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Application of the PseudoIinear-FIow

Model to the Pressure-lWansient

Analysis of Fractured Wells

H. Cinco-Ley,

SPE, and

F. Samaniego-V., SP E, U . of Mexnm/P emex, a nd F. Rodriguez,

SPE,

U. of Mexrco/IMP

sP~ 3 S9

Summary. The themetical basis for the pseudolimar-flow mcdel is established. It is demonstrated by use of an analytical model that

the lin--flow graph (p vs. W) cm be extended to the analysis of pressure data of fractured wells intersected by an intermediate- or

high-conductivity fractme (CP >57r). It appears that the fracture-conductivity effect during the pseudolinem-flow pecicd can be handled

as pseudoskin pressure drop that adds to the pressure drop cawed by tluid-Ioss damage. The combination of the pseudolineac-flow anafy-

sis with other interpretation techniques is illustrated through examples of field cases.

Itrtroduetlon

During the last 2 decades, many analy$is methcnlsl-30 to iaterpret

pressure data to estimate both formation and hydraulic fracture pa-

mneters have been prop.med. In adtiltion to type-awe analysis,

three of the most widely used graphical metbcds of interpretation

are the-l inear-flow graph (p vs. J), the bilinear-flow graph (p

vs. t*), and the pseudoradial-flow gmph (p vs. log r). ft cannot

be overemphasized that each technique appIies to a specific flow

regime. Unfm-htmtely, the general pressure behavior of a fractured

well includes not only these flowperiods but also several intermedi-

ate transition flow periods, as shown in Fig. 1. The only current

graphical analysis methcd avaitable to interpret pressure data fall-

ing in the transition periods is the typwrve-matching technique.

The linear-flow mudel has km applied in the past to early

time/pressure data of iniiite-conducdvity fratires (for practical

purposes, a finite-conductivity fracture can be considered as an

Mtite-conductivity-fmcture case whmever the pressure dmp along

the fracture is negligible-i.e., C kr)a 31J2).This model assumes

i ~~

niform flux inthe formation an nezllmble stom~e ca mcitvwith-

-- . .

in the fmctare.

The application of the li&r-tlow graph was extended empiri-

caffv to cases where the fracture has low or intermediate values

of c_onductivity29This r&@res either knowledge of the formation

permeability frum prefracture testing or a ti-ial-and-errur prueedure.

f.fa&y and Bandyop2dbyay30 used a uniform-flux Z@ticti

mudel to show that the pressure behavior of a ftite:cunductivity

vefiical fracture at intermediate values of tine can be described

by the pressure behavior of the War-flow mcdel plus an extra pres-

sure drop that is a function of the fracture conductivity. They slso

a.wumed tit the fracture penetrates the formation completely in

the horizontal direction (X,lX$=1).

The purpose of this work is tu show through tie analysis of an

irnafydcd solution for tinite-conductivity fractures that the )ine2r-

flow mcdel maybe extended to cuses of h@ fracture conductivity

without assuming a uniform flux along the fracture. Furthermore,

analysis of field cases illustrates the application of the pseudolinear-

flow mcdel and its combmtiun with other interpretation techniques

tu obtaii reliible estimates for fructure and reservoir parametem

PseudrrIhrear.Flow Model

This study considers the model presented in previous

works 15J$2~; i.e., an inftite homogeneous reservoir is prcduced

tbruugh a well intersected by a symmetrical ftite-conductivity ver-

tical fracture of constant properties, as indicated in Fig. 2.

Several autbors16.17.21have demonstrated that me pressure be-

havior of a fractured wefl can b+ expressed in terms of three pa-

nwneters: ditnensionle3s pressure drop, *WD, dimemionless time,

.fD: ~d ~ensio~=s f iac m re ~nduc t i~v, . 2S ~ s~~

m Fig. 1. So

ApWD=f(@,,CD). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...(1)

Cq@ght 989Sccl@lyfPelroleurnng nws

438

For the pericd including the btiear flow, Transition Zupe 1, and

the hear flow, Q. 1can be expressed in the bpkce space as2731

d3(pwD)=m cOril{@s/(cfl)2] *

}Acvzsl(cp)zl x, (2)

where d3(pwD)is the Laplace transform of the dimensionless well-

bore pressure and s is the Laplace space parameter.

The hyperbolic cotangent fiction may be mitten as an inf ini te

serie3.32

cool x)

=(1/x) + (x/3) (x3/45) + .

