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Chapter 8: Gas Injection and Fingering in Porous Media
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be used as “experimental” data for assessing the predictions of averaged equations.
However, such calculations should also be validated by comparison with experimental
results.
8.7 NUMERICAL SIMULATION OF MISCIBLE DISPLACEMENTS
Numerical simulations of unstable miscible displacements have been carried out for
a long time. The main difficulty in this problem lies in the correct reproduction
of the wide range of the relevant length and time scales that typically characterize
these phenomena. At the smallest length scale, the size of the smallest grid block
is determined by the action of physical diffusion or dispersion, while the reservoir
linear size, or the distance between the wells, determines the large scales.
In a pioneering work, Peaceman and Rachford (1962) developed a finite-difference
algorithm for computing an unstable rectilinear miscible displacement. Due to the low
order of their numerical algorithm, a fingering instability did not develop on its own,
but had to be triggered artificially by imposing small random spatial variations in
the permeability. Over the four decades following this early work, a host of novel
numerical approaches for the simulation of miscible displacements has been intro-
duced and tested on problems of varying degrees of difficulty. In what follows, we
briefly discuss some of these methods.

8.7.1 Finite-Element Methods
For a comprehensive discussion of the efforts up until 1983, see the review by Russell
and Wheeler (1983). Douglas et al. (1984) presented the results of computer simu-
lations of unstable miscible displacements for one quarter of the five-spot geometry.
They used a self-adaptive finite-element (FE) method; see also Bell et al. (1985).
Darlow et al. (1984) described the so-called mixed FE methods, which offer certain
computational advantages, in particular for strongly heterogeneous porous media,
by solving for pressure and velocity simultaneously. Ewing et al. (1984) analyzed
a modified method of characteristics for handling the governing equation for the
concentration.
Ewing et al. (1989) performed more detailed FE simulations of miscible displace-
ments in anisotropic and heterogeneous porous media in one quarter of five-spot
geometry, using computational grids of up to size 100×100. They typically observed
dominance of the viscous fingering instability over the permeability-related effects.
In general, the relative importance of these effects is expected to depend on the vis-
cosity ratio and the degree of heterogeneity of the porous medium. Comprehensive
reviews of the above methods, as well as related work (Ewing and Wang, 1994)
are provided by Ewing et al. (1994), along with a discussion of these methods
in light of recent developments in computer architectures; see also Arbogast et al.
(1996).