SPE 144583

A Semi-Analytic Method for History Matching Fractured Shale Gas

Reservoirs

Orkhan Samandarli, Hasan A. Al-Ahmadi, and Robert A. Wattenbarger, SPE, Texas A&M University
Copyright 2011, Society of Petroleum Engineers
This paper was prepared for presentation at the SPE Western North American Regional Meeting held in Anchorage, Alaska, USA, 711 May 2011.
This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been
reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its
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Abstract
This paper presents a semi-analytic method to estimate reservoir parameters by history matching the production data of
hydraulically fractured shale gas wells. The method is based on the analytical solutions to dual-porosity reservoir models.
Algebraic equations developed by Bello and Wattenbarger (2010) for transient flow regimes which can occur in linear dual
porosity model were used in this analysis. As far as practical values of reservoir parameters are involved, only two of the
transient flow regimes would be seen in the field ! bilinear and late linear. If matrix permeability is known, effective fracture
permeability is estimated by doing regression on bilinear flow. Once bilinear flow is matched, linear flow part is also matched
by regressing on fracture half length. Semi-analytic solutions are checked with actual data by making correction for gas
properties change and desorption of gas at low pressures. When a satisfactory match of field data is obtained, reservoir
parameters such as effective fracture and matrix permeability and OGIP can be estimated. Moreover, a future production of
gas can be forecasted with estimated reservoir parameters for given economic constraints.
The method was applied on hydraulically fractured shale gas wells from Barnett play. It gives good results in history matching
and estimation of reservoir parameters. Considering the recent interests in development of unconventional reservoirs, this
method is future promising technique in better understanding those types of reservoirs.
Introduction
Gas Shales are self-sourcing reservoirs which means they act both as a source and reservoir rock. Although as a source rock
they are quite good candidates because of high percentage of Total Organic Carbon (TOC), gas shales are poor reservoir rocks.
Despite some gas shales might have good network of natural fractures, typically matrix permeability in shale ranges from 10 to
100 Nano-Darcy (Cipolla et al. 2009). It is well known that without multi-stage fractured horizontal wells, economic
production of gas from shales would not be possible. In recent years, much effort has been concentrated on understanding the
production behavior of shale gas reservoirs.
Although there are different opinions on how to model the horizontally drilled shale gas wells, one fact is commonly accepted
and proved to be true the main flow regime in the life of the hydraulically fractured shale gas well is linear flow. Arevalo and
Wattenbarger (2001) observed that many hydraulic fractured tight gas wells stay in the linear flow for a long period of time.
The same phenomenon is observed in many shale gas wells. In a log-log plot of rate versus time linear flow exhibits
characteristic half-slope (Fig. 1). Another diagnostic plot to check the existence of linear flow is square root of time plot
where linear flow is a straight line on the plot of normalized rate reciprocal versus square root of time.

SPE 144583

Fig. 1Production history (left) and square root of time plot (right) of Well B-151. Well is in transient linear flow: half slope on log-log
plot and linear behavior on square root of time plot.

