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Scaling of the effective permeability in multifractal porous media
Physica A 191 (1992) North-Holland
289-294
Scaling the effective multifractal media Antoine
in
Saucier
Institutt for energiteknikk,
Box 40, N-2007 Kjeller, Norway
The real space renormalization group method exponents of the effective absolute permeability
is used to calculate analytically in multifractal porous media.
the scaling
1. Introduction
The understanding of fluid flow properties of 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 and these variations occur over wide ranges of scales. In order to reduce the amount of computations the permeability field is usually homogenized before one solves directly the fluid flow equation by numerical methods. Homogenization is an averaging process in which a large array of small scale permeabilities is replaced by a smaller array of lager 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 function k(L) is determined by the structure of 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 [l] 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 [2] and for the pore space geometry as measured on thin sections of core plugs [3]. 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 0378-4371/92/$05.00
0
of multifractal permeability fields. 1992 - Elsevier
Science
Publishers
B.V. All rights
reserved
290
A. Saucier
2. Permeability
I Permeability in multifractal porous media
of deterministic
multifractal
porous media
We shall first recall briefly the main results obtained by Saucier [4,5]. A permeability field constructed with a deterministic multiplicative process involving N= AD weights wl, w2,. . . , wN is considered. For simplicity the process is chosen to be conservative, i.e. C wj = 1. D is the dimension of the space (D = 1,2,3) and A the scale ratio of the multiplicative process (A = 2,3,4, . . .). The permeability field is constructed with II cascade steps. For each cube of a grid of mesh 6, = A-“??, we can define an effective permeability k~‘(6,), i = 1,2,. . ) A”. By definition ky’(6,) is the effective permeability of the ith cube of size 8, for a permeability field constructed with IZcascade steps. The boundary conditions are the same for each cube, i.e. impermeable walls in the x-direction and constant pressure along the walls perpendicular to the x-direction. It was shown by Saucier [4,5] that
q@,) = p”‘(6,) k,-,(a())
(2.1)
)
where ~“‘(6,) is the measure of the cube of size 6, and k,_,(6,) is the permeability of the whole permeability field of size 6, constructed with y1- m cascade steps. Using the real space renormalization group method (in short RSRG) for the calculation of the effective permeability [6], it was shown [4,5] that /~,(a,) satisfies the recurrence relation
kx+l(%)= f(w,k(4A WA(~“O)Y . . . >WNk(%))>
(2.2)
where f is a nonlinear homogeneous function. An analytical approximation off can be obtained with a resistor network modelling of the flow. f being a homogeneous function, eq. (2.2) becomes
kz+,(%)=f(w1, wz>.. . >WN)U%) >
(2.3)
which leads directly to (with &(a,) = 1, 6, = 1, 6, = A-“) (2.4a) where 71 =
-l%Lf(W,, wz, . . .
9
w,>l
.
(2.4b)
If the box of size S,,, is centered about a point X, the usual pointwise scaling p(6,) - s;Cx) is obtained. Using pointwise scaling and the result (2.4a) (and using a,-, = 8,/S,), eq. (2.1) becomes
A. Saucier
k,(S,)
=
I Permeability in multifractal porous media
s$)-~ s; .
291
(2.5)
It is emphasized that eq. (2.5) is a pointwise expression of the scaling of the effective permeability. We will now use eq. (2.1) to derive properties of the spatial average of the effective permeability. Raising eq. (2.1) to the power q, summing over all the N(6,) = 8,” boxes of size 6, and dividing by N(6,) yields
(2.6) On the left-hand side of eq. (2.6) appears a spatial average of [kf’(S,)]’ that will be denoted by ([k,(S,)]q),, and on the right-hand side the summation is the usual generating function x,(6,) of the multifractal permeability field. Using eq. (2.4a) and x,(8,) = 8zq), eq. (2.6) becomes ([k,(sm)]“)s
= Ss;‘”sfpq+T(q) .
(2.7)
By contrast with the pointwise result (2.5), eq. (2.7) describes the scaling of the order-q moment of the spatially averaged permeabilities kz’(6,). We conclude that a multifractal permeability field gives to a multiscaling effective permeability. Indeed, the scaling exponents of ([k,(S,)]“)s are
%(4)= D
-x4
+ T(4)
(2.8)
and are nonlinear in q since I is nonlinear in general. It is interesting to note that the mass exponents r(q) determine directly the permeability scaling exponents e( q).
