Page 123 - Percolation Models for Transport in Porous Media With

116 CHAPTER 6. PORE SIZE DISTRIBUTION

correlation of the PDF with some macro-characteristics could be later inverted, at
least by means of some numerical method.
Realize the first part of the proposed plan, i.e., find an approximate solution
of the direct problem of the EPM. Assume that the experimental measurements
are taken according to the scheme represented in fig. 41. Also, still assume that
only those capillaries are saturated at the height L, which have a radius less than
the critical r(L) as defined by formula (6.14). Now, considering the radius PDF
for capillaries known and the system infinite (since the size of the core is much
greater then the period of the network), find the conductivity of such a medium.
The problem just set is one of the classical problems in percolation theory, the
problem of the calculation of the conductivity for the bonds in a network. ( Conduc-
tivity q1 of a separate capillary correlates with its radius as follows, q1 = a(IC'r 2 l- 1 .)
Solving this problem, but for the case of Bethe's classical network, is possible only
in numerical methods (1, 3]. Due to this fact this solution cannot be used further
for solving the reverse problem. However the necessary analytical correlation a(L)
with the radius PDF for capillaries can be obtained approximately using the effec-
tive medium model (EMM) (29], a model that allows to get approximate results
which agree satisfactorily with those of the exact calculations of the percolation
problems. Notable deviations ("' 20 %) are observed only near the percolation
threshold. Therefore if we consider the measurements of a(L) being taken not too
r(L)
"high," so that J f(r) dr > 1.2~c, the use of the EMM proves to be legal enough.
0
The main idea of the EMM is to substitute the network of random resistances
with a similar network of identical "effective resistances" under the condition for
the conductivity of the whole medium not to change. In this case, as is shown
in (29], the conductivity qm of a single bond in the effective medium is determined
from the equation

00
I fo(qt)(qm - ql)((z/2- 1)qm + q1t dq1 = 0 (6.19)
1
0

Here fo(ql) is the conductivity PDF for bonds, z is the number of the near-
est neighbors for the given network type. In the further calculations, it is more
convenient to pass from the relationship (6.19) for qm to a similar one for rq. In
doing so, we should take into account the correlation between the conductivity
of a capillary and its radius and the equality of random variables when they are
functionally dependent /o(ql) dq1 = f(r) dr. In this case (6.19) can be rewritten
as
oo r(L)
2
(z/2-1)- I f(r)dr+ I f(r)r;[(z/2-1)r;+r ]dr=O
1
r(L} 0