Since this paper is about a semi-analytical method to history match the production of shale gas wells, it would be good to
review the recent similar works in the same topic. Mattar et al. (2008) summarized both empirical and semi-analytical methods
to describe the production data from shale gas wells up to date. Bello and Wattenbarger (2008) presented a linear dual porosity
model for horizontally drilled shale gas wells with multiple hydraulic fractures. Bello and Wattenbarger (2009) presented a
new solution for linear dual porosity model which would incorporate the skin effect early flat region on a log-log plot of
rate versus time of the field data. Ilk et al. (2010) proposed a systematic workflow to analyze the production data from
unconventional reservoirs. Al-Ahmadi et al. (2010) discussed the application of linear flow analysis on field data.
The method described in this paper can be applied to any shale gas well as long as assumptions granted for solution are hold.
The first and most important assumption is that the reservoir is limited to Stimulated Reservoir Volume (SRV) (the reservoir is
as big as the extent of hydraulic factures). For typical shale gas reservoirs in US this is a reasonable assumption because of
ultra-low permeability of the non-stimulated formation. Nevertheless, this would be checked and confirmed to be right in the
following section. The second important assumption is constant bottomhole flowing pressure during production which is the
case most of the time applied in the field. Other assumptions are same as Bello and Wattenbarger (2010) mathematical model
assumptions.
Flow only in SRV versus CLF
Many authors (Freeman et al. (2009), Luo et al. (2010)) have discussed the importance of Compound Linear Flow (CLF) in
horizontal tight/shale gas wells which was initially proposed by Van Kruysdijk and Dullaert (1989). CLF occurs when
hydraulic fractures start to interfere and fluid from non-stimulated reservoir drains normal to the vertical well plane. The
reason for ignoring the CLF is the fact that CLF would not occur in the practical life time of typical shale gas well. To check
this reservoir simulation model was built where horizontal well with transverse hydraulic fractures is simulated for 30 years
(Fig. 2). Table 1 shows the input data (typical Barnett shale well data) used for simulation model. First, sensitivity to matrix
permeability was checked by holding hydraulic fracture permeability constant. It was observed that CLF would not occur for
permeability less than 5.00E-5 md in 30 years (Fig. 3) and occurrence of CLF does not depend on the hydraulic fracture
permeability. Luo et al. (2010) also confirmed these results by streamline simulation. According to results of these studies, we
can confidently say that drainage area of the well in ultra-low permeability reservoirs is limited to SRV only. Therefore, in this
paper whenever Boundary Dominated Flow (BDF) is mentioned, reader should think of the boundary of SRV or no-flow
boundary of hydraulic fractures. Moreover, any OGIP values reported in results should be referred to SRV.

Fig. 2Schematic of horizontal well with transverse multiple hydraulic fractures.

SPE 144583

Table 1Model parameters for CFL analysis.

Hydraulic
Fracture
Matrix

Porosity

Table 2 Model parameters for synthetic data.

Hydraulic
Fracture

Porosity

Width

0.1

ft

Spacing

100

ft

Porosity

0.06

Thickness

200

ft

2500

ft

Perforated Length

150

ft

Fracture Half-Length

ft2

Area of Well Plane

Perforated Length
Fracture Half-Length
Area of Well Plane
# of Macro-fractures
Rock Compressibility
General

0.002

Water Compressibility
Initial Water Saturation
Initial Pressure
Bottom-hole Pressure
Temperature
Gas Gravity

500000

Matrix

25
0.000004
4.2E-06

0.04

ft

Spacing

400

ft

Porosity

0.06

Thickness

# of Macro-fractures
Rock Compressibility

1/psi
1/psi

General

Water Compressibility
Initial Water Saturation

0.3
2950

psi

Initial Pressure

500

psi

Bottom-hole Pressure

160

deg. F

Temperature

0.635

0.002

Width

Gas Gravity

300

ft

10000

ft

500

ft

6000000

ft2

25
0.000004

1/psi

4.2E-06

1/psi

0.3
2950

psi

500

psi

160

deg. F

0.635

Fig. 3Simulation results for CLF analysis. Upper half of the plot is showing rate versus time for different matrix permeability values
simulated. Lower half of the plot is derivative of reciprocal rate with respect to square root of time. Linear flow exhibits flat (zero
derivative) in this kind of plots.

Correction for Time and Adsorbed Gas

In order to use analytical solutions in shale gas wells two corrections have to be done correction for gas properties change
with time and desorbed gas at low pressure. Fraim and Wattenbarger (1987) described a method to correct for gas properties
and modify the time.

(ct )i
d! .... (1)
( p )ct ( p )
0
t

tn = "

Bumb and Mckee (1988) used Langmuir isotherm to describe the adsorption of gas in the surface of coal and shale matrix.
They solved conservation of mass equation and defined an additional compressibility due to adsorbed gas as cgd.

