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ESTIMATION OF MATRIX BLOCK SIZE DISTRIBUTIONIN NATURALLY FRACTURED RESERVOIRS

A Report Submitted to the Department of PetroleumEngineering of Stanford University in

Partial Fulfillment of theRequirements for the

Degree ofMaster of Science.

bYAshok Kumar Belani

August 1988

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Contents

1 ACKNOWLEDGEMENTS

2 ABSTRACT

3 INTRODUCTION

4 LITERATURE REVIEW

5 MODEL FORMULATION

6SOLUTION

7 UNIFORM DISTRIBUTION

8 BIMODAL DISTRIBUTION

9 DISCUSSION

10 CONCLUSIONS

11 NOMENCLATURE

12 BIBLIOGRAPHYA SOLUTIONS

B SOFTWARE PROGRAMS

2

3

4

5

8

10

12

14

15

18

19

22

25

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i

1 ACKNOWLEDGEMENTSI sincerely thank my research advisor, Y. Jalali -Yazdi for his able guidanceand patient help during the course of this research work and during the

preparation of the Report. I have enjoyed working with him.I thank also the Faculty, students and staff of the Petroleum Engi -

neering department for the numerous discussions I had on subjects relatedto the study.

Finally, I thank Schlumberger for having provided partial supportfor my Masters program at Stanford.

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2 ABSTRACTInterporosity flow in a naturslly fractured reservoir is modelled by a newformulation incorporating variability in matrix block size. Matrix block sizeis inversely related to fracture intensity. The size of matrix elements con -tributing t o interporosity flow is expressed as a distribution in the sourceterm of the diffusivity equation. The pressure transient response for uni -form and bimodal distributions of block size is investigated. Both pseudo -steady state and transient models of flow are analysed. It is shown thatfeatures observed on the pressure derivative curve can yield the parame -ters of the distribution. Thus, observed pressure response from fracturedreservoirs can be analysed to obtain the matrix block size distribution inthe volume of the reservoir investigated by the test.

The solution to the uniform distribution can be extended to moregeneral distributions. Other sources of information, like logs and geologicalobservations, can give an estimation of the shape of the distribution, andthis model can be used to compute the reservoir parameters.

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3INTRODUCTION

Flow tests in naturally fractured reservoirs have been analysed using a con -tinuum approach to model the reservoir, i.e., matrix and fracture systemsare assumed continuous throughout the formation.'12 The rock matrix has avery low permeability but stores most of the reservoir fluid in its intergran -ular porosity. The fracture system, on the other hand, has an extremelylow porosity but provides the path of principal permeability.

When a well located in such a reservoir is produced, a rapid pres -sure response occurs in the fracture network due to its high diffusivity. Thiscreates a pressure difference between the matrix and the fractures, which

begins to deplete the fluid from the matrix, commonly termed as inter- porosity flow. As flow progresses, pressures in the matrix and the fracturesequilibrate and the fracture flow response is observed again, with fluid nowcoming from a composite storativity of the matrix and the fractures.

The interaction between the matrix and the fractures is affectedstrongly by the geometrical distribution of the fractures. The parametersused to characterise this interaction are w,, matrix storativity ratio, whichspecifies the relative fluid distribution, and X, the interporosity flow coeffi -cient, which lumps the effects of the flow properties of both media and theirgeometry. Matrix flow can be modelled as pseudo -steady state (PSS): orunsteady state (USS).4*6

Models available in the literature assume fracturing is uniform andhence matrix block size is constant. Geologic studies have shown nonuni-formity in fracture intensity in many reservoirs, from very severe fracturingto very sparse fractures -' Hence, it is necessary to model variability inflow contribution from matrix elements or blocks, depending on their size.

The evolution of the double porosity model is explained in the nextsection.

This work is the development of a robust and general fractured reser -voir model allowing any distribution of matrix block size. Matrix block sizedistibution affects the pressure response significantly. The sensitivity of the

pressure response is studied for the uniform and bimodal distributions and parameter estimation is discussed.

