Page 21 - Percolation Models for Transport in Porous Media With

12 CHAPTER 1. PERCOLATION MODEL

The critical parameter found from numerous computational analyses is
(3 = { 0.4 in the three-dimensional case,
0.14 in the two-dimensional case.
The contribution to the function W is made by all the sites which belong to the
"skeleton" and the "dead ends" of the IC. Relationships (1.3) and (1.6) imply that
near the percolation threshold, the number of sites and bonds of the "skeleton"
of the IC are negligible compared to the total number of sites and bonds inside
the IC. In other words, the principal part of the IC is concentrated in the "dead
ends," which do not affect conductivity.

1.2 Conductivity of a Network with the Random

Distribution of Elements

The existing percolation models can be applied only if the conducting elements
(bonds in the network) are homogeneous. That is, it is assumed in these models
that the intrinsic conductivities of all elements in the network are equal. However
this approach does not work for many kinds of heterogeneous media, e.g., oil and
gas reservoirs. It fails because various groups of structural elements in a medium
with intrinsic conductivities differing by several orders can make a commensurate
contribution to the effective conductivity of a heterogeneous medium. We suggest
a model which allows to describe conductivity of a heterogeneous medium with the
given intrinsic conductivity distribution of structure elements. In essence, it is a
generalization of the Shklovsky - de Gennes model. Our model is able to describe
the case when the network contains conducting elements randomly distributed
with respect to values of intrinsic conductivities.
Consider the network model of a heterogeneous medium whose sites are con-
nected with bonds with different conductivities u. From now on, the general term
"conductivity," if not otherwise specified, will be used to describe both the hydro-
conductivity, or permeability, and the electric conductivity, since the logic behind
the construction of models for the two processes is absolutely the same. Let the
period of the network equall. Suppose that the values of the conductivities (or
values of the parameter which determines them, i.e., radius of the section) of
the bonds are distributed randomly in the network and are characterized by the
distribution function fo(u) which satisfies the normalization condition,
00
J fo(u) du = 1
0
Since fluid conductivity of a conducting bond is ,... r 4 and its electric conduc-
tivity is ,... r , the distribution of bonds with respect to conductivities can also be
2