Page 19 - Percolation Models for Transport in Porous Media With

10 CHAPTER 1. PERCOLATION MODEL

to the closest neighbor. The following invariant holds to within 10 to 15%,

p• I= { 0.16 in the three-dimensional case, (1. )
2
c 0.5 in the two-dimensional case.
The values of P: I for different network types are given in table 1.2.
Structure of the infinite cluster. Shklovsky- de Gennes model. The
conductivity of the network strongly depends on- the IC structure. The regions of
an IC consist of the "skeleton" and the "dead ends." A point is said to belong to
the "skeleton" of the IC if at least two paths originating from it can be followed to
infinity. If there is only one such path, then the point belongs to the "dead end."
A model of the IC structure was proposed independently by B. I. Shklovsky and
P. de Gennes. According to this model, it is possible to represent the structure
of the "skeleton" of an IC as an irregular network with the characteristic period
equal to R, the correlation radius of the IC as determined from the expression

(1.3)

where the correlation radius index
_ { 0.9 in the three-dimensional problem,
11 - 1.33 in the two-dimensional problem.

Numerical experiments show that the relation (1.3) holds for the site percola-
tion as well. In this case, the parameter 11 is the same as for the bond percolation.
Electric conductivity near the percolation threshold. It was shown that
within the framework of percolation theory, the electro- or hydroconductivity of
the IC increases near the percolation threshold with the increase of either of the
probabilities, pb or P 8 , as follows

(1.4)
where the quantity d is determined from the dimension of the problem only:

d = { 1. 7 ± 0.02, D = 3,
1.3 ± 0.02, D = 2.

The numerical experiment shows that the relationship (1.4) holds within the
interval Pg :5 pb :5 Pg + fl.pb, where fl.pb :5 0.1. In the interval Pg + fl.pb :5
pb < 1, the conductivity of the network can be adequately described by the
formula obtained for the model of "effective medium" [29]:
K = Ko[1- (1- pb)/(1- 2/z)],

where Ko is the greatest possible conductivity of a network when there are no
broken bonds in it (Pb = 1). The same relationship holds for the quantity P' in
the "site problem."