Page 124 - Percolation Models for Transport in Porous Media With

6.2 ELECTRIC POROMETRY 117

Now, taking into account the normalization condition for J(r), we obtain the

resultant expression for r q
r(L)
j r [{z/2- 1)r~ + r r f(r) dr = 2/ z (6.20)
2
1
2
0
We shall further set z = 6, which corresponds to the simple cubic network. In
this case the equation (6.20) becomes

r{L)
2
j r (2r~ +r )- f(r)dr = 1/3, (6.21)
1
2
0
which implies, for instance, that for the given network type the percolation thresh-
old ~c determined in the effective medium model equals 1/3. The value of this
quantity found in percolation theory is 1/4. To obtain a reliable criterion of appli-
cability for the EMM take ~c = 1/3. In this case the equation (6.21) can be used for
describing the conductivity of the considered NM within the ranger' < r(L) < oo,
where r' is found from the condition

r'
j f(r) dr = 0.4. {6.22)
0
If the radius PDF for capillaries is known, it is possible to find rq(L) from
(6.22) using (6.19), and therefore find the conductivity

q(L) = €1e?r[rq(L)/l](S* / D.L) (6.23}

(we consider the intervals of the measurement of D.Li equal to D.L all), and thus
solve the direct problem of the EPM for the chosen NM.
Now investigate the reverse problem of the EPM for the given NM. Suppose
that the values of q(L) have been measured in the course of the experiment at
j different heights satisfying the condition (6.22) and determine the radius PDF
for capilaries according to these data. Using formulas {6.14) and {6.23), u(L)
can be easily recalculated into effective radii rq{r(L)). As a result, we obtain a
mathematical problem of solving a Volterra integral equation of the first type,
which is the relationship (6.21) for the unknown function f(r). The difficulty of
this problem lies in the fact that the analytical dependence of the kernel of this
equation on the upper limit or, more exactly, on rq(r(L)), is unknown. Moreover,
the function rq(r(L }}, which is a part of the kernel, has to be found experimentally,
and therefore is always known up to a certain error of measurement. Given these
conditions, the problem of finding the solution of the integral equation (6.21) is