Page 75 - Percolation Models for Transport in Porous Media With

4.2 PLASTICITY AND PERMEABILITIES 67

'lrTp joo f(r) dr (!oo f(r) dr) -1 (4.20)
3JL r r 4
rt Tt
In this case it follows from ( 4.15) and ( 4.20), in accordance with {1.8), that

Q = ~~ (1- {c)- v ;JL "VP J [J J(r) dr] v x
2
0 rt

{ [l j(r)dr- ~ ;~ l f(r) ~] (l f(r) ;: ) -l} j(r,)dr, (4.21)

After introducing the notation Ao = (1/4)v7rl- 2 (1 - {c)- 2 v and r~ = 2rpf"Vp,
we obtain, due to (4.21), the expression for the absolute permeability of the Bing-
hamian plastic with the limiting shear r0 for a given gradient "V P

(4.22)

If we took all the terms of (4.18) into account, then the 17-function in (4.22)
would have had (1- r~/r) as its argument. This would have reflected the obvious
fact that the capillaries with radii r < r~, for given Tp and "VP, become imper-
meable for the Binghamian fluid. However, neglecting the last term in the square
brackets in (4.18) results in an error of~ 30% in calculation of r~.
Flow without the initial pressure gradient. Pseudo-plastic and "dila-
tant" fluids. Flow of a number of fluids, such as colloidal solutions, emulsions,
or thinly dispersed suspensions, is described by a non-linear friction law without
a limiting shearing stress [63]

(4.23)

where J.Lt is some sort of analog to viscosity, and n is an exponent to be found
from experiment. It was discovered [63] that the law ( 4.23) for n < 1 does well
enough reflect the behavior of emulsion and polymeric solutions used, for instance,
in "polymeric" inundation (pseudo-plastic fluids). For n > 1 this law equally well
describes the properties of flowing suspensions, which are widespread in water
extraction and in the of underground lixiviation phenomena (dilatant fluids).