Page 23 - Percolation Models for Transport in Porous Media With

14 CHAPTER 1. PERCOLATION MODEL

or
rc l v(D-1) oo
1
n(rl) = z< -D) [ K, I f(r) dr 'I f(r) dr = P:. (1.8')
r1 rc
The average conductivity of a chain of unit length k composed from succesive
bonds is

(1.9)

If we decrease the threshold value u1 > 0 further, then n grows. The new con-
ducting chains which join the ones that were there for the initial value u 1 contain
the bonds with conductivities u $ 0'1. Therefore the new average conductivity is
less, but still determined by (1.9), where 0'1 denotes the minimum value of conduc-
tivity among the bonds contained in the given chain. The average conductivity of
the chain is uniquely determined from the quantity 0'1. Knowing the distribution
function F(u1 ) of the condw:ting chains with respect to values of u1 , one can find
the total conductivity of the IC

D'c
K = I k(u1) du1, (1.10)
0

where F(ut) is related to the quantity n as follows, F(ut) = -dn/do-1. Using this
relationship, as well as (1.8), (1.9), and (1.10), we obtain

u [ u ]2v-1
1
K = "'fli(D- 1)1< -D) K.v(D-l) I I fo(u) du /o(u1) I~:~) (1.11)
0 D'l
where

The formula (1.11), as well as other known percolation relations, is obtained
under the assumption of no interflows between the conducting parallel chains. This
fact is reflected in the formula (1.11) by the numerical factor "' (of the order of
unity), which depends on the network type. As was pointed out, in the three-
dimensional case the conducting chains of the IC are tortuous. It can be shown
that taking account of this property causes the change of the exponent 2v - 1 in
(1.11) to v + (- 1, where ( = 1 [26]. In the two-dimensional case, no such effect
is observed.