PERCOLATION

HODEL OF AN INHO~{OGENEOUS ANISOTROPIC

A. V. Vasil'ev,

V. I. Selyakov,

~.DIU}! UDC 532.546

and S. A. Terekhov

A model of the variation in capillary conductivity is proposed. The change in the permeability of an inhomogeneous medium under load is investigated on the basis of the percolation model [3] and is numerically modeled for cases of hydrostatic compression and nonlsotropic loading. The validity of the perc01ation approach to the determination of the change in flow properties under load is demonstrated.

The r e l a t i o n between the stress and permeability t e n s o r s o f p o r o u s m e d i a was i n vestigated i n [1, 2 ] , where t h e v a r i a t i o n of the flow properties o f a p o r o u s medium under load was examined in terms of a model of a "crack-and-cavity" medium with cracks of the same width. In this model the permeability is determined essentlally by a single parameter, namely, the width of the cracks. In real media, however, the cracks differ in width; in this sense the medium is inhomogeneous and must be described accordingly. A model of the permeability of an inhomogeneous porous medium based on the principles of percolation theory was developed in [3]. In this model the flow characteristics of the medium are determined by the structure of the infinite cluster formed by the conducting microcapillaries and the conductlvities of the individual microcapillaries composing the infinite cluster. The application of a load leads to a change in the dimensions of the mlcrocapillaries and hence to a change in the permeability of the medlun. I. ~ d e l

of the Hedium

Most reservoir rocks are of the "crack-and-cavity '~ type. In these media the pore space is a system of cavities connected by conducting microcapillaries. The porosity is chiefly determined by the cavity volume, and the flow characteristics by the flow capacity of the mlcrocapillaries. In order to describe the properties of such a medium we assume that the cavities are connected by crack-like capillaries having in cross section the shape of a highly elongated ellipse with semiaxes a and b (b/a << I). The substantial difference in the characteristic dimensions of the microcapillaries leads to considerable relative changes in their widths, which, as shown below, determines the exponential dependence of the permeability of the medium on the effective stresses. The elastic deformation of a porous medium with circular capillaries (or a cross section with commensurable characteristic dimensions) causes only a slight change in the radii of the microcapillaries and hence only a slight change in permeability. Therefore in a model with circular capillaries it is not possible to obtain a strong dependence of the permeability on the state of stress. For reservoir rocks t h e quantity 2a is of the order of the characteristic dimension of the inhomogeneity, i.e., the mean grain size d [4]. Let f(b) be the normalized frequency function of the values of the second geometric parameter of the microcapillaries. Furthermore, for simplicity we assume that the microcapillaries form three systems of mutually perpendicular crack-like channels, that in all these systems the density of the microcapillaries is the same, and that the size distributions of their semiminor axes coincide. Thus the proposed model differs from the model of a "crack-and-cavity" medium described in [1] insofar as it takes into account the distribution of the microcapillaries with respect to the transverse dimension. Let us estimate the characteristic value of the ratio b/a, starting from typical values of the permeability of the medium. The capacity of a single mlcrocapillary with dimensions a and b is proportional to ~ a b 3. The density of the microcapillaries

No.

56

Moscow. Translated from Izvestiya Akademli Nauk SSSR, ~ekhanika Zhldkosti i Gaza, I, pp. 67-75, January-February, 1986. Original article submitted November 12, 1984.

0015-48DS/86/2101-0056512.50

O 1986 Plenum Publlshing

Corporation

n ~ d - 2 , where d is the characteristic dimension of the g r a i n . Therefore, correct {o the coefficient, the permeability of the medium k ~ ~abS/4d ~, Setting d = 2a, k = 100 m D = 10 -9 cm 2 and d = 0.1 cm, we obtain b/a ~ 10 -2 . Thus, in cross section the microcapillaries are highly elongated ellipses and under load their width b changes sharply, causing a significant change in the flow properties. In [3] the following expression was obtained such an inhomogeneous medium: bc

k---- ~v'(P--t)d('-')U"(l_p,)~p o

for the effective

permeability

of

bc

b, ](b)db

l(b,) I(b,) (i.i)