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4 LITERATURE REVIEWThe double porosity concept was introduced in 1960 by Barenblatt et a1.1*2As explained before, it assumed the existence of two porous regions of dis -tinctly different porosities and permeabilities within the formation. Also, acontinuum was assumed, where any small volume contained a large propor -tion of both media. Hence each point in space had associated with i t two

pressure values, Pi in the permeable medium and I? in the porous, less permeable medium. Interporosity flow was assumed to occur in pseudo -steady state condition,

The solution was completed in 1963 by Warren and RootS who de -scribed the reservoir geometry as an orthogonal system of continuous, uni -form fractures, each parallel to the principle axis of permeability. Two

parameters were defined to characterize the double porosity behaviour :

0 The inter - porosity flow coefficient:

where kf is the fracture permeability, rw the wellbore radius and II ageometrical factor with dimensions of reciprocal area.

0 The fracture storativity:

where 4, is the fracture porosity, d,,, the matrix porosity and Cfand C, the corresponding fluid compressibilities.

Pseudo - steady state flow was assumed for the matrixas

a suitableapproximation for late time data. The results were analysed on semilog plots, characterizing the interporosity flow region for different values of Xand w .

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where x = Jm andf ( 4 = w.

Odeh" suggested in 1965 that wellbore pressure response at early times, and hence the be observed.

Kazemi'' (1969) and De Swaan" (1976) removed the pseudo -steadystate assumption and numerically solved the transient problem for flowfrom m a r k to fractures. Kazemi also considered flow directly from thematrix to the wellbore and concluded that the results showed insignificantdifference.

Kazemi also applied the solution to interfexnce tests, solving theequation both analytically and numerically.

Mavor and Cinco18 added wellbore storage and skin to the pseudwsteady sta te flow solution of Warren and Root.

In 1980, Najurieta" proposed an approximate solution for the equa -tion presented by De Swaan. The time domain approximation was of the

same form as the homogeneous reservoir solution. It presented a way togroup parameters to facilitate the solution of the inverse problem.Type curves for analysing wells with wellbore storage and skin in

double porosity reservoirs were introduced by Bourdet and GringartenFOIt was claimed tha t even in the absence of the first straight line on thesemi -log plot, a log -log type curve analysis could yield all reservoir parame -ters. Dimensionless parameters were defined. The idea of computing fissurevolume and matrix block size was presented but was not convincing.

A major contribution was made by Bourdet et a121122 in 1983 whenthe pressure derivative plot was introduced as a tool to analyse pressuretest data. The inter - porosity flow region was identified as a distinct feature

and could be characterized by the A and the w parameters. Both pseudosteady state and the transient matrix flow models could be analysed.At the same time, StreltsovaZS showed that the transient flow in

the matrix did not cause an inflection point on the pressure profile on the

6

storage effects dominatedfirst straight line may not

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semilog plot. A transition straight line was proposed with a slope equal toone half the slope of the early or late time straight lines. This facilitated aHorner plot analysis.

All models presented thus far assumed the orthogonal system ofuniform, continuous fractures, as proposed by Warren and Root. Matrix

blocks in between the fractures were of the same size and shape. Numerous studies in geology and well logging have shown the exis -

tence of nonuniformly fractured reservoirs. The Warren and Root modelis an over -simplification of reality. There is need to develop a model whichhonors the heterogeneity in matrix block properties. Since the matrix -fracture interface area is dependent on the geometry of the matrix blocks,a distribution of matrix block geometries must be considered. The shapeof the blocks does not have a significant effect on the response and hence avariability in block size shall be considered in this work.

Braesterlo concluded block size does not significantly affect the draw -down pressure response of a fractured reservoir. Cinco et al" suggested adiscrete distribution of matrix block sizes with transient interporosity flowand showed the pressure derivative is significantly affected. Jalali -Yazdi andBelani12 show block size variability affects the pressure response markedly.

It would be more appropriate to lump all flow criteria into the flowcoefficient X and consider a distribution of matrix elements with differ -ent flow coefficients. The engineering concept of matrix blocks would bereplaced by t hat of matrix volume elements with a variability in their con -tribution to interporosity flow. This idea has many strengths in modellingthe reservoir and understanding its flow behaviour, and hence will be thesubject of later research.

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which is consistent with the Warren and Root and other single block sizer n ~ d e l s . ~ l ~

For reservoirs with intense fracturing, f(h) is a positively skewed dis -tribution favoring small blocks, and for reservoirs with sparse fracturing,f(h) is a negatively skewed distribution, favoring large blocks.