+[2&B&-1/(2n) ] + . . . . . . . . . . . . . . . . . . . . . . . . . ...(3)

Hence,

s pwD)=T/sc@ti[s/ cp)21 x (1/ti[s/(c*)Zl %

+(~/3)[s/(cD)2]u 1/45{@/(C@)2] }3 + . .). (4)

At large values of time, s-O and the series in RI. 4 can be ap-

proximated by use of two terms:

dXpwD)=m/2sK+m/3c@ . . . . . . . . . . . . . . . . . . . . . . . ...(5)

faplace inversion of this equation yields

pwD=~~(~/3C~). . . . . . . . . . . . . . . . . . . . . . . ...(6)

The fist term on the right side cmresponds to the classic linear-

flow expression discussed extensively in the literature. The sec-

ond term is a result of the fnite conductivity of the ti-acture. Hau-

Iey and Bandyopadbyay30 presented a similar equation; the only

difference isthat their numerical constant in the secund term is unity

instead of the u/3 constant. It is obwousthat this equation also in-

cfude.sthe infite-conductivity case for which the second term on

the right side goes to zeru.

By multiplying Eq. 6 by CP, we obtain

Pw D Cj)=d fD )2 + d~j. . . . . . . . . . . . . . . . . . . . . . 7)

fbis equ ation ind ica tes tha t a graph ofp.D(Cp) VS. W

must

give

a sin de stra ieh t

l ine with an intemeot emml

to

7r13for

.

differ&t vzlue;of Cp~

Fig. 3 shows this type of graph for the pressure behavior of a

fmite 5X falls on the straight Iine defined by RI.

7. Hence, an interesting conclusion is evident the pressure behavior

of a well intersected by a fracture of conductivity > ST is identical

to the behavior of an infinite-conductivity fracture with a skin

that depends on the fracture conductivity. This condition is valid

fOr a @ti the ~ge, t.rjD(C@)2 >1.

Agarwal et af.21 found this type of behavior and provided a

graph for the relationship between fracture conductivity and the in-

tercept inthe linw-flow grapk however, no analytical prouf was

presented. In addition, Lee and Holditchzg emended the use of tie

linear-flow graph to Iinite-cunducdtity fracture.ca.sev theypresented

empirical correlations of both slope and intercept as a function of

fracture conductivity.

It has been demomtmted that the flux distribution afong the frac-

ture is uniform during the lin~-flow periud for an infmite-

SPE Formation Evahatic,n. Septem&T 1989

8/10/2019 15. SPE-13059-PA

2/7

,:W

~:

00

L,,e ] ,.

1

-. ..

TRANS -W

BUN;AR

FLOW

PEQ ODS

,rJ-z-

,.., ]~-z ,~o ,@

104

,.6 ~08

I,.(cr.],

Fig.

IPressure

behavior of a vertically fractured well.

Y

i

lg.

2-Finite.conductlvify vertical fracture [n an infinite

mewolr.

conductivity fracture. Tfds is not so for ftite-conducdvity frac-

tures, as shown in Figs. 4 and 5. These graphs present the flux

distibwion for CP equal to 5X and 10T and for values of time

when E+ 7 applws. The flux distribution tends to be mom uni-

form as flowing time increases. his information was obtained by

use of Cinco-Ley et als 16mcdeI. .4kbougb the pressure behavior

under these conditions is similar to the linear-flow behavior. it is

not the same

because of the dtierence in flux distributions, Con-

sequently we propose that the pressure behavior exhibited by fjdte.

conductivity fractures be called pseudolinear flow.

Fig. 6 shows a heavy line representing the pseudolinear-tlow be-

havior. The beginniig of thisflowperiod occurs at a value of

t.r/D(C@2 = Ii.e., one log cycle of time after the end of the

bdineir flow and two log cycles before the start of the apparent

one-half-slope straight line, The pseudolinm-flow wricd includes

part of the transition mne between the one-qmwter-slope straight

line and the on&Mf-sloIE straight linq therefore, a theoretical basis

exists (Eqs. 2 through 7) for the extension of the linear-flow graph

to cases of intennedtie and high fractore conductivities for pres-

sure data falfing within the transition zone, which can be analyzed

only by type-curve matching with currentfy avadable metbcds.

Pseudollnear-Flow Analysis

Acc.zr&g to Eq. 6, the wellbore pressure behavior under

pseudolinezr-flow conditions is given in terms of real variables by

Apor A+=mL&+bLf, . . . . . . .. . . . . . . . . . . . . . . . . . . ..(8)

where mLf is given by

mLf=(FLf&?B/hf) @/k4@ %

. . . . . . . . . . . . . . . . . . . . . . ..(9)

for oil wells and by

. .

mLf(FLfg9T/~f)(l/ kI@@ %

. . . . . . . . . . . . . . . . . . . . . ..(10)

SPE FormaCionEvaluation.Sc nernber

1989

15

100T

10

Q

o

z

o

0

5J=-

Ig.