SPE 144583

p
.................................................................................................................................................................. (2)
p + pL
VLcuft = 0.031214!bulkVL ...... (3)

VE = VL

cgd =

" gscVLcuft pL
BV p
= g Lcuft L 2
2
! m " g ( p + pL ) ! m ( p + pL )

... (4)

In order to account for adsorbed gas total compressibility factor should be modified in following way.

ct* = c f + cg (1 !S w ) + cw S w+cgd

..... (5)

The value of average pressure always required for normalized time calculations. Solving material balance equation is the
easiest way to get the average pressure for each time step. However, conventional material balance equation cannot be used for
shale gas reservoirs if consideration of adsorbed gas is desired. King (1990) proposed a modified material balance equation
which would take care of violation of nonreactive rock assumption. Following are the summary of main results from Kings
(1990) work which successfully used in our analysis.

z* =

... (6)

#
&
(1 ' S w ) + $$ (m ( p + pL )Tsc zsc !!
"
% VLcuft Tp sc z
p & Gp #
p
! ... (7)
= *i $$1 '
*
G !"
z
zi %

Linear Flow in Dual Porosity Reservoirs Asymptotic Equations

Bello and Wattenbarger (2010) summarized asymptotic equations for transient flow regimes which can occur in linear dual
porosity model. Fig. 4 and Table 3 summarize the regions and asymptotic equations they presented. Most of their work is
related to Region 4 which is transient linear flow from matrix blocks to fracture system. Occurrence of Bilinear flow in the
field data is ignored and early " slope in the log-log plot of rate versus time is estimated as skin effect due to Linear flow
convergence or completion related problems. However, it has been observed that some wells in Barnett, Woodford and
Fayetteville Shales exhibit long periods of bilinear flow and may or may not be followed by linear flow. Therefore, more
general investigation of linear dual porosity model is needed to better understand the occurring flow regimes and parameters
which affect their existence.

Fig. 4Illustration of the five flow regions for a slab matrix dual porosity linear reservoir (yDe = 100); !Ac =10-3, 10-5, 10-7 for values of
" = 10-3. From (Bello and Wattenbarger, 2010)

SPE 144583

Table 3Summary of analysis equations for the constant pwf

inner boundary case (slab matrix, dual porosity). From (Bello
and Wattenbarger, 2010)
Region

Equation

qD =

# !12
tD
2" "
1

(# )
= 3

qD

4
5

" 4 ! 14
qD =
tD
10.133
!1
1
qD =
tD 2
2" "
2

yeD

2" "

!1

tD

or

qD =

1
2" "

!1

tD 2

---------------------------------------------------------------------------

Since early linear flow in hydraulic fractures will last in few hours or few days, Region 1 is not important for practical
considerations. In this work mainly bilinear and matrix transient linear flow will be analyzed. Given the gas flow
dimensionless variables for linear dual porosity model;

tD =

0.00633k F t
.... (8)
!ct Acw

k Acw m( pi ) ! m( pwf
1
= F
qD
1422q gT

!=

)] ...... (9)

12 k m
Acw ..... (10)
L2F k F

yeD =

ye
Acw

...... (11)

and asymptotic equations for Bilinear and matrix transient Linear flow;
1

" 4 ! 14
qD =
tD
10.133
# )
(
= 3

qD

yeD

2" "

...... (12)

!1

t D 2 ........ (13)

substituting Eqs. 8, 9, 10, and 11 into Eq.12 yields to

qg

[m( p ) ' m( p )] &

=
i

wf

14409.126T

12(ct #
$
2 !
% 0.00633 LF "

k F 2 k m 4 Acw
t

......... (14)

SPE 144583

and substituting Eqs. 8, 9, 10, and 11 into Eq.13 yields to

qg

[m( p ) ' m( p )] &

=

12(ct #
$
2 !
2844) 3) T % 0.00633 LF "
i

wf

k m 2 ye Acw
t

....... (15)

