Page 80 - Percolation Models for Transport in Porous Media With

72 CHAPTER 4. MULTIPHASE FLUID FLOW

Thus from (2.1) we obtain the following, by calculating the values of Tkt. rk2,
and rcb and using (4.26)

k,(rk2) = A'/Ko ]b [lbj,(r)dr] "j11(r')
ark2 r 1
X ( K lf(r)dr +lf(r)dr) (K r(r) ~ +lf(r) ~) -~r' (4.30)

rc < Tk2 ~ rc2, Pc2b ~ Pk < Pc

2 2
kb(rk2)=A'/Ko{K 7[K 7f(r)dr+ J f(r)dr]"
0~2 ~ ~2
xf 6(r') ( K l"(r) dr + l/(r) dr) ( •l/(r) ~ + l/(r) ~) -~r'

+l [l J.(r)dr] ~,(r') 1/(r)dr (l/(r) ~) -~r'}, (4.31)

0 < Tk2 ~ rc, Pc ~ Pk < 00
Here A' is the term which appears before the integral in {2.1).
Now consider the case when cosfh cos82 < 0, for instance, cos81 < 0, cos82 >
0. In this case the region 2 will not begin to be filled until the region 1 is completely
filled. The filling of the latter region in this case will be directed towards the
increase of the capillary radii, as was noted before. Depending on the value of "'•
different types of two-phase flow are possible.
If K < ~c. then as Pk increases from -oo to 0, the flow of phase b cannot
begin, since only the capillaries of the first type can be filled with this phase,
whereas there is not enough of them to form an ICB. With further growth of Pk,
the instance will come when the ICB is formed from the sum of the capillaries of
the first and the second type. In this case ( 4.27) becomes

00
K + (1- K) J f(r) dr =~c.

re2