Page 54 - Percolation Models for Transport in Porous Media With

46 CHAPTER 3. PERCOLATION MODEL OF FLUID FLOW

Energy losses on junctions of the "capillary - capillary" ("capillary
- pore") type. A boundary element of a junction of two successive capillaries
in model II or a boundary element between a pore and a capillary in model I
can be taken as a local resistance element. Let the radius of the thin capillary
at the junction equal r and the radius of the thick one, r'. There was a lot of
publications dealing with the determination of the energy losses in question in the
scientific press. A review of them appears, for instance, in [55] or in [53], where
the following semi-empirical formula is proposed

ll.pi(r,r',q) = 0.5Z(r,r',q)p1v (r), (3.16)
2
Z(r, r', q) = (v(r, r', q) + (i(r, r', q).

Here (v ( r, r', q) is the coefficient oflosses caused by viscous forces on the bound-
ary; (i ( r, r', q) is the coefficient of losses caused by inertial forces.
When Re > 300, (i :» (v, and therefore Z ~ (i. The following semi-empirical
formulas of Borde are valid for (i:

(/ (r, r') = Bf[(rfr') 2 - f- 1 (r' /r)] 2 , (3.17)

2
1
(f(r, r') = Bi[l - f- (r' /r)J ,
with (i = (/ when the fluid flows from the capillary of radius r to the one of radius
r'(r < r'), and (i = (f when the flow is in the opposite direction. The coefficients
Bi and Bi are of the order unity. The function f(r' fr) in (3.17) characterizes the
extent to which the stream is compressed. For r' fr > 3 the function f(r' fr) is
almost constant and equal to f ~ 0.61, and as r' fr goes to unity, f(r' fr) also goes
to unity.
When Re:C 10 + 30, (v :» (i, and therefore Z ~ (v· The function (v satisfies
Wust 's formula
(v(r,r',q) = A(r,r')Re- (r,q). (3.18)
1
For r' /r > 1.3 the dependence of A(r, r') in (3.18) on rand r' is weak, and the
value of this function is A ~ 20 + 40. As r' fr goes to unity (r' fr < Ill), A(r, r')
goes down to zero very quickly. The relationship (3.18) holds for both directions
of flow.
In the interval30 ):Re): 20+40 the values of ll.pi(r, r', q) can be found with the
accuracy good enough for practical calculations by substituting (3.17) and (3.18)
into (3.16).