Page 91 - Percolation Models for Transport in Porous Media With

4.4 THREE-PHASE FLOW 83

outlined phenomenon was observed in Leverett's experiments, where it was also
noted that the permeability of the most wettable phase (water) depended only on
the saturation with this phase and was insensitive to the ratio of other phases.
Find the phase permeabilities for the model probability density function (4.9)
when a*fa* «: 1, i.e.,
{4.48)
Assume further that a* = 1. After substituting (4.48) into {4.44) we find the
1
correlation between r1 and r2 and the saturations of the phases, S1 = 1 - r1 ,
1
1
1
S3 = r2 , S2 = 1-S1- S3 = r1 -r2 . Clearly, it is reasonable to find the relation
for k2 right away, since the values of k1 and k3 can be obtained from it by means
of the corresponding passages to limits. We can find the correlation between the
critical radius r~ of the function h(r) and r2 from (1.7), using (4.46) and (4.48).
To simplify the calculations, we take v = 1 instead of the actual v ~ 0.9. In
this case, after substituting r~ into (2.1), using (4.46) and (4.47) and neglecting
the terms ...... (r~ fr2 ) 5 , we find the coefficient of phase permeability for the second
phase,

( 4.49)

To determine the phase permeability k2 , it is necessary to know any two of the
three saturations, which are related through the customary relation, S1 + S2 + S3 =
1.
The values of k1 and k3 are obtained from ( 4.49) by means of the passage to the
limit from a three-phase system to the two-phase one. It helps in the case of k1 to
let S1 approach zero (rt --+ 0). In this case we obtain a two-phase system, where
the part of S1 is formally played by the saturation S2 , while the phase "interface,"
as far as f(r) is concerned, is r2 • The quantity r2 is uniquely determined by the
saturation S3 , which equals 1 - S1 for the considered case. Therefore we find
k1(St) = k2(0, 1-St), or
2 1 { 1- S1 2
( - St +'c)
k 1 (St) = 27 (1 - S1 + 'c)2 1 - 2(1 - St +'c) - 3 1
X [1- jc2- 2St +'c)+ ~(1- s1 + 'c)(1- st)]} 'fJ(St- 'c)

Similarly, in the case of k3 we let S3 approach zero {r2 --+ oo). Now S2 is
formally S3 , while S1 should be replaced by 1- S3. We thus find that k3{S3) =
k2(1 - s3, 0), i.e.,