Page 152 - Percolation Models for Transport in Porous Media With

148 CHAPTER 8. CONDUCTIVITY AND ELECTRIC CURRENT

through an r1-chain during the first impulse, we have

1
Io(rt) = Eou't6.r~F0 - (rt), 'Yrr = uUu~, (8.11}
Fo(rt) = r~( < Tt, r- , Tz > 'Yrr+ < Tz, r- , a* >) (8.12}
2
2
Here 6. ~ 3 + 4 for model II and 6. ~ 6 + 8 for model I. By differentiating
1 0 ( rt) with respect to r1, one can make sure that Io ( rt) is a monotone increasing
function of r1 (for fixed E 0 ). The hierarchy of r1-chains with respect to the size
of the thinnest capillary coincides with the hierarchy of r1-chains with respect to
the value of their average electric conductivity.
Since Ic(r, r) are monotone increasing functions of rand T (this can be verified
by differentiating (8.10} with respect to r and r}, it follows that the threshold
Tc or T: will be exceeded in the r1-capillary of the r1-chain. The minimal field
intensity E.(r1,r} for which this happens can be found by setting (8.10} equal to
(8.12} for r = r1
(8.13}
Similarly, the maximal field intensity E*(rt, r} for which the threshold (8.10}
will be achieved in the thickest non-conducting capillary- the rz-capillary of the
r1-chain- can be found by setting (8.10} for r = rz equal to (8.11}
(8.14}

For Eo < E.(r) neither of the thresholds, Tc and T:, is going to be exceeded
in any non-conducting r1-chain, and therefore the same holds for all capillaries in
the medium that have the same property. Correspondingly, for Eo > E* ( T) one of
the thresholds, Tc or T:, is going to be exceeded in all non-conducting capillaries
of the medium. Thus E.(r} and E*(r) can be called, respectively, the minimal
and the maximal field intensity for the medium.
Consider the case when the relationship E.(rt,T) <Eo < E*(r1,r) is valid
for an r1-chain. Let m > 1 impulses of current have passed through the medium.
Denote by R( r1, m) the radius of the thickest capillary in the r1 -chain where the
threshold (8.10} was achieved as the m-th impulse passed through it. Suppose
that R( r1, m) < r z (for m = 1, R( r1, 1) = r1). In this case in all capillaries with
r $ r1 $ R(rt,m) the threshold {8.10} was exceeded and therefore complete or
partial destruction and ejection of cement took place in these capillaries. Since
u~ > u~, the fluid that replaces the cement increases the electric conductivity
of the capillary. The cement ejected from the capillary gets into the adjacent
pore in model I or a thicker capillary {if filled with fluid} in model II. These
phenomena decrease the electric conductivities of the latter a little. Let the electric
conductivity of the capillary increase by t:(r, m) times after the m-th impulse has
passed through it {1 $ t:(r, m) $ 'Yrr if no additional destruction of capillaries
happens, and t:(r, m) > 'Yrr otherwise). The amplitude of the current flowing