Page 57 - Percolation Models for Transport in Porous Media With

3.2 EFFECTS OF PORE SPACE STRUCTURE ON FLOW 49

Transient regime of flow. When rm(q) ~a* ~ r1 ~ rd(q), Qmm ~ q ~
Qtm
{3.27)
the flow in all capillaries of the r1-chain is transient. The inequality (3.27} can
1
1
be satisfied only for a*r1 ~ d1d2 < 1.5, and therefore the transient flow cannot
take place in any r1-chain of a medium where a*a; 1 > d1d2 . The equation {3.7)
1
becomes
{3.28)
After substituting Qm m and Qt m into (3.28) we find the domain of validity for
{3.28) to be

c::,(rt} ~ G ~ G~(rt),
G::,(ri) =a' < T},gm(T,Qmm),a* > +J(rl,Qmm), (3.29)
G'(rl) =a' < Tt 19m(r,qtm),a* > +J(rt,Qtm)·

By substituting (3.14), (3.19) into (3.29), one can make sure that conditions
(3.10) and (3.11) are satisfied and obtain the inequality dG'm(rt)/drt
< 0.
Turbulent-transient-laminar flow. When a*~ rm(q) ~ rd(q) ~ r 1 (qtm ~
q ~ Qmm), the flow in capillaries with radii r1 ~ r ~ rd(q)) (Re(r) ~ Re2) in
r1-chains is turbulent; in those with radii rd(q) < r ~ rm(q) (Re2 >Re(r) ~Ret),
transient; and in those with radii rm(q) ~ r ~a* (Re1 >Re(r) ~Re(a*)), laminar.
In this case the equation (3.7) becomes
G =a' < TI,9t(r,q),rd(q) >+a' < rd(q),gm(r,q),rm(q) > +

a' < rm(q),gt(r,q),a* > +J(rt,q). (3.30)

After substituting Qtm and Ql m into (3.30), obtain the range of G where (3.30)
holds

Gtm(ri) ~ G ~ Gtmz(rt),
Gtml(rl) =a' < Tt,9m(r,qmm),z' >+a'< z',gl(r,qmm),a* > +
J(rt,Qmm)i
d d-1
I
Z = 1T1 2 •
1
If a*r} < d1d2 , then the turbulent-transient-laminar flow cannot take place
1
in the rrchain. If, moreover, a*a; 1 < d1d2 , then the discussed type can take
1
place in no r1-chain in the whole medium.
Transient-laminar flow. When a* ~ rm(q) ~ r1 ~ rd(q), Qmm ~ q ~ Qlm,
transient flow takes place in those capillaries of the r1-chain with radii rm(q) ~