Page 83 - Percolation Models for Transport in Porous Media With

4.3 MIXED WETTABILITY 75

In the case described by (4.35) we have

2 2 2
1
k4 (rk2 ) = K0 { 7 [ 7 / 4 (r)dr] ~4(r') ( 7/(r)dr + (1- K)
0 r' r'

2 2
x 7~(r) dr) ( 7f(r) :: + (1 - K) 7 f(r) ::) -~r' + (1 - K) 2
arrk2 r' arrk2
x} [l/.(r)dr] "J.(r') lf(r)dr uf(r) ~:) -~.l (4.37)

Tc2t.L < Tk2 < 00, 0 < Pk < Pc2t.L

Note that the cases (4.34) and (4.35) are not necessarily realized separately in
the whole range 0 < Pk < oo. Certain values of a and K can be found, such that
these cases can pass to one another as the capillary pressure changes.
Let now cos/h cos82 < 0, cos81 < 0, cos82 > 0. As Pk increases from -oo to
0, the fraction of capillaries of the first type in the ICA decreases. The probability
density function for these capillaries is
0, T < 0,
{ (4.38)
j 4 (r) = (1- K)j(r)/fl(L, 0 ~ T < Tkt 1
T;:::: Tkt,
f(r)/f1a 1
where Tkl is determined from (4.25). If Tkt < Tc, then the substitution of (4.38)
into {1.7) yields Tc 4 (rkd = Tc 1 while for ka we have

1
1
1
ka(rkl) = A'(l- K)K0 { 7[(1- K) 7/(r) dr +] f(r) dr] v fa(r')
x U. f(r)dr- •lf(r)dr) [(1-•) lf(r) ~: +] f(r);: P•'
ru
r'
0

+l [! /.(r)dr] "/.(Y) l f(r)dr U. f(r);:) -~r'},

0 < Tkl < Tc, -oo < Pk <Pel

When Tc < Tkt < oo from (4.38) and (1.7), we obtain the dependence Tca(rkl)
in the following form
rkl oo
(1 - K) I f(r) dr + I f(r) dr = {c
reo rkl