3 HYDROLOGY Module Final
3 HYDROLOGY Module Final
3 HYDROLOGY Module Final
WOLKITE UNIVERSITY
PREPARED BY
1 DAWIT GIRMA (B.sc)
2 GEMACHU SHUNIYE (B.sc)
3 MELKA LEMA (B.sc)
February, 2017
Wolkite, Ethiopia
Table of Contents
CHAPTER ONE: INTRODUCTION ........................................................................................................... 7
1.1 THE HYDROLOGIC CYCLE ........................................................................................................... 7
1.1.1 Definition ..................................................................................................................................... 7
1.1.2 Hydrologic cycle- detailed definition .......................................................................................... 7
CHAPTER TWO .......................................................................................................................................... 9
2. Rainfall-Runoff Relationships (Application of Different Rainfall-Runoff Models) ................................ 9
2.1 Hydrological Models .......................................................................................................................... 9
2.2 Deterministic Hydrological Models .................................................................................................... 9
2.2.1 Empirical (Black Box) Models .................................................................................................... 9
2.2.2 Lumped Conceptual Models ...................................................................................................... 10
2.2.3 Distributed Process Description Based Models ......................................................................... 10
2.3 Stochastic Time Series Models ......................................................................................................... 11
2.4 Rational Method................................................................................................................................ 11
2.4.1 Runoff Coefficient ..................................................................................................................... 11
2.4.2 Rainfall intensity ........................................................................................................................ 12
2.4.3 Time of Concentration ............................................................................................................... 13
2.5 SCS Curve Number Method ............................................................................................................. 15
2.6 Times-Area Method .......................................................................................................................... 16
2.7 STREAM FLOW HYDROGRAPH ................................................................................................. 17
2.7.1 Hydrograph Analysis ................................................................................................................. 18
2.7.2 Factors affecting flood hydrograph ............................................................................................ 20
2.7.3 Effective Rainfall ....................................................................................................................... 20
2.7.4 Separation of Base Flow and Runoff ......................................................................................... 20
2.8 The Unit Hydrograph (UH)............................................................................................................... 21
2.8.1 Derivation of the Unit Hydrograph from single storms ............................................................. 23
2.8.2 Changing of the Duration of the UH .......................................................................................... 23
2.9 Applications of Unit Hydrograph ..................................................................................................... 25
2.10 Synthetic Unit Hydrographs ........................................................................................................... 25
2.10.1 Snyder‟s method ...................................................................................................................... 26
2.11 UH from a complex storm .............................................................................................................. 28
3. Introduction ............................................................................................................................................. 90
3.1 Steady Radial Flow to a well ............................................................................................................ 90
3.1.1 Steady Flow to a well in Confined Aquifer ............................................................................... 90
3.1.2 Steady Flow to a well in Unconfined Aquifer ........................................................................... 92
3.2 unsteady flow .................................................................................................................................... 93
3.2.1. Unsteady Flow to a well in a confined aquifer ......................................................................... 93
3.2.2. Unsteady Flow to a well in unconfined aquifer ........................................................................ 97
3.3 Unsteady Radial Flow in a Leaky Aquifer- Hantush-Jacob Method and Walton Graphical Method
................................................................................................................................................................ 97
3.4. Multiple Well Systems ................................................................................................................... 100
3.5 Recovery of a well/aquifer .............................................................................................................. 101
3.6 Well Losses and Specific Capacity ................................................................................................. 102
3.6.1 Well Loss ................................................................................................................................. 102
3.6.2 Specific Capacity ..................................................................................................................... 104
CHAPTER FOUR: PUMPING TESTS .................................................................................................... 106
4.1General ............................................................................................................................................. 106
CHAPTER FIVE: Ground water Exploration .......................................................................................... 112
5.1 General ............................................................................................................................................ 112
5.1.1 Topographical maps ................................................................................................................. 112
5.2 Surface investigations of Ground water .......................................................................................... 114
5.2.1 Surface geophysical techniques ............................................................................................... 114
5.3 Subsurface Investigation of Ground water...................................................................................... 122
5.3.1Test drilling ............................................................................................................................... 123
5.3.2Geologic log .............................................................................................................................. 123
5.3.3Drilling-Time log ...................................................................................................................... 124
5.3.4 Geophysical logging ................................................................................................................ 124
CHAPTER SIX: Water Well Design and Construction............................................................................ 126
6.1 Water Well Design .......................................................................................................................... 126
6.1.1Casing Diameter........................................................................................................................ 127
6.1.2Casing Materials ....................................................................................................................... 127
6.1.3Well depth ................................................................................................................................. 127
For multiple stations, the average depth of precipitation over the watershed is determined using the
information from all stations:
– Arithmetic Method
– Thiessen Polygon Method
– Isohyetal Method
Arithmetic Mean
Simplest method
Satisfactory method if gages are uniformly distributed and if individual variations are not great
The arithmetic mean method uses the mean of precipitation record from all gauges in a catchment. The
method is simple and give good results if the precipitation measured at the various stations in a catchment
show little variation.
In the Thiessen polygon method, the rainfall recorded at each station is given a weightage on the basis of
an area closest to the station. The average rainfall over the catchment is computed by considering the
precipitation from each gauge multiplied by the percentage of enclosed area by the Thiessen polygon.
The total average areal rainfall is the summation averages from all the stations.
The Thiessen polygon method gives more accurate estimation than the simple arithmetic mean estimation
as the method introduces a weighting factor on rational basis. Furthermore, rain gauge stations outside the
catchment area can be considered effectively by this method.
The Isohyetal method is the most accurate method of estimating areal rainfall.
The method requires the preparation of the Isohyetal map of the catchment from a network of gauging
stations. Areas between the isohyets and the catchment boundary are measured. The areal rainfall is
calculated from the product of the inter-Isohyetal areas and the corresponding mean rainfall between the
isohyets divided by the total catchment area.
CHAPTER TWO
2. Rainfall-Runoff Relationships (Application of Different Rainfall-Runoff Models)
2.1 Hydrological Models
The two classical types of hydrological models are the deterministic and the stochastic types.
production. Thus, C is not a constant parameter, but varies from storm to storm on the same catchment,
and from catchment to catchment for similar storms.
The other best known among the black box models is the unit hydrograph model which was published by
Sherman (1932), who used the idea that the various time delays for runoff produced on the catchment to
reach the outlet could be represented as a time distribution without any direct link to the areas involved.
Because the routing procedure was linear, this distribution could be normalized to represent the response
to a unit of runoff production, or effective rainfall, generated over the catchment in one time step. The
method is one of the most commonly used hydrograph modeling techniques in hydrology, simple to
understand and easy to apply. The unit hydrograph represents a discrete transfer function for effective
rainfall to reach the basin outlet, lumped to the scale of the catchment.
Other empirical models are developed using linear regression and correlation methods used to determine
functional relationships between different data sets.
The relationships are characterized by correlation coefficients and standard deviation and the parameter
estimation is carried out using rigorous statistical methods involving tests for significance and validity of
the chosen model.
2.2.2 Lumped Conceptual Models
Lumped models treat the catchment as a single unit, with state variables that represent average values
over the catchment area, such as storage in the saturated zone. Due to the lumped description, the
description of the hydrological processes cannot be based directly on the equations that are supposed to be
valid for the individual soil columns. Hence, the equations are semi-empirical, but still with a physical
basis. Therefore, the model parameters cannot usually be assessed from field data alone, but have to be
obtained through the help of calibration.
One of the first and most successful lumped digital computer models was the Stanford Watershed model
developed by Norman Crawford and Ray Linsley at Stanford University. The Stanford model had up to
35 parameters, although it was suggested that many of these could be fixed on the basis of the physical
characteristics of the catchment and only a much smaller number needed to be calibrated.
2.2.3 Distributed Process Description Based Models
Another approach to hydrological processes modeling was the attempt to produce models based on the
governing equations describing all the surface and subsurface flow processes in the catchment. A first
attempt to outline the potentials and some of the elements in a distributed process description based model
on a catchment scale was made by Freeze and Harlan (1969). The calculations require larger computers to
solve the flow domain and points at the elements of the catchment.
Distributed models of this type have the possibility of defining parameter values for every element in the
solution mesh. They give a detailed and potentially more correct description of the hydrological processes
in the catchment than do the other model types. The process equations require many different parameters
to be specified for each element and made the calibration difficult in comparison with the observed
responses of the catchment.
In principle parameter adjustment of this type of model is not necessary if the process equations used are
valid and if the parameters are strongly related to the physical characteristics of the surface, soil and rock.
In practice the model requires effective values at the scale of the elements. Because of the heterogeneity
of soil, surface vegetation establishing a link between measurements and element values is difficult. The
Distributed Process Description Based Models can in principle be applied to almost any kind of
hydrological problem. The development is increased over the recent years for the fact that the increase in
computer power, programming tools and digital databases and the need to handle processes and
predictions of runoff, sediment transport and/or contaminants.
Another reason is the need of the models for impact assessment. Changes in land use, such as
deforestation or urbanization often affect only part of a catchment area.
With a distributed model it is possible to examine the effects of such land use changes in their correct
spatial context by understanding the physical meaning between the parameter values and the land use
changes.
Recent examples of distributed process based models include the SHE model (Abbott et al., 1986), MIKE
SHE (Refsgaard and Storm, 1995), IHDM (Institute of Hydrology Distributed Model; Calver and Wood
1995), and THALES (Grayson et al. 1992), etc.
2.3 Stochastic Time Series Models
Stochastic models allow for some randomness or uncertainty in the possible outcomes due to uncertainty
in input variables, boundary conditions or model parameters. Traditionally, a stochastic model is derived
from a time series analysis of the historical record. The stochastic model can then be used for the
generation of long hypothetical sequences of events with the same statistical properties as the historical
record. In this technique several synthetic series with identical statistical properties are generated. These
generated sequences of data can then be used in the analysis of design variables and their uncertainties,
for example, when estimating reservoir storage requirements.
With regard to process description, the classical stochastic simulation models are comparable to the
empirical, black box models. Hence, stochastic time series models are in reality composed of a simple
deterministic core (the black box model) contained within a comprehensive stochastic methodology.
So, these are the broad generic classes of rainfall-runoff models, lumped or distributed; deterministic or
stochastic.
coefficient of runoff also varies for different storms on the same catchment, and thus, using an average
value for C, gives only a rough estimate of QP in small uniform urban areas. On top of this the Rational
Formula has been used for many years as a basis for engineering design for small land drainage schemes
and storm-water channels.
If the basin contains varying amount of different land cover or other abstractions, a coefficient can be
calculated through areal weighing as shown in equation (2.2). Typical values are given in table 2.1 below.
∑
Where x = subscript designating values for incremental areas with consistent land cover
Table 2.1: Runoff coefficients for rational formula
Business:
Downtown areas 0.70 – 0.95
Neighborhood areas 0.50 – 0.70
Residential:
Single –family areas 0.30 – 0.50
Multi- units, detached 0.40 – 0.60
Multi-units, attached 0.60 -0.75
Suburban 0.25 – 0.40
Apartment dwelling areas 0.50 – 0.70
Industrial:
Light areas 0.50 – 0.80
Heavy areas 0.60 – 0.90
Parks, cemeteries 0.10 – 0.25
Playgrounds 0.20 – 0.40
Railroad yard areas 0.20 – 0.40
Unimproved areas 0.10 – 0.30
Lawns:
Sandy soil, flat 2% 0.05 – 0.10
Sandy soil, average, 2-7% 0.10 – 0.15
Sandy soil, steep, 7% 0.15 – 0.20
Heavy soil, flat, 2% 0.13 – 0.17
Heavy soil, average 2-7% 0.18 – 0.22
Heavy soil, steep, 7% 0.25 – 0.35
Streets:
Asphalted 0.70 – 0.95
Concrete 0.80 – 0.95
Brick 0.70 – 0.85
Drives and walks 0.75 – 0.85
Roofs 0.75 – 0.95
∗ Higher values are usually appropriate for steeply sloped areas and longer return periods because
infiltration and other losses have a proportionally smaller effect on runoff in these cases
2.4.2 Rainfall intensity
Rainfall intensity, duration curve and frequency curves are necessary to use the Rational method.
And it is also the stored precipitation per the duration of the given rainfall.
Table 2.2: Intercept coefficients for velocity versus slope relationship of equation (2.5)
Table 2.4: Runoff Curve Numbers for Urban areas (Average watershed conditions, Ia = 0.2 SR)
Where n, the number of incremental areas between successive isochrones, is given by Tc/ΔT and k is a
counter.
The unrealistic assumption made in the rational method of uniform rainfall intensity over the whole
catchment and during the whole of Tc is avoided in the time – area method, where the catchment
contributions are subdivided in time.
The varying intensities within a storm are averaged over discrete periods according to the isochrones time
interval selected. Hence, in deriving a flood peak for design purposes, a design storm with a critical
sequence of intensities can be used for the maximum intensities applied to the contributing areas of the
catchment that have most rapid runoff. However, when such differences within a catchment are
considered, there arises difficulty in determining Tc, the time after the commencement of the storm when,
by definition, Qp occurs.
2.7 STREAM FLOW HYDROGRAPH
A hydrograph is a graphical plot of discharge of a natural stream or river versus time. The hydrograph is a
result of a particular effective rainfall hyetograph as modified by basin flow characteristics. By definition,
the volume of water under an effective rainfall hyetograph is equal to the volume of surface runoff.
It has three characteristic parts: the rising limb, the crest segment and the falling limb or depletion curve.
With reference to figure 2.3 the effective rainfall hyetograph consisting of a single block of rainfall with
duration D (T is also used in the lecture note alternatively) shown in the upper left part of the figure
produced the runoff hydrograph. The areas enclosed by the hyetograph and the hydrograph each represent
the same volume, V, of water from the catchment. The maximum flow rate on the hydrograph is the peak
flow, qp, while the time from the start of the hydrograph to qp is the time to peak, tp. The total duration of
the hydrograph known as the base time, tb.
The lag time, tL is the time from the center of mass of effective rainfall to the peak of runoff hydrograph.
It is apparent that tp = tL + D/2, using this definition.
Some define lag time as the time from center of mass of effective rainfall to the center of the runoff
hydrograph.
Infiltration characteristics 3
3 land use and cover Evapotranspiration
soil type and geological conditions
lakes, swamps and other storages
There are several methods of base flow separation. Some of them that are that are in common use are:
Straight-line method (Method-I)
The separation of the base flow is achieved by joining with a straight-line beginning of the direct runoff to
a point on the recession limb representing the end of the direct runoff. Point B the end of the recession
limb may be located by an empirical equation for the time interval N (days) from the peak to the point B
is
As 1 mm depth of rainfall excess is considered the area of the unit hydrograph is equal to a volume given
by 1 mm over the catchment.
2. The rainfall is considered to have an average intensity of excess rainfall (ER) of 1/T mm/h for the
duration T-h of the storm.
3. The distribution of the storm is considered to be all over the catchment.
The requirement of uniformity in areal distribution of the effective rainfall is rarely met and indeed unless
the non-uniformity is pronounced, its effect is neglected.
be seen that different rain intensities with the same duration of the rain will produce hydrographs with
different magnitudes but the same base length; however, there will be only one unit hydrograph for the
same duration.
If the UH for a certain duration T is known then the runoff of any other rain of the duration T may be
computed by multiplying the UH ordinates with the ratio of the given rain intensity with unit rain. i.e.
2. The total hydrograph of direct runoff due to n successive amounts of effective rainfall (for instance R1
and R2) is equal to the sum of the n successive hydrographs produced by the effective rainfall (the latter
lagged by T h on the former). This is known as Law of Superposition.
Once a TUH is available, it can be used to estimate design flood hydrographs from design storms. The
law of superposition is demonstrated in Figure 1.6 c above.
3. The third property of TUH assumes that the effective rainfall-surface runoff relationship does not
change with time, i.e., the same TUH always occurs whenever the unit of effective rainfall in T h is
applied on the catchment. Using this time invariance assumption, once a TUH has been derived for a
catchment area, it could be used to represent the response of the catchment whenever required.
2.8.1 Derivation of the Unit Hydrograph from single storms
The derivation of the unit hydrograph of a catchment from single storms proceeds in the following stages:
1. The rainfall records are scanned to find a storm of desired duration that gives a fairly uniform
distribution in time and space. The hyetograph of this storm is constructed using a convenient uniform
interval of time.
2. The base flow is separated from the hydrograph using one of the methods presented in section in the
first chapter.
3. The surface runoff volume is determined as a depth of flow by numerical integration:
∑
Where, Qs = the maximum rate at which an ER intensity of 1/T can drain out of the catchment of area, A
(km2)
T1 = unit storm in hours
S3=u1+u2+u3=u3+S2
.
.
.
Sn=u1+u2+u3+…+un = un+Sn-1
The T2-hour UH is obtained from the difference between two S-curves distanced T2-hours apart,
corrected for the effective intensity as follows. Since the S-curve refers to continuous rain of 1/T1 units,
the difference between the S-curves displaced by T2 hours represents surface runoff from (1/T1) xT2. A
rainfall with duration T2 requires an intensity i2 = 1/T2 to give unit depth. Hence, the S-curve difference
has to be multiplied with the ratio i2/i1= (1/T2)/ (1/T1) = T1/T2 to get a unit depth in T2 hours. Hence, u
(T2, t) follows from:
Note that the base length follows from Tb2 = Tb1 - T1 +T2. The procedure is shown in figure 2.10 below.
scarce. In order to construct UH for such areas, empirical equations of regional validity, which relate the
important hydrograph characteristics to the basin.
2.10.1 Snyder’s method
Snyder (1938), based on a study of a large number of catchments in the Appalachian highlands of eastern
United States developed a set of empirical equations for synthetic-unit hydrographs in those areas.
The most important characteristics of a basin affecting a hydrograph due to a given storm is basin lag.
