Page 98 - Percolation Models for Transport in Porous Media With

5.1 IMMISCIBLE DISPLACEMENT 91

displacing phase.
It is obvious that taking into account all peculiarities of the interaction be-
tween the trees in a three-dimentsional case will require an extremely cumbersome
mathematical apparatus for its description. At the same time, considering the
plane case of this problem will allow to simplify mathematical models, but will all
the same reflect the main features of the phenomenon.
Model of the medium. We shall consider the network model of a hetero-
geneous medium. Conducting bonds (capillaries) of the network are distributed
chaotically in it and their distribution according to the value of effective hydraulic
radius is described by an arbitrary normalized probability density function f(r).
Suppose that before the displacement began the network was completely saturated
with the displaced phase, whose viscosity is J.£1, and the initial pressure, Po. At the
time t = 0 the displacing phase, whose viscosity is p,2 , is supplied under pressure
P to the network boundary x = 0, and the displacement begins.
Consider the case when (P- Po} ~ Pk, where Pk is the capillary pressure.
During the description of the displacement we shall neglect flow tongues formed
at the breakthrough of the displacing phase along a finite sequence of connected
"thick" capillaries, since these "tongues" attenuate quickly. The velocity x f of the
phase interface, beyond which the saturation of the displacing phase is non-zero,
is determined by the average conductivity of an infinite capillary chain, composed
from the largest capillaries whose concentration is high enough for an infinite
cluster to be formed in the medium.
The probability of the radius of a capillary in the network exceeding r 1 is equal
to
00
W(r > rl) = I f(r) dr.

If W(r > rt} exceeds the percolation threshold We in the network, an infinite
cluster composed of capillaries that satisfy the condition r > r 1 is formed in the
network. Thus the capillary chain which determines the location of the front x f
must consist of those capillaries that satisfy the condition r > re, where Te is found
from the condition of the formation of an infinite cluster
00
I f(r) dr =We. (5.1}

The chains containing capillaries with r > r 1 form an irregular network, whose
characteristic period (correlation radius) is determined by the expression

(5.2}
where d is the period of the network, v is the correlation radius index and depends
on the dimension of the problem [4]. The concentration of the conducting chains