3Linear-flow graph for a f[nite-conductivity fracture.

3

,

1

[

I

cm ,

5T

q f~

twD

.. _.~

: *

i. :

0

0 0.2

0.4 0.6 0.8 1.0

XD

?g.

4Flux distribution during the pseudoilnear flow;

:,D =5T.

for gas wells. On the other hand, b=f is defined by

b~f=ALfo(q@/h)(x]cf) . . . . . . . . . . . . . . . . . . . . . . . . . ..(11)

for oil wells and by

bLf=,4Lfg(qT/h)(xf/Cf) , . . . . . . . . . . . . . . . . . . . . . . . . . ..(12)

for gas wefls.

FLfO, FLti AL@

and ALfi

areunitconversion

factors.

Eq.

8

indicates that a graph of pressure data vs. the square root

of time during the pseudoliiea.r flow should exhibit a straight line

of slope mLf and intercept b=f, as shown in Fig. 7.

According to Eqs. 9 and 11 for oil wefls and Eqs. 10 aod 12 for

gas wells, the slope is inversely proportional to fmctuze area hrf

and the intercept is inversely proportional to C)/.xf; hence,

krf= (FLfo@/nI~f)(@@cJ *

. . . . . . . . . . . . . . . . . . . . . ..(13)

and Cf/xf=ALfOqBjJhbL f . . . . . . . . . . . . . . . . . . . . . . . . . . ..(14)

for oil wells, and

hxf= (FLfiqT/m~f)(l/k@cJ ~

. . . . . . . . . . . . . . . . . . . . . . .(15)

and Cflxf=AL&qT/hbL f . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(16)

for gas wells.

The pseudolinear flow starts at about txfD(Cfl)2 = li.e., irl

tWUJSof red variables.

~fipLf=(~@ c f/~)(~f/Cf)2 . . . . . . . . . . . . . . . . . . . . . (17)

439

8/10/2019 15. SPE-13059-PA

3/7

3

lfD

C,D= ]0 T

-.

1

I

o

0 0.2

0,4 0.6

0.8 1.C

XD

lg. 6-Flux distribution during the pseudolinear flow

:m - 10W.

~rtbpLf= 9/m) bL,ImLf) Z. . . . . . . . . . . . . . . . . . . . . . . . . ...(M)

l?qs. 17 and 18am both vafid for.oif and g= ms. M. 18 w=

obtained by considetig f3qs.9 through 12and can be used to check

whotber the prqer pseudolinearflow straight line has been chosen.

Extension and Limitations

Unfortunately, the pressure behavior given by Eq. 8 is exhibited

not only by a finite-conductivity fracturq an inftite-xmductivity

fracture with fluid-loss damage yields an identical behavior. The

Mtor has been considered to k the case when the linear-flow-graph

straight line does not intercept the origin (Ap]~.. > O). Therefore,

a uniqueness problem regarding the ~ of system does exist.

One way to dis

criminate between these cases is based on the ex-

istence of the bifinewflow perid, i.e., a finite-conductivity frac-

mm may exhibit bilinear flow and an infinite-canducfivity damaged

fracture does not. Another possibility is tie use of Eq. 18 to check

the start of the pseudolinear-flow peric@ a highly conductive

damaged fracture becomes e~ident in the event that a straight line

in the squar.acot+f-dme graph begins earlier than indicated by

Eq. 18.

The intorpre.tation of data is more complex when both ftite-

conductivity and fracture fluid-loss damage are taken into account.

It apprs, as demonstrated in the AppendK, that both effects are

c

bu a++

I

Fig. 7Linear-fiow graph for pseudolinear-flow pressure bs-

havior.

::

4

LOW

10+

10+

,.-2 ,.O

,)2

104

,,)6 ,~

t.,.(m),

Fig. 6-Pseudol Inear-flow pressure behavior.

siooless form

PwD=q+%+(7/3cp). . . . . . . . . . . . . . . . . . . . . ..(19)

E+ 8 is sdfJvafid for this case with prqermcdiIication of ffie

defi-

nition of the intercept bLfr as defined by

qBp

Xf

)

T

aOqBp b, k

bLf,=AL@+-

1 . . . . . . . . .. (20)

hCf2kkxfk,

for oil wells and

bLfi=ALfigy+

()

r%qTbs L_l . . . . . . . . . . . .

hCf2khxfks

(21)

for gas wells.

Again, to obtain reliable estimates for both conductivity and

fracture-damage effects, it is essential to have pressure data on the

bilinear-flow p-aid T%iscan be achieved by use of the techniques

dscussed by Wong et aL33 An interesdng point is that a fluid-loss

damaged fracture. of l%dte conductivity does not exhibit a w@l-

defmed bifimwflow stmight fine.