Actually basin lag (also known as lag time) is the time difference between the cancroids of the input
(rainfall excess) and the output (surface runoff) i.e. TL. Physically, it represents the main time of travel of
water particles from all parts of the catchment to the outlet during a given storm. Its value is determined
essentially on the physical features of the catchment, such as size, length, stream density and vegetation.
For its determination, however, only a few important catchment characteristics are considered. For
simplicity,
Snyder has used a somewhat different definition of basin lag (denoted by tp) in his methodology. This tp
is practically of the same order of magnitude as TL and in this section the term basin lag is used to denote
Snyder‟s tp.
The first of the Snyder‟s equation relates the basin lag tp. Defined as the time interval from the midpoint
of the unit rainfall excess to the peak of the unit hydrograph (Figure 2.12 below) to the basin
characteristics as: characteristics are of most important. Unit hydrographs derived from such relationships
are known as synthetic unit hydrographs.
Where, tp in hours
L = basin length measured along the watercourse from the basin divide to the gauging station in
km.
Lc = distance along the main watercourse from the gauging station to the point opposite (or
nearest) the watershed centroid in km
Ct = a regional constant representing watershed slope and storage
The value of Ct in Snyder‟s study ranged from 1.35 to 1.65. However, studies by many investigators have
shown that Ct depends upon the region under study and wide variations with the value of Ct ranging from
0.3 to 6.0 have been reported.
( + )
With tb taken as the next larger integer value divisible by tR i.e. tb is about five times the time to peak.
To assist in the sketching of unit hydrographs,
And
The complexities of rainfall-runoff relationships are such that these simple methods allow only for
average conditions. Flood events can have very many different causes that produce flood hydrographs of
different shapes.
The principal advantages of these simple methods are that they can be developed for stations with only
stage measurements and no rating curve, and they are quick and easy to apply especially for warning of
impending flood inundations when the required answers are immediately given in stage heights.
3.3 Storage Routing
When a storm event occurs, an increased amount of water flows down the river and in any one short reach
of the channel there is a greater volume of water than usual contained in temporary storage. If at the
beginning of the reach the flood hydrograph is (above normal flow) is given as I, the inflow, then during
the period of the flood, T1, the channel reach has received the flood volume given by the area under the
inflow hydrograph. Similarly, at the lower end of the reach, with an outflow hydrograph O, the flood is
given by the area under the curve. In a flood situation relative quantities may be such that lateral and
tributary inflows can be neglected, and thus by the principle of continuity, the volume of inflow equals
the volume of outflow, i.e. the flood
∫ ∫ . At intermediate time T, an amount ∫ has entered the reach and an
amount ∫ has left reach. The difference must be stored within the reach, so the amount of storage, S,
within the reach at time t=T is given by ∫
The principle of hydrologic flood routings (both reservoir and channel) uses the continuity equation in the
form of “Inflow minus outflow equals rate of change of storage”.
i.e.
Where: It = Inflow in to the reach
Ot= Outflow from the reach
dS/dt =Rate of change of storage within the reach.
Alternatively, the continuity (storage) equation can be stated as in a small time interval Δt the difference
between the total inflow volume and total outflow volume in a reach is equal to the change in storage in
that reach, i.e.
̅ ̅
Where, ̅ = average inflow in time
̅ = average out flow in time
= change in storage
= routing period
OR above equation can rewrite as:
+ + +
+ +
The time interval Δt should be sufficiently short so that the inflow and out flow hydrographs can be
assumed to be straight line in that interval. As a rule of thumb Δt ≤ 1/6 of the time to peak of the inflow
hydrograph is required.
The continuity equation (I-Q = dS/dt), forms basis for all the storage routing methods. The routing
problem consists of finding Q as a function of time, given I as a function of time, and having information
or making assumptions about storage, S.
3.4 Reservoir or level pool routing
A flood wave I (t) enters a reservoir provided with an outlet such as a spillway.
The outflow is a function of the reservoir elevation only, i.e., O = O (h). The storage in the reservoir is a
function of the flow reservoir elevation, S = S (h).
Further, the water level in the reservoir changes with time, h = h (t) and hence the storage and discharge
change with time. It is required to find the variation of S, h and O with time, i.e., find S=S (t), O = O (t)
and h = h (t), given I =I (t)
Depending on the forms of the outlet relations for O (h) will be available.
For reservoir routing, the following data have to be known:
1. Storage volume versus elevation for the reservoir
2. Water surface elevation versus out flow and hence storage versus outflow discharge
3. Inflow hydrograph, I= I (t); and
4. Initial values of S, I and O at time t = 0
The finite difference form of the continuity equation (Equation. 3.4) can be rewritten as:
+ +
Where, (I1+I2)/2= I; (O1+ O2)/2 = O and S2-S1=ΔS and suffixes 1 and 2 to denote the beginning and end
of the time interval Δt.
Rearranging Equation (3.4) to get the unknowns S2 and O2 on one side of the equation and to adjust the
O1 term to produce:
+
( + ) ( + )+( )
Since S is a function of O, [(S/Δt) + (O/2)] is also a specific function of O (for a given Δt). Replacing
{(S/Δt) + (O/2)} by G, for simplification, equation (3.5) can be written:
G2 = G1 + Im –O1 or more generally
Gi+1 = Gi + Im, i - Oi
Where: Im = (I1 + I2)/2
To apply this method we need beside It also the G-O relation. The latter is easily established from S-H
and O-H relations, where for equal values of H, S and O are determined; after which the proper interval
Δt the G-O relation is established. Note that G is dependent on the chosen routing interval Δt.
The routing period, Δt, has to be chosen small enough such that the assumption of a linear change of flow
rates, I and O, during Δt is acceptable (as a guide, Δt should be less than 1/6 of the time of rise of the
inflow hydrograph).
So, in short, the method consists of three steps:
1. Inspect the inflow hydrograph and select the routing interval: Δt ≤ 1/6 time to peak
2. Establish the G-O relation
3. Carry out the routing according to equation (3.6)
A useful check on the validity of any level pool routing calculation is that the peak of the outflow
hydrograph should occur at the intersection of the inflow and out flow hydrograph on the same plot. At
that point, I = O, so ds/dt = 0, i.e. storage is a maximum and therefore O is a maximum. Therefore, the
temporary storage is depleted.
At a fixed depth at a downstream section of river reach, prism storage is constant while the wedge storage
changes from a positive value at the advancing flood wave to a negative value during a receding flood.
The total storage in the channel reach can be generally represented by:
S = f1 (O) +f2 (I-O)
And this can then be expressed as:
S = K (x Im + (1-x) Om)
Where K and x are coefficients and m is a constant exponent. It has been found that the value of m varies
from 0.6 for rectangular channels to value of about 1.0 for natural channels.
3.5.1 Muskingum Method of Routing
Using m =1 for natural channels, equation (2.8), reduces to a linear relationship for S in terms of I and Q
as
S= K (x I+ (1-x) O
This relationship is known as the Muskingum Equation. In this the parameter x is known as weighing
factor and take a value between 0 and 0.5. When x=0, obviously the storage is a function of discharge
only and equation (3.9) reduces to:
S = KQ
Such storage is known as linear storage or linear reservoir. When x= 0.5 both the inflow and out flow
are equally important in determining the storage.
The coefficient K is known as storage-time constant and has dimensions of time. K is approximately
equal to the time of travel of a flood wave through the channel reach.
As before, writing the continuity equation in finite difference form, we can write
S2 - S1 = {(I1+I2) Δt}/2 - {(O1+O2) Δt}/2
For a given channel reach by selecting a routing interval Δt and using the Muskingum equation, the
change in storage can be determined.
S1 = K (xI1 + (1-x) O1)
S2 = K (xI2 + (1-x) O2)
Substituting equations (3.12) and (3.13) in equation (3.11) and after rearrangements gives:
O2 = c1I1 +c2I2 +c3O1 and more generally as
Oi+1 = c1Ii+c2Ii+1+c3Oi
Where:-
Note that ΣC=1 and thus when C1 and C2 have been found C3=1-C1-C2. Thus the outflow at the end of a
time step is the weighted sum of the starting inflow and outflow and the ending inflow. It has been found
that best results will be obtained when routing interval should be so chosen that K>Δt>2kx. If Δt < 2kx,
the coefficient C2 will be negative.
3.5.2 Application of the Muskingum Method:
In order to use equation (2.14) for Oi+1, it is necessary to know K and x for calculating the coefficients, C.
Using recorded hydrographs of a flood at the beginning and end of the river reach, trial values of x are
taken, and for each trial the weighted flows in the reach, [xI+ (1-x) O], are plotted against the actual
storages determined from the inflow and out flow hydrographs as indicated in the following figure.
.
CHAPTER FOUR: FREQUENCY ANALYSES (PROBABILITY IN HYDROLOGY)
4.1 General
Water resource systems must be planned for future events for which no exact time of occurrence can be
forecasted. Hence, the hydrologist must give a statement of the probability of the stream flows (or other
hydrologic factors) will equal or exceed (or be less than) a specified value. These probabilities are
important to the economic and social evaluation of a project. In most cases, absolute control of the floods
or droughts is impossible. Planning to control a flood of a specific probability recognizes that a project
will be overtaxed occasionally and damages will be incurred. However, repair of the damages should be
less costly in the long run than building initially to protect against the worst possible event. The planning
goal is not to eliminate all floods but to reduce the frequency of flooding, and hence the resulting
damages. If the socio economic analysis is to be correct, the probability of flooding must be eliminated
accurately. For major projects, the failure of which seriously threatens human life, a more extreme event,
the probable maximum flood, has become the standard for designing the spillway.
This chapter deals with techniques for defining probability from a given set of data and with special
methods employed for determining design flood for major hydraulic structures.
Frequency analysis is the hydrologic term used to describe the probability of occurrence of a particular
hydrologic event (e.g. rainfall, flood, drought, etc.).
Therefore, basic knowledge about probability (e.g. distribution functions) and statistics (e.g. measure of
location, measure of spread, measure of skewness, etc) is essential. Frequency analysis usually requires
recorded hydrological data.
Hydrological data are recorded either as a continuous record (e.g. water level or stage, rainfall, etc.) or in
discrete series form (e.g. mean daily/monthly/annual flows or rainfall, annual series, partial series, etc.).
For planning and designing of water resources development projects, the important parameters are river
discharges and related questions on the frequency & duration of normal flows (e.g. for hydropower
production or for water availability) and extreme flows (floods and droughts).
The shape of the flow-duration curve gives a good indication of a catchment‟s characteristics response to
its average rainfall history. An initially steeply sloped curve results from a very variable discharge,
usually from small catchments with little storage where the stream flow reflects directly the rainfall
pattern. Flow duration curves that have very flat slope indicate little variation in flow regime, the resultant
of the damping effects of large storages.
4.3 Flood Probability
4.3.1 Selection of data
If probability analysis is to provide reliable answers, it must start with a data series that is relevant,
adequate, and accurate. Relevance implies that the data must deal with the problem. Most flood studies
are concerned with peak flows, and the data series will consist of selected observed peaks. However, if
the problem is duration of flooding, e.g., for what periods of time a highway adjacent to a stream is likely
to be flooded, the data series should represent duration of flows in excess of some critical value. If the
problem is one of interior drainage of a leveed area, the data required may consist of those flood volumes
occurring when the main river is too high to permit gravity drainage.
Adequacy refers primarily to length of record, but scarcity of data collecting stations is often a problem.
The observed record is merely a sample of the total population of floods that have occurred and may
occur again. If the sample is too small, the probabilities derived cannot be expected to be reliable.
Available stream flow records are too short to provide an answer to the question: How long must a record
be to define flood probabilities within acceptable tolerances?
Accuracy refers primarily to the problem of homogeneity. Most flow records are satisfactory in terms of
intrinsic accuracy, and if they are not, there is little that can be done with them. If the reported flows are
unreliable, they are not a satisfactory basis for frequency analysis. Even though reported flows are
accurate, they may be unsuitable for probability analysis if changes in the catchment have caused a
change in the hydrologic characteristics, i.e., if the record is not internally homogenous. Dams, levees,
diversions, urbanization, and other land use changes may introduce inconsistencies. Such records should
be adjusted before use to current conditions or to natural conditions.
There are two data series of floods:
(i) The annual series, and
(ii) The partial duration series.
The annual series constitutes the data series that the values of the single maximum daily/monthly/annually
discharge in each year of record so that the number of data values equals the record length in years. For
statistical purposes, it is necessary to ensure that the selected peak discharges are independent of one
another. This data series is necessary if the analysis is concerned with probability less than 0.5. However
as the interest are limited to relatively rare events, the analysis could have been carried out for a partial
duration series to have more frequent events.
The partial duration series constitutes the data series with those values that exceed some arbitrary level.
All the peaks above a selected level of discharge (a threshold) are included in the series and hence the
series is often called the Peaks over Threshold (POT) series. There are generally more data values for
analysis in this series than in the annual series, but there is more chance of the peaks being related and the
assumption of true independence is less valid.
4.3.2 Plotting Positions
Probability analysis seeks to define the flood flow with probability of p being equaled or exceed in any
year. Return period Tr is often used in lieu of probability to describe a design flood. Return period and
probability are reciprocals, i.e.
p = 1/Tr
To plot a series of peak flows as a cumulative frequency curves it is necessary to decide on a probability
or return period to associate with each peak. There are various formulas for defining this value as shown
in table 4.2.
The probability of occurrence of the event r times in n successive years can be obtained from:
Where q=1-p
Table 4.2 plotting position formulae
Method P
California m/N
Hazen (m-0.5)/N
Weibull m/(N+1)
Chegodayev (m-0.3)/(N+0.4)
Gringorten (m-3/8)/(N+1/4)
Consider, for example, a list of flood magnitudes of a river arranged in descending order as shown in
Table 4.3. The length of record is 50 years.
Table 4.3: calculation of frequency T
The last column shows the return period T of various flood magnitude, Q. A plot of Q Vs T yields the
probability distribution. For small return periods (i.e. for interpolation) or where limited extrapolation is
required, a simple best-fitting curve through plotted points can be used as the probability distribution. A
logarithmic scale for T is often advantageous. However, when larger extrapolations of T are involved,
theoretical probability distributions (e.g. Gumbel extreme-value, Log- Pearson Type III, and log normal
distributions) have to be used.
In frequency analysis of floods the usual problem is to predict extreme flood events. Towards this,
specific extreme-value distributions are assumed and the required statistical parameters calculated from
the available data. Using these flood magnitude for a specific return period is estimated.
4.3.3 Theoretical Distributions of Floods
Statistical distributions are usually demonstrated by use of samples numbering in the thousands. No such
samples are available for stream flow and it is not possible to state with certainty that a specific
distribution applies to flood peaks.
Numerous distributions have been suggested on the basis of their ability to “fit” the plotted data from
streams.
Chow has shown that most frequency-distribution functions applicable in hydrologic studies can be
expressed by the following equation known as the general equation of hydrologic frequency analysis:
̅+
Where xT = value of the variate X of a random hydrologic series with a return period T, x = mean of the
variate, σ = standard deviation of the variate, K = frequency factor which depends upon the return period,
T and the assumed frequency distribution.
̅
Thus +
Where x = mean and = standard deviation of the variate X. In practice it is the value of X for a given P
that is required and as such Eq. (4.7) is transposed as
y = -1n (-1n (q)) = -1n (-1n (1-p))
Meaning that the probability of non-exceedence equals:
Noting that the return period T = 1/P and designating; yT = the value of y, commonly called the reduced
variate, for a given T
+
Now rearranging Eq.(4.7), the value of the variate X with a return period T i
̅+
Note that Eq. (4.12) is of the same form as the general equation of hydrologic frequency analysis, Eq.
(4.4). Further Eqs (4.11) and (4.12) constitute the basic Gumbel's equations and are applicable to an
infinite sample size (i.e. N → ∞).
Since practical annual data series of extreme events such as floods, maximum rainfall depths, etc., all
have finite lengths of record; Eq. (4.12) is modified to account for finite N as given below for practical
use.
4.3.5 Gumbel's Equation for Practical Use
Equation (4.11) giving the variate X with the return period T is used as
̅+
Where σn-1 = standard deviation of the sample
∑ ̅
√
̅ = reduced mean, a function of sample size N and is given in Table 4.4; for N → ∞,
̅̅̅̅→ 0.577.
Sn = reduced standard deviation, a function of sample size N and is given in Table 4.5; for N → ∞, Sn →
1.2825.
These equations are used under the following procedure to estimate the flood magnitude corresponding to
a given return period based on annual flood series.
1. Assemble the discharge data and note the sample size N. Here the annual flood value is the variate X.
Find ̅ and σn-1 for the given data.
2. Using Tables 4.4 and 4.5 determine ̅̅̅̅ and Sn appropriate to given N
3. Find yT for a given T by Eq. (4.15).
4. Find K by Eq. (4.14).
5. Determine the required xT by Eq. (4.13).
To verify whether the given data follow the assumed Gumbel's distribution, the following procedure may
be adopted. The value of xT for some return periods T<N are calculated by using Gumbel's formula and
plotted as xT Vs T on a convenient paper such as a semi-log, log-log or Gumbel probability paper. The
use of Gumbel probability paper results in a straight line for xT Vs T plot. Gumbel's distribution has the
property which gives T = 2.33 years for the average of the annual series when
N is very large. Thus the value of a flood with T = 2.33 years is called the mean annual flood. In
graphical plots this gives a mandatory point through which the line showing variation of xT with T must
pass. For the given data, values of return periods (plotting positions) for various recorded values, x of the
variate are obtained by the relation T = (N+1)/m and plotted on the graph described above. A good fit of
observed data with the theoretical variation line indicates the applicability of Gumbel's distribution to the
given data series. By extrapolation of the straight-line xT Vs T, values of xT> N can be determined easily.
The Gumbel (or extreme-value) probability paper is a paper that consists of an abscissa specially marked
for various convenient values of the return period T (or corresponding reduced variate yT in arithmetic
scale). The ordinate of a Gumbel paper represent xT (flood discharge, maximum rainfall depth, etc.),
which may have either arithmetic scale or logarithmic scale.
