Page 87 - Percolation Models for Transport in Porous Media With

4.4 THREE-PHASE FLOW 79

f

Figure 24: Schematic distribution of phases in the pore space under capillary
forces for steady state flow

is greater. The effect is reverse for kb. When a= -1 the kb(Su) curves lie higher
and the kb ( S I), lower than the corresponding plot for a = 1. The characteristic
convexities which appear on these curves with the increase of K go up in the case
of kb(Su) and down in the case of kb(SJ).

4.4 Three-Phase Steady State Flow of

Immiscible Newtonian Fluids

Due to great technical difficulties encountered in any attempt to determine exper-
imentally the phase permeabilities for three-phase flow, theoretical study of the
behavior of the permeabilities is of special importance. The results obtained in
research on the equilibrium two-phase flow presented in §4.1 allow to generalize
these results to the three-phase flow.
Confine ourselves to the study of flow in those media whose pore space structure
is described by model I. Consider equilibrium flow of three different fluids. We
shall assign each of them a number from 1 to 3, so that the number is greater
when the wettable capacity of the fluid is lower.
In constructing a model for three-phase flow, the requirement for the flow to be
equilibrium is significant, just as in §4.1. It means that the flow velocities must be
sufficiently small for the distribution of phases in the pore space to be completely
determined by the capillary forces. In this case, if initial saturations of the phases
are approximately equal (i.e., none of the phases is "trapped"), then any future
change of saturations will be accompanied by the rearrangement of the capillaries
in the network, so that the more wettable phase inflates the capillaries of smaller
radii, and the less wettable one inflates the larger ones.
Estimate the typical saturations (S1 , S2 , S 3 ) of the medium with different
phases which admit flows with different numbers of phases. Equilibrium flow of