A Non Linear Dual Porosity - M. Bai, Q. Ma & J.C. Roegiers

A nonlinear dual-porosity model

Mao Bai, Qinggang Ma, and Jean-Claude Roegiers

School of Petroleum and Geological Engineering, University of Oklahoma, Norman, OK, USA

A nonlinear dual-porosity formulation incorporating a quadratic gradient term in the governingjlow equation is presented. To avoid solving the simultaneous system of equations, decoupling ofjluidpressures in the matrix from the fractures is furnished by assuming a quasi-steady-statejow in the matrix with the pressure d@erence between matrix andfractures as a primary unknown. The nonlinear fracture Jlow equation is linearized using thefunction transformation currently adopted in the nonlinear single-porosity formulation. Analytical solutions are obtained in a radialflow domain using the Hankel transform. Both solution accuracy and eficiency are achieved by using an optimized algorithm when solving the inherent Besselfunctions. The study indicates that the intensity of the dual-porosity effect is strongly conditioned by the magnitude of the initialpressure d&erence between matrix and fracture phases. The model presented appears to be suitable for simulating naturally fractured reservoirs subjected to high injection or production rate, or to sign$cant fracture compressibility.

Keywords: nonlinear dual porosity, matrix, fracture, solution optimization, analytical model

Introduction

In general, naturally fractured reservoirs are geologically prevalent and represent a major source of oil and gas production. The basic characteristic of naturally fractured reservoirs is that the matrix blocks maintain the main storage for reservoir fluids, while the fractures act as the primary vehicle for fluid transportation. As a result, conventional reservoir engineering techniques based on the classical theory of fluid flow through homogeneous porous media are insufficient for reservoir characterization, not to mention production simulation. In essence, the efficient development of naturally fractured reservoirs requires the construction of a more complex fluid flow model.

The first practical model of the fractured porous media was proposed by Barenblatt et al.’ In their model and in subsequent development,’ a naturally fractured formation is represented by a blocky matrix system intercepted by fractures of secondary voids. The high permeability of fractures results in a rapid response along them to any pressure change such as that caused by well production. The rock matrix, with a lower permeability but a higher storativity, has a delayed response to pressure changes that occur in the surrounding fractures. These nonconcurrent responses cause pressure depletion faster in the fractures than in the matrix. The resulting pressure difference between fractures and matrix induces matrix-fracture interporosity flow. This flow takes place

Address reprint requests to Dr. Bai at the University of Oklahoma Energy Center, Suite T301, 100 East Boyd, Norman, OK 73019-062X USA.

Received 13 September 1993; revised 8 March 1994; accepted 1X April 1994

after initial fracture flow and before the matrix-fracture pressure equilibrium. Then fluid flow acts as in a uniform medium with composite properties.

Among numerous analytical and numerical models for describing the aforementioned physical scenario, a nonlinear dual-porosity reservoir model that considers the effects of a quadratic gradient term has not appeared in the literature, primarily due to already complex linear dual-porosity formulation. In the general computation of groundwater hydraulics and petroleum reservoir engineering, the nonlinear quadratic gradient term is usually neglected by assuming small compressibility or small pressure gradients. The implementation of “negligible” compressibility may collapse the flow equation into a steady-state statement. The proposition of small pressure gradients, however, may cause significant errors in predicting pore pressure distribution during certain operations, such as hydraulic fracturing, large drawdown flow, well testing, drill-term testing, and large pressure pulse tesL3 In particular, ultrasensitive pressure measurement devices may be required to detect such small pressure deviations. Neglecting this deviation can be attributed to the linearization-by-deletion approximation rather than to anomalies in the flow system.4 Using the nonlinear approach by maintaining the quadratic gradient term in the flow equation, Odeh and Babu4 pointed out that nonlinear solutions show the difference in pressure change for injection and for production, unlike what is predicted by linear solutions. Finjord’ and Finjord and Aadnoy6 used this nonlinear approach to solve steady-state and quasi-steady-state conditions of reservoir pressure, and Wang and Dusseault7 applied a similar methodology for poroelastic media. The research by Chakrabarty et a1.3 demonstrates a noticeable influence of the nonlinear term for the

602 Appl. Math. Modelling, 1994, Vol. 18, November 0 1994 Butterworth-Heinemann

Nonlinear dual-porosity model: M. Bai et al.

block) should also be subjected to the counteractive influence of fluid pressure in fractures.

