Chapter 9 Flow Regime Identification and Analysis Using Special Methods
9.1 Introduction Transient behavior of oil well with a finiteconductivity vertical fracture has been simulated by Cinco et al. Usually it is assumed that fractures have an infinite conductivity; however, this assumption is weak in the case of large fractures or very lowcapacity fractures. Finiteconductivity vertical fracture in an infinite slab is shown in Figure 91. Transient behavior of a well with a finiteconductivity vertical fracture includes several flow periods. Initially, there is a fracture linear flow characterized by a halfslope straight line; after a transition flow period, the system may not exhibit a bilinear flow period, indicated by a onefourthslope straight line. As time increases, a formation linear flow period might develop. Eventually, in all cases, the system reaches a pseudoradial flow period. Pressure data for each flow period should be analyzed using a specific interpretation method such as A~b versus (t)1/4 for bilinear flow A~b versus (t) 1/2 for linear flow and A~b versus log t
for pseudoradial flow
9.2 Fracture Linear Flow Period 1'4'8 During this flow period, most of the fluid entering the wellbore comes from the expansion of the system within the fracture and the flow is 339
340
Oil Well Testing Handbook
Figure 91.
Finiteconductivity vertical fracture in an infinite slab reservoir (after Cinco and Samaniego, 1978). 2
essentially linear, as shown in Figure 92. Pressure response at the wellbore is given by
pwD
_
3.546
rkfc~c, tDxr ] 0.5
(91)
0.3918q/3( #t ) ~ Pi  P w f  
bfh
kfdpfcft
(92)
Eq. 92 indicates that a loglog graph of pressure difference against the time yields a straight line whose slope is equal to onehalf. A graph of pressure versus the square root of time also gives a straight line whose slope depends on the fracture characteristics excluding the fracture halflength, xf.
Figure
92. Fracture linear flow. 1
Flow Regime Identification and Analysis Using Special Methods
341
The fracture linear flow ends when
O'Ol(kfbf)2 tDxf

(93)
2
This flow period occurs at a time too early to be of practical use.
9.3
Bilinear Flow 1'4'8
It is a new type of flow behavior called bilinear flow because two linear flows occur simultaneously. One flow is linear within the fracture and the other is in the formation, as shown in Figure 93. The dimensionless wellbore pressure for the bilinear flow period is given by 2.45
PWD = {(kfbf )D]O.5 (tDxl)
(94)
1/4
This equation indicates that a graph of PWD v e r s u s (tDxf) 1/4 produces a straight line whose slope is 2.45/[(kfbf)D] ~ intercepting the origin. Figure 94 presents that type of graph for different values of (kfbf) D. The existence of bilinear flow can be identified from a loglog plot of Ap versus At from which the pressure behavior for bilinear flow will exhibit a straight line whose slope is equal to onefourth to the linear flow period in which the slope is onehalf. The duration of this period depends on both dimensionless fracture conductivity, (kfbf) D, and wellbore storage coefficients (dimensionless storage capacity), CfDU. For buildup analysis of bilinear flow period, the pressure drop may be expressed as 44. lq/3# /,,p  h(klbj)O.5(4,~c,k)
(t)0.25 ~
Figure 93. Bilinear flow. 2
(95)
342
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Endofstraightline
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Figure 94. PWDversus [tDx,]0"25for a well with a finiteconductivity vertical fracture (after Cinco and Samaniego, 1978). 2
where Ap is the pressure change for a given test. Eq. 95 indicates that a graph of Ap v e r s u s t 1/4 produces a straight line passing through the origin whose slope, mbf, is given by 44.1q#fl mbf  h(kfbf )O.S(~gCtk)0.25
(96)
Hence, the product h(kfbf) ~ can be estimated by using the following equation:
h(kfbf)O.5 _
44. lq#fl
 mbf (~p#gctk)
(97) 0.25
Figure 95 shows a graph for analysis of pressure data of bilinear flow, while Figure 96 is a loglog graph of pressure data for bilinear flow. Figure 96 can be used as a diagnostic tool. The above equations indicate that the values of reservoir properties must be known to estimate the group h(kfbf) ~ The dimensionless time at the end of bilinear flow period is given by the following equation:
Flow Regime Identification and Analysis Using Special Methods
(kfbf) D >
343
1.6
(kfbf) o <_1.6 I I
mbf
,,=,,==1=1 l
to.25 Figure 95. Graph for analysis of pressure data of bilinear flow (after Cinco and Samaniego, 1978). 2
o
7
Slope = 0 . 2 5
log time
Figure 96. loglog graph of pressure data for bilinear flow analysis (after Cinco and Samaniego, 1978). 2 For (kfbf)D <_ 1.6
E4"'' ] (klbj)O.S  2.5
4
(98)
For (kfbf ) D > 3
0.10
tDebf ~ ( k f b f )2D
(99)
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Oil Well Testing Handbook
For 1.6 _< (kfbf) D <_ 3
tDebf "~ O.0205[(kfbf ) D