3. Permeability of random multifractal porous media A random multifractal permeability field can be constructed by making the weights random variables Wi, j = 1,2, . . . , AD, satisfying the same conservation constraint C Wj = 1. Eq. (2.1) still holds and an ensemble average (denoted here by (. . .)) leads to [4,5]
wz@m)lq) = ~fj:+“q’([k-,(~o)lq) * The calculation of ( (k,(6,))q) kt’(6,) denotes a realization
(3.1)
is more difficult than in the deterministic case. If of the random variable k,(6,), then the RSRG
292
A. Saucier
method
I Permeability in multifractal porous media
leads to
where “2” stands for equality in probability distribution and where the random variables ki’(6,) are independent, identically distributed and independent of the Wj’s. The klf’(S,)‘s being in general different, the homogeneity of f cannot be used directly as in the deterministic case, which complicates the resolution of eq. (3.2). We may however consider a special case where all the ki)(6,,) are equal. This corresponds to a multiplicative process where the Wj’s are strongly correlated spatially. In this case we can take advantage of the homogeneity of f. Indeed, if k,(6,) d enotes the common value of the ki’($,), then (3.2) becomes
k+l(~“)
%Wl,w*,. . .
where &(a,,) is independent averaging yields
3
WN)k(%) >
(3.3)
of the WI’s Raising (3.3) to the power
q
and
which is a simple renormalization equation for the moments of the effective permeability, which leads to (using &(a,) = 1, S,, = 1, S,, = A-“) (3Sa) where (3Sb) Replacing
(3Sa) in (3.1) and using a,_,,, = S,,/S,,, finally gives
([k,(6,)]q)
=
gyjf-y(~)+Q)
The scaling exponent
.
of the effective permeability
(3.6) is therefore
given by (3.7)
It is emphasized that (3.7) holds for a special case of random multiplicative processes where the multipliers Wj are strongly correlated spatially. Comparing
A. Saucier I Permeability in multifractal porous media
293
eqs. (2.8) and (3.7), it is seen that randomness introduces an additional nonlinearity in t(q) via the nonlinear function y(q). The term corresponding to y(q) in the deterministic case (eq. (2.8)) is ‘ylq and is linear in q.
4. Conclusions We have studied the scaling properties of the effective absolute absolute permeability of multifractal permeability fields generated by multiplicative processes. In the deterministic case, the permeability k,,(&,,) of a region of size 6, centered about a point x scales according to
(4.1) where a(x) is the pointwise scaling exponent of the permeability field, and ‘y, is an exponent determined by the weights used in the multiplicative process. The spatial average ( [k,(S,)]q)S of the effective permeability was shown to scale according to ([k,(6,)]q)S
= 8;”
8;-y1q+T(q)
,
(4.2)
where r(q) is the order-q mass exponent of the permeability field. In the random case, the permeability k,(S,) is a scale dependent random variable. For a special case of random multiplicative process, it was shown that ([k#J]q)
=
a;(q) SE--y(q)+‘(q) ,
(4.3)
where y(q) is another nonlinear function determined by the joint probability distribution of the weights used in the multiplicative process. It is emphasized that eqs. (4.1), (4.2) and (4.3) are approximate results derived with the real space renormalization group method. It is interesting to note that the effective transport properties of the porous media are determined, via (4.2) and (4.3), by the multifractal spectrum of the permeability field. From the standpoint of oil reservoir modelling, (4.2) and (4.3) allow 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 necessary to generate realistic effective permeabilities and consequently to predict more accurately the large scale flow.
294
A. Saucier
I Permeability in multifractal porous media
Acknowledgements
I thank Jiri Muller for stimulating discussions and for his careful editing of the manuscript. This work was supported by Fina Exploration Norway.
References [l] B.B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1983). [2] J.L. McCauley, J. Muller and G. Saether, Multifractal spectra of large scale oil-reservoir properties, in: Correlations and Connectivity, H.E. Stanley and N. Ostrowsky, eds. (Kluwer, Dordrecht, 1990) pp. 310-312. [3] J. Muller and J.L. McCauley, Transp. Porous Media 8 (1992) 133. [4] Report IFEIKRIF-91I141. [S] A. Saucier, Physica A 183 (1992) 381. [6] P.R. King, Transp. Porous Media 4 (1989) 37.
