Page 89 - Percolation Models for Transport in Porous Media With

4.4 THREE-PHASE FLOW 81

and mutual orientation of the pores and the channels in the medium are random.
Random structure of bonds is most easily described by a simple cubic network.
Since there are no data at present in favor of any other network type, we shall
consider the network to be cubic, as in the previous studies. In this case, from
{1.1) we find {c = 1/4.
It follows from the relationships (4.44) that in model I, Si = {i· If we sketch
the critical values of saturation obtained in a triangular diagram (fig.25) and draw
dotted lines inside the triangle S1S2Sa, we get a series of domains which differ
from one another by the number of phases taking part in the flow. The numbers
1 in fig.25 denote the domains of one-phase flow, when two phases out of three
are "trapped" (there is no IC for these phases). Given the ratio of the saturations
corresponding to domains 1, only the i-th component flows, where i is the vertex
of the triangle. In the domains marked with number 2, one of the three phases is
"trapped," and two-phase flow of fluids i and j takes place, where i and j are the
indices of the sides SiS; of the triangle adjacent to the domains 2. The domain 3,
where flow of all three phases is possible, is situated in the center of the triangle.
It is clear from the diagram that three-phase flow is only possible in a small
neighborhood of the center of the triangle S1S2Sa. Note that equilibrium four-
phase flow is impossible in the case of a cubic network, since conditions like {4.45)
cannot be satisfied for four phases simultaneously. One of the phases in this
case would have to flow in a disconnected form. However for big coordinational
numbers of the network (z > 6), the percolation threshold of the system drops
and the equilibrium flow of four and more phases is possible, at least in theory.
It appears interesting to compare the obtained theoretical result to the exper-
imental data.
Unfortunately no experimental study of three-phase equilibrium flow for the
case of J.ti = const (i = 1,2,3}, X1coslh > x2cos02 > xacosOa was carried
out before. The only known experiments are those of Leverett (1940) (66], where
viscosities of the phases differed substantially (p,1 ~ 1'2 » J.ta). The results of these
experiments are put on the same triangular diagram of saturations for comparison
(continuous lines inside the triangle S1S2S3 ).
It can be seen that the areas of domains 3 are approximately the same both
in the theoretical investigation and in the experiment; however, the triangle found
experimentally is situated somewhat farther from the vertex Sa. The domains 2
adjacent to the sides S1Sa and S2Sa are deformed in the same direction. This can
be explained by dynamic effects.
For example, due to its significantly smaller viscosity and consequently, its
greater mobility, the third phase can break through the rock on the dynamic stage
of flow and thus isolate the "parts" occupied by the less mobile phases 1 and 2.
As a result, in a broad range of Sa {1 > Sa > 0.35), one-phase flow of the third
fluid takes place. When Sa < 0.35, i.e., when the rock contains a sufficiently small