Page 17 - Percolation Models for Transport in Porous Media With

8 CHAPTER 1. PERCOLATION MODEL

The network consists of sites and conducting bonds between them. Obviously, if
all bonds in the network are broken (i.e., do not conduct), then its conductivity
vanishes. As the concentration of conducting bonds goes up, the latter begin to
merge and form clusters, i.e., conducting unions of bonds. Starting from a certain
threshold value of conducting bond concentration, the bonds begin to form an
infinite conducting cluster (IC), and the conductivity of the network becomes non-
zero. The density of the IC and correspondingly, its conductivity goes up with the
further increase of conducting bond concentration. Quantitative description of the
threshold value for concentration of bonds for different types of networks, as well
as the correlation between the conductivity of a network and the concentration of
bonds, is given by percolation theory.
Thus percolation theory studies formation of connected domains (clusters) from
elements with certain properties, provided that every bond of each element with
another one is arbitrary (though established in a strictly defined way). It is clear
that the phenomena described by percolation theory belong to the so-called crit-
ical processes which are characterized by a particular critical point each. When
this critical point is reached, the principal property of the system, as far as the
process in question is concerned, changes fundamentally. Formation of an IC is in
essence a phase transition of the second kind, which is quantitatively character-
ized by a set of universal critical parameters. The universality of these parameters
means that they do not depend on the specific model, i.e., on network type, but
are determined only by the dimension of the space. This fundamental postulate
of percolation theory is based on analysis of results given by numerical model-
ing of the IC formation in networks of different types. However in the simplest
cases, such as that of a two-dimensional square network, analytical solutions can
be obtained as well [30]. Percolation theory shows also that although distribution
of conducting bonds (sites) in the network is random, there still exists a well-
determined threshold conductivity probability for a bond, when the network as a
whole acquires conductivity. This threshold value depends only on the network
type and the dimension of the problem and does not depend on the specific re-
alization of conducting bonds in the network. In a finite system, however, the
percolation threshold does depend on the specific realization of the conducting
bond distribution, i.e. is a random variable. As the size of the network increases,
the fluctuation of the percolation threshold becomes less, and the value of the
percolation threshold approaches the one predicted by percolation theory. In this
case, t5, the width of the critical region which is most likely (i.e., has overwhelm-
ing probability) to contain the value of the percolation threshold for a network
of finite size, decreases as t5 ~ C fN-vD. Here N is the number of sites in the
network; D is the dimension of the problem; Cis a coefficient (~ 1/2); vis the
critical parameter (the correlation radius) which depends on the dimension of the
problem and will be defined later. Since numerical modeling is carried out for net-