(910)
1.5] 1"53
Figure 97 shows a graphical representation of these equations. From Eqs. 95 and 98 through 910, if (kfbf)D > 3, the dimensionless pressure drop at the end of the bilinear flow period is given by
1.38 (PWD)ebf  (kfbf ) D
(911)
Hence, the dimensionless fracture conductivity can be estimated using the following equation:
1.38 (kfbf )D ~ (PWD)ebf
(912)
(PWD)ebf can be calculated using the following equation: kh(Ap)
(913)
(PWD)ebf "~ 141.2q#/3
where Ap is obtained from the bilinear flow graph. From Eq. 95, a graph of log Ap versus log t (see Figure 96) yields a quarterslope straight line that can be used as a diagnostic tool for bilinear flow detection.
101
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*~ 103 t ......
104~ lo' t
101
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102
(kfbf)o Figure 97.
Dimensionless time for the end of the bilinear flow period versus dimensionless fracture conductivity. 2
Flow Regime Identification and Analysis Using Special Methods
345
9.4 Formation Linear Flow 1'4'8 Figure 98 represents formation linear flow. Figure 99 shows a graph of log [pwD(kfbf)D] versus log [tDx~(kfbf)2]. For all values of (kfbf)D the behavior of both bilinear flow (quarterslope) and the formation linear flow (halfslope) is given by a single curve. Note that there is a transition period between bilinear and linear flows. Bilinear flow ends when fracture tip effects are felt at the wellbore. The beginning of the formation linear flow occurs at (kfbf) 2 ~ 102, that is
tOblf 
(914)
100
Fracture
\
,,.1
Figure 98. Formation linear flow. 2
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A p r o x m a t e start o f s e m i l o g straight line i 10 ~
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I l0 4
I 10 6
10 ~
Figure 99. Type curve for vertically fractured oil wells (after Cinco and Samaniego, 1978). 2
346
Oil Well Testing Handbook
The end of this flow period is given by 1'8 t Delf "~ 0.016
Hence, the fracture conductivity may be estimated as follows"
(kfbf ) D ~
10 (tDblf ) 0"5
(915)
or
O.5
(kfbf)D ~ 1.25 x 10 2 r
(916)
kms)
These equations apply when (kfbf)D >_ 100.
9.5 PseudoRadial Flow 1'4'8 Figure 910 illustrates pseudoradial flow. The dashed line in Figure 99 indicates the approximate start of the pseudoradial flow period (semilog straight line).
9.6 Type Curve Matching Methods ~'7'a Figure 99 can be used as a type curve to analyze pressure data for a fractured well. Pressure data on a graph of log Ap versus log t are matched on a type curve to determine
[r
[Pwz~(keb~)D]M
Fracture Well
Figure 910. Pseudoradial flow. 2
Flow Regime Identification and Analysis Using Special Methods
(t).,
347
[tDx~(kibi)~].
[thai.,
[tbs~dM
Dimensionless fracture conductivity:
[(kyby)~]. Formation permeability for oil:
k

