific elastic storage of the pumped pervious layer
and of the covering semi-pervious layer is very small. This inclination is so small that the
shown contour lines have a very small interval and are freakish. This small undulation of
-_... _ ...
this surface is probably due to numerical errors.
100 100+-____-+______;-____-;
~
- ~
/
10- I
_
10-
IO-~~
_...
./
100 101 102 I~ nlEOUID 100
... LAYER 2
100+-----~------~--~--r_----~ 100~~~~------;------;
IO-'\---.....~f-----+------+----- ..... 10-~

100 1~ nlEOIIID
LAYER
Fig. 6.17. 99% individual confidence intervals of the drawdown obtained by the search
of the parameter combinations resulting from the two-dimensional cross sections through
the approximate joint confidence region (confil)
6.8.7 Approximation of drawdown confidence intervals
In Fig. 6.1799% individual confidence intervals of the drawdown are represented. These
intervals are approximated by the help of hydraulic parameter combinations which
correspond with the bounds of the approximate joint confidence region and are deter-
mined by means of the two-dimensional cross sections (program contil). These bounds
are represented in Fig. 6.11 and are obtained by the program susqln. The largest
confidence interval belongs to the first observed drawdown after the start of the pumping
test in the observation well at 50.1 m from the pumped well in the pumped layer. The
drawdown at the end of the pumping test has smaller confidence intervals than the
drawdowns measured at the start of the test.
- ,..
•~t------+------4-----~
'0-
,.--- '""-
/
I
.0' 2
.I
.0 ' 102 .oJ: TlttE OON)
LAYER
'~t------+------4-------~----~
+-----4
10••• ,...............--"'*"-----:1
.0·2'1o-__...:t"-+______-+-______1-____....,j
.0· .0 ' .0'1 TIlE OIl",

LAYER
of the parameter combinations resulting from the two-dimensional cross sections through
the exact joint confidence region (confill)
In Fig. 6.18 the 99% individual confidence intervals are represented which are
approximated by means of other parameter combinations. These hydraulic parameter
combinations correspond with bounding points of the exact joint confidence region found
by means of two-dimensional cross-section (program confill). These bounds are

represented in Fig. 6.13 and are obtained by the program sumsqr. The differences
between the intervals of Fig. 6.17 and 6.18 are very small. They cannot be detected from
these graphs.
DltAIIIMMI no
I~+-----~-----+----~
-~
~
/
10' I
10' 2
.I
100 ,.' 102 103 nt£ooJO
LAYER
-,Ie
I~~----~------~----~----~ I~~~~-+------~----~
,.-'-\.----"'00......\--------+------+------.1
I'" TU.OIIIO
LAYER
of the parameter combinations resulting from the three-dimensional cross sections through
the approximate joint confidence region (confi3)
The 99 % individual confidence intervals in Fig. 6.19 are obtained by the parame-
ter combination resulting from the three-dimensional cross section through the approxima-
te joint confidence region (program confi3). These bounds are represented in Fig. 6.12
and are obtained by the program susqJ3. The confidence intervals of Fig 6.19 are slightly
larger than the intervals of Fig. 6.17 and 6.18.
The 99% individual confidence intervals in Fig. 6.20 are obtained by the optimiza-
tion of a constrained problem where the minimum and maximum value of the drawdown
is searched for all parameter combinations which correspond with the four-dimensional
ellipsoid in the four-dimensional parameter space (program confi4). This ellipsoid
approximates the bounds of the joint confidence region. Comparing Fig. 6.20 with the
Figs. 6.17, 6.18 and 6.19 one can remark that the so obtained confidence intervals are
considerably larger than the confidence intervals obtained in the first three calculations.
This is particularly true for the drawdowns observed in the semi-pervious layer (layer2).
DRAWDOVII 'I'D _110
... ... t---+_----t-----i
•0-'
'0-
••• 103 TlttEl"lN! ,"

LAYER 2_ _ 110
I.' 102 DISTMCEOtI
DRAVDIMI oe
... ... ~!""oo=_+_----t-----i
.O-~~
.o-T--~_+--~r_--+_--~
~.
101 TncmllD 100
LAYER I
Fig. 6.20. 99 % individual confidence intervals of the drawdown obtained by an

optimization of a constrained problem of which the constrain is the four-dimensional
ellipsoid which approximates the joint confidence region (confi4)
REFERENCES
Banchoff, T. F., 1996, Beyond the third dimension: Geometry, Computer Graphics, and
Higher dimensions: American Scientific Library, New York, Freeman, 211p.
Beisley, D. A., 1990, Conditioning diagnostics: collinearity and weak data in regression:
New York, John Wiley & Sons, 292p.
Carrera, J., 1984, Estimation of aqUifer parameters under transient and steady-state
conditions: Ph.D. dissertation. (unpubi.) Tucson, Dep. of Hydroi. and Water Resour.,
Univ. of Ariz., 258p.
Carrera, J., and Neuman, S.P., 1986a, Estimation of aquifer parameters under transient
and steady state conditions, 1. Maximum likelihood method incorporating prior informa-
tion: Water Resources Research, v. 22, no. 2, p. 199-210.
Carrera, J., and Neuman, S.P., 1986b, Estimation of aquifer parameters under transient
Carrera, J., and Neuman, S.P., 1986c, Estimation of aquifer parameters under transient
and steady state conditions, 3. Application to synthetic and field data: Water Resources
Research, v. 22, no. 2, p. 228-242.
Cooley, R.L., and Naff, R.L., 1990, Regression modeling of groundwater flow: Tech. of
Water Resources Inv., US Geo!. Surv., Book 3., Washington, DC, US Government
Printing Office, 232p.
Dettinger, M.D., and Wilson, J.L., 1981, First-order analysis of uncertainty in numerical
models of groundwater flow, 1, Mathematical development: Water Resources Research,
v. 17, no. 1, p. 149-161.
Draper, N.R., and Smith, H., 1981, Applied regression analysis, second edition: New
York, John Wiley & Sons, 709p.
Haimes, Y.Y., 1977, Hierarchical Analyses of Water Resources Systems. Modeling and
optimization of large-scale systems: New York, McGraw Hill Book Company, 478p.
Hantush, M.S., 1960, Modification of the theory of leaky aquifers: Journal Geophysical
Research, 65, p. 3713-3725.
Hantush, M.S., and Jacob, C.E., 1955, Non-steady radial flow in an infinite leaky
aquifer: Trans. Amer. Geophys. Union, 36, p. 95-100.
Hill, M.C., .1989, Analysis of accuracy of approximate, simultaneous, nonlinear confi-

dence intervals on hydraulic heads in analytical and numerical test cases: Water Resources
Research, v. 25, no. 2, p. 177-190.
Kruseman, G.P. and De Ridder, N.A., 1970, Analysis and evaluation of pumping test
data: Wageningen, Bull. 1LRl 11, 200p.
Lebbe, L., 1988, Execution of pumping tests and interpretation by means of an inverse
model (unpubl., in Dutch): Thesis Geagg. Hoger Onderw. Geologisch Instituut, Univ.
Gent, Gent, 563 p.
Lebbe, L., and De Breuck, W., 1995, Validation of an inverse numerical model for
interpretation of pumping tests and a study influencing accuracy of results: Journal of
Hydrology, v. 172, p. 61-85.
Mardia, K. V., and Zemroch, P.J., 1978, Tables of the F-distribution and related distribu-
tions with algorithms: London, Academic Press, 195p.
McElwee, C.D., and Yukler, M.A., 1978, Sensitivities of groundwater models with
respect to variations in transmissivity and storage: Water Resources Research, v. 3, no.
1, p. 241-244.
Miller, R. G., 1981, Simultaneous Stastistical Inference, second Edition: New York,
Springer-Verlag, 299p.
Neuman, S.P., and Witherspoon, P.A., 1969, Applicability of current theory of flow in
leaky aquifers: Water Resources Research, v. 5, no. 4, p. 1069-1093.
aquifers: Techn. of Water Resources Investigations of the U.S.G.S, Book 3, Chapt. B3,
106p.
Sun Ne-Zheng, 1994, Inverse problems in groundwater modelling: Kluwer Ac. Pub.,
Dordrecht, 337p.
Tarhouni, J., 1994, Modele inverse tridimensionnel d'optimisation de parametres

hydraulique des nappes aquiferes: applications a l'echelie regionale et locale (unpubl. Ph.
D. dissertation): Ghent, Ghent University. 373p.
Theis, C.V., 1935, The relation between the lowering of the pierometric surface and the
rate and duration of discharge of a well using groundwater storage: Am. Geophys. Union
Trans., 16, p. 519-524.
Vecchia, A.V., and Cooley, R.L., 1987, Simultaneous confidence and prediction intervals
for nonlinear regression models with application to a groundwater flow model: Water
Resources Research, v. 23, no. 7, 1237-1250.
Wonnacott, R.J., and Wonnacott, T.H., 1985, Introductory Statistics, Fourth edition.
New York, John Wiley and Sons, 649p.
Chapter 7 / Example of performance and
interpretation of pumping tests
In this chapter a number of pumping tests are interpreted with the help of the inverse
numerical model. In Sect. 6.5.1 it is already shown that it is possible to interpret pumping
tests in groundwater reservoirs in which the flow can be conceptualized according to one
of the classical analytical models (Theis, Jacob, Hantush-Jacob, Boulton, etc.). With most
of these models, the interpreted drawdown is measured in the directly pumped layer. The
hydraulic parameters of the directly pumped layer are derived with a rather high level of
accuracy, at least when the used conceptual model is close enough to the actual flow and
the schematization of the groundwater reservoir approximates enough the assumed one.
Mostly, drawdown observed in the layers adjacent to the directly pumped layers are not
involved in those interpretations. Consequently, the accuracies of the derived hydraulic
parameters of these indirectly pumped layers are far below the accuracies of the hydraulic
parameters of the directly pumped layer. Interpretations following a classical analytical
model will not further be treated in this chapter. In the given examples, observed
drawdowns in pumped layers are simultaneously interpreted with drawdowns observed in
adjacent layer. Not only drawdowns of single pumping tests can be interpreted simulta-
neously but also all drawdowns of a multiple pumping tests.
Here, the performances and the interpretations of two multiple pumping tests in a
multi-layered groundwater reservoir are shown. The first is a double pumping test made
in layered Quaternary sediments; the second is a triple pumping test made in layered
Tertiary sediments. The interpretation of a double pumping test made in a laterally
anisotropic aquifer formed by fractured rocks is also included in this chapter. The
deduction of a rather small vertical conductivity is demonstrated by the observed
evolution of the vertical drawdown gradient in the concerning semi-pervious layer. In the
last part it is shown that the inverse numerical model cannot only be considered as a tool
for the interpretation of pumping tests but also for the interpretation of artificial recharge
tests. In this last test the rises of the hydraulic head were observed at two different levels
and at two different distances from the infiltration pond.
7.1 Double pumping test in layered groundwater reservoir

formed by Quaternary sediments
The studied groundwater reservoir is formed by Quaternary sediments in the Scheidt

Valley some seventeen kilometers south of Ghent city. These Quaternary sediments were
deposited in a valley which was eroded in the Ypresian clay substratum. The width of the
valley ranges between 4 and 6 km. In the central part of the valley the same sequence of
sediments are found over a width of ca. 2 km. These sediments form two pervious layers
which are separated by a semi-pervious layer. The upper pervious layer is covered by a
304 Chapter 7 / Example of perfonnance and interpretation of pumping tests
LITHOLOGICAL CROSS SECTION

':'~eI
(m)
LAYERS
IN
5822 NUMERICAL
MODEL
semi-pervious layer lay.. 5
+5
layer4
layar3
layer 2
·5~~~-----------=~~-f~+-------~~----~-------
layer 1
10 20 30 40 50
'-~~~==7=.~'-----1
Legend cro •• section
OfiMsand
o medium sand
ttl silty fine sand
0I.,;gh1ly sihy
E3 !': :a~ amount of shells
G with a smaIJ amount of shells 5B:(4)/
E3 with a 1arge amount of clay /5828(3) "

[!] with a smal amount of clay • 0 5B25 (3'\5B 26(3)
.
/, 5830(3)· •
EJ clay GBl (1) 5B 23(2) \
~cI.Y 1 0 •
5B 20(1) 58 24(2) \
/.
Legend location \
•
5B21(1)
o pumping wei
• observation weil \
S8 rotary drilfang
GB cored drW;"g
22 number of welJ
(1) Iocatw,n of scre.n in tayer of numerical model
Fig. 7.1. Lithological cross section, layers in numerical model and location of pumping
and observation wells of double pumping test in the Scheidt Valley. Source: Lebbe, L.
and De Breuck, W. 1994. The aplication of an inverse numerical model for the interpreta-
tion of single or multiple pumping tests. in Peters, A et al. (eds.), Computational Methods
in Water Resources X, 769-776. p.769-776. p.773. Copyright \C) 1994, Kluwer Academic
Publishers, reprinted by perm is ion of Kluwer Academic Publishers.
Chapter 7 / Example of performance and inJerpretation of pumping tests 305
semi-pervious layer in which the water table is situated. The lower boundary of the
groundwater reservoir can be supposed to be impervious. Two pumping tests are
performed in which the pervious layers are pumped separately. During each pumping test
the drawdown is observed on four different levels and at different distances from the
pumped well. Here, it is demonstrated that with relatively short-lasting pumping tests, the
vertical conductivities of the semi-pervious layers can be derived with accuracies which
are slightly smaller than those obtained for the horizontal conductivities of the pumped
layers. Also, the specific elastic storage of the pervious and of the semi-pervious layer
can be determined as well as the storage coefficient near the watertable.
7.1.1 Lithostratigraphical cross section
The lithostratigraphical cross section (Fig. 7.1) is based on the description of core
samples and on the results of geophysical borehole logs. In our particular case the
resistivity measured with the short-normal device characterized the layering of the
groundwater reservoir quite well. The reservoir is bounded below by a thick layer which
consists mainly of clay at a level of -9.5 m TAW (all levels are given in meters versus
the Belgian Datum Level, TAW, 0 m TAW is 2.33 m below the mean sea level).
Between -9.5 and -4 occurs a pervious layer, which consists mainly of fine to medium
fine sands with very thin layers of gravel and silt. Above this pervious layer lies a semi-
pervious layer. This layer is composed of a sequence of fine layers (1 to 2 cm thick) silty
fine sands, silt with fine sand and occasionally peat bearing silt. Between the levels -2 and
+ 3 occurs a second pervious layer which consists of gravelly sand with a small amount
of fine silt layers which are seldom larger than one centimeter thick. The top of the
groundwater reservoir is formed by a semi-pervious layer. This layer occurs between the
levels +3 and +8. The lower part consists of silty fine sand. The top is composed of silt
and peat bearing alluvial clay.
7.1.2 Location of pumping and observation wells
The location of the two pumping wells and the observation wells are also shown in Fig.
7.1. The observation wells have a screen of one meter length and a diameter of 40 mm.
Three observation wells are installed in the lower pervious layer, two in the middle semi-
pervious layer, four in the upper pervious layer and finally two in the upper semi-
pervious layer. All screens of the pumping wells and the observation wells are surrounded
by calibrated sand. The annular space between the riser pipes and the semi-pervious
layers are sealed with bentonitic clay.
7.1.3 Performance of double pumping test
During the first part of the double pumping test, water was withdrawn from well GB1(1)
(Fig. 7.1). The discharge rate was measured frequently and was equal to 221 m3 /d. By
means of an electrical sounder the hydraulic heads were measured in all observation
306 Chapter 7 / Example of perf017tlance and interpretation of pumping tests
wells. The pumping test lasted 16 hours. During the second part of the double pumping
test, water was withdrawn from well SB30 of which the screen is situated in the upper
pervious layer. The discharge rate was equal to 191 m3 /d and the test lasted 23 h.
7.1.4 Discretization of groundwater reservoir in numerical model
In the numerical model the groundwater reservoir is schematized in five layers. Layer 1
corresponds with the lower pervious layer, layer 2 with the middle semi-pervious layer,
layer 3 with the upper pervious layer and layer 4 and 5 with the upper semi-pervious
layer. The middle semi-pervious layer is only represented by one layer and was not
further discretized in more than one layer because of its small thickness. The observation
wells are located in the middle of the layers 1, 2, 3 and 4. The pumped wells have their
screens over the entire thickness of layer 1 and 3.
7.1.5 Hydraulic parameters derived from observed drawdown
Studying the sensitivities and the variance-covariance matrices generated with the initial
estimates of the parameters, the following parameterization is made. Eleven hydraulic
parameters or groups of hydraulic parameters can be derived from all the observed
drawdowns of the double pumping test by means of the inverse numerical model: the
horizontal conductivities of the pervious layers Kh(l) and Kh(3) , their specific elastic
storages S,(1) and S,(3), the hydraulic resistances of lower and upper part of the middle
semi-pervious layer, c(1) and c(2), and its specific elastic storage and horizontal conducti-
o vity, S,(2) and Kh(2), and the storage coefficient near the watertable or the specific yield
So. The remaining parameters are classified in two groups of deducible parameters. The
first group of deducible parameters comprises the specific storages of the upper semi-
pervious layers or of the layers 4 and 5 of the numerical model, S,(4-5). The second
group of deducible parameters comprises the hydraulic resistances of the upper semi-
pervious layer, c(3) and c(4) , and the horizontal conductivities of the layers 4 and 5 of
the numerical model, Kh(4) and Kh(5), are included in this last group. It is assumed that
the upper semi-pervious layer behaves as an isotropic one.
7.1.6 Interpretation with ordinary least square method
The estimated values of the hydraulic parameters are given in Table 7.1 along with their
marginal and conditional standard deviations and their 98 % marginal and conditional
confidence factors. The accuracies of the hydraulic resistances of the semi-pervious layers
have similar accuracy ranges as the specific elastic storages of the pumped layers S,(1)
and S,(3). These accuracies are only slightly smaller than the accuracy ranges of the
horizontal conductivities Kh(1) and Kh (3) of the pumped layers. The specific elastic
storages of the semi-pervious layers S,(2) and S,(4-5) and the storage coefficient near the
watertable So can roughly be estimated from the measured drawdowns of the double
pumping test. The conditional and marginal confidence interval of the horizontal conduc-
Chapter 7 / Example of performance and interpretation of pumping tests 307
tivity of the middle semi-pervious layer Kh(2) are both larger than the square root of ten.
Consequently, this parameter can hardly be considered as deduced from the double
pumping test data. The accuracies with which the hydraulic parameters can be derived are
rather large because of the remaining residuals after the optimization. The accuracies can
further be ameliorated by the application of the biweighted least square method described
by Wonnacott and Wonnacott (1985) (see Sect. 6.3.1).
Table 7.1. Optimal values of hydraulic parameters deduced from the double pumping test
with the ordinary least square method along with their marginal and conditional standard
deviation, Sm and Sc and 98 % marginal and conditional confidence factors
Hydraulic parameters Units Value Sm Sc 98%Cfm 98%Cfc