As long as perforated length (xe) and fracture spacing (LF) is available from completion data and Acw =2* xe*h, in order to
calculate gas flow rate, only unknowns for bilinear and linear flow are kF, km, and ye. In some cases coring is done and value of
matrix permeability is known or well estimated. If field data includes both bilinear and linear flow then regression can be done
on field data and unique solution can be found 2 equations, 2 unknowns, zero degree of freedom.
Estimates of matrix permeability are important. As it is clear from Eq. 15 low values of matrix permeability will overestimate
the SRV. Therefore, most of the time there are 3 unknowns which make the solution non-unique. On the other hand, powers
for each parameter are different which means that they have different impact on the rate. For example, in Linear flow gas flow
rate is more sensitive to fracture half length than matrix permeability. Before using Eq. 14 and Eq. 15 to history match the
field data, an analysis was carried out on synthetic data which was generated in commercial simulator.
History Matching with Synthetic Data
A reservoir simulation model was built in commercial simulator and one segment of horizontal well (Fig. 2) was simulated
because of identical fractures assumption and symmetry. Table 2 shows general input data used for this model. Only fracture
and matrix permeability values were changed to obtain desired flow regimes. Well Model-1, Well Model-2, and Well Model-3
were simulated to analyze bilinear, linear, and bilinear followed by linear cases, respectively.
A Least Absolute Value (LAV) regression was used to history match the production data of synthetic data with linear dual
porosity model. A computer program was developed in EXCELs VBA (DP-History Matching.xls) to run regression for
history matching. In order to run History Matching by taking into account the gas properties change over time and desorption
of gas at low pressures following algorithm is applied.
1.
2.
3.
4.
5.
6.
7.

Input initial guesses for variables. In case of known permeability variables are going to be kF and ye.
Calculate the volumetric OGIP (free + adsorbed) with initial guesses.
Read real time and rates as an input and calculate cumulative gas production for each time.
Perform Material Balance Calculation with modified parameters to get average pressure.
Calculate normalized time which should be equivalent to analytical time.
Run Linear Dual Porosity Model to calculate rate for each input time
Calculate the objective function which is ;

ObjFunc =

!q

meas ,i

" qcalc,i ................................................................................................. (16)

i =1

8.
9.

Run Regression for first iteration and update unknown variables.

Check if objective function is less than tolerance. If not, then repeat steps 2-8.

First, two unknown problem was checked to confirm uniqueness of solutions with simulated data. Table 4 summarizes
important results of this analysis.
Well Model-1
This well exhibits bilinear flow for 400 days followed by boundary dominated flow (Fig. 5). The first 100, 200, 300 and 400
days of production were used in History Matching program to calculate effective fracture permeability and fracture half length.
Since all four cases gave approximately same results, only results of 400 days case is shown in Fig. 5. Besides calculating the
fracture permeability with less than 1.5% error, fracture half-length also was estimated as good as with 6% error. On the other
hand, asymptotic equation for bilinear flow (Eq. 14) clearly shows that rates in bilinear flow do not depend on fracture half
length. Does it mean that bilinear flow can be matched with any fracture half length? This was the next thing we checked. Two
cases , ye=100 ft and ye=2500 ft were run in Linear Dual Porosity Model and it was observed that Bilinear flow can be matched
with any ye as a guess as long as this guess is higher than actual value. This result, actually, is not a surprise. If we look at the
theory of bilinear flow we will see that it lasts as long as the flow inside the fracture is transient. Once the transient flow ends
bilinear flow terminates. Longer the fracture half length, with constant values for other parameters, longer should be the
transient flow inside the fractures.

SPE 144583

Fig. 5Production history of Well Model-1. Bilinear flow (1/4 Slope) followed by BDF (left). History matching results for Well Model-1.
Only 400 days (green points) were used in history matching program as an input (right).