141.2qpfl [PwD(kfbf )D]M h[(Ap)]M [(kfbf)D]M
(917)
Fracture halflength:
O'O00264k(t)M[(kfbf)Z]M xf

r
[(k~f)~].
(918)
Fracture conductivity: (919) End of bilinear flow:
Beginning of formation linear flow:
Beginning of pseudoradial flow:
[tbss.]M
Pressure Data Analysis If large span pressure data are available, the reliable results can be obtained using the specific analysis graphs. Now we will discuss various cases where all the pressure data fall on a very small portion of the type curve and a complete set of information may not be obtained.
348
Oil Well Testing Handbook
Field Case Studies Case 1: Bilinear Flow Type of Analysis 4 Pressure data exhibit onefourthslope on a loglog graph; and when a loglog graph of pressure data indicates that the entire test data are dominated by bilinear flow (quarterslope), the minimum value of fracture halflength xf can be estimated at the end of bilinear flow, i.e., for (kfbf)D > 3, using the following equationl'8: 0.0002637 (kfbf )2tebf)
Xf ~
0.25
dp#ctk
(920)
By definition, the dimensionless fracture conductivity is
ksbs
(kfbf)D~f
(921)
where kfbf is calculated using Eq. 925 and slope mbf can be found from bilinear flow graph which is a rectangular graph of pressure difference against the quarter root of time. This graph will form a straight line passing through the origin. Deviations occur after some time depending on the fracture conductivity. The slope of this graph, mbf, is used for the calculation of the fracture permeabilityfracture width product (kfbf). The dimensionless fracture conductivity is correlated to the dimensionless effective wellbore r a d i u s ,rw/r ' f, as shown in Table 91. Then, the skin can be calculated from the following relationship: sIn
(rT~w)
(922)
Generally, wellbore storage affects a test at early time. Thus it is expected to have pressure data distorted by this effect, causing deviation from the onefourthslope characteristic of this flow period. It is important to note that pressure behavior in Figure 911 for both wellbore storagedominated and bilinear flow portions is given by a single curve that completely eliminates the uniqueness matching problem. Figure 911 is a new type curve and is used when pressure data exhibit onefourthslope on a loglog graph. The end of wellbore storage effects occurs when F2(tDxs) 2 X 102, yielding C4 teWS