+ ( S I(+)++) 'S.)+-'++, b:

bI

~ere, ~is the fraction of conducting capillaries (0<~1), P=2, 3 is the dimensionality of the problem (plane or three-dimensional), ~p i s t h e c o r r e l a t i o n radius index, w h i c h d e p e n d s on t h e d i m e n s i o n a l i t y o f t h e p r o b l e m (~2 = 1 . 3 3 , ~3 = 0 . 9 ) , d is a quantity characterizing the lattice period (in the case of a "crack-and-cavity" medium i t i s o f the order of the grain size), B is a numerical coefficient of the order of unity, and Pc i s t h e f l o w t h r e s h o l d for the conducting link problem of percolation theory. Th e q u a n t i t y

b c is

determined

from the

condition

~ ](b) db=P. b,

load if

Expression (1.1) makes it possible to determine the the corresponding change in the frequency function

change in permeability f ( b ) i s known.

under

Let us consider the deformation of a microcapillary when t h e l o a d a c t i n g on an a r e a element with normal perpendicular t o t h e s e m i m a j o r a x i s a c h a n g e s b y an a m o u n t A ~ , . F r o m the solution of the two-dimensional p r o b l e m o f a c r a c k i n an i n f i n i t e medium upon t h e application of additional stresses at infinity [5] we o b t a i n f o r t h e c h a n g e i n c r a c k w i d t h Ab t h e e x p r e s s i o n

Ab=4/n ( t - v 2) (holE) d where ~ is Poisson's

ratio,

(1.2)

and E is Young"s modulus.

2. Chan~e in Permeabilit[ L e t us c o n s i d e r the case of hydrostatic compression. ; ~ e n Ao ~ 10 ~ a a n d E ~ 104 ~a, as may b e s e e n f r o m ( 1 . 2 ) , the dimensions of all the microcapillaries change by t h e s a m e a m o u n t Ab = a d , a ~ 10 - 3 . T h i s l e a d s t o a c h a n g e i n t h e f r e q u e n e y f u n c t i o n f(b), generally speaking, o w i n g t o tw o f a c t o r s : a change in the fraction of nonzero ] i n k s z, w h i c h may d e c r e a s e o w i n g t o t h e t o t a l c l o s u r e o f some o f t h e c a p i l l a r i e s , and the reduced dimensions and flow capacity of the other capillaries with sufficiently large values of b. To a l a r g e e x t e n t , the effect o f t h e s e m e c h a n i s m s on t h e c h a n g e i n flow properties w i l l d e p e n d on t h e f o r m o f t h e f r e q u e n c y f u n c t i o n f(b). Thus, for example, for the function f ( b ) = 8 ( b -- b o ) , w h e r e 8 ( x ) i s t h e D i r a c d e l t a f u n c t i o n , an d b 0 - Ab, a p p l i c a t i o n o f t h e l o a d may l e a d t o t h e c l o s i n g o f a l l t h e c a p i l l a r i e s so that t h e medium b e c o m e s i m p e r m e a b l e . On t h e o t h e r h a n d , f o r s m o o t h e r d i s t r i b u t i o n s the decrease in the fraction of conducting microcapillaries p l a y s an u n i m p o r t a n t - r o l e in the variation of the effective permeability as compared with a "shift" in the distribution function. Before application of the load the fraction of capillaries w i t h d i m e n s i o n s on t h e interval f r o m b t o b + db i s e q u a l t o f ( b ) d b . Upon t h e a p p l i c a t i o n of hydrostatic pressure the dimensions of the capillaries d e c r e a s e b y Ab a n d b e c o m e e q u a l t o b ' = b -Ab. Now f ( b ) d b i s t h e f r a c t i o n of capillaries with the dimension b'. S i n c e d b / d b ' = 1, the microcapillary frequency function in the deformed state f'(b') has the form f'(b')= fCb' + Ab). Let us consider the change in the effective permeability of the medium f o r t h e f r e q u e n c y f u n c t i o n f ( b ) = 2bSb - 3 , w h e r e b ~ b 0 >> Ab. The s m a l l n e s s of Ab/b 0 ensures the constancy of the conducting link fraction z. T h e n t h e p o s t deformation frequency function f'(b') has the form:

57

]' (b') =2bo~(b'+.Ab) -3

(2.1)

w h e r e b ' ~> b 0 -- h b . The p e r m e a b i l i t y o f t h e medium on a p p l i c a t i o n of the load is determined from expressions (2.1) and (1.1) using relation (1..2) f o r 5 b . Since for the three-dimensional problem in (1.1) a = 1.9 + 0.1, setting a = 2 , we c a n c a r r y o u t t h e c a l c u l a t i o n s analytically. In order to simplify the calculation o f I , we make u s e o f t h e s m a l l n e s s o f t h e r a t i o 5b/b a n d e x p a n d t h e f a c t o r s o f t h e t y p e (1 + h b / b ) +n i n t h e i n t e g r a n d i n a series in powers of Ab/b. Correct to the linear terms for the relative change in permeability we o b t a i n :

