Page 127 - Percolation Models for Transport in Porous Media With

120 CHAPTER 6. PORE SIZE DISTRIBUTION

respect to those presented in §6.2 if the percolation model of transfer phenomena
(described in part 1), instead of the approximate effective medium model, is used.
The scheme of the experimental measurement of the specific electric conductiv-
ity uy for the medium is as follows. A specimen with cross-section S* and height
L is placed in a container with mercury, and the latter is injected in the speci-
men under pressure p. After making a current I pass through the specimen and
measuring the voltage drop U on it its resistance Ro = U /I, the specific electric
conductivity O'y = L/(S* Ro) is found. After carrying out a series of such mea-
surements for different values {Pi} of pressure and relating them, according to the
Laplace formula, to the minimal capillary radius ri = 2xcos0/Pi where mercury
can pass, we obtain the dependence uy(ri). This correlation can be either rep-
resented in the form of an interpolated curve or tabulated for further numerical
processing on a computer.
Specific electric conductivity can be calculated using the percolation model of
a heterogeneous medium according to (2.4), using the formula
[1

f(r)
av(r;) = A" l [ l f(r) dr r /(r) ~ r 1 dr f(r) dr (6.27)

where A" is a numerical factor, rc is the critical percolation radius.
If we now consider the function uy(ri) known and the function f(r) unknown,
then (6.27} becomes a nonlinear integral equation for f(r). After differentiating it
once with respect to ri and making some transformations we obtain

f(r;) = -1/A"(dav/dr;) l f(r) ~; [! f(r)dr] -• [! f(r)dr] -I (6.28}

or, if we introduce the following notations

~(r;) = -1/A"(davfdr;) [! f(r)dr] -u [! f(r)dr] -I, (6.29)

a nonlinear inhomogeneous Volterra equation of the second type in the standard
form

(6.30)
r;