Page 20 - Percolation Models for Transport in Porous Media With

1.1 BASIC CONCEPTS 11

Table 1.2:

c I P%f
Network Type z ps
Plane
Square 0.79 0.59 0.47
Triangular 0.91 0.50 0.46
Hexagonal 0.61 0.70 0.43
Solid
Simple Cubic 0.52 0.31 0.16
Body Centered Cubic 0.68 0.25 0.17
Face-Centered Cubic 0.74 0.20 0.15
Diamond Type 0.34 0.43 0.15

The Shklovsky - de Gennes model allows to relate the quantity d to the corre-
lation radius index. Since, for instance, the electric current flows only through the
"skeleton" of the IC, the electric conductivity of the network is determined only
from the conductivities of the parallel capillary chains within the "skeleton" of
the IC. The number n of the capillary chains reaching a unit surface of the cross-
section perpendicular to the chosen direction equals R-(D- 1 ). The conductivity
of the network E ""na1 , where a 1 is the specific conductivity of a chain. Using
the relationship {1.3), we obtain a formula for the specific electric conductivity of
the network
{1.5)
Here, a0 is the specific electric conductivity of the network when pb = 1.
After comparing the relationships {1.4) and {1.5), we find that d = v in the two-
dimensional case and d = 2v in the three-dimensional case. This fact supports
the validity of the Shklovsky - de Gennes model. In the three-dimensional case,
capillary chains can be tortuous. However this feature changes only the formula
d = 2v to d = v +(,where ( = 1 [26]. In the two-dimensional case this effect does
not appear.
Note that the formula {1.5), as well as other percolational relations, was ob-
tained up to a numerical factor of the order of unity.
Density of an IC. Research has showed that the value of W, the number of
sites (bonds) which belong to the IC, obeys the exponential law

{1.6)