9 small

hk/k=-~/4~Ab/bo,

(2.2)

Substituting the expression f o r Ab ( 1 . 2 ) i n ( 2 . 2 ) , we o b t a i n a n e q u a t i o n changes in permeability c o r r e s p o n d i n g t o s m a l l c h a n g e s ha i n h y d r o s t a t i c

Ak/k_-~ In the ferentials :

limit

for

small

ha r e l a t i o n

( i - v 2) ( d/bo) (2.3)

dk/k=-Dda/E, stress

~----( 1 5 - - i 0 1 n 2 ) / n for the pressure:

( Ao/E)

(2.3)

can be regarded

as a r e l a t i o n

between dif-

D = ~ ( ' l - v 2)d/b0

(2.4)

For a porous "crack-and-cavity" medium t h e Y o u n g ' s m o d u l u s E d e p e n d s on t h e [1]. This dependence can be represented in the form [1]:

state

E (o,) = A [ l - e x p ( - B o , ) ] + 6

of

(2.5)

H e r e , Gi i s t h e p r i n c i p l e stress acting along the xi axis; A + C and C are the maximum a n d m i n i m u m v a l u e s o f t h e Y o u n g ! s m o d u l u s , r e s p e c t i v e l y ; " B is a certain constant that determines the rate of transition of t h e Y o u n g ' s m o d u l u s from t h e minimum t o t h e maximum v a l u e . ( H e r e a n d i n w h a t f o l l o w s b y t h e s t r e s s we mean t h e s o - c a l l e d " e f f e c t i v e ~' s t r e s s [6], which is equal to the difference between the external load and the intracore pressure.) For hydrostatic compression o i = Pf, and expression (2.5) t a k e s t h e f o r m E ( P f ) = A l l -- e x p ( - - B P f ) ] + C. Substituting this expression in (2.4) and integrating, we o b t a i n t h e d e p e n d e n c e o f t h e p e r m e a b i l i t y on the effective pressure Pf:

It should be noted that a "double exponential" was p r e v i o u s l y u s e d f o r t h e a n a l y t i c description of the difference in the behavior of the permeability under loading and unloading [7]. In our case the presence of a "double exponential" is linked with the change in the compressibility as loading progresses. The p e r m e a b i l i t y behavior described by (2.S) is already manifest in the loading stage.

Let us consider the limiting cases when B(Pf -- Pf) << 1 and B(Pf --P~) >> i. For a small change in effective pressure, correct to terms of the first order in BAPf from (2.S) we obtain

~exp(,Bpjo)]Ap,}

k It

should

be noted

that

when P f = 0 e x p r e s s i o n

(2.7)

simplifies

(2.7)

to

k/ko=exp ( - D AP ~/C)

(2.8) O

D e p e n d e n c e ( 2 . 8 ) shows t h a t f o r s m a l l i n i t i a l pressures Pf the change in the permeability o f t h e i n h o m o g e n e o u s medium i s d e t e r m i n e d b y t h e m i n i m u m v a l u e o f t h e O modulus of elasticity C. If, however, the initial conditions are such that Pf is not small, then the change in permeability, in accordance with (2.7), is determined by the maximum v a l u e o f t h e m o d u l u s A + C. In the

second limiting

case,

~=exp{.

63

when BAPf >> 1,

expression

(2.6)

A+cDBI[BAP'+~exp(-Bp~~

takes

the

form:

(2.9)

k

k

/

g.g

/J.2 0

0

Zg AP,.,.MI2a FiE.

1

Z5

1

Fig.