Kh(3) mid 1,19.10' 0,0166 0,0095 1,1996 1,1097
Kh(1) mid 1,83.10' 0,0167 0,0097 1,2008 1,1122
S,(3) m-' 6,26_10-5 0,0188 0,0144 1,2288 1,1710
e(l) d 6,87.10' 0,0266 0,0160 1,3385 1,1917
e(2) d 4,51.10' 0,0286 0,0191 1,3682 1,2329
c(3),c(4) d 4,50.10' 0,0489 0,0210 1,7091 1,2588
Kh(4),Kh(5) mid 5,07.10-2
S,(I) m-' 4,15.10-' 0,0357 0,0248 1,4789 1,3124
S,(4-5) m-' 9,04.10' 0,0650 0,0323 2,0390 1,4248
S,(2) m-' 9,63.10-6 0,0729 0,0568 2,2234 1,8637
So m 3 /m 3 1,44.10-2 0,0864 0,0715 2,5780 2,1895
Kh(2) mid 4,71.10-' 0,1494 0,1167 5,1652 3,5935
The observed and calculated drawdowns are shown in Fig. 7.2. They correspond
with the optimal parameter values. The calculated and observed drawdowns in the layers
1, 2 and 3 are very similar. The highest discrepancy occurs in the layer 4 during the
pumping test on the upper pervious layer. From this figure one can easily deduce the
level-, distance- and time-dependency of the residuals. The residuals of the drawdowns
observed in the pumped layer show in both tests a distance-dependency. In the wells
located close to the pumped wells, all observed drawdowns are larger than the calculated
drawdowns. In the observation wells at large distances to the pumped wells, it is the
inverse. In the undirectly pumped layers the residuals are less distance-dependent. This
slight distance-dependency of the residuals points to the fact that there is a small differ-
ence between the modelled and the real flow. In the numerical model it is assumed that
the layers are lateral homogeneous and of a constant thickness. It is, however, well
known iliat ilie Quaternary sediments at the test site shows small lateral variations in
thickness and in constitution within the area influenced by ilie pumping test. The slight
distance-dependance of the residual can raise the question if the results of the inverse
model must be denied as a whole (see Sect. 6.3.3). Another possibility is that these
results are accepted with some reservation and that the slight distance-dependancy of the
residuals is interpreted as an indication of lateral heterogeneity as Herweijer (1997) did.
308 Chapter 7 / Example of perfonnance and interpretation of pumping tests
-a l!iJ
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x~ ~ 1
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LAYER 1 LAYER 1
Fig. 7.2. Measured (x-signs) and calculated (solid curves) in time- and distance-draw-
down graphs for the double pumping test in the Scheidt Valley (left two columns for
pumping test in lower pervious layer, Q(l)= 221 m3 /d, right two columns for pumping
test in upper pervious layer, Q(3)= 191m3 /d). Source: Lebbe, L. and De Breuck, W.
1994. The application of an inverse numerical model for the interpretation of single or
multiple pumping tests. in Peters, A et al. (eds.), Computational Methods in Water
Resources X, 769-776. p.769-776. p.775. Copyright c 1994, Kluwer Academic Pub-
lishers, reprinted by permission of Kluwer Academic Publishers
In Fig. 7.3 the residuals of this double pumping test are represented versus the
cumulative frequency or the probability along with the best fitted normal distribution.
StUdying this figure one can conclude that the distribution of the residuals approximates a
normal distribution with a mean equal to zero.
Chapter 7 / Example of peiformance and interpretation of pumping tests 309
fjIr + + ~
110
~
-' ~ 110
80 ~ 80
70 J7 TO
/1
""'"
~ 80 60
e
§
iii
50
~
50
....
~
.
<C ::>
ID
.. 02
7
~O
...
0
II:
"-
'" '"
30 i'... 30
20 /J 20
10 / d 10
i+_
V ~ ~ I--
-.4 -.3 -.2 -. I 0 .1 ·2 .3 .4 .5
RESIDUAL
~AN OF NORI1AL OISTRIBIiTiON IS .001
STANDARD DEVIATION OF NOR11AL DISTRIBIiTlON IS .17~
Fig. 7.3. Residuals of double pumping test in the Scheidt Valley versus cumulative
frequency or probability (crosses) along with the best fitted normal distribution versus
probability and frequency (continuous lines)
Table 7.2. Condition indexes ." and n matrix of marginal variance decomposition
proportions corresponding with the ordinary least square solution of the double pumping
test in the Quaternary sediments
~ 1.0 1.4 1.9 2.7 3.1 3.6 4.8 7.8 20.1 12.5 10.8
Parameter
K.(3) 0.0667 0.2361 0.0030 0.0637 0.0003 0.0934 0.0166 0.0570 0.2018 0.2608 0.0005
Kb(l) 0.0033 0.0290 0.3655 0.0008 0.0803 0.3417 0.0312 0.0032 0.0482 0.0851 0.0118
S,(3) 0.0914 0.0799 0.0727 0.0003 0.3819 0.1019 0.0001 0.0404 0.1144 0.1137 0.0032
c(l) 0.0074 0.0175 0.0004 0.0218 0.4192 0.0418 0.2491 0.0083 0.0015 0.2048 0.0283
c(2) 0.0020 0.0025 0.0403 0.1190 0.0089 0.3700 0.1561 0.0035 0.0052 0.2198 0.0727
c(3),c(4) 0.0010 0.0000 0.0015 0.0814 0.0022 0.0466 0.0004 0.1542 0.0441 0.4740 0.1946
Kb (4),K b (5)
S,(I) 0.0013 0.0012 0.0020 0.0023 0.0588 0.0588 0.6487 0.0786 0.0060 0.1410 0.0042
S,(4-5) 0.0000 0.0000 0.0000 0.0000 0.0001 0.0004 0.0009 0.0001 0.0535 0.2908 0.6542
S,(2) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0026 0.9889 0.0069 0.0016
So 0.0000 0.0000 0.0003 0.0000 0.0005 0.0012 0.0001 0.3875 0.0109 0.4951 0.1043
K.(2) 0.0000 0.0000 0.0013 0.0209 0.0009 0.0087 0.0306 0.2034 0.0664 0.5108 0.1570
The condition indexes and the matrix of marginal variance-decomposition

proportions of this first interpretation phase are represented in Table 7.2. Four condition
indexes suggest weak to moderate dependencies. The largest condition index (20.1) is
310 Chapter 7 / Example of performance and interpretation of pumping tests
responsible for 98.89% of the marginal variance of S.(2) and 20.18% of the marginal
variance of Kb(3). The condition index 12.5 is respectively responsible for 49.51 %,
47.40%, 29.08%, 26.08%, 21.98% and 20.48% of the marginal variances of So, (c(3)-
c(4)-Kb(4)-Kb(5», S,(4-5), Kb(3), c(2) and c(1). The condition index 10.8 is responsible
for 65.42 % of the marginal variance of S.(4-5) and 19.46 % of the marginal variance of
(c(3)-c(4)-Kb(4)-Kb(5». The condition index 7.8 is responsible for 38.75% of the marginal
variance of So and 15.42% of the marginal variance of (c(3)-c(4)-K;,(4)-Kb(5».
7.2 Double pumping test in a laterally anisotropic aquifer

formed by fractured rocks of Palaeozoic and Mesozoic age
In this case study a double pumping test is performed. In the two tests water is pumped
from two different wells in the same series of fractured rocks. During the first short
lasting test water is withdrawn from one well and the drawdown is measured in one
observation well. During the second pumping test water is withdrawn from another
pumping well while the drawdown is measured in three observation wells which are
situated in the same fractured rocks. So, the drawdowns due to a pumping in the fractured
rocks are observed in four different directions of which two directions are, however, very
close to each other. The results of the inverse model are the optimal values of five
hydraulic parameters and their joint confidence region. Four hydraulic parameters
describe the flow and the storage change of the directly pumped fractured rock: the
effective horizontal conductivity, the angle defining the principal direction of maximum
horizontal conductivity, the lateral anisotropy and the specific elastic storage of the
fractured rock. The fifth deduced hydraulic parameter determines the vertical inflow from
the adjacent layers. By plotting the contours of the sum of the squared residuals for
different combinations of drawdowns observed in different wells, it can be examined
which combinations of observation wells are sufficient to derive a unique solution.
The lithostratigraphical cross section of the pumping test site and the location of the wells
are represented in Fig. 7.4. The pumped layer consists mainly of phyllites of Silurian
age, the upper part of which is fractured and therefore has high secondary hydraulic
conductivity. These phyllites are overlain by fractured chalk of Turonian age, which in
turn is overlain by Palaeocene age deposits which belong to the Landen Formation. The
lower part of this formation is clay and sandy clay and can be considered as a semi-
pervious layer. The upper part consists of clayey fine sands, considered as a pervious
layer with rather small hydraulic conductivity. Above the Landen Formation occurs the
Kortrijk Formation of the leper Group (Marechal and Laga, 1988). This unit consists
mainly of clay which is considered as a semi-pervious layer with very small hydraulic
conductivity. The thickness of the clay varies between 103 and 125 m. Above the clay,
Quaternary deposits occur which differ strongly in thickness and composition and form a
pervious layer limited on the top by the watertable.
Chapter 7 / Example of perjonnance and interpretation of pumping tests 311
sw KDlffHfI.n<
NE
mTAW
,
.100
,
0 0 0 Pc
Be
0
Ys
Yc
· 100
10 20 30 .-0 50 60km
= aqu~"r Om TAW-ia<f Iow...a level
a Ouat"mary sediments Ye Ypr.."" clay
Be BanDnian day Ls Landen"" sand
L" Lllldian sand Le Land"nian day
Ps Panisalian sand Cc C(etac8OUs : Cam~ian chalk
Pc Paniselian clay (non-lissured)
Y,. Y""IItS'an sand C1 eretaCeOus: Turonian cha. (fissured)
The Netherlands
Gennany
France
l..J...U...LJ out~ne of Brabant Massif

boundary FlanderslWallonia
A A' profile
o, saun
,
Fig. 7.4. Geological cross section through the western part of Flanders and the location
of the study area
312 Chapter 7 / Example of petj'ormance and intetpretation of pumping tests
The screens of the four available production wells are located in the upper part of the
Brabant Massif and in the lower part of the overlying chalk. All wells were equipped with
a submersible pump and a flow meter. During the first part of the double pumping test,
water was withdrawn from well PI (Fig. 7.5) at a discharge rate of 604 m3 /d. The
drawdown was measured only in well P2 during the first 200 min. During the second part
of the double pumping test, water was pumped from well PP at a discharge rate of 211
m3 /d for 2820 min. The drawdown was measured in the wells PI, P2, and P3. After
pumping ceased, the residual drawdown was measured for 600 min.
In the numerical model, the groundwater reservoir is discretized in nine layers (Fig. 7.5).
The fractured rocks of the Brabant Massif and the overlying fractured chalk, which are
pumped, correspond with layer 1 of the numerical model. Layers 2, 3, and 4 correspond
with the covering semi-pervious layer. Layer 5 corresponds with the clayey fine sands in
the upper part of the Landen Formation, which is a pervious layer. Layers 6, 7, and 8
correspond with the clay of the Kortrijk Formation which is considered as a semi-
pervious layer with very small hydraulic conductivity. The uppermost layer of the
numerical model, layer 9, is the pervious layer formed by the Quaternary deposits.
7.2.4 Drawdowns used as input data
The drawdowns observed during the second part of the double pumping test were directly
introduced in the inverse model. The observed drawdowns of the first part of the pumping
test were adjusted and multiplied by the ratio of the discharge rates of both pumping tests.
As a consequence, drawdowns observed in wells at four different directions to the
pumped well, are available (Fig. 7.5). The adjusted observations of the first part, where
well PI is pumped and P2 is used as an observation well, were then considered to be
made at an imaginary well P4 during the second part. The distance between well P4 and
the pumped well PP is the same as the distance between PI and P2. The line P4-PP is
parallel with the line PI-P2. These adjustements of the drawdowns was allowed in the
case that the pumped aquifer behaves as a homogeneous laterally anisotropic medium.
One of the interpretation goals is the examination of the reliability of this assumption.
7.2.5 Hydraulic parameters derived from observed drawdowns
The following five hydraulic parameters or groups of parameters are derived from the
observed drawdowns and residual drawdowns: the effective horizontal hydraulic conduc-
tivity of the pumped layer (layer 1), Kt,.(I), the lateral anisotropy of the pumped layer,
vim, the angle, fJ, measured eastward between the north and the principal direction with
maximum hydraulic conductivity, the specific elastic storage S,(1) of the pumped layer
Chapter 7 / Example of peifonnance and interpretation of pumping tests 313
LITHOSTRATIGRAPHICAL CROSS SECTION LAYERS IN

level NUMERICAL MODEL
(mTAW) P 1 PP P3
o
LAYERS
-50
LAYER 7
-100
LAYERS
LAYERS
LAYER 4
-150
I -200
LAYER 1
P3 legend of cross section

N
o sand
~ clay
§ shale
llJJ chalk
x .-
./
o• LDcalion
Location of pumped wen pp
01 observation weff•
• Location 01 imaginary wen P4
luxembUrG
100 200m
Fig. 7.5. Lithostratigraphical and hydrogeological cross section and location of pumping
and observation wells at the pumping test site of Kortemark and the schematization of the
groundwater reservoir in the numerical model
314 Chapter 7 / Example of peiformance and interpretation of pumping tests
layer and the hydraulic resistances c(I), c(2), c(3) and c(4). The latter are considered in
one group because they were assumed to be proportional to the thickness of their repre-
sentative part of the covering semi-pervious layer. So, the vertical hydraulic conductivity
is supposed to be level independent. At the end of the interpretation a sixth hydraulic
parameter was introduced: the specific elastic storages S,(2), S,(3), S,(4) and S,(5) of the
covering semi-pervious layer. They are considered in one group. Consequently, it is
assumed that the elasticity of this semi-pervious layer is also level-independent.
The sensitivities of the used drawdowns to the other hydraulic parameters were so
small that these parameters can be considered as unidentifiable. The assumed values for
these unidentifiable parameters are 0.125 mid for Khe(5), 2xlO"" mid for Kbe(6-8), 1 mid
for Khe(9), lxlO"" mid for K.(6-8), 1.6xlO-5 m- I for S.(2-8), lxlO"" m- I for S,(9), and 0.1
for the specific yield. The estimate of Khe (5) is based on values of two pumping test in the
upper part of the Landen Formation. These pumping test where made at Poperinge ca. 29
km south-west of Kortemark (Lebbe et aI., 1989) and at Oostduinkerke ca. 28 km north-
west of Kortemark (Lebbe et aI., 1989). The estimates of the other parameters were
preliminary results of a calibrated regional groundwater flow model (Lebbe et aI., 1988).
7.2.6 First interpretation results
The optimal values for the five hydraulic parameters are 0.441 mid for the
effective horizontal hydraulic conductivity, Khe(l); 2.046 for the "lateral anisotropy, -vim;
1.106 radians for the angle, 0, measured eastward between the north and the principal
direction with the maximum hydraulic conductivity; 6.94xlO-7 m- I for the specific elastic
storage S,(l); and 4.34xlO-3 mid for the vertical conductivity of the covering semi-
pervious layer (layers 3, 4 and 5 of the numerical model).
DRAWODWIUD DltAtI;JC)W ot)
'0' ,.'
,00 .
,
10"" 10-1
,.- ,.-
",.
'0· 101 Tll'I!mnn ,00 '0' 103 !#PAllO'
OIST»ICI!:O'D
LAYER 1
Fig. 7.6. Calculated (continuous lines) and observed drawdowns (crosses) in time-draw-
down and apparent distance-drawdown graphs and the optimal values of the hydraulic
parameters of the first interpretation phase
Chapter 7 / Example of perfonnance and interpretation of pumping tests 315
The calculated and observed drawdowns are represented in time-drawdown and in
apparent distance-drawdown curves (Fig. 7.6). There is a fair agreement between the
calculated and the observed drawdowns. The residuals as defined in Eq. 6.1 are represen-
ted versus the cumulative frequency or the probability along with the best fitted normal
distribution versus the probability and frequency (Fig. 7.7). Figure 7.7 shows that the
residuals approximate quite closely a normal distribution with a mean equal to zero.
...en
It)
en
<0
en ...
<Xl
III
<Xl
....
It)
en
<0
en
0
...
en
It)
0 0
0
0
0 0 o·
0 0 0 + + +
go I V""'" 90
80 /t 80
70
!.~/ 70
~ 60 ~ 60
E
~
:::; 50 ~!f"
iii
~ /1
""
~o
... ff
0
II:
30 30
/
~
vy
"'"'" "'-.
20 20
10 10
~ r;"-
RESIDUAL
Fig. 7.7. Residuals versus cumulative frequency or probability (crosses) along with the
best fitted normal distribution versus probability and frequency (continuous lines)
Because of this normal distribution of the residuals and due to the fact that none of
the time-drawdown curves of any well is in disagreement with the optimal solution as
proved in Sect. 7.2.9, one can conclude that the model of flow towards a pumped well in
a homogeneous and laterally anisotropic aquifer covered by a semi-pervious layer is a
reasonable approximation of the real flow around the pumped well in the fractured rocks
at Kortemark. Based on numerical simulation of flow in fractured rocks, Long et al.
(1982) concluded that fractured rocks do not always behave as a homogeneous porous
medium that is laterally anistropic and has a symmetric permeability tensor. They found,
however, that fractured rocks behaves more like an equivalent porous medium that is
laterally anisotropic medium and has symmetric permeability tensor in the case that the
tested sample size is large and that the rock fractures has the following four characteris-
tics: the fracture density is rather large; the fracture aperture is constant rather than
distributed; and the fracture orientation is distributed rather than constant. Smith and
316 Chapter 7 / Example of peifonnance and interpretation of pumping tests
Schwartz (1984) states that increased fracture interconnectedness inhances the validity of
the equivalent continium. Based on the results of the interpretation and on the results of
Long et aI., (1982) and Smith and Schwartz (1984), one can made two conclusions. The
first is that the volume of fractured rock that is influenced by the pumping test of
Kortemark, is large enough so that this sample can be considered as an equivalent porous
medium. A second conclusion is that the tested fractured rocks shows probably pre-
sumably the three characteristics mentioned by Long et aI., (1982). These characteristics
of the fracture geometry can only be confirmed by an extensive study.
7.2.7 Two-dimensional cross sections through exact joint confidence region
Ten two-dimensional cross sections through the exact joint confidence region are shown
in Fig. 7.8. Each cross section is parallel to two parameter axes and goes through the
optimal values. They correspond with another combination of three hydraulic parameters
which are put equal to their optimal values. The other two parameters change around
their optimal values. For each cross-section, 2F parameter combinations are considered.
The way these parameter combinations are chosen, is described in Sects. 6.6.4, 6.6.5,
and 6.6.7. For each parameter combination the pumping test is simulated and the
residuals are determined along with the sum of the squared residuals. With the thus
obtained matrix of the sum of the squared residuals, the cross section through the exact
joint confidence regions are drawn.
Once the matrix of the sums of the squared residuals is obtained, the results are
plotted in graphs (Fig. 7.8). The ratios of the hydraulic parameter values to their optimal
values are put on the logarithmic axes of the graphs. The optimal values of the above
mentioned parameters are always located in the middle of the respective axes. The
optimal values of the remaining three parameters are shown in the left lower part of the
figure. In the graphs four contour lines of the sums of the squared residuals are shown.
They correspond with the bounds of the 90%-, the 99%-, the 99.9%- and the 99.99%
confidence region. The corresponding values of the sum of the squared residuals are
calculated with Eq. 6.22. As Draper and Smith (1981) stated, these contours are exact
confidence contours in the linear as well as in the nonlinear case. However, in the
nonlinear case the level of probability is approximate.
The shapes of the lines of the graphs in Fig. 7.8 deviate slightly from the pure
elliptical shape and are not fully symmetric with respect to the optimal values.
Consequently, the problem is not purely linear in the considered parameter space.
Because these deviations are rather small, the problem only deviates slightly from the
purely linear case. Cooley and Naff (1990) describe a method to rank the degree of non-
linearity. This method is based on a modification of Beale's measure of non-linearity. On
the bounds of the 99.99% joint confidence region given by the two-dimensional cross
section, the modified Beale's measure was everywhere smaller than O.OIlF(p,n-p,l-a).
Therefore, the model is classified as effectively linear within these bounds.
Each cross section

