Page 99 - Percolation Models for Transport in Porous Media With

92 CHAPTER 5. NON-STEADY STATE TWO-PHASE FLOW

Figure 28: Diagram of the "tree" formation in the "forest growth" model

with r > r1 {the number ofthe conducting chains intersecting a unit surface) N(rt)
equals R 1 -z(rt), where z is the dimension of the problem. As for the density of
the concentration of r 1-chains, where r1 changes from r1 to (r1 + drt) equals
[ l
n(rt) = -dN(rt)fdr1 and, using (5.1), (5.2), is determined by the expression

v(z-1)-1
1
n(rt); v{l- z)f(r1 ) l f(r)dr d -• {5.3)

"Forest growth" model. Consider the interaction between trees during their
growth in more detail. At the micro level the average flow velocity along an r-chain
under the conditions p 1 ~ p2 and Xf « l, where lis the characteristic size of the
region of the applied pressure difference, is determined from the Hagen-Poiseuille
formula
{5.4)

This chain joins two opposite ends of the specimen, and therefore in order
of magnitude its length coincides with that of the specimen, and the pressure
difference applied to the ends equals the pressure difference applied to the given
specimen. As is evident from {5.4), the maximum average displacement velocity
in an r c-chain is
(5.5)

Knowing the flow velocity along an rc-chain {5.5), we can determine the posi-
tion of the phase interface at any instant t; behind it, saturation of the displacing
phase is non-zero
Xj = Vmt.
Since the average flow velocity along the chain for a given pressure gradient is
uniquely related to its effective radius, it is possible to go over from the capillary