Substituting equation (1) into (2) yields

constant discharge case with high injection rate and small reservoir transmissivity.

It becomes obvious that the extension of current nonlinear technology for studying the behavior of homogeneous reservoirs to investigate the dual-porosity behavior of naturally fractured reservoirs may provide more accurate calculation of temporal and spatial pressure profiles. This extension appears particularly important when interpolating test data subjected to high flow rate and large reservoir compressibility, which is often true because the pressure is frequently measured at the bottom of the hole where the excess pressure and induced stress tend to be of significant magnitude.

In this paper, a nonlinear dual-porosity model incorporating a quadratic gradient term in the fracture phase is presented. Similarly to the technique used by Streltsova’ the matrix pressure is decoupled from the fracture pressure, and simultaneous solution of the dual-porosity system is hence avoided. This concise approach shows an advantage over the traditional dual-porosity formulation because the acquisition of matrix pressure is frequently of little practical interest.

k, ap2 ; V2p2 + c2 : (Vp2)’ = n2c2 x + n1c1

a(p, - p2)

at

The last term on the right side of the above equation indicates that rate change of pressure difference between matrix and fractures is indeed a source of fluid storage change in fractures.

It may be noted that in equation (1) no spatial derivation is involved; the function relationship between p1 and p2 can be determined if the initial pressures are known.

Equation (1) can be rewritten as

a(pl - p2)

at = --a(P, - P2)

where

a*k, a-

wlcl

Solution of nonlinear dual-porosity formulation If the initial pressure difference between matrix

and fractures is known, and pT2 = py - pi, then the

The approach presented can be considered an extension and a modification of the dual-porosity model

following relationship can be derived from equation using the Laplace transform:

(3)

introduced by Warren and Root.2 It is assumed that flow Pl = P2 + PT2e-at (5) can occur between fractures and matrix. However, flow to the well takes place only through fractures. At the matrix-fracture interface, the matrix flow contribution to fracture flow is proportional to the pressure difference between the matrix and the fracture. As a result, the pressure difference between matrix and fractures is considered a unique driving force for the interporosity flow in the governing equations for the matrix phase, and the quadratic gradient term is included in the fracture phase to account for nonlinear pressure variations. Therefore,

It is understood intuitively that the initial pressure difference between matrix and fractures is a unique mechanism for activating interporosity flow. Because most field test measurements are completed at the well bore, where the generation of pressure difference between matrix and fractures is almost instantaneous after initial pumping (Figure Z), equation (5) holds, The solution in equation (5) indicates that the pressure difference between matrix and fractures is created at the initial stage, and afterward the effect of this difference will decay exponentially until the final pressure eoualization is

cc*k, - 7 (pl - p2) = nlcl a(plai “I (1)

a*k, : V2p2 + c2 5 (VP,)' = n2c2 2 - ~

P ~ (Pl - P2) t 0.8

1 (2) Tfl

where subscripts 1 and 2 represent matrix and :

fractures, respectively; p is the fluid pressure, k is the LO.6 :

permeability, p is the dynamic viscosity, n is the porosity, a : ; - fracture

c is the compressibility, t is the time, and a* is the .3 OQQae matrix

characteristic dimension of the matrix blocks. For T 0.4 7 B L- __-w-@

a orthogonal fractures, CI * = 60/s* in the Warren and Root model2 where s* is the fracture spacing.

2 a_--- Qc 2 ____-----

It should be noted that the right side of equation (1) 0.2 f

is different from the Warren and Root model. In their model, flow in the matrix is prohibited. The justification of their assumption was questioned by Moench.g In our 0.0 - I I view, the rate of fluid accumulation in the matrix should 0.01

Tim% ‘(day) 1

not vanish, in particular at the matrix-fracture interface where the matrix compressibility (block and fluid in the Figure 1. Early time fluid pressure profiles.

Appl. Math. Modelling, 1994, Vol. 18, November 603


reached. Equation (5) also implies that at the place where the initial pressure difference is not generated, the fluid flow should behave in a similar fashion as in a homogeneous reservoir. In terms of interactive flow between matrix and fractures, the difference between our model and the model proposed by Streltsova’ is that the initial fluid pressure in fractures is not explicitly expressed at the initial stage for the latter model.

