Page 106 - Percolation Models for Transport in Porous Media With

5.2 VISCOSITIES AND INTERFACIAL TENSION 99

microscopic point of view, v, is the velocity averaged over the r-chains [14] with
r > r 1 , where r 1 is the minimum radius capable of admitting the displacing phase.
Using the technique of the effective radius distribution for chains, we can calcu-
late the concentration of r-chains in the direction of the flow n( r) [25] and after
summing up over all chains, we obtain

During the flow of the front, in some vicinity, the formation of "traps" is
observed, i.e., the restraint of a displacing phase in the chains, where flow runs at a
velocity less than the speed of overlapping of the chains with "branches" of rapidly
growing "trees" (Figs. 28,29). In essence, this is the process of infinite cluster
formation. The skeleton of this cluster, according to the Shklovsky-de Gennes
model is a network of irregular form with the characteristic period (correlation
radius)

The correlation radius represents the characteristic size of traps, and the char-
acteristic time of their closure is

(5.15)

Here two processes compete. The maximum velocity of the interphase move-
ment is realized in the thickest chains (the rc- chains). At the same time R(rc) -+
oo; therefore, they cannot interact with forming of traps. It is obvious that phys-
ically the situation which is realized corresponds to the minimal time of restraint
T = r., whereas from the condition of achieving the minimum value, dr/dr = 0,
the corresponding minimum radius of chains r * in the skeleton of the IC of the
displacing fluid at the moment of the trap formation can be determined.
In the chains with r < r * the ratio of the volume occupied by the displacing
fluid to the total volume of the chains at the moment of restraint is

f = V(r)/V(r.) = r fr~,
2
where V(r) is the flow velocity in the r-chain. In this case the mass of the IC of
the displacing phase is