Page 18 - Percolation Models for Transport in Porous Media With

1.1 BASIC CONCEPTS 9

Table 1.1:

pb
Network Type z I c I P~z
Plane
Square 4 0.5 2.0
Triangular 6 0.35 2.1
Hexagonal 3 0.75 2.0
Solid
Simple Cubic 6 0.25 1.5
Body Centered Cubic 8 0.18 1.4
Face-Centered Cubic 12 0.12 1.4
Diamond Type 4 0.39 1.6

works of finite dimensions, the results given by percolation theory are valid only
for networks with sufficiently large numbers of sites (N ~ 10 ). In this case the
4
size of the network can be considered macroscopic with respect to the size of an
elementary cell, and the percolation threshold is defined as the limit of the mean
value of the percolation threshold, as the number of sites in the network goes up.
We shall now briefly present the major results of percolation theory.
Percolation threshold. Let P" characterize the probability of conductivity
in a bond between any two sites, and P 8 , the probability of conductivity in the
sites. Then if P" ~ P~, where P~ is the threshold value of the conductivity
probability, then an IC is formed in the network. If P" < P~, then there is no
IC and the conductivity vanishes. The quantity P~ depends on z, the number of
closest neighbors of a site in the network, and on the dimension D of the network,
i.e., it depends on the network type. With good precision, the following invariant
can be indicated,

P:z = Df(D -1). (1.1)
The values of P:z for different network types are given in table 1.1.
A similar result is obtained when the conductivity of the network is considered
in terms of site percolation. If the probability P 8 of the site conductivity satisfies
the condition ps ~ P;, where P; is the threshold conductivity probability of a
site, then an IC is formed in the network.
The quantity P; depends on f, the charge coefficient, i.e., on the network type.
The charge coefficient equals the fraction of the volume covered by a set of balls
constructed around each site of the network with radius equal to half the distance