Proceedings Eighth Uorkshop Gwthernal Reservoir Engineering Stanford Univermity. Stanford, Calffornir. December 19.982 SGP-TU-60 w

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WELL TEST ANALYSIS FOR NATURALLY FRACrmRED RESERVOIRS

Giwanni Da Prat

INTEVEP, S .A. Apartado 76343

CARACAS VENEZUELA 1070

ABSTRACT pressure transient solutions for constant rate production and transient rate analysis for con stant pressure production are presented for a naturally fractured reservoir. The results 02 tained for a finite no-flow outer boundary -- are surprising. Initially, the flow rate shows a rapid decline, then it becomes nearly constant for a certain period, and finally it falls to zero. Ignoring the presence of a c o ~ stant flowrate period in a type-curve match can lead to erroneous estimates of the dimension1 ess matrix pressure and fracture pressure dis- tributions are presented for both the constant rate and constant pressure production cases.1: terference test for-constant rate productioncan be interpreted for long times by means of the line-source solution. For the constant pres sure production case, the pressure awayfrmthe wellbore does not correlate well with the line source solution.

INTRODUCTION

Naturally fractured reservoirs or reservoirs with double porosity behavior as they are corn_ monly referred consist of heterogeneous porous media where the opening (fissures andfractures) vary considerably in size. Fractures and open ings of large size form vugs and interconnected channels, whereas the fine cracks form blocksy5 tems which are the main body of the reservoir. The porous blocks store most of the fluid the reservoir and are often of low permeability, whereas the fractures have a low storagecapacity and high permeability. Most of the fluid flow will occur through the fissures with the blocks acting as fluid sources. These systems have been studied extensively in the petroleum literature. One of the firstsuch studies was published by Pirson(1.953).In1959 Pollard presented one of the first pressuretran sient models available for interpretationofwell test data from two-porosity systems. The mostcorn_ plete analysis of transient flow in two-porosity systems was presented by Barenblatt and Zheltap (1960) - The Warren mad Root (1963) study is widely considered to be the forerunner of modern interpretation of two-porosity systems. The be havior of fractured systems has long been a toe ic of contrwersy. Warren and Root and mzemi (1969) have indicated that the graphical techni que proposed by ~ollard is susceptible to error

in

caused by approximations in the mathematical model-nevertheless, the pollard method is still used. The most complete study of two-porosity systems appears to be the Mavor and Cinco- Ley Study in 1979. This study considers wellbore storage and skin effect, and also considers prz duction, both at constant rate and at constant pressure. However, little information is pre- sented concerning the effect of the size of the system on pressure build-up behavior. In this paper a literature review is presented bn the basic solutions which can be applied in dealing with pressure transient analysis natural ly fractured reservoirs.

PARTIAL DIFFERENTIAL EQUATIONS

The basic partial differential equations for flu id flow in a two-porosity system were presented by Warren and Root in 1963. The model has been extended by Mavor and Cinco Ley (1979) toinclude wellbore storage and skin effect. Da Prat (1981) extended the model and developed a method todg termine the permeability thicknes product.Kh. Deruyck et all(1982) applied with success the karren and Root model to study interference data from a geothermal field. The basic partial differential equations are:Da Prat (19_31).

and

(1 - w ,ap, = UPDf- P*) a tD

and t aie defined in the nomenclature. 'f D D

where w is the dimensionaless fracture storage parameter :

w (3) (W e c c q +

and con&ols the interporosity flow and is given by

2 k m r bu- w (4)

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A complete mathematical definition requires a9 ditional equations which represent the appro - piate initial and boundary conditions.

The initial boundary conditions is: -

For a well producing at a constant pressurethe inner boundary condition is:

(6 I

Where S is the skin factor For a well producing at a constant flow rate the condition is: .

u where :

The skin effect con ition s:

'fVD 'fD - [2 1 ' I rD=l Two outer boundary conditions are considerated in this study: an infinitely large reservoir and a closed outer boundary. For an infinitely large reservoir, we have:

lim pfD (rD, t,) = 0 (10) r + W For the closed outer boundary, the conditionis D

(11) = o - apD arD I rD ZreD

The dimensionless flow-rate into the wellbore

141.2 q y B ,. kfh (pi - pwf) (13) METHOD OF SOLUTION

A common method for solving Eqs. 1 and 2 is to use the Laplace transformation. The equations are transformed into a system of ordinary di - fferential equations which can be solved analy tically. The resulting solution in the trans- formed space is a function of the Laplace va - riable s, and the space variable,rD. To obtain the solution in real time and space the inver- se Laplace tranform is used. In the present work, the inverse was found by using an algo - rithm for approximate numerical inversion of the Laplace space solution; This algorithm was presented by Stehfest (1970).

