Page 157 - Percolation Models for Transport in Porous Media With

8.4 CONDUCTIVITY AND PERMEABILITY CHANGE II 153

the following approximate expressions can be obtained based on (8.20)
')'.,.e- 1 r~c(r1)exp[AnFJ(r1)r} ];
2
ro(rl) ~ { (8.22)
2
1
11'- 1 r~c(rl)[1- A~r1 l- 1 F 0 - (rl)r} ]

1
2
2
An = BnTcE0 ; A~ = Bnp'- T:lE0 ;
Bn = 411' C2P2~t(cr~coAo')'.,.crD- 1
2
The condition (8.21) is satisfied when
(BnTc) 1 1 2 Fo(r1 )r} 1
Eo<E1= { (8.23)
1
(Bnp'- T:l) 1 1 2 (r1l- 1 ) 1 1 2 Fo(r1 )r} 1
The quantity E1 qualitatively defines the concept of smallness of the current
amplitude introduced in §8.3. The upper expressions in (8.22), (8.23) are valid
when the "temperature mechanism" of cement destruction is realized ((8.21) is
already valid when Eo/ E1 is of the order several times unity). The lower ones are
valid when the "gradient mechanism" is realized.
Suppose that for t > r~c(rl) in all capillaries with r1 ~ r ~ R(r1, t) one of
the thresholds (8.7) was exceeded, and therefore partial or complete destruction of
cement took place there. Considerations similar to those stated in §8.2 in deriving
the relationship (8.15) allow to determine the amplitude of the current that passes
through the r1-chain at the instant t

(8.24)

where

F2(r17R,t) = r~(< r1,f- (r,t)r- ,R >
1
2
2
+')'.,. < R,r- ,rz > + < Tz,r- ,a* >)
2
After substituting (8.24) in (8.20), we obtain R(r1 , t) as an implicit function
of r1 and t
1
1
2
')'.,.e- r~c(R) exp[Anr~c(R)rk" (r1 )Ff(r1, R, t)r} ],
t = t(R, r 11 t) = { (8.25)
1
2
1r- 1 r~c(~)[l- A~r~c(R)rk" Ff(r11 R, t)r} ]
Obviously, fort= r 0(rl) we have R(r1,r 0(rl)) = r1, and (8.25) becomes (8.22)
and (8.24), (8.11). Denote by Tq the time when the threshold {8.7) is achieved
in the thickest capillary of the r1-chain, i.e., R(r1,rq(r1)) = rz. Suppose that