Page 158 - Percolation Models for Transport in Porous Media With

154 CHAPTER 8. CONDUCTIVITY AND ELECTRIC CURRENT

~~-----++------~~~--~t~gt~/~
Figure 56: Variation of the permeability of the medium depending on the duration
of treatment with periodic current

starting from t;::: Tq(rl) the r1-chain begins to conduct. The function F2(rt,R,t)
depends on its argument rather weakly and is close to Fo(rl). Therefore, after
setting R = rz, t = To(rl), we can obtain the following value from (8.25) with
good accuracy
Tq(r!) ~ t(rt, rz, To(rl))

The relationship (8.22), with regard to the condition (8.21}, shows that To(rt)
decreases rapidly as r1 grows (perhaps, with small violations of monotonicity),
while Tq(rl) > To(r1) and decreases more rapidly than To(rtO as r1 grows. There-
fore the thick rz-chains are the first of all r1-chains to become conducting and
increase the electric conductivity, and a*-chains are the last to do so. This results
in the conservation of the initial hierarchy of r1-chains with respect to the values
of their average electric conductivity during the electric treatment.
It follows from (8.21), (8.25) that for the "temperature mechanism" R(rt,t)
grows slower than In t, while for the "gradient mechanism" R(rt, t) is upper bounded.
If we also take into account the fact that large capillaries affect the average hy-
draulic and electric conductivity of an r1-chain to a small degree, we can notice that
as time passes, K(t) and I;(t) steady. (According to (8.18), the major contribu-
tion to the alteration ofthe specific electric conductivity I;(t) and the permeability
K(t) of the medium is made by the thick r1-chains.)
We shall proceed to determine the starting time t' for changes of I; and K in
the medium and the time t" when these changes stop

t'(Eo) = min(To(rt,Eo)), t"(Eo) = max{Tq(rt.Eo)) (8.26)

Here the minimum and maximum are taken over all non-conducting r1-chains
(a* :5 r1 :5 rz).
Results of the numerical calculation of K(t) for a function of the form (8.19)
and the same values of the parameters as in §8.2, are presented in fig. 56.