Page 58 - Percolation Models for Transport in Porous Media With

50 CHAPTER 3. PERCOLATION MODEL OF FLUID FLOW

r 2:: Tt (Re2 2::Re(rt) 2::Re(r) 2:: Ret), whereas in those with radii a* 2:: r 2:: rm(q)
(Ret > Re(r) 2::Re(a*)) the flow is laminar. Equation (3.7) in this case is

G =a' < Tt.9m(r,q),rm(q) >+a' < rm(q),gz(r,q),a* > +J(rt.q). (3.31)

After substituting qm m and Ql m into (3.31) we find the domain of validity for
(3.31) to be

Laminar flow. When r1 2:: rm(q), q 2:: Qtm, laminar flow takes place in all
capillaries (Ret ~Re(rt) ~Re(r) ~Re(a*)) in the Tt-chain. Equation (3.7) here is
(3.32}

The validity condition for (3.32) can be obtained after substituting qz into
(3.32}
(3.33)
If ( 3.33) holds for all Tt -chains (a* ::::; Tt ::::; r c), then the flow is laminar in the
whole medium.
Laminar flow under large pressure gradients.
If the condition
qzm ~ q ~ qz = max(go(r)r J.t-t),
4
is satisfied for all capillaries in the r 1 -chain, then the flow in all capillaries is close
to Poiseuille. After substituting qz into (3.32} we get the boundaries of the domain
where Poiseuille flow in an Tt-chain takes place
(3.34)

Gz(rt) =a' < rt,Yt(r,qz),a* > +J(rt,ql)·
After substituting (3.19} into (3.32) we get the following relations for Poiseuille
flow
2
G = JL(k0t(rt) + V(rt))q + 0.5pf I'(r1)q , (3.35)
ko(rl) = (1rj8)(a' < r 11 r- 4 ,a* >)- 1 .
Here ko(r1) is the conductivity of the rt-chain when Poiseuille flow takes place
in all of its capillaries if there are no pressure losses on junctions. If (3.34) holds
for all Tt-chains, then it follows from (3.8) that
rc
1
K(G) = JLG- j E(rt)( vh + GF-t(rt)- 1) dn(rt), (3.36)
a,