289-294
Scaling the effective multifractal media Antoine
in
Saucier
Institutt for energiteknikk,
Box 40, N-2007 Kjeller, Norway
The real space renormalization group method exponents of the effective absolute permeability
is used to calculate analytically in multifractal porous media.
the scaling
1. Introduction
The understanding of fluid flow properties of 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 and these variations occur over wide ranges of scales. In order to reduce the amount of computations the permeability field is usually homogenized before one solves directly the fluid flow equation by numerical methods. Homogenization is an averaging process in which a large array of small scale permeabilities is replaced by a smaller array of lager 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 function k(L) is determined by the structure of 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 [l] 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 [2] and for the pore space geometry as measured on thin sections of core plugs [3]. 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 0378-4371/92/$05.00
0
of multifractal permeability fields. 1992 - Elsevier
Science
Publishers
B.V. All rights
reserved
290
A. Saucier
2. Permeability
I Permeability in multifractal porous media
of deterministic
multifractal
porous media
We shall first recall briefly the main results obtained by Saucier [4,5]. A permeability field constructed with a deterministic multiplicative process involving N= AD weights wl, w2,. . . , wN is considered. For simplicity the process is chosen to be conservative, i.e. C wj = 1. D is the dimension of the space (D = 1,2,3) and A the scale ratio of the multiplicative process (A = 2,3,4, . . .). The permeability field is constructed with II cascade steps. For each cube of a grid of mesh 6, = A-“??, we can define an effective permeability k~‘(6,), i = 1,2,. . ) A”. By definition ky’(6,) is the effective permeability of the ith cube of size 8, for a permeability field constructed with IZcascade steps. The boundary conditions are the same for each cube, i.e. impermeable walls in the x-direction and constant pressure along the walls perpendicular to the x-direction. It was shown by Saucier [4,5] that
q@,) = p”‘(6,) k,-,(a())
(2.1)
)
where ~“‘(6,) is the measure of the cube of size 6, and k,_,(6,) is the permeability of the whole permeability field of size 6, constructed with y1- m cascade steps. Using the real space renormalization group method (in short RSRG) for the calculation of the effective permeability [6], it was shown [4,5] that /~,(a,) satisfies the recurrence relation
kx+l(%)= f(w,k(4A WA(~“O)Y . . . >WNk(%))>
(2.2)
where f is a nonlinear homogeneous function. An analytical approximation off can be obtained with a resistor network modelling of the flow. f being a homogeneous function, eq. (2.2) becomes
kz+,(%)=f(w1, wz>.. . >WN)U%) >
(2.3)
which leads directly to (with &(a,) = 1, 6, = 1, 6, = A-“) (2.4a) where 71 =
-l%Lf(W,, wz, . . .
9
w,>l
.
(2.4b)
If the box of size S,,, is centered about a point X, the usual pointwise scaling p(6,) - s;Cx) is obtained. Using pointwise scaling and the result (2.4a) (and using a,-, = 8,/S,), eq. (2.1) becomes
A. Saucier
k,(S,)
=
I Permeability in multifractal porous media
s$)-~ s; .
291
(2.5)
It is emphasized that eq. (2.5) is a pointwise expression of the scaling of the effective permeability. We will now use eq. (2.1) to derive properties of the spatial average of the effective permeability. Raising eq. (2.1) to the power q, summing over all the N(6,) = 8,” boxes of size 6, and dividing by N(6,) yields
(2.6) On the left-hand side of eq. (2.6) appears a spatial average of [kf’(S,)]’ that will be denoted by ([k,(S,)]q),, and on the right-hand side the summation is the usual generating function x,(6,) of the multifractal permeability field. Using eq. (2.4a) and x,(8,) = 8zq), eq. (2.6) becomes ([k,(sm)]“)s
= Ss;‘”sfpq+T(q) .
(2.7)
By contrast with the pointwise result (2.5), eq. (2.7) describes the scaling of the order-q moment of the spatially averaged permeabilities kz’(6,). We conclude that a multifractal permeability field gives to a multiscaling effective permeability. Indeed, the scaling exponents of ([k,(S,)]“)s are
%(4)= D
-x4
+ T(4)
(2.8)
and are nonlinear in q since I is nonlinear in general. It is interesting to note that the mass exponents r(q) determine directly the permeability scaling exponents e( q).