Table 4.4: Reduced mean ̅n in Gumbel's extreme value distribution, N = sample size
It is seen that for a given sample and T, 80% confidence limits are twice as large as the 50% limits and
95% limits are thrice as large as 50% limits.
In addition to the analysis of maximum extreme events, there also is a need to analyze minimum extreme
events; e.g. the occurrence of droughts. The probability distribution of Gumbel, similarly to the Gaussian
probability distribution, does not have a lower limit; meaning that negative values of events may occur.
As rainfall or river flows do have a lower limit of zero, neither the Gumbel nor Gaussian distribution is an
appropriate tool to analyze minimum values. Because the logarithmic function has a lower limit of zero, it
is often useful to first transform the series to its logarithmic value before applying the theory. Appropriate
tools for analyzing minimum flows or rainfall amounts are the Log-Normal, Log-Gumbel, or Log-Pearson
distributions.
4.3.7 Log-Pearson Type III Distribution
This distribution is widely used in USA. In this distribution the variate is first transformed into
logarithmic form (base 10) and the transformed data is then analyzed. If X is the variate of a random
hydrologic series, then the series of Z variates where
Z = log x are first obtained. For this z series, for any recurrence interval T, equation (4.4) gives
̅+
Where Kz = a frequency factor which is a function of recurrence interval T and the coefficient of skew Cs,
The flood-frequency analysis described above is a direct means of estimating the desired flood based
upon the available flood-flow data of the catchment.
The results of the frequency analysis depend upon the length of data. The minimum number of years of
record required to obtain satisfactory estimates depends upon the variability of data and hence on the
physical and climatological characteristics of the basin. Generally a minimum of 30 years of data is
considered as essential. Smaller lengths of records are also used when it is unavoidable. However,
frequency analysis should not be adopted if the length of records is less than 10 years.
Flood-frequency studies are most reliable in climates that are uniform from year to year. In such cases a
relatively short record gives a reliable picture of the frequency distribution. With increasing lengths of
flood records, it affords a viable alternative method of flood-flow estimation in most cases.
A final remark of caution should be made regarding to frequency analysis. None of the frequency
distribution functions have a real physical background. The only information having physical meaning are
the measurements themselves. Extrapolation beyond the period of observation is dangerous. It requires a
good engineer to judge the value of extrapolated events of high return periods. A good impression of the
relativity of frequency analysis can be acquired through the comparison of result obtained from different
statistical methods. Generally they differ considerably.
Example 4.1
Annual maximum recorded floods in a certain river, for the period 1951 to 1977 is given below. Verify
whether the Gumbel extreme-value distribution fit the recorded values. Estimate the flood discharge with
return period of (i) 100 years and (ii) 150 years by graphical extrapolation.
Example 4.2:
Data covering a period of 92 years for a certain river yielded the mean and standard deviation of the
annual flood series as 6437 and 2951 m3/s respectively. Using Gumbel's method, estimate the flood
discharge with a return period of 500 years. What are the (a) 95% and (b) 80% confidence limits for this
estimate?
Example 4.3: For the annual flood series data given in Example 3.1, estimate the flood discharge for a
return period of (a) 100 years (b) 200 years and (c) 1000 years by using Log-Pearson Type III
distribution.
Droughts
Droughts are extended severe dry periods. To qualify as a drought, a dry period must have duration of at
least a few months and be a significant departure from normal. Drought must be expected as part of the
natural climate, even in the absence of any long term climate change. However, “permanent” droughts
due to natural climate shifts do occur, and appear to have been responsible for large scale migrations and
declines of civilizations through human history. The possibility of regional droughts associated with
climatic shifts due to warming cannot be excluded.
As shown in Figure 3.3, droughts begin with a deficit in precipitation that is unusually extreme and
prolonged relative to the usual climatic conditions (meteorological drought). This is often, but not
always, accompanied by unusually high temperatures, high winds, low humidity, and high solar radiation
that result in increased evapotranspiration.
These conditions commonly produce extended periods of unusually low soil moisture, which affect
agriculture and natural plant growth and the moisture of forest floor (Agricultural drought). As the
precipitation deficit continues, stream discharge, lake, wetland, and reservoir levels, and water-table
decline to unusually low levels (Hydrological drought). When precipitation returns to more normal
values, drought recovery follows the same sequence: meteorological, agricultural, and hydrological.
Meteorological drought is usually characterized as a precipitation deficit.
4.5.2 Low flow frequency analysis
As noted earlier, the objective of low flow frequency analysis is to estimate quantiles of annual d-day-
average minimum flows. As with floods, such estimates are usually required for reaches without long-
term stream flow records. These estimates are first developed by analyzing low flows at gauging stations.
Low flow analysis at gauging stations:
For gauged reach, low flow analysis involves development of a time series of annual d-day low flows,
where d is the averaging period. As shown in the table below, the analysis begins with a time series of
average daily flows for each year. Then the overlapping d-day averages are computed for the d values of
interest. For each value of d, this creates 365-(d-1) values of consecutive d-day averages for each year.
The smallest of these values is then selected to produce an annual time series of minimum d-day flows. It
is this time series that is then subjected to frequency analysis to estimate the quantiles of the annual d-day
flows.
Example 4.4: Computation of d-consecutive Day averages for low flow analysis. Values in bold are
minimum for the period shown
The probability of occurrence of an event (x≥xT) at least once over a period of n successive years is called
the risk, ̅ . Thus the risk is given by ̅ = 1 - (probability of non-occurrence of the event x≥xT in n years)
It can be seen that the return period for which a structure should be designed depends upon the acceptable
level of risk. In practice, the acceptable risk is governed by economic and policy considerations.
Safety Factor: In addition to the hydrologic uncertainty, as mentioned above, a water resource
development project will have many other uncertainties. These may arise out of structural, constructional,
operational and environmental causes as well as from non-technological considerations such as economic,
sociological and political causes. As such, any water resource development project will have a safety
factor for a given hydrological parameter M as defined below. Safety factor (for the parameter M) equals
(SF) m= =
The parameter M includes such items as flood discharge magnitude, maximum river stage, reservoir
capacity and free board. The difference (Cam - Chm) is known as Safety Margin.
Example 4.5: A bridge has an expected life of 25 years and is designed for a flood magnitude of return
period 100 years. (a) What is the risk of this hydrological design? (b) If 10% risk is acceptable, what
return period will have to be adopted?
Exercise: Annual flood data of a certain river covering the period 1948 to 1979 yielded for the annual
flood discharges a mean of 29,600m3/s and a standard deviation of 14,860m3/s. for a proposed bridge on
this river near the gauging site it is decided to have an acceptable risk of 10% in its expected life of 50
years. (a) Estimate the flood discharge by Gumbel's method for use in the design of this structure (b) If
the actual flood value adopted in the design is 125,000m3/s what are the safety factor and safety margin
relating to maximum flood discharge? (Answers (a) 105,000m3/s and (b) (SF) flood = 1.19, Safety Margin
for flood magnitude = 20,000m3/s)
Stationary time series: If the statistics of the sample (mean, variance, covariance, etc.) as calculated by
equations (5.2)-(5.4) are not functions of the timing or the length of the sample, then the time series is said
to be stationary to the second order moment, weekly stationary, or stationary in the broad sense.
Mathematically one can write as:
+
In hydrology, moments of the third and higher orders are rarely considered because of the unreliability of
their estimates. Second order stationary, also called covariance stationary, is usually sufficient in
hydrology. A process is strictly stationary when the distribution of Xt does not depend on time and when
all simultaneous distributions of the random variables of the process are only dependent on their mutual
time-lag. In another words, a process is said to be strictly stationary if its n-th (n for any integers) order
moments do not depend on time and are dependent only on their time lag.
Non-stationary time series:
If the values of the statistics of the sample (mean, variance, covariance, etc.) as calculated by equations
(5.2)-(5.4) are dependent on the timing or the length of the sample, i.e. if a definite trend is discernible in
the series, then it is a non-stationary series. Similarly, periodicity in a series means that it is non-
stationary.
Mathematically one can write as:
+
White noise time series:
For a stationary ties series, if the process is purely random and stochastically independent, the time series
is called a white noise series. Mathematically one can write as:
0 for all L
+
Gaussian time series:
A Gaussian random process is a process (not necessarily stationary) of which all random variables are
normally distributed, and of which all simultaneous distributions of random variables of the process are
normal. When a Gaussian random process is weekly stationary, it is also strictly stationary, since the
normal distribution is completely characterized by its first and second order moments.
5.4 ANALYSIS OF HYDROLOGIC TIME SERIES
Records of rainfall and river flow form suitable data sequences that can be studied by the methods of time
series analysis. The tools of this specialized topic in mathematical statistics provide valuable assistance to
engineers in solving problems involving the frequency of occurrences of major hydrological events. In
particular, when only a relatively short data record is available, the formulation of a time series model of
those data can enable long sequences of comparable data to be generated to provide the basis for better
estimates of hydrological behavior. In addition, the time series analysis of rainfall, evaporation, runoff
and other sequential records of hydrological variables can assist in the evaluation of any irregularities in
those records. Cross-correlation of different hydrological time series may help in the understanding of
hydrological processes.
Tasks of time series analysis include:
(1) Identification of the several components of a time series.
(2) Mathematical description (modeling) different components identified.
If a hydrological time series is represented by X1, X2, X3,... Xt,… then symbolically, one can represent the
structure of the Xt by:
X ⇔ [Tt, Pt, and Et]
Where Tt is the trend component, Pt is the periodic component and Et is the stochastic component. The
first two components are specific deterministic features and contain no element of randomness. The third,
stochastic, component contains both random fluctuations and the self-correlated persistence within the
data series. These three components form a basic model for time series analysis.
The aims of time series analysis include but not limited to:
(1) Description and understanding of the mechanism,
(2) Monte-Carlo simulation,
(3) Forecasting future evolution,
Basic to stochastic analysis is the assumption that the process is stationary.
The modelling of a time series is much easier if it is stationary, so identification, quantification and
removal of any non-stationary components in a data series is under-taken, leaving a stationary series to be
modeled.
If the end points of the mass curve are joined by a straight line AB, then its slope represents the average
discharge of the stream over the total period for which the mass curve has been plotted. If a reservoir is to
be constructed to permit continuous release of water at this average value of discharge for the period, then
the capacity required for the reservoir is represented by the vertical intercept between the two straight
lines A1B1 and A11B11 drawn parallel to AB and tangent to the mass curve at the lowest tangent point C
and the highest tangent point D, respectively. If the reservoir having this capacity is assumed to contain a
volume of water equal to AA1 at the beginning of the period, then the reservoir would be full at D and it
would be empty at C.
However, if the reservoir was empty in the very beginning, then it would be empty again at point E and
also during the period from F to K. On the other hand if the reservoir was full in the very beginning it
would be full again at points F and K, and between points A and E' there will be spill of water from the
reservoir. In the earlier discussions the rate of demand has been assumed to be constant.
However, the rate of demand may not be always constant, in which case the demand curve will be curve
with its slope varying from point to point in accordance with the variable rate of demand at different
times. In this case also the required capacity of the reservoir can be determined in the same way by super
imposing the demand curve on the mass curve from the high points (or beginning of the dry period) till
the two meet again. The largest vertical intercept between the two curves gives the reservoir capacity. It is
however essential that the demand curve for the variable demand coincide chronologically with the mass
curve of stream flow, i.e. June demand must coincide with June inflow and so on.
Example 6.2: Reservoir Capacity determination by the use of flow duration curve
Determine the reservoir capacity required if a hydropower plant is designed to operate at an average flow.
Solution: The average flow is 340.93m3/s
i) First option: Storage is same as the hatched area under flow duration curve.
In a detailed study, the sediment size distributions also have to be determined for question 1. Question 2
may also involve determining the location of the deposits and the concentration and grain size distribution
of the sediments entering the water intakes.
In general, there are two approaches to the sedimentation problem:
1. The reservoir is constructed so large that it will take a very long time to fill. The economic value of the
project will thereby be maintained.
2. The reservoir is designed relatively small and the dam gates are constructed relatively large, so that it is
possible to remove the sediments regularly by flushing. The gates are opened, lowering the water level in
the reservoir, which increases the water velocity. The sediment transport capacity is increased, causing
erosion of the deposits.
A medium sized reservoir will be the least beneficial. Then it will take relatively short time to fill the
reservoir, and the size is so large that only a small part of the sediments are removed by flushing. The
flushing has to be done while the water discharge in to the reservoir is relatively high. The water will
erode the deposits to a cross-stream magnitude similar to the normal width of the river. A long and
narrow reservoir will therefore be more effectively flushed than a short and wide geometry. For the later,
the sediment deposits may remain on the sides.
The flushing of a reservoir may be investigated by physical model studies.
Another question is the location of sediment deposits. Figure 5.5 shows a longitudinal profile of the
reservoir. There is a dead storage below the lowest level the water can be withdrawn. This storage may be
filled with sediments without affecting the operation of the reservoir.
Qs is the sediment load, Qw is the water discharge and a and b are constants, obtained by curve fitting
PART II
CHAPTER ONE: GROUNDWATER RESOURCES & OCCURRENCE
The study of groundwater flow is equally important as studying the surface water resources since about
22% of the world‟s fresh water resources exist in the form of groundwater. Further, the subsurface water
forms a critical input for the sustenance of life and vegetation in arid zones. Because of its importance as
significant source of water supply, various aspects of groundwater dealing with the exploration,
development and utilization have been extensively studied by workers from different disciplines, such as
geology, geophysics, geochemistry, irrigation engineering, hydraulic engineering and civil engineering
etc.
Hydrological cycle: Almost all water on earth participates in a continuous movement, the so called the
hydrologic cycle. The hydrologic cycle represents the sequence of events when water drops from the
atmosphere to the earth and hydrosphere (water bodies such as rivers, lakes, seas and oceans covering the
earth‟s surface) and then goes back to the atmosphere. In other words hydrologic cycle can be defined as
the circulation of water evaporated from the sea through the atmosphere to the land and then via surface
and subsurface routes back to the sea.
River flow to the sea shows the existence of this cycle. No resulting rising of sea level occurs and the
rivers do receive water again and again. Evidently a return flow of water exists from the sea to the sources
of the rivers. Ocean water vapour and transportation takes place through the atmosphere to the continents.
Evaporation occurs above as well as on the land surface. The sun generates the energy to sustain the cycle
Major processes involved in the hydrologic cycle include the following.
Precipitation - any form of water which falls on the surface of the earth
Evaporation - the transfer of water into the atmosphere from a free water surface, a bare soil or
interception on a vegetal cover
Transpiration - the process by which water in plants is transferred to the atmosphere as water
vapour.
Infiltration - the process of water entry into a soil from rainfall snow melt, or irrigation
Percolation – the process of water entry into the saturated zone or the groundwater table.
Surface runoff - the flow of water over the land surface
Groundwater flow - The movement of water in the subsurface
Part of the rain falling over the land surface infiltrates into the soil and the remaining flows down as
surface runoff. From the point of view of water resources engineering, the surface water forms a direct
source which is utilized for a variety of purposes. However, most of the water that infiltrates into the soil
travels down to recharge the vast groundwater stored at a depth within the earth. In fact, the groundwater
reserve is actually a huge source of fresh water and is many times that of surface water. Such large
water reserve remains mostly untapped though locally or regionally, the withdrawal may be high.
Table 1.1: Variation of groundwater density based on temperature and total dissolved solids (TDS)
concentration.
Temperature (oC) Density (Kg/m3) TDS (mg/l) Density at 4oC(Kg/m3)
0 999.87 0 1000
4 1000 1000 1000.70
5 999.90 5000 1003.60
10 999.75 100000 1072
20 998.27
Density differences as a result of variation in TDS concentration are more pronounced than variations
resulting from changes in temperature. Groundwater is contained in rocks. Rocks may be classified as
consolidated and unconsolidated. Consolidated rocks include granites, basalt, gneiss, sandstone, shale etc.
1.2.1. Unsaturated Zone/ Zone of aeration
Unsaturated Zone: This is also known as zone of aeration. In this zone the soil pores are only partially
saturated with water. The space between the land surface and the water table marks the extent of this
zone. Further, the zone of aeration has three sub zones: soil water zone, capillary fringe and intermediate
zone.
The soil water zone lies close to the ground surface in the major root band of the vegetation from which
the water is lost to the atmosphere by evapotranspiration. Capillary fringe on the other hand hold water by
capillary action. This zone extends from the water table upwards to the limit of the capillary rise. The
intermediate zone lies between the soil water zone and the capillary fringe.
The thickness of the zone of aeration and its constituent sub-zones depend upon the soil texture and
moisture content and vary from region to region. The soil moisture in the zone of aeration is of
importance in agricultural practice and irrigation engineering. This part is however concerned only with
the saturated zone.
Groundwater is the water which occurs in the saturated zone. All earth materials, from soils to rocks have
pore spaces although these pores are completely saturated with water below the groundwater table or
phreatic surface (GWT). From the groundwater utilization aspect only such material through which water
moves easily and hence can be extracted with ease are significant. Natural variations in permeability and
ease of transmission of groundwater in different saturated geological formations lead to the recognition of
Aquifer, Aquitard, Aquiclude and Aquifuge.
a) Aquifer: This is a water-bearing layer for which the porosity and pore size are sufficiently large that
which not only stores water but yields it in sufficient quantity due to its high permeability.
Unconsolidated deposits of sand and gravel form good aquifers (e.g. sand, gravel layers).
b) Aquitard: It is less permeable geological formation which may be capable of transmitting water (e.g.
sandy clay layer). It may transmit quantities of water that are significant in terms of regional
groundwater flow.
c) Aquiclude: is a geological formation which is essentially impermeable to the flow of water. It may be
considered as closed to water movement even though it may contain large amount of groundwater due
to its high porosity (e.g. clay).
d) Aquifuge: is a geological formation, which is neither porous nor permeable. There are no
interconnected openings and hence it cannot transmit water. Massive compact rock without any
fractures is an aquifuge.