Well Model-2
Linear flow is observed for 3000 days which is followed by BDF (Fig. 6). The first 500, 1000, 2000, and 3000 days were
analyzed and all of them gave approximately same results for fracture half length. Fig. 6 shows History Matching results for
3000 days case. However, values of fracture permeability were different in each case. This is expected result since from
asymptotic equation (Eq. 15) we know that linear flow depends on matrix permeability and fracture half length. On the other
hand, same question is rising again Does it mean that linear flow can be matched with any fracture permeability? In order to
check this two cases, kF=50 md and kF=1200 md were run in the program. It was observed that linear flow will match
regardless of kF as long as guess for kF is higher than actual value. This is also not a surprise. Since low values of kF will result
in Bilinear Flow everything else being constant, it will lower the rates compared to Linear Flow and slope will be different.

Fig. 6Production history of Well Model-2. Linear flow (1/2 Slope) followed by BDF (left). History matching results for Well Model-2.
Only 3000 days (green points) were used in history matching program as an input (right).

Well Model-3
This Model was run to analyze the sensitivity of found parameters with wells having bilinear flow followed by linear (Fig. 7).
Late bilinear and early linear flows were analyzed in History Matching program (Fig. 7). Values of fracture permeability and
fracture half-length were calculated with 5% and 3 % error, respectively. As it was discussed earlier, if two flow regimes exist
for two unknowns then uniqueness of solution is guaranteed.

Fig. 7Production history of Well Model-3. Bilinear flow (1/4 slope) followed by Linear flow (1/2 Slope) (left). History matching results
for Well Model-3. Only 8000 days late bilinear and early linear (green points) were used in history matching as an input (right)

SPE 144583

Table 4Summary of 2 unknown History Matching Problem

Model

Well - Model 1

Well - Model 2

Well - Model 3

Period
100
200
300
400
500
1000
2000
3000
Bilinear
Linear
Late Bilinear + Early Linear

Calculated
kF, md
ye, ft
25.74
413
25.48
458
25.37
472
25.31
477
490
539
539
534
643
527
790
521
12.16
524
24.8
519
12.57

515

Actual
kF, md
25
25
25
25
300
300
300
300
12
12
12

ye, ft
500
500
500
500
500
500
500
500
500
500
500

What if matrix permeability cannot be estimated with high confidence? In this case regression should be done on matrix
permeability, too. According to asymptotic equations, solution to this problem will be non-unique. Now, we will look at the
same wells but with three unknowns: km, kF, and ye.
Well Model-1
The History Matching program was run for different values of matrix permeability and regression was done on kF and ye.
Although matrix permeability was wrong, excellent match was obtained with synthetic data. However, values found by
regression for fracture permeability were not right answers. For lower values of km higher values of kF were found and vice
versa which is obvious from asymptotic equation (Eq. 14). Then the program was modified to history match with three
unknown variables and regression on km, kF, and ye was done simultaneously. Although the excellent visual match was
observed, values found for unknowns were not right answers. Table 5 summarizes results of this analysis with Fig. 8.

Fig. 8History Matching results for Well Model-1. Regression was done on kF and ye for early 400 days by assuming different km
values (5 times greater and 5 times smaller than actual value). Then regression was done on all 3 parameters to get best match.

Well Model-2
Like in previous model, first regression was done on kF and ye with different guesses for matrix permeability. Then matrix
permeability also was added to unknowns and modified program was run to regress on km, kF, and ye at the same time. With
different guesses for km, excellent matches observed with calculated and measured rates for linear flow. Results of three
unknown problem also were good for linear flow (Fig. 9). However, values found for fracture half-length were not right which
would affect the evaluation of reserves and estimation of SRV.

SPE 144583

Fig. 9History Matching results for Well Model-2. Regression was done on kF and ye for early 3000 days by assuming different km
values (3 times greater and 3 times smaller than actual value). Then regression was done on all 3 parameters to get best match.

Well Model-3
In this case Bilinear and Linear flows were matched at the same time for different guesses for matrix permeability. Then one
time regression was done on three parameters to get best estimates for km, kF, and ye. Like in previous two cases excellent
matches were obtained for history matched part (Fig. 10). However, model parameters for km, kF, and ye were not calculated
right.