Substituting equation (5) into (2), and for the case of radial flow,

a2P2 1 aP2 -+;I_+cz ar2 (6)

where

pn2 c2 P=,

2

~*klPL e=-

kz

and where p:l = -py2 = p!j - p;; this sign modification is due to the fact that the initial pore pressure in the fractures is always found to be higher than the pore pressure in the matrix as a result of large fracture compressibility, as demonstrated in Figure I.”

A more detailed study of the method presented, and qualitative comparison with the singe-porosity solution as well as with the solution of Warren and Root,’ is provided in the Appendix, through analysis of a simple one-dimensional linear flow case.

The solution of the presented model is more general if it is solved in a dimensionless form. Hence, we introduce the following parameters:

l Dimensionless fluid pressures

I p

D2 = 2&h(pz - P;,

w

Po _ 2nkzhp;, - Dl2

w

(8)

where h is the reservoir thickness, p$ is the initial fluid pressure in fractures, and q is the flow rate. Dimensionless radial distance and time

r rD = ~

rw

kzt to,t=p brt pn2c2 ri

(9)

where rw is the well radius. 0 CoefJicients * 4w2

Ic2- 2rck2 h

k e* = pr2 po 2

w 4, k,

hn2c2

Equation (6) can be dimensionless form:

a2pDl i aPDz

h-f, + < at-, +

reformulated into the following

CT apD2 ’ (0 arD

apD2

= at + e*e-c*D (11) D

Equation (11) is apparently nonlinear due to the existence of the quadratic gradient term. To achieve an analytical solution, linearization of equation (11) can be made by applying the following function transformation, which has been used in nonlinear single-porosity modelling:4

(12) ‘2

Substituting PD2 in equation (12) along with its corresponding derivatives, equation (11) can be reformulated into a linear equation:

(13)

The boundary and initial conditions for equation (13) for characterizing a finite reservoir with a constant-pressure outer boundary and a constant- pumping-rate inner boundary are

(14)

I P&lfD=O = 1

Because equation (13) is now a linear system, its solution may be expressed as

AP$,(rD, tD) = A WD) + A WD, tD) (15)

AI/ in equation (15) can be obtained by solving an ordinary differential equation

d2V 1 dV q+;r=O

D D D

with nonhomogeneous boundary conditions

(16)

= -c*p* 2 D, (17)

The solution of (16) is

I/ = c,*p&(ln rDe - ln rD) + 1 (18)

AU in equation (15) can be determined by solving a partial differential equation:

a2u i au au (19)

604 Appl. Math. Modelling, 1994, Vol. 18, November


Similarly, substituting equations (18) and (27) into (15), and rearranging, gives

P& = 1 - c$rc(in rDe - In rD) f j= 1

with homogeneous boundary conditions and nonhomogeneous initial conditions:

(20)

Using the Hankel transform defined as

s

IDr

u(j, tD) = UkD? tD)@j(rD)rD drD

1

(21)

where

(22)

and where Dj is a coefficient to be determined later, cDj(rD) can be obtained by solving the eigenvalue problem and can be expressed as

(23)

where ~j are the roots of the following equation:

Jl(~j)U~jrDJ - Jo(AjrDe)Yl(Aj) = 0 (24)

Applying the Hankel transform to equation (19), yields

dO -220 = ~ + f&;Oe-5’” dt,

B can be obtained from equation (25) as

(25)

-<to )I Substituting equation (26) into (22), one has

U = f Djexp ;Ifto - 8%; ~ emrrD @jr,)

i= 1 5

(27)

Substituting equations (18) and (27) into (15) and applying the initial condition given in equation (14), yields

P& = f Dj exp j=l

+ 1 + c:P&(ln rDe - In rD) = 1

Rearranging, one obtains

(28)

1 + f Djexp

P& = j=l

= 1

1 - cr(ln rDe - In rD) (29)

Using the orthogonal properties, the coefficient Dj can be determined from equation (29) as

Dj = -&(ln rDe - In rD) exp (30)

x exp [

y(e-<‘” - 1) - J_j2tD 1 1 mj(rD)

_ 1 - c$(ln rDe - In rD)

(31)

The final expression for pore pressure in the fractures can be obtained by substituting equation (31) into (12); therefore

P,, = L In 4 i

1 - cgn(ln rDe -In rD) fJ j=l

x exp [

7 (edCto - 1) - Ajzt, 1 i air,)

- $ In [l - c:(ln rDe - In rd (32)

It is readily verifiable that P,, expressed in equation (32) satisfies the limiting boundary and initial conditions.

