Flow Regime Identification and Analysis Using Special Methods

Flow Regime Identification and Analysis Using Special Methods

Chapter 9 Flow Regime Identification and Analysis Using Special Methods 9.1 Introduction Transient behavior of oil well with a finite-conductivity ve...

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Chapter 9 Flow Regime Identification and Analysis Using Special Methods

9.1 Introduction Transient behavior of oil well with a finite-conductivity 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 low-capacity fractures. Finiteconductivity vertical fracture in an infinite slab is shown in Figure 9-1. Transient behavior of a well with a finite-conductivity vertical fracture includes several flow periods. Initially, there is a fracture linear flow characterized by a half-slope straight line; after a transition flow period, the system may not exhibit a bilinear flow period, indicated by a onefourth-slope straight line. As time increases, a formation linear flow period might develop. Eventually, in all cases, the system reaches a pseudo-radial 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 pseudo-radial 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 9-1.

Finite-conductivity vertical fracture in an infinite slab reservoir (after Cinco and Samaniego, 1978). 2

essentially linear, as shown in Figure 9-2. Pressure response at the wellbore is given by

pwD

_

3.546

rkfc~c, tDxr ] 0.5

(9-1)

0.3918q/3( #t ) ~ Pi -- P w f - -

bfh

kfdpfcft

(9-2)

Eq. 9-2 indicates that a log-log graph of pressure difference against the time yields a straight line whose slope is equal to one-half. 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

9-2. Fracture linear flow. 1

Flow Regime Identification and Analysis Using Special Methods

341

The fracture linear flow ends when

O'Ol(kfbf)2 tDxf

--

(9-3)

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 9-3. The dimensionless wellbore pressure for the bilinear flow period is given by 2.45

PWD = {(kfbf )D]O.5 (tDxl)

(9-4)

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 9-4 presents that type of graph for different values of (kfbf) D. The existence of bilinear flow can be identified from a log-log plot of Ap versus At from which the pressure behavior for bilinear flow will exhibit a straight line whose slope is equal to one-fourth to the linear flow period in which the slope is one-half. 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 9-3. Bilinear flow. 2

(9-5)

342

Oil Well Testing Handbook i

,

i

i

i

i

i

i

,

i

,

,

~

~

i

~

2---

--

--

Endofstraightline

i

................. ; .................. i........ i i i

i i i

/

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I

i

J

i

j'-~.-- . . . . . . . --~-:-,

j

/'.

i

--

!

_

(kfbf)d-O.17r,

/

,

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,

0

~

S ;

1---

0

"

0.2

i 0.6

0.4

0.8

1

( t DxI )0.25

Figure 9-4. PWDversus [tDx,]0"25for a well with a finite-conductivity vertical fracture (after Cinco and Samaniego, 1978). 2

where Ap is the pressure change for a given test. Eq. 9-5 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

(9-6)

Hence, the product h(kfbf) ~ can be estimated by using the following equation:

h(kfbf)O.5 _

44. lq#fl

-- mbf (~p#gctk)

(9-7) 0.25

Figure 9-5 shows a graph for analysis of pressure data of bilinear flow, while Figure 9-6 is a log-log graph of pressure data for bilinear flow. Figure 9-6 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 9-5. Graph for analysis of pressure data of bilinear flow (after Cinco and Samaniego, 1978). 2

o

---------7

Slope = 0 . 2 5

log time

Figure 9-6. log-log 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

(9-8)

For (kfbf ) D > 3

0.10

tDebf ~ ( k f b f )2D

(9-9)

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Oil Well Testing Handbook

For 1.6 _< (kfbf) D <_ 3

tDebf "~ O.0205[(kfbf ) D

-

(9-10)

1.5] -1"53

Figure 9-7 shows a graphical representation of these equations. From Eqs. 9-5 and 9-8 through 9-10, 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

(9-11)

Hence, the dimensionless fracture conductivity can be estimated using the following equation:

1.38 (kfbf )D ~ (PWD)ebf

(9-12)

(PWD)ebf can be calculated using the following equation: kh(Ap)

(9-13)

(PWD)ebf "~ 141.2q#/3

where Ap is obtained from the bilinear flow graph. From Eq. 9-5, a graph of log Ap versus log t (see Figure 9-6) yields a quarter-slope straight line that can be used as a diagnostic tool for bilinear flow detection.

10-1

i/

i

i

t 7------

*~ 10-3 -t ......

104~ lo-' t

10-1

/---i

.

.

.

.

.

.

I ...... 1

-I 101

102

(kfbf)o Figure 9-7.

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 9-8 represents formation linear flow. Figure 9-9 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 (quarter-slope) 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 --

(9-14)

100

Fracture

\

,,.1

Figure 9-8. Formation linear flow. 2

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~- ..........