To perform a complete analysis of both flow-rate and pressure

data, the value of formation permeability must be avaifable from

prefmctwe-test interprotadon or from analysis of pressure data fM-

ing in or tier the Transition Flow Pericd 2 shown in Fig. 1.

EXMIIDI~S of Am lcatlon

.

The application of&e pseudofinew-flow mcdel is ilfmhatedthrough

tie anafysis of tests psrformed on massivo-hydmufic-fkwtumf

(MHF) gas wells. One of these cases was presented by Bosdc et

al. N and the other by Cmix-Ley and Sammdego-V .27

440

.wE

FormationEvdualion.Swemb=r ~~9

8/10/2019 15. SPE-13059-PA

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0

Fig. 9-Billnear-Now graph for Bostic et al.sw example. I

3

2

I

o

mLf= i . ot iXlo4pS i2 Cp-l MSCF~-l ~ fA

I

b,, =2X105p i2Cp- MSC~-

/,

mw

1

..

.- 0

. .

} b~f

50 100

150

200

fi,hours

Fig. 10Linear-flow graph for Bostic et al.sw example.

TABLE 1TEST INFORMATION AND RESULTS OF ANALYSIS FOR CASE 1U

Reservoir Data

p{, psia

5,100

:tim~:a

1,275

0.0027

h: ft

56

*OR

0.085

725

pd. Cp 0.0254

Cg,, psi1

1.465XI0-4

Analysis Results

Type Curve Siiinear Flow Pseudolinear flow

C,lxt, md-fflft 0.103

0.0965

x,~k, ft-md * 131.2 140.5

C,W rndx-n 13.5

13.6

13.0s

>107T

k, md 50.00328

C,, md-ff 2236

xl. ff

>2,291

TASLE 2TEST INFORMATION AND RESULTS OF ANALYSIS FOR

CASE 2=

Res6tvoir Data

q, Mscf/D

7,350

t hours 2,640

h, ft

118

fp

0.1

0.023

~ OR 690

P g. OP

0.0252

ct . psi-f 0.129 x10-3

PM, pda

1,320

Analysis Results Type Curve Bilinear Flow Pseudolinear Flow

C,, md-ft 14/3 154 137

x,wk ft .md k

58.9

54.6

C,/x,, md-tift 0.3978 0.4;85 0.427

cm

15.9

16.74 17

Case 1. This correspmds to the fmld case presented by Bosfic ef

=[,34 BofJ ~m~~um.btitip test dam and rate were pr~s~ to

e3timate the iotl~ce function for the presznre behzvior of the fmc-

tnmd well; i.e., they applied the supfxposition principle to the in-

formation tn obtain the pseudopressure behavior corresponding tn

the unit of flow 22tc. Thi2 technique allows the analysis of m,ex-

fended pecicd of dine. Flow-rnta information for

a @od

of 3 years

waa reported in nddition tn 134 hours of pressure-buildup meas-

urements.

Fig. 8 shows the type-mme matching for this case. Although

an exceUentmatch is obfaincd, the duration of the flow-rate @ml

w23 not long enough to zllow the estimmion of the dimensinnfess

fracture conductivity, C@; however, a minimum value for this pa-

mete i Cp 2107.

The match points in Fig.8 zre f)M= 1 honr [t. D Cf)2]M=

I.2 x 10-3, ~Fa)M=

106

(psi2/cp)/@f3cf/D) [I .6(x 109NaZ/,

Pa.s)/(std m3/d)], and f.pWD.C@)M=5.6.

SP E Formation Evaluation. 3eutember 1989

441

8/10/2019 15. SPE-13059-PA

5/7

)02

m

Lil\lIIIl l

107 t .0_lo5 1_10 :2___l _ LO-l_ _I__l_ _ _ L_ pl _.

,.-2

,.-1

I

t.tD(cfD)2 ]01

10 2

At [ho. rsl

Fig. 1 I-lype-cttrve matching for Clttce-Ley and Samwtlego-V.sm example.

By applying Eqs.

38

through 40 in Ref. 27 and considering the

information of Table 1, we can estimate

Cf/xf =

@hDcjD)Mlh(PFdM

=(1

,424x725x5.6){(56X106)

=0. 103md-ftht[0.103

md .tim],

fi=(cf lxf){fi(t)M1+Pci[tx

fD(cD)21M} h

(

2.637x 10-4 X1

)

,%

= 0.103

0.35 X0.0264 X1.465 XIO-4X1.2X10-3

=131. 2 f t-mdf i [39. 99 m.md%],

and C i=xfl(Cf/Xf) =131.2 xO. 103

=13.5 md% [4.1 md~. m].

