Page 85 - Percolation Models for Transport in Porous Media With

4.3 MIXED WETTABILITY 77

was expressed for a > 0 by the following

rk2 oo
S 1 = K. I I ( r) dr + I I ( r) dr (4.39)
rk1 rk2
2
Su~<r >-' (•.lf(r)r dr+l f(r)r dr); (4.40)
2
2
and for a ~ 0, by the following
l(r)dr + (1- "') 1
sf="' 1 oo l(r)dr ( 4.41)
rk1
0 rk2

2
2
S11 ~< r > -• [ •lf(r) r dr +(I-•).l /(r) r dr] {4.42)
2
Plots of ka and k& for K. = 0.5 and the model probability density function

l(l)(r) = { 0, r ~ [1,3],
5
4.05r- , r E [1, 3]
are depicted in fig.21.
It can be seen that as a decreases, the values of ka fall, whereas the values of k&
grow. The same effect was obtained in [9] as a result of the numerical simulation
of the capillary displacement on a two-dimensional network of capillaries. In [9],
K. = 0.5; a log normal distribution was taken as the function l(r); and it was
assumed that the lengths of capillaries correlate with their radii, l ....., rk, where k is
the varied parameter. Therefore the above-mentioned congruence of tendencies in
the changes of ka and k& caused by change of a is merely qualitative. Furthermore
it takes place only when saturation is calculated according to model II (see fig. 21,
a). If S = S1, then the nature of variation of ka and k& with the change of a shifts
to the opposite, as is evident from fig.21, b, where plots of ka(SI) and k&(SI) are
presented, calculated for the same function l(r) and values of a and "'·
To study the impact the form of l(r) has on the functions ka and kb, calcu-
lations were performed for "' = 0.5, a = -1 and 1 and for three model functions,
1< 1 >(r) and the following two others,

1 < 2>(r)={O, _ 2 r~[1,3],
1.5r , r E [1, 3]

l(a)(r) _ { 0, r ~ [0, 5],
- 2rexp(-r 2 ), rE[0,5]