Optimized solution algorithm

To study the convergence and error of the solution, the following special case related to equation (22) is considered. At t, = 0 and rD = 1, a specific solution can be written as

U(1, 0) = f @j(l) = k j=O

(33)

where the expression of Oj is given in equations (23) and (24).

Because the solution procedure using the Hankel transform involves obtaining the roots from equation (24), acquisition of an accurate solution with less computational intensity but reasonable convergence rate often becomes difficult, due to the infinite summation of the Bessel function. For example, calculation of Qj in equation (33) using the conventional technique needs a summation up to 20,168 terms to achieve an accuracy of 0.3%.

The following introduces an optimized solution algorithm that is proved to be superior to the conventional solution technique. For the case of r - 10,000, the following approximation is applic- De -

able :

( JO(Aj) x l JAjl << 1


Ahhear dual-porosity model: M. Bai et al.

Table 1. Comparison of conventional and optimized methods (a)

i

Equation (39) Equation (33)

@I U-l/k @j U-l/l-C

5 0.0014931 0.2064300 0.2064300 10 0.0030631 -0.1441615 0.0383547 20 0.0062046 -0.1012827 0.0585922 50 0.0156294 -0.0637849 0.0768556

100 0.0313374 - 0.0449834 0.0861511

0.2065467 0.2065467 -0.1441660 0.0383974 -0.1012935 0.0586319 - 0.0638217 0.0768876 -0.0450722 0.0861585

I4rhl >> 1

IAjrhl >> 1 (34)

Using the above equations to approximate equations (23) and (24), Aj must be within the range 0.001 and 0.1, or j within the range of 4 and 320. As a result, equation (24) is simplified as

4<j< 320 (35)

lj can be derived from equation (35) as

n.=4j-l I 4r,,X

4<j<320 (36)

Similarly, equation (23) can be approximated by

Q,j(l) = (&y”[Sin ('odj - i) -21n$cos(r,.i,-~)]

7L

4<j<320

Substituting equation (36) into (37), yields

(37)

2(-l)j 2 112 @Al) = ~ _______

71

[

4j - 1

1 4<j<320 (38)

Substituting equation (38) into (33), one has *

j=l

+i ji5 C-1)’ & [ 1

l/2 1

=- (39) rc

where N is an integer between 4 and 320. The comparison between the results using equation (39) and

the exact solution (equation (33)) starting from the summation of the fifth term and subtracting l/rr from the results, is given in Table 1. An excellent match is achieved.

Equation (39) can be written alternatively as

U( 1,0) = U, + Err, (40)

where U, can be obtained from equations (23) and (33). Err, is the error due to truncation,

ErrJ=$ 2 (-I)jJ!: 524 (41) j=J+ 1 4j 1

It should be noted that the terms in equations (39) and (41) are alternately positive and negative. It is possible to accelerate the convergence rate by using the following scheme. Let

U(1, 0) = CU,(I, 0) + U,(L ow = (U, + U, + ,)/2 + Err? (42)

where solutions U,(l, 0) and U,(l, 0) are obtained by taking J terms and J + 1 terms in equation (39), respectively. Taking the average of the two solutions gives the average truncation error from the respective error Err, and ErrJ+ 1 as

or written in a short form

(43)

fiN ErG=~,=~+l C-1)’ [(4j + 3)“’ - (4j - l)“‘]

J(4j - 1)(4j + 3)

(44)

Applying the Taylor series expansion to the above equation, the following expression for the error term can be obtained:

(45)

In contrast with the conventional technique using equation (33), the optimized solution can be represented

by J 1 u(l, 0) = 1 @j(l) + ;@J+l(l)

j= 1 L



Table 2. Comparison of conventional and optimized methods (b) Table 3. Basic parameters for analytical modelling

Conventional Parameter Symbol Value Unit

J lJ,- l/z error (%) error (%)

5 0.0978644 30.75 0.0040368 1.27 11 0.0670815 21.07 0.0014221 0.45 21 0.0488370 15.34 0.0005810 0.18 51 0.0314432 9.88 0.0001748 0.05

101 0.0223432 7.02 0.0000756 0.02 201 0.0157700 4.95 0.0000311 0.01 501 0.0098664 3.10 0.0000301 0.01