From these results and from the minimum value of CfD, we obtain

kS(Cf/xf)(lllOr)

s 0.tM328 md,

cf2 c*/d t lh z i

>236 md-ft ~1.9 md.m],

and

Xf>Xfl/_

22,291 ft [698 m].

Fig. 9 shows the bti-t low

graph for this example. The hiieac-

tlow straight line includes ~ints up to ==40 hours; it Irasa slope

m =8.3x104 psi2/cp-Mscf-D-hr A [139.3 X1012kpazfpa. s-std

Y

m .d.h] and goes through the origin. Application of the bilinear-

flow-anatysis equation yields

Cfl=[Fb~qT/mbf (+ c,) kh]z

[

444.75 X1 x725

2

8.3 X 104 x(O.035X0.0264 X 1.465 X 10-4)* x56

= 13.Og md~-fl [3.99 md~.m].

Fig. 10 shows the Iinear-flow graph as applied to this case. The

straigh-line portion has a slope mzf= 1.025x 104 psi2/cp-Mscf-

D-hr% [17.2X1012 kFa2/Fa.s.std m3 .d.h*J and an intercept

fJLf=2X105 psi2/cpMwf-D [335.76x 1012k%2iPa.s.2td m3d].

The pseudofintwwflow analysis yields

x+

=FLtiqT/mLf h(@cJ %

40.925x1x725

1.025 X 104 x56 x(0,035 XO.0264X l.465x 10-4)*

=140.5 md%t [42.8 md%m].

AA7

On

the other hand,

Cf/xJ=ALkqT/hbLf

=(1,491.2x

1x725)/(56x2x 105)

=9.65 x10-2 md-ftfft [9.65 x10-2 md,-timl.

Eq. 18 can be used to check whether the proper straight line has

been drawn. The stwt of the pseudolinear-tlow period occurs at

ftiLf=(9/T)KZ x I@)/( 1.025 x 104)12= I SOPOhOurs,

which is, for practical purposes, the case in Fig. IO.

Information of hDtbthe bilinear- and linear-flow graphs can be

combined to estimate Cflxf as follows:

Cf/xf=(~bfi/FLjg) [qTmw/fi(mbf)2]

(444.75)2X1 x725x 1.025x 104

.

40.925 x56x(8.3 X104)2

=0.0931 md-fWI [0.0931 md+mlm].

Table 1pre+ents the results obtied by use of different techniques.

The agreement of values for parameters is excellent and the maxi-

mum difference is about 10%.

Case 2. This example represwm Wefl A in Ref. 27. A pressure-

buifdup test was run inan MI-IFgas well for 120houm after a flow-

~g @d of 2,1W ho~. The information for this testis present-

ed in Table 2.

Fig. 11 shows the application of the type-curve-matching tech-

nique. As can be seem,both bilinear and pseudoliiear-flow peri-

cds are present in the test. 71e w curve md bilkwflow graphs

are shown here; the analysis and a canplek discussion are presented

in Ref. 27.

Fig. 12 indicates a welldefied bibw-flow straight line that

goes through the origin of the graph. Fig. 13 presents the linear-

flow graph and shows that pressure data exhibit a straight-line por-

tion where slope mLf=5.65x107. psi2/cp-ti% [26.86X 109

kpa2~a.S.h 4J md intercept

bLf=

15x 107 psizkp 171.3x 109

kpa2/Fa.s]. Application of the pseudolinear-flow @YS~ Yield3

40.925x7,350x690

x+=

5.65x107X 118x(O.1 xO.02.32X 1.29 X10-4)X

=54.6 md%-ft [16.6 mdfi .m]

and Cf/xf=(l ,491.2X 7,350X6s0)/(118

X

15x 107)

=0.427 md-tllfl [0.427 md m/m].

SH5

Formation Evd..ticm, SeWernber1909

8/10/2019 15. SPE-13059-PA

6/7

,0

t

lg. 12BObtear-llow graph for Cbtco-Ley and Samaniegl

1.s==7example.

Combining these resufts gives

C = 21.6 md~-fi [6.6 mdfi.m].

Table 2 summarizes tbe resuffs obtained by different techniques.

The agreement ap~s to be excelfent and the differences are

< 10% bt alf cases.

Concius lons

1. The tbearetiwd basis for the use of the linear-flow graph to

analyze pressure data of finite-conductivity fractures has been es-

tablished.

2. The effect of fmckre conductivity on the Iin--flow graph

a- to be Similar to the effect of a skin damage aromd the

fiacmre.

3. Tim effect of interrrdiate and high fracture conductivities is

additive to the effect of a skin-fracture damage during the

pseudofinear-flow pericd. Both effects can be evaluated if pressure

data in the bti~-flow period am available.

