Page 118 - Percolation Models for Transport in Porous Media With

6.1 MERCURY INJECTION 111

0 ~~:':--:'=---:':-....U.-'
13 N IS 15 17 /8 ·tnlr,!
Figure 39: Results of the study on stability of the dependence X(rJk) with respect
to small changes of initial values

f

Figure 40: Plots of the function f( -logrk) obtained for curves I and II (see fig.
39)

the curves I and II are observed to approach each other. This fact confirms the
conclusion about the stability of the dependence X(rk) with respect to the small
changes of the initial conditions based on (6.5) and (6.8).
The plots of f( -ln rk) relations for the cases I and II shown in fig. 39 are
presented in fig. 40. These curves diverge by more than 100% almost everywhere,
a fact that speaks of the essential unstability of the dependence f(rk) with respect
to the small changes of the initial conditions.
Stability of the dependence X(rk) and unstability of f(rk) are typical not only
for percolation models of mercury injection. The problem arises from the fact
that obtaining the dependence f(rk) requires applying to the dependence X(rk)
a differential operator which is unstable in the general case. To overcome this
unstability, one can use the regularized differential operator [76].
At the same time, the permeability and the electric conductivity of a porous
medium are expressed in terms of integrals of the J ¢(r) f(r) dr type, where ¢(r)