Page 102 - Percolation Models for Transport in Porous Media With

5.1 IMMISCIBLE DISPLACEMENT 95

zone is equal to zero. Finally, when x 1 > x > x(Vt) we have the dynamic stage of
the flow.
Now estimate the values of saturation for the porous medium in these zones.
When x < x(Vt) the fraction of the displacing phase in the V -chain at the moment
of blocking of the V 1-chain by the Yo-chain is t: = V /Vt, and therefore the fraction
of the trapped displaced phase in this chain is 1 - t:.
After averaging over the region of characteristic size R(Vo), using {5.1)-(5.3),
we have
Vt
I {1 - V/Vt) n(V) B(V) dV
0
So(x) = .::..-..,..,.------- {5.9)
Vm
I n(V) B(V) dV
0
where B(V) = 1 in the pore model, and B(V) = V in the capillary model [26].
In the region x > x(V 1 ) trees grow practically without interaction. Around
each trunk a "crown" is formed, i.e., a zone where the first phase has already
been displaced from {Fig.30). Assuming that crowns have a triangular form and
grow around the tree trunks with constant rates, we find the distribution of the
saturation of the medium with the displacing phase in this region. The displace-
ment zone begins to form around a V -chain after the length of the tree trunk in
the course of its growth exceeds the considered coordinate x(Vt ). The transverse
dimension of the region is proportional to the quantity

l(V) = Xj(V/Vm)- X,
which characterizes the distance between the tree top and the plane with coordi-
nate x. Therefore the saturation by the displaced phase in the region x1 > x >
x(Vt) equals
Vm
S = 1 - {3 J n(V) l(V) B(V) dV {5.10)
V(x2)
Here the coefficient {3 is determined from the condition of matching solutions
(5.9) and {5.10) for x = x(Vt ); x 2 is the current coordinate, measured from the
level x = x(Vt) (Fig.30), and V(x 2 ) is the velocity of the phase interface in the
chain, where it manages to move up to the levetx2 • The velocity is determined
from relationship (5.8)

(5.11)

Thus, the system (5.6) - (5.11) allows to determine the distribution of the
saturation of the displaced phase for an arbitrary position of the front x 1, if the
probability density function Q>(V) is known.