Page 150 - gas transport in porous media

Chapter 8: Gas Injection and Fingering in Porous Media
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where σ k is the standard deviations of the permeability distribution, and k m is its mean.
The second characteristic quantity is a permeability correlation length ξ k , which is
the length scale over which the permeabilities are correlated.
Moissis et al. (1987) found that the permeability distribution near the inflow end
of their two-dimensional model determines the number of the fingers, their initial
locations, and the relative growth rates. The locations where the fingers form are
controlled by the maxima in the permeability distribution near the inflow end. The
initial number of the fingers depends strongly on the correlation length of the per-
meability distribution. Highly-correlated porous media have fewer maxima in their
permeability distribution and, consequently, fewer fingers form in such porous media.
As mentioned above, some of the initial fingers grow faster than the rest and eventu-
ally dominate the displacement. Due to the shielding effect, the long fingers suppress
the growth of small fingers. The smaller fingers may merge later on with the larger
ones, leaving upswept areas that may be fairly extensive. This process results in a
number of large fingers which grow quite independently of each other, at least until
the breakthrough, and are referred to as the active fingers. The number of the initial
fingers, that of the active fingers, and the time of their formation all depend on C k , ξ k ,
and the viscosity ratio.
Thenumberoftheactivefingersisadecreasingfunctionofthepermeabilitycorrela-
tion length ξ k since, as discussed above, the initial number of the fingers is smaller for
larger ξ k . Moreover, the effect of downstream permeability distribution (see below)
is more significant for large ξ k , resulting in more merger of the fingers, which reduces
their number. For a given value of C k , the growth rate of the fingers increases with
ξ k up to a limiting value. For large values of ξ k , the permeability variation tends to
generate fingers of large wavelengths which grow relatively slowly. However, once
these fingers grow beyond the zone in which their growth is approximately described
by the linear stability analysis (see Section 8.9), it is easier for the displacing gas to
develop flow channels which accelerates the subsequent growth rate of the fingers.
The number of active fingers is also a decreasing function of the coefficient of
permeability variations C k .As C k increases, the difference in permeability between
high- and low-permeability regions increases. In the initial stages of the displacement,
this difference favors the growth of the longest fingers, which tend to grow in the
high-permeability regions. Thus, for highly heterogeneous porous media, the longest
fingers grow much faster than the rest and dominate the displacement relatively early,
shielding the smaller fingers. Later on during the displacement, the large difference
in permeabilities from region to region facilitates merging of the fingers. The net
result of these two mechanisms is the reduction of the number of active fingers. Since
almostallthedisplacingfluidflowsthroughoneoratmostafewfingers, breakthrough
occurs early and, therefore, the sweep efficiency is poor.
The effect of downstream permeability variations on the finger formation and
growth is negligible for random (uncorrelated) porous media, but it becomes increas-
ingly more important as the correlation length ξ k increases. This effect is more
pronounced for small viscosity ratios. The cause of this behavior can be traced to
the scale of the permeability variations. For a finger to be significantly affected by