- Email: [email protected]
Effective permeability of multifractal porous media
North-HollandPhysica A 183 (1992)381-397
PHYSICA
Effective permeability of multifractal porous media A n t o i n e Saucier lnstitutt for Energiteknikk, Box 40, N-2007 Kjeller, Norway
Received 28 November 1991 Revised manuscript received 24 January 1992 We show how the real space renormalization group method can be used to calculate analytically the scaling exponents of the effective absolute permeability in multifractal porous media. The permeability fields considered are deterministic and random multifractals constructed with multiplicative processes. We discuss the implications of the results on the understanding of fluid ftow in oil reservoirs.
I. Introduction
T h e understanding of fluid flow properties inside oil reservoirs is complicated by the high variability of the permeability field. One typically observes intermittent variations of the absolute permeability over several orders of magnitude at a given scale (say the core plug length scale, i.e. about 10 cm), and these variations occur over wide ranges of scales, i.e. from 10 cm to the size of the whole reservoir (usually several kilometers). A n o t h e r difficulty comes from the fact that the three-dimensional structure of the permeability field remains poorly known because the information comes from rather sparse drilling wells, while geology and seismic data provide only rough guidelines a b o u t stratification, heterogeneity sizes, etc. But on the other hand, even if all the information was actually available, say for example a complete threedimensional picture of the permeability field with a 10 cm resolution, then the processing of this large amount of data would exceed the capacity of computers or else be prohibitively expensive. For a given permeability field the flow can be obtained in principle by solving a diffusion equation derived from Darcy's law and continuity. In order to reduce the a m o u n t of computations the permeability field is usually h o m o g e n ized before one solves directly the fluid flow equation by numerical methods. 0378-4371/92/$05.00 © 1992- Elsevier Science Publishers B.V. All rights reserved
382
A. Saucier / Permeability o f multifractal porous media
Homogenization is an averaging process in which a large array of small scale permeabilities is replaced by a smaller array of larger scale effective permeabilities. The homogenization procedure allows to generate effective permeabilities k(L) at any scale L (L being the mesh of the grid). The dependence of k(L) on L is determined by the structure of the permeability field. For example, if the permeability field is an uncorrelated noise then k(L) converges rapidly to a constant value as L---~w [1], which means that the homogenized permeability field is essentially constant at large scale. By contrast, quite different behaviors can be obtained for correlated permeability fields. Indeed such fields may be variable even after homogenization, i.e. the variance of k(L) can remain large even for L large. A realistic large scale modelling of the reservoir properties therefore requires a minimum understanding of the variability of the permeability field. A whole spectrum of different variable fields exists, ranging from the simplest uncorrelated fields to the most complex correlated structures. It is likely that real permeability fields in reservoirs are correlated and anisotropic because of the nature of the sedimentation process. Indeed the genesis of sedimentary rocks usually occurs in the presence of a fluid flow (e.g. air, water, oil, etc.) and a gravitational field that creates correlations and anisotropies in the permeability field. When dealing with poorly known geophysical fields, such as the permeability field in oil reservoirs, it is sensible to make the simplest and most natural assumptions as possible. Along similar lines Mandelbrot [2] has argued that structures involving self-similarity are among the simplest ones and that they can often provide qualitatively reasonable approximations for many irregular geophysical fields. From the experimental standpoint, some preliminary evidence of multiscaling was found for permeability fields as measured from well data ]3] and for the pore space geometry as measured on thin sections of core plugs [4]. Multifractal permeability fields are among the simplest correlated structures with no characteristic length scale and therefore they offer a good starting point for the study of flow through inhomogeneous porous media. In this perspective, we examine in this paper the scale dependence of the effective absolute permeability of multifractal permeability fields. Both deterministic and random multifractal fields are considered. The real space renormalization group method (in short R S R G ) is used in the calculation of the effective permeability and is shown to be particularly suitable for the study of self-similar permeability fields. Simple renormalization equations can be derived for the effective permeability k(L) at scale L. Their solution allows to link the statistics of the effective permeability field at one scale, say the core plug scale, to the statistics of the effective permeability field at larger scales.
A. Saucier I Permeability of multifractal porous media
383
2. The RSRG method for the calculation of the effective absolute permeability The R S R G method used in this paper for the calculation of the effective permeability was described by King [1]. We shall briefly recall the main idea. The values of permeabilities are distributed on a D-dimensional cubic grid. The grid blocks are then grouped into blocks of 2 t~ and the effective permeability of each group is assigned to a new coarser grid (fig. 1). This coarse-graining process can be applied again to the new larger scale grid and repeated many times until the scale of the whole field is reached. The effective permeability ke, of a cubic group of 2 ° blocks is defined via Darcy's law, assuming that the pressure is uniform along both vertical boundaries while the horizontal boundaries are impermeable (fig. 2), i.e.
(U)n_
kef f
Ap
It
L
kef f
It
L
Ap (u). =--f(kI, k~
k2D )
(2.1)
where u is the x-component of the fluid velocity, ( u ) . is the value of u averaged over a surface normal to the x-direction, It is the viscosity, Ap is the pressure difference across the group of 2 D blocks and L is the size of this group. The pointwise fluid velocity is given by u = -(k/it) Vp and the pressure field p is the solution of the continuity equation V - [ k ( x ) Vp] = 0
(2.2)
with the boundary conditions of fig. 2. We focus only on isotropic media and therefore it is sufficient to consider the flow in only one direction. The function f defined by (2.1) does not have in general a simple analytical
Fig. 1. In the RSRG method, here illustrated in 2 dimensions, a grid containing 4 x 4 blocks with different permeabilities is replaced by 2 × 2 homogeneous blocks of equivalent permeabilities. This process is repeated many times and at each step the size of the grid blocks doubles.
A. Saucier / Permeability of muhifractal porous media
384
Y
impermeable boundary
kl
k2
FI© w
P=O
k3
P =AP
k4 x
impermeable boundary
L
Fig. 2. Sketch of the boundary conditions used for the calculation of the effective permeability of a group of 2 I~ cubic blocks.
form in terms of the ki's. However, an estimate of f can be obtained by using the resistor network approach. In this approximation each block is replaced by an elementary resistor network whose particular geometry is sometimes referred to as a tesselation [5]. In fig. 3 we have illustrated an example of tesselation for a two-dimensional flow, namely the resistor cross. The equivalent resistance between the midpoints of the edges is 1/k (more generally in a D-dimensional space it is 1 / k L °-z, where L is the size of the block, assuming that /z = 1). Once the 2 t~ resistor crosses have been hooked together, the boundary conditions are set so that the sides of the blocks are at constant voltage (fig. 4), which corresponds to a uniform pressure on both vertical boundaries (fig. 2). The inverse of the equivalent resistance of this circuit then yields an estimate g(k~, k 2, k3, k4) of the effective permeability
f ( k l , k2,
k3, k4).
1/(2k)
1/(2k)
~
1/(2k)
Fig. 3. In the cross tesselation approximation, each h o m o g e n e o u s block of permeability k is replaced by a resistor cross. T h e equivalent resistance between the edges is 1/k for a twodimensional flow. If the m e d i u m was anisotropic the resistances on the horizontal and vertical arms of this cross would take different values.
A. Saucier / Permeability of multifractal porous media
L_J
385
!.........
~" i i
i
1 Fig. 4. Graph of the equivalent circuit obtained when four crosses are linked together with constant pressure (or voltage) on both sides. Around each node, here labelled 1, 2, 3, 4, the value of each of the four resistors enclosed in the dotted square is 1/2k~.
The R S R G m e t h o d can be criticized in several ways: Firstly, many different tesselations are a priori possible and accordingly they lead to different estim a t e s of kerr, i.e. different functions g; secondly, the approximations involved are difficult to quantify and in practice the accuracy of the predictions remains unknown [6]. Nevertheless, as shown by several numerical experiments perf o r m e d on various test cases, this method has proved to be surprisingly accurate [5, 1, 7-10].