TRANSIENT PRESSURE SOLUTIONS-CONSTANT FLOWRATE PRODUCTION

The solutions for the dimensionless wellbore pressure from either an infinite or a closed

outer boundary system have appeared in several places in the literature (see Mavor and Cinco- Ley (1979) and Da Prat (1981). Fig. 1 showsthe solution for p for an infinite system for two values of &%nd several values of 1 (Ci,= o and skineffect = 0 ) . At early times, pfvD de- pends on $ and W. For a given value of l , as time increases a period is reached whereinthe pressure tries to stabilize due to flow from the matrix. After this transition period, the solution becomes the same as that for a homog= neous system. Fig. 2 shows the solution fprthe dimensionless wellbore pressure for a well lo- cated in a closed outer boundary system. early times the (assuming skin effect = o and value of A.Then sition period to finally meet the solution for a homogeneous system. Fig. 3 shows the wellbare storage ehect on the Horner plot for a rosity system where (IF 0.01 and 1 = 5.10 . It is seen that low values of CD have a large im- pact on the initial straight line making impo- ssible the evazuation of 0. For large valuesof C , not only the initial straight line,butalso &e transition zone is obscured by wellbore - storage.

The analysis of interference tests in two pore sity - systems has been the subject of study for many years. These tests can be used to pro vide information such as-mobility thicknesspro duct, k h , and the porosity-compressibility - thickness product, @Cth. Kazemi (1969), based on the two-porosity model of Warren and Root, presented results for the fracture-pressuredis- tribution in the reservoir. In Kazemi's work, the wellbore response to an interference test is dependent on the pressure variations in the fractures, rather than in the matrix. Hence, - this study is limited to the case where the wells used in the tests are completed in the fractures. Streltswa-Adams (1976) considered both fracture and matrix pressure distribution and pointed out the importance of differencing matrix flow and fracture flow in the analysis of test m d e on fractured formations.Recently Deruyck et a1 (1982) presented a systematic approach for analyzing interference tests in reservoir with double porosity behavior and field tests conducted in a geothermal reser - voir are discussed to illustrate the method. Fig. 4 shows the solution for the dimensionless fracture pressure us t& for several values of W and 8. The parame er e is equal to 1 r2D. It was found to be a correlating group. Fig. 5 shows the solution for the dimensionless matrix pressure for several values of the parameters 8(= 1%~) and 0.

TRANSIENT RATE SOLUTIONS-CONSTANT PRESSURE PRO- DUCTION Although decline curve analysis is widely used, methods specific to naturally fractured reser- voirs do not appear to be available. Fig. 6 shows the dimensionless flowrate functions for a well produced at a constant pressure from a two-porosity system where (I)c 0.01 and A= 10-5.

At

FI

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:iJ

c

c i

Fetkovich (1980) observed that for hanogeneous systems a t the onset of depletion (a type of pseudosteady state) a l l solutions*for various values of t develop exponential rate decline, ana convergg t o a single curve. This statement is not true for two-porosity systems as ashown in Fig. 6. It can be seen that the solutionsdo not converge t o a single line. Log-log type - curve matching t o analize rate-time data canbe applied t o naturally fractured systems. Howe - ver, the relationship between and t is controlled by o and A, as w e l l % by othep pa- rameters. Thus more than one type-- may be necessary. Figs. 7 to 20 show the solutionsfor different values of o and A.

In the case of hanogeneous systems, interfere; ce t e s t s have been used with success in many cases, Earlougher (1977) ., For wells producing a t constant inner pressure, Ehlig-Econanides (1979) observed that, unlike the constant ra te solution, the pressure distribution for cons - tant inner pressure does not correlate withthe line-source solution. A different solution re- su l t s for each value of rD. Considering the ho mogeneous reservoir solutxon, a particular ca- se of the naturally fractured reservoir solu - tion, it can be expected that the same depen - dence on r Fig. 2 1 shows p r = 1000, and %eralDvafues of w and 1. Fig. 29 shows the Solution for the dimensionless ma_ t r i x pressure p vs tp 2, for a well produ- cea a t a consta- inner Dpressure.