2

r"

2

Here, the terms exp(--BAPf) ~ 1 have been omitted from the tial. As shown b y a s t u d y o f t h e e l a s t i c properties of reservoir quantities A an d C a r e c o m m e n s u r a b l e . Therefore the second term i n ( 2 . 9 ) can be n e g l e c t e d a s c o m p a r e d w i t h BA P f . Thus, for the tain the following expression for the pressure dependence of the

exponent of the exponenrocks [2], the in the square brackets c a s e BAPf >> 1 we o b permeability:

k / ko=exp [-D (A +C) -'AP~] i.e., the elasticity

change in permeability A + C.

is

determined

by t h e

maximum v a l u e

(2.10) of

the

modulus

A G r a p h o f d e n e n d e n c e ( 2 . 6 ) w i t h ~ = 0 . 2 5 , d / b 0 = 102 h a s b e e n p l o t t e d for the values C = 3"103 ~a, A = 6 " 1 0 3 ~D~a, an d B = 0 . 1 7 ( ~ a ) - 1 t y p i c a l of [2]. For comparison, the broken curve represents the ~raph of the function

k/ko=exp

of

in Fig. 1 reservoirs

(-~APt)

(2.11)

f o r y = 4 . 2 " 1 0 - 2 (~Wa) - 1 9 As s h o w n i n [ 8 ] , on t h e b a s i s o f a n a n a l y s i s of extensive experimental data, such a function satisfactorily approximates the actual change in permeability with change in effective pressure. The t h e o r e t i c a l dependence obtained (2.8) is in good agreement with the phenomenological data. It should be noted that the coefficient D / ( A + C) i n ( 2 . 6 ) d e p e n d s b o t h on t h e e l a s t i c properties of the material of the reservoir r o c k and on t h e form o f t h e d i s t r i b u t i o n function f(b). However, this affects only the numerical value of the coefficient and does not change the form of the o functional d e p e n d e n c e o f k / k 0 on ~ P f = P f -- P f . T h i s i s f u l l y c o n f i r m e d by t h e e x p e r i mental data presented i n [ 8 ] , w h e r e i t i s shown t h a t f o r d i f f e r e n t rocks only the coefficient y changes in relation (2.11). 3.

Anisotropy

of Flow Properties

Let us c o n s i d e r t h e c a s e i n w h i c h t h e s t a t e o f s t r e s s at a given point in the medium c h a n ~ e s i n s u c h a way t h a t t h e c h a n g e s i n t h e p r i n c i p a l values of the stress tensor are different, and i n p a r t i c u l a r the case of a fairly high initial pressure o Pf. Then in accordance with (2.5) the elastic properties o f t h e medium a r e d e t e r mined by a value of the modulus of elasticity c l o s e t o t h e maximum A + C, an d t h e Y o u n ~ ' s modulus remains almost constant as the state of stress changes. Therefore in order to describe the change in the state of stress it is possible to use the linear theory of elasticity. As an e x a m p l e o f a n o n i s o t r o p i c chan~e in the state of stress o f t h e medium l e t us c o n s i d e r t h e p r o b l e m o f a s p h e r i c a l c a v i t y o f r a d i u s R i n an i n f i n i t e medium w i t h o at infinity the pressure P f , t o w h o s e s u r f a c e an e x c e s s l o a d Ao 0 i s a p p l i e d . The s o l u t i o n o f t h i s p r o b l e m i s w e l l known:

ao,J=-A~o(R/r) s, Ao+l=-'/~a,r I l e r e , Az~ a n d Az~ a r e t h e c h a n g e s i n t h e r a d i a l effective stress tensor, AGO i s t h e e x c e s s l o a d o n t h e r and ~ a r e t h e r a d i a l and a z i m u t h a l c o o r d i n a t e s . at

In order to determine the chance in the e a c h p o i n t a s a medium w i t h t h r e e n u t u a l l y

(3.1)

an d a z i m u t h a l c o m p o n e n t s o f t h e wall of the spherical cavity, a nd

flow properties we c o n s i d e r t h e r e s e r v o i r perpendicular systems of conducting

59

microcapillaries, each of which is inhomogeneous. The permeability in the radial direction is determined by the system of conducting channels oriented along a radius drav~ from the center of symmetry. These microcapillaries are subjected to tension in a plane perpendicular to the radius, which causes their cross--sectional area and hence flow capacity to increase. Expression (2.7) makes it possible to determine the change in permeability in the radial direction from the change in the azimuthal component of the effective stress tensor known from (3.i):

k-~= exp

-~

I + --exp(-BPJ~

(-AoJ)

Here, for the pressure we have substituted A~$, since it is p r e c i s e l y the latter that determines the change in the permeability k r with change in the state of stress. The flow properties of the medium in the azimuthal direction are determined by its compression in the radial direction. The dimension b of the microcapillaries will decrease, reducing the azimuthal permeability. The corresponding value of the coefficient of permeability ~ can be obtained by substituting A~r~ from (3.1) in (2.7): ~= ko

exp

~

I +--exp(-BPj C

~

(-Ao/)