3. P r e l i m i n a r y
considerations
3.1. Multifractal permeability fields and multiplicative processes T h e permeability fields considered in this p a p e r take constant values k i on each cube of a regular cubic grid, and these c u b e s - or grid b l o c k s - have the s a m e volume AV. We associate a density function p(x) to the permeability field with the simple relation p(x)-
k(x) ,av
(3.1)
T h e measure tz {R} of any spatial domain R is then simply defined by /~{R} = f p(x) d V .
(3.2)
R
/~ {R} is therefore the sum of the permeabilities of all the grid blocks contained in R. It is emphasized t h a t / ~ { R } is not in general the effective permeability of the porous m e d i u m enclosed in R.
A. Saucier / Permeability of multifractal porous media
386
T h e support #~ of the permeability field can be covered with a regular cubic grid of mesh 6. On this grid each D-dimensional cube of size 6 is denoted by B ( 6 ) . A multifractal permeability field can be characterized by the behavior of its generating function Xq(6 ), defined by Xq(6) = E [/~{Bi(6)}] q ,
(3.3)
i
where the summation extends over all the cubes of the covering. For multifractal fields the generating function exhibits a scaling behavior over a wide range of scales, i.e.
)¢q(6 ) ~ 6 "lql ,
(3.4)
where in general the mass exponents ,c(q) depend non-linearly on q. In this study the multifractal permeability field is constructed with a multiplicative process [11-17]. These processes are simple models of multifractals which have been used in many fields of physics and geophysics. The process involves 212 weights w i/> 0, i = 1 . . . . . 2 t). The first two steps of the construction process are shown in fig. 5 for a two-dimensional process. The unit cube B(60) of size 6o is divided into 2 D identical cubes B~il(6~) of size 6~ = 6o/2 (here 6,, = 6o2 m) and each of these sub-cubes is assigned a weight w~, i = l . . . . . 2 z~. This process is then repeated on each s u b - c u b e : B(i)(61) is divided into 2 D cubes of size 62, which are assigned the weights w~wj, j = 1 . . . . . 2 °. This process is continued n times down scale until a scale 6,, = 2 " is reached. 6,, is the homogeneity scale - or inner scale - of the permeability field, i.e. the scale at which it becomes homogeneous. The weights obtained at scale 6,, are interpreted to be the permeabilities of the blocks of size 6,,. For illustration the permeability field obtained after 7 cascade steps with a two-dimensional multiplicative process is shown in fig. 6. For simplicity we choose to consider here a conservative multiplicative process, i.e. a process where the total measure is conserved in the construction. This is done by imposing on the weights the constraint Z wj = 1, which implies t h a t / x {B(6o) } remains equal to unity at each cascade step. It also implies that the measure of a cube B(6,,) takes the form
u { B ( ¢ . ) } -- ,46,) w(62).., w(6,.),
(3.5)
where w(6k) denotes a multiplier at scale 6 k (w(6k) can be equal to any of the 2 ° values of wj). To different cubes B(6m) therefore correspond different strings of multipliers w(61) w(62)--. W(6m).
~ I.e. the region of space where the permeability is non-zero.
A. Saucier / Permeability of multifractal porous media
387
n=O
8o=1
wl
w2 n=l
~1 = 112
w3
w4
wlwl
wlw2
w2wl
w2w2
wlw3
wlw4
w2w3
w2w4
w3wl
w3w2
w4wl
w4w2
w3w3
w3w4
w4w3
w4w4
82 = 1/4
n = 2
Fig. 5. The first two steps of construction of a deterministic multifractal two-dimensional permeability field (from top to bottom). R a n d o m multiplicative processes can also be constructed by making the weights random variables Wi, i = 1 . . . . . 2 °. The Wj's are independent from each other from one cascade level to the other, but are correlated at a given level. Indeed the conservation constraint Z Wj = 1 creates correlations between the Wj's. This random process builds a collection of permeability fields. Each realization of this process could be regarded as a block of material taken from a different location in some multifractal porous medium. Such random multifractal fields can be characterized by the scaling behavior of the expectation value of their generating function, which satisfies
( )(,q((~) ) ~ • r(q),
(3.6a)
A. Saucier / Permeability of multifractal porous media
388
Fig. 6. Two-dimensional permeability field constructed with 7 cascade steps and four different weights (w~ = 0.35, w 2 = 0.05, w3 = 0.15, wz = 0.45). Dark areas represent regions of low permeability while the bright areas are regions of high permeability.
where the mass exponents
T ( q ) a r e t h e n g i v e n b y [17]
~-(q) = - D - log2( ( W q ) ) .
3.2.
Homogeneity
(3.6b)
o f the e f f e c t i v e p e r m e a b i l i t y f u n c t i o n
T h e e f f e c t i v e p e r m e a b i l i t y k ( 6 ) o f a c u b i c r e g i o n o f size ~5 c a n b e e x p r e s s e d in t e r m s o f t h e M p e r m e a b i l i t i e s k ~ ) ( 6 , ) ( u = 1, 2 . . . . . M ) at t h e h o m o g e n e i t y s c a l e 6 , , i.e. t h e r e e x i s t s a f u n c t i o n F s u c h t h a t
k ( ~ ) = F(k(l)(t~n) . . . . . F is a h o m o g e n e o u s
r(ak~')(6,),
k(M)(t~n)) .
function such that for any real number
(3.7) a
. . . , t~k~M)(6,)) = a F ( k ~ l ) ( 6 , ) , . . . , k ~ M ) ( 6 , ) ) .
(3.8)
A. Saucier / Permeability of multifractal porous media
389
(3.8) follows from the linearity of Darcy's law. Indeed, given a pressure field p(x) that satisfies the continuity equation (2.2) with the boundary conditions of fig. 2, then for any real number a the field p(x) is also solution of V . [ak(x) Vp] = 0 with the same boundary conditions. Hence p(x) is invariant under the transformation k(x)--> ak(x). Using the differential expression of Darcy's law for the x-component of the velocity field, i.e. u = - [ k ( x ) / l x ] Oxp, and replacing it in the definition (2.1) of the effective permeability, yields
k~. = ~
1
(k(x) O . p ) . .
(3.9)
When k(x)--+ a k ( x ) , Oxp remains unchanged and therefore (3.9) implies that kcff---~ akcf,. (3.8) is the expression of this general result obtained in the special case where k(x) takes discrete values in M cubes. It is emphasized that (3.8) also holds for the effective permeability Fnetwork(k(l)(~n) k(M)(c~n) ) of capillary networks as a consequence of the linearity of the flow equations in tubes (or resistors). (3.8) implies in particular that the function f defined by (2.1) satisfies . . . . .
(3.10)
f ( a k l , . . . , ak2o ) = a f ( k I . . . . , k2o ) .
(3.10) also holds for the estimate of f obtained with the resistor network modelling. When the RSRG procedure is iterated n times the effective permeability k(60) obtained at the largest scale 60 can also be expressed in terms of the permeabilities k~V)(6n), ~, = 1 , . . . , 2 no, at the homogeneity scale 6n = 2-". Indeed, introducing the notation N(n)=-2 "°, k(6o) can be written in the form
k(6o) = f(k(')(61 ) , . . . , = f(f(k~l}(~2) .....
kN('))(6, )) k~N~'~(~2) ) . . . . .
f(k~N~2~-3~(~ 0 . . . . .
kCN~2~(~a)))
= etc... --= FRSR~(k(I)(t5 ), k(2)(~,) . . . . .
k(N("))(6~)) .
(3.11)
By definition FRsRc is an estimate of the exact effective permeability F. The nested structure shown in (3.11) and the property (3.10) of the function f imply that FRSRG also satisfies the homogeneity property (3.8). The homogeneity of the effective permeability therefore holds for Darcy flows, for flows in capillary networks and for the estimate FRSRG obtained with the RSRG method.
