Page 60 - Percolation Models for Transport in Porous Media With

52 CHAPTER 3. PERCOLATION MODEL OF FLUID FLOW

calculations. Consider (3. 7) 1 taking the first three terms in the expansion

G = Go(rt) + q1M(rl) + q~N(rl}1

Go(rl} =a' < rl!go(r) 1 a* >1
2 2 (3.39)
M(rl} =a' < r1 1 U~/ (r)r- 1 a* > 1
1
N(rl) = 1r/8[2k0 (rt) + V(rl)J.
In writing out (3.39) 1 we took into account the fact that the contribution of
the quadratic term of q to (3.19) can be neglected in comparison with other terms
in the equation (3.39). First negative terms appear in (3.39) only with the term
proportional to qr 1 and the term proportional to £A does not appear at all.
The value G0 ( rt) can be called the minimal pressure gradient of the r1-chain 1
since when G < G0(rt} 1 the flow in this r1-chain stops because the thinnest cap-
illaries get completely filled with the bounded fluid. After differentiating Go(rl)
with respect to r 11 one can verify that dGo(rt)fdrl < 0 1 since g0(r) is a mono-
tone decreasing function of r. Thus if we decrease G 1 then at the point when
G = G0(a*) the flow stops in the a*-chain 1 and when G < Go(a*) 1 it stops in the
chains with r 0(G) ;::: r;::: a* as well, where ro(G) is found from the equation

Go(ro) =G. (3.40)

If r0 = rc, then the flow stops in the largest r1-chain, i.e., the medium becomes
impermeable. Therefore G0(rc) can be called the minimal pressure gradient for
the given medium. Taking account of the fact that q(r1, G) is found from (3.7) for
those r1-chains with r0 (G) ~ r1 ~ r c and the fact that for the rest of the r1-chains,
a* ~ r1 ~ r0 (G), we obtain the following relationship according to (3.8)
rc
K(G) = j k(r1,G)dn(rl} (3.41)
ro(G)
which is valid in the interval

(3.42)

If G > Go(a* ), then ro( G) in (3.41) must be taken equal to a*. The relationship
(3.34) shows that when G varies within the domain (3.42) the following process
takes place. First of all, the fluid flow begins or stops in the set of r 1-chains
[see (3.40)] and second of all, the permeability k(r1,G) of other conducting r 1-
chains changes. These two factors cause strong dependence of the permeability
on G. If the medium is heterogeneous, then the relationship (3.41) can have a
rather complicated form, and the domain (3.42) can appear very lengthy, a fact
which causes difficulties in determining its boundaries in an experiment. When