(3.3)

The anisotropy of the coefficient of permeability under load was noted in [2] for the model of a "crack-and--cavity" medium with cracks of the same initial width. In the present study, within the framework of the percolation approach, we analyzed the effect of the nonuniform composition of the conducting microcapillaries with respect to their flow capacity on the change in the permeability tensor of the saturated medium. It was found that the exponential dependence o f t h e permeability tensor components on the effective stresses is determined by the deformation (conpression or expansion) under load of the narrowest microcapillaries, which limit the effective permeability of the medium. Taking the inhomogeneity of the conducting capillaries into account leads only to a change in the coefficient D in expression (2.6) as compared with the case of identical width. Q u a l i t a t i v e l y , the anisotropic change i n permeability remains the same as in [2]. Graphs of k r (curve I) and k~ (curve 2) as functions of r' = r/R are presented Fig. 2. The calculations were based on Eqs. (3.1), (3.2), and (3.3) for the o previous values of the constants A, C, B, and D with A~ 0 = 25 r~a and Pf = i0 ~ a . Thus, as a result of stress redistribution it is possible to change even when the effective pressure remains the same. 4. Numerical~Iodeling

of Conductivity

in

for the flow properties

of Inhomo~eneous

Medium Under Load In order to check the applicability of the percolation model of the permeability of an inhomogeneous m e d i u m u n d e r load we modeled the medium numerically and determined the effective conductivity. Two cases of change in conductivity at each point in space were considered. In the first an increase of conductivity at each point in one direction was accompanied by a decrease in conductivity in the perpendicular direction, which corresponds to nonisotropic loading, w h i l e in the second the conductivity at each point in space decreased for all directions, which corresponds to the case of hydrostatic compression of the medium. A comparison of the results of the numerical analysis for the case of hydrostatic compression with the approximate analytic consideration of the change in effective conductivity revealed satisfactory agreement within the limits of applicability of the analytic approach. In the case of nonisotropic loading in modeling the effective conductivity we varied the degree of nonisotropicity, i.e., the ratio of the conductivities in two mutually perpendicular directions. The conductivity was modeled for the two-dimensional link problem using an array of I00 • i00 elements. During modeling we studied the conductivities for different values of the nonzero element fraction ~, which was varied on the interval from 0.6 to 1.0.

~0

TABLE N

2 3 4 5 6 7 8 9 i0

1 ~i

0,9

0,8

0,7

0.6

/%/

t,35 1,51 t,62 t,69 i,74 1,77 i,80 1,82 1,85

i,33 1,48 i,56 i,63

1,26 t,38 ,44 1,49 i,5i i,54 t,55 1,56 1,58

i,43 i,57 i,7i i,71 t,86 1,86 t,86 1,86 2,00

1,33 1,50 t,6i i,67 t,72 1,72 i,78 i,78 i,83

t,33 t ,50 t,60 i,67 i.7t t,75 i,78 1,80 1,82

i,67

i,70 i,74 i,74 i,78

For each value of ~ by means of a random number generator we assigned in the array of the conducting elements with different conductivity values, ivity frequency function p(~) being selected in the form:

the position the conduct-

(4.1)

At first, it was assumed that the conductivities of the elements in two mutually perpendicular directions were equal, and the system of conducting links obtained was used to solve the steady-state boundary value problem

(4.2) y~at

V~aZ

The s o l u t i o n was c o n s t r u c t e d by t h e s t a b i l i z a t i o n effective conductivity of the region in the direction

of

method. We t h e n the x axis.