390
A. Saucier / Permeability of multifractal porous media
4. Effective permeability of deterministic multifractal permeability fields W e shall n o w s t u d y t h e effective p e r m e a b i l i t y o f the d e t e r m i n i s t i c m u l t i f r a c tal p e r m e a b i l i t y field o f s e c t i o n 3.1 as a f u n c t i o n o f scale. M o r e specifically, we a s k t h e f o l l o w i n g q u e s t i o n : G i v e n a cubic ball B x ( 8 ) o f size 6 c e n t e r e d on a p o i n t x, h o w d o e s t h e effective p e r m e a b i l i t y o f the m e d i u m e n c l o s e d in B x ( 8 ) v a r y with 8 w h e n x is fixed? A n d h o w d o e s it v a r y if 8 is fixed b u t w h e n x v a r i e s ? In t h e f o l l o w i n g k,,(6,,,) will d e n o t e the effective p e r m e a b i l i t y o f the p o r o u s m e d i u m c o n t a i n e d in Bx(8,, ,) for a p e r m e a b i l i t y field c o n s t r u c t e d with n cascade steps (here 8,,=2 " and m
k,, 0,,) = w ( 8 , ) . . , w( m)
v = 1,2 .....
N(n - m),
(4.1) w h e r e t h e kl,"),,,(8,,_,,,) a r e the p e r m e a b i l i t i e s at the h o m o g e n e i t y scale g e n e r a t e d b y t h e s a m e m u l t i p l i c a t i v e p r o c e s s , b u t with o n l y n - m c a s c a d e steps. S u b s t i t u t i n g (4.1) into (3.7) a n d using (3.8), we get k,,(8,,) = w(Si),
"" w ( 6 , , , ) F ( k l , 1 ' m ( 8 . . . . . ), . . . ,
k ,(u~, _ , , m)) (8,
m))"
(4.2)
But by definition k,, _ ,, (8 o) = F ( k , _ ,, ( ,, _ m ),
80=
(4.3)
- m ))
I
Wl
W2
51 = 1/2 Wl
etc...
W2
= = = =
Wl
=m
W2
=
v
Fig. 7. Here is represented a one-dimensional multiplicative process (D = 1) constructed with 4 cascade steps and two different weights w~ and w 2. The inner scale is 84 = (1/2)48o. If we consider the part of the permeability field enclosed in the horizontal bracket, it is seen that all the values of permeability in this interval share the same multipliers at scales 8~ and 82, namely w(8~)= w2 and w(82) = wt. The permeabilities in the horizontal bracket can therefore be written in the form k(f)(82) = w ( 6 , ) w(82) kl2")(82), where k(2")(82) is a permeability generated by only the part of the cascade process indicated by the vertical brackets, i.e. with only two cascade steps.
A. Saucier / Permeability of multifractal porous media
391
hence using (4.3), eq. (4.2) becomes kn(~m) = W ( ~ I ) ' ' ' W(~m) kn_m(¢5o) ,
(4.4)
and using (3.5) finally yields k . ( ~ , . ) =/~{B~(6m) } k. m(6O).
(4.5)
k,(60) is the effective permeability of the whole field constructed with n cascade steps. We will now show that kn(6o) can be determined by using a recurrence relation between k,+1(60) and kn(6o). In the following ~n will denote the permeability field constructed with n cascade steps. Firstly, 2 D copies of ~ , are produced and scaled down by a factor 1/2 (fig. 8a). Secondly, for each copy, # i , i = 1 . . . . ,2 °, all the permeabilities k{n~)(6,) at the homogeneity scale 6, are multiplied by w~. According to (3.8) the new effective permeabilities of these scaled-down copies are n o w wikn(t~o), i = 1 . . . . . 2 D. The new fields are next arranged in a 2 D array with the same spatial order as the wj's in the multiplicative process. This can be seen to yield the new permeability field ~,+1 (fig. 8b) constructed with n + 1 cascade steps. Thirdly (fig. 8c), the RSRG method is used to compute the effective permeability of ~,+~ according to k.+,(a0)
w2k.(a0) . . . . .
(4.6)
(3.8) then implies kn+l(~0) =f(w
1, w 2 . . . . .
WN(1)) k n ( ~ 0 ) ,
(4.7)
which is a simple renormalization equation for the effective permeability k,(6o). Iterating (4.7) yields = (flw,.
(4.8)
w2 .....
With ko(6o) = 1 and eliminating n using 6, = 2 ", (4.8) becomes k,(6o) =- 6~, ,
(4.9a)
where y = - l o g z ( f ( w l , w2 . . . . .
Wu~l))).
(4.9b)
(4.9a) shows that the effective permeability of the whole permeability field
392
A. Saucier / Permeability of multifractal porous media
kn(1)
k.O)
kn(1)
Xl/2
(a) kn(1)
Wl kn(1)
kn(1)
w2 kn(1) (b)
ws kn(1)
w4 kn(1)
RSRG
kn+l(1)
(c)
Fig. 8. ,~,, d e n o t e s a two-dimensional permeability field constructed with n cascade steps. (a) F o u r copies of 0% are p r o d u c e d and scaled d o w n by a factor 1/2. (b) F o r each copy # i , i = 1, 2, 3, 4, all the values of permeabilities are multiplied by w i. The new effective permeabilities are then w,k,,(1), i = 1, 2, 3, 4. The four new fields are next arranged in a 2 x 2 array, which yields the permeability field ,.%,,+~ constructed with n + 1 cascade steps. (c) The R S R G m e t h o d is used to c o m p u t e the new effective permeability according to k,,~ , ( 1 ) = g ( w t k , , ( 1 ), w~_k,,(1), w3k,,(1 ), w~k,,(1)).
scales with the inner scale 6,. According to (4.9b) the scaling exponent 3' is determined by the function f and by the 2 D weights w/#2 The ball Bx(6m) enclosing the porous medium is centered about a point x. For multifractal measures one usually defines a pointwise scaling exponent a(x) such that /x{Bx(6)) - 6 ~x) as 6 - + 0 [18, 19]. When the inner scale 6, is finite #2 M o r e details on the sensitivity of y to the choice of a specific tesselation can be found in the r e p o r t no. I F E / K R / F - 9 1 / 141.
A. Saucier / Permeability of multifractal porous media
393
this statement takes the approximate form /x{Bx(8)} ~ 8 ~(x) when 8 , < 8 becomes
(4.10)
< 6 o. Using (4.9a), (4.10) and 8 , _ m = 8,18,,,, eq. (4.5) finally
kn(Sm) ~ --m8 °t(x)-Y t] 7--n
(4.11)
when 8, < 8 < 80. (4.11) describes how the effective permeability of a piece of medium contained in Bx(8 ) varies with 8. kn(Sm) simply scales with 8 m for a fixed 8n. The scaling exponent is partly determined by the location x of the piece of medium, which accounts for a(x), and the scaling properties of the medium embodied by the 2 D weights w i, which account for - % The dependence on the homogeneity scale 8 n is also scaling with an exponent % This appears to be the first analytical derivation of the effective absolute permeability of a multifractal permeability field.
5. Effective permeability of random multifractal fields A permeability field can be correlated over wide ranges of scales, eventually in a multifractal manner, but random variations are expected from sample to sample, e.g. for blocks of porous medium coming from different locations. From this standpoint the effective permeability k ( L ) at scale L can be regarded as a random variable that depends on a length scale parameter L. We choose to characterize the random variations of k ( L ) by the scaling behavior of its moments, i.e. we consider the quantities ([k(L)] q) as a function of L and the real parameter q. It will be shown that for random multifractal permeability fields these moments satisfy ( [ k ( L )] q) ~ L ~q) ,
(5.1)
where ~(q) is a nonlinear function of q. In this section the formalism of section 4 is adapted to random multifractal permeability fields and the expression of ~(q) is derived using the R S R G method. We consider permeability fields generated with the random multiplicative process introduced in section 3.1. The random version of eq. (4.4) is simply
k.(Sm) L w(81). • • W(Sm) k. ~.(8,,),
(5.2)
394
A. Saucier / Permeability of multifractal porous media d
where " = " stands for equality in probability distribution and k n m(60) is a random variable independent of the W(6k)'s. Raising (5.2) to the power q and averaging yields ([k,,(6m)] q) = ( w q ) m ( [ k n _ m ( 6 o ) ] q ) .
