Page 61 - Percolation Models for Transport in Porous Media With

3.3 RESULTS AND COMPARISON WITH EXPERIMENT 53

G ~ G0(a*) the dependence K(G) becomes weaker. Since the second and the third
terms in (3.39) are small if the condition (3.38) is satisfied and the inequality
V(rt)k0(r1 ) < < r > z-l is also satisfied, for sufficiently homogeneous media
(ud < r >- 1 < 0.1) the dependence k(rt. G) near the permeability threshold G0 (rc)
of the medium can be found to have approximately the following form
1 2
k(rt.G) ~ 1/8ko(rt)(l- Go(rt)G- ) GNm(rt)[1-1/2V(rt)ko(rt)-
1
1/2GNm(rt)(1- Go(rt)G- )], (3.43)
Nm(rt) = 4N(rt)M- (rt) ;: G0 (rt).
1
2
It is evident from (3.43) that the conductivity of an r 1-chain near G0(rt) is
substantially less than its Poiseuille conductivity ko(rt).
Introduce the Reynolds number for a medium

1
Re = tf < 11 > p1p.- • (3.44)
Here < 11 > is the average flow velocity, and the quantity lfl is chosen using
either the average grain size in the medium, or the average pore diameter. If we
substitute K(G) = p. < 11(G) > a- 1 in (3.44), we obtain

(3.45)

If we consecutively substitute into (3.45) the above boundary conditions for
the pressure gradient for different types of flow and the corresponding values of
permeability determined from {3.8), the critical Reynolds numbers for the medium
can be obtained.

3.3 Results of Numerical Calculations and Com-

parison with Experiment

The experiment studied the flow of water (PJ = 10 3 kg/m 3 , p.f PI = 10- 6 m 2 fs) in
a medium whose pore space was simulated by a cubic network (Pg = 0.25, K. = 0.5)
with pores situated in sites and connected with rough capillaries. The experimental
relations found in [53] were used in the construction of the laws to describe the
flow in capillaries and in the determination of the pressure losses on the capillary -
capillary (capillary- pore) junctions: in (3.14), i = 3,j = 6, Km = 3·10- 7 p}p.- 1 =
300 kg-fm 5 ; in (3.15), s = 2,h = 5,Kt = 10- 3 pf = 1 kg/m 3 ; in (3.16) and (3.17),
a'= 1.5,Bi = Bf = l,A0 = 30.
The dependence g0(r) = C0r- 2 was taken from [50], where the value Co= 10 5
Nfm has been found experimentally [56]. To illustrate the dependence of the
function K(G) on the form of the relationship f(r), a numerical experiment was