Page 70 - Percolation Models for Transport in Porous Media With

62 CHAPTER 4. MULTIPHASE FLUID FLOW

in negative powers of r (Weierstrass's theorem)

00
/(r) = L aifri
i=1
where ai are the coefficients of the expansion. By choosing the principal term in
the sum, which makes the major contribution to /(r), and taking it as a model
probability density function, we can obtain analytical expressions for the phase
permeabilities and the capillary pressure. Take a model probability function as in
the empirical dependence {3.46)

f(r) = 0, (a*< r, r >a*), /(r) = a*a* 1 2 , (a*$ r $a*) (4.9)
a*- a* r
Here a* defines the maximum possible capillary radius; obviously, a* cannot
exceed the size of a grain in the medium. The quantity a* defines the minimum
radius of a conducting capillary. The existence of such a limiting radius can be
related, for instance, to the fact that each capillary has a double electric layer
which hinders fluid flow through thin capillaries because of the exceedingly large
viscosity generated there.
After using, for determinedness, the relationship {4.8) in the case {4.9), we find
the following correlation between the saturation of the medium with the wettable
fluid and the quantity rk
{4.10)
The phase permeabilities found from relationships (4.3)- {4.5) and (4.10) are
presented in fig.17, a, where the curves k1(Sl) and k2{Sl) are denoted by numbers
1 and 2, respectively. In the same figure, the functions k1(S1 ) and k2(Sl) in the
case when saturation is defined by the relationship {4.7) (model I) are drawn in
dotted lines. Fig.17, b, contains the phase permeability curves obtained in the case
when
/(r) = ( V27r(Jd r)- exp[-(logr- p.') /(2q~)],
1
2
a form of the dependence /(r) used in numerical calculations in [60]. As in [60],
the parameters were set as follows, qd = 0.25, p.' = 2, z = 6. The results of the
numerical simulation of the two-phase flow using the network models [60] presented
in fig.17, b, {dots) show that the outcomes of analytical and numerical calculations
for the same function /(r) coincide satisfactorily. For the limiting cases (S1 -+ 1
and S1 -+ a*fa*), setting v = 1, we can obtain the asymptotic expressions for the
relative phase permeabilities by expanding the corresponding relations in powers
of the small parameter e1 =a*/ a* ~ 1. After taking only the first terms in the
expansions we obtain the following

(4.11)