Fig. 10History Matching results for Well Model 3. Regression was done on kF and ye for late Bilinear and early linear - 8000 days
by assuming different km (4 times greater and 4 times less than actual value). Then regression was done on all 3 parameters to get
best match.

10

SPE 144583

Table 5Summary of 3 unknown History Matching Problem

Model

Well - Model 1

Well - Model 2

Well - Model 3

Actual

Calculated
km, md

kF, md

ye, ft

km, md

kF, ft

ye, md

Assumed = 1.40E-4

50.15

1000

7.00E-04

25

500

Assumed = 3.50E-3

11.56

348

7.00E-04

25

500

Regressed = 6.41E-4

26.33

492

7.00E-04

25

500

Assumed = 6.00E-4

205

306

2.00E-04

300

500

Assumed =6.67E-5

923

926

2.00E-04

300

500

Regressed = 2.89E-4

645

434

2.00E-04

300

500

Assumed = 1.25E-5

21.45

999

5.00E-05

12

500

Assumed = 2.00E-4

4.83

283

5.00E-05

12

500

10

411

5.00E-05

12

500

Regressed = 7.71E-5

Field Examples from Barnett Shale

Two wells from Barnett Shale were analyzed in this paper with proposed method.
Well B-151
Different authors (Bello and Wattenbarger (2009, 2010), Al-Ahmadi et al., 2010) used matrix permeability, km=1.5E-4 md for
Barnett wells in their analysis. In our history matching analysis we started as if matrix permeability is known and then showed
possible values for it. As we already discussed above, Well B-151 is showing bilinear flow followed by linear. Late data also
indicates that BDF or interference between fractures has started. First, only linear flow period is matched with proposed model
with known permeability. If we look at Fig. 11 it is clear that fracture half-length is underestimated. Asymptotic equation
(Eq. 15) should give us a clue that initial assumption for matrix permeability, km=1.5E-5 md is wrong for this well. Therefore,
multi-parameter regression was done on this well in order to history match the linear flow period (Fig. 12). Although three of
the matrix permeability values match linear period quite well only one of them is an actual solution. Since we have BDF in this
well, we know that km=5.00E-5 md is the right solution because it correctly determines when curve will bend down. However,
in most of the time we do not see the BDF in the production history of the well. In that case there is a range of numbers for
both matrix permeability and fracture half length which can estimate the linear flow part. Once the correct solution is obtained,
regression can be done on kF to match the early bilinear part of the flow. Since the matrix permeability found from this
analysis is believed to be actual solution, there is only one kF value which can match the early few points of bilinear Flow
(Fig. 13).

Fig. 11History matching results for Well B-151. Matrix permeability is assumed to be known as 1.5E-4 md and regression was done
on kF and ye (left). A part of linear flow (green points) were used in history matching. From figure (right) it is clear that assumed km
(1.5E-4 md) is overestimated because found ye is less than actual value (Curve bending down sooner).

SPE 144583

11

Fig.12History matching results for Well B-151. Regression was done on kF and ye for different assumptions of km. Linear period is
used in history matching. It was concluded that km =5.00E-5 md is giving the best match. Therefore, matrix permeability for this well is
estimated as 5.00E-5 md.

Fig. 13History Matching Results for Well B-151 with known matrix permeability, km =5.00E-5 md. Regression also was done on kF to
match the early bilinear flow.