Dynamic viscosity Fluid flow rate Reservoir thickness Well radius Outer radius

2 x lo-4 0.01 300 0.3

10,000

Pa,s m3/s

m m

Table 4. Parameters for studying effect of pressure difference

Conventional solution

is odd number)

, 1 10 100 1000

Number of tern

Figure 2. Comparison of solution methods

Comparing the errors given by equations (41) and (49, it can easily be seen that the optimized solution produces an error at least an order of magnitude lower than that by the conventional method using direct summation. A comparison of the two methods shown in Table 2 indicates that fewer than five iterations are needed to achieve 2% accuracy by the optimized method, in contrast with over 500 iterations by the conventional method for the same accuracy. The optimized method obviously offers a converging speed significantly higher than the conventional technique. This can be visualized from Figure 2.

Analytical modelling

The utility of the nonlinear dual-porosity model as described by equation (32) for fluid pressure in fractures can be envisioned well through a simple analytical model. The basic calculating parameters, listed in Table 3, describe an oil-producing reservoir with constant-pressure outer boundary and constant- flow-rate inner boundary conditions. Pressure-transient analysis was made at the well radius (I~ = l), where the effect of well bore storage can be neglected. The fracture spacing is assumed to be 1 m. In addition, all temporal pressure changes are illustrated in the form of drawdown.

Parameter Symbol Matrix Fracture Unit

Permeability Porosity Compressibility

k,. kz nl, “2 Cl, cz

1 o-16 10-l IO-8

to-‘4 lo-5 IO-'

m2

(Pa)-’

-plZ=l KPa ~000 p12=3 KPa 1 \I/, A&A** p12=5 KPa \ 7\t, n*.+~d p12=7 KPa W-H p12=10 KPa i QD+W pl2=30 KPa l -.A p12=50 KPa \ *+-A* pl2

I^ -,_

#+#w p12 -*- P12

lrl -21n -' 1 In IO' Ill3 IO' 101 IO' IO

D<m&ibni& time

Figure 3. Temporal pressure for various initial pressure difference

(a).

In the dual-porosity modelling, the determination of the initial pressure difference between matrix and fractures py2 becomes a critical step to evaluate the magnitude of the dual-porosity effect. A quantitative determination of py2 value requires a comprehensive study of the effect of matrix-fracture geometry and will be presented in future endeavors. For simplicity, only a range of hypothetical py2 values was selected for this qualitative analysis.

Using the parameters listed in Table 4, Figure 3 represents the temporal pressure changes for various py2 values. A general trend is that pressure is depleted earlier for larger initial pressure differences. The larger difference implies a faster flow in the fractures, in contrast with the smaller difference, which tends to render a uniform intergranular flow. The fluid inter- change between the matrix and the fractures appears to occur only at later times for smaller pyZ values (reduction



‘:-:;“-‘Illll;a

e, 2 10 -‘I 0 ‘2 10 +1

s! 10 -o_ - plZ=l KPa

.r( i w~~plz=lo KPa n _ . ..-*p12=30 KPa

10 -‘: AQUA* p12=50 KPa i em+- p12=70 KPa

10 -I - ~.=0-0p12=.1 MPa

e-1 p12=.5 MPa

“““‘I “““Ii “am ““Y “““‘I P,,l,NlN, I I

1o-z1o -’ 1 10 loz 10” 10’ lo6 10’ 10’ 10’ Dimensionless time

Figure 4. Temporal pressure for various initial pressure difference

(b).

10 --._ ‘\ . .

P i-r--= ‘. -......_-----._ 1 ______---_;,

1 10 102 10” 10’ lo6 10” 10’ 10’ IO0 Dimensionless time

Figure 5. Comparison of single-porosity and dual-porosity models.

of the slope of pressure change). There is a noticeable jump of dimensionless pressure between pyz = 10 KPa and pyz = 30 KPa. It appears that the magnitudes of py2 values do not affect the onset cycle of interporosity how; however, a smaller pyz magnitude represents a smaller pressure drawdown in the fractures, or a larger matrix fluid storage supply. The uniform pressure drawdown resumes as pyz diminishes. As discussed previously, a larger py2 magnitude corresponds to a more dominated fracture flow in the dual-porosity media. This becomes more apparent if the scenario in Figure 3 is redrawn in a log-log form for some pyz values (Figure 4). Although the dual-porosity behavior never disappears for large pyz values theoretically, the fractured reservoir practic- ally behaves in a similar fashion as a homogeneous one with a speedier depletion history for larger pyz values.