A more general formulation of the source integral can account forvariability in several matrix properties. For instance, a random variationin block permeability km and block size h, results in :

1

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6 SOLUTIONEquations (1) and (2) are solved for slab matrix blocks for the initial and

boundary conditions stated in the Appendix. The wellbore pressure re -sponse in the Laplace space is:

where s is the Laplace variable related to dimensionless time:

and the argument z = \/sgo.For PSS:

For USS:

The dimensionless parameters are defined below :

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r

hhD =

-.h-, (16)

The interporosity flow coefficient X depends on h and hence is included inthe integral in Eqns. (7) and (8); these equations collapse to the single

block size case if f (hD) is a Dirac delta function.The pressure response is markedly governed by the distribution func -tion in Eqns.(7) and (8). Geologic studies of outcrops do not express ob -served fracture intensities in terms of block size, hence it is difficult tochoose any particular shape of block size distribution. This work solves thecases of Uniform and Bimodal distributions.

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7 UNIFORM DISTRIBUTIONA distribution of interest is the uniform or rectangular distribution whereall block sizes (hmin to hmaz) have an equal chance of occurrence:

1f h) =

hmaz - hmin'with mean block size:

hmin + hmazhmean = 2

9

and variance:

The applicability of the uniform distribution is two -fold:

1. It should be used when the matrix block size distribution is unknown.

2. A sum of uniform distributions of small variance spread can approxi -mate any distribution and hence the pressure response for other dis -tributions can be obtained.

For PSS flow, Eqns.(7), (12) - (14) yield:

For USS flow, Eqns. (8), (12) - (14) yield:

Eqn. (18) does not have a closed form analytical solution and requiresnumerical integration.

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The block sizes hmin and hmaz correspond to interporosity flow co -efficients X and Amin respectively:

The Xmoz/Xminratio governs the variance of the uniform distribution. Asthis ratio approaches unity, the uniform distribution approaches a Diracdelta function and hence the pressure response approaches the single blocksize response.

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8 BIMODAL DISTRIBUTIONGeneral tectonic stresses over a region can cause fracturing at a macroscale and associated breaking of the rock at a finer scale. This results intwo controlling sets of matrix block sizes, which can be represented by a

bimodal distribution. If the two modes of the distribution are equally prob -able (same height), then:

where hl < hz < hs < h,,, correspond to A 1 > A 2 > A s > Amin , respec -tively. The PSS solution for this distribution function is:

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9 DISCUSSICNFigure (3) shows the PSS pressure and pressure derivative response for uni -form block size distributions of different variance spread (X ratio). X,,, iskept constant at lo- and Xmin varies from lo- to lo- by an order ofmagnitude at each step. Figure 3 indicates:

1. In the limit Xmin approaches X (or vice versa), the single block sizeresponse is obtained (Warren and RootS).

2. The interporosity flow region on the derivative curve shows distinctly

the effect of the variation in block size. The contribution from each block size affects the pressure at a different point in time depend -ing on the interporosity flow coefficient, causing a stretching of thederivative curve. The change from the characteristic 'peaked valley'to a stretched valley with more features, is hence dependent on the

block size distribution.

3. The beginning of the late t h e semi -log straight line (P = 0.5) isinversely related to Xmin (slowest contributing block), with an approx -imate relation:

The solution was investigated for other values of X varying Xratio. Figure (4) shows the pressure derivative response for X,, of lo-and X ratios of 1 to 10000. Similarly, Fig.(5) shows the response for X of

and the same X ratio values. Identical derivative profiles are obtainedfor a given w and X ratio; only the placement of the profile in time isgoverned by the magnitude of the flow coefficients.

Figure (6) illustrates the response for a range of matrix storativitywm) for Am,, = lo- and Amin = lo-'. The 'stretched' transition curve

characterizes the variance of the distribution function as noted in Fig.(3).

Also, PAmin and Ti (the coordinates of the point of inflection) increase withdecreasing w,.The response was closely studied for the inflection points in the three

cases of Figs. 3,4,5. The dat$is presented in Tables 1,2,3.

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Figure (7) is a correlation of slope ratio (SR) and w, and X,in/X-,.SR is the ratio of semi -log slope at the point of inflection and the earlytime or late time semi -log slope, SR=PAmin/0.5.13114 SR is independent ofthe magnitude of Ami, and X,,,.

