Page 29 - Percolation Models for Transport in Porous Media With

20 CHAPTER 1. PERCOLATION MODEL

which satisfy the condition 0'1 (O) :::; u :::; 0'1 (t) have already had their conductivities
changed.) To obtain the condition for the conductivity change, define the energy
discharge t:1 in the bond u 1 (t) contained in the chain u1 (0). After neglecting the
effects of heat exchange between the chains and the non-conducting skeleton of
the medium, we obtain

2
2
E
( ))
E It 2(
( ))
0'1(t)
2(
E -- = -(-) l; 0'1 0 + -(t) l; 0'1 T dr, (1.18)
O'm 0'1 t 0'1
to
The first term in the right side of the relationship (1.18) corresponds to Joule's
heat which has discharged in the bond before the instant to = eu;, 1
x (''Vr/J*)- 2 , when the conductivity of the chain began to change. The second
term describes the energy discharge during the period when the conductivity of
the chain was changing. In this case, t~c is the time needed for all elements of the
chain to acquire infinite conductivity.
Consider the change of the conductivity in the case when the probability density
f 0(u) of the bonds with respect to intrinsic conductivities is described by the model
function
(1.19)

In this case the solution (1.18) can be obtained in the analytical form. As it can
be deduced from the relationship (1.16), the value 'ilr/J* f E = nf(n -1) is the same
for all chains and is assumed at those bonds which have the least conductivity for
the given chain.
For an arbitrary chain, (1.18) implies

1
An= 2(n -1)

This means that according to (1.9), the average conductivity of all chains
increases in the like fashion, i.e., proportional to 1- (n- 1)(tft0 - 1)-~"n, so
that to is the same for all chains. Therefore the effective conductivity of the
ITl
medium I: (t) "' J I:(u(t)) du, and after integrating
0
0

H bonds are cylindrical capillaries, then the electric conductivity u of such a
bond is related to its permeability coefficient k in the following way, k "' u 2 • Thus
we obtain the time dependence of the permeability K(t) of the medium

K(t)fK(O) = [1- (n -1)(tfto- 1)17(t- to)t ~nn
2