65,415.24
(kfbf )5h4c/)ct
(923)
Flow Regime Identification and Analysis Using Special Methods
349
Table 91 The Values of Effective Wellbore Radius as a Function of Dimensionless Fracture Conductivity for a Vertical Fractured Well 2 Dimensionless fracture conductivity, (kfbf )o
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 20.0 30.0 40.0 50.0 100.0 200.0 300.0
r~w
xf 0.026 0.050 0.071 0.092 0.115 0.140 0.150 0.165 0.175 0.190 0.290 0.340 0.360 0.380 0.400 0.410 0.420 0.430 0.440 0.450 0.455 0.460 0.465 0.480 0.490 0.500
If Figure 911 is used as a type curve, the following information may be obtained: [F1 ~WD)]M,
[Fz(tDx,)]M,
(Ap)M,
(t)M
Hence, we can estimate the following: Wellbore storage constant for oil"
C 
0.234q#3(t) M IF1 (PWD)]M (Ap) M [Fz(tDxl)] M
(924)
350
Oil Well Testing Handbook 10 2 I
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F2(tox~)= (CDf)4,3 (kfbf)D tDxf F i g u r e 911. T y p e c u r v e for w e l l b o r e s t o r a g e u n d e r b i l i n e a r f l o w c o n d i t i o n s (after Cinco and Samaniego,
1978). 2
Fracture conductivity for oil:
0"4~~tk(141"2q#/3[Fl(PWD)g]} 3 klbl 
(Ap) M
(925)
Case 2: Pressure Data Partially Match Curve for the Transition Period Between Bilinear and Linear Flows Cinco and Samaniego 2 (1978) have presented a new set of type curves that are given in Figure 99. Figure 99 shows a graph of log [PwD(kfbf)D] versus log [tDxi(kfbf)2]. The main feature of this graph is that for all values of (kfbf) D the behavior of both bilinear flow (quarterslope) and the formation linear flow (halfslope) is given by a single curve. The type curve match is unique because the transition period has a characteristic
Flow Regime Identification and Analysis Using Special Methods
351
shape. This comment is valid for dimensionless fracture conductivity, (kfbf)D _> 57r. From the type curve match of pressure data for this case in Figure 99, we obtain
Hence, for oil
(kfbs) xj
141.2q#/3[PwD(kubu)D]M h (Ap)M
(926)
Fracture halflength and fracture conductivity for oil are given by 0.5
(kfbf'] [0.0002637
(t)M
"1
(927)
and
[kfbf ] kfbf  (xf) k xf I
(928)
Since the formation permeability is generally known from prefracture tests, the dimensionless fracture conductivity can be estimated by using the following equation:
kfbf l (kfbf)D = xf k Then using Table 91, find the value of rw/X ' f, " since xf is known r w can be calculated. Estimate skin factor from Eq. 922. If all pressure data fall on the transition period of the curve, type curve matching (Figure 99) is the only analysis method available.
Case 3: Pressure Data Exhibit a HalfSlope Line on a loglog Graph (See Figure 912) There is no unique match with Figure 99; however, the linear flow analysis presented by Clark 4 can be applied to obtain fracture halflength if formation permeability is known. In addition, a minimum value for the dimensionless fracture conductivity, (kfbf)D, can be estimated using Eq. 929. If the wellbore storage effects are present at early times in a test
352
Oil Well Testing Handbook I
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m a x i m u m t i m e at the beginning of linear flow
0
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_ ,.9 9999~~ ] 9
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99999
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Figure 912.
Pressure data for a halfslope straight line in a loglog graph (after Cinco and Samaniego, 1978). 2
for this case, the analysis can be made using the type curve presented by Ramey and Gringarten. 3 O.5
(kfbf) D  1.25 • 102 (tezf)
\tb~/
(929)
Using Table 91, find rw/X ' f , 9 then using Eq. 922, estimate skin factor, s. Case 4: Pressure Data Partially Falling in the PseudoRadial Flow Period s
Figure 913 is a graph Of PWD versus tDew which is the dimensionless time defined by using r 'w instead of xf. This curve provides an excellent tool for type curve analysis of pressure data partially falling in the pseudoradial flow period because the remaining data must follow one of the curves for different fracture conductivities. Table 91 must be used to determine (kfbf)D when using Figure 913. The type curve in Figure 913 involves the following steps: 1. Plot a loglog graph of the pressure data; neither a onefourthslope nor a halfslope is exhibited by the data. 2. Apply Figure 913 to match pressure data. 3. Estimate reservoir permeability from pressure match point
141.2q#fl ~WD ) M k 
h
(Ap) M
(930)
353
Flow Regime Identification and Analysis Using Special Methods pwD=[ kh ((Pi)  (Pwf))] / [141.2q/zfl]
lO, I
i
(kfbf) o
I
i
 
End of bilinear flow
0. 27r ~7r0" 57r27r
1
~ ~
~
Beginning of semilog straight line
End of linear flow lO1

