Page 79 - Percolation Models for Transport in Porous Media With

4.3 MIXED WETTABILITY 71

Practically the mentioned property means that when Pk increases from -oo to 0
the capillaries with cos (J < 0 are filled, and only after this process comes to an
end, when 0 < Pk < oo, are the capillaries of second type (cos(} > 0) being filled.
Moreover for capillaries with cos (J < 0, we have imbibition instead of displacement:
in this case the formula ( 4.25) yields not the least, but the largest radius to be filled
by phase b (the unshaded part of region 1 in fig. 20, a). When cos91 cos92 > 0, the
capillaries of both types are being filled simultaneously, though they have different
critical radii as found from (4.25).
Now turn to the calculation of the coefficients of relative phase permeability
ka. and kb for such a micro heterogeneous medium. First, study the behavior
of the quantity kb. We shall begin with the case when cos91 > 0, cos92 >
0, cos91 :$ cos92 • The last condition can always be made valid by means of the
simple interchange of the indices 1 and 2. Denote a = cos9t/ cos92 • Then it
follows from (4.25) that Tkt = ark2, where 0 <a :$ 1. The distribution function
of capillaries filled with phase b (the shaded region in fig. 20, a) is

rk2 oo
nb = "'f f(r)dr + f f(r)dr (4.26)

ru rk2
Now analyze the relation (1.7). When Pk increases from -oo to 0 all capillaries
are filled with phase a, and Tkt and rk2 equal oo. When we pass to the domain of
Pk > 0, Tkt and rk2 are calculated using formula (4.25). As Pk increases, the radii
Tkt and rk2 decrease until they reach the values rc2 and ret = are2 (see fig. 20,
a), where re2 is found from the following condition
rc2 ex>
K J f(r) dr + J f(r) dr = {c (4.27)
arc2 re2
At this point, an infinite cluster of capillaries of both types filled with phase b
(ICB) is formed in the medium for the first time. With the further increase of Pk,
rkt and rk2 continue to decrease, and the substitution of /b(r) in (1.7) yields the
following condition for reb

rk2 oo
K J f(r)dr + J f(r)dr = {e (4.28)
reb rk2
The quantity reb is analogous to re. The only difference is in that the former
is not a constant but a function of rk2· It varies in the range ret :$reb :$ re, where
reb= re when rk2 = re. Further increase of Pk causes decrease of the values of Tk2
and rkt, but does not change reb= re.