Page 55 - Percolation Models for Transport in Porous Media With

3.2 EFFECTS OF PORE SPACE STRUCTURE ON FLOW 47

3.2 Effect of Pore Space Structure on Laws for
Macroscopic Flow

The foregoing relations (3.13), (3.14), (3.15) for flow types in the capillaries and
(3.16) for the pressure losses on the junctions allow to specify the form of the
equation (3.7) which relates the flow q through an arbitraryr1-chain (a*~ r1 ~ re)
to the pressure gradient G in the medium.
Consider the second term, J(r11 q), in (3.7). In model I each capillary of radius
r joins with the pore of size lp ~ r both at its entrance and exit. After setting
r' = 0.5lp and passing in (3.17) and (3.18) tor' ~ r(A(r, r'O ~ Ao, €(r' fr) ~ €o),
we obtain the following from (3.16) and (3.6)

(3.19)

V(rt) = "'( 01 < r1,r- 3 ,a* >,
I'(rt) = d 01 < r1,r-\a* >, (3.20)
"Ya = 0.5 o:' l- 1r- Ao,
1
1
da = o:' z-17r-2€()2[Bi + (1- €o)2 Bi].
For model II, after averaging (3.16) over r' in accordance with (3.4) with further
substitution of the averaged result into the second term of (3.5) (and then averaging
the latter with respect to r), we shall arrive back at (3.19), but with different
functions V ( rt) and I' ( rt)

3
3
V(rt) = "Yb < r1, [< r1, A(r, r')/r' , r > +r- < r, A(r, r'), a* >],a* >,
I'(rt) =db< r11[< r-l,(I(r,r')/r' ,r > +r- < r,({(r,r'),a* >],a*>, (3.21)
4
4
"Yb = 0.250: 7r- 1 l- 1 , db= 0: 11"- 2 1- 1 .
1
1
Introduce the following notations
Tm(q) = d1q, rd(q) = d2q, (3.22)

di = (0.511" ReiPt/tL)- , i = 1, 2,
1
*d-1 *d-1 d- 1 l d- 1
Qt =a 2 , Qt m = a 1 1 Qm m = T1 2 1 Q m = T1 1 ·
We shall now specify the form of equation (3.7), taking into account the rela-
tionship (3.19) for various pressure gradients, starting with the interval of large
g's.
Turbulent flow. When rd(q) ~a*, q ~ Qt· In this case, according to (3.22)
and (3.19), Re(rt) ~ Re(r) ~ Re(a*) ~ Re2 , and therefore the flow in all capillaries
of the r1-chain is turbulent.