For a description or mathematical treatment of groundwater flow the geological formation can be
schematized into an aquifer system, consisting of various layers with distinct different hydraulic
properties. The aquifers are simplified into one of the following types (see Fig. 1.5).
a) Unconfined aquifer (also called phreatic or water table aquifer): Such type of aquifer consists of
a pervious layer underlain by a (semi-) impervious layer. This type of aquifer is not completely
saturated with water. The upper boundary is formed by a free water-table (phreatic surface) that is in
direct contact with the atmosphere. In most places it is the uppermost aquifer.
b) Confined aquifer: Such an aquifer consists of a completely saturated pervious layer bounded by
impervious layers. There is no direct contact with the atmosphere. The water level in wells tapping
these aquifers rises above the top of the pervious layer and sometimes even above soil surface
(artesian wells).
c) Semi-confined or Leaky aquifers: consists of a completely saturated pervious layer, but the upper
and/or lower boundaries are semi-pervious. They are overlain by aquitard that may have inflow and
outflow through them.
d) Perched aquifers: These are unconfined aquifers of isolated in nature. They are not connected with
other aquifers.
The following are some of the groundwater flow parameters or aquifer properties which are important in
the storage and transmission of water in aquifers.
1. Porosity (n)
The porosity, n is the ratio of volume of the open space in the rock or soil to the total volume of soil or
rock.
Vv
n *100 (1.1)
VT
Where:
Vv = the pore volume or volume of voids
VT = the total volume of the soil
Porosity is also the measure of water holding capacity of the geological formation. The greater the
porosity means the larger is the water holding capacity. Porosity depends up on the shape, size, and
packing of soil particles. Porosity greater than 20% is considered large; 5-20% medium and less than 5%
is small.
Table 1.2: Variation of porosity based on the rock type
Type of rock Range of porosity Type of rock Range of porosity
Unconsolidated % Consolidated %
Gravel 0.2-0.4 Basalt 0.05-0.5
Sand 0.2-0.5 Lime stone 0.05-0.5
Silt 0.3-0.5 Sand stone 0.05-0.3
Clay 0.3-0.7 Shale 0.0-0.1
While porosity gives a measure of the water storage capability of a formation, not all the water held in the
pores is available for extraction by pumping or draining by gravity. The pores hold back some water by
molecular attraction and surface tension. The actual volume of water that can be extracted by the force of
gravity from a unit volume of aquifer material is known as the specific yield, Sy. The fraction of water
held back in the aquifer is known as specific retention, Sr.
2. Specific yield (Sy)
When water is drained by gravity from saturated material, only a part of the total volumes is released. The
ratio of volume of water in the aquifer which can be extracted by the force of gravity or by pumping wells
to the total volume of saturated aquifer is called Specific yield (Sy).
V
S y w *100 (1.2)
VT
Where:
Sy= Specific yield,
Vw=the volume of extractible water,
VT = the total volume of the soil.
All the water stored in the water bearing formations can‟t be extracted by gravity drainage or pumping; a
portion of water remains held in the voids of the aquifer by molecular and surface tension forces.
For unconfined aquifers the specific yield (Sy) is defined as the amount of water stored or released in an
aquifer column with a cross-sectional area of 1m2 as a result of a 1m increase or decrease in hydraulic
head.
Table 1.3: Common values for Sy
Type of Rock Range Mean
Medium gravel 0.17-0.44 0.24
Fine gravel 0.13-0.40 0.28
Medium sand 0.16-0.46 0.32
Fine sand 0.01-0.46 0.33
Silt 0.01-0.39 0.20
Clay 0.01-0.18 0.06
Tuff 0.02-0.47 0.21
Sandstone 0.02-0.30 0.21
sandstone (non-cemented) 0.12-0.30 0.27
Siltstone 0.01-0.28 0.12
storage coefficients as 0.001.Calculate the annual rechargeable ground water storage from the area.
Calculate the average well yield.
Solution
Annual rechargeable ground water storage=A x∆piezo. Level x Storage coefficient
=930x106x (12-5) x0.001
=6.51x106m3=6.51Mm3
Average well yield/day x pumping days x numbers of wells
=Annual fluctuation in piezo.head (average well yield /day)
= Annual fluctuation volume of water/ (pumping days x No. of wells)
=6.51Mm3/ (250x40) =6.51x106m3/1000
=651m3/day=27.125m3/h=7.5lt/sec
2) In an unconfined aquifer covering 2 sq.km, the original water table was 12.3m below ground level.
Pumping of 1Mm3 of water from the area dropped the water table to 15.1m below GL. Calculate specific
yield and retention of the aquifer if porosity of aquifer material is 23%.
Solution
Volume of water pumped out=Aquifer area x change in ware table x specific yield
Sy =106/(2 x 106m2 x (15.1-12.3)m)=0.1786 or 17.86%
Sr=n-Sy=23-17.86=5.13%
4. Coefficient of permeability (k)
Coefficient of permeability is also called hydraulic conductivity reflects the combined effects of the
porous medium and fluid properties. It is an ease with which water can flow through a soil mass or rock
and usually it is the capacity of geological formation to transmit water. Coefficient of permeability is
primarily dependent on the soil property and water contained in it. Unconsolidated rocks are permeable
when the pore spaces between grains are sufficiently large.
K=ki.kw (1.5)
Where:
K = Coefficient of permeability,
ki = Intrinsic permeability; depending on rock properties (such as grain size & packing),
kW = Permeability depending on fluid properties (such as density and viscosity of water)
Further for unconsolidated rocks, from an analogy of laminar flow through a conduit the coefficient of
permeability K can be expressed as:
K = C dm2 ( / ) = C dm2 (g / ) (1.6)
Where:
dm = Mean pore size of the porous medium (m),
= unit weight of the fluid (kg/m2s2),
= density of the fluid (kg/m3),
g = acceleration due to gravity (m/s2),
= dynamic viscosity of the fluid (kg/ms),
C = a shape factor which depends on the porosity, packing, shape of grains and grain-size distribution
of the porous medium. Thus for a given porous material K 1/ where = kinematic viscosity = / = f
(temperature).
Eq (1.6) can be split into two components: intrinsic permeability (ki) and permeability due to fluid
properties (kw). ki = C dm2 and kw = / = g/.
n3
According to Kozeny-Carman‟s formula K i Cd m
2
2
(1 n)
5. Transmissivity (T) and Vertical Resistance (C):
Transmissivity is the product of horizontal coefficient of permeability and saturated thickness of the
aquifer. For an isotropic aquifer (Kx = Ky = K):
T = KB (1.7)
Where:
T = aquifer Transmissivity (m2 / day),
B = aquifer thickness (m).
The vertical resistance of an Aquitard is defined as the ratio of the thickness of the aquitard and its
permeability in the vertical direction (kz):
C = D / KZ (1.8)
Where:
C = vertical resistance (days),
D = thickness of the Aquitard (m).
Values for the transmissivity of aquifers and vertical resistances of an aquitard are usually determined
from pumping tests. There are different stratifications in aquifers may be stratification with different
permeability in each stratum. Two main kinds of stratifications (flow situations in stratified aquifers) are
possible in aquifers; horizontal and vertical stratifications.
6. Storage Coefficient (S)
The amount of water stored or released in an aquifer column with a cross sectional area of 1m2 for a 1m
increase or drop in head is known as storage coefficient. Storage coefficient of unconfined aquifer is
equal to the specific yield.
In confined or semi-confined aquifers water is stored or released from the whole aquifer column mainly
as a result of elastic changes in porosity and groundwater density. Common values for the storage
coefficients for confined and semi-confined aquifers range form 10-7 to 10-3.The volume of water drained
from an aquifer, Vw may be found from the following equation.
Vw=SxAxh
Where A is horizontal area and h is fall in head
7. Specific Storage (Ss)
In a saturated porous medium that is confined between two transmissive layers of rocks, water will be
stored in the pores of the medium by a combination of two phenomena; water compression and aquifer
expansion
For confined aquifer, the relation between the specific storage and the storage coefficient is as follows:
S = Ss*b (1.11)
Where:
S = Storage coefficient (dimensionless),
b = aquifer thickness (m)
Specific Storage is also called elastic storage coefficient and is given by the following expression.
Ss=g (+n) (1.12)
Where:
=fluid (water) density,
g=gravitational acceleration,
=aquifer compressibility,
n= porosity,
=water compressibility.
Elastic storage is the only storage occurring in semi-confined and confined aquifers.
8. Leakage Factor (L)
The leakage factor or characteristics length is a measure of the spatial distribution of the leakage through
an Aquitard into a leaky aquifer and vice versa.
L KDc
K and D are the hydraulic conductivity and thickness of the leaky aquifer respectively while c is hydraulic
resistivity. Large values of L indicate a low leakage rate through the Aquitard.
Groundwater in its natural state is invariably moving. This movement is governed by established
hydraulic principles. The rate of groundwater movement depends upon the slope of the hydraulic head
(hydraulic gradient), and intrinsic aquifer and fluid properties.
The flow through aquifers, most of which are natural porous media, can be expressed by what is known
as Darcy‟s law, which is one of the established hydraulic principles.
Henry Darcy, a French hydraulic engineer, observed that the rate of laminar flow of a fluid (of constant
density and temperature) between two points in a porous medium is proportional to the hydraulic gradient
(dh/dl) between the two points (Darcy 1856). The equation describing the rate of flow through a porous
medium is known as Darcy‟s Law and is given as:
Where
Q = volumetric flow rate [L3T-1]
K = hydraulic conductivity [LT-1]
A = cross-sectional area of flow [L2]
The experimental verification of Darcy‟s law can be performed with water flowing at a rate Q through a
cylinder of cross-sectional area A packed with sand and having a piezometric distance L apart (as shown
in fig below).
Total Energy head, or fluid potentials, above the datum plane may be expressed by Bernoulli equation as:
+ + + + +
Where p is the pressure, v is the velocity of flow, g is the acceleration of gravity, z is the elevation and hL
is the head loss and w is the specific weight of water. Subscripts refer to the points of measurement.
Since the velocity of flow in porous media is very small, the velocity head can be neglected (v2/2g ≈0)
and thus the head loss can be obtained as:
( + ) ( + )
Therefore, the resulting head loss is defined as the potential loss with in the sand cylinder. This head loss
is due to the energy loss by frictional resistance dissipated as heat energy. It follows that the head loss is
independent of the inclination of the cylinder
Fig 2.1 Pressure distribution and head loss in flow through a sand column
2.1.2 Specific Discharge
Specific discharge is also called as the Darcy Velocity. It is the discharge Q per cross-section area, A. The
specific discharge is designated by q.
Q h
Form Darcy‟s equation, q = k
A
h dh
Taking the limit as 0 i.e. lim it K k
0
dl
-q=
The Darcy velocity (v) or the specific discharge (q) assumes that flow occurs through the entire x-section
of the material without regard to solids & pores. Actually, the flow is limited to the pore space only so
that is the average interstitial velocity.
Where n = porosity
+ + + +
∗
Va > v
To define the actual flow velocity (Va), one must consider the microstructure of the rock material. The
actual velocity is non-uniform, involving endless accelerations, decelerations, and changes in direction.
Thus the actual velocity depends on specifying a precise point location within the medium.
2.1.3 Validity of Darcy’s law
In general the Darcy‟s law holds well for
i) Saturated & unsaturated flow.
ii) Steady & unsteady flow condition
iii) Flow in aquifers and aquitards.
iv) Flow in homogenous & heterogeneous media
v) Flow in isotropic & an isotropic media.
vi) Flow in rocks and granular media.
Darcy‟s law is valid for laminar flow condition as it is governed by the linter law.
( )
In flow through pipes, it is the Reynolds number(R) to distinguish b/n laminar flow & turbulent flow.
+
For the flow in porous media, v is the Darcy velocity and D is the effective grain size (d 10) of a
formation/media. D10 for D is merely an approximation since measuring pore size distribution a complex
research task.
Experiments show that Darcy‟s law is valid for NR < 1 and does not go beyond seriously up to NR =10.
This is the upper limit to the validity of Darcy‟s laws.
Fortunately, natural underground flow occurs with NR < 1. So Darcy‟s law is applicable. Deviations from
Darcy‟s law can occur where steep hydraulic gradients exist; such as near pumped wells. Turbulent flow
can contain large underground openings.
2.2 HYDRALIC CONDUCTIVITY
The famous Darcy‟s law (1856), which describes the flow of fluids through (inter-granular) porous media,
was derived experimentally after hundreds of laboratory tests with the apparatus. In its basic form, this
linear law states that the rate of fluid flow (Q) through a sand sample is directly proportional to the x-
sectional area of the flow (A) and the loss of hydraulic head b/n two points of measurements ( h ), and it
is inversely proportional to the length of the sample L.
( )
Therefore, K is the proportionality constant of the law & called hydraulic conductivity and has the unit of
velocity. It can also be called as the coefficient of permeability.
A medium has a unit hydraulic conductivity if it will transmit in unit time a unit volume of GW at
prevailing kinematic viscosity through a cross section of unit area, measured at right angles to the
direction of flow, under unit hydraulic gradient.
Generally, hydraulic conductivity is a coefficient of proportionality describing the rate at which water can
move through a permeable medium. The density and kinematic viscosity of water must be considered in
determining the hydraulic conductivity.
The general hydraulic equation of continuity of flow, which results from the principle of conservation of
mass, is (for incompressible fluids).
Where
h Is the hydraulic gradient
L
h Is the head loss along the distance L.
The hydraulic gradient is given by i.
i = h (dimensionless)
L
v = Ki (another form of Darcy‟s equation)
The relationship between intrinsic permeability (Ki) and hydraulic conductivity (K) is expressed through
the following formula.
Ki = Kµ/ ρg
Viscosity of a fluid is the property which describes its resistance to flow. The dynamic viscosity (µ) and
the density of fluid (ρ) are related through the kinematic viscosity (ν):
ν = µ/ρg
Therefore, knowing the kinematic viscosity, which is a function of temperature, the intrinsic permeability
can be determined from field experiments having the value of hydraulic conductivity.
Ki = K ν/g
Although it is much better to express the permeability in units of area (m2 or cm2), for reasons of
consistency and easier use in other formulas, in practice it is more commonly given in Darcy‟s (which a
tribute to oil industry).
Fig 2.2 the relation b/n kinematic viscosity and temperature variation (After Maidement and Chow)
2.2.2 Determinations of Hydraulic Conductivity
Hydraulic conductivity in saturated zones can be determined by variety of techniques. These include,
analytical or empirical methods, laboratory methods, tracer tests, augur hole tests and pumping tests of
wells.
Loess 0.08 V
Peat 5.7 V
Schist 0.2 V
Slate 0.00008 V
Till, predominantly sand 0.49 R
Till, predominantly gravel 30 R
Tuff 0.2 V
Basalt 0.01 V
Gabbro, weathered 0.2 V
Granite, weathered 1.4 V
Most commonly used relationship of such a formula has the following general formula.
Where
fs = the grain shape factor
fn = the porosity factor
d= the characteristic grain diameter.
Few formulas give reliable estimates of results because of the difficulty of including all possible
variables in porous media.
It should be clearly understood that these empirical formulas have various limits of application and give
just approximate values of hydraulic conductivity for small point samples. Since they are derived for
different experimental materials and conditions, it is very common that several formulas applied to the
sample will yield several very different values of hydraulic conductivity (K). For preliminary works the
value of C is often taken as 100 and d is the effective grain size (d10)
ii) Laboratory methods
In the laboratory, hydraulic conductivity is determined by permeameters in which flow is maintained
through a small sample of material while measurements of flow rate and head loss are made. A
permeameter is a laboratory device used to measure the intrinsic permeability and hydraulic conductivity
of a soil or rock sample.
There are two types of permeameters:
a) Constant head permeameter
b) Variable head permeameter
The constant head permeameter is the one which can measure the hydraulic conductivities of consolidated
and unconsolidated formations under low heads. Water enters the medium cylinder from the bottom and
is collected as overflow after passing upward through the material/sample (See figure 2.3).
Here water is added to the fall tube; it flows upward through the cylindrical sample and collected as an
overflow. The test in falling head permeameter consists of measuring the rate of fall of the water level in
the tube and collecting volume of water overflow through time.
The flow rate in the tube is
Qtube = atube x dh/dt
Where a is the area of the tube and dh/dt is the rate of fall of head in the tube.
And the rate of flow in the sample is governed by Darcy‟s law.
Thus the flow rate through the sample is
Qsample = -KiA
h2 t
aL dh / h KA dt
h1 0
Field determination of hydraulic conductivity can be made by measuring the time interval for a water
tracer to travel b/n two observation wells or test holes. The tracer can be a die such as sodium flourescein
or salt.
Consider the unconfined aquifer case below where the GW flow is from point A to point B.
The tracer is injected in hole A as a slug after which samples of water are taken from hole B to determine
the time passage of the tracer. B/c the tracer flows through the aquifer with the average interstitial
velocity, va, then;
va = Kh/ (nL)
Where K is the hydraulic conductivity, n is the porosity, L is the distance b/n two points and h is the
difference in head causing flow b/n the points.
But va = L/t
Where
t: is the travel time interval of tracer b/n two holes
K= nL2/ht
Though the method is simple, the results of this method may face serious limitations in the field. Such
as:-
1. The holes need to be close together; otherwise, the travel time interval can excessively be
long. For this requirement, the value of K is highly localized.
2. Unless the flow direction is accurately known, the tracer may miss the d/s hole entirely.
Multiple sampling holes may help, but costly.
3. If the aquifer stratified with layers having different hydraulic conductivities, the first
arrival of the tracer will result in conductivity considerably larger than the average for the
aquifer.
iv) Auger hole method
This method is relatively simple method and most adaptable to shallow water table conditions. The value
of K obtained is essentially that for a horizontal direction in the immediate vicinity of the hole.