9
corresponds with
another combination ........
of three hydraulic
....•
parameters that are
put equal to their
optimal values
~ ~
s. (1) K.(l)
a S.(1)~ 1.'.'0' 7 m-'

,ch.f'): 4 .... ,0"m/ct
e 9" 1.1.,0'tadlan
"".(1)= "."IIII1Q"mld .....
K)2-41= 1.211111O·mld K~12-4)= 1.21110'mld
00•
."
,,'iii =2.0-10'
...
b 9" 1.,.,D'radlan
K".(1)= ....... 'O·'mld 5," '.Sw10· 1 m· ' ~
ie.ll)
" .. {2-4,= '.21110·mld K•.c2-4)" 1,Zll10'mld
C ../m= 2.0-10' 9 9= 1.11C10 1 ,..dlan 9 .. '.,x1O'nd/an

5,(1)= I.Swi0· ' m-' 5,(1)= '.,.,0·'m- 1
j;ic Z.O-lO'
K ..12 -4)'" 1.2x10'mld .c".(1)= 4 .... ,0·'mld K".(1)= ~.4.'D-'mld
d ./;;;= 2.0.,0' h 9: 1.,.,0'radlan 9: 1.11111108radlan

S,(1)", •. ,.,a· m-
7 1 lift=- 2.0X'O· Iii", 2.0-10'
K".(11= "."11110"mld " ...(2-4,= '.ll1i0'mJd S.(1': 1.'1III1G· t m-'
·".ar·... n:· ar 1..l·I~.J ''1.,1'
Fig. 7.S. Ten two-dimensional cross sections through the 90%-, 99%-, 99.9%-, 99.99%
exact joint confidence regions in the five-dimensional parameter space. These contours
show some bounds of these regions which correspond with the first interpretation phase.
Axes are ratios of parameter values to optimal parameter values (dimensionless).
These two-dimensional cross sections through the joint confidence region can also
help in the regression diagnostic. The graphs in the upper row of Fig. 7.8, show that the
angle (), which determines the principal direction of maximum transmissivity, is found
with the highest precision. This parameter showed the smallest correlation with the other
hydraulic parameters, which is confirmed by the correlation matrix. It is the only
hydraulic parameter where the smallest possible interval (between 0.949 and 1.054 times
the optimal value) can be considered so that the bounds of the 99.99% joint confidence
region do not fall outside the chosen windows of the cross sections. The anisotropy v'm,
318 Chapter 7 I Example of peifonnance and interpretation of pumping tests
the specific elastic storage S.(I), and the effective horizontal hydraulic conductivity Kh.(1)
are determined with smaller precision. The first graph of the third row in Fig. 7.8 shows
a strong correlation between the specific elastic storage S,(I) and the effective horizontal
hydraulic conductivity Khe(1). These parameters have the highest correlation (0.8479).
Because of this high correlation, a large difference exists between their marginal and
conditional standard deviation as can be derived from Table 7.3. The two-dimensional
cross section through the joint confidence region, which is parallel to these parameters
and goes through the optimal point, gives a typical long thin ellipse. The hydraulic
parameter that is determined with the smallest precision is the vertical hydraulic conduc-
tivity K.(2-4) of the covering layer. It is the only hydraulic parameter for which it is
necessary to consider the largest interval, between 0.487 and 2.054 times the optimal
value. The intervals of the other hydraulic parameters are all the same, except for the
parameter determined with the highest precision (Fig. 7.8).
Table 7.3. Results of the first interpretation phase of the double pumping test in the
laterally anisotropic aquifer formed by the fractured rocks
Hydraulic Unit Optimal Sm Sc

parameter value
Khe (1) mid 0.441 0.0184 0.0082
.,fm - 2.046 0.0131 0.0063
() radian 1.106 0.0053 0.0042
S,(I) m- I 6. 94xlO-7 0.0207 0.0068
K.(2-4) mid 4. 34xlO-3 0.0328 0.0198
7.2.8 Three-dimensional cross sections through approximate joint confidence region
Ten three-dimensional cross sections are shown through the 99.9% approximate JOInt
confidence region in Fig. 7.9. These three-dimensional cross sections can be called
conditional confidence spaces. Their bounds are parts of the bounds of the joint confi-
dence region. Each cross section is parallel to three parameter axes and goes through the
optimal values. Each time they correspond with another combination of two parameters
which are put equal to their optimal values. The bounds of these conditional confidence
spaces are represented by fifteen closed polylines which are plotted in graphs with three
orthogonal axes. The way these graphs are obtained is described in Sect. 6.6.6. The
residuals for each considered parameter combination are here calculated by means of the
linearization method (Eq. 6.23). Therefore, the represented cross sections are three-
dimensional cross section through the approximate joint confidence region. These cross
sections are centered exactly around the optimal values and have a pure ellipsoidal shape.
These three-dimensional cross sections further enlarge the regression diagnostic
possibilities. The conditional confidence spaces of the parameter combinations which
deteriorate the sensitivity matrix are ellipsoids with large principal axes of which the
directions deviate strongly from the parameters axes. The ellipsoid in the upper left
corner of Fig. 7.9 is an example of such an ellipsoid. The parameter axes of this
conditional parameter space are the specific elastic storage Ss(l), the effective horizontal
hydraulic conductivity Khe(l) and the vertical hydraulic conductivity Kv(2-4). The
ellipsoids with principal axes more aligned with the parameter axes results in well-
conditioned parts of the sensitivity matrix, e.g., the ellipsoid in the first column of the
last row in Fig. 7.9. However, interpreting these graphs, one should always consider the
different chosen intervals for the different parameters. The two- and three-dimensional
cross section can be seen as an additional tool in the collinear diagnostic. They supple-
ment, especially visually, the information given by the eigenvalues and eigenvectors of
the variance-covariance matrix, the correlation coefficients, the condition indexes and the
matrix of marginal variance-decomposition proportions.
7.2.9 Condition indexes and matrix of marginal variance-decomposition
The condition indexes and the matrix of marginal variance-decomposition proportions of

this first interpretation phase are represented in Table 7.4. The two condition indexes 6.0
and 9.8 suggest weak dependencies. These two condition indexes are responsible for
96.66 % of the marginal variances of the hydraulic parameter Khe(l) and for 96.65 % of
the marginal variance of Ss(l). The large marginal variances or large marginal standard
deviations of those two parameters are due to a correlation between these parameters, as
can be deduced from the cross sections through the joint confidence region. The highest
condition index influences most the marginal variance of the least sensitive parameter
K.(2-4) which has the highest marginal and conditional standard deviation. Also remark-
able is that the marginal variance of the angle (J is for a large proportion due to the
smallest condition index. As derivable from Table 7.3, this parameter has the smallest
conditional and marginal standard deviation and the smallest ratio between those two
standard deviations.
Table 7.4. Condition indexes ." and II matrix of marginal variance-decomposition

proportions corresponding with the first interpretation phase of the double pumping test in
the laterally anisotropic aquifer
." 1.0 1.4 2.7 6.0 9.8

Parameter
.Jm 0.0001 0.0534 0.3001 0.3908 0.2547

e 0.4720 0.0451 0.0846 0.3078 0.0905
Khe(l) 0.0011 0.0203 0.1120 0.3667 0.4999
S,(l) 0.0019 0.0196 0.0120 0.5459 0.4206
Kv(2-4) 0.0001 0.0005 0.0095 0.0801 0.9099
Vl
t5
a K.,(l) = 4.4xIO· 1 mid e .[ii,=2.0xIo" h ~ = 1.lxlO· radian
i,(2-4) = 4.3xIO·J mid i.,(l) = 4.4xIO- 1 mid K.,(l) = 4 .4xlO- 1 mid
i ~= 1.lxlO· radian
b C,(l) = 6. 9xIO·' m· 1 f .[ii,= 2. OxlO' S,(l) = 6.9xlO·' m-I
i,(2-4) = 4.3xIO·J mid S,(l) = 6.9xIO·' m-I g
~
j ~ = 1.lxlO· radian ~
c C,(l) = 6 .9xIO·' m· 1 g ~ = 1. lxl 0"radian .[ii,= 2. OxlO'

..
'I
K.,(2-4) = 4.3xIO·J mid "-
K".(l) = 4.4xIO· ' mid
~
Each 3D cross section corresponds with another ~l\"
d .[ii,= 2.0xlO'
K,(2-4) = 4.3xlO·J mid combination of two hydraulic parameters
.sa.
that are put equal to their optimal values ~
~
9
I
~
~
~
g
~.
.sa.
""~
• 12 ~ .
....'2
.., .. ....'2 , ~
... • 8.(1) ... 8.(l)
......
... . T.W _M
'l~ • ~(l)
~
"1/0. / / ...
Fig. 7.9. Ten 3D cross sections through the 99.9% approximate joint confidence region in the five-dimensional parameter space.
These contours show bounds of this region which correspond with the first interpretation phase (upper half).
Q
~
~
..
'I
.....
~
.11
.11
~~
~
...... .",
.'0 1(__• • "V • .~ '"
~
i
1I
'"
"'"
l'"
~
i
ii
g.
1I
~
.",
.11 ~~.
~
......
.12 ~
.'01( .'Y . .~ ...~lm~ /
Fig. 7.9. Ten 3D cross sections through the 99.9% approximate joint confidence region in the five-dimensional parameter space. W
N
......
These contours show bounds of this region which correspond with the first interpretation phase (lower half).
7.2.10 Observational constraints for unique solution
Theoretically, the effective hydraulic conductivity, the anisotropy and the angle of the
principal direction can be deduced if the drawdown is observed in wells that are located
in three different directions with respect to the pumped well (Hantush, 1966). According
to this theory, it should be possible to obtain a unique solution from the observed draw-
downs of the second pumping test. However, in this example, the direction of the wells
PI and P3 with respect to the pumped well PP are very close to each other (see Fig. 7.5).
A first attempt was made to estimate the hydraulic parameters only from the data of the
second pumping test or the data of the observation wells PI, P2 and P3. During the
iterations of the inverse model with the data of the second pumping test alone, the
parameter values varied widely. An oscillation in the inverse iterative process occurred
and no satisfactory convergence was obtained. With this observation, the question arises
whether the two directions represented by the pumped well PP - observation well PI and
pumped well PP - observation well P3 are too closely aligned.
To obtain an answer to this question, an extensive number of calculations was
made. The dependence of the contours of the sums of the squared residuals to the input
data was studied. In Fig. 7.10, the results of these calculations are shown in 15 different
graphs. In each graph the contours of the sums of the squared residuals are plotted in a
same cross section through the parameter space. This cross section is parallel to the two
parameter axes, the anisotropy, --1m, and the angle of the principal direction of maximum
hydraulic conductivity, O. The effective hydraulic conductivity and the specific elastic
storage of the pumped layer are set equal to the optimal values of the first interpretation,
as well as the hydraulic resistance of the covering semi-pervious layer. The other
parameters are equal to the previously estimated values. For each graph, 61 2 simulations
with the numerical model are made. Each time, the sum of the squared residuals is
calculated. In each cross section, the value of the anisotropy changes from 1 to 7 with a
step of 0.1 and the value of the angle 0 varies from a to 7r radians with a step of 7r/60
radians. In all graphs the contours of the sum of the squared residuals, S(hp), are drawn.
The largest contoured value is eight. The smaller contoured values are half the former
value, or 4, 2, 1, 0.5, 0.25, 0.125.
The four graphs in the upper row of Fig. 7.10 represent the S(hp) contours of the
observed drawdowns in each well separately. The sum of the squared residuals shows
each time a C -shaped area with a large number of local minima. If the observations of
only one well are available, then an infinite number of solutions are possible, which
correspond with the central line through the area with the local minima. The S(hp)
contours of the four graphs in the upper row of Fig. 7.10 are typical for an ill-condi-
tioned problem. Ill-conditioning could indicate a model that is overparameterized, that is,
having more parameters than needed, or inadequate data that will not allow the estimation
of the parameters postulated (Draper and Smith, 1981). Here, the ill-conditioning of the
problem is due to inadequate data. The data of only one observation well are insufficient
to estimate the parameters characterizing the anisotropy of the pumped layer. The areas of
the minima of the S(hp) are situated in about the same parameter space for the wells PI
and P3. The minima for the observation wells P2 and P4 are at a completely different
location in the parameter space than those for the wells PI and P3.
.
0
P',P2
~C
2
~ ,..frn
r
3
:·c
P1.P2.P3
..
..
~
~
~ ..frn ..rm
e ~tao) P1. P2. PJ. P. Pl, P3 I'2. P4 1
( x zene with ona minimum
..
>8
or saverallccal minima
2
~. .
o I
..rm
I
..rm
234 567 1 2 3 4 5 6 71 2 3 4 5 6 7
Fig. 7.10. Contour lines of the sums of the squared residuals for different combination of
observed drawdowns of the wells PI, P2, P3 and P4. In each graph the same conditional
cross section is considered in the parameter space. The cross section is parallel with the
anisotropy vm and the angle () and goes through the optimal values of the first interpreta-
tion phase.
Six different combinations of two observation wells are possible. They are
indicated as (PI, P2), (P2, P3), (P3, P4), (P4, PI), (PI, P3) and (P2, P4). The S(hp)
contours are represented in the four graphs of the second row and in the two graphs in
the right part of the last row of Fig. 7.10. In three of these graphs, (PI, P2), (P2, P3),
324 Chapter 7 I Example of peifonnance and interpretation of pumping tests
and (P2, P4), two areas with a global minimum can be distinguished. In these cases, two
possible solutions exist. The two graphs (P3, P4) and (P4, PI) show an elongated banana-
shaped area where S(hp) is flat around the minimum. This is an example of an unstable
problem. In the graph (PI, P3), there is a .-shaped area with several local minima. This
area is in the same location of the parameter space as in the graphs PI and P3 and is only
slightly smaller. Therefore, the information given by the drawdowns of these two
observation wells is only complementary for only a very small part.
The available data of the four wells provide the possibility of four different
combinations of data of three observation wells, (PI, P2, P3), (P2, P3, P4), (P3, P4, PI)
and (P4, PI, P2). Their S(hp) contours are represented in four graphs of the third row of
Fig. 7.10. Two of these graphs (P2, P3, P4) and (P4, PI, P2) show a single minimum. A
well-defined unique solution exists. This agrees with the theoretical consideration that the
effective hydraulic conductivity, the anisotropy, and the angle of the principal direction
can be deduced from drawdowns observed in wells that are located in three different
directions. The graph (PI, P2, P3) shows, however, two global minima. The shape of the
contours in this graph does not differ much from the shape of the contours in graph (PI,
P2). Although the data of three wells are simultaneously interpreted, two different
solutions are still possible. Therefore, well P3 does not give enough information to
complement the data of well PI and that the directions, pumping well PP - observation
well P3 and pumping well PP - observation well PI, are too closely aligned. This is con-
firmed by the comparison of the graphs (P3, P4) and (P3, P4, Pi) are similar.
The first graph (PI, P2, P3, P4) of the last row represents the S(hp) contours
when the data of all four observation wells are put together. This graph shows one global
minimum. A well-defined unique solution is obtained. These optimal values of the aniso-
tropy and the angle of the principal direction deduced from the data of the four observa-
tion wells coincides with all the minimum S(hp) areas of all the graphs. This is in
particular the case for the four graphs in the upper row of Fig. 7.10. Therefore, it is
concluded that none of the observed time-drawdown curves is in disagreement with the
proposed model of groundwater flow to a well in a homogeneous and laterally anisotropic
aquifer which is covered by a semi-pervious layer.
7.2.11 Second interpretation phase and post-optimization
During the second interpretation phase, six groups of parameters are considered as
derivable parameters, instead of five as in the first phase. These parameters are: the
effective horizontal hydraulic conductivity Khe(l), the lateral anisotropy v'm, the angle (J
and the specific elastic storage S.(l) of the pumped layer and the vertical hydraulic
conductivities Kv(2-4) and the specific elastic storages 5.(2-4) of the covering semi-
pervious layer. The other hydraulic parameters are unidentifiable and their assumed
values are the same as in the first interpretation phase.
Table 7.5. Results of the second interpretation phase of the double pumping test in the
laterally anisotropic aquifer formed by the fractured rocks and the two post-optimizations
Hydraulic With minimal values of the With estimated values for the With maximal values for the
parameters unidentifIable parameters unidentifiable parameters unidentifiable parameters
Sym- Unit Optimal s,.. s, Optimal sm s, Optimal sm s,