determined

the

In order to determine the effect of the nonisotropicity of the conductivity on t h e average conductivity o f t h e m e d i u m , f o r e a c h v a l u e o f ~ u s i n g t h e same c o n d u c t i v i t y a r r a y we c a r r i e d out a series of calculations i n w h i c h we v a r i e d t h e l o c a l v a l u e s o f t h e conductivities of the links in the directions o f t h e x and y a x e s . Th e c o n d u c t i v i t i e s w e r e v a r i e d i n s u c h a way t h a t t h e i r values satisfied the conditions ~x + ~y = 2 ~ ( x , y ) , ~ x / ~ v = N, w h e r e ~ ( x , y) i s t h e v a l u e o f t h e c o n d u c t i v i t y of the links at a given point o f t ~ e a r r a y f o r an i s o t r o p i c m e d i u m (N = 1 ) , a n d N i s an i n t e g e r that determines the degree of nonisotropicity of the conducting properties o f t h e i n h o m o g e n e o u s medium i n mutually perpendicular directions. The c a l c u l a t i o n s were carried out for ten different v a l u e s o f N f r o m 1 t o 10. The c a l c u l a t e d dependence of ~/~0 on the ratio ~/~_ = N for ~ = 0.9 is uresented i n F i g . 3. H e r e , 60 i s t h e e f f e c t i v e conductivity of the array in the direction of the x a x i s f o r N = 1, a n d ~ i s t h e e f f e c t i v e conductivity i n t h e same d i r e c t i o n ~ o r N > 1. A comparison of the results of the calculations for various values of ~ at constant N shows that the ratio ~/~0 remains approximately constant for all values of ~ (see Table 1). The column on t h e e x t r e m e r i g h t o f T a b l e i g i v e s t h e r a t i o of the local conductivities M = ~ x / ~ x (N = 1) i n t h e d i r e c t i o n of the x axis (the other columns give the.relative values of the effective conductivity of the array as a whole). A comparison of these d a t a s h o w s t h a t an i n c r e a s e in the conductivities in the direction of the x axis is associated o n l y w i t h an i n c r e a s e in the local conductivity along the x axis at each point o f t h e a r r a y and i s n o t a s s o c i a t e d with a change in the conductivity in the direction of the y axis. The r e s u l t s obtained lead to the following conclusions. Firstly, since the effective conductivity o f t h e medium i s p r o p o r t i o n a l t o ~x, t h e r e w o u l d a p p e a r t o be no cross flows (in the direction of the y axis) between conducting chains. Thus t h e structure of the infinite cluster is a system of parallel chains, the connections between which can be neglected. Secondly, changes in the local conductivitSes are not associated with changes in the structure of the infinite cluster. This confirms the

@1

/.5

/

/

/.5 Fig. 3 validity of under load.

the

assumption

made i n

0.5 0

N

~ O.J

0.6

Fig. 4

calculating

the

change

in

the

permeability

of

the

medium

I n m o d e l i n g an i s o t r o p i c change in conductivities we u s e d an a r r a y o f 150 x 150 elements. Again the conducting elements were assigned by means of a random number generator, the nonzero values of the conductivity of the individual elements being distributed w i t h d e n s i t y p ( ~ ) = 2~ - 3 (~ > 1 ) , w h i l e t h e z e r o e l e m e n t f r a c t i o n w as a s s u m e d t o b e 20%. The e f f e c t i v e conductivity o f t h e m e d i u m was f o u n d , a s i n t h e c a s e o f a nonisotropic change of conductivities, by s o l v i n g p r o b l e m ( 4 . 2 ) . Th e v a r i a t i o n of the conductivity was s u c h t h a t e a c h p o i n t t h e c o n d u c t i v i t y was r e d u c e d by t h e same a m o u n t A in both directions. Calculations were carried out for ten different values of 5 from 0 to 0.9 with a step of 0.1. Th e r e s u l t s of the calculations are presented i n F i g . 4, w h e r e c u r v e 1 c o r r e s p o n d s to the change in the array-average element conductivity with increase i n A, a n d c u r v e 2 to the change in the effective conductivity o f t h e medium d e t e r m i n e d by a v e r a g i n g t h e f l o w at the boundary of the region. On t h e i n t e r v a l of variation of A investigated both curves are closely a p p r o x i m a t e d by s t r a i g h t lines. In order to study the effect of the zero element fraction on t h e s t r u c t u r e of the infinite cluster and h e n c e on t h e e f f e c t i v e conductivity o f t h e m e d i u m we c a r r i e d out another series of calculations f o r t h e c a s e o f an i s o t r o p i c change in conductivities. The c o n d u c t i v i t y array for the second series of calculations was o b t a i n e d f r o m t h a t u s e d in the first by r a n d o m l y e l i m i n a t i n g a further 20% o f t h e c o n d u c t i n g e l e m e n t s , thus bringing the zero element fraction to 40~. The r e s u l t s of this second series of calculations are presented i n F i g . 4. Curve 3 corresponds to the change in the arrayaverage value of the element conductivity, an d c u r v e 4 t o t h e c h a n g e i n t h e e f f e c t i v e conductivity o f t h e medium. A comparison of the calculations f o r t h e two s e r i e s sh o w s that as the conducting element fraction decreases the difference between the arrayaverage value of the conductivity and t h e e f f e c t i v e conductivity o f t h e medium i n creases, which is associated with a decrease in the number of conducting chains in the structure of the infinite cluster. On t h e b a s i s o f t h e r e s u l t s o f [ 3 ] , i n w h i c h an e x p r e s s i o n was o b t a i n e d f o r t h e effective conductivity o f an i n h o m o g e n e o u s m e d i u m , w h o s e l o c a l c o n d u c t i v i t y values are distributed with density p(~), it is possible t o o b t a i n an a n a l y t i c estimate for the change in the effective conductivity o f t h e medium. ~"ne f o r m o f t h e e x p r e s s i o n for the effective conductivity is similar t o t h e form o f Eq. ( 1 . 1 ) . The o n l y c h a n g e i s i n t h e expression for I. ~Tritten directly in terms of the conductivities of the individual l i n k s ~, i t t a k e s t h e f o r m :

