Page 150 - Percolation Models for Transport in Porous Media With

146 CHAPTER 8. CONDUCTIVITY AND ELECTRIC CURRENT

where
1
1
1
u
A(y') = I 2 (x}(1 - exp(y'- (1- x)- }dx,
0
, 4 t - 2
y=KtT 1 Ao = A(O}
Here w(/ 0 , r, t) is the energy (expressed in degrees) per unit volume released in a
capillary as the impulse passes through it
(8.5}

The z-component of the temperature gradient (we shall denote it by T') can be
obtained by differentiating the expression (8.3} with respect to z and integrating
it with respect to z' and r'. T' assumes its maximal value at the center of the
contribution of the second term in (8.3} can be neglected. After denoting r1 =
r
capillary junction (r = 0, z = 0}. If r2 ~ r1 then T' ~ 0. If r2 ~ 1 then the
r,
we find
1
1
T'(t) =w(/o,r,t)B(y'}A0 0'- , O' = {411"Ktt) 1 1 2 ,
1
1
2
1 2
1
B(y') =I u {x}[1- exp(-y'- {1- x)- }]{1- x)- 1 dx {8.6}
0
In this case the conditions of achieving any of the threshold values Tc or T:
can be written in the following form,
T(t) = Tc, T'(t) = r: {8.7}
Consider the dependencies ofT and T' on the duration T of the impulse for a
fixed energy density in the impulse w(Io, r, t) = wo. It follows from {8.4} that for

1
1
T(r) < wo, T'(r) ~ woBoA0 0'- , Bo = B{O} {8.8}
Evidently, for short impulses, the maximum temperature that can be achieved
in a capillary is bounded by the value w0 of energy density in the impulse. At
the same time, the temperature gradient grows proportionally to w0r- 1 1 2 . In
other words, if wo < Tc then the "temperature mechanism" of cement destruction
cannot be realized for any r, while the "gradient mechanism" is realized for r <
1
(woBoT:- Ao) 2 /{411"Kt) according to {8.7) and {8.8}.
Using the asymptotics for the functions A(y') and B(y') for y' ~ 1 (r ~ Tk),
we obtain
Tk (en)
(71< )3]
1 (Tk )2
T(r) = Cowo [-:;:-In 'YoT + 2 -:;:- - o -:;:- , (8.9)
1 1
T'(r) = C~w0B0A0 r- [ (~) - 11"- 1 (~) + o (~) ]
112
312
512
1 2