Page 186 - Percolation Models for Transport in Porous Media With

184 CHAPTER 11 GAS COLMATATION IN ELECTRIC ACTION

T(t, oo) = qp{1- exp[-(y')- 1 ]} {11.10)
T~(t, z) = qpvft/(47rK-t) [Abo)- Abo+~)] {11.11)
Abo)= 2e-"Y 0 - 2y'?FY0[1- ~(J10)], 'Yo= z f(r y'), ~ = 1/y'
2
2
The integral exponential function is
"'
1
Ei(-x) = J C e-{~ (11.12)
00
Using the asymptotics for the functions (11.6) and (11.12), the following asymp-
totics forT, t, and T' can be obtained based on (11.9)- {11.11)
T() () { 1- (To/t)exp(-To/t), t ~To, y' ~ 1
t = Qp t (Toft) ln[et/bTo)]- Toft, t ~To, y' ~ 1

where 'Y = 1.78 is the Euler- Masceroni number; To= r /{4K-t)i
2
T(t) (t) { 1 - e-1/y'' y' ~ 1, t « To
~ qp 1/y'- 1/(2y' ), y' ~ 1, t ~To
2
.p/?-exp(-z fr y') (1- ~~), 1 ~ y' ~ (zfr) 2
2
2
T'(t) ~ q 0 (t)r { z z
2 exp(-z 2 /r 2 y')),
r (1-
"-t rz v;y; 1 ~ (z/r) 2 ~ y'
It follows from (11.10) and (11.2) that

8Tjor < 0; 8Tj8r1 > 0 {11.13)

We thus conclude that the greatest rate of the temperature change is assumed
in the thinnest r 1-capillary of the r 1-chain. Also, the capillaries with radii equal
to r have greater values of t in the larger r 1-chains. Therefore the greatest rate
of the temperature change in the medium is observed in the thinnest capillaries of
the thickest r1-chains, i.e., in the r c-Capillaries of the Tc-chains.

11.2 Movement and Growth of Bubbles in Cap-
illaries

Inequalities (11.13) imply that in the thin capillaries of all q-chains the temper-
ature grows faster than in the thick capillaries of the same chains. If we take into
account the temperature dependence x(T) of the interfacial tension, we can state
that the thin ("hot") capillaries become stable attraction centers for the bubbles
coming from thicker ("cold") capillaries.