Table 5. Parameters for studying effect of porosity

Parameter Symbol Matrix Fracture Unit

Permeability Porosity Compressibility Pressure

k,, kz “1, “2

cl, c2 0

Pl?

IO-l4 10-13 m2 IO-' 10-5 lomg 10-s

10 10 (P&l

The utilities of the presented nonlinear dual- porosity model can be explicitly demonstrated by comparison with the single-porosity model or the linear dual-porosity model. An equivalent nonlinear single-porosity formulation can be directly obtained from equation (11) assuming 8* = 0. Figure 5 shows the comparison of the single-porosity model and the dual-porosity model (a) (Table .5), as well as the dual-porosity model (b) (Table 4 when pyz = 1 KPa). It can be observed that the drastic pressure drop at a late period occurs with all cases, which is attributed to the effect of a quadratic gradient term in the formulation. Depending on the individual cases, the initial fluid pressures for the dual-porosity model can be either higher or lower than the pressure for a single-porosity model. The patterns of “s” shape interporosity flow curves for the dual-porosity model also vary in duration and in magnitude.

Because of the difficulty in accommodating the nonflow outer boundary pressure condition, a direct comparison between the present model and the conventional dual-porosity model such as Warren and Root’s model’ cannot be made. However, Figure 6 depicts the comparison between the linear dual- porosity model by Bai et al” and the current model (Table 4 when p12 - O - 5 KPa). As discussed previously, comparably higher initial fluid pressure and more drastic late pressure drop warrant the differences between the linear and nonlinear dual-porosity models.

__ linear dual-porosity ----~~- nonlinear dual-porosity

Figure 6. Comparison of linear and nonlinear dual-porosity models.


Conclusion

A nonlinear dual-porosity model incorporating the quadratic gradient term in fracture space is presented. This may be suitable for characterizing naturally fractured reservoirs involving high injection or production rates, or large reservoir compressibilities. The decoupling of fracture pressure from matrix pressure using the sequential solution technique (similar to that proposed by Streltsova7), results in a single equation including only fracture pressure; therefore, the solution of a simultaneous system of equations as is common in the conventional dual-porosity model is avoided. The solution of the nonlinear dual-porosity model is simplified using a function transformation identical to that required to solve a nonlinear single-porosity model. The solution is obtained via Hankel transform and applied to the pressure-transient analysis for a producing naturally fractured reservoir with constant-pressure outer boundary and constant-flow-rate inner boundary conditions, commonly encountered in petroleum production. Utilizing an optimized solution algorithm in calculating the inherent Bessel functions, this efficient solution technique produces more accurate results at significantly faster convergence than commonly adopted methods.

Analytical study indicates that the initial fluid pressure difference between matrix and fractures plays a critical role in the determination of temporal pressure changes. The larger pressure differences correspond to a shorter pressure depletion history. The appearance of dual-porosity behavior occurs at a later period of similar time cycles, irrespective of the magnitude of the initial pressure difference. The comparison between the present nonlinear dual- porosity model and a single-porosity model, as well as the linear dual-porosity model, demonstrates that the nonlinear effect may create a more drastic pressure drop at later stages.

Nomenclature

Subscripts 1 and 2 represent matrix and fractures, respectively.

4 A, B Cl, c2 c; h

Jo, JI k,, k, nl, n2 Plv P2

PL P!

PY2

P&l P DZ

;k

4

Q

flow cross-section area constants compressibility coefficient reservoir thickness Bessel functions permeability porosity pore pressure initial pore pressure initial pore pressure difference dimensionless initial pore pressure difference dimensionless pressure in fractures transformed dimensionless pressure in fractures initial dimensionless pressure in fractures flow rate equivalent flow rate

YD

re

rDe

rw s

s*

t

kJ

AV

r,, F Lx ci*

; 4j ‘j

8, e*, 8, 5 P


dimensionless radius reservoir outer radius dimensionless reservoir outer radius well radius transform parameter fracture spacing time dimensionless time linear function linear function Bessel functions coefficient characteristic dimension of matrix de1 operator coefficient function in Hankel transformation function roots in Hankel transformation coefficients coefficient fluid dynamic viscosity

Acknowledgment

Support from the National Science Foundation under contract EEC-9209619 is gratefully acknowledged. The first author also wishes to express appreciation for technical advice received from Dr. D. Elsworth.