Well B-130
From Log-Log plot of rate versus time we can see that well exhibits long period of Linear flow (Fig. 14). However, at the late
times, well has a problem in keeping up the rates stable. After doing analysis for liquid loading it was concluded that most of
the data points at the late production of the well are below critical rate. From Fig. 14 it is clear that if we ignore rates below
critical then the well is still in transient. The period which is showing clearly # slope (Linear) in Log-Log plot was used in
history matching program. Since Well B-130 is from the same county as Well B-151, matrix permeability value found for
Well B-151 was used as known parameter for Well B-130. Fig. 15 shows the results of history matching where regression was
done only on fracture half length. Although fracture permeability is also shown as a result, it cannot be found exactly because
Bilinear flow was not observed in this well. Value of 20 md should be considered as a lower limit for this well. The actual
value of fracture permeability can be any number bigger than 20 md. Although excellent match was obtained (Fig. 15) it does
not mean that parameters found from History Matching are exactly same parameters that exist in the reservoir. It should be
noted that results are based on the assumption that matrix permeability is known. If matrix permeability is less that the
assumed value, then fracture half-length would be bigger which will result in bigger OGIP, too. Therefore, results should be
tuned for new production data or any other updates on parameters to decrease the uncertainty as much as possible.

12

SPE 144583

Fig. 14Production history of Well B-130 (left). Square root of time plot for Well B-130 (right). Linear flow is exhibiting linear behavior
shown with black line. Blue points are rates below critical value not to have liquid loading.

Fig. 15History matching results with known matrix permeability km=5.00E-5 md (left). Blue points are showing rates below critical
value not to have liquid loading (right). From plot it is clear that assumed value for km is good enough not to underestimate ye or
OGIP.

Conclusions
The main objective of this paper was describing a method to history match the production data of Shale gas wells by using the
solutions proposed initially by Bello and Wattenbarger (2010). Correction for gas properties change and desorption of gas at
low pressures is well handled in order to use analytical solutions. Early results show that the method works quite well and can
be applied in field scale. Following are the main points drawn from this research.

Contribution from outside of SRV will not be felt in practical life time of a shale gas well with permeability values
less than 5.00E-5 md.
In order to match bilinear flow matrix permeability and fracture permeability is needed. On the other hand, for linear
flow main unknowns are matrix permeability and fracture half length.
With known matrix permeability unique solution can be obtained by history matching if other parameters are also
known with high confidence.
In the absence of good estimate for matrix permeability, history matching results are non-unique and minimum OGIP
can be estimated from results.
In case of non-uniqueness, results should be updated as more information from field is available.
Future forecasting is possible with given economic constraints.

Nomenclature

Acw
cf

= formation compressibility, 1/psi

cg

= gas compressibility, 1/psi

c gd

= additional compressibility due to adsorbed gas, 1/psi

ct
cw
G

= total compressibility, 1/psi

= cross sectional area to the flow, ft2

= water compressibility, 1/psi

= original gas in place, Mscf

SPE 144583

13

Gp

= cumulative production, Mscf

kF

= effective fracture permeability, md

km
LF
m( pi )
m( pwf )

OGIP
p
p

pL
p sc
qD
qg
Sw

SRV

tD
tn

Tsc
VE
VL
VLcuft
xe
ye
yeD
z
z sc
zi*
z*

= matrix permeability, md
= Fracture Spacing, ft
= pseudo initial pressure, psi2/cp
= pseudo bottomhole flowing pressure, psi2/cp
=original gas in place, Bscf
= pressure, psi
= average pressure, psi
=Langmuir pressure, psi
= pressure at standard conditions, psi
= dimensionless rate
= gas rate, Mscf/Day
= water saturation
= stimulated reservoir volume
= dimensionless time
= normalized time, days
= temperature at standard conditions, R
= volume of gas adsorbed, scf/ton
= Langmuir volume, scf/ton
= Langmuir volume, scf/ft3
= perforated length, ft
= fracture half length, ft
= dimensionless fracture half length
= compressibility factor
= compressibility factor at standard conditions
= modified compressibility factor at initial pressure
= modified compressibility factor at average pressure

Greek Symbols

g
!
!
! sc
!g
!bulk

= gas viscosity, cp
= porosity
= interporosity coefficient
= gas density at standard conditions, gr/cm3
= gas density at average pressure, gr/cm3
= matrix density, gr/cm3

14

References

SPE 144583

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