bol value value value
K",(I) mid 0.443 0.0220 0.0083 0.439 0.0210 0.0083 0.427 0.0191 0.0075
..rm dim.less 2.023 0.0132 0.0061 2.021 0.0132 0.0062 2.002 0.0147 0.0079
0 radian 1.101 0.0054 0.0042 1.100 0.0054 0.0042 1.098 0.0059 0,0044
S,(I) m'! 6.99x107 0.0207 0.0075 6.97x107 0.0214 0.0075 6.74xl07 0.0259 0.0069
K,,(2-4) mid 4.38xl0' 0.0992 0.0147 4.36xl0' 0.1037 0.0147 4.67xl0' 0.1098 0.0148
S,(2-4) m'! 1.62xl0' 0.1181 0.0347 1.63xl0' 0.1209 0.0349 1.53xl0' 0.1239 0.0388
To study the influence of the choice of these unidentifiable parameter values on

the deduced values of the hydraulic parameters, two post-optimizations are made with the
inverse model. In the first post-optimization the unidentifiable parameter values are
changed so that the drawdown above the covering semi-pervious layer increases. This
change is obtained by dividing the horizontal and the vertical hydraulic conductivities and
the specific elastic storages of layers 5, 6, 7, 8 and 9 by the same factor, 3.2, as well as
the specific yield. In the second post-optimization the initially chosen values of these
unidentifiable parameters are multiplied by the same factor 3.2.
The results of the second interpretation and the two post-optimizations are given in
Table 7.5. Here the deduced values are given with their marginal and conditional standard
deviation. As shown by comparing the results of the first interpretation phase (Table 7.3)
with the results of the second, the marginal standard deviation of the vertical hydraulic
conductivities Kv(2-4) and the specific elastic storages S,(2-4) are much larger in the
second interpretation phase. The reason for these considerable enlargements of these
marginal standard deviations are indicated in Table 7.6. In this table, the condition
indexes corresponding with the second interpretation phase are given along with the
matrix of marginal variance-decomposition proportions. In this interpretation phase, one
large condition index exists that corresponds with a moderate to strong dependency
(Belsley, 1990). This index is responsible for, respectively, 96 % and 98 % of the marginal
variances of the parameters Kv(2-4) and S,(2-4). A comparison of the results of the second
interpretation phase and the two post-optimizations indicates that the estimated values of
the unidentifiable parameters have a negligible influence on the optimal values of the
deducible parameters.
Table 7.6. Condition indexes 71 and n matrix of marginal variance-decomposition

proportions corresponding with the second interpretation phase of the double pumping test
in the laterally anisotropic aquifer along with the estimated values for the unidentifiable
parameters
71 1.0 1.4 2.6 5.7 8.4 40.1

Parameter
..Jm 0.0001 0.0545 0.2255 0.5713 0.1485 0.0002
e 0.4502 0.0455 0.0516 0.3904 0.0421 0.0202
Kbe(l) 0.0008 0.0140 0.0690 0.1057 0.5580 0.2525
S,(1) 0.0019 0.0195 0.0103 0.3708 0.5668 0.0307
Kv(2-4) 0.0000 0.0001 0.0010 0.0082 0.0305 0.9602
S,(2-4) 0.0000 0.0000 0.0007 0.0043 0.0128 0.9822
7.2.12 Conclusions
The introduction of apparent distances makes it possible to apply a numerical axi-

symmetric model in an inverse model for an interpretation of drawdowns measured in a
laterally anisotropic aquifer. Results from pumping test at Kortemark has shown that the
aquifer in the fractured top of the Brabant Massif and the fractured chalk behaves as an
equivalent laterally anisotropic porous medium at the scale of the influenced area. At that
site, the pumped aquifer presumably has a fracture system with a rather large fracture
density, a fracture aperture that is constant rather than distributed, and a fracture orienta-
tion which is distributed rather than constant.
The precision with which the hydraulic parameters are deducible can be derived
from the exact or approximate joint confidence region in the p-dimensional parameter
space. The marginal standard deviation helps us to derive the maximum and the minimum
values of a certain parameter on the bounds of the approximate joint confidence region.
The conditional standard deviations help us to locate the intersections of the bounds of the
approximate joint confidence region with the parameter axes, which go through the
optimal values. Small differences between both standard deviations of one hydraulic
parameter means that the considered parameter estimate is rather independent from the
other estimates. The existence of large differences between both standard deviations is an
indication of a relation between different hydraulic parameters.
An insight in the location of many points on the joint confidence region bounds
was obtained from two- and three-dimensional cross sections through the exact or the
approximate joint confidence region. By studying these cross sections, further insight was
obtained in the correlation between two or three columns of the sensitivity matrix J.
These cross sections are an additional tool, especially visual, in the collinear diagnostics.
They supplement the information obtained from the correlation coefficients, the condition
indexes, and the matrix of marginal variance-decomposition proportions.
Chapter 7 / Example of petj'0171lance and interpretation of pumping tests 327
By plotting the contours of the sums of the squared residuals for different
combinations of data measured in different wells, it was possible to examine which wells
provide enough complementary information to deduce a unique solution for the parame-
ters which describe the lateral anisotropy of an aquifer. These graphs also show that all
the observed data are consistent with the model of the homogeneous and laterally aniso-
tropic aquifer. None of the observed time-drawdown curves is in disagreement with the
proposed model.
7.3 Triple pumping test in a layered groundwater reservoir

formed by Tertiary sediments
The inverse numerical model is demonstrated by the simultaneous interpretation of a triple

pumping test in a layered groundwater reservoir. Here, it is demonstrated how the
lithological information obtained during the drilling and the geophysical well logging can
be incorporated in the numerical model. The three different pervious layers which are
separated by semi-pervious layers are pumped separately. During each test the drawdowns
are measured on four different levels and at different distances from the pumped well. By
the simultaneous interpretations of all the observed drawdown of these three tests the
vertical conductivities of the semi-pervious layers can be deduced with accuracies
comparable to those of the horizontal hydraulic conductivities of the pumped pervious
layers.
The pumping test site is located near the Geological Institute of Ghent University. The
lithostratigraphical cross section (Fig. 7.11) is based on the description of samples
collected during drilling activities and on the results of geophysical borehole logs (caliper,
spontaneous potential, point resistance, natural gamma and resistivity measurements with
the long- and the short-normal device). In our particular case the point resistance log
characterises quite well the layering of the groundwater reservoir (Fig. 7.11).
The groundwater reservoir is bounded below at -40.6 m TAW by a 95 m thick
mainly clayey layer (Kortrijk Formation, Yc in Fig. 7.11). This information is deduced
from alSO m deep borehole near the pumping site. The geometrical (all levels are given
versus the Belgian Datum Level, 0 m TAW is 2.33 m below the mean sea level),
lithological and hydrogeological characteristics of the overlaying layers are given below:
- from -40.6 to -24.6, sandy and silty clay with intercalations of thin clayey fine sand
beds (Ydl complex), this semi-pervious complex corresponds to the Kortemark Member,
- from -24.6 to -20.6, clayey glauconitic fine sand (Yd2), hydraulic head in this pervious
Yd2 layer was +4.6,
- from -20.6 to -14.6, sandy clay to clay (Yd3) forming a semi-pervious layer
- from -14.6 to -5.1, slightly clayey glauconitic fine sand containing small shell fragments
(Yd4), hydraulic head in this pervious layer was +7.3,
- from -5.1 to -3.6, very sandy clay (Yd5) forming a semi-pervious layer
328 Chapter 7 I Example of peifonnance and inJerpntation of pumping tests
- from -3.6 to + 1.4, slightly clayey glauconitic fine sand containing small shell fragments
(Yd6), the hydraulic head was in Yd6 only a few centimeters higher than in Yd4,
- from + 1.4 to +2.4, stiff silty clay (Plm, Merelbeke Member) forming a semi-pervious
layer,
- from +2.4 to + 10.4, alternation of thin sandy clay beds and very clayey fine sand
containing sandstone concretions (PIc, Pittem Member). The watertable is situated in this
layer at +8.3. The layers Yd2 to Yd6 belong to the Egem Member.
--:.- ._-_......
rn
~ C" 9 ~"'O
E] C'Oy WlI'O)' ClOy
~ ' CI""-CI.t':'l ""e COI"IC. 't: toOft!.
Fig. 7.11. Lithological cross-section through triple pumping test site (Ghent) and location
of pumping and observation wells (Screens indicated in black are situated in cross-sections
and screens in dashed lines are not)
The location of the pumping and observation wells and their screened intervals are shown
in Fig. 7.11. The pumping wells have screens in the pervious layers Yd2 (pumping well
PP2), Yd4 (PP4) and Yd6 (PP6). Diameters of these wells (screen and riser pipe) are in
each case 125/116 mm; the drilling diameter was 250 mm.
In each pervious layer three observation wells were installed at different distances
from the pumping well. The observation wells are indicated as PBx.y where the index x
indicates the layer of location and y corresponds with the distance to the pumping well in
the concerned pumped layer (1 for 6.3m, 2 for 12.5m and 3 for 25m). All observation
wells have the same diameter of 63/57 mm; the drilling diameter was 110 mm.
Pumping and observation wells in the same pervious layer were located on a line.
In the semi-pervious layer PIc (Pittem Member) only one observation well (PB8.1) was
drilled near the centre of gravity of the triangle formed by the three pumping wells; the
screen length was 0.5 m. During the completion of the observation and pumping wells the
annular space between riser pipe and semi-pervious layers was sealed with neat cement.
7.3.3 Performance of triple pumping test
The first part of the triple pumping test started at 15hOO of the 10th of October 1989. The
well PP6 was pumped at a constant discharge rate of 18.0 m3/d. Above the submersible
pump an inflatable packer was installed to eliminate the effect of wellbore storage.
Drawdowns in the observation wells were measured by means of electronic pressure
transducers connected to a datalogger. Above each pressure transducer an inflatable
packer was installed so that the effect of well bore storage in the observation wells could
be eliminated during pumping and recovery tests. The first test was stopped at IlhOO on
the 13th of October 1989. The residual drawdowns were measured in the same way as the
drawdowns until 13hOO on the 14th of October 1989.
The second part of the triple pumping test started at 16hOO of the 18th of October
1989. The well PP2 was pumped at a constant discharge rate of 39.9 m3 /d. This test was
performed in the same way as the first. The pumping was stopped at 16hOO on the 20th of
October 1989. The residual drawdowns were measured until 22hOO on the 22th of
October 1989.
The third part of the triple pumping test started at 14hOO on the 24th of October
1989. The well PP4 was pumped at a constant discharge rate of 74.2 m3 /d. The pumping
was stopped at 14hOO on the 26th of October 1989. The residual drawdowns were
observed until 8hOO on the 27th of October 1989.
In the numerical model the groundwater reservoir is discretized in nine layers. Layer 1
corresponds with the lower 12.5 m of the semi-pervious layer Ydl. The upper 3.5 m of
the semi-pervious layer Ydl is considered as the horizon between layer 1 and layer 2 of
the numerical model. Consequently, the storage decrease and the horizontal flow in these
upper 3.5 m of semi-pervious layer Ydl are neglected. In other words the specific elastic
storage and the horizontal hydraulic conductivity of this layer are assumed equal to zero.
Layers 2, 4 and 6 of the numerical model correspond with the pervious layers
Yd2, Yd4 and Yd6. Layers 3 and 5 of the numerical model correspond with the semi-
pervious layers Yd3 and Yd5. These layers are not subdivided in more than one layer in
the numerical model because of two reasons. The first is that the storativity of the layers
is expected to be of the same order of magnitude as the storativity of the pumped
pervious layer. The second reason is that there are no observations of the drawdown in
these layers and consequently these drawdowns should not be calculated accurately. The
semi-pervious layer Plm is considered as the horizon between layers 6 and 7. Again the
storage decrease and the horizontal flow of this 1 m thick layer is neglected in the
numerical model.
The uppermost three layers of the numerical model 7, 8, and 9 correspond with
that part of the semi-pervious layer PIc which is situated below the watertable. The
thickness of these three layers decreases upwards to the watertable. This allows one to
calculate accurately the drawdowns in the observation well PB8.1 of which the screen is
located in the middle of layer 8 of the numerical model.
7.3.5 Hydraulic parameters derived from observed drawdowns
Studying the sensitivities, the variance-covariance matrix and the partial correlation
coefficients generated with the initial estimates of the parameter values the following
parametrization is made. Twelve groups of hydraulic parameters can be derived by means
of the inverse model from the observed and residual drawdowns of the three pumping
tests. The horizontal conductivities of the pumped pervious layers 2, 4 and 6, Kn(2),
Kh (4), Kh (6) can be deduced separately. The horizontal conductivities of the semi-pervious
layers 1, 3, 5, 7, 8 and 9 are unidentifiable. As can be proved, the introduction of rough
estimates of their values was sufficient. These values do not have a significant influence
on the values deduced for the other parameters. The estimated and introduced values for
layers 1, 3 and 5 are respectively 0.04 mid, 0.002 mid and 0.02 mid. Because layers 7, 8
and 9 of the numerical model represent the same lithostratigraphical layer PIc, it is
assumed that the horizontal conductivities of those layers are all equal to 0.25 mid.
The hydraulic resistance between layers 1 and 2, c(1), is separately identifiable.
The hydraulic resistance between layers 2 and 3, c(2), and between layers 3 and 4, c(3)
are identifiable as a group. The hydraulic resistance c(2) is taken about one and a half
time larger than the hydraulic resistance c(3). This information was derived from the
geophysical well logs like the natural gamma ray indicating a more clayey lower half of
layer Yd3. The hydraulic resistances c(4) and c(5) are considered together. The sum of
their values is equal to the hydraulic resistance of the semi-pervious layer 5. It is assumed
that both halfs of this layer have the same hydraulic resistance. The hydraulic resistance
c(6) can be deduced separately. The hydraulic resistances c(7) and c(8) have only a
limited influence on the observed drawdowns of observation well PB8.1 located in layer 8
of the numerical model. The sensitivities of the drawdowns to these two hydraulic
resistances show, however, a correlation with the sensitivities of the drawdowns to the
storage coefficient near the watertable which is more sensitive. These hydraulic resistan-
ces cannot be deduced from the observations and are estimated equal to 36 d and 6 d
which correspond with a vertical hydraulic conductivity of 0.13 mid.
The specific elastic storages of layers 1 and 2, Ss(l) and Ss(2), are considered in
one group of hydraulic parameters. It is assumed that those two layers have the same
elasticity. Also the specific elastic storages Ss(3) and Ss(4) are assumed to have the same
value and form one group. The specific elastic storages Ss(5), Ss(6), Ss(7), Ss(8) and Ss(9)
are included in one group and have the same value. The last hydraulic parameter
deducible from the observations is the storage coefficient near the watertable So.
7.3.6 Interpretation of results
The inverse process was started with the ordinary linear square (OLS) method. When the
optimum was reached, the residuals were examined. The residuals show a rather high
number of outliers. Therefore, the inverse process was continued by means of the
biweighted least square (BWLS) method. A possible explanation of the occurrence of the
outliers is given in the following part.
The optimal values of the hydraulic parameters obtained with the BWLS method
are given in Table 7.7. This table includes also the estimated values of the hydraulic
parameters which could not be deduced. Table 7.8 shows the optimal values of the
deduced hydraulic parameters along with their marginal and conditional standard
deviations. The accuracies of the hydraulic resistances or the vertical conductivities of the
semi-pervious layers situated between the pumped pervious layers, c(2-3) and c(4-5), are
hardly smaller than those of the horizontal hydraulic conductivities of the pumped
pervious layers. These accuracies are in the same order as those of the specific elastic
storages Ss(3-4) and Ss(5-9).
The accuracies of the hydraulic parameters of the layers adjacent to the pumped
ones, such as the layers 7, 8, 9 and 1, are much smaller. From this last group of
hydraulic parameters the best accuracy is obtained for the hydraulic resistance c(6) and
for the storage coefficient near the watertable So. This can be attributed to the availability
of observations in well PB8.1, which is located in layer 8 of the numerical model. The
hydraulic parameters deduced with the smallest accuracy were the specific elastic storage
Ss(1) and the hydraulic resistance c(1). The reason herefore is that layer 1 neighbours
only on one side a pumped layer and that no drawdowns were observed in this layer.
To examine if the sensitivities of the observed drawdowns to the newly deduced
values have changed significantly, the sensitivities of all groups of hydraulic parameters
are calculated in a last step. With those sensitivities, residuals and weights, the variance-
covariance matrix was calculated. From this variance-covariance matrix the marginal and
the conditional standard deviation of all these parameters were calculated (Table 7.9).
From this table one can see that the attainable accuracy of the horizontal conductivities of
the layers 7, 8 and 9, Kb(7-9), is of the same order as the one of the least sensitive
hydraulic parameter which was deduced, c(l). This change in sensitivity is not important
enough to continue the inverse process. Only the values of the least sensitive parameters
which were deduced can change a little by the introduction of this parameter (this is
demonstrated for another pumping test at the end of Sect. 7.5.6 and in Table 7.12).
Table 7.7. Optimal values of the hydraulic parameters deduced with the BWLS method
from the observations (underlined) of the triple pumping test and the estimated values of
the unidentifiable hydraulic parameters (not underlined)
RADIUS OF WELLSCREEN, R, IN M, ------------------ 0.100

