Page 78 - Percolation Models for Transport in Porous Media With

70 CHAPTER 4. MULTIPHASE FLUID FLOW

a I

f

r
Figure 20: Probability density function for a medium with mixed wettability of
the pore space surface, and the conditional breakup of this function

in a two-dimensional network.
Consider a porous medium formed by a regular network of capillaries with the
probability density function f(r) (see fig.20} in an impermeable skeleton. A two-
phase equilibrium flow in such a medium is determined by the capillary forces on
the phase interface. These forces depend on the coefficient x of surface tension at
the phase interface and the contact angle 9.
In homogeneous media the parameters x and 9 are constant. Assume now that
the medium is heterogeneous, and x and 9 can take one of the two values x1 or
X2 and 9t and 92 each, with probabilities K. and 1 - K. for the first and the second
values, respectively. It means that the fraction K. of all capillaries in the network
(the conditional region 1 in fig. 20, a) is characterized by values Xt. 911 when they
belong to the phase interface, while the remaining fraction (1- K.) (the conditional
region 2 in fig. 20, a) is characterized by values x 2 , 92 •
For determinedness, consider henceforth the capillary displacement of phase a,
which initially occupies all the capillaries, by phase b under the increase of the
capillary pressure Pic = Pa - Pb from -oo to +oo. When the capillary pressure
equals Pic the displacing phase b can occupy only those capillaries, whose radii are
greater than the critical one defined by the Laplace's formula. In the given case
we have a specific critical radius for each capillary type

(4.25)

As it follows from (4.25), the parameters Xi and 9i appear only in the product
Xi cos 9i. Since Xi > 0, and -1 < cos 9i < 1, it seems natural to decrease the
number of external parameters by setting x1 = x2 = x and considering further
only the dependence on cos 9i
An important property of the problem in question is that if cos 91 and cos 92
have opposite signs, then the values of Pic for the two outlined capillary types
must have opposite signs, too, since the critical radius ((4.25)) cannot be negative.