]-'

Here,

~1 i s

mined from the

the

analog

of

the

quantity

relation ~c

62

h 1 in

(1.1),

an d

~c ( a n a l o g

o f b c)

is

deter-

All the other notation remains as before. For small values of the ratio 8 = A/~ after evaluating the integrals (in the approximation linear in 8), we obtain the expression for the change in the effective conductivity of the medium

, 1s327/2• ~'/~ A~,=-3.34d- • ~ ( ~ ] A Thus, for small 5 the linearly w i t h h. Numerical preserved f o r 6 ~ 1.

effective conductivity calculations show t h a t

o f t h e i n h o m o g e n e o u s medium v a r i e s the linear dependence is also

T h i s m o d e l os t h e p e r m e a b i l i t y o f an i n h o m o g e n e o u s m e d i u m e x p l a i n s the universality of the exponential dependence of the flow properties on t h e e f f e c t i v e stresses, as manifested in the fact that the coefficient y in (2.11) depends on the elastic properties of the reservoir r o c k and on t h e s t r u c t u r e o f t h e p o r e s p a c e , b u t d o e s n o t depend on

the effective pressure in the medium. The sharp pressure dependence of the flow properties is due to the considerable difference between the characteristic dimensions of the microeapillary cross sections. Within the framework of the model proposed we obtained the anisotropic change in the flow characteristics of the medium in the case of a complex state of stress. In this case even under conditions of constant effective pressure owing to the stress redistribution there is an increase in the radial and a decrease in the azimuthal component of the permeability tensor. Numerical modeling of the change in the conductivity of a n inhomogeneous medium has confirmed the suitability of the percolation model for calculating the coefficient of permeability of an inhomogeneous medium. Hodeling the nonisotropic loading of the medium has confirmed the concept of the structure of the infinite cluster as a system of parallel conducting chains. LITERATURE CITED I~ T. N. Kreohetova

2.

3.

4. 5. 6. -7. 8.

and E. S. Romm, "Use of a nonlinear-elastlc crack-capillary model of a porous medium for studying the effect of pressure on the physical properties of oil and gas reservoir rocks," in: Problems of Nonlinear Geophysics [in Russian], Hoscow (1981), pp. 86-87. T. N. Krechetova and E. S. Romm, "Relation between the principal components of the stress and permeability tensors of porous media," Izv. Akad. Nauk SSSR, }~ekh. Zhidk. Gaza, No. 1, 173 (1984). V. I. Selyakov, "Effective permeability of an inhomogeneous medium," in: All-Union ~eminar on Modern Problems and ~athematical ~ethods of the Theory of Flow in Porous Media [in Russian], MoScow (1984), Abstracts of Proceedings, Moscow (1994), pp. 99-101. L. }4. }~armorshtein, Trapping and Screening Properties of Sedimentary Rocks Under Various Thermodynamic Conditions [in Russian], Nedra, Leningrad (1975). V. Novatskii, Theory of Elasticity [in Russian], Mir, Moscow (1975). K. Terzaghi, Theoretical Soil ~echanics [Russian'translation], Gosstroiizdat, Moscow (1061). A. T. Gorbunov, Exploitation Of Anomalous Petroleum Deposits [in Russian], Nedra, Moscow (1931). V. N. Nikolaevskii, K. S. Basniev, A. T. Gorbunov, and G. A. Zotov, ~lechanics of Saturated Porous ~ d i a [in Russian], Nedra, ~oscow (1970).

63