Page 153 - Percolation Models for Transport in Porous Media With

8.2 CONDUCTIVITY AND PERMEABILITY CHANGE I 149

through the r1 -chain when the next, ( m + 1 )-th impulse is generated is determined
by the expression
(8.15)
where
F1(r1.m) = r~(< r1.r f(r,m),R(r1.m) >
2
+ < R(r1,m),r- ,r.z > 'Yu+ < r.z,r- ,a* >)
2
2
Therefore the critical radius R( rt. m + 1) is determined from the equation

(8.16)
Since the amplitude of the current {8.15) increases as m grows and Ic(R, r) is
a monotone increasing function of R, it follows that R( r1. m + 1) > R( r1. m).
Thus the considered process is self-supporting. If the sequence R(r1. m) con-
verges to Ro(rl) < r.z, which is determined from {8.16) by substituting R(rl) for
R(r1. m), then for the given E0 , cement is destroyed only in thin capillaries with
r1 ~ r < Ro(rl) < r.z, and the r1-chain remains non-conducting. If it turns out
that for some m, R( r1, m + 1) > r .z, then in all non-conducting capillaries of the
r1-chain (r1 ~ r ~ r.z), destruction (perhaps, partial) of cement takes place, f falls
within the interval 'Yu > f(r,m) > 1, and the r1-chain becomes conducting.
We shall now directly calculate how the changes of relative permeability K / K 0
and electric conductivity E/'£0 depend on the duration of the treatment with
impulse current. (Ko and '£0 are the values of the mentioned quantities before
electric action was started.)
Suppose the hydraulic conductivity of an r-capillary after the m-th impulse
have passed through it is determined by the expression
(8.17)

where w(r, m) = 1 for all capillaries with r.z ~ r ~ a*; w(r, m) = 0 for the
non-conducting capillaries where the threshold {8.10) was not reached; and 0 <
w(r, m) ~ 1 if the threshold was exceeded, and no additional destruction of capil-
laries took place (otherwise w(r, m) > 1).
Based on (1.11), we can represent the specific electric conductivity of the
medium E{m) and its permeability K(m) in the following form

rc
E(m) = E0 1 I Io(rl, m) dn(rl);

rc
K(m) =I k(r1,m)dn(rl); (8.18}
a.