Page 92 - Percolation Models for Transport in Porous Media With

84 CHAPTER 4. MULTIPHASE FLUID FLOW

It is impossible to compare the theoretical calculation of k2(S1. 8 3 ) to the ex-
perimental data. in the domain ABC of three-phase flow because there are no
experimental data of such kind. However good qualitative and quantitative agree-
ment of theory with experiment in the domains of one- and two-phase flow (do-
mains 1 and 2 on the phase diagram 818 28 3 ) presented in §§1.2 and 4.1, as well
as in the case of Leverett's experiments mentioned above, supports the proposed
theoretical description of the equilibrium three-phase flow.

4.5 Stability of Percolation Methods for Calcu-

lation of Phase Permeabilities

The essential characteristic of a. medium in the developed approach to determi-
nation of the phase permeabilities, as well as other coefficients of transfer, is the
radius probability density function J(r) of capillaries. For actual media, this func-
tion is determined by one of the existing porometric methods [47, 48) and is always
known up to a. certain error. Therefore it is important to estimate the effect that
the error in establishing the function j(r) has on the results of the phase perme-
ability calculations.
The expressions for calculating the phase permeabilities K 1 and K 2 for two-
phase flow obtained in §4.1 have similar structure, and therefore it suffices to study
the behavior of any one of them, e.g., K 2 , when f(r) varies.

Here A' is a pre-integral term which does not affect stability of ( 4.50). In the
three-dimensional case the correlation radius index v = 0.9 ± 0.1. In this interval
K2 is a continuous function of the given parameter with a continuous derivative.
Therefore, without loss of generality, we can investigate stability of this function
for any of the values 0.8 ~ v ~ 1.0, for instance, v = 1.
Take two functions j+(r),J-(r) E C[a*,a*), where C[a*,a*) is the class of
continuous functions on [a*, a*] normalized on unity. We shall mark the values
correspon[di~g to lt~J:e two functions by the same signs. Introduce the norm
(-)
2
II · II = 2 dr . Using Cauchy-Bunyakovsky inequality, we can write the
following estimate for w(r) = j+(r)- f-(r)

V. w(r) dr :> Jlw(r)idr :> (J w (r)dr 11 · dr) 112
2