(5.3)
Eliminating m in (5.3) with 6 m = 2 - " and using (3.6b) leads to
([k.,(6..)]o)
')+'~ ( [ k , _ , . ( 6 o ) ] q ) . = 6m
(5.4)
Similarly, the random version of (4.6) is
k.+,(6,,) ~ f(W,k.(6,,). W2k.(6,,) . . . . . WN~,~k.(6,,)).
(5.5)
Using the homogeneity of f, (5.5) becomes k,,+,(60) a=f(W,, W2, . . . , WN~,) ) k,(6,,),
(5.6)
where k,,(6o) is independent of the Wi's. Raising (5.6) to the power q and averaging yields ([k,,+,(6,,)] q) = ( [ f ( W ~ , W 2 , . . . , WN~i))]q)(Ik,,(6o)] q) ,
(5.7)
which is a simple renorrnalization equation for the moments of the effective permeability. Iterating (5.7) leads to ( [ k , ( 6 o ) ] q) = ([f(W,, W 2 , . . . ,
Wu~1))lq)"([k,,(6,,)] q) .
(5.8)
Using ko(6o) = 1 and eliminating n with 6, = 2 " yields (5.9a)
([k,,(60)] q) = 6~ (q) ,
where y( q) = - l o g 2 ( ( [ f ( W 1 , W 2 . . . .
, Wu(1))]q)).
(5.9b)
Replacing (5.9a) in (5.4) and using 6,_ m = ~,/6,,, finally gives ( [ k n ( 6 m ) ] q) = 6D+T(q)-'r(q)6"£ (q) .
(5.101
The scaling exponents of the effective permeability, as defined by (5.1), are
A. Saucier / Permeability of multifractal porous media
395
therefore given by sO(q) = D + ~-(q) - 3'(q).
(5.11)
Hence a multiscaling permeability field gives rise to a multiscaling effective permeability field. It is interesting to discover that the mass exponents r(q) determine directly-although not completely-the permeability scaling exponents ~:(q). The permeability exponents y ( q ) are not trivially related to the mass exponents r(q) because f is a non-linear function. (5.10) appears to be the first analytical derivation of the effective permeability of random multifractal permeability fields.
6. Conclusions
We have studied the scaling properties of the effective absolute permeability of multifractal permeability fields generated by multiplicative processes. In the deterministic case, the permeability k(6) of a region of size 6 centered about a point x was found to scale according to k(6) - 6,~x)-y,
(6.1)
where oz(x) is the local pointwise scaling exponent of the permeability field, and y is an exponent determined by the weights used in the multiplicative process. In the random case, the permeability k(~) is a scale dependent random variable that was shown to satisfy
([k(6 )] q) ~ 6o+,~q)-~,~q)
(6.2)
where ~-(q) is the order-q mass exponent of the multifractal permeability field and y ( q ) is another nonlinear function determined by the joint probability distribution of the weights used in the multiplicative process. It is emphasized that both (6.1) and (6.2) are approximate results derived with the real space renormalization group method. It is interesting to discover that the effective transport properties of the porous media are determined, via (6.2), by the multifractal spectrum of the permeability field. From the standpoint of oil reservoir modelling, (6.2) allows to relate the statistics (e.g. variance) of the permeabilities measured at one scale, say the core plug scale, to the statistics of the effective permeabilities at the scale of the grid blocks used in the flow simulators. A proper description of the scaling properties of the permeability field in oil reservoirs is therefore
396
A. Saucier / Permeability of multifractal porous media
necessary to g e n e r a t e realistic effective permeabilities and c o n s e q u e n t l y to predict m o r e accurately the large scale flow. M e a s u r i n g b o t h ~-(q) and y ( q ) requires three-dimensional information a b o u t the permeability field. Such data is usually not directly available for oil reservoirs since most o f the information c o m e s f r o m one-dimensional vertical cuts t h r o u g h the reservoir, i.e. wells. R e c o v e r i n g three-dimensional information f r o m o n e - d i m e n s i o n a l cuts is a nontrivial p r o b l e m where the k n o w l e d g e of the anisotropy, stratification and h e t e r o g e n e i t y of the reservoir properties plays an i m p o r t a n t role. This p r o b l e m might not be hopeless if the permeability field was locally isotropic, or else if the anisotropy could in some way be characterized. T h e role of a n i s o t r o p y in fluid flow t h r o u g h p o r o u s media and the possibility o f extracting information about anisotropy from one-dimensional cuts are t h e r e f o r e questions of interest for reservoir characterization.
Acknowledgements I t h a n k Jiri Muller for stimulating discussions, e n c o u r a g e m e n t and for his careful editing of the manuscript. This work is part of a c o m m o n project b e t w e e n the Institutt for E n e r g i t e k n i k k , Fina Exploration N o r w a y and Petrofina.
References [1] P.R. King, The use of renormalization for calculating effective permeability, Transp. Porous Media 4 (1989) 37-58. [2] B.B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1983). [3] J.L. McCauley, J. Muller and G. Saether, Multifractal spectra of large scale oil-reservoir properties, in: Correlations and Connectivity, J.E. Stanley and N. Ostrowsky, eds. (Kluwer, Dordrecht, 1990) p. 310. [4] J. Muller and J.L. McCauley, Implication of fractal geometry on fluid flow properties of sedimentary rocks, Transp. Porous Media, in press. [5] N. Shah and J.M. Ottino, Effective transport properties of disordered, multiphase composites: application of real-space renormalization group theory, Chem. Eng. Sci. 41 (1986) 283-296. [6] J. Bernasconi, Real-space renormalization of bond-disordered conductance lattices, Phys. Rev. B l g (1978) 2185-2191. [7] S. Mohanty and M.M. Sharma, A recursive method for estimating single and multiphase permeabilities, SPE 20477, 65th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in New Orleans, LA, September 23-26, 1990, pp. 127-136. [8] P.R. King, A.H. Muggeridge and W.G. Price, Renormalization calculations of immiscible flow, Transp. Porous Media, submitted in August 1991. [9] A.O. Grindheim and J.O. Aasen, An evaluation of homogenization techniques for absolute
A. Saucier / Permeability of multifractal porous media
[10] [11] [12] [13] [14] [15] [16] [17] [18]
[19]
397
permeability, presented at Lerchendal Petroleum Engineering Workshop. The Norwegian Institute of Technology, Trondheim (1991). A. Aharony, E.L. Hinrichsen, A. Hansen, J. Feder, T. Jcissang and H.H. Hardy, Effective renormalization group algorithm for transport in oil reservoirs, Physica A 177 (1991) 260-266. A.M. Yaglom, Effect of fluctuations in energy dissipation rate on the form of turbulence characteristics in the inertial subrange, Dokl. Akad. Nauk SSSR 166 (1966) 49-52. B.B. Mandelbrot, Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier, J. Fluid Mech. 62 (1974) 331-358. C. Meneveau and K.R. Sreenivasan, Simple multifractal cascade model for fully developed turbulence, Phys. Rev. Lett. 59 (1987) 1424. D. Schertzer and S. Lovejoy, Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes, J. Geophys. Res. 92 (1987) 9693-9714. P. Meakin, Diffusion-limited aggregation on multifractal lattices: a model for fluid-fluid displacement in porous media, Phys. Rev. A 36 (1987) 2833-2837. E.A. Novikov, The effects of intermittency on statistical characteristics of turbulence and scale similarity of breakdown coefficients, Phys. Fluids A 2, No. 5 (1990). A. Saucier, Cascade processes and fully developed turbulence, Ph.D. dissertation, McGill University (Physics Department), Montr6al, Canada (1991). C.H. Halsey, M.H. Jensen, L.P./Kadanoff, I. Procaccia and B.I. Shraiman, Fractal measures and their singularities: The characterization of strange sets, Phys. Rev. A 33 (1986) 1141-1151. J.D. Farmer, E. Ott and J.A. Yorke, The dimension of chaotic attractors, Physica D 7 (1983) 153-180.