Figure (8) is a correlation of TD, and Ami as indicated by Eqn.(22).Figure (9) indicates the time at which the transition curve begins,

TDB,is a dominant function of Amoz but also varies with the variance of thedistribution function (X ratio).

Figure (10) illustrates the time coordinate of the point of inflection,Ti, depends on the magnitude of the interporosity flow coefficients. How -ever, the ratio T D ~ / TAis a function of w, and X ratio and not of the Xvalues.

Figure (11) illustrates the effect of wellbore storage on the pressureresponse. Flow tests where early time dat a may be lost should be run longenough to obtain TD,.

Figure (12) exhibits the pressure response for the unsteady statemode of interporosity flow. The Xdistribution is the same as that of Fig.(3)

but w, is 0.99. The features of Fig.(lO) are similar to those of the pseudo -steady state response (Fig.3), although less pronounced.

The relations illustrated above can be used to estimate w,, Amin,and X,,, from pressure transient data. Alternatively, use of the proposedsolution (Eqns. 5,17,18) in nonlinear regression of pressure da ta yields thereservoir parameters.

Figures (13) and (14) illustrate the pressure response for a bimodaldistribution with the parameters, A1 = lo-', X 2 = 0.8 x lo-', X 3 =and Amin = 0.8 X lo-'. Figure (15) exhibits the response for a bimodaldistribution with X1 = lo-', X 2 = 0.8 X lo-', X3 = lo-' and Amin = 0.8 x

Figures (16) and (17) exhibit the response for a bimodal distribution

Figure (18) exhibits the response for a bimodal distribution with X1 = lo-',X 2 = 0.8 x lo-', X 3 = and Ami, = 0.8 x

When the two modes of the distribution are very close, the responseis similar to tha t of the unimodal uniform distribution corresponding tothe larger Xmode. As the separation between the two modes increases, the

pressure response deviates from that of the unimodal distribution. Beyonda certain degree of separation, the derivative plot character due to thehigher X mode is suppressed.

with X1 = lo-', X 2 = 0.8 X lo-', X3 = and = 0.8 X

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The solutions of the unimodal and-5iinodal distributions can be ex -tended to multimodal distributions, which may be obtained from geologicinformation. The procedure would be t o estimate a shape of the distri -bution from well -log data and compute the parameters using the pressureresponse.

..

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10 CONCLUSIONS1. A robust formulation of pressure transient response in nonuniformly

fractured reservoirs is presented.

2. The matrix block size distribution for a uniformly fractured reservoiris a Dirac delta function and results in a sharp pressure response.

3. The pressure response of a nonuniformly fractured reservoir becomesless pronounced with an increase in the variance of the matrix blocksize distribution.

4. The pressure derivative curve for uniform distribution c a n be anal-ysed to estimate the reservoir parameters.

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11NOMENCLATURE

formation volume factor,RB/STB

compressibility, psi-fracture total compressibility,

matrix total compressibility,

wellbore storage coefficient, di -mensionless

block size distribution function,

block size distribution function,dimensionless

joint probability distributionfunction, f t - .m d-a parameter in the Bessel func -tion argumentmatrix block size variable, f tmatrix block size, dimensionless

fracture thickness, ftminimum block size, uniform dis -tribution, ftmaximum block size, uniform dis -tribution, f tmean block size, uniform distri -

bution, ft block size bounds for bimodal dis -tribution, f tconstant matrix block size, ftfracture permeability, md

ps i- '

psi- '

f -

matrix permeability, md

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modified Bessel function, secondkind, zero ordermodified Bessel function, secondkind, first orderfracture pressure, dimensionlessmatrix pressure, dimensionlessLaplace transformed wellbore

pressure responsefracture fluid pressure, psiinitial pressure, psimatrix fluid pressure, psi

pseudo -steady statevolumetric flow rate, STB/Dflow contribution of matrix size h,

hour -flow contribution of matrix size hand permeability k m , hour -cumulative matrix flow contribu -tion, hour

radial coordinate, ft

radial coordinate, dimensionlesswellbore radius, ftLaplace parameterskin factor, dimensionlessminimum slope ratio, dimension -lesstime, hourstime, dimensionlesstime transition period begins, di -mensionless

time transition period ends, di -mensionlesstime of minimum slope, dimen -sionless

