Page 84 - Percolation Models for Transport in Porous Media With

76 CHAPTER 4. MULTIPHASE FLUID FLOW

o~~-'~~~~ u~--~~~---J
' 41 ~ 1
Figure 21: Curves for the phase permeabilities calculated for models I (b) and
II(a). Code: a

If (1 - ~~:) < {c, then as Pic < 0 increases, for some Pic = Pea < 0, the ICA
must cease to exist, since the capillaries of the second type are not sufficient for
the ICA formation. If, however, (1- ~~:) > {c, then the ICA does not disappear up
to Pic = 0 (r1c1 = oo), and the flow of phase a stops only when Pic > 0 , i.e., when
Pea= Pc2a, and Tela= oo. Thus we obtain the following expression for ka in the
interval Tc <Tiel <Tela

2 1
ka(r~cl) = A'(1- ~~:) K0 7o [ 7°/a(r)dr] v !a(r') ( j f(r)dr

0 ~ ~
-K J"(r)dr) [{1-K) lf(r) ~ +] /(r) ~ Pr',

rc <Tiel <Tela. Pel <Pic < min{O,Pca}
If {1 - ~~:) > {c, then in the case when Pic > 0, only those of the capillaries of
the second type are filled with phase a, whose radii r < r1c2, where r~c2 is found
from (4.25) if Pic is known. As Pic increases, the quantity ka(r~c2) keeps falling until
it vanishes at the point r1c2 = rc2a 1 where rc2a is found using formula (4.36). In
the interval 0 <Pic < Pc2a (rc2a < r1c2 < oo), ka(r1c2) is defined by the relationship
{4.37), and the function rc4 (r~c2) is defined by {4.35).
Thus the obtained formulas describe the change of ka and kb for the discussed
model of the micro heterogeneous medium in the whole range of the parameters
describing micro heterogeneity, 0 ~ ~~: ~ 1, -1 ~a~ 1.
Calculations of the relative phase permeabilities ka(Sb) and kb(Sb) were carried
out for different probability density functions f(r) and different values of param-
eters a and ~~:, based on the formulas obtained above. Here saturation with phase
b (Sb) was defined as before for the two limiting cases, model I and model II, and