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the permeability variations, the length scale over which the variations are significant
must be of the same order of magnitude as the finger’s width or larger. In porous
media with a correlated permeability distribution, the existence of high- and low-
permeability regions causes flow channeling and, thus, affects finger growth mainly
by causing additional merging of the fingers. For uncorrelated porous medium, on
the other hand, the fingers’ width is always larger than the length scale over which
the permeability variations are significant.

8.4.3 Viscosity Ratio

As discussed above, one of the main characteristics of finger growth is that the large
initial number of fingers is reduced by their merging and formation of a smaller
number of active fingers. The effect of the viscosity ratio on this process is important:
At low viscosity ratios, there are no fingers that grow dramatically faster than the
rest. Consequently, finger merging and growth suppression occur to a much lesser
extent, resulting in a larger number of active fingers with growth rates that do not vary
much. At high viscosity ratios, on the other hand, the active fingers start to outgrow
the others earlier, and more finger merging and suppression of growth occur. This
results in a smaller number of long active fingers. Note also that, as the viscosity ratio
increases the fingers become more unstable, their shapes become more irregular, and
exhibit some tip splitting.

8.4.4 Dispersion
Strong longitudinal dispersion helps formation of a thick (more diffused) displace-
ment front, hence making the fingers less susceptible to flow disturbances that are
caused by small-scale heterogeneities. Moreover, since the front is more diffused, the
waves that correspond to small values of the concentration grow faster and, there-
fore, the breakthrough occurs earlier. While it may appear that the fingers should
grow faster when longitudinal dispersion is larger, it is in fact difficult to define fin-
ger lengths or growth rates in this case. The effect of transverse dispersion on the
fingers was already described. We will come back to the effect of dispersion later in
this chapter, when we discuss quantitative modeling of fingers.

8.4.5 Aspect Ratio and Boundary Conditions

We define the aspect ratio 0 of a porous medium as the ratio of its length and
width, 0 = L x /L y .If 0 increases (narrower porous media, somewhat like a slim
tube commonly used in laboratory studies of CO 2 injection), the initial fingers are
closer to one another and, consequently, their interaction is stronger. This results in
suppression of growth of the smaller fingers and their merging at the initial stages
of the displacement, hence yielding, even at early times, a small number of active
fingers. At very large aspect ratios, only one active finger will form.