Page 74 - Percolation Models for Transport in Porous Media With

66 CHAPTER 4. MULTIPHASE FLUID FLOW

permeabilities, the term in (1.8} which appears before the integral cancels out, and
therefore the knowledge of l is not necessary.
Flow with the initial pressure gradient. The Binghamian plastic. The
friction law with limiting shearing stress in the case of the flow of a visco-plastic
fluid is defined by its structure in the field of the surface forces at the points of
contact with the solid surface. The most general form of such law was given by
Wilkinson
(4.16)

where A and B are constants found from experiment. For the calculation of the
permeability in accordance with the given algorithm, the specific form in which
the law <P( T 8 ) is written does not matter. However if A and B are arbitrary
constants, then after integrating, the expression (4.13} becomes very complicated
and inconvenient for further use. Therefore it seems reasonable to consider, for
illustration, a rather frequently encountered specific case of (4.16}, when A = r,,
Bhs « 1, i.e., the Binghamian plastic. Similar rheological properties can be found
in heavy viscous oils which contain components with high molecular weights.
In the outlined case the function <P( T 8 ) has the following form

(4.17}
Substitution of ( 4.17) into ( 4.17} yields the well-known relation of Bingham

(4.18)

(4.18} implies that even in the considered case (i.e., Vp* fVp < 1}, the ratio of
the third term in the square brackets to the second one is "' 0.1. Therefore it is
possible to neglect the third term in the square brackets and write the following
for the local V p
8JL 4
Vp(q) = - 4 q + - Vp* (4.19}
3
11"T
After substituting (4.19} into (4.14) we obtain the value of the macroscopic
gradient dr l (

. 8JL 00 dr 8~ 00 00 ) -1
Vp{rt) = [ -;- q(rt) [ f(r) r' + -f [ f(r)-; [ f(r) dr

and

q(r,) = ~ Vp{r,) i f(r)dr (i f(r)::) _,