INITIAL TIME, Tl, IN MIN, ---------------------------- 0.100
LOGARITIIMIC INCREASE OF TIME AND OF RADIUS OF RINGS
LOG A, ------------------------------------------------- 0.100
LATEST CALCULATED TIME, T2, IN MIN, ------------------------------ 0.100
NUMBER OF LAYERS, N, ------------------------------------- 9
NUMBER OF RINGS, M, ------------------------------------------- 48
THICKNESS OF THE SUCCESSNE LAYERS, IN M
THICKNESS OF LAYER 1, IN M, --------------------------- 12.Soo
THICKNESS OF LAYER 2, IN M, ------------------------------------- 4.000
THICKNESS OF LAYER 3, IN M, ------------------------------------------ 6.040
THICKNESS OF LAYER 4, IN M, -------------------------------- 9.400
THICKNESS OF LAYER S, IN M, ----------------------------------------- I.SOO
THICKNESS OF LAYER 6, IN M, ----------------------------- 4.8000
THICKNESS OF LAYER 7, IN M, ---------------------- 3.2000
THICKNESS OF LAYER 8, IN M, -------------------------- 2.100
THICKNESS OF LAYER 9, IN M, -------------------------------- O.SOO
-------- NUMBER OF HYDRAULIC PARAMETER -----/NRI---------
HYDRAULIC CONDUCTNITY, 1<,.(1), IN MIDAY, ----I 1/----- 0.040
HYDRAULIC CONDUCTNITY, 1<,.(2), IN MIDAY, -------/ 2/----- 1.287
HYDRAULIC CONDUCTNITY, 1<,.(3), IN MIDAY, - - I 3/----- 0.002
HYDRAULIC CONDUCTNITY, 1<,.(4), IN MIDAY, -------/ 4/------- 1.104
HYDRAULIC CONDUCTNITY, I<,.(S), IN MIDAY, -----/ S/------ 0.020
HYDRAULIC CONDUCTNITY, K,.(6), IN MIDAY, -----/ 6/-------- 0.8S8
HYDRAULIC CONDUCTNITY, K,.(7), IN MIDAY, -----/ 7/----- 0.2S0
HYDRAULIC CONDUCTNITY, 1<,.(8), IN MIDAY, ------/ 8/------ 0.2S0
HYDRAULIC CONDUCTNITY, 1<,.(9), IN MIDAY, ----I 9/----- 0.2S0
HYDRAULIC RESISTANCE, c(I), IN DAY, ----------------/10/------ 283.2
HYDRAULIC RESISTANCE, c(2), IN DAY, -------------/ 11/------- 48SI.
HYDRAULIC RESISTANCE, c(3), IN DAY, -------------/12/------- 3234.
HYDRAULIC RESISTANCE, c(4), IN DAY, -----------------/13/------ 40.22
HYDRAULIC RESISTANCE, c(S), IN DAY, ---------------/14/------- 40.22
HYDRAULIC RESISTANCE, c(6), IN DAY, -----------------/ IS/-------- 4172.
HYDRAULIC RESISTANCE, c(7), IN DAY, -----------/16/------ 39.00
HYDRAULIC RESISTANCE, c(8), IN DAY, ---------/171----- 6.2S7
SPECIFIC ELASTIC STORAGE, S,(I), IN M", -------/18/----- 0.130E-04
SPECIFIC ELASTIC STORAGE, S,(2), IN M", ------------/ 19/------- 0.420E-04
SPECIFIC ELASTIC STORAGE, S,(3), IN M", - - - - - - / 20/----- 0.36OE-04
SPECIFIC ELASTIC STORAGE, S,(4), IN M", --------/21/----- 0.360E-04
SPECIFIC ELASTIC STORAGE, S,(S), IN M", ----------/22/---- 0.S6OE-04
SPECIFIC ELASTIC STORAGE, S,(6), IN M", -------/23/---- 0.S60E-04
SPECIFIC ELASTIC STORAGE, S,(7), IN M", -------/ 24/---- 0.S6OE-04
SPECIFIC ELASTIC STORAGE, S,(8), IN M", ------------/ 2S/----- 0.S60E-04
SPECIFIC ELASTIC STORAGE, S,(9), IN M", -------------/ 26/------- 0.S60E-04
STORAGE COEFFICIENT NEAR THE WATERTABLE, S,,/ 27/----- 0.009189
Chapter 7/ Example of pet/onnance and interpretation of pumping tests 333
Table 7.S. Optimal values of the hydraulic parameters deduced with the BWLS-method
deduced from the observation of the triple pumping test and their marginal and condi-
tional standard deviations
Hydraulic parameter(s) Units Value sm Sc
Kh (4) mid 1.10.100 0.0020 0.0009

Kb (6) mid 8.58.10- 1 0.0019 0.0015
Kh (2) mid 1.29.100 0.0027 0.0023
S,(5-9) m-I 5.61.10-5 0.0034 0.0030
c(4-5) d 4.02.101 0.0030 0.0017
c(2) d 4.85.103 0.0045 0.0018
c(3) d 3.23.103
S.(3-4) m- l 3.62.10-5 0.0048 0.0024
c(6) d 4.17.103 0.0088 0.0047
S,(2) m-I 4.19.10- 5 0.0071 0.0030
So m3 /m 3 9.19.10-3 0.0125 0.0055
S,(I) m-I 1.31.10-5 0.0236 0.0139
c(l) d 2.83.102 0.0316 0.0205
As a conclusion one can state that with the triple pumping test the hydraulic
resistances of the semi-pervious layers which are located between the pumped pervious
layers can be derived with an accuracy which is hardly lower than the accuracy of the
horizontal hydraulic conductivities of the pumped layers. Only a few hydraulic parameters
of the layers neighbouring the directly pumped layers can be derived with much smaller
accuracy. The accuracy of those hydraulic parameters can be improved by measuring
drawdowns in these layers. Obviously, above reached accuracies for the vertical hydraulic
conductivities of the semi-pervious layers can never be obtained by means of a simple
pumping test, in which only one pervious layer is pumped and drawdowns are only
measured in the pumped layer.
334 Chapter 7/ Example of peifonnance and interpretation of pumping tests
Table 7.9. Optimal values deduced with the BWLS-method (underlined) and estimated
values of all possible groups of hydraulic parameters corresponding with the final result
of the triple pumping test along with their marginal and conditional standard deviation
Hydraulic parameter(s) Units Value Sm Sc
Kh(4) mid 1.10.10° 0.0028 0.0009

Kh(6) mid 8.58.10- 1 0.0055 0.0015
Kh(2) mid 1.29.10° 0.0036 0.0023
S,(5-9) m- I 5.61.10-5 0.0034 0.0029
c(4-5) d 4.02.101 0.0030 0.0018
c(2) d 4.85.103 0.0104 0.0018
c(3) d 3.23.103
S,(3-4) m- I 3.62.10-5 0.0050 0.0024
c(6) d 4.17.103 0.0118 0.0047
S,(2) m- I 4.19.10- 5 0.0075 0.0030
So m 3 /m 3 9.19.10-3 0.0223 0.0054
S,(1) m- I 1.31.10-5 0.0253 0.0147
KP-9) mid 2.50.10- 1 0.0665 0.0178
c(l) d 2.83.102 0.0412 0.0271
Kh(l) mid 4.00.10-2 0.0824 0.0201
c(7) d 3.90.102 0.3821 0.1591
c(8) d 6.26.103
Kh(5) mid 2.0.10 2 1.0712 0.3740
Kh(3) mid 2.0.10-3 2.8307 0.7435
7.3. 7 Outliers
In Fig. 7.12 the calculated and observed drawdowns are shown. In this figure one can see
that the calculated and measured drawdowns for the layers 4, 6 and 8 during the pumping
test on the layers 4 and 6 are similar. This is not so for the drawdowns measured in layer
2 during these two last mentioned pumping tests. There is also a rather high discrepancy
between the measured and calculated drawdowns of the layers 4 and 6 during the
pumping test on layer 2.
The residuals or the differences between the measured and the calculated draw-
downs are due to measurement and/or simulation errors. The first errors are the differen-
ces between the measured and the real drawdowns; the second are the differences between
the real and the simulated drawdowns. A distinction between these two kinds of errors
can only be made if the real drawdown is clearly defined. The real drawdown is the one
which occurs at a well defined place and time in the undisturbed aquifer. Here the
omission of the Noordbergum effect (Verruijt, 1968) can be regarded as a simulation
error. The errors due to leakage through seals and due to flow through wells that short
circuits the semi-pervious layer(s) can be considered as measurement errors.
In the numerical model of groundwater flow (Lebbe, 1988) only the deformation
in the vertical direction is considered. The Noordbergum effect is due to a three-dimensi-
onal deformation of the groundwater reservoir which is caused by the changes of the
gradients of the stresses in the three dimensions. This effect is rather important close to
the pumping well in the non-pumped pervious layers. During the pumping test on layer 2
the hydraulic head rose 30 mm in well PB6.1 after 24 minutes of pumpage; this was the
highest value observed. During the pumping test in layer 4 the largest effect (54 mm) was
reached in well PB8.1 also after 24 minutes after the start of the pumpage. The largest
effect was also observed in PB8.1 after 16 minutes of pumpage in layer 6 and was equal
to 14 mm.
Most of the outliers are caused by measurement errors due to leakage through
cement seals in the semi-pervious layer Yd3. This is a rather thick semi-pervious layer
with a small vertical hydraulic conductivity of about 7.4 10-4 mid. A possible explanation
is the poor construction of the cement seals around the riser pipes of the pumped and
observation wells in layer Yd2. Especially during the pumping test in well PP2 the
leakage through the seal in Yd3 around the pumped well must be considerable. The
maximum leakage through this seal was estimated at 1.6 m3 /d with the help of the
differences between the observed and calculated drawdown in layer 4 during pumpage on
layer 2 and from the drawdown in layer 4 when layer 4 is pumped. A second interpreta-
tion of the pumping test was executed by means of the inverse model. The only difference
with the first interpretation phase was the discharge rate during the pumping test in well
PP2. Here, a discharge rate of 38.3 m3/d was assumed in layer 2 and of 1.6 m3 /d in layer
4 whereas in the first interpretation phase a discharge rate of 39.9 m3 /d was assumed in
layer 2 and 0 m3 /d in layer 4. In Fig. 7.12 the observed and calculated drawdowns during
the pumping test in well PP2 are shown corresponding with the solution of this second
interpretation phase. The observed distance dependance of the drawdowns in layer 4 and
6 agrees now more with the calculated one. The deduced values of the second interpretati-
on phase are given in Table 7.10. The largest differences between the two interpretation
phases are the values of the hydraulic resistances of the layers Yd3 and P1m. The first
changes from 8,085 d to 10,085 d and the second from 4,172 d to 3,922 d. The other
values of the hydraulic parameters deduced with a rather high accuracy do not change
considerably. With this second interpretation phase it becomes possible to evaluate the
influence of the errors due to leakages through the seals.
[J<fWMj(H) [JlfJ.ID,I'I(M) w
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.t .' .t\ .' III' TJI'E<Mi~) LRym 2 .100STOCE(tn
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Fig. 7.12. Measured (crosses) and calculated (continuous curves) drawdown in time- and distance-drawdown graphs for triple pumping test
E .'
(left two columns for pumping test in layer Yd2, Q(2)= 39.9 m3/d, middle two columns for pumping test in layer Yd4, Q(4)= 74.2 m3/d,
and right two columns for pumping test in layer Yd6, Q(6)= 18.0 m3/d)
Table 7.10. Hydraulic parameters deduced from all observations of the triple pumping
test where a leakage of 1.6 m 3 /d was assumed through the seal in Yd3 around well PP2
during the pumping on this well
--------- NUMBER OF HYDRAULIC PARAMETER ------/NRI----------------

HYDRAULIC CONDUCTIVITY, K,,(I), IN MIDAY, --------/ 1/------- 0.040
HYDRAULIC CONDUCTIVITY, K,,(2), IN MIDAY, --------/ 2/-------- 1.287
HYDRAULIC CONDUCTIVITY, K,,(3), IN MIDAY, -------/ 3/-------- 0.002
HYDRAULIC CONDUCTIVITY, K,,(4), IN MIDAY, --------/ 4/------- ~
HYDRAULIC CONDUCTIVITY, K,,(5), IN MIDAY, -------/ 5/-------- 0.020
HYDRAULIC CONDUCTIVITY, K,,(6), IN MIDAY, ------/ 6/-------- 0.858
HYDRAULIC CONDUCTIVITY, K,,(7), IN MIDAY, ------/ 7/------ 0.250
HYDRAULIC CONDUCTIVITY, K,,(8), IN MIDA Y, -------/ 8/-------- 0.250
HYDRAULIC CONDUCTIVITY, K,,(9), IN MIDAY, -------/ 9/------- 0.250
HYDRAULIC RESISTANCE, c(I), IN DAY, --------------/10/------ 283.2
HYDRAULIC RESISTANCE, c(2), IN DAY, -----------------/ 11/-------- 4851.
HYDRAULIC RESISTANCE, c(3), IN DAY, -----------------/12/------- 3234.
HYDRAULIC RESISTANCE, c(4), IN DAY, -----------------/13/------ 40.22
HYDRAULIC RESISTANCE, c(5), IN DAY, ----------------/14/-------- 40.22
HYDRAULIC RESISTANCE, c(6), IN DAY, ---------------/15/------ 4172.
HYDRAULIC RESISTANCE, c(7), IN DAY, ----------------/16/-------- 39.00
HYDRAULIC RESISTANCE, c(8), IN DAY, ---------------/171------ 6.257
SPECIFIC ELASTIC STORAGE, S,(I), IN M', ------------/ 18/------
SPECIFIC ELASTIC STORAGE, S,(2), IN M·', ---------------/ 19/-------- 0.420E-04
SPECIFIC ELASTIC STORAGE, S,(3), IN M·', ------------/ 20/------- 0.360E-04
SPECIFIC ELASTIC STORAGE, S,(4), IN M·', -------------/ 21/--------
SPECIFIC ELASTIC STORAGE, S,(5), IN M', --------------/ 22/--------
SPECIFIC ELASTIC STORAGE, S,(6), IN M·', -------------/23/-------
SPECIFIC ELASTIC STORAGE, S,(7), IN M·', -----------/ 24/-------
SPECIFIC ELASTIC STORAGE, S,(8), IN M·', --------------/ 25/------
SPECIFIC ELASTIC STORAGE, S,(9), IN M·', -------------/ 26/--------
STORAGE COEFFICIENT NEAR THE WATERTABLE, S,,/ 27/-----
Beside the leakage through the seals there was also a hydraulic short circuit in well
PBl-8 during the pumping tests. This well, which was made for testing geophysical well
logging equipment, has a screen over the entire thickness of the groundwater reservoir.
The Noordbergum effect as well as the leakage through the seals and through the test well
caused a rather high amount of outliers. By means of the biweighted least square method
the influence of these outliers on the interpretation could be minimized.
7.3.8 Conclusions
The inverse model based on a numerical model allows a detailed schematization of the
groundwater reservoir. Consequently, the simulated flow approximates closely the real
flow. The lithostratigraphical information gathered from drilling and geophysical logs can
be used in an optimal way. Compared to the classical methods where time-drawdown or
distance-drawdown curves are separately interpreted, the inverse numerical model allows
the interpretation of all the drawdowns observed in the directly pumped as well as in the
indirectly pumped layers at the same time. The inverse numerical model has the addi-
tional advantage that all observed drawdowns and residual drawdowns of a multiple
pumping test can be interpreted simultaneously. All observations contribute to determine
one set of values of hydraulic parameters along with their marginal and conditional
338 Chapter 7 / Example of perjonnance and interpretation of pumping tests
standard deviations. From these standard deviations one can deduce the vertical hydraulic
conductivities of semi-pervious layers with an accuracy comparable to that of horizontal
hydraulic conductivities of pumped pervious layers.
A practical problem in constructing reliable observation wells for pumping tests
arises when the riser pipes of these wells must be placed in semi-pervious layers with
small vertical hydraulic conductivities. Making seals in these layers to prevent leakage is
difficult. In the case study this leakage was responsible for rather high amount of outliers.
By the application of the biweighted least square method the influence of these outliers on
the interpretation results was minimized although these leakages still have an influence on
these results as was demonstrated by the second interpretation phase.
It can be stated that vertical hydraulic conductivities and specific elastic storages of
semi-pervious layers cannot be deduced accurately from classical pumping tests. The
reason is that in these tests only one pervious layer is pumped and the drawdowns are
measured in this layer. Far more accurate deductions are possible if drawdowns are
measured in the adjacent pervious and semi-pervious layers. The measurements in the
semi-pervious layers can be influenced by leakage through seals especially when small
vertical hydraulic conductivities are encountered. In that case separate pumping tests in all
the pervious layers and drawdown measurements in these layers are advised.
7.4 Single pumping test to determine the conductivity

