Page 125 - Percolation Models for Transport in Porous Media With

118 CHAPTER 6. PORE SIZE DISTRIBUTION

ill-posed, and classical methods for its solution are not applicable. To find the
PDF from (6.21), some kind of a regularized method, stable towards small errors
in input data, should be used.
One of the possibilities lies in the reducing the integral equation (6.21) to
a system of linear algebraic equations and solving the latter by means of the
regularization method. In this study, the method of approximating functions was
used for passing to a system of algebraic equations, where the sought PDF was
expanded over some system of linearly independent functions. Due to the fact
that a priori information about the behavior of the PDF is usually very limited,
it is hard to prefer one system of approximating functions to another. In this case
we can use Weierstrass's theorem on the expansion of any analytical function in
power series

(6.24)
-oo
It can be assumed that f(r) is non-vanishing only in the interval [a., a*]. After
taking only a finite number of terms in the expansion (6.24) and substituting it in
(6.21), we obtain the following

(6.25)
-n
where
r(L)
Fik = j r2+i(r + 2(r~)k)- dr.
1
2
0
The immediate use of the system (6.25) would have led to a system of equations
for an infinite number of unknowns, which does not have a unique solution.
When j > 2n + 1 the relation (6.25) represents an overdetermined system
of algebraic equations for the unknown coefficients {ai} of the expansion and
an inaccurately assigned matrix Fik. Regularization method [76] can be used to
find a normal pseudosolution of this system, and the regularization parameter
should be chosen consistent with the errors in the input data. However the error
caused by using the NM for describing the pore space structure is impossible to
estimate quantitatively, and therefore the error of the assignment for the matrix
is in fact unknown. Hence the customary condition of the residual cannot be used
to choose an optimal regularization parameter. Instead, quasi-optimum criteria or
relations [76] that do not require knowledge of the input data errors can be used.
If the condition of obtaining /(r) as an analytical expression is not necessary
and it is sufficient to represent the sought f(r) as a plot, then the following pro-
cedure can be suggested.