Page 67 - Percolation Models for Transport in Porous Media With

4.1 FLOW OF IMMISCIBLE FLUIDS 59

By substituting h(r) into (2.1), we can find K2, the permeability ofthe macro-
scopic volume under consideration, to the displacing fluid. Using the relationships
for the absolute permeabilities K 0 and K 2 of the medium, one can find the an-
alytical expression for the relative phase permeability k2(rk) of the displacing
fluid. Relative phase permeabilities are defined by the relationships ki = Ki/ Ko
(i = 1, 2), and therefore can be found from (2.1) and (4.2)

Note that the resultant expression for relative phase permeability contains only
the radius probability density function for capillaries and the percolation invariant
~c = P: which depends only on the network type and the dimension of the problem.
Thus the resultant expression ( 4.3) is valid not only for a cubic network, but also
for networks of other (in fact, any) types and dimensions. The network type
affects the value of ~c. while the dimension of the problem affects the value of the
exponent v.
To determine the relative phase permeability k1 , consider the structure of the
ICD in a similar fashion. Obviously the ICD contains those capillaries from which
the wettable fluid can by no means be displaced for a given value of t:!.p, i.e., the
capillaries of radii r < rk. Furthermore the ICD contains the capillaries with no
displacing fluid in them whose radii exceed rk. The fraction of such capillaries is

00
a(rk) = {- W({), { = K-j f(r)dr
rk
Here the quantity~ defines the fraction of capillaries which satisfy the condition
r > Tk, and the function W(~) defines the fraction of those of such capillaries
contained in the ICG.
Near the percolation threshold { = {c, the asymptotics {1.6) W(~) "' I~- ~ci.B
is valid, where f3 = 0.4 when D = 3. Hence~- W(~) ~ ~c "'(1 + 2) · 10- 1 when
I~- ~cl « 1. However, starting from the values~- ~c ~ 10- 1 and further, up to
~ = 1, the dependence W(~) becomes linear (W(~) =~)very quickly. This means
that a( rk) = 0 in the outlined range of {. Thus the coefficient a( rk) is of the order
of 10- 1 only in the closest vicinity of the percolation threshold when the ICG is
formed, while further on, with the increase of~. it goes to zero very quickly.
Consider equilibrium flow, i.e., the process of the ICG formation when the
percolation threshold { ~ {c is crossed, is not being analyzed. The study is carried
out under the assumption that the dynamic stage of growth of the chains forming
the ICG is finished, and that the cluster is already a sufficiently stable formation
in the space. Formally this means that the quantity ~ is finitely separated from