Page 27 - Percolation Models for Transport in Porous Media With

18 CHAPTER 1. PERCOLATION MODEL

with the mean value of the conductivity of the medium~~ M =< u >. As for
the one-dimensional case, the effective conductivity is determined from the mean
value of the inverse conductivity of the medium, i.e., from the average resistance
~ ~< 1/u >-1, and does not coincide with the mean value < u >.
The dependence of the effective conductivity of a medium on the parameter
K. defined for the second and third distribution functions by the formula (1.11) is
shown in fig. 3. The parameters v, l, and 'Y were the same as in the previous
case. Fig. 3, a, b, shows the results of the calculation for f 0(u) No. 2, and fig.
3, c, shows the results for f 0(u) No. 3. Values of~ obtained in the numerical
calculation are marked by circles on the same plots.
Distribution function No. 2 differs from the one regularly used in percolation
theory only in that the non-conducting bonds are substituted in the network with
the bonds with small, but non-zero conductivities. Numerical modeling shows that
introduction of the bonds with small conductivities into the network instead of the
corresponding "zero" ones changes the nature of the dependence ~(K.) drastically.
Within the framework of classical percolation theory, where the second term in
the relation for fo(u) No. 2 is of the form (1- K.)6(u- 0), the given dependence is
characterized by a curve resembling the dashed line in fig. 3, b. After comparing
the curves in fig. 3, b, one can notice that although according to the classical
theory, in the region K. ~ P~, ~(K.) = 0, in the considered generalization of this
theory, ~(K.) > 0 for all K.. Besides, the behavior of the curves ~(K.) in the interval
P~ < K. ~ 1 varies. This is most noticeable near K. = P~. The plots presented in
fig. 3 show that satisfactory agreement of the analytical dependence (1.11) with
the results of the numerical experiment does take place.
Note that the obtained relation {1.11) includes the limiting case (1- K.)6(u-
w- ) ~ (1-K.)6(u-O). In this case the calculations using (1.11) yield the classical
1
percolational relation shown on fig. 3, b, by the dashed line and described by the
formula (1.5).

1.3 Effect of Electric Current on Conductivity
of Heterogeneous Media

When electrical current passes through successive capillaries of radii r 1 and r 2
the ratio of current densities in them is proportional to (r2/r1 ) 2 , while the ratio
of energy discharge densities is "' (r2/rt) 4 • For heterogeneous media, e.g., rocks,
the ratio ( r2/ r1) can be ~ 10 3 and more. This fact shows how far from unifor-
mity can the energy discharge density be in a medium. High densities of energy
discharge in thin capillaries can cause changes to the intrinsic conductivities. Spe-
cific mechanisms causing such changes can be very different, e.g., the increase of
pressure in capillaries, pressure gradients at the micro level, etc. Since in actual