The value of K is given by
K = C/864 (dy/dt)
C = Constant (dimensionless)
K = hydraulic conductivity (m/day)
Lw y
The factor 864 yields k values in m/day. And the value of C which is defined based on and
rw Lw
can be obtained from tables.
The most reliable method of estimating aquifer hydraulic conductivity is the pumping test of wells. Based
on observations of water levels near pumping wells an integrated K value over sizable aquifer section can
be obtained. It is the superior method where the sample is not disturbed
Aquifer flow can be one dimensional, two dimensional or more. Darcy‟s equation can be used to calculate
one dimensional flow in aquifers. To obtain the volume rate of flow in aquifer, Darcy‟s velocity is
multiplied by cross sectional area of an aquifer normal to the flow.
Q = Av = -AKdh/dl = Aki i is the hydraulic gradient (slop of water table or piezometric surface)
T = Kb
Where
The saturated thickness for confined aquifer is fairly constant and hence the value of T is constant;
however, the saturated thickness for unconfined aquifers is variable as the water table varies. Hence the
transmissivity for unconfined aquifers vary as a function of the water table variation.
Homogeneity and Isotropy
If hydraulic conductivity is consistent throughout a formation, regardless of position, the formation is
homogeneous. If hydraulic conductivity within a formation is dependent on location, the formation is
heterogeneous. When hydraulic conductivity is independent of the direction of measurement at a point
within a formation, the formation is isotropic at that point.
If the hydraulic conductivity varies with the direction of measurement at a point within a formation, the
formation is anisotropic at that point. Figure 2-4 is a graphical representation of homogeneity and
isotropy.
Geologic material is very rarely homogeneous in all directions. A more probable condition is that the
properties, such as hydraulic conductivity, are approximately constant in one direction. This condition
results because: a) of effects of the shape of soil particles and b) different materials incorporate the
alluvium at different locations. As geologic strata are formed, individual particles usually rest with their
flat sides down in a process called imbrication. Consequently, flow is generally less restricted in the
horizontal direction than the vertical and Kx is greater than Kz for most situations. Layered heterogeneity
occurs when stratum of homogeneous, isotropic materials are overlain upon each other
n
qx = q
i 1
i i(K1b1 + K2b2 +………….+ Knbn) (1)
If the whole aquifer system is taken as taken as homogeneous; then the total flow is:
qx = Kxi(b1 +b2+ ………+ bn) (2)
Where Kx is the horizontal hydraulic conductivity for the entire system
Consider an aquifer system consisting of n horizontal layers each individually isotropic, with different
thickness values. If there is a vertical flow through the system, the flow q per unit horizontal area for the
top layer can be expressed as: qz = K1Δh1/b1
Where Δh1 is the total head loss across the first layer
Solving for Δh1, Δh1 = qzb1/K1
Similarly for the second layer and n layer:
The statement, horizontal hydraulic conductivity ( Kx) is grater than the vertical hydraulic conductivity (
Kz) can also shown with the help of the above derivations. Mathematically, harmonic mean is less than
arithmetic mean; thus, Kx > Kz .
For two dimensional flows in anisotropic media, the approximate value of K must be selected for the
direction of flow. For directions other than horizontal (Kx) and vertical (Kz) the K value Kβ can be
obtained from:
1/ Kβ = cos2β/Kx + sin2 β /Kz Where β is the angle b/n Kβ and the horizontal
Kx
β
Kβ
Kz
Fig 2.7 Hydraulic conductivity in other direction
2.4.3 Average Hydraulic Conductivity
The hydraulic conductivity in horizontal direction (Kx) and in the vertical direction (Kz) defined
previously were the average hydraulic conductivities in their respective directions.
However, it is not customary to determine the hydraulic conductivities of each layer and determine the
average hydraulic conductivities, unless in rare circumstances as limited in research and academic
purposes.
Field methods such as pumping test, auger hole methods, tracer tests etc… allow the computation of
average hydraulic conductivity of a formation.
The overall average hydraulic conductivity is computed from the geometric mean or the arithmetic mean
of the logarithm of the average horizontal and vertical hydraulic conductivities.
K av K x .K z or logKav = (logKx + logKz)/2
General Flow Equations
a. Confined aquifer. The governing flow equation for confined aquifers is developed from application of
the law of mass conservation (continuity principle) to the elemental volume shown in Figure 2-8.
Continuity is given by:
Rate of mass accumulation = Rate of mass inflow - Rate of mass outflow
h h h h
Ss kx ky kz ----------------- (1)
t x x y y z z
Equation above is the general flow equation in three dimensions for a heterogeneous anisotropic material.
Discharge (from a pumping well, etc.) or recharge to or from the control volume is represented as
volumetric flux per unit volume (L3/T/L3 = 1/T):
h h h h
Ss W kx ky kz
t x x y y z z
Where
W=volumetric flux per unit volume [1/T]
Assuming that the material is homogenous, i.e K does not vary with position, equation [1] can be written
as
h h h h
Ss Kx Ky Kz --------------- (2)
t x x y y z z
If the material is both homogenous and isotropic, i.e Kx=Ky=Kz, then equation 2 becomes :
h h h h
Ss K
t x x y y z z
Using the definitions for storage coefficient,(S=bSs), and transmissivity ,(T=Kb), where b is the aquifer
thickness and equation 3 becomes
2 h 2 h 2 h S h
-------------------------------------- (4)
x 2 y 2 z 2 T t
If the flow is steady-state, the hydraulic head does not vary with time and equation 4 becomes
2h 2h 2h
0 ----------------------------------- (5)
x 2 y 2 z 2
b Unconfined aquifer- In an unconfined aquifer, the saturated thickness of the aquifer changes with time
as the hydraulic head changes. Therefore, the ability of the aquifer to transmit water (the transmissivity) is
not constant:
h h h h
kxh kyh kzh S y --------------------- (6)
x x y y z z t
Where
SY= specific yield [dimensionless]
For a homogeneous, isotropic aquifer, the general equation governing unconfined flow is known as the
Boussinesq equation and is given by:
h h h S y h
h h h
x x y y z z K t
-------------------------- (7)
If the change in the elevation of the water table is small in comparison to the saturated thickness of the
aquifer, the variable thickness h can be replaced with an average thickness b that is assumed to be
constant over the aquifer. Equation -7 can then be linearized to the form:
h 2 h 2 h 2 S y h
------------------------------------ (8)
x y z Kb t
Regional ground water flow occurs in ground water basins which usually occupy large areas. Regional
flow may influenced by local groundwater flow phenomena. For example, the construction of canals may
influence the regional natural flow. Another example concerns wells. Pumping from well May also affects
the regional flow and this can often be observed on groundwater head contour lines parallel to the canals
on the maps. Estimates on local flow of groundwater can be obtained using groundwater head contour
maps, flownets or numerical groundwater models. Traditionally, however, these local flow problems were
also solved by analytical methods. In these methods the differential Darcy and continuity equations are
solved in a direct way; either separately or combined. We will briefly discuss the analytical methods by
presenting cases of the flow between canals or two water bodies and the flow to a well.
Confined Groundwater Flow between Two water Bodies
Figure(x-x) shows a very wide confined aquifer of depth B connecting to water bodies. A section of the
aquifer of unit width is considered. The piezometric head at the upstream end is ho and at a distance x
from the upstream end the head is h.
2h
0
x 2
On integrating twice
h C1 x C2
On substitution of the boundary condition h=ho at x=0
h C1 x ho
ho h1
Also at x=L, h=h1 and hence C1
L
ho h1
Thus h ho x
L
This is the equation of the hydraulic grade line, which is shown to vary linearly from ho to h1
dh
q kB
dx
ho h1
q KB
L
Unconfined flow by Dupit’s assumptions
For a similar flow situation in unconfined aquifer, direct analytical solution of a Laplace equation is not
possible; the water table being a stream line and a problem become a free surface flow problem with non-
linear boundary. A simplified method of solution was first developed by Dupuit (1863) in the study of
steady flow to wells and ditches and further developed by Forchiemer.
a. Velocity of flow is proportional to hydraulic gradient (slope of water table). I.e. The hydraulic
grade line is equal to the free surface slope and does not vary with depth.
b. Flow is horizontal and uniform everywhere in the vertical section (see figure 4.2). I.e. The
curvatures of the free surface is very small so that the streamlines can be assumed to be
horizontal at all directions
Fig: 2.9 steady flow in an unconfined aquifer between two water bodies with vertical boundaries.
The gradient of water table in unconfined aquifer flow is not constant; it increases in the direction of
flow.
All the velocities in Dupuit assumption are horizontal, while the same velocities of magnitude have a
vertical component in actual condition. Thus, this demands a greater saturated thickness for the same
given discharge. At the d/s boundary, a discontinuity in flow forms because no consistent flow pattern can
connect a water table directly to the d/s free water surface.
From Darcy‟s law, assuming flow to be horizontal and uniform everywhere in a vertical section, the
discharge per unit width is given by:
dh
q Av (h.1)( K )
dx
dh
( Kh )
dx
qdx Khdh
qL
2
K 2
h2 h1
2
q
K 2
2L
h2 h1 ----------------------------------- (1)
2
q
K 2
2x
h2 h1 ------------------------------------ (2)
2
h h1
2 x 2
2
h2 h1
2
This indicates that for a given discharge, the water table is in parabolic form.
One dimensional Dupit’s Flow with Recharge
Consider an unconfined aquifer on a horizontal impervious base situated b/n two water bodies with a
difference in surface elevation, as shown in figure(x-x). Further, there is a recharge at a constant rate of R
m3/s per unit horizontal area due to infiltration from the top of the aquifer. The aquifer is of infinite length
and hence one dimensional method of analysis is adopted. A unit width of aquifer is considered for the
analysis.
2h 2R
x 2
K
R
h 2 x 2 C1 x C 2
K
Where C1 and C2 are constants of integration
The boundary conditions are:
i. At x=0,h=ho hence C2 =ho2
R 2
At x=L,h=h1 hence h1 ho L C1 L
2 2
ii.
K
2 RL2
ho h1 2
K
C1
L
RL2
( ho 2
h1 2
)
Rx 2 K
h
2
x ho 2 -------------------(*)
K L
The water table is thus an ellipse represented by equation (*). The value of h will in general rise above
ho, reaches a maximum at x=a, and falls back to h1 at x=Las shown in the figure above. The value of a is
dh
obtained by equation 0 and is given by
dx
L K ho 2 h12
a
2 R 2L
The location x=a, is called the water table divided. In figure (xx) the flow to the left of the divide will be
to the upstream water body and the flow to the right of the divide will be to the downstream water body.
dh
qx Kh
dx
2 RL2
ho h12
Rx K
K
K 2L
L K
qx R x
ho 2 h12
2 2L
It is obvious the discharge qx varies with x. At the upstream water body=0 and discharge
RL K
qo qx 0 (ho 2 h12 )
2 2L
At the downstream water body x=L and
RL K
q1 qx L (ho 2 h12 ) RL qo
2 2L
2.5 Groundwater flow directions
2.5.1 Flow nets
Flow net is a net work flow lines and equipotential lines intersecting at right angles to each other.
The imaginary path which a particle of water follows in its course of seepage through a saturated soil
mass is called flow line. An equipotential line is the line which joins points with equal potential head.
Equipotential lines are lines that intersect the flow lines at right angles. At all points along the
equipotential line, the water would rise in a piezometric tube to a certain elevation known as piezometric
level.
For specified boundary conditions, flow lines and equipotential lines can be mapped in to two dimensions
to form a flow net. The two sets of lines form an orthogonal pattern of small squares. See fig. below.
Fig 2.10
Portion of an
orthogonal
flow net
formed by
flow and equipotential lines
Basically flow net is constructed to quantify the flow rate through a medium. Consider the portion of a
flow net shown in figure above. The hydraulic gradient is given by:
i = -dh/ds
and the constant flow rate , between two adjacent lines is given by
q = -K.dm.dh/ds for unit thickness. But for the squares of the flow net, the approximation ds ~ dm can be
made. Therefore, the above equation reduces to q = Kdh
Applying this to an entire flow net, where the total head loss h is divided in to n squares b/n two adjacent
flow lines, then
dh = h/n
If the flow field is divided in to m channels by flow lines, then the total flow rate is:
Q = mq = Kmh/n
Thus the geometry of the flow net, together with the hydraulic conductivity and head loss, enables the
total flow to be computed directly.
Flow of water in earth mass is in general three dimensional. Since the analysis of three dimensional flows
is too complicated, the flow problems are usually solved in the assumption that the flow is two
dimensional.
The three common types of boundaries of GW flow are:
i) Impermeable ( No flow boundary)
ii) Constant head boundary( head not varies)
iii) Water table ( Variable head boundary)
i) Draw carefully the boundaries of the region to scale and sketch a few stream lines on the
drawing
ii) Identify the components of flow nets which are regarded as the boundaries of the flow region
(at least two in many flow nets).
iii) Begin and end stream lines at equipotential surface and they must intersect these
equipotential surface at right angles.
iv) As a first trail, use not more than four to five channels.
v) Follow the principle of „Whole‟ to part.
vi) All the flow and equipotential lines should be smooth and there should be no any sharp
transition b/n straight and curved lines.
2.5.2 Flow in relation to GW Contours
Contour maps of water levels (both unconfined and confined aquifers) are made in the majority of hydro
geologic investigations and, when properly drawn, represent a very powerful tool in aquifer studies.
Although commonly used for determination of GW flow directions, contour maps, when accompanied
with other data, also allow for the analysis and calculation of the flow velocity, particle travel time,
hydraulic conductivity and transmissivity.
At least several data sets collected in different hydrologic season should be used to draw GW contour
maps for the area of interest. In addition to recordings from piezometres, monitoring and other wells,
every effort should be made to record elevations of water surface in the nearby surface streams, lakes,
seas, ponds and other bodies including cases when these bodies seem “too far” to influence GW flow
pattern. In addition, one should gather information about hydro-meteorological conditions in the area for
preceding months paying attention to the presence of extended wet or dry periods. All of this information
is essential for making a correct contour map.
2.5.2.1 Ground water Flow Direction
The direction of GW flow in a localized area of an aquifer can be determined if at least three recordings
of water table (piezoelectric surface) elevations are available. Figure below illustrates the principle of
finding the position of water table in three dimensions using data from three monitoring wells. In a map
or two dimensional views, the water table is represented by the contour lines which connect points with
the same hydraulic head, h. The fastest way to construct contours is by linear triangulation as shown in
the figure (three well methods). The direction of GW flow is indicated by the arrow drawn perpendicular
to and „down‟ the contour lines. This is also the direction of the dip of the water table approximated by
the plate. It is very important to understand that the flow direction determined this way is representative
only of the local area covered by the three monitoring wells. Depending on the hydrologic condition this
direction may change in nearby aquifer portion. For that reason more observation points are usually
established during a hydro geologic investigation to construct a reliable contour map/s.
Contouring Methods
Manual contouring
Contouring with computer programes
Manual contouring
Manual contouring is practically always used in GW studies, either as the only method or in conjunction
with computer based methods. A complete reliance on software contouring could lead to erroneous
conclusions since computer programes are unable to recognize interpretations apparent to a GW
professional such as presence of geologic boundaries, varying porous media, influence of surface water
bodies or principles of GW flow. Thus manual contouring and/or manual reinterpretation of a computer
generated maps are essential and integral parts of hydro geologic studies.
Manual contouring is essentially based on triangular linear interpolation combined with the hydro
geologic experience of the interpreter. The first draft map is not necessarily an exact linear interpolation
b/n data points. Rather it is an interpretation of the hydro geologic and hydrologic conditions with
contours that roughly follow numeric data on water table (or piezometric surface) elevations. Whenever
possible the contours should be drawn to satisfy principles described in flow net analysis. This means that
almost inevitable local “irregularities” in water table elevations should not blur the overall tendency of
GW flow( remember that GW contour map is also a flow net without stream lines shown and the a
graphical solution of the two dimensional flow field). Unless there is a valid hydro geologic explanation
(example: presence of pumping well) these depressions are probably the result of erroneous contouring or
data. Similarly, “mysterious” local mounds in water table should be carefully examined.
One of the most important aspects of constructing contour maps in alluvial aquifers is to determine the
relationship between ground and surface waters. Sometimes water flows from river to an aquifer (which
in hydraulic contact with a river) and vice versa. If the flow is from aquifer to river/stream, then it is
called gaining (effluent) stream. However, if the flow of water is from river to an aquifer, then the stream
is said to be losing (influent) stream. In some complicated situations, the two basic cases can co-exist.
Fig
2.12
Conto
uring
using
the
three
well
methods
Contouring with computer programes
In this computer era, we can have to different programes which undertake the contouring of groundwater
levels. Some of them may be Arc View GIS, Surfer Golden software, AutoCAD and so on….
Most of them need elevation of GW levels as an input. But the drawn contours should be checked against
the true conditions in some situations since software usually depend on the given data and not able to
catch up the natural condition in the field.
Since ancient times, wells have been dug or drilled into the subsurface to access groundwater. Prior to the
development of drilling technologies, buckets were used to collect water from shallow hand-dug wells.
Modern groundwater wells can be thousands of meters deep and allow extraction of large quantities of
water with electric pumps.
While wells are used in a number of different applications, they find extensive use in water supply and
irrigation engineering practice. Drinking water for example is obtained in many communities from
groundwater wells. As water is extracted from a well, the water level within the well drops and the water
in the surrounding aquifer flows towards the well causing a lowering of the water level extending outward
from the well. The drop in water level is greatest immediately adjacent to the well and decreases radially
outward creating a feature called the cone of depression. As pumping continues, the cone of depression
extends out farther gathering water from a larger cylindrical volume surrounding the well. The expansion
of the cone of depression will continue until the volume of water intercepted or drawn by the well equals
the pumping rate. Besides aquifer water, the water drawn by a well can also be recharge from the ground
surface, adjacent aquifers, streams, lakes or oceans.
Impermeable boundaries formed by low hydraulic conductivity materials (bedrock, faults, etc.) will halt
the progression of the cone of depression at their location.