5~
'10~ '20~ 100~ 102 I 103
I 102
I 101
t 1
I 101
I 102
103
tOr,w = [0.0002637kt] / [Ol~gCt(rw)2]
Figure 913. Type curve for a finiteconductivity vertical fracture (after Cinco and Samaniego, 1978). 2
4. Using information from time match estimate effective wellbore radius
r~w 
[0.0002637k (At_!M ] ~ O # g C t (tDr~,)MJ
5 9 By using [(kfbf)D]M in Figure 913, obtain F! W
Ir,w]
(931) ( r w' / X f ) T a b l e
91, 9 hence
(932)
L rf J Table 91
6. Estimate the skin factor as follows:
in(rw)
(933)
7. Calculate fracture conductivity as follows: kfbf  ( k f b f ) D k x f
(934)
8. The pressure data falling in the pseudoradial flow period also must be analyzed using semilog methods to estimate k, r w, and s. The following three field examples illustrate the application of several of the methods and theory previously discussed.
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Oil Well Testing Handbook
Example 916 P r e s s u r e D a t a A n a l y s & f o r P s e u d o  R a d i a l F l o w A buildup test was run on this fractured oil well after a flowing time of 1890hours. Reservoir and test data are as follows: qo = 220stb/day; h  49 ft; ct  0.000175 psii; rw  0.25 ft; Pwf  1704 psi; q5  0.15 (fraction); #o  0.8 cP; ct  17.6 x 10 6 psi 1. Identify type of flow period and determine the following using type curve matching and semilog analysis techniques, and estimate reservoir parameters. Solution Figure 914 shows a loglog graph of the pressure data; from this graph we can see that neither a onefourthslope nor a halfslope is exhibited by the data. Figure 914 shows that the pressure data match the curve for ( k f b f ) D = 27r given in Figure 913 and the last 14 points fall on the semilog straight line. Match points from Figure 914 are given below. Pressure match points: ( A p ) M 100 psi, (PWD)M  0.34 Time match points: ( A t ) M 1 hour, (tDr,w)M = 0.19 =
 
( P W D ) M  0.45,
[tDr,w]M  1.95
F r o m the pressure : k .~_
=
ch using Eq. 930, estimate reservoir permeability:
141.2q#/3 (PWD ) M
h
(Ap)M
141.2 • 220 x 0.8 x 1.2 0.34 49 100  2.07 mD 101 Beginning of semilog straight line
(k: b:)o 0.27r 1
103r
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tOrw) O.'9
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At (h) 102
I
103
102
I
101
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I 102
tor'w
Figure 914. Type curve matching for Example 112.
103
Flow Regime Identification and Analysis Using Special Methods
355
U s i n g the i n f o r m a t i o n f r o m time m a t c h in Eq. 931 0.0002637k (At)M ] ~
let=
~ # g C t (tD/w)MJ [0
=
0.0002637x2.07 1 ]~  36.9 ft .15 x 0.8 x 17.6 x 10 6 0.19
36.9/0.403  88.9 ft. F r o m Table 91, rw/X ' f  0.403; hence, xf The skin factor is estimated by using Eq. 933"
rw)
sfln
7w w
(0.25~ ln\36.9]4.99
F r o m Eq. 934, the fracture c o n d u c t i v i t y is
k~bj (kfb~)Dkxf = 27r x ( 2 . 0 7 ) ( 8 8 . 9 ) 
l156.2mDft
Semilog analysis: F i g u r e 915 is a semilog g r a p h for this example. The correct semilog straight line has a slope m  307 psi/cycle a n d (Ap) 1h r   4 7 psi. The form a t i o n p e r m e a b i l i t y can be calculated f r o m Eq. 52: 162.6q#/3
mh
k
162.6 x 220 x 0.8 x 1.2 307 x 49 = 2.28 m D
=
800 1 700
.........
600
..........
500

E :
sf= 4 . 8 xf=6 0 . 7 6
....................... <~
400
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300
oi9 " ~
0
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1 Ap= 47Ps:l ......................
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, 10 1
1
At ( h o u r s )
Figure 915.
Semilog plot.
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103
356
Oil Well Testing Handbook
Table 92 Summary of Analysis Results Analysis results
Type curve matching solution Semilog solution
Permeability (mD) Fracture skin factor, s f Effective wellbore radius, r~w(ft) Fracture halflength, x f (ft) Fracture conductivity (mD ft)
2.07 4.99 36.89 88.7 1156 27r
(kfbf)D
2.28 4.8 30.37 60.76
Using Eq. 53, the fracture skin factor is [(A~lhr ~  l o g
sf1.151
k dp#octr2
k
3.23
[
2.28x 10  6 = 1.151 30~  4 7 _ log O. 15 x 0.8 x 17.6
X
0.252 k 3.23 ]
= 1.151[0.227  6.92 + 3.23]   4 . 8 Find the effective wellbore radius by rearranging Eq. 932: r !w 
rw e
Sf

