Page 131 - Percolation Models for Transport in Porous Media With

124 CHAPTER 6. PORE SIZE DISTRIBUTION

1,0

0,5

0 5 10 r
Figure 44: Result of establishing the disturbed model function /in(r)
f

7,0

Figure 45: Result of establishing the function /in(r) in the case of inadequate
choice of the system of approximating functions (monotone decreasing exponential
functions)

while in the second case (fig. 44), errors of~ 1% were introduced into /in(r) after
solving the direct problem. It can be seen that when the system of approximating
functions is chosen adequately and the errors in input data are not too large the
desired PDF is recovered with good precision.
To study the infl.l!ence of errors brought in by inadequate choice of approxi-
mating functions, the PDF was determined for the same /app(r)

* _
f ( ) _ 6(a*- r)(r- a*) , a* - _ 1 , a - 4
2 r -
(
a*- a* ) 3
The results are presented in fig. 45 (the dotted line represents the exact PDF
for comparison). It is clear that although on the whole the original PDF is re-
covered approximately correctly, it is altogether impossible to approximate the
exact PDF by means of the chosen system of functions with adequate precision.
Therefore substantial errors are observed in the vicinities of the endpoints of the
range. It follows that in order to adequately choose /app(r), additional a priori
information concerning the qualitative behavior of f(r) is necessary. In this case,
one can expect a high degree of accuracy in determining J(r), and the result can
be represented in the analytical form.