Page 56 - Percolation Models for Transport in Porous Media With

48 CHAPTER 3. PERCOLATION MODEL OF FLUID FLOW

Equation (3. 7) becomes

G = o:' < Tt.9t(r,q),a* > +J(rt.q). (3.23)
The least absolute value of the gradient Gt(rt) when (3.23) still holds can be
found by substituting qt into (3.23)

Gt(rt) = o:' < Tt.9t(rt.qt),a* > +J(rt,qt) G > Gt(rt). (3.24)

Substituting (3.15), (3.19) into (3.23), (3.24) confirms (3.10) and (3.11) and
results in the following inequality
dGt(rt)fdrt < 0.

Hence, as G goes up, the first to transfer to turbulent flow are the rc-chains
and the last are the a*-chains. Moreover for heterogeneous media, the interval
where Gt(a*) ~ G ~ Gt(rc) can be rather lengthy. If G > Gt(a*), then the
flow in all capillaries in the medium is turbulent. After substituting (3.15) with
s = 2, h = 5 [53) into (3.23) and neglecting the first term in (3.19), we find from
(3.8) that
rc
1 2
K(G) = ~tG- 1 j<o:'Kt < r 1 ,r- ,a* > +0.5rho 1 I'(rt))- 1 1 2 dn(rt). (3.25)
5
a.
The relationship (3.25) expresses the well-known quadratic law of fluid flow.
Turbulent - transient flow. When rm(q) ~ a* ~ rd(q) ~ rt. qt ~ q ~
qtm, the flow in the capillaries of the r 1-chain with radii r from r 1 :5 r :5 rd(q)
(Re(rt) ~ Re(r) ~ Re2) is turbulent, and the flow in the capillaries with radii
from rd(q) < r :5 a* (Re2 > Re(r~Ret) is transient. Equation (3.7) becomes

G = o:' < rt,9t(r,q),rd(q) > +o:' < rd(q),gm(r,q),a* > +J(rt,q). (3.26)

The range of G where (3.26) is valid can be found by substituting qt and qtm
into (3.26)
Gt(rt) ~ G ~ 9tm(rt),
Gtm(rt) = o:' < Tt,9t(r,qtm),za > +o:' < Za,9t(r,qtm),a* >
+J(rllqtm),
d *d-l
Za = 2a 1 .
Using (3.14), (3.15), and (3.19), one can confirm the validity of (3.11) and
obtain the following inequality, dGt m ( rt) f dr1 < 0. Therefore the first to transfer
to the turbulent - transient flow with the increase of G are the rc-chains, and the
last are the a*-chains.