Page 24 - Percolation Models for Transport in Porous Media With

1.2 RANDOM CONDUCTIVITIES 15

Since the formulas obtained in this work assume no interflows between the
parallel conducting chains, the question arises about the accuracy of the results
obtained.
Note that in the case of equal conductivities for all conducting channels, formu-
las of the (1.11) type turn into critical percolation relations E""' K""' (Pb- P~)d.
It was shown in [29] that relations of this sort describe with good accuracy (10
to 20%) the change of network conductivity near the critical point for the fol-
lowing range of the bond conductivity probability, P~ ~ pb ~ P~ + t::,.pb, where
t::,.pb ~ 0.1. For a given distribution of conducting bonds with respect to values
of their intrinsic conductivities, verification of formula (1.11) with a numerical
experiment is necessary.
Numerical experiment. In order to verify the fundamental relationship
(1.11) obtained above, its results were compared to those of the following numerical
experiment. Consider stationary flow in an arbitrary medium described by the
elliptic equation
div(uVQ>) = 0. (1.12)
The equation determining the distribution of potential 4> in a square network
with a period l

2
l- [ui+l/2,j(l/>i+l,j- 1/>i,j)- D"i-1/2,j(l/>i,j- 1/>i-l,j)]+ (1.13)
2
+l- [ui,j+lf2(4>i,j+l- 1/>i,j)- ui,j-lf2(4>i,j- 1/>i,j-1}] = 0
is the difference analog of the relationship (1.12). Here i = 1, 2, ... , N;
j = 1, 2, ... , N are natural numbers that locate the site in the network; D"i±l/2,j±l/2
is the intrinsic conductivity of the bond at the point i ± 1/2,j ± 1/2 (set by a ran-
dom number generator); and 1/>i,j is the value of the potential at the point i,j
(fig. 2). Conductivities of bonds were assigned by a random number generator
provided their probability density was determined by a given function fo(u). The
boundary conditions were as follows

1/>o,j = 1, 1/>N,j = 0, (1.14)
1/>i,N- 1/>i,N-l = 0, 1/>i,l - l/>i,2 = 0. (1.15)

The condition (1.15) requires no flow through the boundary of the network in
the jth direction, and the condition (1.14) indicates constancy of the potential on
those boundaries of the network, through which the flow comes.
The equation (1.13) has been solved numerically using the relaxation method.
To speed up calculations, initial distribution was set to be

1/>i,i = 1- ifN.