applies for two-porosity systems. vs t /r 2 for the case of D

INTERFERENCE EXAMPLE

The Jse of type-curve matchhg w i l l be i l l u s - trated w i t h a simulated injection tes t . ~ u r i n g an interference test, water was injected into a w e l l for 400 hours. The pressure response in a w e l l 250 f t away was observed during the in- jection process. Reservoir properties and the observed pressure data are given in table 1. - The pressure change was graphed as a function of time on tracing paper and then placed over Fig. 4 (see Fig. 23). FrCnn the match, 0 and 8 can be obtained as parameters. Frcm Fig. 23, - e= 1 and (P 0.01- The fracture permeability is given by:

141.2 qB p (3) E141.2(-100) ( 1 ) (1) 0.4 kf = h Ar? 480 (-10)

= 1.2 ma Fran the time - tDhg match, the to t a l stor= t i v i ty is obtained:

= 0.000264 (1.2) 100

(250) 1.5 (- 1

= 3.38 psi-

L,

FrauWn 1 and the total storativity, the frac- ture storativity is:

-6 = 10 e A- K

also fran the relation o = a 2 r2 = kf .

e kf C ( k m m - =

r2

Thus, i f Inn can be obtained from core analysis, then the shape fac to ra i s equal to:

The shape factor a can provide information about the effective block size in the system.

- DISCUSSION AND CCNCLUSIONS

This work presents basic solutions that can be used t o analyse pressure or flawrate transient data fran a naturally fractured reservoir. The model used assumes pseudo steady s ta te flow - fran matrix t o fractures. According t o the li- terature the solutions assuming transient flow have been presented by many authors: Raghavan and Ohaeri(1981) presented declining rate type curves for a constant producing pressure De - ruyck e t a1 (1982) presented solutions for the pressure distribution throughout a reservoir with double porosity behavior consideringboths pseudo-steady state flow and transient interpg rosity flow. The ahthors concluded based on f ie ld examples that the pseudo-steady stateand the transient interporosity flow models are shown t o yield consistent interpretations.

Fran the solution presented in this work t h e f z llowing conclusions can be derived:

1) The i n i t i a l decline i n flawrate is often not representative of the f inal state depletion.

2) The fracture permeability kf; t o t a l stora t ivi ty ,

{ (I@) + (4 c) 1; and the shape factor aan can be obtained fran type curve m a t - ching . Both dimensionless matrix pressure and f r a c ture pressure are necessary for proper and lys i s of interference tes ts .

of

m f

3)

4) For constant-rate production interference

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5)

0= Ax 2, correlating group, dimensialess. D AQWOWLEDMENT

My sincere thanks t o Henry Ramey Jr. of S t a n - ford university for h i s guidance during t h i s study and INTEVEP, S.A. for permission t o pu- blish this paper.

tests can te -1ised:at long-tiaei w i i g the line-source a o l u t i a ~ .

~n analyzing interference t e s t s for cons - tant-inner-pressure production, a different solution for the pressure distribution re - su l t s for each value of radial distance,rD. The pressure function does not correlate with the line-source solution. REFERENCES

NOMENCLATURE

A= mainage area, f t 2 B= formation volume factor, RB/STB C= canpressibility, psi" h= formation thickness, f t k- permeability, md pws= wellbore pressure during the build-up pe-

p= average reservoir pressure, psi pD= dimensionless wellbore pressure

riod, psi -

k- h S

(Pi - Pw' 141.2 qm

= dimensionless fracture pressure 'fD pa- dimensionless matrix pressure q= volumetric rate, B/D 9 ~ ' ; dimensionless flawrate

r= r n r =

r 5 e eD"

r

S= t=

tD=

W

D

radius, f t wellbore radius, ft dimensionless radius , rfiw

Barenblatt, G.I. and Ple l tw, IU. P.: (1960) - "on the Basic low -ations of Homogeneous Li - quids i n Fissured &ksn (in Russian), =. Akad. Nauk SSSR.

Da P r a t , G.: (1981) "Well T e s t Analysis for N& tural ly Fractured Fksser~oirs~, Ph.D Disserts - tion, Stanford University.

Deruyck, B., Bourdet, D., ~a mat, G., m y , - H. (1982) "Interpretation of interference tese in reservoirs w i t h double porosity Bhavior - Theory and f ie ld examples", paper SpE 11025prg sented at the 1982 annual f a l l meeting, SPE of Aime, New Orleans, LA, Sept. 26-29.

Earlougher, R.C. , Jr. (1977) "Advances i n Well T e s t Analysis, xonograph Series, SPE of AIME, Dallas, 5. FetkOVich, M.J.: (1980) "Decline Curve Analy - sis Using Type-Curves", J. Pet. Tech. 1065.