4. The pseudolinear-flow analysis provides estimates for C ,

C/Xf, and Z+ and can be used to compare resdts from other

methods of interpretation.

5. Neither the bilinear- nor tbe pseudolinear-flow graph can be

u3d @esbtM@ fo~tim p$-~ty. Hence, ffds-~~ mu~

be obtained from prefracture tests or ftom long-time pressure data

falling in or beyond the transition between the pseudolinear-flow

perkd and the Pseudoradial-flow pericd.

Nomenclature

A = unit conversion factor

ALh = unit conversion factor for linear flow of gas,

1,491.2 [418.361

.4Lfi = unit c0nver3 i0n factor for linear flow of oil,

147.86 [1928.9]

b = intercept

bf =

fracture width

bpL = intercept on Cartesian pseudolinear graph of

fransient-test pre.wtre data.

B = FVF, RB/STB [res m3/stock-tank m3]

B& =

Bemoufli numbers

c = 3YS@ItI total compressibility, psi-1 ptPa-l]

Cf = fracture conductivity

C@ = dimensionless fracture conductivity

f= function uti in Fq 1

F =

unit conversion factor

Ftig = unit conversion ftior for bilii flow of gas,

444.75 .[24.57]

WE Fwmmkm Evduatkm, .%ptunber 1989

.,~~

; 40

_

mL,=563xf07psi2c7F

L

2468 10

m , how,

Fig. 13Linear-flow graph for Cinco-Ley and maniegc

V.SZ7

example.

FM. = unit conversion factor for btiear flow of oil,

,.

44.1 [34.971

FLfi = unit conversion factor for linear ffow of gas,

40.925 [11.498]

FLfO = unit conversion

factor for Iinear flow of oil, 4.064

[53.01]

h = formation thickness. ft M

k =

permeabfity, md -

p.D) =

Laplace transform of dimensionless weflbore

pressure

m = slope

IIZPL= S1OF

of stight liie for pseudoliiez flow,

psizkp pcPa2/Pa.s]

p =

pressure, psi m]

Ap = pressure change, psi P@

APWD = dimensionless pressure drop

~ = ~eff floWrate, STI+V [stock-tank m31dJ or

MscflD [Std m3/d]

~ = sti fa~or or fapkice space variable

t = time, hours

tX,D = dimensionless time

T = reservoir temperature, R KI

x = variable used in Eq. 3

XD = dimensionless variable

j+ = ~xte~ mdls

f = fia*@ fdf-length in the z dii~fion

a = pressure (or pseudopressure) unit convemion

f3.ctOr

% = Pr~~ (OIpseudopressure) unit conversion

factor, 1,424 [399.5]

~. = p~s~~ (or psedopressme) nit mnvetsion

factor, 141.2 [1S42]

i3 = time unit conversion factor, 2.637 x10-4

g = fluid viscosity, cp pa.s]

.$ = p6r0sity, fraction

+ = real gas potential, psizlcp Ma2/Pa.s]

A* = real gas potential change, Psizkp &Paz/Pa.s]

Subscripts

b = bilinear

bf = biliiear flow

D = dmensiordess

e = external

f = fracture, flowing

g=ga3

h = hydrocarbon

443

8/10/2019 15. SPE-13059-PA

7/7

~ = ~ti~

L= linear

Lf = Jinear flow

M = match

~ =o i J

p = production, prcducing

PL = pseudoJinear

s = dmage

w . wellxxe.

Snpmaipt

=average

Referwrcss

1. S@t, J,O.: The Effect of Vertical Frmiwres cmTransient Pre%ure

B&vim of Wells, ,s JPT@cc. 1%3) 1365-69; Trans., AR@ 22S.

2. Russell, D.G. and Tmitt, N.E.: Transient Pressure Behavior inVer-

ticallyFraduredP.&xvoim,

JFT(&I.

1954)1159-70; Trans.,AIME,

231.

3. Lee,W.J. Jr.: AmJysisof HydmuJicaflyFracturedWellswithFrw-

sure BuildupTess,>, pqer SPE 1820presented atthe 1967SPE An-

nual Meeting, Housmn, Oct. 1-4.

4. Clark, K.K.: eTransient Pressure Testing of Fmtwed Water Jnjt -

tion Wetls,v

JPT

(June 1968) 639-43; Trans., AJME. 243.

5. Milfheim. K.K. and Cichwuicz, L.: iTesdng and Analyzing Low-

Permeabilitv Fractured GasWells,

JPTITeb. 196S)193-9ti Trans..

AfME, 243-.

6. Watfwdmrger,R.A. a ndRamey, H.J. Jr.: Welf Test Jmerpretation

of Vertically Fractured Gas WeIfs,, .JPT (T&Y1969)625-3Z

Tram.,

AJME, 246.

