removed by the 1P irreducibility condition. Due to the diffusive nature of the original equation, we can expect
that for large times and large lengthscales, the mean pres- sure will have typical variations over lengthscales given
by eqn 16 very much larger than S
↔
r9
, t9
. We can
thus approximate =hPir ¹ r9, t ¹ t9 by =hPir, t in eqn 25 under the integral sign, producing the following
approximation: ] P
h i r
, t
]t ¼ =
· D
= P h i
r
, t
þ =
r
: X
↔
q ¼ 0
, s ¼ 0
:=
r
P h i
r
, t
þ Q
Fc
t
d r
d t
26
This equation is a diffusion-like equation corresponding to a large time and large distance approximation of the origi-
nal integro-differential equation. Once again, it appears an effective diffusion tensor equal to
D
eff
¼ D
1 þ
X
↔
q ¼ 0
, s ¼ 0
¼ D
1
þ h H
↔
i
q ¼ 0
, s ¼ 0
27 We used eqn 23 to obtain the last equality, which shows
that this result is consistent with eqns 16 and 17 of the effective diffusivity tensor obtained using the second-
moment approach. Such an equality could not be easily obtained without using eqn 25.
Another interesting physical feature of eqn 25 is that the vectorial
quantity V
r, t
defined by
V r
, t
¼ ¹
D =
r9
P h i
r ,
t ¹
R
t
dt R
dr9 S
↔
r9
, t9
: =
r9
P h i
r ¹ r9
, t ¹ t9
can itself be interpreted as the average flux vector at point r at time t
12
. This corresponds to a generalized form of Dar- cy’s law in which the local flow rate is a linear function
operating on the whole pressure gradient field variations. Using this interpretation, eqn 25 appears to be no more
than the usual transient mass balance equation associated with this flux, i.e.
] P h i
r
, t
]t ¼ ¹ =
r
: V
r ,
t þ
Q Fc
t
d r
d t
The last approximation leading to eqn 26 thus becomes very easy to interpret using the preceding expression of the
flux Vr, t.
Another interesting property of the kernel S
↔
r9, t9 is that, owing to its more localized nature than the original H
↔
tensor, eqn 25 must be robust when considering finite size domains implying Dirichlet or Neuman boundary con-
ditions. In particular, it is reasonable to think that if the boundaries local radii of curvature are greater than the cor-
relation length l
c
than eqn 25 could still work, although it was derived for an infinite medium. The preceeding results
and the general framework which we have set up show that this affirmation is probably correct. A rigorous derivation
would certainly involve a considerable amount of work. Analogous conclusions were given by Rubin and Dagan
by means of a second-order perturbation expansion
15
.
7 A SECOND-ORDER CALCULATION
When computing the S
↔
kernel up to the order j
2
, we only need compute the value of the graph
. We obtain X
↔
r9
, t9
j
2
==P r9
, t9
C r9
28 where j
2
Cr9 denotes the covariance function of the
permeability field which is assumed to be isotropic, with Cr9 ¼ 0 ¼ 1. Computing the FLT of S
↔
, for q ¼ 0 and
s ¼ 0, using Parseval’s equality and the isotropy hypothesis, in two dimensions, we obtained
X
↔
q ¼ 0
, ¼ ¹
1 2D
j
2
1
29 Using eqn 21, the effective diffusion tensor is thus given
by
D
eff
¼ D
1 ¹
1 2D
j
2
1 30
This is a well-known result, as it corresponds to the steady- state second-order expression for the effective permeability
of a heterogeneous stationary and isotropic medium
11,16
. More generally, a wave vector dependant effective dif-
fusivity may be introduced by the definition D
eff
q, s ¼ 0 ¼
D 1 þ
S
↔
q,s ¼ 0 which corresponds to the static limit. For large wave vectors q, one can compute q.S
