Page 130 - Percolation Models for Transport in Porous Media With

6.4 RESULTS AND DATA PROCESSING 123

f
1,0

0 !0 r
Figure 43: Result of establishing the model function fin(r) by means of solving
the reverse problem with exact output data

model and described in §6.2 was tried for the following initial model PDFC

a*a* 1 a < r <a*
a :2"•
ft(r) = { a* - * r * - - ' (a* = 1, a* = 11) {6.35)
0, r < a*, r > a*.

In the direct problem with a given function fin ( r) and a chosen set of limits su-
perior {(r(L))k}, the transcendental equation {6.21) was solved using the Newton
- Ruthson method and {(rq(L))k} were found. Also, the validity threshold r' of
the effective medium model was found previously from the condition (6.22), and
all (r(L))k were chosen from the segment [r', a*].
The obtained dependence {{rq(L))k} was used as the initial data for solving
the reverse problem of the EPM by means of solving the system (6.25) using the
regularization method with the quasi-optimality criterion. The following system of
functions, decreasing with the increase of the radius, was chosen as approximating,

The following equation, which reflects the normalization condition for the PDF,
was added to the system (6.25),

The results of solving the reverse problem of determining the model PDF (6.35)
are presented in figs. 43, 44 (the dotted line indicates the true curve fin(r)). In
the first case (fig. 43), exact input data were used for solving the initial problem,