Page 16 - Percolation Models for Transport in Porous Media With

Chapter 1

Percolation Model of Micro

Heterogeneous Media

Conductivity of a medium (coefficients of permeability and electric conductivity)
depends significantly on the pore space structure. In the case of stochastic distri-
bution of conducting channels in the medium, it is possible to describe the topology
of the pore space in terms of the percolation theory [25-27, 29, 30]. However the
existing percolation models can be applied only if the conducting structural bonds
in the medium are sufficiently homogeneous. This is due to the fact that all the
theoretical relationships in percolation theory were obtained under the assump-
tion that there are only two types of structural bonds in the medium, namely the
conducting and the non-conducting ones. It is also assumed that the intrinsic con-
ductivities of all conducting bonds are equal. At the same time, in the majority of
actual media, a commensurate contribution to effective conductivity can be made
by groups of conducting bonds whose intrinsic conductivities are notably different.
Rocks, which may have many different types of pore space structure, represent an
example of such media.

1.1 Percolation Theory. Basic Concepts

Percolation theory and a number of its applications to various problems of math-
ematical physics are presented in enough detail in the reviews [31-36]. We shall
now mention only the basic ideas of percolation theory, those which we will need
in the future, as we build models to describe conductivities of media with different
types of pore space structure.
Consider specifically the problem of flow through a periodic network (we can
consider solid and intersecting spheres, ellipses, covering graphs, or continual flow).

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