Page 28 - Percolation Models for Transport in Porous Media With

1.3 EFFECTS OF ELECTRIC CURRENT 19

media, capillaries can be bonded not only successively, but also in parallel, cor-
rect estimates of the energy discharge density at the micro level must be made
using the network model of heterogeneous media (see §1.2). In analyzing current
flow through a network of conductors, we will use the more habitual notation for
electric conductivity, k(at), rather than E(at)·
When electric current flows through a chain of successive resistances, the max-
imum voltage is achieved on a bond of minimum conductivity a 1 in the chain. The
current through the chain is proportional to E(at).
Let E be the gradient of the potential applied to the network. Then the
maximum local gradient of the potential in the chain satisfies the relationship

After looking through chains with different a1, one can find the maximum
gradient of the potential in the network

(1.16)

Consider a model probability density function of the form

(1.17)

After substituting (1.17) into (1.16), we find that

VtjJ* /E = (a3/a2 -1)ln- (a3/a2)
1
It is evident now that with the increase of the variance in the probability density
function, the heterogeneity of the local gradient of the potential in the network goes
up sharply. The relationship (1.16) allows to determine the conductivity of the first
bond in which a change of conductivity has occured. To determine further change
of the conductivity of the bonds in the network, consider a chain characterized
by a parameter, say, a 1 . Suppose that the conductivity a 1 of those bonds, whose
energy discharge reaches the value e:a 1 fum, increases up to the level where it has
practically no effect on the conductivity of the chain, i.e., a 1 :» E(a1). Note that
the quantity e:a;,l, where am = const, depends only on the physical"properties of
the material constituting the bonds, and is thus constant for the given medium.
The increase in the conductivity of the network results in the increase of the current
density in the chains. The latter phenomenon may cause further change in the
conductivity of the network.
Suppose that in the network characterized by the parameter a 1(0), a 1(t) is the
conductivity of the bond which changes conductivity at the instant t. (The bonds