PHYSICA
Effective permeability of multifractal porous media A n t o i n e Saucier lnstitutt for Energiteknikk, Box 40, N-2007 Kjeller, Norway
Received 28 November 1991 Revised manuscript received 24 January 1992 We show how the real space renormalization group method can be used to calculate analytically the scaling exponents of the effective absolute permeability in multifractal porous media. The permeability fields considered are deterministic and random multifractals constructed with multiplicative processes. We discuss the implications of the results on the understanding of fluid ftow in oil reservoirs.
I. Introduction
T h e understanding of fluid flow properties inside oil reservoirs is complicated by the high variability of the permeability field. One typically observes intermittent variations of the absolute permeability over several orders of magnitude at a given scale (say the core plug length scale, i.e. about 10 cm), and these variations occur over wide ranges of scales, i.e. from 10 cm to the size of the whole reservoir (usually several kilometers). A n o t h e r difficulty comes from the fact that the three-dimensional structure of the permeability field remains poorly known because the information comes from rather sparse drilling wells, while geology and seismic data provide only rough guidelines a b o u t stratification, heterogeneity sizes, etc. But on the other hand, even if all the information was actually available, say for example a complete threedimensional picture of the permeability field with a 10 cm resolution, then the processing of this large amount of data would exceed the capacity of computers or else be prohibitively expensive. For a given permeability field the flow can be obtained in principle by solving a diffusion equation derived from Darcy's law and continuity. In order to reduce the a m o u n t of computations the permeability field is usually h o m o g e n ized before one solves directly the fluid flow equation by numerical methods. 0378-4371/92/$05.00 © 1992- Elsevier Science Publishers B.V. All rights reserved
382
A. Saucier / Permeability o f multifractal porous media
Homogenization is an averaging process in which a large array of small scale permeabilities is replaced by a smaller array of larger scale effective permeabilities. The homogenization procedure allows to generate effective permeabilities k(L) at any scale L (L being the mesh of the grid). The dependence of k(L) on L is determined by the structure of the permeability field. For example, if the permeability field is an uncorrelated noise then k(L) converges rapidly to a constant value as L---~w [1], which means that the homogenized permeability field is essentially constant at large scale. By contrast, quite different behaviors can be obtained for correlated permeability fields. Indeed such fields may be variable even after homogenization, i.e. the variance of k(L) can remain large even for L large. A realistic large scale modelling of the reservoir properties therefore requires a minimum understanding of the variability of the permeability field. A whole spectrum of different variable fields exists, ranging from the simplest uncorrelated fields to the most complex correlated structures. It is likely that real permeability fields in reservoirs are correlated and anisotropic because of the nature of the sedimentation process. Indeed the genesis of sedimentary rocks usually occurs in the presence of a fluid flow (e.g. air, water, oil, etc.) and a gravitational field that creates correlations and anisotropies in the permeability field. When dealing with poorly known geophysical fields, such as the permeability field in oil reservoirs, it is sensible to make the simplest and most natural assumptions as possible. Along similar lines Mandelbrot [2] has argued that structures involving self-similarity are among the simplest ones and that they can often provide qualitatively reasonable approximations for many irregular geophysical fields. From the experimental standpoint, some preliminary evidence of multiscaling was found for permeability fields as measured from well data ]3] and for the pore space geometry as measured on thin sections of core plugs [4]. Multifractal permeability fields are among the simplest correlated structures with no characteristic length scale and therefore they offer a good starting point for the study of flow through inhomogeneous porous media. In this perspective, we examine in this paper the scale dependence of the effective absolute permeability of multifractal permeability fields. Both deterministic and random multifractal fields are considered. The real space renormalization group method (in short R S R G ) is used in the calculation of the effective permeability and is shown to be particularly suitable for the study of self-similar permeability fields. Simple renormalization equations can be derived for the effective permeability k(L) at scale L. Their solution allows to link the statistics of the effective permeability field at one scale, say the core plug scale, to the statistics of the effective permeability field at larger scales.
A. Saucier I Permeability of multifractal porous media
383
2. The RSRG method for the calculation of the effective absolute permeability The R S R G method used in this paper for the calculation of the effective permeability was described by King [1]. We shall briefly recall the main idea. The values of permeabilities are distributed on a D-dimensional cubic grid. The grid blocks are then grouped into blocks of 2 t~ and the effective permeability of each group is assigned to a new coarser grid (fig. 1). This coarse-graining process can be applied again to the new larger scale grid and repeated many times until the scale of the whole field is reached. The effective permeability ke, of a cubic group of 2 ° blocks is defined via Darcy's law, assuming that the pressure is uniform along both vertical boundaries while the horizontal boundaries are impermeable (fig. 2), i.e.
(U)n_
kef f
Ap
It
L
kef f
It
L
Ap (u). =--f(kI, k~
k2D )
(2.1)
where u is the x-component of the fluid velocity, ( u ) . is the value of u averaged over a surface normal to the x-direction, It is the viscosity, Ap is the pressure difference across the group of 2 D blocks and L is the size of this group. The pointwise fluid velocity is given by u = -(k/it) Vp and the pressure field p is the solution of the continuity equation V - [ k ( x ) Vp] = 0
(2.2)
with the boundary conditions of fig. 2. We focus only on isotropic media and therefore it is sufficient to consider the flow in only one direction. The function f defined by (2.1) does not have in general a simple analytical
Fig. 1. In the RSRG method, here illustrated in 2 dimensions, a grid containing 4 x 4 blocks with different permeabilities is replaced by 2 × 2 homogeneous blocks of equivalent permeabilities. This process is repeated many times and at each step the size of the grid blocks doubles.
A. Saucier / Permeability of muhifractal porous media
384
Y
impermeable boundary
kl
k2
FI© w
P=O
k3
P =AP
k4 x
impermeable boundary
L
Fig. 2. Sketch of the boundary conditions used for the calculation of the effective permeability of a group of 2 I~ cubic blocks.
form in terms of the ki's. However, an estimate of f can be obtained by using the resistor network approach. In this approximation each block is replaced by an elementary resistor network whose particular geometry is sometimes referred to as a tesselation [5]. In fig. 3 we have illustrated an example of tesselation for a two-dimensional flow, namely the resistor cross. The equivalent resistance between the midpoints of the edges is 1/k (more generally in a D-dimensional space it is 1 / k L °-z, where L is the size of the block, assuming that /z = 1). Once the 2 t~ resistor crosses have been hooked together, the boundary conditions are set so that the sides of the blocks are at constant voltage (fig. 4), which corresponds to a uniform pressure on both vertical boundaries (fig. 2). The inverse of the equivalent resistance of this circuit then yields an estimate g(k~, k 2, k3, k4) of the effective permeability
f ( k l , k2,
k3, k4).
1/(2k)
1/(2k)
~
1/(2k)
Fig. 3. In the cross tesselation approximation, each h o m o g e n e o u s block of permeability k is replaced by a resistor cross. T h e equivalent resistance between the edges is 1/k for a twodimensional flow. If the m e d i u m was anisotropic the resistances on the horizontal and vertical arms of this cross would take different values.
A. Saucier / Permeability of multifractal porous media
L_J
385
!.........
~" i i
i
1 Fig. 4. Graph of the equivalent circuit obtained when four crosses are linked together with constant pressure (or voltage) on both sides. Around each node, here labelled 1, 2, 3, 4, the value of each of the four resistors enclosed in the dotted square is 1/2k~.
The R S R G m e t h o d can be criticized in several ways: Firstly, many different tesselations are a priori possible and accordingly they lead to different estim a t e s of kerr, i.e. different functions g; secondly, the approximations involved are difficult to quantify and in practice the accuracy of the predictions remains unknown [6]. Nevertheless, as shown by several numerical experiments perf o r m e d on various test cases, this method has proved to be surprisingly accurate [5, 1, 7-10].
