Page 115 - Percolation Models for Transport in Porous Media With

108 CHAPTER 6. PORE SIZE DISTRIBUTION

J
I

x,
I
0 s' s;/3
Figure 38: Dependence of the fraction of super-critical capillaries filled with mer-
cury on the saturation of the medium with it

After substituting the value Sc of saturation at the time of the breakthrough
as measured in the experiment and the value Xc = X(Sc) from (1.1) in (6.5), we
VdS' l
obtain the following relationship for the constants z, V, and l of the network,

D
z(D -1) = - exp -! VS' + 71'lr~(S') (6.6)
Sc
1
[
Simulation of the intervening stage. At the second stage of mercury
injection, the non-wetting phase fills the infinite cluster of the super-critical capil-
laries. Concentration N(rk) of such capillaries per unit volume of the specimen is
0.5 Zttzl- 3 W(X), where ltz is the correction due to the geometry of the network
and W(X) is the density of the infinite cluster. Saturation S(rk) at this stage is
equal to N(rk)v'(rk)/~. where~ is the porosity of the medium, or, taking account
of (6.1),
00
S( ) = 11'Z/tz W(X) I 2j( ) d (6.7)
r r
r
rk
mt2X
2
rk
After integrating by parts in (6.7) using (6.2) and differentiating both sides of
the obtained relation with respect to rk, we obtain the equation (6.4) for the func-
tion X(rk) again, where now g(rk, X) = 2Sml 2 X[11'Z/tz
1
W(X)J- •
It is also possible, as at the first stage, to pass to the new independent variable
S in this equation:

dX [r~(S) ( X dW )] - 1
dS = 2X mZ2 71'ZttzW(X) + 2S W(X) dX - 1 (6.8)

Since when X is equal to the threshold value Xc, the density W(X) of the
infinite cluster vanishes, it follows from the equation (6.7) that S(Xc) = 0. This