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Figure 8.2. Tip splitting (top) and shielding (bottom) in viscous fingering Sahimi et al.

(large-scale permeability variations and channeling). In what follows, we describe
the effect of some these factors on fingering phenomena.

8.4.1 Displacement Rate
Compared to immiscible displacements, miscible displacements are much less sensi-
tive to the displacement rate. In immiscible displacements at high flow rates smaller
and more numerous fingers are formed than at low rates. In contrast, fingers that
emerge during miscible displacements are only mildly sensitive to the displacement
rate. This is due to the fundamental role that dispersive mixing plays which help the
smaller fingers to merge.
The mild sensitivity of the fingers’patterns during unstable miscible displacements
to the displacement rate has practical implications. To control the stability of an
immiscible displacement it is often enough to control the displacement rate. This is
clearly not the case with miscible displacements. While it is possible to obtain a stable
displacement by using a rate less than a critical rate, in most practical situations this
would imply using a rate which would not be economical.

8.4.2 Heterogeneity Characteristics
As mentioned earlier, one factor that plays a fundamental role in finger formation is
the heterogeneity of the porous medium. Once a finger starts to grown, its subsequent
growth is closely linked to its interaction with the heterogeneity and, in particular, to
the spatial variations of the porosity and permeability. Permeability variations have
been found to play an important role in finger initiation and growth (Stalkup, 1983;
Moissis et al., 1987). The spatial variation of the permeability is usually described by
two parameters. One is the coefficient of permeability variation C k defined by
σ k
C k = (8.5)
k m