3. P r e l i m i n a r y
considerations
3.1. Multifractal permeability fields and multiplicative processes T h e permeability fields considered in this p a p e r take constant values k i on each cube of a regular cubic grid, and these c u b e s - or grid b l o c k s - have the s a m e volume AV. We associate a density function p(x) to the permeability field with the simple relation p(x)-
k(x) ,av
(3.1)
T h e measure tz {R} of any spatial domain R is then simply defined by /~{R} = f p(x) d V .
(3.2)
R
/~ {R} is therefore the sum of the permeabilities of all the grid blocks contained in R. It is emphasized t h a t / ~ { R } is not in general the effective permeability of the porous m e d i u m enclosed in R.
A. Saucier / Permeability of multifractal porous media
386
T h e support #~ of the permeability field can be covered with a regular cubic grid of mesh 6. On this grid each D-dimensional cube of size 6 is denoted by B ( 6 ) . A multifractal permeability field can be characterized by the behavior of its generating function Xq(6 ), defined by Xq(6) = E [/~{Bi(6)}] q ,
(3.3)
i
where the summation extends over all the cubes of the covering. For multifractal fields the generating function exhibits a scaling behavior over a wide range of scales, i.e.
)¢q(6 ) ~ 6 "lql ,
(3.4)
where in general the mass exponents ,c(q) depend non-linearly on q. In this study the multifractal permeability field is constructed with a multiplicative process [11-17]. These processes are simple models of multifractals which have been used in many fields of physics and geophysics. The process involves 212 weights w i/> 0, i = 1 . . . . . 2 t). The first two steps of the construction process are shown in fig. 5 for a two-dimensional process. The unit cube B(60) of size 6o is divided into 2 D identical cubes B~il(6~) of size 6~ = 6o/2 (here 6,, = 6o2 m) and each of these sub-cubes is assigned a weight w~, i = l . . . . . 2 z~. This process is then repeated on each s u b - c u b e : B(i)(61) is divided into 2 D cubes of size 62, which are assigned the weights w~wj, j = 1 . . . . . 2 °. This process is continued n times down scale until a scale 6,, = 2 " is reached. 6,, is the homogeneity scale - or inner scale - of the permeability field, i.e. the scale at which it becomes homogeneous. The weights obtained at scale 6,, are interpreted to be the permeabilities of the blocks of size 6,,. For illustration the permeability field obtained after 7 cascade steps with a two-dimensional multiplicative process is shown in fig. 6. For simplicity we choose to consider here a conservative multiplicative process, i.e. a process where the total measure is conserved in the construction. This is done by imposing on the weights the constraint Z wj = 1, which implies t h a t / x {B(6o) } remains equal to unity at each cascade step. It also implies that the measure of a cube B(6,,) takes the form
u { B ( ¢ . ) } -- ,46,) w(62).., w(6,.),
(3.5)
where w(6k) denotes a multiplier at scale 6 k (w(6k) can be equal to any of the 2 ° values of wj). To different cubes B(6m) therefore correspond different strings of multipliers w(61) w(62)--. W(6m).
~ I.e. the region of space where the permeability is non-zero.
A. Saucier / Permeability of multifractal porous media
387
n=O
8o=1
wl
w2 n=l
~1 = 112
w3
w4
wlwl
wlw2
w2wl
w2w2
wlw3
wlw4
w2w3
w2w4
w3wl
w3w2
w4wl
w4w2
w3w3
w3w4
w4w3
w4w4
82 = 1/4
n = 2
Fig. 5. The first two steps of construction of a deterministic multifractal two-dimensional permeability field (from top to bottom). R a n d o m multiplicative processes can also be constructed by making the weights random variables Wi, i = 1 . . . . . 2 °. The Wj's are independent from each other from one cascade level to the other, but are correlated at a given level. Indeed the conservation constraint Z Wj = 1 creates correlations between the Wj's. This random process builds a collection of permeability fields. Each realization of this process could be regarded as a block of material taken from a different location in some multifractal porous medium. Such random multifractal fields can be characterized by the scaling behavior of the expectation value of their generating function, which satisfies
( )(,q((~) ) ~ • r(q),
(3.6a)
A. Saucier / Permeability of multifractal porous media
388
Fig. 6. Two-dimensional permeability field constructed with 7 cascade steps and four different weights (w~ = 0.35, w 2 = 0.05, w3 = 0.15, wz = 0.45). Dark areas represent regions of low permeability while the bright areas are regions of high permeability.
where the mass exponents
T ( q ) a r e t h e n g i v e n b y [17]
~-(q) = - D - log2( ( W q ) ) .
3.2.
Homogeneity
(3.6b)
o f the e f f e c t i v e p e r m e a b i l i t y f u n c t i o n
T h e e f f e c t i v e p e r m e a b i l i t y k ( 6 ) o f a c u b i c r e g i o n o f size ~5 c a n b e e x p r e s s e d in t e r m s o f t h e M p e r m e a b i l i t i e s k ~ ) ( 6 , ) ( u = 1, 2 . . . . . M ) at t h e h o m o g e n e i t y s c a l e 6 , , i.e. t h e r e e x i s t s a f u n c t i o n F s u c h t h a t
k ( ~ ) = F(k(l)(t~n) . . . . . F is a h o m o g e n e o u s
r(ak~')(6,),
k(M)(t~n)) .
function such that for any real number
(3.7) a
. . . , t~k~M)(6,)) = a F ( k ~ l ) ( 6 , ) , . . . , k ~ M ) ( 6 , ) ) .
(3.8)
A. Saucier / Permeability of multifractal porous media
389
(3.8) follows from the linearity of Darcy's law. Indeed, given a pressure field p(x) that satisfies the continuity equation (2.2) with the boundary conditions of fig. 2, then for any real number a the field p(x) is also solution of V . [ak(x) Vp] = 0 with the same boundary conditions. Hence p(x) is invariant under the transformation k(x)--> ak(x). Using the differential expression of Darcy's law for the x-component of the velocity field, i.e. u = - [ k ( x ) / l x ] Oxp, and replacing it in the definition (2.1) of the effective permeability, yields
k~. = ~
1
(k(x) O . p ) . .
(3.9)
When k(x)--+ a k ( x ) , Oxp remains unchanged and therefore (3.9) implies that kcff---~ akcf,. (3.8) is the expression of this general result obtained in the special case where k(x) takes discrete values in M cubes. It is emphasized that (3.8) also holds for the effective permeability Fnetwork(k(l)(~n) k(M)(c~n) ) of capillary networks as a consequence of the linearity of the flow equations in tubes (or resistors). (3.8) implies in particular that the function f defined by (2.1) satisfies . . . . .
(3.10)
f ( a k l , . . . , ak2o ) = a f ( k I . . . . , k2o ) .
(3.10) also holds for the estimate of f obtained with the resistor network modelling. When the RSRG procedure is iterated n times the effective permeability k(60) obtained at the largest scale 60 can also be expressed in terms of the permeabilities k~V)(6n), ~, = 1 , . . . , 2 no, at the homogeneity scale 6n = 2-". Indeed, introducing the notation N(n)=-2 "°, k(6o) can be written in the form
k(6o) = f(k(')(61 ) , . . . , = f(f(k~l}(~2) .....
kN('))(6, )) k~N~'~(~2) ) . . . . .
f(k~N~2~-3~(~ 0 . . . . .
kCN~2~(~a)))
= etc... --= FRSR~(k(I)(t5 ), k(2)(~,) . . . . .
k(N("))(6~)) .
(3.11)
By definition FRsRc is an estimate of the exact effective permeability F. The nested structure shown in (3.11) and the property (3.10) of the function f imply that FRSRG also satisfies the homogeneity property (3.8). The homogeneity of the effective permeability therefore holds for Darcy flows, for flows in capillary networks and for the estimate FRSRG obtained with the RSRG method.
