Page 100 - Percolation Models for Transport in Porous Media With

5.1 IMMISCIBLE DISPLACEMENT 93

- ~·~--------~~

r,

\
-.1...-l~---...r.....-a:·fl
Figure 29: Blocking of the movement of the displacing fluid along the adjacent
thinner capillary

radius distribution function f(r) to the fluid velocity distribution function for r-
chains cj>(V). For example, in the case f(r) = Ar- 3 7J(r - rn), we have cj>(V) =
BV- 2 7J(Vm- V) 17(V- Vn), where 77(*) is the stepwise Heaviside function, and the
coefficients A and Bare determined from the normalization conditions
00 00
I f(r)dr = 1; I cj>(V)dV = 1.
0 0
In this case it should be taken into account that the "cutting-off" of the func-
tion cj>(V) in the region of large V will occur at the point V = Vm, determined not
only by the properties of the medium, but also by the nature of the phenomenon.
Consider the interaction between trees growing with different rates behind the
displacement front, and first of all obtain the condition for the blocking of a trunk
by branches of other, faster-growing trees. Two competing factors influence this
process: on the one hand, more slowly growing trees must be restrained by the
more quickly growing trees; on the other hand, the characteristic distance between
such trees is great, a property that decreases the restraint probability.
Blocking may take place if along the branches of the tree formed by a V 0-chain
the displacing phase reaches the Vt -chain earlier than along the trunk of the latter
(Fig.29). Consequently, the blocking condition has the form

x +aR(Vo) x (5.6)
Vo =vt·

Here R(V 0 ) is the correlation radius of the IC formed by the capillaries, where
the minimal flow velocity is V 0 ; a is a coefficient of the order unity introduced as
a result of the fact that the condition (5.6) is written for an arbitrary V 1-chain in
the mean. Besides, in writing (5.6) it was supposed that branches grow with the
same rate as the trunk.