Page 59 - Percolation Models for Transport in Porous Media With

3.2 EFFECTS OF PORE SPACE STRUCTURE ON FLOW 51

1
E(rt) = P!JL-l I'- (rt)(V(rt) + k0 {rt)),
1
F{r1) = o.sp,I'(rt) E (rt).
2
Define the average radius < r > of a capillary in the medium and the variance
u3 of the function f ( r)

< r >=< a*,r,a* >, u3 =< a*,(r- < r >)\a*>.

If the medium is described by model II and is sufficiently homogeneous ( u d < <
r >- 1<: 0.1), then k0 -< r >\and according to (3.21), V(rt) < r > 4 , I'(rt) < <
r > 5 <: r1l- 1 • In this case F(rt) ~ G. After expanding the square root in (3.36)
in powers of GF- 1 (rt) and taking the first three terms, we obtain the following

rc rc
K(G) =I ko(ri)dn(rl)-I V(r1)k~(rt)dn(r1)-

rc
o.sp,GJL- 2 I I'(rt)kg(rl)dn(rl). {3.37)
a.
The first integral in {3.37) is the permeability of the medium in the case when
Poiseuille flow takes place in every capillary, and therefore this term does not
depend on G. The presence of the second and third integrals in {3.37) is due to
the decrease in the permeability of the medium because of, respectively, the viscous
and the inertial pressure losses on heterogeneities like the capillary junctions. For
homogeneous media, the relationship {3.37) is valid in the whole domain described
by {3.34). However for heterogeneous media described by models I and II {when
O'd < r >- 1> 1) the contribution of the third term to {3.35) can become notable;
in the domain G <: Gml(rt), where Re(rt) <: Ret, the term can even become
dominant. Thus in heterogeneous media, permeability can decrease by several
times merely because of the inertial pressure losses on the capillary - capillary
(capillary- pore) junctions, with no turbulence in capillaries. As for homogeneous
media, it is known that any notable decrease in permeability is possible only as a
result of turbulence in capillaries.
Laminar flow under very small pressure gradients. If the condition

1
4
q « qo =min (go(r)r JL- ) (3.38)
is satisfied for all capillaries in an r 1-chain, then after expanding (3.13) in powers
1
of the small parameter q~ g0 ( r )r\ we get the right side of (3. 7) as a series in q1.
The number of terms to be taken in the series (3.7) depends on the form of go(r),
f(r), the size of the range G under consideration, and the required accuracy of