Page 37 - Percolation Models for Transport in Porous Media With

28 CHAPTER2. ONE-PHASE FLOW IN ROCKS

and the porosity. This relation was found from {2.10) with f = 0.68, l = 10- 3 m.
Experimental data (46] for the permeability of sandstone, which agree satisfactorily
with the relationship {2.10), are presented on the same plot for comparison.
The proposed model allows to explain some experimental facts. For example,
it is known that for small porosities, a dependence like K ""' <PP between the
coefficient of permeability and porosity takes place, where p can be as big as 10,
while Causeni 's formula implies p = 3. Within the framework of percolation theory
for grained media this fact follows naturally, since near the percolation threshold,
a sharp dependence of permeability on porosity takes place. If formula (2.10) is
approximated by the exponential relation K ""' <PP then for small <P, p becomes
just ~ 10. Note that formula {2.10) implies the lack of a single-valued correlation
between the coefficient of permeability and porosity. In order to definitely find the
coefficient of permeability for a grained medium, it is necessary to set the value of
<Po which defines the structure of the grain packing. It is therefore possible that
the scatter of experimental points near the percolation threshold is caused not
by errors of measurement, but by the fact that structural porosities of sandstone
cores vary.

2.3 Conductivity of Cavernous Media

Consider a medium with the globular pores of equal size and with randomly dis-
tributed centers. From the point of view of percolation theory, pores are conduct-
ing sites. Bonds between the pores can be formed only when the latter intersect.
Consider an arbitrary "reference" point. Other pores can intersect with it if they
are no more than 2Rp away, where Rp is the radius of the pore. If centers of two
pores are 2Rp{1- €) apart {0 $ € $ 1) (see fig. 7), then their intersection forms a
channel which connects them, of radius

{2.11)

In this case, the number of channels whose radii exceed the value r 1 is deter-
mined by the number of pore centers inside a sphere of radius 2Rp{1-€) (excluding
the "reference" pore)

{2.12)

where n° is the concentration of the pore centers. Using the fact that porosity
of the medium <P ~ {4/3)7rn°R: (with no account of the pore overlap), the for-
mula (2.12) can be rewritten as N = 8m(1-€) 3 -1. The greatest possible number
z of nearest neighbors of the "reference" pore can be estimated taking € = 0,