Page 195 - Percolation Models for Transport in Porous Media With

11.3 MODIFICATION OF PERMEABILITY 193

The shut-o:tf a capillary cross-section can happen in two ways. First of all, it
can be caused y merging of the bubbles when their densities become sufficiently
large and the istance between them becomes of the order of their radii
2
l a "' n 0 R:j a, 10Aw: R:j 1 (11.48)
-1/3
Second of all, the capillaries can be shut by solitary growing bubbles, i.e., when

(11.49)

Consider the case when the initial radii of all bubbles in the fluid are the same
and equal to a0 ; we shall also neglect the changes of these initial radii during the
bubble's movement from a pore to a capillary. Furthermore, we shall neglect the
change of the current density, as the growing bubbles shut the cross-section of the
capillary. Estimates show that it drops not more than by a factor of two up till
an all but total shut-off of the capillary by the growing bubble.
According to (11.13)

ae(t)/8r < 0, 8e(t)f8rt > 0
Therefore the greatest rates the bubble growth are to be observed in the thinnest
r 1-capillaries of the largest r 1-chains. This, however, does not mean that the
cross-sections of these capillaries are going to be the first to be shut off.
The condition of a capillary shut-off due to the merging of the bubbles (11.48),
with regard to (11.44}, has the following form

(11.50)
When (11.49} is satisfied we have

1
H9.Aox0 [t/To(r)] R:j 4n.r 3 (11.51)
After taking into account the change of the density for bubbles, we obtain

1
2
A= n.(4/3}7rag(l/r} , Ae = E3u' P! l- 1
It is possible to calculate the time dependence for the permeability of the
medium by changing order relationships with equalities in (11.51}, (11.52) and
solving them with respect to t(r, r1 }, which is the period of shut-off for a capillary
of radius r in an r 1-chain. For small (11.47} and large (the lower expression in
(11.45)) heat losses, respectively, we obtain the following explicit relationships for
t(r, rt},

4
2
t(r,rt} R:j xo/(10HAe4> (rt))(r/rt} , t «:To (11.52)
10
2
4
2
t(r,rt} R:j "(pr /(4eK-t) exp [ ~~~~(rt} (r/rt} r- ], t >To,