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x 1 9 x 2 x s =

unsteady stateBessel function argumentDirac delta functioninterporosity flow coefficient, di -mensionlessinterporosity flow coefficients, bi -modal distribution, dimension -lessmaximum interporosity flow coef -ficient, dimensionlessminimum interporosity flow coef -

ficient, dimensionlessviscosity, cpnormal coordinate to fracture -matrix interface, f tnormal coordinate to fracture -matrix interface, dimensionlessvariance of the matrix block sizedistribution, f t2fracture porosity, dimensionlessmatrix porosity, dimensionless

fracture storativity ratio, dimen -sionlessmatrix storativity ratio, dimen -sionless

SI METRIC CONVERSION FACTORS

bbl X 1.589873 E O 1 = msCP x 1.0' E 0 3 = pa.s

ft x 3.048' E-01 = m psi X 6.894757 S O 1 = kpa

psi- x 1.450 E0 1 = kpa-l'

Conversion factor is exact.

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12 BIBLIOGRAPHY1. Barenblatt, G.E., Zheltov, I.P., and Kochina,I.N.: Basic Concepts

in the Theory of Homogeneous Liquids in Fissured Rocks, J. Appl.Math. Mech. 24, (1960), 2861303.

2. Barenblatt, G.E.: On Certain Boundary -Value Problems for theEquations of Seepage of a Liquid in Fissured Rocks, J. Appl. Math.Mech., 27, (1963), 513 - 518.

3. Warren, J.E. and Root, P.J.: Behaviour of Naturally Fractured Reser -voirs, SOC. Pet. Eng. J., (Sept. 1963), 245 - 55.

4. de Swaan, O.A.: Analytical Solutions for Determining NaturallyFractured Reservoir Properties by Well Testing, SOC. Pet. Eng.J., (June 1976).

5. Kazemi, H.: Pressure Transient Analysis of Naturally FracturedReservoirs with Uniform Fracture Distribution, SOC. Pet. Eng. J.,(Dec. 1969), 451 - 62.

6. Isaacs, C.M.: Geology and Physical Properties of the Monterey For -mation, California, paper SPE 12733 presented at the 1984 CaliforniaRegional Meeting, Long Beach, April 11 - 13.

7. McQuillan, H.: Small Scale Fracture Density in Asmari Formationof Southwest Iran and its Relation to Bed Thickness and StructuralSetting, AAPG Bulletin, V.57, No. 12, Dec. 1973), 2367 - 2385.

8. McQuillan, H: Fracture Patterns on Kuh -e Asmari Anticline, South -west Iran, AAPG Bulletin, V.58, No. 2, (Feb. 1974), 236246.

9. Stearns, D.W., and Friedman, M.: Reservoirs in Fractured Rock,AAPG Memoir (1972), 82 - 106.

10. Braester, C.: Influence of Block Size on the Transition Curve for

a Drawdown Test in a Naturally Fractured Reservoir, SPEJ 1984,498 - 504.

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11. Cinco -Ley, H., Samaniego -V, F., and Kucuk, F.: The Pressure Tran -sient Behavior for Naturally Fractured Reservoirs with Multiple BlockSize, paper SPE 14168, presented at the 60th Annual Fall TechnicalConference and Exhibition, Las Vegas, NV, Sept. 22-25, 1985.

12. Jalali -Yazdi, Y., and Belani, A.: Pressure Transient Modeling of Nonuniformly Fractured Reservoirs, Proceedings of Advances in Geother -mal Reservoir Technology, Lawrence Berkeley Laboratory, June 14-15, 1988.

13. Jalali -Yazdi, Y., and Ershaghi, I.: Pressure Transient Analysis ofHeterogeneous Naturally Fractured Reservoirs, paper SPE 16341,

presented at the SPE California Regional Meeting, Ventura, Cali -fornia, April 8 - 10, 1987.

14. Jalali -Yazdi, Y., and Ershaghi, I.: A Unified Type Curve Approachfor Pressure Transient Analysis of Naturally Fractured Reservoirs,

paper SPE 16778 presented a t the 62nd Annual Fall Technical Con -ference and Exhibition, Dallas, TX, Sept 27 - 30, 1987.