of Tertiary silty clay
Here, it is shown how the vertical conductivity of a semi-pervious layer is deduced from
the observed evolution of the vertical drawdown gradient in this layer. The considered
semi-pervious layer is situated between a pervious layer and a layer with very small
conductivity. The screen of the pumped well is installed in the pervious layer so that a
considerable drawdown is created in it and in the lower part of the semi-pervious layer.
By the measurement of the propagation of the drawdown at two different levels in the
semi-pervious layers, it was possible to deduce its vertical conductivity by means of the
inverse model based on the AS2D numerical model. With the numerical model the rise of
the drawdown at the different levels in the semi-pervious layer can be simulated very
accurately.
The lithostratigraphical cross section of the single pumping test, which is performed at
Kemzeke, is given in Fig. 7.13. This cross section is based on description of samples
during the drilling activities and the results of geophysical measurements in borehole
DBl. The considered groundwater reservoir belongs to two formations. The lower part is
composed of the Members of Bassevelde, Watervliet and Ruisbroek belonging to the
Zelzate Formation; the upper part is composed of the Members of Belsele-Waas and
Terhagen belonging to the Boom Formation (Marechal and Laga, 1988).
The considered groundwater reservoir is bounded below by the heavy clay of the
Onderdijke Member. The top of this clay is located around the level -58 m TAW.
Between the levels -58 and -46.6 occurs the Bassevelde Member which consists of
glauconite and mica bearing silty fine sand. This sands can be considered as a pervious
layer. The Watervliet Member occurs between the levels -46.6 and -43.1 and consists of
sandy clay with some glauconite and mica. This layer can be considered as a semi-
pervious layer. Between the levels -43.1 and -25.9 occurs the Ruisbroek Member which
consists of clay bearing sands with some clay rich bands. This member is considered as a
pervious layer.
The Belsele-Waas Member occurs between the levels -25.9 and -14.1. This
member consists of clay with two thick bands at its base which contain a large amount of
silt. This is the semi-pervious layer of which the vertical conductivity is determined with
the single pumping test. Between the levels -14.1 and + 1.6 occurs the Terhagen Member
which consists of clay. This member can be considered as a semi-pervious layer with a
very small hydraulic conductivity. The water table, which forms the top of the ground-
water reservoir, is situated in the upper part of the Terhagen Member.
The location of the pumping and observation wells is represented in Fig. 7.13. The
pumping well has a screen between the levels -41.4 and -31.4. Three observation wells
have a screen in the upper part of the pumped layer. The screens of these observation
wells are situated between the levels -36.2 and -34.2. The distances from the pumped
well to these observation wells SBl, SB3 and SB5 are respectively equal to 15.6, 50.3
and 159 m. Three observation wells are placed above the directly pumped layer: well SB2
with a screen between -28.4 and -26.4, well DrBI with a screen between -21.2 and -19.2
and well SB4 with a screen between -16.4 and -14.4. The distance between the pumped
well and the wells SB2, DrBl, SB4 are approximately the same, respectively 12.45 m,
12.25 m and 11.70 m.
7.4.3 Performance of single pumping test
The pumping test started at 13h40, 8 October 1990. Water was pumped by means of a
submersible pump with a constant discharge rate of 223.3 m3/d. In all observation wells
the drawdowns were measured by means of pressure transducers connected with a
datalogger. Inflated packers are installed above the pressure transducers in the observation
wells to reduce as much as possible the storage decrease due to the drawdown in these
wells. The pumping test was stopped at 13h40, 10 October 1990. So, the pumping test
lasted exactly 2 days. The residual drawdown was measured during 1 day after the stop of
the pump.
LITHOSTRATIGRAPHICAL CROSS SECTION
S B2
~EL o.B1
"' TA'M 5B. SBl 581 DB1 SII4
I.,,\YERSI
- t.
NUMERICAl. I.4ODEL
LAYER
-.. -- LAYER 17
-. LAYER .e
\.AveR I S
LAVER , ,,
BELSElE-WAAS MEMBER
, ~YER 13
-25 LAYER '2
I LAV ER 11
30 PERV/oUSI.AYER t.AVER 10
.... YER.
RUISBROEK MEMBER
.... YeR 7
LAYEAe
.... YEA.
I
SEMI-PERVIOUS I.AYER - WATERVLIET ME"'BER LAYER .
LAYEA>
.... YEA2
PERVIOUSI.AYER
BASSEVELDE MEMBER YER.
,
-60 - SEMI-PERV1OUS LOYER Wlrn VERY lOW CONDUCTMTY
DEROUKE EMBER /
/'
/ , / /
r-:::"J
LEG£NO
watllftable
~
... tcr ••n
Fig. 7.13. Lithostratigraphical and hydrogeological cross-section and location of the

pumping and observation wells at the pumping site of Kemzeke and the schematization of
the groundwater reservoir in the numerical model
The groundwater reservoir is subdivided in eighteen layers in the numerical model (Fig.
7.13). The Bassevelde Member which forms the lowest considered pervious layer, is
discretized in three layers in the numerical model. The thicknesses of the layers 1, 2, and
3 are respectively 6.1 m, 3.6 m and 1.8 m. These thicknesses decrease towards the
directly pumped layers to simulate as accurately as possible the vertical flow near the
pumped well. Layer 4 of the numerical model coincides with the semi-pervious layer
which is formed by the Watervliet Member.
Layer 5 up to and including 11 of the numerical model coincide with the Ruis-
broek Member. This member forms a pervious layer which is partially pumped. The
descritization of this member in the numerical model is in function of the location of the
screens of the pumped well and the observation wells. The lower part of the Ruisbroek
Member which is situated under the screen of the pumped well corresponds with layer 5
of the numerical model. Layer 6, 7 and 8 correspond with the directly pumped part of the
Ruisbroek Member. Layer 7 coincides with an important band in the Ruisbroek Member
which is rich in clay. The well screens of the observation wells SB1, SB3 and SB5 are
situated in layer 8 of the numerical model. Layer 8 coincides also with the layer of the
Ruisbroek Member with the smallest clay content. Layers 9, 10, and 11 of the numerical
model correspond with the upper part of the Ruisbroek Member which is situated above
the screen of the pumped well. The thicknesses of these layers decrease towards the
directly pumped layer and are chosen so that the screen of observation well SB2 is located
in the middle of layer 11.
The Belsele-Waas member, the lower more sandy part of the Boom Formation, is
subdivided in five layers, layer 12 up to and including 16 of the numerical model. The
screens of the observation wells DrBl and SB4 are located in the middle of layer 14 and
16. The layers 17 and 18 of the numerical model correspond with the Terhagen Member.
Layer 18 coincides with the upper part of the Terhagen Member with the largest clay
content.
7.4.5. Hydraulic parameters derived from observed drawdown
By the performance of sensitivity analyses and the study of the senSItIVity matrix, the
correlation matrix, and the variance-covariance matrix, the hydraulic parameters or
groups of hydraulic parameters are determined which can be identified by means of the
observations. By the lack of observations in the Bassevelde Member, it was not possible
to derive the hydraulic parameters of this member and of the Watervliet Member which
are located under the pumped layer. The used values for the hydraulic parameters of these
members are those which are determined in the double pumping test in Assenede (Lebbe
and Boughriba, 1995) in the same lithostratigraphical layer some 23 km west of the
investigation site. The assumed values for the Bassevelde Member are 1.4 mid for Kh(l-
3), 8.75xlO-2 mid for Kv(l-3) and 2x1o-s m- 1 for the specific elastic storage S,(1-3). For
the Watervliet Member following conductivities are assumed 4xlQ4 mid for Kv(4) and
342 Chapter 7 I Example of perfonnance and interpretation of pumping tests
6.4xlO- 3 mid for Kb(4). The same elasticity is assumed for the Watervliet Member as for
the Bassevelde Member or 2xIa-5 m- I for S,(4).
The hydraulic parameters of the Terhagen Member cannot be determined from the
observations. The assumed values for the horizontal conductivity Kb(17-I8) is 8xlO-4 mid,
lxlO-4 mid for the vertical conductivity Kv (17-18) and 0.002 for the storage coefficient
near the water table. As can be deduced from the results of the two post-optimizations
(see Sect. 7.4.6), these estimated values have a very small influence on the calculated
drawdown in the layers 8, 11, 14 and 16. Consequently, these values have also a very
small influence on the values of the parameters which can be deduced from the observa-
tions.
Seven groups of hydraulic parameters are derived from the observations. The first
group of derivable hydraulic parameters consists of the horizontal conductivity of the
lower part of the Ruisbroek Member or of the layers 5, 6, 7, and 8 of the numerical
model. In this group it is assumed that the horizontal conductivity of the layers 5 and 6
are the same, that the horiwntal conductivity of layer 7 (the clay rich band) is 10 times
smaller than the conductivity of layer 8 (the layer with the smallest clay content of the
Ruisbroek Member) and that this hydraulic conductivity is 1.333 times the horizontal
conductivities of the layers 5 and 6. These estimates of the mutual proportions of the
horizontal conductivity are based on the variation of the natural gamma radiation with
depth. Better estimates of the mutual proportions of the horizontal conductivities can be
made during pumping test with vertical flow logging in the pumped well as was demon-
strated by Gaus and Lebbe (1996). By these measurements of the vertical flow at different
levels in the screened part of the pumped well, it is possible to derive the discharge rates
delivered within the considered level intervals (Repsold, 1989). The mutual proportions of
the discharge rates are approximately equal to the mutual proportions of transmissivities
of the considered level intervals. Here, it is demonstrated how the soft information of the
natural gamma radiation or the harder information of the vertical flow logging can easily
be incorporated into the interpretation of a pumping test using an inverse numerical
model.
The second group of derivable hydraulic parameters are the specific elastic storage
of the Ruisbroek Member, S,(5-11). It is assumed that the elasticity is the same for all the
layers of this member. The third group of hydraulic parameters consists of the specific
elastic storages of the layers belonging to the Boom Formation, S,(12-18). Here, it also
assumed that the elasticities of all these layers are the same.
The following three groups concern the vertical conductivity between the different
layers in the numerical model. The fourth group of hydraulic parameters consists of the
hydraulic resistances c(12), c(13) and c(14). Based on the gradual upward increase of the
natural gamma ray, it is assumed that the vertical conductivity decreases gradually in the
upward direction. The vertical conductivity which determines c(14) is 1.1 times smaller
than the vertical conductivity which determines c(13) and 1.25 times smaller than the
vertical conductivity which determines c(12). The fifth group of hydraulic parameters
consists of the hydraulic resistances c(15) and c(16). Here, it is assumed that the vertical
conductivity which determines c(16) is 1.6 times smaller than the vertical conductivity
Chapter 7 / Example of petformance and interpretation of pumping tests 343
which determines c(15). The sixth group of hydraulic parameters consists of the hydraulic
resistances c(9), c(10) and c(U) in the upper part of the Ruisbroek Member above the
screen of the pumped well. Within this sixth group it is assumed that the vertical
conductivities are the same.
The seventh and last group is the ratio between the horizontal and the vertical
conductivity of the layers 5 up to and including 16. Here, it is assumed that the
transversal anisotropy of all these layers is the same. This is realised by assuming initially
a constant ratio between the horizontal and vertical conductivity of these layers. In the
first group of hydraulic parameters the hydraulic resistances are linked with the horizontal
conductivities. Because the hydraulic resistances are inversely proportional with the
conductivity of the layer, a negative sign is assigned to their parameter number (see also
Sect. 6.2.2). In the fourth, fifth and sixth group of hydraulic parameters the correspon-
ding horizontal conductivities are also considered with their negative parameter numbers.
By considering the hydraulic resistances of the first group and the horizontal conductivi-
ties of the fourth, fifth and sixth group in a seventh group, it is possible to deduce the
best average value for the ratio of the vertical to the horizontal conductivity of the
Ruisbroek and the Belsele-Waas Member. Within this seventh group a negative sign is
attached to the numbers of the horizontal conductivities whereas the numbers of the
hydraulic resistances are considered without negative sign.
7.4.6 Interpretation results
With the ordinary least square method a good fit is obtained between the observed and the
calculated drawdowns (Fig. 7.14). There is a particular good agreement between the
calculated and the observed drawdown in the pumped layer (layer 8). This agreement is
better than in the pumped layers in the Quaternary sediments (see Sect. 7.1.6). This is
due to the facts that the Tertiary sediments of Flanders are laterally more homogeneous
than the Quarternary sediments. A second reason is the large contrast between the
horizontal conductivities of the pumped and the adjacent semi-pervious layers. The
difference between the calculated and the observed drawdowns are rather large for only
one observation well. It is the well DrB 1, which is situated in the middle of the Belsele-
Waas Member and in layer 14 of the numerical model. These rather large residuals are
probably due to the long screen of the observation well (2 m), which is situated in a zone
with a relatively large vertical hydraulic gradient during the observation times.
The optimal values obtained by the ordinary least square method are represented in
Table 7.11. This table includes also the estimated values of the hydraulic parameter which
could not be deduced from the observations. The marginal and conditional standard
deviations of the deduced hydraulic parameters are represented in Table 7.12. In Table
7.13 the condition indexes and the matrix of the marginal variance decomposition of the
OLS solution are represented.
Studying these tables one can distinguish three different kinds of deduced hydraulic
parameters. The hydraulic parameters of the pumped pervious layer (Ruisbroek Member)
belong to the hydraulic parameters of the first kind. These parameters are the horizontal
344
_
Chapter 7 / Example oj petfonnance and interpretation oj pumping tests
... ...........
,
.00 ,oOt---\---t---j
..- . ,.-"~ ---+--+----j

