Page 128 - Percolation Models for Transport in Porous Media With

6.3 MERCURY ELECTRIC POROMETRY 121

It easily follows both from the physical meaning of the percolation model of
transfer phenomena in a heterogeneous medium and from the immediate analysis
of the relation (6.28) that the latter expression, as well as (6.30), is valid within
the interval 0 < Ti < Tc·
In measuring a 11 ( r i) some intervals r i < r < r i + ~i can be obtained where elec-
tric conductivity of the medium does not change, i.e., da 11 (ri)fdri
= 0. If ~i do not contain rc, then in these intervals A(ri) = <f>(ri) = 0, and
from (6.30), f(ri) = 0 when ri < r < ri + ~i· This means that there are no
capillaries with Ti E ~i in the medium. If, however, some ~i contains an rc to-
gether with its neighborhood, then after substituting A(ri) = <f>(ri) = 0 in (6.30),
we also obtain f(ri) = 0. In this case, the fact that the integral of f(r) is in the
denominator of (6.29) formally causes a zero-over-zero indeterminacy. However
the equality da 11 fdri = f(ri) = 0 near the initially introduced rc simply means
that there are no capillaries with Ti close to rc. Therefore it suffices to decrease
rc up to the closest ri's, for which da 11 /dri f. 0, and take it as the new Tc· This
changes neither the meaning nor the contents of all formulas, but allows to get rid
of the formal indeterminacy. These cases can be encountered only in those media
which possess an f(r) function with two global maximums, i.e., in those having
two different types of porosities (e.g., porous and capillary or block and inter block)
described by the same f(r).
Taking account of the new definition of rc, estimate A(ri) and <f>(ri). It follows
from (6.27) and (6.29), noting that 0 ~ f(r) ~ Mo < oo, that

(6.31)

Here Nc is the average value of f(r) in the first segment r1 ~ r ~ rc (rc- r1 =
t5').
Since the contraction mappings principle is valid for the Volterra integral equa-
tion of the second type for every finite A(ri) and <f>(ri) [77, 78], the relationships
(6.31) speak in favor of the existence and the uniqueness of the solution to the
equation (6.30) and justify the use of the method of successive approximations
with an arbitrary initial function j( 0 )(r) for solving this equation.
In this case, once we have a normalized function j(n)(r) as the n-th approx-
imation, to obtain the next approximation according to (6.30), we first calculate
its non-normalized value
j<n+l)(ri) = A(n)(ri) J (6.32)
rc
f(n)(r) ~~ + <f>(n)(ri)

r;