15. Odeh, A.S.: Unsteady State Behaviour of Naturally Fractured Reser -voirs, SOC. Pet. Eng. J. (March 1965) 60 - 66.

16. Kazemi, H., Seth, M.S., Thomas, G.W.: The Interpretation of Inter -

ference Tests in Naturally Fractured Reservoirs with Uniform Frac -ture Distribution, SOC. Pet. Eng. J. (Dec 1969) 463 - 72.

17. de Swaan, 0.: Analytical Solutions for Determining Naturally Frac -tured Reservoir Properties by Well Testing, SOC. Pet. Eng. J. (June1976).

18. Mavor, M.J., Cinco, H.: Transient Pressure Behaviour of NaturallyFractured Reservoirs, Paper SPE 7977 presented at the 1979 SPECalifornia Regional Meeting, Ventura, April 18 - 20.

19. Najurieta, H.L.: A Theory for Pressure Transient Analysis in Natu -

rally Fractured Reservoirs, J. Pet. Tech. (July 1980) 1241 - 50.

20. Bourdet, D. and Gringarten, A.: Determination of Fissured Vol -ume and Block Size in Naturally Fractured Reservoirs by Type -Curve

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Analysis, paper SPE 9293 presented at the 1980 SPE Annual Tech -

nical Conference and Exhibition, Dallas, Sept. 21 -24.21. Bourdet, D. et al.: A New Set of Type Curves Simplifies Well Test

Analysis, World Oil (May 1983).

22. Bourdet, D. et al.: Interpreting Well Tests in Fractured Reservoirs,World Oil (Oct 1983).

23. Streltsova, T.D.: Well Pressure Behaviour of a Naturally FracturedReservoir, SOC. Pet. Eng. J. (Oct.1983) 769 -80.

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A SOLUTIONSCombining equations (1) and (2) from the text,

Pseudo - Steady State

Considering the direction of flow that of the normal to the matrix -fracture interface, material balance yields:

- -2pm 4mCmp apm-k m at

2)

For pseudo - steady state, the pressure gradient is a constant in space, hence:

Integrating twice with respect to e

at D = 1,

B = Pf A = - h C ( t )

From Darcy's law at the matrix -fracture interface,

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which yields,

Averaging the expression for Pm from 0 to h, Eqn.(5)

Substituting the dimensionless parameters as defined before in the text, theequation for PSS flow becomes,

and in the matrix,

To solve the equations so obtained, (11) and 12), we specify the followinginitial and boundary conditions:

Boundary conditions in the radial direction:

Transforming this set of equations to Laplace space, and rearranging,

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where,

The solution to this equation is the double porosity solution,

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A similar procedure can be followed for the unsteady state case, using thematrix flow equation,

and the material balance equation

a 2 p m +mCmp a p m

a2 k m at *- = (17)

The initial and boundary conditions are exactly the same as the PSS case. Substituting dimensionless parameters, and transforming t o Laplace do -main,

and,

Solving equation(l9) with the boundary conditions in D and substitutingin eqn(l8), we get a form similar to PSS

where,

The solution is the same as eqn(l5) with a different g(s).

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CC

CC

CCCC

CC

1CC

4

56

CCc

CCCCC

CC

CCC

CC

9

CC8

1211

13

1410

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

SUBROUTI NE PWD TD, N, PD, dpd, d2pd) THI S FUNTI ON COMPUTES NUMERI CALLY THE LAPLACE TRNSFORMI NVERSE OF F S) .

I MPLI CI T REAL* 8 ( A- H, - 2DI MENSI ON G 50) , V 50) , H 25)common m, r ed, sk, cbar , sl mi n, sl max, omegm, ssn, f , sf

NOW I F THE ARRAY V 1) WAS COMPUTED BEFORE THE PROGRAMGOES DI RECTLY TO THE END OF THE SUBRUTI NE TO CALCULATEF S) .I F N. EQ M GO TO 17

M=NDLOGTW - 0. 6931471805599NH=N/ 2

THE FACTORI ALS OF 1 TO N RE CALCULATED I NTO A RRA Y G.G (1) 1DO 1 I =2, N

CONTI NUEG I ) =G I - l ) *I

TERMS WI TH K ONLY ARE CALCULATED I NTO ARRAY H.H 1) =2. / G NH- l )DO 6 I =2, NH

FI - II F ( I - NH) 4, 5, 6H(I)=FI**NH*G(2*I)/(G(NH-I)*G(I)*G(I-l))GO TO 6H(I)=FI**NH*G(2*I)/(G(I)*G(I-l))

CONTI NUE

THE TERMS (-1) *NH+ ARE CALCULATED.FI RST THE TERM FOR 111

SN=2* NH- NH/ 2*2) - 1

THE REST OF THE SN' S ARECALCULATED I N THE MAI N RUTI NE.

