Page 122 - Percolation Models for Transport in Porous Media With

6.2 ELECTRIC POROMETRY 115

After passing to the limit L -+ 0 we have

Consider the function X =< r 2 > L / < r 2 > L=O which can be expressed in
terms of u(L) as well as f(r)
u(L)L I
r(L)
2
2
X= lim[u(L)L]= f(r)rdr/<r >L=O (6.16}
L-+0 o
Obviously lim [u(L)L) = uoS* (uo is the specific electric conductivity of the
L-+0
completely saturated core). Therefore X = u(L)Lu0 (S*)- 1 is uniquely related
1
to r(L) and can be measured experimentally.
On the other hand, after differentiating the equality (6.16) with respect to r(L)
00
and taking account of the normalization condition J f(r) dr = 1 we find
0

1
f(r(L)) ; r- (L )(dX/dT) { l r-'(L )(dX/dr) dr(L)} - (6.17)
2

Taking into account the correlation between r and L determined by the ex-
pression (6.14) and the correlation between X and u(L), it is possible to pass to
the new variable Lin (6.17}

(G' = 2xcos0fpg)

I(~) = ~: ~L[u(L)L] { l L'~[u(L)L]dL} 1 (6.18}
-

Thus upon measuring the integral conductivity u(L) experimentally for the
corresponding sequence of the values of L, we can determine the radius PDF for
capillaries uniquely using the formula (6.18).
If we give up the severe constraints on the network type and orientation and
consider these parameters arbitrary, then the ICP approximation proves to be
unacceptable, and a percolation model should be used. However the direct and
reverse problems of electric porometry in this case become substantially more
complex.
One of the possible approaches to the determination of the PDF for capillaries
in this more general setting is as follows. The NM is considered in an approxima-
tion, admitting an analytical solution of the direct problem, so that the analytical