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(~ib I

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~/~~_

--

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"'_

~

.1027r

i~ . - ~ ~ ~

;

....

~ .............. ;

7r

r ;

'

i

0~ . . . . . . .

, 10-2 10 -4

I 10 -2

A p r o x m a t e start o f s e m i l o g straight line i 10 ~

i 10 2

I l0 4

I 10 6

10 ~

Figure 9-9. 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

(9-15)

or

O.5

(kfbf)D ~ 1.25 x 10 -2 r

(9-16)

kms)

These equations apply when (kfbf)D >_ 100.

9.5 Pseudo-Radial Flow 1'4'8 Figure 9-10 illustrates pseudo-radial flow. The dashed line in Figure 9-9 indicates the approximate start of the pseudo-radial flow period (semilog straight line).

9.6 Type Curve Matching Methods ~'7'a Figure 9-9 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 9-10. Pseudo-radial 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

(9-17)

Fracture half-length:

O'O00264k(t)M[(kfbf)Z]M xf

-

r

[(k~f)~].

(9--18)

Fracture conductivity: (9-19) End of bilinear flow:

Beginning of formation linear flow:

Beginning of pseudo-radial 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 one-fourth-slope on a log-log graph; and when a log-log graph of pressure data indicates that the entire test data are dominated by bilinear flow (quarter-slope), 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

(9-20)

By definition, the dimensionless fracture conductivity is

ksbs

(kfbf)D--~f

(9-21)

where kfbf is calculated using Eq. 9-25 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 permeability-fracture 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 9-1. Then, the skin can be calculated from the following relationship: s--In

(rT~w)

(9-22)

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 one-fourth-slope characteristic of this flow period. It is important to note that pressure behavior in Figure 9-11 for both wellbore storagedominated and bilinear flow portions is given by a single curve that completely eliminates the uniqueness matching problem. Figure 9-11 is a new type curve and is used when pressure data exhibit one-fourth-slope on a log-log graph. The end of wellbore storage effects occurs when F2(tDxs)- 2 X 102, yielding C4 teWS

--

65,415.24

(kfbf )-5-h4c/)ct

(9-23)

Flow Regime Identification and Analysis Using Special Methods

349

Table 9-1 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 9-11 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

(9-24)

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 9-11. 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

(9-25)

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 9-9. Figure 9-9 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 (quarter-slope) and the formation linear flow (half-slope) 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 9-9, we obtain

Hence, for oil

(kfbs) xj

141.2q#/3[PwD(kubu)D]M h (Ap)M

(9-26)

Fracture half-length and fracture conductivity for oil are given by 0.5

(kfbf'] [0.0002637

(t)M

"1

(9-27)

and

[kfbf ] kfbf - (xf) k xf I

(9-28)

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 9-1, find the value of rw/X ' f, " since xf is known r w can be calculated. Estimate skin factor from Eq. 9-22. If all pressure data fall on the transition period of the curve, type curve matching (Figure 9-9) is the only analysis method available.

Case 3: Pressure Data Exhibit a Half-Slope Line on a log-log Graph (See Figure 9-12) There is no unique match with Figure 9-9; however, the linear flow analysis presented by Clark 4 can be applied to obtain fracture half-length if formation permeability is known. In addition, a minimum value for the dimensionless fracture conductivity, (kfbf)D, can be estimated using Eq. 9-29. If the wellbore storage effects are present at early times in a test

352

Oil Well Testing Handbook I

I

i

i

i i

i i

tblf,

m a x i m u m t i m e at the beginning of linear flow

0

..............

u '

7-

..~7"

_ ,.9 9999~~ ] 9

tbtf

i i

99999

o 9 1 4 9 1 7 96 t..a 9 ~ ..........

.....

!1/2 [

.............. 1

tel/

'e/f, m , n , m u m v a l u e of end of half slope

I

I

I

i

i

,

I

I log time, t

Figure 9-12.

Pressure data for a half-slope straight line in a log-log 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 • 10-2 (tezf)

\tb~/

(9-29)

Using Table 9-1, find rw/X ' f , 9 then using Eq. 9-22, estimate skin factor, s. Case 4: Pressure Data Partially Falling in the Pseudo-Radial Flow Period s

Figure 9-13 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 pseudo-radial flow period because the remaining data must follow one of the curves for different fracture conductivities. Table 9-1 must be used to determine (kfbf)D when using Figure 9-13. The type curve in Figure 9-13 involves the following steps: 1. Plot a log-log graph of the pressure data; neither a one-fourth-slope nor a half-slope is exhibited by the data. 2. Apply Figure 9-13 to match pressure data. 3. Estimate reservoir permeability from pressure match point

141.2q#fl ~WD ) M k -

h

(Ap) M

(9-30)