THE ARRAY V ( I ) I S CALCULATED.DO 7 I = , N

FI RST SET V I ) =OV I ) =O

THE LI MI TS FOR K ARE ESTABLI SHED. THE LOWER LI MI T I S Kl = NTEG I +1/ 2) )

K1= I +1) 2

THE SUMMATI ON TERM I N V 1) I S CALCULATED.DO 10 K=Kl , K2

I F 2*K- I ) 12, 13, 12

I F (I - K) 11, 14, 11V I ) =V I ) +H

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implicit double precision (a - h, - 2double precision k0,klargmin = dsqrt (3*slmin/omegm/s)argmax = dsqrt(3*slmax/omegm/s)hratio = dsqrt (slmin/slmax)

argl = omegm*argmin* (datan (argmin) datan (argmax) ) / (1-hratio)fs = 1.0 - omegm - arglsfs=s*fsx-dsqrt ( sf )y=dsqrt (sf )kl = dbskl(x)

kO = dbskO (x)plapdZ=s* (kO + (sk*x*kl) ) / (1* (x*kl) +(cbar*s* kO+sk*x*kl) ) ) )returnend

C fs = (1.0-omegm) - (omegm*argmin* (datan (argmin) datan (argmax) ) / (1-hratio)

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C returnC end

function f (x)implicit real*8 (a-h,o-z)f=dtanh (x) xreturnend

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A

M a t r i x B l o c ks

n n m m F r a c t u r eh A hB h C h D

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Td

Fig. 3 - Lambda Range le -5 to le -9

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1 F

Derivative Curves Uniform Distribution

O

1 10 100 loo0 loo00 leM5 le* l e d 7 le+08

Td

Fig. 4 - Lambda Range le -4 to le -8

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10

1

0.1

Derivative Curves Uniform DistributionI I111111 I I l l n l I I I11111 I I I rn

I

, -

I

1 bda(max)

1 0 0 l o l o o le+05 l e a le+07 le+OS le* le+10

Td

Fig. 5 - Lambda range 1e -6 to le - 10

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Td

Fig. 6

.tion

l e 4 8

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SR

Ucn

o*800

0.600

0.400 -

0.200 - iL Omega=O.7- mega=0.8

Omega=O.9- Omega - 0.95Omega=0.99

0.000

3 41 10 100 1000 10000Lambda Ratio

TDE

Fig. 7

Fig. 8

Omegad.7Omega=0.8Omega=0.9

Omega=0.95Omega=0.99

l /Lambda min)

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103

102

101

100

TDB

lambda(max)=le-5

Omega=0.7- mega=0.8Omega=O.9- mega=0.95

I , , Omega=0.991

I . . . . I . ... . . . I . , ...,10 100 1000 10000

Lambda Ratio Fig. 9

TD

lo7

06 1

103o 4

Fig. 10

1 1 0 100 1000 10000

Lambda Ratio

lambda(max)=le-5

Omega=0.7- mega=0.8- mega=0.9-

mega=0.95

I Omega=0.99

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.form DistributionI I I I I I H ~

- -----

---

----

--

:a(m>= 0.95da(min) =: le -8

da(max) := le -5

I I 1 1 1 1 1 1 I I I IIIU,10 100 lo00 loo00 l e 4 5 le+M le47 l e 4 8 le-

Td

Fig. 11

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y State Soln.

1 10 1 l o l o o l e 4 5 le l e 4 7 l e 4 8 le-

Td

Fig. 12

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BIMODAL DISTRIBUTION

h l - l ambda max1h4 - l ambda min1

Fig. 13

PD -TD Derivative Curves BIMODAL Distribution

c

on ga(m) =

L

1.95

100 lo00 l oo00 l e 4 5 le& l e 4 7 l e 4 8 le-

Td

Fig. 14

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Fig. 15

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PD -TD Derivative Curves BIMODAL Distribution

T

o ega(m) =

100 lo00 loo00 l e 4 5 le l e 4 7 l e 4 8 l e a

Td

Fig. 18