Page 69 - Percolation Models for Transport in Porous Media With

4.1 FLOW OF IMMISCIBLE FLUIDS 61

I

Figure 17: Phase permeability curves calculated using model exponential (a) and
log-normal (b) distribution functions

In model II the quantity S1 is defined by the volume of capillaries which conduct
the first fluid. In this case

(4.8)

Thus the formulas (4.3) and (1.7) for a1 = rc, pb = ~c and (4.5)- (4.8) define
parametric dependencies of the relative phase permeabilities on saturation of the
medium for the two models mentioned above.
Note that the probability density function for capillaries, together with (4.8)
or (4.7), defines Leverett's function Pk = (Sl), which sets the correlation between
the capillary pressure and the saturation of the specimen during equilibrium flow.
The phase permeabilities and Leverett's function make up the complete set of
data necessary for calculating the two-phase flow. To define the function J(r)
in the form most convenient for future computations, investigate the behavior
of Leverett's function Pk = (S1) (its typical form is presented in fig.16). Using
the following formula for the derivative, dSl/dr = dSddPk · dpk/dr, and taking
account of the fact that Pk "' r- 1 , we obtain the following estimate from either
f(r) "'A' I
(4.7) or (4.8)
dS1 (r) I~
dpk r 2
where A'= r- 2 in the case (4.8) and A'= 1 when S1 is defined by formula (4.7).
Since today there are virtually no reliable ways of experimental determination
of the probability density function for capillaries with respect to values of intrinsic
conductivities, it seems reasonable to carry out qualitative analysis of the phase
permeabilities for a model probability density function. Since J(r) -+ 0 as r-+ oo,
in the general case this function can be represented in the form of an expansion