3. Permeability of random multifractal porous media A random multifractal permeability field can be constructed by making the weights random variables Wi, j = 1,2, . . . , AD, satisfying the same conservation constraint C Wj = 1. Eq. (2.1) still holds and an ensemble average (denoted here by (. . .)) leads to [4,5]
wz@m)lq) = ~fj:+“q’([k-,(~o)lq) * The calculation of ( (k,(6,))q) kt’(6,) denotes a realization
(3.1)
is more difficult than in the deterministic case. If of the random variable k,(6,), then the RSRG
292
A. Saucier
method
I Permeability in multifractal porous media
leads to
where “2” stands for equality in probability distribution and where the random variables ki’(6,) are independent, identically distributed and independent of the Wj’s. The klf’(S,)‘s being in general different, the homogeneity of f cannot be used directly as in the deterministic case, which complicates the resolution of eq. (3.2). We may however consider a special case where all the ki)(6,,) are equal. This corresponds to a multiplicative process where the Wj’s are strongly correlated spatially. In this case we can take advantage of the homogeneity of f. Indeed, if k,(6,) d enotes the common value of the ki’($,), then (3.2) becomes
k+l(~“)
%Wl,w*,. . .
where &(a,,) is independent averaging yields
3
WN)k(%) >
(3.3)
of the WI’s Raising (3.3) to the power
q
and
which is a simple renormalization equation for the moments of the effective permeability, which leads to (using &(a,) = 1, S,, = 1, S,, = A-“) (3Sa) where (3Sb) Replacing
(3Sa) in (3.1) and using a,_,,, = S,,/S,,, finally gives
([k,(6,)]q)
=
gyjf-y(~)+Q)
The scaling exponent
.
of the effective permeability
(3.6) is therefore
given by (3.7)
It is emphasized that (3.7) holds for a special case of random multiplicative processes where the multipliers Wj are strongly correlated spatially. Comparing
A. Saucier I Permeability in multifractal porous media
293
eqs. (2.8) and (3.7), it is seen that randomness introduces an additional nonlinearity in t(q) via the nonlinear function y(q). The term corresponding to y(q) in the deterministic case (eq. (2.8)) is ‘ylq and is linear in q.
4. Conclusions We have studied the scaling properties of the effective absolute absolute permeability of multifractal permeability fields generated by multiplicative processes. In the deterministic case, the permeability k,,(&,,) of a region of size 6, centered about a point x scales according to
(4.1) where a(x) is the pointwise scaling exponent of the permeability field, and ‘y, is an exponent determined by the weights used in the multiplicative process. The spatial average ( [k,(S,)]q)S of the effective permeability was shown to scale according to ([k,(6,)]q)S
= 8;”
8;-y1q+T(q)
,
(4.2)
where r(q) is the order-q mass exponent of the permeability field. In the random case, the permeability k,(S,) is a scale dependent random variable. For a special case of random multiplicative process, it was shown that ([k#J]q)
=
a;(q) SE--y(q)+‘(q) ,
(4.3)
where y(q) is another nonlinear function determined by the joint probability distribution of the weights used in the multiplicative process. It is emphasized that eqs. (4.1), (4.2) and (4.3) are approximate results derived with the real space renormalization group method. It is interesting to note that the effective transport properties of the porous media are determined, via (4.2) and (4.3), by the multifractal spectrum of the permeability field. From the standpoint of oil reservoir modelling, (4.2) and (4.3) allow 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 necessary to generate realistic effective permeabilities and consequently to predict more accurately the large scale flow.
294
A. Saucier
I Permeability in multifractal porous media
Acknowledgements
I thank Jiri Muller for stimulating discussions and for his careful editing of the manuscript. This work was supported by Fina Exploration Norway.
References [l] B.B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1983). [2] J.L. McCauley, J. Muller and G. Saether, Multifractal spectra of large scale oil-reservoir properties, in: Correlations and Connectivity, H.E. Stanley and N. Ostrowsky, eds. (Kluwer, Dordrecht, 1990) pp. 310-312. [3] J. Muller and J.L. McCauley, Transp. Porous Media 8 (1992) 133. [4] Report IFEIKRIF-91I141. [S] A. Saucier, Physica A 183 (1992) 381. [6] P.R. King, Transp. Porous Media 4 (1989) 37.