Page 50 - Percolation Models for Transport in Porous Media With

42 CHAPTER 3. PERCOLATION MODEL OF FLUID FLOW

First of all, point out two limiting kinds of porous media, namely the one
where the characteristic dimensions of sites (pores) are much greater than the
characteristic cross-sections of bonds (capillaries); and the other one where the
mentioned dimensions are of the same order of magnitude. Call the first of the
described models, model I and the second one, model II.
Second of all, assume the porometric curve f(r) to be defined on an arbitrary
interval [a*, a*] and to vanish outside this interval.
As it was shown in chapter 1, the conductivity of the infinite cluster depends
substantially on its structure, or, more precisely, on the structure of its "skeleton"
composed of chains formed by the conducting capillaries. To investigate properties
of the "skeleton," it is reasonable to build an hierarchy of the conducting chains
using the radius r1 of the thinnest capillary in the chain. We shall call such chains
r1-chains.
When r1 = rc, the r1-chains contain only the largest capillaries (rc :5 r :5 a*).
These will be called the rc-chains. When r1 =a*, r1-chains contain capillaries of
all possible radii (a* :5 r :5 a*). These will be called the a*-chains.
Finally, for the conductivity k(rt) of the chain, as in (1.9), introduce the no-
tation of the mean value of an arbitrary function '1/J(r) over an r1-chain as follows
• -1

< r, ¢,a>; i ¢(r)f(r) dr (l f(r) dr) (3.1)

Consider the steady state one-dimensional flow under a given pressure gradient
1
G with absolute value Ap L 0 , where Ap is the pressure difference in the medium
at a distance £ 0 > l. In this case, for an arbitrary r1-chain of model II, we have

1
G = £0 ~t::.pci + ~6.Pii) · (3.2)
(
Here 6.pci, 6.p;i are the pressure differences in the i-th capillary and the junc-
tion of the i-th and the (i + 1)-st capillaries, respectively. Summation in (3.2) is
taken in the direction of the pressure increase, i.e., against the flow. Denote by q
the flow of the fluid through an r 1-chain. Assume that 6.p;i depends only on the
radii rand r' ofthe i-th and the (i + 1)-st capillaries and on the flow q, while 6.pci
can be expressed in terms of the absolute value of the pressure gradient Uc{r, q) in
the i-th capillary as is customary

6.pci = Uc(r,q)l, 6.p;i = 6.p;(r,r',q). (3.3)

The average pressure losses on the exit from a capillary of radius r in an r 1-
chain 6.p;(r, q) are found by means of averaging 6.p;1 over r' as in {3.1), using
(3.3)
(3.4)