Knowledge of the drop in water level and pattern of groundwater flow resulting from well pumping is
necessary for assessing environmental impacts in many situations.
Excessive drops in groundwater levels over regional scales can result in adverse impacts to stream flows,
vegetation and the use of shallow wells. At sites of groundwater contamination, the cone of depression
can expand outward from the pumping well and “capture” the contaminated water.
3.1 Steady Radial Flow to a well
When a well is pumped, water is removed from the aquifer surrounding the well, and the water table or
piezometric surface, depending on the type of aquifer, is lowered. The drawdown at a given point is the
distance the water level is lowered. A drawdown curve shows the variation of drawdown with distance
from the well. In three dimensions the drawdown curve describes a conic shape known as the cone of
depression. Also the outer limit of the cone of depression defines the area of influence of the well.
3.1.1 Steady Flow to a well in Confined Aquifer
The radial flow equation which relates the well discharge to drawdown for a well completely penetrating
a confined aquifer can be derived by referring fig 3.1. In this case the flow is assumed to be two-
dimensional to a well centered on a circular island and penetrating a homogeneous and isotropic aquifer.
Since the flow is horizontal everywhere in confined aquifer case, the Dupuit‟s assumption applies without
error. Using the plane polar coordinates for the well and its surrounding; the well discharge at any
distance r from the well equals
dh
Q AV 2rbK
dr
for steady radial flow to a well. Rearranging and integrating for boundary conditions at the well,h=hw and
r=rw, and at the edge of the islands, h=ho and r=ro yields
Qdr 2rbKdh
ro ho
dr
Q 2bK dh
rw
r hw
2Kb(ho hw )
Q
ln( ro )
rw
With the negative sign neglected.
For a well penetrating an extensive confined aquifer, this equation shows that h increases as r increases.
Yet, the maximum h is the initial uniform hO. Thus from theoretical point of view, steady radial flow in an
extensive aquifer does not exist b/c the cone of depression must expand indefinitely with time. However,
from practical stand point, h approaches ho with distance from the well, and the drawdown vary with the
logarithm of the distance from the well
If the values of head(h) are known (h1 and h2) at the respective positions of distance r1 and r2
respectively from the well, then the flow equation can be written as :-
2bK (h2 h1 )
Q
r
ln( 2 )
r1
Where r2 > r1 and h2 > h1 (Refer Todd, fig 4.5)
The above equation is known as an equilibrium equation or Them Equation enables one to determine the
values of hydraulic conductivity (K) and Transmissivity (T) of a confined aquifer from pumping test data.
Because any two points define the logarithmic drawdown curve, the method consists of measuring
drawdowns in two observation wells at different distances from a well pumped at constant rate.
The term „steady flow’ in well hydraulics, hence, refers to the state of flow in which the change b/n two
consecutive drawdowns/water levels become negligibly small. That is the time after long period of
pumping, after which aquifer flow and rate of pumping become almost equal.
Exercise: - Prove that equation for flow of water in confined aquifer towards a well is given by:
2T ( s1 s 2 )
Q
r
ln( 2 )
r1
An equation for steady radial flow to a well in an unconfined aquifer also can be derived with the help of
the Dupuit assumptions. As shown in figure 3.2 the well completely penetrates the aquifer to the
horizontal base and concentric boundary of constant head surrounds the well.
(ho 2 hw )
2
Therefore, Q K
ro
ln
rw
Converting heads and radii at two observation wells (as shown in figure)
(h2 h1 )
2 2
Therefore, Q K
ln r 2
r1
This equation fails to accurately estimate the drawdown curve near the well because large vertical flow
components contradict with the Dupuit‟s assumption; however, estimates of hydraulic conductivity (k)
values for a given heads are good. In practice drawdowns should be small in relation to the saturated
thickness of the confined aquifer. Then the average Transmissivity can be estimated from the equation:-
T K
h1 h2
2
Where drawdowns are appreciable, the heads h1 and h2 in the above equation can be replaced by (ho-s1)
and (ho-s2), Then the Transmissivity for the full thickness expressed as:
ln r2
Q
T = Kh0 =
s 1
r
s
2 2
2 s1 I s 2 2
2h0 2h0
3.2 unsteady flow
3.2.1. Unsteady Flow to a well in a confined aquifer
When a well penetrating an extensive confined aquifer is pumped at a constant rate, the influence of the
discharge extends outward with time. The rate of decline of head times the storage coefficient summed
over the area of influence equals the discharge. Since the water must come from a reduction of storage
within the aquifer, the head will continue to decline as long as the aquifer is effectively infinite; therefore,
unsteady flow exists. The rate of decline, however, decreases continuously as the area of influence
expands.
2 h 2 h S h
x 2 y 2 T t
or in polar coordinates the above equation ,to represent the radial flow in to a well ,takes the form
2 h 1 h S h
r 2 r r T t
Where “h” is head, r is radial distance from the pumped well, S is storage coefficient, T is transmissivity
and t is the time since beginning of pumping.
Theis obtained a solution for the above equation based on the analogy between groundwater flow and heat
induction. By assuming that the well is replaced by a mathematical sink of constant strength and imposing
the boundary conditions h=ho for t=0 and h → ho as r →∞ for t>=0, the following solution is derived.
Q du
s
4T u
e u
u
(3.2)
Equation (3.3) is widely used in practice and preferred over equilibrium equation because
1. a value of S can be determined
2. only one observation well can suffice
3. Shorter period of pumping generally is required
4. No assumption of steady state flow condition is required.
The assumption made to steady state case holds good except the flow is taken as unsteady state.
Because of the mathematical difficulties encountered in applying equation (3.3) several investigators
developed simpler approximate solutions that can be readily applied for field purposes. Three of the
methods namely Theis, Cooper and Jacob and Chow are discussed in the subsequent sections.
i. Prepare the logarithmic plot of W (u) Vs u or W (u) Vs 1/u, known as type curve
ii. Plot values of drawdown(s) against values (r2/t) on logarithmic paper of the same size as for the
type curve.
iii. Superimpose the observed time-drawdown data on the type curve, keeping the coordinate axes
of the two curves parallel and adjusting the graphs till most of the plotted points of the observed
data fall on the segment of the type curve.
iv. Select any convenient match point and record the coordinates i.e. obtain the values of W (u), u,
r2/t and s on the match point.
v. Determine the values of T and S by inserting the values (step 4) in the above equations.
N.B The match point doesn‟t have to be on the type curve. In fact calculations are greatly simplified if
the point is chosen where W (u) = 1 and 1/u=10.
Figure 3.3 the non-equilibrium reverse type curve (Theis curve) for a fully confined aquifer
Figure 3.4 Field data plot on logarithmic paper for Theis curve-marching technique
The analysis presented here is of a pumping test in which drawdown at a piezometer distance, r from the
abstraction well is monitored over time. This is also based upon the Theis analysis
Q u2 u3
s [0.5772 ln u u ................]
4T 2.2! 3.3!
From the definition of u it can be seen that u decreases as the time of pumping increases and as the
distance of the piezometer from the well decreases. So, for piezometers close to the pumping well after
sufficiently long pumping times, the terms beyond lnu become negligible. Hence for small values of u,
the drawdown can be approximated by:
Q r 2S
s 0.577216 ln
4T 4Tt
2.303Q 2.25T
s log 10 2 t
4T r s
It follows that a plot of s against log t should be a straight line. Extending this line to where it crosses that
t axis (i.e. where s is zero and t=to) gives
2.25Tto
1
r 2S
The value of T can be obtained by nothing that if t/to=10, then log t/to=1 ;therefore replacing s by s ,
where s is the drawdown differences per log cycle of t and the value of T can be computed from :
2.30Q
T
4s
The gradient of the straight line (i.e. the increase per log cycle, Δs) is equal to
2.30Q
s
4T
Note: - The procedure in this method is first to solve for T and then solving for S. To avoid large errors
the straight-line approximation is also restricted for small values of u (u<0.01)
In these method measurements of drawdown in an observational well near a pumped well are made and
the observational data are plotted on semi logarithmic paper in the same manner as for the Cooper-Jacob
method. On the plotted curve arbitrary point is chosen and the coordinates, t and s are noted. Then tangent
line to the curve at the chosen point is drawn and the drawdown difference per log cycle time is
computed. The value of F (u) is computed using the following formula.
s
F (u )
s
Having the value of F(u) the corresponding value of W(u) and u can be obtained from graph of F(u),W(u)
and u (after Chow).
3.2.2. Unsteady Flow to a well in unconfined aquifer
The first and by far the simplest approach is to use the same flow situation as for the case of confined
aquifer provided the basic assumptions are satisfied. In general, if the drawdown is small in relation to the
saturated thickness (unconfined aquifer) good approximations are possible with the methods developed
for the confined aquifer.
If the drawdown in the monitoring well does not exceed 25% of the saturated thickness, the Theis
equation can be applied to unconfined aquifers with certain adjustments. For the drawdown that is less
than 10% of the aquifer‟s pre-pumping thickness, it is not necessary to adjust the recorded data since the
error introduced by using the Theis equation is small. When the drawdown is kept between 10% and 25%,
it is recommended to correct the measured values using the following equation derived by Jacob:-
S‟ = s- s2/2h
Where
s‟ = is the corrected drawdown
s = measured drawdown in monitoring well
H = the saturated thickness before pumping started
This correction is needed since the Transmissivity of aquifer changes during the test as the saturated
thickness decreases (remember that for unconfined aquifers, T = Kh where h is the saturated thickness
liable for variation)
If the drawdown in the monitoring well is more than 25%, the equation (Theis and Theis based) should
not be used in the unconfined aquifer analysis.
There are different methods of analysis for unconfined aquifer, when the drawdown due to pumping is
remarkably large. Neuman, Boulton, Hantush etc., methods which are but beyond the scope of the class.
3.3 Unsteady Radial Flow in a Leaky Aquifer- Hantush-Jacob Method and Walton Graphical
Method
Leaky aquifer bounded to and bottom by less transmissive horizons, at least one of which allows some
significant vertical water “leakage” into the aquifer.
Unsteady radial flow for leaky aquifer can be represented in the following equation:
2 h 1 h e S h
r 2 r r T T t
Where
r is the radial distance from a pumping well (m)
e is the rate of vertical leakage (m/day)
When a leaky aquifer, as shown in figure (3.6), is pumped, water is withdrawn both from the aquifer and
from the saturated portion of the overlying semipervious layer. Lowering the piezometric head in the
aquifer by pumping creates the hydraulic gradient within the semi pervious layer; consequently,
groundwater migrates vertically downward in to the aquifer. The quantity of water moving downward is
proportional to the difference between the water table and piezometric head.
Steady-state flow is possible to a well in leaky aquifer because of the recharge through the semipervious
layer. The equilibrium will be established when the discharge rate of the pump equals the recharge rate of
the vertical flow in to the aquifer, assuming the water table remains constant. Solution for this special
steady-state situation area available, but a more general analysis for unsteady flow follows.
The Hantush and Jacob (1955) solution for leaky aquifer presents the following equations (see Figure 3-
6):
s
Q
4T
W u, r → T
B
Q
4s
W u, r
B
r 2S 4Ttu
Where s, Q and r are as shown in the figure and u →S 2
4Tt r
Where
Where
b' is thickness of the aquitard (m)
K‟ is hydraulic conductivity of the aquitard (m/day)
1. Type curves W (u, r / B) Vs 1/u for various values of 1/u and r/B, see Figure 3.7.
2. Field data are plotted on drawdown (s) vs. time on full logarithmic scale.
3. Field data should match one of the type curves for r/B (interpolation if between two lines)
4. From a match point, the following are known values W (u, r / B) , 1/u, t, sw, and r/B
5. Substitute in Hantush-Jacob equation:
T
Q
4s
W u, rB
4Ttu
S
r2
r r
( frommatch) then
B T
K b
' '
2
r
Tb
'
K' B
r2
where
Q is the pumping rate (m3/day)
t is the time since pumping began (day)
r is the distance from pumping well to observation well (m)
b‟ is the thickness of aquitard (m)
K‟ is the vertical hydraulic conductivity of confining bed (aquitard) (m/day)
B is the leakage factor (m)
Where the cones of depression of two nearby pumping well over lap, one well is said to interfere with
another because of the increased drawdown and pumping lift created. For a group of wells forming a well
field, the drawdown can be determined at any point if the well discharges are known, or vice versa. From
the principle of superposition, the drawdown at any point in the area of influence caused by the discharge
of several wells is equal to the sum of the drawdowns caused by each well individually. Thus,
ST sa sb sc sd .... sn
Where
S T is the total drawdown at a given point,
sa sb sc sd .... sn : are the drawdowns at the point caused by the discharge of wells a, b, c
…n respectively.
The summation of drawdowns may be illustrated in a sample way by the well line of Figure 3.8; the
individual and composite drawdown curves are given for Q1 Q2 Q3 clearly, the number of wells and
the geometry of the well field are important in determining drawdowns. Solutions of well discharge for
equilibrium or non-equilibrium equation. Equations of well discharge for particular well patterns have
been developed.
In general, wells in a well field designed for water supply should be spaced as far apart as possible so
their areas of influence will produce a minimum of interference with each other. On the other hand
economic factors such as cost of land or pipelines may lead to a least-cost well layout that includes some
interference. For drainage
Fig 3.8: Individual and composite drawdown curves for three wells in a line.
3.5 Recovery of a well/aquifer
At the end of a pumping test, when the pump is stopped, the water levels in the pumping and observation
wells will begin to rise. This is referred to as groundwater levels, while measurements of draw down
below the original static water level (prior to pumping) during the recovery period are known as residual
draw downs. A schematic diagram of change in water level with time during after the pumping is shown
the figure (fig 3.9).
It is a good practice to measure residual drawdowns b/c analysis of the data enable transmissivity to be
calculated, thereby providing an independent check on pumping test results. Also, costs are nominal in
relation to the conduct of pumping test. Furthermore, the rate of recharge to the well during recovery is
assumed constant and equal to the mean pumping rate, whereas pumping rates often vary and are difficult
to control accurately in the field. During recovery, one can measure the water level variation in the
pumped well itself.
If the well is pumped for a known period of time and then shut down, the draw down there after will be
identically the same as if the discharge had been continued and a hypothetical recharge well with the flow
were superposed on the discharging well at the instant the discharge is shut down. From this principle,
Theis showed that, the residual draw down s‟ can be given as:
2 2
s'
Q
W (u) W (u' ) Where u r S and u' r S
4T 4Tt 4Tt '
and t and t‟ are defined in figure. For r small and t‟ large, the well functions can be approximated by the
first two terms of the This equation and can be written as
2.30Q t
s' log 10
4T t'
Thus, a plot of residual draw down s‟ versus the Logarithm of t/t‟ forms a straight line. The slope of the
line equals 2.30Q/4 T so that for ∆s‟, the residual draw down per log cycle of t/t‟, the transmissivity
becomes
2.30Q
T
4s'
Note: No comparable value of S can be determined by this recovery test method.
Fig 3.9 Draw down and Recovery curves in an observation well near a pumping well
3.6 Well Losses and Specific Capacity
3.6.1 Well Loss
The total DD (sw) at the well face is made up of:
a. Head loss resulting from laminar flow in the formation,
b. Head loss resulting from turbulent flow in the zone close to the well face where Re > 1.
c. Head loss through the well casing and screen
The components under (ii) and (iii) are contributing to the so called well loss.
Therefore, well loss can be expressed as the difference between the actual measured DD in the pumping
well and the theoretical DD which is expressed by the Theis equation and as the result of GW flow
through the aquifer in the undisturbed zone only.
The additional DD, or well loss, which is always present in pumping wells, is created by a combination of
various factors such as: improper well development (drilling fluid left in the formation, mud cake along
the bore hole is not removed, fines from formation are not removed, poorly designed gravel pack and well
screen), turbulent flow near the well and others.
To summarize, the drawdown at a well includes not only that of the logarithmic drawdown curve at the
well face, but also a well loss caused by flow through the well screen and flow inside of the well to the
pump intake.
Because the well loss is associated with turbulent flow, it may be indicated as being proportional to an nth
power of the discharge, as Qn , where n is a constant greater than one. Jacob suggest that a value n=2
might be reasonably assumed.
Taking account of the well loss, the total draw down sw at the well may be written for the steady state
confined case
Q r
sw ln 2 CQ n (1)
2T r1
Where C is a constant governed by the radius, construction and condition of the well. For simply let
ln( r2 r1 )
B (2)
2T
So that the total drawdown in a well can be represented by:
sw saquifer s wellloss BQ CQ n (3)
Therefore, as shown in figure 3-10, the total drawdown sw consists of the formation loss BQ and the well
loss CQn
Figure 3.10: Relation of well loss CQn to draw-down for a well penetrating a confined aquifer
Consideration of Equation (2) provides a useful insight to the relation between well discharge and well
radius. From equations confined aquifer it can be seen that Q varies inversely with ln r2 r1 if all other
variables are held constant. This shows that the discharge varies only a small amount with well radius.
For example, doubling a well radius increases the discharge only 10 percent. When the comparison is
extended to include well loss, however, the effect is significant. Doubling the well radius doubles the
intake area, reduces entrance velocities at almost half, and (if n=2) cuts the frictional loss to less than a
third. For axial flow within the well, the area increases four times, reducing this loss an even greater
extent.
It is apparent that the well loss can be a substantial fraction of total drawdown when pumping rates are
large, as illustrated in Figure 3.11. With proper design and development of new wells, well losses can be
minimized. Clogging or deterioration of well screens can increase well losses in old wells. Based on field
experience
Figure 3.11 Variation of total drawdown w s, aquifer loss BQ, and well loss CQn with well discharge.
