Page 51 - Percolation Models for Transport in Porous Media With

3.1 FLOW AT THE MICRO LEVEL 43

After changing sums in (3.2) to mean values of the summands in the r 1-chain
for a fixed q, we obtain the following, using notations in (3.3) and (3.4)

where a' = LL0 ~ 1.5, L is the length of the chain.
1
For model I, as in (3.5), we have

Here !l.Pck and !l.ppk are the pressure drops on the k-th capillary of length l
and on the pore of size lp following this capillary, respectively. Summation is taken
in the direction of the pressure increase. !l.p~k is the pressure drop on the entrance
to the k-th capillary, and !l.p§k is the pressure drop on its exit. Estimates show
that since r 1; 1 « 1, the pressure drops !l.ppk in the pores, as well as the pore size
lp dependencies of !l.pfk and !l.p~k• are negligible. In this case the pressure drops
on the entrance to and the exit from a capillary depend only on its radius r and
flow q. Denote them by !l.p~(r, q) and !l.pf (r, q), respectively. As in (3.5), we have

G =a" < r1, Yc(r, q), a* > +a"z- < r1, (!l.p~(r, q) + !l.pf (r, q)), a* >,
1
1
a"= a'l(l + lp)- . (3.6)
Denote the first terms in {3.5) and {3.6), i.e., the average pressure losses per
unit length of a capillary, by C{r1 , q). The second term in {3.5) and {3.6) (notation:
J(r1,q)) describes the average pressure losses per unit length of heterogeneities,
such as junctions of capillaries and sites. For model I, it is a capillary - pore
junction; for model II, a capillary - capillary junction.
In this case expressions {3.5) and {3.6) can be written in the usual form

{3.7)
The relationship (3. 7) is an equation for flow q through an r 1-chain with fixed
G, and its solution is q(r1,G). If the number of r 1-chains is found from {1.8),
then the conductivity k{r1,G) of the r 1-chain and the permeability K(G) of the
medium are expressed as follows
re
k{r1,G) = Jtq(r1,G)G- , K(G) = j k(r1,G}dn(rt), {3.8}
1

where Jt is the dynamic viscosity of the fluid.