,.- '.-.~--+---f--_j
,00
,00
"AlfDClWOI)
,.' ,.0 103 rUlE '"III
LAYER 16
,00
-........
loG ,., 102 OIlTMCE "0
, !
INxx
,.-,
.
, / ,.- ~
,.0
7 ~1T=iOTtiiii1
,.'
,00
'.' I~ TltEUUl1
LAYER 14
100
,"'j.----i'-'-""'<--".__---l
10- 1
,.-
, .. 102 103 rll.lN(11
LAYER
,00
,.-
,.-
,00 I~ TlttEUnl' toO
LAYER 8
Fig. 7.14. Observed (crosses) and calculated (solid lines) drawdown in time- and
distance-drawdown graphs for pumping test of Kemzeke (Q(6)=53.48 m3/d, Q(7)= 9.36
m3/d and Q(8)= 160.45 m3/d) according to the OLS solution
conductivities of the lower directly pumped part of the Ruisbroek Member ~(5-6), Kb(7)
and Kh(8), the specific elastic storage of the Ruisbroek Member S,(5-11) and the hydraulic
resistance in the upper part of the Ruisbroek Member c(9), c(10) and c(ll). Their
marginal and conditional standard deviations are small. The absolute value of their partial
correlation coefficients with respect to the other hydraulic parameters ranges between
0.0071 and 0.6582.
The deduced hydraulic parameters of the second kind are the hydraulic parameters
of the Belsele Member, the covering semi-pervious layer where the vertical hydraulic
gradient was observed. These hydraulic parameters show a large correlation between each
other. They are the hydraulic resistances c(12), c(13) and c(14) of the lower part of the
Belsele-Waas Member, the hydraulic resistances c(15) and c(16) in the upper part of the
same member and the specific elastic storages of this member S,(12-18). Because of these
large mutual correlations between their sensitivities, there is a large difference between
their conditional and marginal standard deviations. The partial correlation coefficient
between the parameters c(12), c(13), c(14) and c(15), c(16) is 0.9250, between c(12),
c(13), c(14) and S,(12-18) is -0.9484, and between c(15), c(16) and S,(12-18) is -0.9879.
These large correlation results also in one large condition index (25.6). The marginal
standard deviations of these hydraulic parameters are rather large. As can be seen from
Table 7.13 their marginal variance can be explained for more than 94 % by this largest
condition index.
The last deduced hydraulic parameter, the mean ratio between the horizontal and
the vertical conductivity of the Ruisbroek and the Belsele-Waas Member, Kb(5-16) is a
special case. The marginal and the conditional standard deviation are both large. The
marginal variance of this hydraulic parameter can be attributed for 99.85 % of the second
largest condition index (14.0). This is caused by rather small sensitivities of the observa-
tion with respect to this hydraulic parameters. However, this rather small sensitivities
show no correlation with sensitivities with respect to the other parameters.
Two post-optimization were performed. In the first post-optimization the conduc-
tivities of the Terhagen Member are divided by a factor l(f.5 or 3.162 along with the
storage coefficient near the watertable or the specific yield. In the second post-optimiza-
tion the values of the same parameters used during the first optimization are multiplied by
a factor of 3.162. In Table 7.12 the results of the two post-optimizations are represented
with the first optimization. Comparing these three solutions one can remark that the
differences between the optimal values and the standard deviations of the hydraulic
parameters of the Ruisbroek Member are very small. The differences between the optimal
values and the standard deviations of the hydraulic parameter of the Belsele-Waas
Member are somewhat larger. The largest differences concern the optimal values of the
hydraulic resistance in the upper part of the Belsele-Waas Member. However, the
deduced value of this last mentioned parameter differs only with a factor of 1.06 for the
optimization with the maximum values of the hydraulic parameters of the Terhagen
Member and a factor of 1.02 with the minimum values.
w
01
""
Table 7.11. Optimal values of deduced hydraulic parameters with OLS method from observations (underlined) and estimated
values of other hydraulic parameters which were not deduced with inverse model (not underlined) for single pumping test
at Kemzeke.
~
RADIUS OF WELLSCREEN, R, IN M, ------- 0.100 - NUMBER OF HYDRAULIC PARAMETER -INRI- HYDR. RESISTANCE, c(IO), IN DAY, ------/28/- ~ i..
INITIAL TIME, n, IN MIN, --------------- 0.100 HYDR. CONDUCTIVITY, K,.(I), IN MlDAY, --I 11- 1.4000 HYDR. RESISTANCE, c(11), IN DAY, -----/291- ~ .....
LOGARlTIIMIC INCREASE OF TIME AND HYDR. CONDUCTIVITY, K,.(2), IN M/DAY, --I 2/- 1.4000 HYDR. RESISTANCE, c(12), IN DAY, -------/301- 4610.2 "-
OF RADll OF RINGS, LOG A, ------------ 0:100 HYDR. CONDUCTIVITY, K,.(3), IN M/DAY, --I 3/- 1.4000 HYDR. RESISTANCE, c(13), IN DAY, ------/311- 5119.8 ~
LATEST CALCULATED TIME, T2, IN MIN, --- 0.100 HYDR. CONDUCTIVITY, K,.(4), IN MlDAY, -I 4/- .00600 HYDR. RESISTANCE, c(14), IN DAY, ------132/- lliU
NUMBER OF LAYERS, N, ----------- 18 HYDR. CONDUCTIVITY, K,.(5), IN MlDAY, -I 51- b.2lli HYDR. RESISTANCE, c(15), IN DAY, ------/331- 9182.5 I
NUMBER OF RINGS, M, -------------- 48 HYDR. CONDUCTIVITY,K,.(6), INMlDAY, --I 6/- b.2lli HYDR. RESISTANCE, c(16), IN DAY, ------/34/- 45370.
~
TIlICKNESS OF TIlE SUCCESSNE LAYERS HYDR. CONDUCTIVITY, K,.(7), IN MIDAY, --I 7/- ~ HYDR. RESISTANCE, c(17), IN DAY, -----/351- 104750.
NUMBERED FROM LOWER TO UPPER HYDR. CONDUCTIVITY,K,.(8), IN M/DAY, --I 8/- 3.3755 SPEC. ELAS. STORAGE, S,(I), IN M", ----/36/- 2.ooE-05 ~o
TIlICKNESS OF LAYER I, IN M, ----------- 6.100 HYDR. CONDUCTIVITY, K,.(9), IN M/DAY, --I 9/- .16892 SPEC. ELAS. STORAGE, S,(2), IN M", ----/37/- 2.ooE-05
TIlICKNESS OF LAYER 2, IN M, ----------- 3.600 HYDR. CONDUCTIVITY, K,.(IO), IN M/DAY, -/10/- .16892 SPEC. ELAS. STORAGE, S,(3), IN M", ---/38/- 2.00E-05
TIlICKNESS OF LAYER 3, IN M, ---------- 1.800 HYDR. CONDUCTIVITY, K,.(l1), IN M/DAY, -/1lI- .16892 SPEC. ELAS. STORAGE, S,(4), IN M", --/391- 2.ooE-05
j
TIlICKNESS OF LAYER 4, IN M, ------------ 3.500 HYDR. CONDUCTIVITY, K,.(12), IN MlDAY, -112/- .00537 SPEC. ELAS. STORAGE, S,(5), IN M", ----/40/- 4. 14E-05 "
TIlICKNESS OF LAYER 5, IN M, ------------- 1.500 HYDR. CONDUCTIVITY, K,.(13), IN MIDAY, -/13/- .00488 SPEC. ELAS. STORAGE, S,(6), IN M", -----/411- ~ 1
TIlICKNESS OF LAYER 6, IN M, ------------- 2.000 HYDR. CONDUCTIVITY, K,.(14), IN M/DAY, -114/- .00429 SPEC. ELAS. STORAGE, S,(7), IN M", -----/42/- 4.14E-05
TIlICKNESS OF LAYER 7, IN M, ------------ 2.500 HYDR. CONDUCTIVITY, K,.(15), IN MIDAY, -115/- .00372 SPEC. ELAS. STORAGE, S,(8), IN M", ----/431- ~
i'
TIlICKNESS OF LAYER 8, IN M, -------------- 4.500 HYDR. CONDUCTIVITY, K,.(16), IN M/DAY, -116/- .00232 SPEC. ELAS. STORAGE, S,(9), IN M", ---/44/- 4.14E-05 ~
TIlICKNESS OF LAYER 9, IN M, ------------ 1.000 HYDR. CONDUCTIVITY, K,.(17), IN MIDAY, -117/- .00098 SPEC. ELAS. STORAGE, S,(10), IN M", ---/45/-4.14E-05 i
TIlICKNESS OF LAYER 10, IN M, --------------1.800 HYDR. CONDUCTIVITY, K,.(18), IN M/DAY, -118/- .00098 SPEC. ELAS. STORAGE, S,(l1), IN M", -----/46/-4.14E-05
TIlICKNESS OF LAYER II, IN M, ------------- 2.900 HYDR. RESISTANCE, c(I), IN DAY, -------119/- 55.400 SPEC. ELAS. STORAGE, S,(12), IN M", ---/47/-1.36E-05 t·
TIlICKNESS OF LAYER 12, IN M, -------------- 2.300 HYDR. RESISTANCE, c(2), IN DAY, ------/20/- 30.900 SPEC. ELAS. STORAGE, S,(13), INM", ---/48/-1.36E-05 ~
TIlICKNESS OF LAYER 13, IN M, -------- 2.300 HYDR. RESISTANCE, c(3), IN DAY, ---------/211- 4375.0 SPEC. ELAS. STORAGE, S,(14), IN M", --/49/-1.36E-05 "='
TIlICKNESS OF LAYER 14, IN M, --------------2.300 HYDR. RESISTANCE, c(4), IN DAY, ----/22/- ~ SPEC. ELAS. STORAGE, S,(15), IN M', ----/50/-1.36E-05
TIlICKNESS OF LAYER IS, IN M, ------------- 2.400 HYDR. RESISTANCE, c(5), IN DAY, --------/23/- llJ.2Q SPEC. ELAS. STORAGE, S,(16), IN M", --/511-1.36E-05
~~.
TIlICKNESS OF LAYER 16, IN M, ------------2.500 HYDR. RESISTANCE, c(6), IN DAY, --------/24/- 64.820 SPEC. ELAS. STORAGE, S,(17), IN M", -----/52/-1.36E-05
~
TIlICKNESS OF LAYER 17, IN M, ------------7.400 HYDR. RESISTANCE, c(7), IN DAY, -----------/25/- 64.820 SPEC. ELAS. STORAGE, S,(18), IN M", -----/53/-1.36E-05
TIlICKNESS OF LAYER 18, IN M, --------------- 6.700 HYDR. RESISTANCE, c(8), IN DAY, --------/26/- 43.400 SPECIFIC YIELD, So, ------------/ 54/- 0.002000
&
HYDR. RESISTANCE, c(9), IN DAY, -----------/27/- 88.380
Chapter 7 / Example of petfonnance and interpretation of pumping tests 347
Table 7.12. Results of the interpretation with the ordinary least square method of the
single pumping test at Kemzeke and the two post-optimizations
Hydraulic With minimrun values of the With estimated values for the With maximal values for
parameters unidentifiable parameters unidentifiable parameters the
unidentifiable parameters
Symbol Unit Optimal Sm s, Optimal Sm S, Optimal 8m s,

value value value
Kh(5-ti) mid 2.534 0.0079 0.0064 2.532 0.0079 0.0065 2.526 0.0079 0.0067
Kh(7) 0.253 0.253 0.253
Kh(8) 3.378 3.375 3.369
S,(12-18) m- l 1.35xlO""' 0.0439 0.0030 1.36xl0" 0.0437 0.0030 1.39xlO"" 0.0432 0.0030
S,(5-11) m- l 4.14xI0-' 0.0069 0.0053 4.14xI0-' 0.0069 0.0054 4. 14x\(j' 0.0069 0.0058
c(12) d 2301. 0.0443 0.0049 2282. 0.0440 0.0049 2233. 0.0433 0.0051
c(13) 4648. 4610. 4510.
c(14) 5161. 5119. 5009.
c(9) d 43.37 0.0136 0.0115 43.40 0.0136 0.0115 43.47 0.0136 0.0116
c(lO) 88.33 88.38 88.52
c(ll) 148.2 148.3 148.5
c(15) d 5714. 0.0456 0.0182 5597. 0.0488 0.0183 5274. 0.0425 0.0183
c(16) 9373. 9182. 8652.
Kh/K,,(5-16) dim.1ess 10.67 0.0408 0.0173 10.66 0.0408 0.0174 10.64 0.0410 0.0177
Table 7.13. Condition indexes 1/ and n matrix of marginal variance decomposition

proportions corresponding with the ordinary least square solution of the single pumping
test at Kemzeke
"Parameter
1.0 1.3 1.7 4.2 4.5 14.0 25.6
Kh(5-6),Kh(7),Kh(8) 0.0813 0.0934 0.0126 0.0005 0.0001 0.4157 0.3964

S,(12-18) 0.0012 0.0023 0.0003 0.0073 0.0002 0.0012 0.9874
S,(5-11) 0.0000 0.0265 0.4762 0.0077 0.1150 0.2871 0.0876
c(12),c(13),c(14) 0.0006 0.0020 0.0004 0.0201 0.0009 0.0025 0.9730
c(9),c(1O),c(ll) 0.0000 0.0002 0.0042 0.0500 0.8643 0.0061 0.0752
c(15),c(16) 0.0000 0.0001 0.0000 0.0464 0.0046 0.0030 0.9458
Kh/K.,(5-16) 0.0001 0.0000 0.0001 0.0004 0.0000 0.9985 0.0009
348 Chapter 7 I Example of performance and interpretation of pumping tests
7.5 Artificial recharge test in a natural bare dune valley

in the Flemish coastal plain
The inverse model can not only be applied in the interpretation of single and multiple
pumping test but can also be used for the interpretation of infiltration test and injection
test. In this last example an interpretation of an infiltration test is made. This test was
executed as a part of a preliminary studies for optimizing the water supply in the Western
Flemish Coastal Plain (Lebbe et aI., 1995).
7.5.1 Lithological cross section and discretization of groundwater reservoir
The lithostratigraphical cross section of the recharge area is given in Fig. 7.15. The
considered groundwater reservoir is bounded below by 120 m thick layer consisting
mainly of clay (Kortrijk Formation). The top of this layer occurs at -20. The geometrical,
lithological and hydrogeological characteristics of the overlaying Quaternary deposits are:
- from -20.0 to +3.00 medium to fine and sand forming a lower pervious layer,
- from +3.00 to +4.00 silt-bearing fine sand forming a semi-pervious layer,
- from +4.00 to +8.00 well-sorted medium to fine dune sands forming the upper
pervious layer. The lower pervious layer is discretized in five layers (layer 1 up to and
including 5), with a decreasing thickness in the upward direction. Layer 6 of the
numerical model coincides with the silt-bearing fine sands. The upper three layers (7, 8,
and 9) represent the dune sands.
7.5.2 Location of observation wells and peiformance of artificial recharge test
The artificial recharge test was performed from 6 January 1992 to 26 January 1992.
Water was infiltrated in a natural bare dune valley of the water catchment of Koksijde.
The surface of the recharge area was left unmodified, so that the recharge area has no
regular geometric boundaries. The water which was pumped towards the recharge pond
was measured by means of a counter. At the start of the test, several readings per day
were made; at the end, only one reading per day was performed. The recharge rate
fluctuates around 1798 m3/d.
Four observation wells were installed (Fig. 7.15). Two observation wells, PI and
P2, are at the edge of the recharge ponds, and two observation wells, P3 and P4, are
about 140 m from the recharge pond. The wells PI and P3 have screens a few meters
below the semi-pervious layer and P2 and P4 have screens in the thin semi-pervious
layer. During the first two weeks of the test, hydraulic heads in wells PI and P2 were
measured by means of pressure transducers connected to a datalogger. In observation
wells P3 and P4, hydraulic head was measured daily. After the removal of the pressure
transducers in observation wells PI and P2, measurements were made daily.
For the interpretation different simplifying assumptions are made. So, it was
assumed that the recharge pond has a circular shape. The screens of the well P2 and PI
are located in layer 6 and 4 at 16 m from the centre of the recharge pond. The screens of
P3 and P4 are located in layer 4 at 147 m from the same centre.
HYD "OOIOLOQ IC SICTIOH
A
. ...,
--
P2' .. ,
.I I
·8
.4
.-
•
0
-
•4
·s
.........
.2
IS
~ ~_"'": <XXX.X
~ "'_Ino"'"
Fig. 7.15. Hydrogeologic cross section, location of observation wells of artificial

recharge tests in the Koksijde dune area and schematization of the groundwater reservoir
in the numerical model
7.5.3 Hydraulic parameters derived from observed rises of hydraulic head
Four hydraulic parameters or groups of hydraulic parameters were determined: the

horizontal conductivity of layers 1 to 5 and 7 to 9, indicated as K.,(1-5,7-9); the specific
yield or the storage coefficient near the water table So; the hydraulic resistance between
layers 5 and 6, indicated by c(5); the hydraulic resistance between the layers 6 and 7;
indicated by c(6). The other hydraulic parameters have a negligible influence on the
observed hydraulic-head changes. Because they are not readily identifiable, values for
these hydraulic parameters were estimated.
During the first interpretation phase, uniform weights equal to one were given to
all observed heads; the ordinary least square (OLS) method was applied. The values of
the hydraulic parameters were derived by minimization of the sum of the squared
residuals . Because the observed rises are of a different order of magnitude, such as the
drawdowns in a pumping test, it is also here suitable to define the residuals as the
difference between the logarithms of the calculated and the observed rises of the hydraulic
head. During the study of the residuals of the OLS solution, it was revealed that this
distribution shows some outliers. Therefore, a second interpretation phase was under-
taken, in which the BWLS method was applied. After optimization with the BWLS
method, the weighted residuals showed a normal distribution.
The parameters values and their respective confidence factors of the OLS and
BWLS solution are given in Table 7.14. These two solutions are given to illustrate which
of the estimated parameters is the most influenced by the outliers. The optimal values of
the hydraulic resistance c(6) obtained by both solutions show the largest difference. The
other differences between the deduced values are small.
In Fig. 7.16, the observed and calculated rise of the hydraulic head are repre-
sented in relation to the start of the recharge test and in relation to the distance from the
centre of the recharge canal according to the BWLS solution. This figure shows that the
difference between the observed and calculated rises are rather small for the two observa-
tiQn wells close to the recharge site and after 300 min of recharge until the end of the
recharge test. The largest residuals correspond with the first three daily observations in
the well at 140 m from the recharge site.
Table 7.14. Values of the hydraulic parameters deduced from the artificial recharge test
and their respective conditional and marginal confidence standard deviations, Sc and Sm,
for the ordinary (OLS) and the biweighted (BWLS) least square method
Hydraulic parameters OLS solution BWLS solution

Symbol Units Value Sc Sm Value Sc Sm
Kh(I-5,7-9) mid 10.7 0.0014 0.0027 11.1 0.0005 0.0019

c(6) d 11.7 0.0036 0.0068 8.21 0.0028 0.0130
So m 3 /m 3 0.0270 0.0081 0.0133 0.0329 0.0016 0.0104
c(5) d 2.38 0.0076 0.0140 2.04 0.0039 0.0072
Kh(6) mid 0.40 - - 0.40 - -
K.(I-5,7-9) mid 9.0 - - 9.0 - -
S,(1-9) m· l 8.10-4 - - 8.10-4 - -
Chapter 7 / Example of peiformance and interpretation of pumping tests 351
10 1
IW'" !
10 0
~ i
,
V
X
I
!
E I
I 10- 1 T=10 MIN
-0 10- 1
m
..c: ,
IT=10 3MINI
T=10 MIN
"i ,
.~ 102 103 104 10 5 10 0 101

::;
~ LAYERS
"0
>.
..c:
'0
Q)
U) 101'~------4--------+------~
a:
103 104 10 5 100 10 1 10 2

