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Chapter 8: Gas Injection and Fingering in Porous Media
Batycky, 1997) relies on the consistency of treating the convective flux independently
of gravity flux within a given time step of the simulation. For small time steps, the
operator-splitting approximation is fairly accurate, whereas large time steps may
lead to significant errors. Jessen and Orr (2004) used the operator-splitting tech-
nique to develop a compositional simulator suitable for displacement processes with
significant gravity segregation.
8.7.4 Spectral Methods

To overcome the limitation in the accuracy of many of the above grid-based methods,
significantly more accurate spectral methods have also been developed (see, e.g.,
Tan and Homsy, 1988; Zimmerman and Homsy, 1992a,b; Rogerson and Meiburg,
1993a,b). These methods possess superior accuracy for smooth flows, that is, for
flows without discontinuities and singularities (Gottlieb and Orszag, 1977), and avoid
problems associated with numerical diffusion. In addition, the spectral simulations
are carried out for the governing equations that are formulated in terms of the vorticity
and the stream functions. Since one no longer needs to solve for the pressure distribu-
tion, such formulation of the governing equations often leads to higher computational
efficiency. Moreover, this formulation offers the advantage that it satisfies the con-
servation of mass identically. Thus, these methods allow for detailed investigation
of mobility- and gravity-driven fingering processes in rectilinear displacements at
relatively low levels of physical dispersion.

8.8 STOCHASTIC MODELS OF MISCIBLE DISPLACEMENTS

In addition to the above methods, several models of miscible displacement pro-
cesses in porous media, for both the laboratory and field-scale porous media, have
been developed in which probabilistic concepts have been utilized. Many of such
models (DeGregoria, 1985; Paterson, 1984; Sherwood and Nittman, 1986; Siddiqui
and Sahimi, 1990a,b) are applicable when the effect of dispersion can be neglected.
We describe briefly two stochastic models that do take into account the effect of
dispersion.
The first model that we describe is due to King and Scher (1987, 1990). Consider
first the case of miscible displacements without dispersion. For point injection of
fluids the governing equations are

∂C 2
+ u ·∇C = δ (x) (8.13)
∂t
υ
u = =∇ × (ψˆz) (8.14)
Q
ˆ
where Q is the injection rate (volume per unit thickness per unit time), ψ is the stream
ˆ
2
function, and δ (x) is the two-dimensional Dirac delta function. The injected volume