Page 90 - Percolation Models for Transport in Porous Media With

82 CHAPTER 4. MULTIPHASE FLUID FLOW

amount of the third phase for the difference in the mobility of the phases to stop
determining the nature of their distribution in the pore space, the theoretical and
the experimental diagrams of different domains of three-phase flow coincide with
accuracy of R~ 10%. This fact speaks of a good agreement between theory and
experiment for the discussed case of equilibrium three-phase flow.
We shall now present the calculations of the coefficients of relative phase per-
meability in different domains of the flow. Obviously, in the domains 1 of the
triangular diagram, only the relative phase permeabilities of the corresponding
i-th phases do not vanish (they equal 1). In the domains 2, as it was mentioned
above, two-phase flow of the i-th and the j-th phases takes place. Coefficients
of phase permeability for these phases can be found using relationships ( 4.3) and
(4.5). Generalization of these relationships to the case of three-phase flow per-
mits to determine the relative phase permeabilties in the domain 3. Using the
conditions (4.43) enables one to cut out the part fi(r) of the general probability
density function f(r) corresponding to the fraction of the capillaries containing the
i-th phase. In the case of three-phase flow fi(r) have the following form (before
normalizing)
0,
{
h(r) = f(r),
( 4.46)

After substituting /i(r) for f(r) in (2.1), we find the absolute phase perme-
abilities Ki(r1, r2) as functions of the quantities r1 and r2, which characterize the
domains of saturation of the capillaries with the i-th phase. Here the satura-
tions of phases are set by the relationships ( 4.44). Coefficients of relative phase
permeability are calculated using the formula

(4.47)

They are completely defined by the radius probability density function for
capillaries and by the percolation threshold of the system, which depends on the
network type (coordinational number z).
Thus the expressions (4.44), (4.45), (2.1), (4.46), and (4.47) allow to calculate
the relative phase permeabilities in the domain 3 using the relationship ( 4.43).
In the special case, r1 = r2, we have h(r) = 0, and two-phase flow is realized
directly along the side S1S3 of the triangle S1S2S3. If, however, T! = T2 and
some ~i < ~c. then two-phase flow takes place in the domains 2 with a trapped
i-th phase. Note that the phase permeabilities k1 (r1 = 0, r 2 = r 1 ) = k1 (rl)
and k3(r1 = r2,r2 = oo) = k3(r2) are actually functions of merely r1 and r 2 ,
respectively, and are calculated uniformly for all domains on the diagram. The