Time, min LAYER 4 Distance, m
Fig. 7.16. Measured (crosses) and calculated (continuous curves) rise of hydraulic head in
relation to time since start of recharge test and in relation to distance from the centre of
recharge pond, according to the BWLS solution
REFERENCES
Belsley, D.A., 1990, Conditioning diagnostics, collinearity and weak data in regression:
John Wiley & Sons, New York, 292p.
Carrera, J., and Neuman, S.P., 1986, Estimation of aquifer parameters under transient
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Draper, N.R., and Smith, H.E., 1981, Applied regression analysis, second edition: John
Wiley & Sons, New York, 709p.
Gaus, I., and Lebbe, L., 1996, Use of geophysical borehole parameters during the
interpretation of a double pumping test at Ronse (in Dutch): Water, nr. 86, p. 3-11.
Hantush, M.S., 1966, Analysis of data from pumping tests in anisotropic aquifers:
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Herweijer, J. e., 1997, Sedimentary heterogeneity and flow towards a well. Assessment
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Hill, M.e., 1989, Analysis of the accuracy of approximate, simultaneous, nonlinear

confidence intervals on hydraulic heads in analytical and numerical test cases: Water
Lebbe, L., 1988, Execution of pumping tests and interpretation by means of an inverse
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563 p.
Lebbe, L., and Boughriba, M., 1995, Interpretation of a double pumping test by means
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53-63.
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Index
accuracy of observed drawdown, 264 condition number, 255

adjustment factors, 245 condition index, 5, 255, 256, 309, 319, 325,
alternating direction implicit method, 131-132 326, 343, 345, 347
analytical models of pumping test, 6, 55 conditional confidence interval, 252, 307
angle () indicating max. hor. conductivity, 184, conditional standard deviation, 5, 252, 307, 318,
187, 235 325, 333, 334
anisotropy, 14, 184 conditional confidence factor, 253
aquiclude, 25 conditional standard deviation, 347, 350
aquifer loss, 190 conditional confidence (3D) space, 254
aquifer, 24 conditional confidence (2D) area, 254
aquitard, 25 confidence intervals:
Archie formula, 35 - confidence band, 277
artifical recharge test, 1,7,303,348-351 - confidence limits, 277
- site: Koksijde, 348-351 - individual confidence intervals of simulated
AS2D numerical model, 3, 117 drawdowns, 5, 7, 229, 277-284,269-299
axi-symmetric two-dimensional model, 3, 117 - simultaneous confidence band, 277
- simultaneous confidence interval, 277
- upper and lower bound, 281
Bassevelde Member, 338-347 confined aquifer, 25
Bdsele-Waas Member, 338-347 consistent data, 327
biweighted least square (BWLS) method 246, Cooper-Jacob interpretation method, 65-67
307, 331, 337, 338, 350
Boom Formation 338-347
Boulton delay index 49, 103 Darcy velocity, 9
boundaries: Darcy's law:
- vertical and forming an edge, 208-210 - simple, 9,
- vertical and forming a right angle, 208 - generalized, 19
- vertical and parallel, 207 De Glee interpretation method, 75-77, 82
- vertical and straight, 207-210 dependencies:
Brabant Massif, 310-327 - weak, 255
- strong, 255
diffusivity, 44, 45, 235
C-value of well loss, 191, 235 discharge rate magnitude, 264
central limit theorem, 232 discretization, 231, 256, 257, 306, 312, 329,
chalk, 310 341, 348
characteristics of joints, 35 double pumping test, 7, 303,303-310,310-327
clayey sediments, 338-347 drainable porosity, 50
coefficient of consolidation, 44 drainage curve, 47
collinear diagnostic, 5, 252, 255, 256 drawdown: 57
compressibility: - due to multiple well fields, 6, 194
- fluid, 40 - in pumping wells, 190
- matrix, 42 - observation distance, 264
compression test, 45 - observation time, 264
computational errors, 231 - acuracy of observation, 264
conceptual errors, 231 drying curve, 47
conceptualization of flow, 6, 261-263,286 dune valley, 348-351
356 Index
effective horizontal conductivity, 184 flow conceptualization, 6, 261-263,286

effective grain size, 30 forward problem, 229
effective porosity, 10 fractured rocks, 7, 35, 184, 310-327
Egem Member, 327-338 fragmented or separate analyses of drawdown, 2,
eigenvalues of variance-covariance matrix, 5, 113, 261
249, 256, 319 fresh water head, 10, 12
eigenvectors of variance-covariance matrix, 5,
249,256, 319
elastic storage coefficient, 44 gamma ray, 32
electrical resistivity, 34 Gauss distribution, 232
elevation head, 11 geophysical data: 256
equivalent effective conductivity, 18 - electrical resistivity, 34
equivalent horizontal conductivity, 15 - gamma ray, 32
equivalent anisotropy, 18 - Humble formula, 35
equivalent vertical conductivity, 17 - Repsold relation, 32
Ernst formula, 30 Ghent, 327-338
errors: global minimum, 323, 324
- measurement, 231 grain size distribution, 30
- computational, 231 grid parameters, 117, 118, 135-141, 145, 163,
- conceptual, 231 164, 166
Hantush-l interpretation method, 78-79

field capacity, 50 Hantush-II interpretation method, 80-81
file: Hantush-Jacob interpretation method, 77
- file name1.dap(u) 168-169, 187, 192,235, Hazen formula, 30
268, 269, 274, 275, 277, 282, 283, 284 head:
- file name1.dapu.result, 173 - elevation head, 11
- file name1.dar, 173 - fresh water head, 10, 12
- file name1.d(p)pc, 269, 270, 274, 275, 276, - hydraulic head, 10
277, 282, 283, 284 - pressure head, 11
- file name1.eta(dfdp), 233, 235, 268, 271, 273, - velocity head, 11
274, 275, 277, 282, 283, 284 - suction head, 46
- file name1.lss, 268 - tension head, 47
- file name1.lst, 170, 173 Hessian matrix J'J, 270, 275
- file name1.ml(p)l, 195-196 heterogeneity, 13
- file name1.ml(p)2, 195-196 Humble formula, 35
- file name1.oup(u), 170, 173, 195 hydraulic conductivity: 9
- file name1.pap(u), 164-165, 166-167, 178, - tensor, 20
187, 192, 235, 268, 269, 270, 274, - ellipsoid, 21
275, 276, 277, 282, 283, 284 - secondary, 310
- file name1.papu.resinv, 268,269, 270 hydraulic gradient, 9
- file name1.par, 268, 269, 270 hydraulic head, 10
- file name1.pas(e), 233, 234, 235, 268, 269, hydraulic resistance, 26, 221, 234
274,275,276,277,282,283,284 hypothetical problem, 257, 284
- file name1.prl(n), 274, 276, 282, 283
- file name1.pr3(n), 275, 283
- file name1.solpuS.lst, 268 identifiability, 5, 246-247
- file name1.susqpr.inp, 282,283 leper Group, 310
- file name1.s(s)ql, 274, 276, 277, 282 III-conditioned problem, 322
- file name1.s(s)q3, 275, 283 imbibition curve, 47
- file name1.s(u)sq, 276, 283 "impervious" layer, 24
- file plot.plt, 272, 274, 275, 276, 277 inadequate data, 322
finite-difference grid, 117-119 inclusive standard deviation, 32
Flemish coastal plain, 348-351 instability, 5, 246-247
Index 357
interquartile range (IQR), 246 marginal confidence factor, 253

intrinsic permeability, 12 marginal confidence interval, 251, 307
inverse problem of type 1, 229 marginal standard deviation, 5, 251, 307, 318,
inverse problem of type 2, 229 325,333,334, 347, 350
inverse problem of type 3, 229 marginal variance, 255
inverse problem of type 4, 229 mass conservation equation, 36, 39
inverse models: matrix of marginal variance-decomposition
unsteady state flow, 7, 51 proportions, 5, 255, 256, 309, 319, 325,
326, 343, 347
mean actna! velocity, 9
joint confidence region: 5, 249-255, 256 mean drawdowns, 119-125
- approximated by p-dim. ellipsoid, 249 measurement errors, 231
- approximated by linearization method, 254 Merelbeke Member, 327-338
- exact, 254 Mesozoic age, 310
model of Neuman and Witherspoon, 104-109,
112-113
Kemzeke, 338-347 model of Boulton, 6, 99-104, 111-112, 157,236
Koksijde, 348-351 model of Hantush, 6, 83-95,110-111,145,236
Konzeny-Carman relation, 34 model of Boulton-Cooley, 6,99-104,111-112,
Kortemark Member, 327-338 157,236
Kortemark, 310-327 model of Hantush-Weeks, 6, 95-99, 154,
Kortrijk Formation, 310, 327, 348 187-190, 192-194, 236
model of Jacob-Hantush, 6, 67-83, 110, 141,
236,262
Landen Formation, 310 model of Thiem, 55-59, 110
Laplace's equation, 38 model of Theis, 6, 59-67, 110, 134, 236, 261
lateral anisotropy 184, 187, 235, 310-327 Mohr's circle, 20
lateral bounds: 203 moisture content, 47
- constant head boundary, 205 Monte Carlo method, 278
- discontinuous conductivity change, 6, 211-218 multiple pumping test, 4, 7, 267, 268, 303, 337
- impervious boundary, 203
laterally anisotropic aquifer, 6, 7, 184-190,
310-327 negative side of vertical plane, 211-212
leakage factor, 26 non-uniqueness, 5, 246-247
least square method: nonlinear regression, 245
- OLS method, 246, 286, 306, 307, 309, 331, normal distribution, 232, 233, 248, 249, 257,
341,343,347,350 267,270-271,279,286,287,290
- BWLS method, 246, 307, 331, 337, 338, 350 numbering of hydraulic parameters, 234-235
- weighted least square (WLS) method, 246
lithostratigraphical cross-section, 304, 305, 310,
313, 327, 328, 338, 340, 348, 349 observation well, 10, 256
lithostratigraphical data: 256 observation time of drawdown, 264
- effective grain size, 30 observation distance of drawdown, 264
- Ernst formula, 30 observation well: 10, 256
- grain size distribution, 30 - location well screen, 304, 305, 312, 313, 328,
- Hazen formula, 30 329, 339, 340, 348, 349
- inclusive standard deviation, 32 - with long screen, 231
- Kozeny-Carman relation, 34 Onderdijke Member, 339, 340
- specific surface, 31 optimal values, 245, 307, 314, 318, 325, 333,
- Vander Gun relation, 46 334, 346, 350
local tuiuima, 322, 323, 324
lower quartile, 246 ordinary least square (OLS) method, 246, 286,
306, 307, 309, 331, 341, 343, 347, 350
outliers, 246, 331, 334-335, 337, 338, 350
overparametrized, 322
358 Index
packers, 231 pumping wells:

Palaeocene age, 310 - C-value of well loss 191, 235
Palaeozoic age, 310 - drawdown 6, 190
panunererization, 256, 306, 312, 330, 341, 349 - location well (screen) 304, 305, 312, 313, 328,
partial correlation coefficients, 5, 251, 256, 319, 329, 339, 340
326, 330, 345 - partially penetrating 67, 98, 111, 154, 193,
partially penetrating pumping wells, 67, 97, 98, 261,341
111, 154, 193,261,341 - power N of well loss 191, 235
permeamerer, 27 - storage 4, 60, 134, 178
pervious layer, 24 - well loss 190, 235
pF-curves, 49 - well efficiency, 191
phreatic aquifer, 25 pumping rest sire:
phyllires, 310 - Ghent, 327-338
Pittem Member, 327-328 - Kemzeke, 338-347
positive side of vertical plane, 211-212 - Kortemark, 310-327
post-optimization process, 247, 257, 324, 325, - ScheIdt Valley, 303-310
342, 345, 347
power N of well loss, 191, 235
pressure head, 11 Quarernary sediments, 303,307, 309, 310, 311,
probability density function of hydraulic head, 312, 343, 348
278
program etabdi, 267, 270-272, 287
program intinp, 162, 164-165, 168, 178,233 recharge rest, 1, 7, 303, 348-351
program inpurS, 233, 234, 267, 269-270, 282, recovery rest, 175
283, 284, 286 Repsold relation, 32
program mulp12, 198 residual drawdown, 175
program multpl, 194-202, 219, 224, 226 residuals, 230-232, 248
program outpuS, 162, 172-173,219,222,226, rocks of Paleaozoic and Mesozoic age, 310-327
282,283,285,287 Ruisbroek Member, 338-347
program package contil, 281, 282, 296 rule of superposition, 176, 194
program package contill, 281, 282, 297
program package conf"J3, 281, 283, 298
program package conti4, 281, 284, 298, 299 S-shaped tirne-drawdown curve, 104, 110, 111
program ple103, 267, 275, 283 scanning curves, 47
program plprcr, 267, 272-273, 273, 274, 276, ScheIdt Valley, 303-310
277, 288, 290 schernatization of groundwarer reservoir, 256
program plssq3, 267 semi-confined aquifer, 25, 82
program plssql, 267, 274 semi-pervious layer, 24
program plsus3, 275, 283 sensitivity factor, 232, 234, 235
program plsusd, 268, 275, 278 sensitivity matrix: 232-244, 256
program plsusq, 268, 274, 275, 276, 282, 283 - finire-difference approximation, 232
program sidapS, 162, 173,221 - graphical representation, 236, 241-244
program sipurS, 162, 170-172, 194, 197 - numerical representation, 237-241
program solpuS, 235, 267, 268-269, 282, 283 separare or fragmented analyses of drawdown, 2,
program sumsq2, 268, 276-277, 278, 293, 294, 113,261
295 Silurian age, 310
program sumsqr, 268,276,277,283,292 simultaneous inrerpretation of drawdown, 7, 303
program susqI3, 267, 274-275, 283, 290, 291, single pumping rest, 4, 7, 303, 338-347
298 singular value of sensitivity matrix, 255
program susqlo, 267, 273-274, 276, 282, 290 specific yield, 36
pumping rest, 29 specific elastic storage, 36, 39, 234
Index 359
specific surface, 31 well characteristics: 191

specific storage, 39 - well efficiency, 191
stability, 5, 246-247 - well loss, 190, 235
step-drawdown pumping test, 29 - C-value of well loss 191, 235
storage coefficient near the water table, 36, 46, - power N of well loss 191, 235
49, 234 well efficiency, 191
subsidence, 7, 218-227 well loss, 190, 235
suction head, 46 well-posed inverse problem, 229
synthetic problem, 257, 284 well test, 28
wetting curve, 47
tensiometer, 47
tension head, 47 Ypresian clay, 303
Terhagen Member, 338-347
Tertiary sediments, 303, 327-338, 338-347
Theis interpretation method, 65, 67 Zelzate Formation, 338-347
transmissivity, 25
triple pumping test, 7, 303, 348-351
Turonian age, 310
two-dimensional axi-symmetric model, 117
unconfined aquifer, 25
unidentifiability, 5, 246-247
uniqueness, 5, 246-247
upper quartile, 246
upscaling, 18
validation of inverse numerical model, 257-260

Van der Gun relation, 46
variable discharge rate, 4, 6, 175-183
velocity head, 11
velocity:
- Darcy velocity, 9
- mean actual velocity, 9
verification of AS2D numerical model: 133,
161-162
- with Boulton-Cooley, 157
- with Hantush model, 145
- with Hantush-Weeks model, 154
- with Jacob-Hantush model, 141
- with Theis model, 134
verification of iteration process, 133
vertical boundaries, 207
Walton interpretation method, 81-82

water table, 46
Watervliet Member, 338-347
weight in BWLS method, 246
weighted least square (WLS) method, 246
well bore storage 231

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Hydraulic Parameter Identification

Generalized Interpretation Method for Single and Multiple Pumping Tests

With 112 Figures and 34 Tables

ISBN-13: 978'3.642.64264-7 Springer. Verlag Berlin Heidelberg New York

C Springer.Veriag Berlin Heidelberg 1999

Production: ProduServ GmbH Verla8$service, Berlin

This book is intended for all hydrogeologists or geohydrologists, environmental and

The executable codes are available on http://allserv.rug.ac.be/-luclebbe. The FORTRAN

Chapter 1 / Introduction .................................................................................... .

Chapter 2 / Hydraulic Parameters .............. ......................................................... 9

Chapter 3 / Evolution of analytical models of pumping tests

5.5 Drawdown in groundwater reservoir with lateral bounds ..................................... 203

6.7 Confidence interval for optimal estimated drawdown .......................................... 277

Chapter 7 / Example of performance and interpretation of pumping tests........... 303

7.3 Triple pumping test in layered groundwater reservoir formed by

1.1 Previous literature on pumping test interpretation

1.2 Proposed generalized interpretation method

In the present work, which is a sequel to Lebbe (1988), a two-dimensional axi-symmetric

The interpretation of a properly planned pumping test can be considered as a rather

1.4 Arrangement of subject matter

Hill, M.C., 1989, Analysis of the accuracy of approximate, simultaneous, nonlinear

2.1 Hydraulic parameters describing water conducting properties

2.1.1 Darcy's law

where n., is the effective porosity (UL-3).

A. Cross secllon ~-:-'-_-'--'-L.:::.Z=O

2.1.2 Hydraulic head andfresh water head

hf=h z+ (1 t-d-hz) (pg) obs (2.3)

where hf is the fresh water head (L),

2.1.3 Intrinsic permeability

The hydraulic conductivity, the constant of proportionality in Darcy's law, is a function

where K is the hydraulic conductivity (LTl),

2.1.4 Heterogeneity and anisotropy of geological formations with respect to hydraulic

The hydraulic conductivities of a geological . formation usually show varIation through

A geological formation is heterogeneous if the hydraulic conductivity depends on the

There are probably as many types of heterogeneous configuration as there are

A quantitative description of the degree of heterogeneity in geological formation is

A geological formation is isotropic at a point if the hydraulic conductivity K is indepen-

Consider first a vertical plane through a horizontally bedded sedimentary forma-

With the aid of the two adjectives, "homogeneous" or "heterogeneous" on one

Kh(x,y,z)=Kv(x,y,z)=f(x,y,z) Kh(x,y ,z) =f1(x,y ,z)

Geological formations which have a layered heterogeneity behave on a large scale

Qh=-K.D. Llh (2.5)

where D; is the thickness of the individual layers, then:

Qc=LJ=lQi=-LJ=l (KiD) At (2.6)

where D is the total considered thickness of the formation.

Fig. 2.3. Horizontal flow through a layered heterogeneous formation

Fig. 2.4. Vertical flow through a layered heterogeneous formation

equivalent vertical conductivity of the whole formation, K V •

where D is the sum of the thickness of the individual layers Ei.1 d i .

or from Eqs. (2.9) and (2.10) one can derive that

and because QV=Q. =Q2= ... =Q.

2.1.5 Generalized law of Darcy

Kxx Kxy Kxz

2.1.6 Hydraulic conductivity ellipsoid

Consider an arbitrary flow q, in a homogeneous, anisotropic medium with principal

where Ks is unknown, although it presumably lies in the range Kx-K..

Given x, y, Kxx, K~,Kxy.What are

q, can be separated in the components qx and <Iz, where

Geometrically, ax/as=coso and azlas=sinO. Substituting these relationships together with