Kazemi, H.: (1969) APressure Transient m l y - sis of Naturally Fractured Reservoirs", T-., AIME. 256, 451-461. MVW, M.J., and ChCO-Ley, E.: (1979) "Tran - sient Pressure Behavior of Naturally Fractured Reservoirsn, Paper SPE 7977, presented a t the 1979 California Regional Meeting, SPE of AIME, Ventura, California, Apr. 18-20.

reservoir outer boundary radius, f t

dimensionless outer boundary radius , rehw skin effect time, hours dimensionless t i m e

tn =$, dimensionless tAD A

&= shubin time, h r

Subscripts

f a fracture m= matrix De dimensionless

Greek

p= viscosity, cp I$= porosity, fraction a=

A =

interporosity, flow shape factor, f t ' l

akm rw2 , dimensionless kf

( 4 4 f , dimensionless w =

Pirson, J. Silvan: (1953) "Performance of Frac tured O i l Reservoirs", Bull. her . Assn. Pe-5. a01. 7 (21, 232-244. - Pollard, P.: (1959) nEvaluation of Acid Trat - ments from Pressure Build-Up Analysis", Trans., AIME, 216, 38. Raghavan, R., and Ohaeri, C.U.: (1981) "unstea- dy Flow t o a W e l l Produced at constant Pressure in a Fractured Reservoir", paper SPE 9902 pre - sented a t the SPE 1981 California Regional we- ting, Bakersfield, California, March 25-26.

Streltsova-Adams , T.D. : (1976) nHydrodynamics of Groundwater Flow i n a Fractured Formation*, Water Resources Research, z, No. 13, 405. Stehfest, H.: (1970) "Algorithm 368, Numerical Inversion of Laplace hansforms", D-5, Ccmmuni- cations of the ACM, 2, No. 1, 47-49. Warren, J.E. , and Root, P.J.: (1963) "The Behc vior of Naturally Fractured Fksservoirs" , @E J . 245-255.

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Ehlig-Econddes, C.A. t (1979) "Well Tes t Ana- l y s i s fo r Wells Produced a t a Constant Pressu- re", Ph.D. Dissertation, Stanford University.

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TABLE I

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t p, ' AP = Pi-Pw hours (psis) (psi 1 0 0 4.7 11 -11 6.1 11 -1 1 115 12 -12 16.5 12.5 -12.5 26 5 13.5 -13.5 46 155 -1% 5. 65 17 -17 98 195 -19.5 135 22 -22 200 26 -26 265 29 -29 400 33.5 -33.5

-

I n . . ! ' " ! " ' ,kl too m4 IO'

t D

vs tD fo r a w e l l produced at cons - Fig. 1. 'fD t a n t flowrate f ran an in f in i t e ly large

10

~ i g . 2. pfD vs tD for a w e l l produced at cons- tant flowrate fran a bounded, two-porl! s i t y system (reD = 50)

t + A I A I -

Fis. 3. Wellbore storage e f f ec t i n horner build- up, i n f i n i t e two porosity system (W O . O l , X = 5.10 -', S = 0 )

.1 1.-

2

%r Fig. 4. vs t /r 2 fo r a w e l l producing a t

ks t an tDf lgwra te from an i n f in i t e ly large two-porosity system.

Fig. 5. - p,,,,, v s t /r 2 f o r a w e l l producing at a canstant flawrate f ran a naturally fraz tured reservoir.

D D

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1

101

I. r,p-al*

Fig. 6. Dimensionless flowrate functions far a well produced at a constant pressure fran a two-porosity system (UP 0.01, A= loe5).

x

c

c

I

E

10"

b

16'

I d '

%

16'

I 1 ! ' ! ' ' ' I 1 I ! ' ' ' 102 10' 1 0' 10.

10-1

*D

I

10-2

10' m' 10. 10' lo

W = 0.1, x = 10- 5 Fig. 19. qD vs tD

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L C 102 10. 108 108

t D

-9 w = 0.1, h = 10 Fig. 20. qD vs tD

10-2 lo- 100 10

Fig. 21. p f D v s tD/r 2 for a w e l l produced at castant d e r pressure fran an inf i - nitely large, two porosity system.

a

Fig. 22. pm vs t /r 2 D D for a w e l l producing at at constant flowrate fran a naturally fractured reservoir.

10-3 10-2 10-1 ioo 10 102 0

02

FS. 23. Type-curve matching for interference

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example.

.

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