7. van EverdinSen, A.F. and Meyer, L.J.: %dysis ofBuildupCurves

Obtained After Well Treabnent, JPT(April 1971)513-24 Trans.,

AIME. 251.

8. Raghavan, R., Cady, G.C., and Ramey, H-J. Jr.: kWeUTest Analy-

sis forVeniwJJy Fractured WeJls, JPT(Aw. 1972)1014-21JTrans.,

AJME, 253.

9. Gringanen,

A.C ., RameY,H . J . J r .,

amAffaghavM, R.: Unsteady-

State Pressure Distributions Created by a Well With a Single Jrdinite-

Conducdvily VerdcaI Fracture,

SPEJ

(Aug. 1974) 347-Q Trans.,

AJME, 257.

10. Gringarten, A.C,, Sanmy, H.J. S,., amdR@avan, R., ,A@ied Pres-

wreAnalysis for Frac.turedW.US,,,JPT(IUIY 1975) 887-92; Trm.,

AJME, 259.

11. P.amey, H.J. Jr. and Gringarten, A.C.: &EHect of 3tigb-VolumeVer-

tical Fractures on Geotiermd Steam WelJlkbavior, Proc., Second

United Nations Synqmium on the Use and Development of GeoOIer-

mal E.ergy, 8an Framisco(May 20-29, 1975).

12. Ragbavan, R.: W.me PI@caJ Considerationsin tbeAmlY?.isofPres-

sure Da a,s, JPT(Oct.

1 ?76)1256-6% Trans., AJME,2J3L

13, Holdkh,S, A.andMome,R. A.: TbeEffectso fNon-DarcyFlow

onthe Bebavior of HydraticaJJy Fractured WeUs,''JPT(Oct. 1976)

1169-78.

14. Ramey, H.J. Jr. et al.: pressure Transient Tesfing of Hydraulically

Frachued Wells, paper presented at the 1977American Nuclear .%x.

Meedg, Golde, CO, ApriJ 12-14.

15. CincQLey, H. andSamaniegc.V., F.: CWfeciof WeJlt.xe Storageand

Damage on tbe Transient Pressure Behavior of Vertically %mtured

Welk,,, paper SPE 6752 presented at the 1977SPE AnnwdTechnical

Conference andExbibition,Denver n.+ 0-f?

16. Cin.o-laY, H., Samaniego-V., F.,

., - . . .

and Dominguez. N.: Transient

pressure Behavior for a WeJJrnti a Fiite-Conducdvity VerdcaJFmc-

ture,,SPEf(Aug. 197S)253-54.

17. Barker. B.md Ramey, H.J. Jr.: TmnsientFiowmFiniteCondm-

fivity Vertical Fractures, paper SPE 7489 presented at the 1978SPE

&nti Tmhtiti Conferememd~tition, Houston, Oct. 1-3.

1S. Scott, J.O.: LANew Methcd for Determining Flow CJIaractmisticsof

Fractured Wells in Tight Formations: pap?x75-T-2 presented at fbe

lW8AGATmstission Cotieren=, MOrdreal,T-179-186,

19. Ragbavan, R., Uraiet, A., and Thomas. G.W.: Vertical FraUure

Height Effe.2onTransiemtFJowSehavicn: SPEJ(Aug. 1978)263-77.

20. R& bavan,R. andHadinoto,.N.: AIwJysis of Pressu= Data for Frac-

mred Wellx Tht Constant-Pressure OuteJ Boundary,,y

SPEJ

(ApriJ

197s) 139-5@ J-m., AlME, 265.

21. Aganwl, R.G., Carter, R.D., andPoJJcck,C.B.: rEvahmtionandPer-

formancePm&don of fmw-PermeabililvGas Wells SdmuJ@d b Mas-

sive Hydraulic

Fmcruring,,7

JPT WI.& 1979)362-T2 Tram., iDvfE,

267.

22. Bandyopadhyay, P. and H8nleY,E.J.: An Jmpmved Fmsure Tmn-

sient Metbcd for Evaluating Hydraulic Fmcmre Effectiveness inLaw

444

Perm&2bili ty Reservoirs, . paper presented at tie 1979 Congxso ,

Pammericam & Jngeniwfa de Pekfk,

Mexica City,

March

19-23.

23. Namdnba m,T.N. and Pabm, W.A.: A Purely NumericaJApproach

for .4naJyzingFluid Flow to a WeU Jntmcepdnga VexticalFrwture,

paw SPE 7983pzesentedat tbe 1979SFE California RegionaJMeet-

ing, Vmtura, April 18-20.

24. Guppy, K.H. et af.: NoII-Darcy Flow in We 3s WMI Finite-

Cond.cdvity VerdcaJ Frmturm, SP EJ ( Oc t . 19S2) 681-98.