Evaluation of Well Loss
To evaluate well loss a step-drawdown pumping test is required. This consists of pumping a well initially
at a low rate until the drawdown within the well essentially stabilize. The discharge is then increased
through a successive series of steps as shown by the time-drawdown data in Figure 3.12. Incremental
drawdowns Δs for each step are determined from approximately equal time intervals. The individual
drawdown curves should be extrapolated with a slope proportional to the discharge in order to measure
the incremental drawdowns.
sw
B CQ
Q
Therefore, by plotting sw/Q versus CQ (see Figure 3.13) and fitting a straight line through the points, the
well loss coefficient C is given by the slope of the line and the formation loss coefficient B by the
intercept Q=0
Figure 3.12 Step-drawdown pumping test analyses to evaluate well loss. (a) Time-drawdown data from
step-drawdown pumping test.
Figure 3.13 Step-drawdown pumping test analyses to evaluate well loss. (b) Determination of B and C
from graph s Q w versus Q
3.6.2 Specific Capacity
It is the ratio of discharge to drawdown in a pumping well. It is the measure of the productivity of a
well. The larger the specific capacity, the better the well is.
Any significant decline in the specific capacity of a well can be attributed either to a reduction in
transmissivity due to a lowering of the groundwater level in an unconfined aquifer or to an increase in
well loss associated with clogging or deterioration of the well screen.
Fig 3.14 Variation in specific capacity of a pumping well with discharge and time
If a pumping well is assumed to be 100 percent efficient ( CQ 0 ), then the specific capacity from
n
equation (*) can be presented in the graphic form of Figure 3.15. Here specific capacity at the end of one
day of pumping is plotted as a function of S, T, and a well diameter of 30 cm. This graph provides a
convenient means for estimating T from existing pumping wells; any error in S has a small effect on T.
Figure 3.15 Graph relating specific capacity to transmissivity and storage coefficient from the non-
equilibrium equation.
Well Efficiency
Q Sw BQ
Ew 100 100
Q BQ Sw
4.1General
Pumping tests (or aquifer tests) are in situ methods that can be used to determine hydraulic parameters
such as transmissivity, hydraulic conductivity, storage coefficient, specific capacity, and well efficiency.
1. Constant-rate test. A constant-rate pumping test consists of pumping a well at a constant rate for
a set period of time (usually 24 or 72 hr), and monitoring the response in at least one observation
well. The number and location of observation wells is dependent upon the type of aquifer and the
objectives of the study. Values of storage coefficient, transmissivity, hydraulic conductivity (if
aquifer thickness is known), and specific capacity can be obtained.
2. Step-drawdown test. During a step- drawdown test, the pumping rate is increased at regular
intervals for short time periods. The typical step- drawdown test lasts between 6 and 12 hr, and
consists of three or four pumping rates. Because step -drawdown pumping tests are typically
much shorter than constant-rate pumping tests, transmissivity and storativity values are not as
accurate for these tests. The primary value of the step-drawdown test is in determining the
reduction of specific capacity of the well with increasing yields.
3. Recovery test. A recovery test consists of measuring the rebound of water levels towards pre-
existing conditions immediately following pumping. The rate of recovery is a valuable source of
data which can be used for comparison and verification of initial pumping test results.
The results from properly conducted tests are the most important tool in groundwater investigations. The
following measurements for both well tests and aquifer tests are important:
1. The static water levels just before the test is started
2. Time since the pump started
3. Pumping rate
4. Pumping levels or dynamic water levels at various intervals during the pumping period
5. Time of any change in discharge rate
6. Time the pump stopped
At the end of the test, you will have the following data:
- A pump rate
Measurements of water levels after the pump is stopped are extremely valuable in verifying the aquifer
coefficient calculated during the pumping phase of the test.
Constant-rate and step-drawdown tests are the most suitable methods to analyse highly productive
aquifer. Other types of tests such as pressure testing by the drill-stem method, testing by injection which
is most limited to sanitary drainage fields can be used when aquifer has a low hydraulic conductivity
which limits the yield from virtually nothing to 5.5 to 10.9 m3/s
Pumping test will not produce accurate data unless the tests are carried out methodically, carefully
recording time, discharge and depth measurement. Certain preliminary tests should be taken to assure the
reliability of the pumping data recorded during the actual test.
Never begin the actual pumping test until the water level in the aquifer has returned to the normal (pre-
test) static level following preliminary testing. About 24 to 72 hours should be allowed, depending on the
type of the aquifer. Beginning a pumping test when the static water level is below normal may eliminate
early data that shows discharge or recharge boundaries. Without the early data, it may be impossible to
obtain the correct transmissivity and storage parameters to the aquifer
The accuracy of drawdown data taken during a pumping test depends on the following:-
1. Maintaining a constant yield during the test
2. measuring the drawdown carefully in the pumping well and in one or two properly placed
observation wells
3. Taking drawdown readings at appropriate time intervals
4. determining how changes in stream levels and tidal oscillations affect the drawdown data
5. comparing recovery data with drawdown data taken during the pumping portion of the test
6. Continuing the test for 24 hours for confined aquifer and 72 hours for unconfined aquifer
during constant-rate tests. For step drawdown tests, 24 hours is usually sufficient for either type of
aquifer.
Pump failure during the test is expensive and even if the test is quickly resumed after repairs or
refuelling the data are of questionable value. Therefore, the pump should be in good repair and
the fuel supply should be adequate for the full term of the pumping test.
A test well is a well where pumping is conducted. An observation well is a well in which observation of
variation in head is being observed from a certain reference point.
While locating a test well the following considerations should be taken in to account.
Test well should be located away from any other recharging/discharging well if such are not used
for the testing purpose.
The well should be located away from high traffic sites where variations in head may occur
The pumped water should be discharged sufficiently away from the well site.
The well should be far from recreational areas, buildings and areas of cultural value
The site should be easily accessible for machinery, labour and construction material transport.
A test well and observation well (one or more) can be used for obtaining pumping test data. However, due
to economic reasons usually one test well is used as an observation well. If it is intended to drill a test
well, a test well is constructed first and if the potential for yield is sufficient, then the test well is changed
to a production well later on. During drilling, the geologist can have the geologic well log and can
determine the potential aquifer zone depth wise for further development of the water well.
Drawdown data can be taken from both the pumping well and appropriately placed observation
wells, but the accuracy of data taken from the pumping well is usually less reliable because of the
turbulence created by the pump.
Drawdown data from an observation well are required to calculate the storage coefficient
accurately whereas transmissivity value may be calculated on the basis of drawdown taken from
either a pumping well or observation well.
Observation wells should be large enough to allow accurate and rapid measurement of water
levels. But small-diameter wells are best, because the volume contained in a larger-diameter
observation well may cause a time lag in drawdown
When observation wells are too close to the pumped well the drawdown readings may be affected
by the stratification of the aquifer. Stratification distorts the distribution of hydraulic head and
drawdown in the vicinity of the pumped well during the aquifer. Recall that the vertical hydraulic
conductivity of stratified formation is usually much less than its horizontal hydraulic
conductivity. This means that changes in head caused by pumping occur more slowly in the
vertical direction than in the horizontal direction
The distorted pattern of drawdown caused by stratification is eliminated at distance equal to three
to five times the aquifer thickness.
For unconfined aquifers, observation wells should be paced not farther than 30 to 91 m from the
pumped well. Whereas for thick confined aquifer that are considerably stratified, observation
wells should be placed within 91 to 214m of pumped well.
The appropriate number of observation wells depends upon the amount of information desired
and up on the funds available for the test program.
The data obtained by measuring the drawdown at a single location outside the pumped well
permit the calculation of the average hydraulic conductivity, transmissivity and storage
coefficient of the aquifer. If two or more observation wells are placed at different distances the
test data can be analysed by studying both the time-drawdown and distance-drawdown
relationship. Therefore using both these analytical methods provides greater assurance that the
calculated transmissivity and storage coefficient values are calculated
It is usually advantages to have as many observation wells as condition allow because the
hydraulic conductivity may vary in one or more directions away from the pumping well.
Observation wells placed in a circle around the pumping well will reveal this trend
Early test data are extremely important and as much information as possible must be collected in the first
10 minutes of pumping for every observation well. The reason for this is that, as the cone of depression
moves out ward from the well, it may encounter in homogeneities in the ground which cause either
acceleration or a deceleration of drawdown with increasing time
The following format displays the format/form for collecting pumping and recovery test data in the field
with.
Gradually the search for groundwater also focussed on the identification of geological formations
and rock types. Formations including river alluvial deposits, coastal and tectonic basin
sediments, and certain sedimentary formations and volcanic sequences were found to contain
groundwater in large quantities, which could be exploited by sinking wells into them. To identify
and exploit the resources in these formations in a proper manner sound professional help was
needed. The professional hydro-geologist or geo-hydrologist came on the scene to assist in this
task. Although many wells are drilled without professional guidance the hydro-geologist plays an
increasingly important role in groundwater exploration programmes.
Studies and investigations within the context of groundwater exploration programmes focus on
the:
Identification of areas where groundwater can be exploited;
Determination of the quantity and quality of the water resources that can be
exploited without depleting the groundwater system;
Location of springs and selection of suitable sites for the installation of individual
production wells and/or well fields;
Determination of the depths of wells to be drilled, and the abstraction rate (well
yield) and the quality of the groundwater that we may anticipate when production
wells are installed.
5.1.1 Topographical maps
Studying topographical maps is perhaps the very first activity that we carry out in our 'search' for
groundwater. Where is our investigation area located? How large will it be? What will the terrain
look like? A topographical map showing the topography, the river systems, natural vegetation
and cultivated land, roads, towns and villages may give an answer to these questions. By careful
study of the map and cross-checking it in the field features may be identified which inform the
hydro-geologist about potential groundwater resources.
In addition to carrying out a groundwater related analysis the maps mentioned may also serve as
base maps for the presentation of data collected in groundwater investigation programmes.
There are usually several features of interest to the hydro-geologist on a topographical map.
Some of those are:
Springs, seepage zones and areas with shallow groundwater tables may be shown
on the map indicating the presence of groundwater resources.
These features may be shown directly on the map or may be inferred from features
like dense vegetation, the presence of villages etc
Depressed flat areas like river valleys, coastal areas, inter-mountain basins etc may
indicate the presence of groundwater resources;
River- and other drainage patterns may also point to rock types and availability of
groundwater resources;
The slope of the terrain shown on the map usually indicates the direction of shallow
groundwater flow.
Rivers- and drainage patterns have a strong relation with groundwater. Rivers and streams can be
observed in the field carrying surface runoff and groundwater discharge towards lakes and seas.
The course of rivers or streams and their typical patterns can also be observed on topographical,
geological or other brands of maps. The density of a river system or stream pattern is a measure
of the permeability of the formations underlying an investigation area. If the pattern of streams is
very dense than the permeability of the formations is very low (aquifuges): most of the
precipitation will become surface runoff thereby creating this dense pattern. When the pattern is
not very dense then the permeability of the formations is high (aquifers): much of the
precipitation will infiltrate and there will hardly be any surface runoff to create a dense pattern.
Thus, the density of the stream and river pattern indicates where the permeable formations and
the water groundwater resources are located. These formations may be considered for the
exploitation of groundwater.
Geological maps
Geological maps and sections are valuable tools for the professional hydro-geologist. Geological
maps show the different geological units in an area. Usually tectonic information including the
presence and direction of folds and faults is shown as well. Geological sections show the same
features, but then represented in 'vertical cuts' through the area. By assigning hydro-geological
interpretations to the various units, the geological material can be converted into „simple‟ hydro-
geological maps and sections. At a later stage other groundwater data (e.g. well data) may be
added to the maps to produce „detailed‟ hydrogeological maps and sections
Geological maps are usually prepared on the basis of aerial photo material and field
reconnaissance surveys
Hydro-meteorological data
Meteorological data can be considered to make assessments on the prospects for groundwater
development in an investigation area. Consider precipitation and evapotranspiration data. When
the precipitation surplus (the amount of precipitation minus the amount of evapotranspiration) is
large then the recharge into the groundwater basin can be large. That is, when the geological
formations are sufficiently permeable. Thus, the best prospects for the development of
groundwater resources are associated with areas where precipitation surpluses are large and the
geological formations are of a permeable nature.
Remote sensing studies
Satellites circling around the earth take pictures of the earth surface. The full light spectrum may
be used when taking these pictures or certain wavelengths of the spectrum may be selected. True
and false colour pictures emphasizing certain particulars of the earth surface or even from a
shallow zone below the surface are being generated. Groundwater related features on these
pictures can be detected by bare eye and, if the correct set of pictures has been ordered, by using
a stereoscope (SPOT images).
the earth crust. For oil and mineral exploration the first few thousand meters of the crust may be
considered. Groundwater exploration, which is our concern, usually does not extend below 500
m depth. With respect to groundwater exploration practices, the following geophysical
techniques are most widely used: the electrical resistively and seismic refraction techniques.
Details of these techniques will be discussed respectively in the following sections.
5.2.1.1Electrical Resistivity Method
These techniques are based on the injection of an electrical current into the earth by means of
two current electrodes (Figure 2). The potential differences that are created between these
electrodes are measured at another pair of intermediate electrodes: the measuring or potential
electrodes. Readings of current strengths at the current electrodes, and potential differences at the
measuring electrodes enable us to determine rock resistivities. These resistivities can be related
to subsurface rock types, rock water contents and groundwater quality (pore water resistivity).
This information can be used to identify permeable rocks (aquifers), which can be used for
groundwater exploitation.
The most widely used electrode layouts for geo-electrical measurements are the so-called
Wenner and Schlumberger layouts. Schematically these set-ups are presented in Figure 3. Note
that the Wenner configuration is symmetrical with the four electrodes always at equal distances
from each other. The Schlumberger configuration is also symmetrical, but the distance between
the measuring electrodes differs from the spacing between the measuring and current electrodes.
Figure 4.4 Common electrode arrangements for resistivity determination. (a) Wenner. (b)
Schlumberger.
The resistivity is called apparent because each resistivity value for a certain depth is really an
average resistivity for all the materials above this depth. This apparent resistivity value results
from the three-dimensional resistivity structure of the geologic materials and is weighted average
of all materials encountered by the current flow. As resistivity values change they indicate a
change in subsurface conditions. The depth of electrical penetration is governed by the spacing
of the electrodes-the larger the separation, the deeper the penetration. Therefore the electrode
spacing can be progressively increased to determine the variation in resistivity.
The apparent resistivity in schlumbergers arrangement will be given as
2 2
L b
a
2 2 V
b I
Where: L and b is the current and potential electrode spacing respectively
Theoretically, L>>b, but for practical application good results can be obtained if L>=5b
When apparent resistivity is plotted against spacing (a for wenner and L/2 for schulumberger) for
various spacing at one location, a smooth curve can be drawn
Apparent resistivity values are usually plotted along the vertical axes and the corresponding 'half
the current electrode distances, C1C2/2, along the horizontal axes on double-logarithmic paper.
Direct plotting has the advantage that measurement errors become visible in the field, and not
later in the office. Make sure that the various segments (curves for a particular half potential
electrode distance) of the plot are smooth and re-do outlying points.
In the field we measure the potential differences, ΔV, and the injected currents, I, for various
electrode spacings. Then apparent resistivity values are computed. We end up with a whole
series of apparent resistivities for a range of electrode spacings. When we use the variable
electrode distance technique, we are not so much interested in these apparent resistivities, but in
the resistivities and thicknesses of individual layers.
Therefore the next step is to translate the apparent resistivity in to layer resistivities, and into
layer thicknesses h1, h2, h3 etc, and to set up the hydrogeological interpretation of the layer
resistivities.
The translation (interpretation) of apparent resistivities into layer resistivities and layer
thicknesses is done by means of curve matching techniques which are strictly speaking only
valid for sets of horizontally stratified earth layers. In these techniques field curves showing
apparent resistivities against 'half current electrode spacings' are matched with curves, which are
computed from mathematical expressions. The mathematical expressions relate apparent
resistivities to layer resistivities, layer thicknesses, and electrode spacings. In case field and
mathematical curves fit, the layer resistivities and thicknesses at a measurement location are
similar to those used for the computation of the mathematical curves.
Traditionally, sets of curves computed on the basis of mathematical expressions were drawn on
paper, and curve matching with field curves was done manually. However, the approach towards
resistivity interpretation has changed tremendously. Nowadays the computation and presentation
of mathematical curves, and the storage and presentation of the field curves is largely done on
the personal computer.
For the case of 2 layers the final result of the derivation for the mathematical expression will be
presented. The equation relates apparent resistivities to resistivities of the first and second layer,
the thickness of the first layer, and to current electrode spacings. The deepest layer, which in this
case is the second layer, is always assumed to be of infinite thickness. Also note that an electrode
configuration following the Schlumberger arrangement has been assumed.
The derivations for the mathematical expressions relating the apparent resistivities to layer
resistivities, layer thicknesses and electrode spacings are based on the method of images
including the creation of image poles. Central in the derivation is the mirroring (n times) of the
current electrode with strength I by the earth atmosphere boundary and the rock layer interfaces
resulting in a complex expression for the potential differences (beyond the scope of this chapter)
In case φ1= φ2, equation above reduces to the equation for a 1-layer case.
If we insert the condition into the equation it follows indeed that the apparent resistivity φa is
equal to the resisitivity of one single layer: φ1. From the equation it also follows that for small
values of C1C2 the φa approaches to φ1.
Consider the shape of the curves for the 2-layer case. The mathematical expression for φa as
expressed in equation above shows that hundreds of curves could be calculated for as many
combinations of selected values for the layer resistivities φ1, φ2, and the thickness of the first
layer h1. By plotting on double-logarithmic scale the ratios of selected apparent resistivity values
and the resistivities of the first layer (φa / φ1) along the vertical axes, and the ratios of 'half the
current electrode distances' and the thicknesses of the first layer (C1C2/2h1) along the horizontal
axes, the curves could be reduced to a set that fits on one sheet of paper.
The waves are called elastic because as the waves pass a point in the rock, the particles
are momentarily displaced or distorted but immediately return to their original position or
shape after wave passes;
Three types of waves can be created: Compressional waves (P), Shear waves (V), and
surface waves. Compressional waves are the first to arrive at the geophones and therefore
are the most useful in seismic surveys;
In general, the higher the density and elasticity of the rock unit, the faster the P wave will
be transmitted. The velocity is much less and the energy is dissipated more quickly if the
material is unconsolidated or poorly consolidated.
