Page 93 - Percolation Models for Transport in Porous Media With

4.5 STABILITY 85

F;r 8
1 {\

Figure 26: Effect of the oscillatory perturbation of the function f(r) (a) on the
permeability change (b)

Introduce the notations, Ko = K2(rk = 0); t/Jo = (Kt- K(;)/A',
a* a*
¢11 = (Kt- Ki)/A'; s+ =I j+(r) ~:' s_ =I ,-(r) ~:

r;T" r;;
a* a*
S"!:. =I j+(r) ~:; f* =I lw(r)ldr

After substituting the critical radii r"[ and r; into (1. 7) we can write the following
r+ r+
1 2
j j+(r) dr ~(a*- a*) 1 11 w(r)ll, j f-(r) dr ~(a*- a*) 1 1 2 11 w(r)ll (4.52)

Suppose rc+ ~ r;. Then the following estimate can be written using (4.51)
and (4.52) for Tk < r; {the intervening computations are left out)

I¢1II ~ Aoe* (4.53}
1
where Ao = S.+ +4(8"!:.)- 1 +a; 4 (S_S~)- 1 .
0
When rk ~ r; Ki = and
(4.54}
where A1 = s; • In the case of r; > rj (4.53) and (4.54) remain valid up to the
1
interchange of the indices "+" and "-" in the expressions for Ao and At. Thus
(4.53) holds for all rk and we finally obtain the following
(4.55)