Page 132 - Percolation Models for Transport in Porous Media With

6.4 RESULTS AND DATA PROCESSING 125

f

Figure 46: Result of solving the inverse problem of establishing the original func-
tion fin(r) without using the system of approximating functions

If, however, the condition of obtaining the recovered f(r) as an analytical
expression is unnecessary, then a procedure of reducing the integral equation {6.21)
to a system of linear algebraic equations, based on substituting the integral with
an integral sum according to some quadrature formula, can be used as described
in §6.2. When the trapezoid formula is used, the relationship {6.21) is reduced
to the system {6.26). After solving this system using the regularization method
the values of the desired PDFC at the chosen set of points {ri} are obtained. The
results of solving the reverse problem of the EPM on recovering the original PDFC
h(r), based on the system {6.26), are presented in fig. 46 {the dotted line indicates
h(r)). It can be seen from the figure that the consistency of the original and the
recovered functions is good enough. This result was obtained for the errors of the
order 6 8 ~ 1%.
To define the PDFC properly in the interval of small radii, it is necessary
to use the combined mercury electric porometry method and invert the exact
percolational dependence uy{ri) according to the iterative procedure described
in §6.3. To verify its efficiency, the dependencies uy(ri) obtained by the direct
calculations from the formula {6.27) for the given fin(r) were taken as the initial
data. Then these functions were determined by means of the described iterative
procedure and the obtained distributions !out{r) were compared to the original
fin(r).
In all cases j< 0 l(r) = const was taken as the zero approximation. Calculations
showed that the iterative process converges quickly enough, and the number of
iterations before relaxation is ~ 5 - 10. When the nature of the dependence
f(r) in {6.34) is chosen correctly the consistency of the recovered and the original
functions is very good (~ 0.1%). Introduction of a significant error into {6.34)
and, consequently, into the quantity Zc causes the distortion of the behavior of
f(r) near rc (~ 50%). Nevertheless on the whole the function f(r) is recovered
satisfactorily in this case as well, the most precise case being in the interval of
small r. This fact that is crucial for many applications.
Results of recovering for the following original distributions