390
A. Saucier / Permeability of multifractal porous media
4. Effective permeability of deterministic multifractal permeability fields W e shall n o w s t u d y t h e effective p e r m e a b i l i t y o f the d e t e r m i n i s t i c m u l t i f r a c tal p e r m e a b i l i t y field o f s e c t i o n 3.1 as a f u n c t i o n o f scale. M o r e specifically, we a s k t h e f o l l o w i n g q u e s t i o n : G i v e n a cubic ball B x ( 8 ) o f size 6 c e n t e r e d on a p o i n t x, h o w d o e s t h e effective p e r m e a b i l i t y o f the m e d i u m e n c l o s e d in B x ( 8 ) v a r y with 8 w h e n x is fixed? A n d h o w d o e s it v a r y if 8 is fixed b u t w h e n x v a r i e s ? In t h e f o l l o w i n g k,,(6,,,) will d e n o t e the effective p e r m e a b i l i t y o f the p o r o u s m e d i u m c o n t a i n e d in Bx(8,, ,) for a p e r m e a b i l i t y field c o n s t r u c t e d with n cascade steps (here 8,,=2 " and m
k,, 0,,) = w ( 8 , ) . . , w( m)
v = 1,2 .....
N(n - m),
(4.1) w h e r e t h e kl,"),,,(8,,_,,,) a r e the p e r m e a b i l i t i e s at the h o m o g e n e i t y scale g e n e r a t e d b y t h e s a m e m u l t i p l i c a t i v e p r o c e s s , b u t with o n l y n - m c a s c a d e steps. S u b s t i t u t i n g (4.1) into (3.7) a n d using (3.8), we get k,,(8,,) = w(Si),
"" w ( 6 , , , ) F ( k l , 1 ' m ( 8 . . . . . ), . . . ,
k ,(u~, _ , , m)) (8,
m))"
(4.2)
But by definition k,, _ ,, (8 o) = F ( k , _ ,, ( ,, _ m ),
80=
(4.3)
- m ))
I
Wl
W2
51 = 1/2 Wl
etc...
W2
= = = =
Wl
=m
W2
=
v
Fig. 7. Here is represented a one-dimensional multiplicative process (D = 1) constructed with 4 cascade steps and two different weights w~ and w 2. The inner scale is 84 = (1/2)48o. If we consider the part of the permeability field enclosed in the horizontal bracket, it is seen that all the values of permeability in this interval share the same multipliers at scales 8~ and 82, namely w(8~)= w2 and w(82) = wt. The permeabilities in the horizontal bracket can therefore be written in the form k(f)(82) = w ( 6 , ) w(82) kl2")(82), where k(2")(82) is a permeability generated by only the part of the cascade process indicated by the vertical brackets, i.e. with only two cascade steps.
A. Saucier / Permeability of multifractal porous media
391
hence using (4.3), eq. (4.2) becomes kn(~m) = W ( ~ I ) ' ' ' W(~m) kn_m(¢5o) ,
(4.4)
and using (3.5) finally yields k . ( ~ , . ) =/~{B~(6m) } k. m(6O).
(4.5)
k,(60) is the effective permeability of the whole field constructed with n cascade steps. We will now show that kn(6o) can be determined by using a recurrence relation between k,+1(60) and kn(6o). In the following ~n will denote the permeability field constructed with n cascade steps. Firstly, 2 D copies of ~ , are produced and scaled down by a factor 1/2 (fig. 8a). Secondly, for each copy, # i , i = 1 . . . . ,2 °, all the permeabilities k{n~)(6,) at the homogeneity scale 6, are multiplied by w~. According to (3.8) the new effective permeabilities of these scaled-down copies are n o w wikn(t~o), i = 1 . . . . . 2 D. The new fields are next arranged in a 2 D array with the same spatial order as the wj's in the multiplicative process. This can be seen to yield the new permeability field ~,+1 (fig. 8b) constructed with n + 1 cascade steps. Thirdly (fig. 8c), the RSRG method is used to compute the effective permeability of ~,+~ according to k.+,(a0)
w2k.(a0) . . . . .
(4.6)
(3.8) then implies kn+l(~0) =f(w
1, w 2 . . . . .
WN(1)) k n ( ~ 0 ) ,
(4.7)
which is a simple renormalization equation for the effective permeability k,(6o). Iterating (4.7) yields = (flw,.
(4.8)
w2 .....
With ko(6o) = 1 and eliminating n using 6, = 2 ", (4.8) becomes k,(6o) =- 6~, ,
(4.9a)
where y = - l o g z ( f ( w l , w2 . . . . .
Wu~l))).
(4.9b)
(4.9a) shows that the effective permeability of the whole permeability field
392
A. Saucier / Permeability of multifractal porous media
kn(1)
k.O)
kn(1)
Xl/2
(a) kn(1)
Wl kn(1)
kn(1)
w2 kn(1) (b)
ws kn(1)
w4 kn(1)
RSRG
kn+l(1)
(c)
Fig. 8. ,~,, d e n o t e s a two-dimensional permeability field constructed with n cascade steps. (a) F o u r copies of 0% are p r o d u c e d and scaled d o w n by a factor 1/2. (b) F o r each copy # i , i = 1, 2, 3, 4, all the values of permeabilities are multiplied by w i. The new effective permeabilities are then w,k,,(1), i = 1, 2, 3, 4. The four new fields are next arranged in a 2 x 2 array, which yields the permeability field ,.%,,+~ constructed with n + 1 cascade steps. (c) The R S R G m e t h o d is used to c o m p u t e the new effective permeability according to k,,~ , ( 1 ) = g ( w t k , , ( 1 ), w~_k,,(1), w3k,,(1 ), w~k,,(1)).
scales with the inner scale 6,. According to (4.9b) the scaling exponent 3' is determined by the function f and by the 2 D weights w/#2 The ball Bx(6m) enclosing the porous medium is centered about a point x. For multifractal measures one usually defines a pointwise scaling exponent a(x) such that /x{Bx(6)) - 6 ~x) as 6 - + 0 [18, 19]. When the inner scale 6, is finite #2 M o r e details on the sensitivity of y to the choice of a specific tesselation can be found in the r e p o r t no. I F E / K R / F - 9 1 / 141.
A. Saucier / Permeability of multifractal porous media
393
this statement takes the approximate form /x{Bx(8)} ~ 8 ~(x) when 8 , < 8 becomes
(4.10)
< 6 o. Using (4.9a), (4.10) and 8 , _ m = 8,18,,,, eq. (4.5) finally
kn(Sm) ~ --m8 °t(x)-Y t] 7--n
(4.11)
when 8, < 8 < 80. (4.11) describes how the effective permeability of a piece of medium contained in Bx(8 ) varies with 8. kn(Sm) simply scales with 8 m for a fixed 8n. The scaling exponent is partly determined by the location x of the piece of medium, which accounts for a(x), and the scaling properties of the medium embodied by the 2 D weights w i, which account for - % The dependence on the homogeneity scale 8 n is also scaling with an exponent % This appears to be the first analytical derivation of the effective absolute permeability of a multifractal permeability field.
5. Effective permeability of random multifractal fields A permeability field can be correlated over wide ranges of scales, eventually in a multifractal manner, but random variations are expected from sample to sample, e.g. for blocks of porous medium coming from different locations. From this standpoint the effective permeability k ( L ) at scale L can be regarded as a random variable that depends on a length scale parameter L. We choose to characterize the random variations of k ( L ) by the scaling behavior of its moments, i.e. we consider the quantities ([k(L)] q) as a function of L and the real parameter q. It will be shown that for random multifractal permeability fields these moments satisfy ( [ k ( L )] q) ~ L ~q) ,
(5.1)
where ~(q) is a nonlinear function of q. In this section the formalism of section 4 is adapted to random multifractal permeability fields and the expression of ~(q) is derived using the R S R G method. We consider permeability fields generated with the random multiplicative process introduced in section 3.1. The random version of eq. (4.4) is simply
k.(Sm) L w(81). • • W(Sm) k. ~.(8,,),
(5.2)
394
A. Saucier / Permeability of multifractal porous media d
where " = " stands for equality in probability distribution and k n m(60) is a random variable independent of the W(6k)'s. Raising (5.2) to the power q and averaging yields ([k,,(6m)] q) = ( w q ) m ( [ k n _ m ( 6 o ) ] q ) .