References

1

2

3

4

5

6

I

8

9

10

11

Barenblatt, G. I., Zheltov, I. P., and Kochina, I. N. Basic Concept in the Theory of Homogeneous Liquids in Fractured Rocks. J. Appl. Murh. Mech. 1960, 24(5), 12861303 Warren, J. E. and Root, P. J. Behavior of naturally fractured reservoirs. Sot. Per. Eng. J., 245-55; Trans., AIME, 1963, 228 Chakrabarty, C., Farouq, Ali S. M., and Tortike, W. S. Analytical solutions for radial pressure distribution including the effects of the quadratic-gradient term. Water Resow. Res. 1993, 29(4), 1171-1177 Odeh, A. S. and Babu, D. K. Comparison of solutions of the nonlinear and linearized diffusion equations. SPE Rex Eng. 1988, 3(4), 1202-1206 Finjord, J. Curling up the slope: effects of quadratic gradient term in the infinite-acting period for two-dimensional reservoir flow. SPE 16451, Richardson, Tex., 1986 Finjord, J. and Aadnoy, B. S. Effects of auadratic gradient term in steady-state and quasisteady-state solutions for reservoir pressure. SPE Form. Eval. 1989. 4(3). 413-417 ,. Wang, Y. and Dusseault, M. B. The effect of quadratic gradient terms on the borehole solution in poroelastic media. Water Resour. Res. 1991, 27(12), 3215-3223 Streltsova, T. D. Well testing in Heterogeneous Formations. John Wiley & Sons, New York, 1988 Moench, A. F. Double-porosity models for a fissured groundwater reservoir with fracture skin. Water Resow. Res. 1984,20,831X346 Bai, M., Elsworth, D., and Roegiers, J-C. Modeling of naturally fractured reservoirs using deformation-dependent flow mechan- isms. 34th U.S. Symp. on Rock Mech., Univ. of Wisconsin at Madison, 1993 Bai, M., Ma, Q., and Roegiers, J-C. Dual-porosity behavior of naturally fractured reservoirs. Int. J. Num. Anal. Methods Geomech., (in press)



Appendix: Comparative study

In a one-dimensional linear flow system, equation (6) reduces to

a2P2 ~ = p 2 - (J1 pyzep ax2 (AlI

cr*k, where p is referred to equation (7), 81 = __

k, ’ py2 = py - p:.. The boundary and initial conditions are

(49

where A, is the flow cross-section area. Using the Laplace transform defined as

s

‘lz Pz(x, t) = p2(x, t)emS’ dt 0

equation (Al) in Laplace space reduces to

(A3)

(A4)

where s is a transform parameter. The solution of the nonhomogeneous ordinary

differential equation (A4) is given by

where A and B are arbitrary constants. Based on the boundary condition that p2 is finite, it

is apparent that B = 0 in equation (A5). Therefore,

(‘46)

where “A” can be determined by satisfying Row rate boundary conditions from equation (A2) in the Laplace space; then one has

A=& (A7)

and

Inverting equation (A8) yields the pore pressure in the fractures:

p2=,“t.[2Jeip(-~)-x&erfc(~J)]

0, PY2

+ BE ~ (1 - e-“‘) + pz

If x = 0, then

p2 = 2Q J

4+ !!!!& (1 _ e-at) + p; Bn Pa

(AlO)

p2 in equation (AlO) can be compared with the single-porosity solution such as

(~2)s = 2Q J ; (All)

It is of interest to note that the second term on the right side of equation (AlO) acts as a damping factor to reduce the pressure change at certain periods of time (note: py2 = -(p; - py)), as a consequence of the composition of linear change of time and nonlinear change of time from the first and second terms on the right side of equation (AlO).

For comparison purposes, Warren and Root’s model2 was used to solve the same problem in a similar fashion. The corresponding pore pressure in the fractures in Laplace space is given as follows:

6412)

(A13)

Comparing equation (A8) with (A12), it is apparent that when the second term in equation (A13) is omitted, and py2 in equation (A8) is changed to py, then the two equations become identical. It can be inferred, therefore, that the model presented seems to be a simpler version of Warren and Root’s model2 under the present circumstance.


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A Non Linear Dual Porosity - M. Bai, Q. Ma & J.C. Roegiers