Page 129 - Percolation Models for Transport in Porous Media With

122 CHAPTER 6. PORE SIZE DISTRIBUTION

and then, after normalizing,

00
,(n+l)(r) = [c<n+l)rl j(n+l)(r), c<n+l) = J J<n+l>(r)dr (6.33)

0
The use of relationships (2.1'), (6.29), (6.32), (6.33), where the function u 11(ri)
is considered known from the experiment, permits to determine /(r) in the interval
0 ~ r ~ rc- 6'. For r > rc- 6', the function /(r) is not determined directly in
the given approach, since "pressure scanning" becomes impossible because the IC
breaks up. Therefore some a priori suppositions regarding the behavior of /(r)
in this interval are necessary; in the absolute majority of cases f(r) decreases
monotonely in this interval. For r > rc- 6' one can take f(r) ""'r-i, where j > 1.
In this case, for example, for j = 2, taking into account (2.1') we have

2
f(r) = {c(rc - 6')/r , Tc - 6' < r < 00 (6.34}
Consequently the quantity z, which is also defined in the interval rc- 6' < f < oo,
must be calculated using relationships like (6.34).
In the example given, Zc = ({c/3)(rc- 6')- 2 •
Thus the relationships (2.1}, (6.29), (6.32)- (6.34) represent a closed algorithm
for determining f(r) when the dependence u 11 (ri) is known from experiment.

6.4 Numerical Calculations and Core Data Pro-

cessing with the Electric Porometry Method

To determine the efficiency of the methods for recovering of the PDF for cap-
illaries (PDFC) using the electric porometry data described in §§6.2 and 6.3, a
series of numerical experiments has been carried out. The experiments have been
performed in the following order. At first, for the chosen PDFC /in(r), the direct
problem of electric porometry was solved and the specific electric conductivity of
the medium found for different degrees of its saturation with electrolyte. Then a
uniformly distributed random error 6 8 was introduced into the obtained data, and
the determination of the PDFC was carried out using one of the known methods.
The functions /out(r) thus obtained were then compared to the original /in(r). To
find out the properties of the outlined procedures for determination and eliminat-
ing additional systematic errors, introduced by the use of different models (the
effective medium model and the percolation model}, solutions of the direct and
the reverse problems were carried out for the same model.
The method for determining the PDFC based on the use of the effective medium