(5.3)
Eliminating m in (5.3) with 6 m = 2 - " and using (3.6b) leads to
([k.,(6..)]o)
')+'~ ( [ k , _ , . ( 6 o ) ] q ) . = 6m
(5.4)
Similarly, the random version of (4.6) is
k.+,(6,,) ~ f(W,k.(6,,). W2k.(6,,) . . . . . WN~,~k.(6,,)).
(5.5)
Using the homogeneity of f, (5.5) becomes k,,+,(60) a=f(W,, W2, . . . , WN~,) ) k,(6,,),
(5.6)
where k,,(6o) is independent of the Wi's. Raising (5.6) to the power q and averaging yields ([k,,+,(6,,)] q) = ( [ f ( W ~ , W 2 , . . . , WN~i))]q)(Ik,,(6o)] q) ,
(5.7)
which is a simple renorrnalization equation for the moments of the effective permeability. Iterating (5.7) leads to ( [ k , ( 6 o ) ] q) = ([f(W,, W 2 , . . . ,
Wu~1))lq)"([k,,(6,,)] q) .
(5.8)
Using ko(6o) = 1 and eliminating n with 6, = 2 " yields (5.9a)
([k,,(60)] q) = 6~ (q) ,
where y( q) = - l o g 2 ( ( [ f ( W 1 , W 2 . . . .
, Wu(1))]q)).
(5.9b)
Replacing (5.9a) in (5.4) and using 6,_ m = ~,/6,,, finally gives ( [ k n ( 6 m ) ] q) = 6D+T(q)-'r(q)6"£ (q) .
(5.101
The scaling exponents of the effective permeability, as defined by (5.1), are
A. Saucier / Permeability of multifractal porous media
395
therefore given by sO(q) = D + ~-(q) - 3'(q).
(5.11)
Hence a multiscaling permeability field gives rise to a multiscaling effective permeability field. It is interesting to discover that the mass exponents r(q) determine directly-although not completely-the permeability scaling exponents ~:(q). The permeability exponents y ( q ) are not trivially related to the mass exponents r(q) because f is a non-linear function. (5.10) appears to be the first analytical derivation of the effective permeability of random multifractal permeability fields.
6. Conclusions
We have studied the scaling properties of the effective absolute permeability of multifractal permeability fields generated by multiplicative processes. In the deterministic case, the permeability k(6) of a region of size 6 centered about a point x was found to scale according to k(6) - 6,~x)-y,
(6.1)
where oz(x) is the local pointwise scaling exponent of the permeability field, and y is an exponent determined by the weights used in the multiplicative process. In the random case, the permeability k(~) is a scale dependent random variable that was shown to satisfy
([k(6 )] q) ~ 6o+,~q)-~,~q)
(6.2)
where ~-(q) is the order-q mass exponent of the multifractal permeability field and y ( q ) is another nonlinear function determined by the joint probability distribution of the weights used in the multiplicative process. It is emphasized that both (6.1) and (6.2) are approximate results derived with the real space renormalization group method. It is interesting to discover that the effective transport properties of the porous media are determined, via (6.2), by the multifractal spectrum of the permeability field. From the standpoint of oil reservoir modelling, (6.2) allows to relate the statistics (e.g. variance) of the permeabilities measured at one scale, say the core plug scale, to the statistics of the effective permeabilities at the scale of the grid blocks used in the flow simulators. A proper description of the scaling properties of the permeability field in oil reservoirs is therefore
396
A. Saucier / Permeability of multifractal porous media
necessary to g e n e r a t e realistic effective permeabilities and c o n s e q u e n t l y to predict m o r e accurately the large scale flow. M e a s u r i n g b o t h ~-(q) and y ( q ) requires three-dimensional information a b o u t the permeability field. Such data is usually not directly available for oil reservoirs since most o f the information c o m e s f r o m one-dimensional vertical cuts t h r o u g h the reservoir, i.e. wells. R e c o v e r i n g three-dimensional information f r o m o n e - d i m e n s i o n a l cuts is a nontrivial p r o b l e m where the k n o w l e d g e of the anisotropy, stratification and h e t e r o g e n e i t y of the reservoir properties plays an i m p o r t a n t role. This p r o b l e m might not be hopeless if the permeability field was locally isotropic, or else if the anisotropy could in some way be characterized. T h e role of a n i s o t r o p y in fluid flow t h r o u g h p o r o u s media and the possibility o f extracting information about anisotropy from one-dimensional cuts are t h e r e f o r e questions of interest for reservoir characterization.
Acknowledgements I t h a n k Jiri Muller for stimulating discussions, e n c o u r a g e m e n t and for his careful editing of the manuscript. This work is part of a c o m m o n project b e t w e e n the Institutt for E n e r g i t e k n i k k , Fina Exploration N o r w a y and Petrofina.
References [1] P.R. King, The use of renormalization for calculating effective permeability, Transp. Porous Media 4 (1989) 37-58. [2] B.B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1983). [3] J.L. McCauley, J. Muller and G. Saether, Multifractal spectra of large scale oil-reservoir properties, in: Correlations and Connectivity, J.E. Stanley and N. Ostrowsky, eds. (Kluwer, Dordrecht, 1990) p. 310. [4] J. Muller and J.L. McCauley, Implication of fractal geometry on fluid flow properties of sedimentary rocks, Transp. Porous Media, in press. [5] N. Shah and J.M. Ottino, Effective transport properties of disordered, multiphase composites: application of real-space renormalization group theory, Chem. Eng. Sci. 41 (1986) 283-296. [6] J. Bernasconi, Real-space renormalization of bond-disordered conductance lattices, Phys. Rev. B l g (1978) 2185-2191. [7] S. Mohanty and M.M. Sharma, A recursive method for estimating single and multiphase permeabilities, SPE 20477, 65th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in New Orleans, LA, September 23-26, 1990, pp. 127-136. [8] P.R. King, A.H. Muggeridge and W.G. Price, Renormalization calculations of immiscible flow, Transp. Porous Media, submitted in August 1991. [9] A.O. Grindheim and J.O. Aasen, An evaluation of homogenization techniques for absolute
A. Saucier / Permeability of multifractal porous media
[10] [11] [12] [13] [14] [15] [16] [17] [18]
[19]
397
permeability, presented at Lerchendal Petroleum Engineering Workshop. The Norwegian Institute of Technology, Trondheim (1991). A. Aharony, E.L. Hinrichsen, A. Hansen, J. Feder, T. Jcissang and H.H. Hardy, Effective renormalization group algorithm for transport in oil reservoirs, Physica A 177 (1991) 260-266. A.M. Yaglom, Effect of fluctuations in energy dissipation rate on the form of turbulence characteristics in the inertial subrange, Dokl. Akad. Nauk SSSR 166 (1966) 49-52. B.B. Mandelbrot, Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier, J. Fluid Mech. 62 (1974) 331-358. C. Meneveau and K.R. Sreenivasan, Simple multifractal cascade model for fully developed turbulence, Phys. Rev. Lett. 59 (1987) 1424. D. Schertzer and S. Lovejoy, Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes, J. Geophys. Res. 92 (1987) 9693-9714. P. Meakin, Diffusion-limited aggregation on multifractal lattices: a model for fluid-fluid displacement in porous media, Phys. Rev. A 36 (1987) 2833-2837. E.A. Novikov, The effects of intermittency on statistical characteristics of turbulence and scale similarity of breakdown coefficients, Phys. Fluids A 2, No. 5 (1990). A. Saucier, Cascade processes and fully developed turbulence, Ph.D. dissertation, McGill University (Physics Department), Montr6al, Canada (1991). C.H. Halsey, M.H. Jensen, L.P./Kadanoff, I. Procaccia and B.I. Shraiman, Fractal measures and their singularities: The characterization of strange sets, Phys. Rev. A 33 (1986) 1141-1151. J.D. Farmer, E. Ott and J.A. Yorke, The dimension of chaotic attractors, Physica D 7 (1983) 153-180.