25. Economies, M.J. eral.: Pressure Buifdup Amdysis of Gmthemxd

Seam WeUsWith a PamlleJepi@ McdeLJPT(April 1982)925-29.

26.

Bennen, C.O. et al.: %flne.ce of Fracture Heterogeneity and W%

La@ on fbe Respemc of Verdcally Fmtured WeJJs, SPEI (APIO

1983) 219-30.

27. Cinmley, H. and Samaniegc-V., F.: Transient pressure Analysis

for Fractured WeUs;

JPT

(Sept. 1981) 1749-66.

2S. Cincc-L+J, H. and Sameniego-V., F.: Tmnsi.@ pressure Analysis:

Finite Cmducdvity FracimmCase vs. DamagedFracture Case, pw

SPE 10179 presented at tbe 19S1SPE Annual TednicaJ Conference

and Exhibition, &n Antonio, Oct. 5-7.

29. Lee, W.J. andHolditch,S.A.: Frdchuc Evah2iti0nwithFIwsure Tran-

sient Testing in J.ow-PermeabiJify* Reservoir s,,,

JPT

(Sept. 19S1)

1776-92.

30. Hsnley, E.J. and Bandyopadbyay, P.: Pressure Transient Behavior

of theUtionn Fbu F& capacity Fmcmre,-, _ SPE 8278present-

ed at the 1979SPE Am@ Technical Conference and Exhibition, Las

Vegas, Sqt. 23-26.

31. L%, S. and Bmckenbrougb, J. : A New Soludm for Finite Conduc-

tivity VetdcaJFraclures with Red Time and Laplace Space parameter

Esdmadon,

SPEFE

(Feb. 19S6) 75-8S.

32. AbmmowiU, M. and Stegun, I.A.: Hand600k

of ,?%fhematicd Func-

tions, Nait.

Bureau of Standards, Washington, DC (1964).

33. WOW, D.W., Barrington, A.G., andCimday, H.: Applimdon of

theFressueDetivative Functionin tbe Fmssnre-Tramient Testing of

FracturedWdJS,S,SPEFE (Oct. 1986) 470-S0.

34. Bosdc, J.N., Agarw8J,R.G,, and Carter, R.D.: Combined Analysis

of Postfmct@ng Performance andpressure Buildup Data for Evacuat-

ing an MHF Gas WeU,>JPT (Ott. 1980) 1711-19.

App.sndIx-Pseudollffear.Flow Behavior

Conslderhrg Fracture Dsmage

Consider the fmnducdvity ffacture described earlier to & sur-

rounded by a zone of width

b,

sn d permca bt it y k, les s t bsn t h e

formation permeability, k. l%is intmxfucss an extra pressure drop

on tie fluids ffowing from the i imnadan to the tiactum, as describd

by Cmcc-f-ey &d Samaniego-V.

15

For tbw t30w

conditions, F.q,.

2 can be written

fK.pwD =

T cot h{(I /cfD )[2/( lA@ +C 2h r)(s fi)ll ~ , . (A-l)

s cp(l/cfi )[2/(uw ) + (z hr )(s fi)] *

where Sfi is the fracture skin ,factor, deIined as

Sfi=(Z/2)(f@[(HkS)-11. . . . . ... . . . . . . . . . . . . ... . . (A-2)

Theseriesexpansion for the hyperbolic cotangent ~ven by Eq. 3

cm be used as in E.q. 4, therefore, for Jarge vafues of time, the

folJowirig approximation is obtained

d3(pWD)=(r/2sX) +(s@)+(tr/3C@). . . . (A-3)

Lsplace inversion of this equation yields J?q. 19.

.S1

Metric Conversion Factors

bbl

X

1.5S9 873 E01 =m3

Cp x 1.0*

E-03 = Pa.s

ft x 3.04S* E01 = m

t13 X 2.S31 6S5

J3-02 = m3

rod X 9.S69233

E04 = pmz

psi x 6.S94 757 E+(Y3 == J&a

psi-l X 1.450377

E-01 = kPa-l

psiz x 4.7538

E+OI = kPa2

R

X

519

E+WI =

K

.Cunvwabnwor 1sexact. SPEFE

Ofigl.ti2FEnmnutiptr=61vdIorrewuviept.16,1934.Paperacceptedor

p.bllca

tlonW 12,19$3.Revlsdnmnus+rk.tmmi?64NcM.11, t9e8.

P aw PE 12059)

flint

rmsentedatme raw

SPE Annum Tech.k l Oo.fermce

and

Exhlbluon held In Hou8M.,

6WL W-19.

SPE

Formation Ev?.luatkn, Sqtembex 19S9