Three distinct paths are taken by compressional waves in the ground: direct, refracted and
reflected (see Figure 5). A single seismic impulse can be recorded as three separate
arrivals at the geophone. In practice, however, only the first arrival can be readily
recognized.
Figure 4.6: Waves from a seismic disturbance can travel as surface, reflected, and refracted
waves. In water well exploration, analyses of refracted and reflected waves can determine the
depth to bedrock at a potential drilling site
When elastic waves cross a geological boundary between two formations with different
elastic properties, the velocity of wave propagation changes and the wave paths are
refracted.
Seismic methods use artificially seismic waves travelling through the ground. By
studying the arrival times of seismic waves at various distances from energy source, the
depth to bedrock can be determined.
These methods are useful in determining depth to bedrock, depth to water table and in
some cases general lithology.
Seismic refraction method is used to determine the thickness of unconsolidated materials
overlying bedrock. The loose material transmits seismic waves more slowly than
consolidated bedrock. By studying the arrival times of seismic waves at various distances
from energy source, the depth to bedrock can be determined.
The energy source can be a small explosive charge set in a shallow drill hole. One or two
sticks of dynamite are sufficient for depths to bedrock in excess of 30 to 50 m.
The seismic wave is detected by geophones placed in the earth in a line extending away
from the energy source. Waves initiated at the surface and refracted at the critical angle
by a highvelocity layer at depth will reach the more distant geophones more quickly than
waves that travel directly through the low-velocity surface layer.
Interpretation
A seismograph records the travel time for the wave to go from energy source (short point) to
geophone. This time should be plot against distance from shot point to geophone;
Figure 6 illustrates the travel paths of compressive seismic waves traveling through a two layer
earth. The seismic velocity in the lower layer is greater than that in the upper layer. As the
energy travels faster in the lower layer, the wave passing through it gets ahead of the wave in the
upper layer. At the boundary between the two layers, part of the energy is refracted back upward
from the lower-layer boundary to the surface;
Figure 4.7 Travel paths of a refracted seismic wave and a direct wave. The direct wave will reach
the first five geophones first, but for the more distance geophones the first travel is from a
refracted wave. Numbers inside symbols refer to distances travelled by wave paths going toward
the indicated geophone.
The angle of refraction of each wave front is called the critical angle ic, and is equal to the arc
sin of the ration of the velocities of the two layers:
The velocities computed from the reciprocals of the slope are called apparent velocities. If the
lower layer is horizontal, they represent the actual velocity
Energy travels directly through the upper layer from the source to the geophone. This is
the shortest distance, but the waves do not travel as fast as those traveling along the top of
the lower layer. The latter go farther, but with a higher velocity. Figure 6 shows the
positions of waves traveling to each geophone. Geophones 1 through 5 first receive
waves that have traveled through only the upper layer. The sixth and succeeding
geophones measure arrival times of refracted waves that have gone through the high-
velocity layer as well. The figure shows the position of the trailing wave front at each
time the leading front reaches each geophone;
A graph is made of the arrival time of the first wave to reach geophone versus the
distance from the energy source to the geophone (travel-time or time-distance curve).
Figure 7 shows the time-distance curve for the shot in figure 6. The reciprocal of the
slope of each straight-line segment is the apparent velocity in the layer through which the
first arriving wave passed. The slope of the first segment is 10 milliseconds per 10 m so
that the reciprocal is 10 m per 10 ms or 1000 m/s.
Figure 4.8: Arrival time-distance diagram for a two-layered seismic problem. Numbers refer to
geophones
The projection of the second line segment backward to the time-axis (X=0) yields a value known
as the intercept time, Ti. As shown in Figure 7.
The depth to the lower layer can be also found from the equation:
Where: - X is the distance from the shot to the point at which the direct wave and the refracted
wave arrive simultaneously. This is shown in Figure 7 as the x-axis distance where the two line
segments cross.
A more typical case in the hydrogeology is a three-layer earth, the top layer being unsaturated,
unconsolidated material. In the next layer below the water table, the unsaturated deposits are
saturated, which yield a higher seismic velocity. The third layer is then bedrock. Under such
conditions the seismic method can be used to find the water table.
The three-layer seismic case with V1<V2<V3 is shown in Figure 8. The first arriving waves
show three line segments. The reciprocal of the slope of each line is the seismic velocity of the
respective layers. The intercept time for each of the two deeper layers is the projection of the line
segment back to the time-axis. Indicated on the Figure is the distance X1, from the shot to the
point at which waves from layers 1 and 2 arrive simultaneously and the distance X2, to the point
at which waves from layers 2 and 3 arrive simultaneously. The thickness Z1 of layer 1 is found
from the values of V1 and V2 and either Ti1 or X1 using the above equations and the thickness
of the second layer Z2 is found from:
Figure 4.9 A. Diagram of arrival time versus distance for a three-layered seismic problem B.
Wave path for a three-layered seismic problem
5.3 Subsurface Investigation of Ground water
Detailed and comprehensive study of groundwater and conditions under which it occurs can only
be made by subsurface investigation.
Whether the information needed concerns an aquifer (its location, thickness, composition,
permeability, and yield) or groundwater (its location, movement, and quality), quantitative data
can be obtained from subsurface examinations.
Test drilling furnishes information on substrata in a vertical line from the surface.
It may even be possible to do some preliminary pumping tests or slug tests on the bore hole to
estimate hydraulic properties of the aquifer or aquifers and to calculate potential well yields.
Test wells normally are of relatively small-diameter holes and can be drilled at a fraction of the
cost of full-sized wells.
When a test well indicates a favorable location, it can often be converted into a production (or
pumping well) by re-drilling or reaming to increase its diameter.
Test holes can also serve as observation wells for measuring water levels or for PS.
Two types of logs can be kept by the driller ad drilling progresses
5.3.2Geologic log
It is constructed from sampling and examination of well cuttings collected at frequent intervals
during the drilling of a well or test hole. Such logs furnish a description of the geologic character
and thickness of each stratum encountered as a function of depth, there by enabling aquifers to
be delineated.
Considering all types of logs, the geologic log is probably the most important, but preparation of
a good geologic log can be difficult.
One problem is that well cuttings are small and mixed with mud. Drill cuttings are often a
mixture of material from the bottom of the hole, drilling mud, and material from higher layers
that was still in the hole or that caved in from the wall.
Thus, the sample must be carefully analyzed. Often it is better to look for changes in samples
than at actual composition. For example, if the bailer initially yielded primarily fine materials,
and sand begins to show up, a sand layer may be reached. If gravel chips show up, gravel layer
may be reached, etc…
Similarly, a reduction in sand content of the bailer material may indicate that the bottom of a
sand bed has been reached.
Experienced drillers with good knowledge of local subsurface conditions often „know‟ what king
of material the bit is in from its rate of advance and how it bounces, churns, sounds, or other wise
reacts to the material.
II. To determine
3. length & setting of well casing
4. optimum length & setting of water well screens
5. porosity & water content of the formation
6. Resistivity of the formation
7. Lithology & formation boundaries
8. Thermal gradient
9. Relative quantity of water into or out of zones of water entry into the hole
10. Approximate permeability of lithologic sections penetrated by the hole.
11. Accurate depth referenced for use with other type of logs
12. Depth and thickness of thin beds or aquifers
13. Average bore hole diameter.
III. To locate
14. Position of cement grout behind casing
15. Point of entry of different quality water through leaks in casing or opening in rocks
16. Cemented & cased intervals
17. Depth of lost circulation
18. Active gas flow
19. Fissures and solution opening in open holes
20. Leaks or perforated sections in cased holes
Although well design may appear to be a straight forward procedure, local hydro-geologic condition and
practical considerations complicates many well designs
Important hydro-geologic information required for the design of efficient high-capacity wells includes
1. Stratigraphic information concerning the aquifer and overlying sediments
2. Transmissivity and storage coefficients values for the aquifer
3. Current and long term water balance conditions in the aquifer
4. Grain-size analysis of unconsolidated aquifer materials and identification of rock or mineral types
if necessary
5. Water quality.
Dimensional factors, strength requirements, and cost associated with well construction and maintenance
also plays apart in establishing the particular design parameters.
Every well consist of two main elements, the casing and intake portion. The casing serves as a housing for
the pumping equipment and as a vertical conduit for water flowing upward from the aquifer to the pump
intake. The reasons for using casing in water well or borehole is summarized as follows
1. To prevent the collapse of the walls of the borehole (i.e. structural support against caving in)
serving as a lining.
2. To exclude, along with grouting, pollutants either from surface or subsurface from entering the
water sources
3. To provide a channel for conveying the water to the surface.
4. to provide a channel for conveying the water into the well for injection purpose
5. To provide a housing for the pump mechanism
6. To provide a channel for conveying a cement grout in the well for cementation purpose
7. Serving as a reservoir for a gravel pack
Some of the borehole length serving as a conduit may be left uncased when the well is constructed in
consolidated rock. The intake portions of wells in unconsolidated and semi-consolidated aquifer is
generally screened to prevent sediment from entering with the water and to serve as a structural retainer to
support the loss formation materials. At the same time, the screen must not obstruct the flow of water in
to the well. The design of the well screen requires careful consideration of the hydraulic factors that
influence well performance.
Standard design procedure involve choosing the casing diameter and material, estimating the well depth,
selecting the length, dimeter, and material of the screen, determining the screen slot size, and choosing the
completion method. In addition the choice of the particular well design hinges on the type of drilling rigs
that are available.
6.1.1Casing Diameter
Choosing the proper casing diameter for the well is important because it may significantly affect the cost
of the structure, depending on the type of drilling equipment used. The diameter must be chosen to satisfy
two requirements: (1) the casing must be large enough to accommodate the pump, with enough clearance
for installation and efficient operation, and (2) the diameter of the casing must be sufficient to assure that
the uphole velocity is 1.5 m/sec or less. Excessive head loss will occur in the system if the uphole velocity
is greater than 1.5 m/sec.
The size of the pump required for the desired yield is the controlling factor in choosing the size of the
casing. It is recommended that the casing diameter be two pipe sizes larger than the nominal diameter of
the pump. Table 1 shows the casing sizes recommended for various pumping rate.
Table 1: Maximum discharge rates for certain diameters of standard-weight casing, based on an uphole
velocity of 1.5 m/s
Casing size (mm) Maximum discharge Casing size (mm) Maximum discharge
Actual inside (m3/day) Actual inside (m3/day)
diameter diameter
102 1,090 337 11,700
127 1,690 387 15,500
152 2,450 438 19,800
203 4,250 489 24,700
254 6,700 591 36,100
305 9,590
6.1.2Casing Materials
Selection of casing material is based on water quality, well depth, cost , borehole diameter and drilling
procedure .The types of casing used in water well construction are steel, thermoplastic, fibreglass,
concrete and asbestos cement. Steel is used most commonly, but thermoplastic materials are gaining
larger share of the water well casing market, especially in areas where groundwater is highly corrosive
and well are less than 305 m deep.
6.1.3Well depth
The depth of a well is usually determined from information from the driller‟s log of a test hole, sample
logs of other nearby wells in the same aquifer, geophysical analysis of the formation, and data taken
during the drilling of the production well.
Generally, a well should be completed to the bottom of the aquifer because:
1. More of the aquifer thickness can be utilized as the intake portion of the well, resulting in
higher specific capacity
2. More drawdown can be made available, permitting greater well yield
3. Sufficient drawdown is available to maintain well yield even during periods of sever drought or
over pumping
6.1.4Well screen length
The optimum length of the well screen is based on the thickness of the aquifer, available drawdown, and
nature of the stratification of the aquifer. In virtually every aquifer, certain zones will transmit more water
than others. Thus, the intake part of the well must be placed in those zones having the highest hydraulic
conductivity.
Determination of the most productive layers can be made by one or more of the following techniques:-
1. Interpretation of the driller‟s log and comments on drilling characteristics such as fluid loss,
penetration rate and pull down
2. Visual inspection and comparison can be made of samples representing each layer. The relative
transmissivity of each layer is estimated from the observed coarseness, lack of silt and clay and
thickness of the layer
3. Sieve analysis can be made from samples taken from the various layers in the aquifer. Comparison
of grain-size curve can indicate the relative hydraulic conductivity of each samples
4. Laboratory methods of hydraulic conductivity determination based on darcy‟s law
5. Borehole geophysical logging techniques can help to locate zones having the highest hydraulic
conductivity
For a well in unconfined aquifers, selection of screen length is a compromise b/n two factors. On one
hand, higher specific capacity is obtained by using the longest screen possible. This reduces convergence
of flow and entrance velocity, thereby increasing specific capacity. On the other hand, more available
drawdown results from using the shortest screen possible. This two conflicting aims are satisfied, in part
by using an efficient well screen that minimizes the loss in specific capacity as drawdown increases.
Slot openings should be continuous around the circumference of the screen, permitting maximum
accessibility to the aquifer so that efficient development is possible.
Slot openings should be spaced to provide maximum open area consistent with strength
requirements to take advantage of the aquifer hydraulic conductivity.
Individual slot openings should be V- shaped and widen inward to reduce clogging of the slots
and sized to control sand pumping (see Figure 1)
Figure 6.1 V-shaped slot openings reduce clogging where straight cut, punched or gauze-type openings
can be clogged by elongate or slightly oversize particles
In general, a larger slot-size selection enables the development of a thicker zone surrounding the screen,
therefore, increasing the specific capacity. In addition, if the water is encrusting, a larger slot size will
result in a longer service life. However, the use of a larger slot size may necessitate longer development
times to produce a sand-free condition.
A more conservative selection of slot size (for instance, a 50% passing value) is selected if there is
uncertainty as to the reliability of the sample; if the aquifer is overlain or underlain by fine-grained, loose
materials; or if development time is expensive.
For formations of fine sands and silts the aquifer must be stabilized. It is not usually practicable to have
very small slot sizes, and so an artificial gravel pack is selected which forms the correct size of pore
opening, and stabilizes the sand in formation. The use of a pack in a fine formation enables the screen
opening to be considerably larger than if the screen were placed in the formation by itself. There is a
consequent reduction in head loss. If the grading in the aquifer is small, several grading in the aquifer is
very small, several grading of gravel pack may be required to retain the formation, and provide practical
screen opening sizes.
The gravel pack adjoining the screen consists of larger sized particles than the surrounding formation, and
hence larger voids are formed at and close to the screen allowing water entry nearly free from head loss.
Necessary conditions for a gravel pack are:
Sand-free operation after development,
Highest permeability with stability (low resistance),
Low entrance velocities,
Efficient service life, i.e. resistant to chemical attack
In gravel packed wells, zones immediately around the well screen is made more permeable by removing
some formation material and replacing it with specially graded material. The filter-pack material should
consist of clean well-rounded grains of a uniform size. These characteristics increase the permeability and
porosity of the packed material. The chemical nature of the filter pack is as important as its physical
characteristics. Filter pack material s consisting of mostly of siliceous rather than calcareous, particles
preferred (up to 5 % of calcareous material is common allowable limit)
These are produced by the development of the formation itself. Development techniques are used to draw
the finer fraction of the unconsolidated aquifer through the screen leaving behind a stable envelope of
coarser and therefore more permeable material. Suitable aquifers are coarse grained and ill sorted,
generally with a uniformity coefficient greater than 3.
Slot size recommended for the screen is between D10 and D60 (often D40). Choice of slot size is then
dependent upon the reliability of the sample and nature of aquifer (e.g. thin and overlain by fine material,
formation is well sorted). Not recommended if slot size is less than 0.5 mm. (see Figure 4)
Fig 6.4 Natural development removes most particles near the well screen that are smaller than the slot
openings, thereby increasing porosity and hydraulic conductivity in a zone surrounding the screen.
6.1.8.2Artificial Gravel Pack
Also known as gravel filter pack (see Figure 5), graded envelope, the gravel pack is intended to fulfill the
following functions:
To support the aquifer formations and prevent collapse into the casing;
To laterally restrain the casing, effectively strengthening the casing;
To prevent the movement of fine aquifer material into the well.
The normal approach is to use a filter pack when:
The uniformity coefficient < 3;
The aquifer is fine, with D10 of the formation < 0.25 mm.
Fig 6.5: The basic differences between the arrangement of the sand and gravel in natural and artificial
gravel packed wells.
(a) The principle of the natural or „developed‟ well with each zone correctly graded to the next so that the
whole pack is stabilized. (b) An artificial gravel packed well in which the correct size relationship is
established between the size and thickness of the gravel pack material and the screen slot width. Such a
well can be effectively developed and will be efficient and stable. (c) Undesirable result of using gravel
that is too coarse. The aquifer sand is not stabilized and will eventually migrate into the well. This
unstable condition will persist regardless of how thick the gravel pack may be, thus causing a continued
threat of sand pumping.
In theory, a pack thickness of 2 or 3 grains is all that is required to retain formation particles. In practice
around 10 cm is used to ensure an envelope around the well. Upper limit of thickness of the gravel pack is
20 cm; otherwise, final well development becomes too difficult and cost of drilling escalates. Packs with a
thickness of less than 5 cm are simply formation stabilizers, acting to support the formation, but not
effective as a filter.
A common consensus is that a gravel pack will normally perform well if the uniformity coefficient is
similar to that of the aquifer, i.e. the grain size distribution curves of the filter pack and the aquifer
material are similar (see Figure 6). The grain size of the aquifer material should be multiplied by a
constant of approximately (4-7) with average (5) to create an envelope defining the filter grading.
6.2Well construction
Well construction methods are many and varied ranging from simple digging with hand tools to high
speed drilling with sophisticated equipment. Well construction, in terms of operations, basically includes:
The drilling operation
Installing the casing
Installing the well screen and artificial gravel packing
Grouting when needed to provide sanitary protection and well head construction
Developing the well to insure sand free operation at maximum yield
Installing the pump