Page 94 - Percolation Models for Transport in Porous Media With

86 CHAPTER 4. MULTIPHASE FLUID FLOW

a I

Q 2 r
Figure 27: Effect of the shift of< r > for the function f(r) (a) on the perme-
ability change (b)

Therefore as llw(r)ll-+ 0, IIK:i- Kill-+ 0, i.e., the phase permeabilities are
stable with respect to small perturbations of f(r). Obviously the relative phase
permeability k2 = K 2 / K 0 is also stable. Indeed, we can follow the same line used
in deriving the relationship (4.53) for r~c = 0 to obtain

and then

and finally
llk:i- k:ZII ~-'*(a*- a! 12 11 w(r)l
which proves the stability of the relative phase permeabilities with respect to small
variations of f(r).
To illustrate the analysis presented above, a series of calculations of K2(r1c)
was carried out for different f(r)'s. Two examples from this series are presented
in figs.26 and 27. Shown in figs.26, a, and 27, a, is the nature of the perturbations
(dotted lines correspond to f- ( r)) of the original function j+ ( r) = 2r exp( -r 2 )
defined on [0, 3]. In the first case the perturbation is oscillatory, but does not
change the average radius < r > of capillaries and the variance O'd of the distribu-
tion f(r). In the second case, a shift of< r > takes place. However in both cases
the total deviation on the whole range of f(r) is the same, f.* !::!! 0.1. Presented
in figs.26, b, and 27, b, are the corresponding plots of K2 (r~c). They imply that
in the first case K2 (r~c) does not really change, whereas in the second case a no-
table difference between K:i and K2 appears. Calculations show that the phase
permeability is as sensitive to changes in the variance of f(r).
In the outlined cases the numerical value of the coefficient A' A0(a* - a*) 3 1 2 ,
which appears in the estimate (4.55), is virtually always greater than its exact