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 lO-1

-

5~

'10~ '20~ 100~ 10-2 I 10-3

I 10-2

I 10-1

t 1

I 101

I 102

103

tOr,w = [0.0002637kt] / [Ol~gCt(rw)2]

Figure 9-13. Type curve for a finite-conductivity 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 9-13, obtain F! W

Ir,w]

(9-31) ( r w' / X f ) T a b l e

9-1, 9 hence

(9-32)

L rf J Table 9-1

6. Estimate the skin factor as follows:

in(rw)

(9-33)

7. Calculate fracture conductivity as follows: kfbf - ( k f b f ) D k x f

(9-34)

8. The pressure data falling in the pseudo-radial 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.

354

Oil Well Testing Handbook

Example 9-16 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 psi-i; 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 9-14 shows a log-log graph of the pressure data; from this graph we can see that neither a one-fourth-slope nor a half-slope is exhibited by the data. Figure 9-14 shows that the pressure data match the curve for ( k f b f ) D = 27r given in Figure 9-13 and the last 14 points fall on the semilog straight line. Match points from Figure 9-14 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. 9-30, 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

,

,

' '

i ,

, ,

~ ^_ I

I

,

/~

zTr---~

~

1007r ~ / ~'t" "

1

-~, ,~ i

~

~,r,,--,

~

~

'

/ \ i ~ ~ ~ - - ~

,'t

i

10-' 1

~ Ii 1

,

Matchpoin

ts (Z~)~t= 100psi

tOrw) -O.'9

1 i

10 -l

, ,

',

,

',

10 2

10 3

,

10

At (h) 10-2

I

10-3

10-2

I

10-1

I 1

I 101

I 102

tor'w

Figure 9-14. Type curve matching for Example 11-2.

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. 9-31 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 9-1, rw/X ' f - 0.403; hence, xf The skin factor is estimated by using Eq. 9-33"

rw)

sf-ln

7w w

(0.25~ -ln\36.9]--4.99

F r o m Eq. 9-34, 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 9-15 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. 5-2: 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

ft

.................

4 .......................

..................

4 .............

..................

i ....

/

I

!iili

i- . . . . . . . . . . . . . . . . . . . .

........

|---

i- . . . . . . . . . . . . . . . . . . . . . . .

1

Slope, m = 307 psi

.....

'i....................... ~ ....................... '~....................... i i

i i

200

...................... ~............... ~r.__.i. ......................... ~.................... t--~ ; 0o, I i ; I i e~ i n 9 i.; . . . . . . . . . . . . . . . . . . . . . . .' ..... _,_~.J_ . . . . . . . . . i.' ............................................. 1--i o'7 i i I

100-'

-'- :':~176

300

oi9 " ~

0

i ....

1 Ap= 47Ps:l ......................

; .l

, 10 -1

1

At ( h o u r s )

Figure 9-15.

Semilog plot.

i

r'

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103

356

Oil Well Testing Handbook

Table 9-2 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 half-length, 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. 5-3, the fracture skin factor is [(A-~-lhr ~ - l o g

sf--1.151

k dp#octr2

-k-

3.23

[

2.28x 10 - 6 = 1.151 3--0~ - 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 re-arranging Eq. 9-32: r !w -

rw e

--Sf

-

0.25 e -(-4"8) -p 30.37 ft

Finally, the fracture half-length 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 9-2. 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

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(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 finite-conductivity 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 Finite-Conductivity Vertical Fracutre," Soc. Pet. Eng. J. (Aug. 1981), 253-264. 2. Cinco, H., and Samaniego, F., "Effect of Wellbore Storage and Damage on the Transient Pressure Behavior for a Well with a Finite-Conductivity Vertical Fracture," Soc. Pet. Eng. J. (Aug. 1978), 253-264. 3. Ramey, H. J., Jr., and Gringarten, A. C., "Effect of High-Volume Vertical Fractures in Geothermal Steam Well Behavior," Proc. Second United Nations Symposium on the Use and Development of Geothermal Energy, San Francisco, May 20-29, 1975. 4. Clark, K. K., "Transient Pressure Testing of Fractured Water Injection Wells," J. Pet. Technol. (June 1968), 639-643; 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), 362-372. 6. Raghavan, R., and Hadinoto, N., "Analysis of Pressure Data for Fractured Wells: The Constant-Pressure Outer Boundary," Soc. Pet. Eng. J. (April 1978), 139-150; 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), 887-892; 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, 1014-1020.

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2. Raghavan, R., "Pressure Behavior of Wells Intercepting Fractures," Proc. Invitational Well-Testing Symposium, Berkeley, CA, Oct. 19-21, 1977. 3. Wattenbarger, R. A., and Ramey, H. J., Jr., "Well Test Interpretations of Vertically Fractured Gas Wells," J. Pet. Technol. (May 1969), 625-632; Trans. AIME, 246.