Page 108 - Percolation Models for Transport in Porous Media With

5.2 VISCOSITIES AND INTERFACIAL TENSION 101

s
.
1.1111 ------------------------------ :1
!
... .
.
.
..

uo _,. ....... -w -uo -1•
Zog C
Figure 35: The dependence of the residual saturation of the displacing phase on
the capillary number (for a fixed viscosity ration: log M = 2.9}

Thus, using (5.18} we can determine from the capillary number the minimal
radius of capillaries rk accessible for the non-wetting fluid. Two situations are
possible. In the first case r. < rk. Then the capillary pressure does not allow the
non-wetting fluid to enter the capillaries with radii r < rk, and traps are formed
by the rk-chains. At the closure of these traps, restraint does not occur since other
chains, being the more rapidly growing ones, have managed to grow beyond the
region formed by this conditional trap. For such formation of an IC, equilibrium
fluid flow takes place. To take into account the influence of the surface tension
forces on the structure of the IC, it is necessary to substitute rk for r. in (5.16}.
For the model function f(r) used above, calculations ofthe relation S(C) were
carried out for a fixed M (In M = 2.9}. The obtained theoretical relation is shown
in Fig.35 by line 1. Line 2 in this Figure corresponds to the results of the numerical
experiment [18] for In M = 4.0. A good consistency of transfer regions from
capillary fingering to stable piston-type displacement in theoretical and calculated
data can be noticed from the figure. Some difference may be explained by the
corresponding differences in the values of M and in the forms of functions f(r)
used in the calculations.
The second situation occurs when r. > rk. In this case the process is essentially
non-steady state. When the fore front of the displacement passes, there appear
traps where the displaced phase is retained, the fraction of the trapped phase being
greater than the critical value necessary for forming an IC of the displaced phase.
Therefore the phase retained in the traps does not lose connectedness and flows
away through its IC, accompanied by the replacement of phase 2 by phase 1 in
r-chains with rk < r < r •. This means that the skeleton ofiC 2 remains as it was
(IC 2 only loses its "dead ends"), whereas new chains are added to the skeleton of
IC 1 to increase its conductivity.
We may suggest the following model of relaxation. Fluid 2 is displaced from
the capillary chains retained in the trap starting from r., up to rk. When fluid 2 is