Page 73 - Percolation Models for Transport in Porous Media With

4.2 PLASTICITY AND PERMEABILITIES 65

the formula [62]
3 To
q = - 3 - 1' 8 ¢( 1's) drs, (4.13)
81rR9 J 2
To
0
Here q is the volumetric flow velocity in a capillary; ¢( 1' 8 ) is defined by the
friction law .:Ys = ¢(r 8 ); R 9 is the hydraulic radius of the capillary (in the case of
a circular cylindrical capillary of radius r, R 9 = r/2); 'Vp is the pressure gradient
in the capillary; Tp is the maximum shearing stress generated on the surface of the
contact between the fluid and the capillary if the traditional condition of" sticking"
(i.e., vanishing of the fluid velocity of the fluid) is granted.
Using (4.13), it is possible to express the pressure gradient of the fluid in the
capillary as a function of the flow q, 'Vp = 'Vp(q).
According to the second assumption, the pore space structure is simulated by
a regular network with pores situated in its sites and pore channels of circular
cylindrical form the edges (bonds) of the network. All the inferences made from
this model in the percolation approach, which takes account of the hierarchy in
summation of the selected conducting chains, are the same as those outlined in
§1.2. Among them, the relationship (1.8) holds for the distribution function n(rl)
of the rrchains, i.e., for those chains of the conducting capillaries with minimal
radius r1 chosen in the direction of the exterior 'V P when capillaries in the network
in general are distributed according to some probability density function f(r).
During the flow through a selected r1-chain with flux q, a local gradient 'Vp,
defined by (4.13), acts inside each of the capillaries. The macroscopic pressure
gradient 'V P, averaged over the chain, equals the following
'17 P(r,) ~ l ( 4.14)

'17p(q) /(r) dr (! f( r) dr) -t

which results in the correlation q(rl) = q('VP(rl)).
The total flux
Tc
Q = j q('V P(rl)) dn(rl) (4.15)

0
Since the gradient 'V P is the same for all r1-chains, ( 4.15) describes the corre-
lation between the flux Q through a porous medium and the applied gradient 'V P
(or some function of this gradient). The coefficient in this relation is naturally
set to equal the coefficient of permeability divided by viscosity (perhaps, raised to
some power). Thus the suggested plan of calculations (4.13) - (4.15) using (1.8)

allows to determine the permeability of the medium if the functions f(r) and¢( 7' 8 )
and the values of z and l are known. In the course of computing the relative phase