Page 114 - Percolation Models for Transport in Porous Media With

6.1 MERCURY INJECTION 107

mercury [75].
The initial stage of displacement. Suppose that at the initial stage dis-
placement takes place along non-intersecting tortuous paths. In this case the
fraction of the super-critical capillaries being filled with mercury is
00
X(rk) = 1- F(rk) = J f(r) dr. (6.2)
rk
The probability {i of the first i capillaries being super-critical and the ( i + 1 )-th
capillary having radius rk (i.e., subcritical) is Xi(rk) F(rk)·
The average volume v'(rk) of a super-critical capillary is found as the condi-
tional expectation of the random variable 1rr 2 l (capillary volume) when rk :::; r <
oo. The volume of mercury that has reached the first i super-critical capillaries in
a chain equals iv'(rk)· After averaging this volume with regard to the expression
for probability {i, we obtain the mean volume V(rk) of mercury which has passed
into the chain of total volume V = V(rk = 0). Since at the considered stage the
saturation of the pore space with mercury isS= V(rk)/V, using (6.2) we obtain
the following correlation

(6.3)

After using (6.1) to pass to the variable rk, in the dependence S(LlP), which
has been measured by means of the mercury porometry method, we obtain the
experimentally determined dependence S(rk).
After integrating the right side of (6.3) by parts using (6.2) and then differen-
tiating both sides of the obtained equation with respect to rk, we obtain

(6.4)

After passing in (6.4) to the new independent variableS we obtain an ordinary
differential equation for the function X(S) with the initial condition X(S = 0) = 0.
X(S) = 1 -ex+ I s• : ~;;l(S') l (6.5)
Its solution is

v

From (6.5) and (6.2) we obtain the desired dependence f(rk) = -(1
- X)V(V S + 1rlrn- 1 (dSjdrk).
At the instant when mercury appears in the outer cross-section, the fraction
X of the super-critical pores equals the percolation threshold Xc, and an infinite
cluster of the super-critical poresis formed. The value Xc is determined from the
relationship (1.1).