Page 88 - Percolation Models for Transport in Porous Media With

80 CHAPTER 4. MULTIPHASE FLUID FLOW

s, .sj
Figure 25: Triangular diagram of the domains where different numbers of phases
flow, as a function of the ratio of saturations (the Gibbs- Rosebome triangle)

any phase is possible only in the case when the capillaries filled with this phase
form an IC. An IC is formed in the network if and only if the fraction of capillaries
containing the given phase exceed a certain threshold value {c·
Estimate the probability of a capillary containing the i-th phase. In the course
of the equilibrium flow, any phase can access only those capillaries where the
capillary pressure does not exceed the pressure Pi in the phase. Obviously the
phase with index 3, whose wettable capacity is least, fills the largest capillaries
(r > r 2 , fig.24). The thinnest capillaries (r < rt) contain the phase of index 1,
i.e., the most wettable one. Finally, the capillaries of radii rt < r < r2 contain the
phase of index 2. The values rt and r2, which break f ( r) into zones saturated with
different phases (see fig. 24), are determined from the phase equilibrium condition

(4.43)
where Xi.i+l is the coefficient of surface tension on the interface of the j-th and
the (j + 1)-th phases; 8;,;+1 is the contact angle for the j-th and the (j + 1)-th
fluids.
It was shown in §4.1 that (with a 85% accuracy) the i-th IC contains all capil-
laries whose radii satisfy the condition of the i-th phase penetrating through them.
Therefore the probability {i of the capillary to contain the i-th phase is defined by
the following expressions

~ ~ 00
{t = J f(r) dr; e2 = J f(r) dr; ea = J f(r) dr (4.44)
0 ~ ~
Consequently the i-th phase flows if

(4.45)

To determine the value of ~c quantitatively, it is necessary to know the network
type that simulates the pore space structure most adequately. Actual location