Page 81 - Percolation Models for Transport in Porous Media With

4.3 MIXED WETTABILITY 73

if the lower limit in the first term of ( 4.27) is set to equal 0 when a < 0, and
the normalizing condition for f(r) is taken into account. The function rcb(rk2) is
still determined from the condition (4.28) and assumes values from 0 to rc, as rk2
varies between rc2 and 0. Thus in this case the displacement of phase a by phase
b is identical to the case described above, with cos fh > 0, cos 02 > 0, and the
formulas (4.29)- (4.31) for kb remain valid. Note only that since a< 0, the lower
limits of integration in (4.30) and (4.31), formally, are in the domain of r < 0,
where, obviously, /(r) is defined to vanish.
Now let ~t ~ ec· In this case, the ICB formation from the capillaries of the first
type is possible as early as the imbibition stage. When -oo < Pk < 0,

(4.32)

After substituting (4.32) into (1.7) we obtain the following condition for deter-
mining Tcb(rkt)

For reb to be greater than zero, it is necessary that rk1 exceed its limiting value
rclb which is defined by the condition for the start of the flow for phase bin the
capillaries of the first type,
rctb
K, J f(r)dr = ec
0
Thus when Pk < 0, it can be found from (4.32) and (2.1) that

kb(rkt) = 0, 0 < Tkt < rc1b, -oo < Pk < Pclb,
k,(r.,) =A'/ Ko l[lf•(r)dr] ~•(•'1'/(r) dr (l f(r) ~) -~r',

Tclb ~ ru < oo, Pclb ~ Pk < 00

When the capillary pressure exceeds zero some of the capillaries of the second
type being filled with phase b associate with the ICB, while all capillaries of the
first type are by that time filled with this phase. The technique of the further
calculations of the change of kb(rk2) is identical to the one used for the case when
K, > ec· It is still described by formulas (4.30) and (4.31), where rc2 = 00 and Tcb
is found from the condition (4.28).
We shall use the same scheme to study the quantity ka. Let cos01 > 0,
cos02 > 0, cos02 ~ cos01. The distribution function for phase a (the shaded