Page 196 - Percolation Models for Transport in Porous Media With

194 CHAPTER 11 GAS COLMATATION IN ELECTRIC ACTION

when the bubbles merge in the case described by {11.48). When the bubbles grow
in the case described by {11.49), the relationships are as follows

Aoxo (lfao) 3 ( r ) 4 r
t(r,rt) ~ HAe 4>2(rt) r1 l' t ~To
{11.53)

t »To

It is clear from the expressions presented above that the shut-off time for a
capillary goes up sharply as its radius goes up and r 1 decreases. This phenomenon
is especially notable when the intensive heat outflow to the skeleton of the medium
takes place {the lower expressions in {11.52), {11.53)). When r = r1 and the
bubbles merge we have

{11.54)

when the bubbles grow we have

t ~To
{11.55)

The lower expressions in {11.48), {11.55) imply that when intensive heat ex-
change takes place t( Tt) is minimal in the large r1 -chains. As some of the ways for
the fluid and current flow are shut off in these chains, a considerable rearrange-
ment of the current in the medium takes place there, and the study of the further
development of colmatation becomes substantially more difficult. Therefore we
shall confine ourselves to the case of the steady-state heat exchange {11.47), when
the medium receives heat as a whole. In this case, according to the upper expres-
sions in {11.54), {11.55), if the heat exchange with the skeleton of the rock is not
too intensive, the values of t( r 1 ) either decrease as r 1 goes up, or undergo little
change. This corresponds to the shut-off of the thinnest r 1-chains by the bubbles.
However no notable increase of the current in the large r 1 -chains is likely to be
detected, and the hierarchy of the r 1-chains with respect to their conductivities
is preserved. Conductivities of those r 1-chains, for which t <: t(rt), is determined
primarily by the conductivities of the r 1-capillaries, since the shut-off periods for
other capillaries in the r 1-chain are much greater than t(rt). After the r 1-capillary
is shut off, the current in the chain vanishes, and further heating of the capillaries
in the chain (and consequently, growth of bubbles there) stops. Actually, a certain
heat exchange with the medium does take place. The r 1-capillaries shut off by
the bubbles gradually (over a period of To(rt) ~ rU4K-t) cool down; part of the
vapor in the bubbles condenses; the bubbles diminish in size; and the r 1-capillary