0.25 e (4"8) p 30.37 ft
Finally, the fracture halflength is calculated as: ln(xf)  ln(2rw) 
sf 
ln(2 x 0.25)  (4.8)  0.6931 + 4.8  4.11
Hence xf

e ln(xy)  e 4"11  60.76 ft
Summary of analysis results is given in Table 92. The results provided by both the type curve analysis and semilog analysis methods are reasonable. F r o m these examples it is demonstrated that type curve matching analysis, when applied properly, provides an excellent diagnostic tool and a technique to estimate both reservoir and fracture parameters.
9.7 Summary 9 Prefracture information about the reservoir is necessary to estimate fracture parameters. 9 The type curve analysis methods must be used simultaneously with the specific analysis methods to produce reliable results.
Flow Regime Identification and Analysis Using Special Methods
357
(pwf ) versus t 1/4, (pwf ) versus t 1/2, and (pwf) versus log t
9 It provides new techniques for analyzing pressure transient data for wells intercepted by a finiteconductivity vertical fracture. This method is based on the bilinear flow theory which considers transient linear flow in both fracture and formation. These new type curves overcome the uniqueness problem exhibited by other type curves.
References 1. Cinco, H., Samaniego, F., and Dominguez, N., "Transient Pressure Behavior for a Well with a FiniteConductivity Vertical Fracutre," Soc. Pet. Eng. J. (Aug. 1981), 253264. 2. Cinco, H., and Samaniego, F., "Effect of Wellbore Storage and Damage on the Transient Pressure Behavior for a Well with a FiniteConductivity Vertical Fracture," Soc. Pet. Eng. J. (Aug. 1978), 253264. 3. Ramey, H. J., Jr., and Gringarten, A. C., "Effect of HighVolume Vertical Fractures in Geothermal Steam Well Behavior," Proc. Second United Nations Symposium on the Use and Development of Geothermal Energy, San Francisco, May 2029, 1975. 4. Clark, K. K., "Transient Pressure Testing of Fractured Water Injection Wells," J. Pet. Technol. (June 1968), 639643; Trans. AIME, 243. 5. Agarwal, R. G., Carter, R. D., and Pollock, C. B., "Evaluation and Prediction of Performance of Low Permeability Gas Wells Stimulated by Massive Hydraulic Fracturing," J. Pet. Technol. (March 1979), 362372. 6. Raghavan, R., and Hadinoto, N., "Analysis of Pressure Data for Fractured Wells: The ConstantPressure Outer Boundary," Soc. Pet. Eng. J. (April 1978), 139150; Trans. AIME, 265. 7. Barker, B. J., and Ramey, H. J., Jr., "Transient Flow to FiniteConductivity Vertical Fractures," Ph.D. Dissertation, Stanford University, Palo Alto, CA, 1977. 8. Gringarten, A. C., Ramey, H. J. Jr., and Raghavan, R., "Applied Pressure Analysis for Fractured Wells," J. Pet. Technol. (July 1975), 887892; Trans. AIME, 259.
Additional Reading 1. Raghavan, R., Cady, G. V., and Ramey, H. J., Jr., "Well Test Analysis for Vertically Fractured Wells," J. Pet. Technol. (1972) 24, 10141020.
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2. Raghavan, R., "Pressure Behavior of Wells Intercepting Fractures," Proc. Invitational WellTesting Symposium, Berkeley, CA, Oct. 1921, 1977. 3. Wattenbarger, R. A., and Ramey, H. J., Jr., "Well Test Interpretations of Vertically Fractured Gas Wells," J. Pet. Technol. (May 1969), 625632; Trans. AIME, 246.