Page 145 - gas transport in porous media

Sahimi et al.
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in the more permeable layer moves faster, and mixes with the oil not only by longi-
tudinal dispersion in the direction of flow, but also transverse (perpendicular) to the
direction of flow in the less permeable layers. Dispersion processes in porous media
are usually modeled based on the convective-diffusion (CD) equation:
2
∂C ∂ C 2
+ υ.∇C = D L + D T ∇ C (8.3)
T
∂t ∂x 2
where υ is the macroscopic mean velocity (the macroscopic flow is assumed to be in
2
the x direction), C is the mean concentration of the solute, and ∇ is the Laplacian
T
in the transverse directions. Thus, the basic idea is to model the dispersion process
as anisotropic diffusional spreading of the concentration, with the effective diffusiv-
ities being the longitudinal dispersion coefficient D L and the transverse dispersion
coefficient D T .
The longitudinal dispersion coefficient D L is usually larger than the transverse dis-
persion coefficient D T by a factor that, depending on the morphology of the porous
medium, ranges anywhere from 5 to 24. This large difference between D L and D T
implies that more dispersive mixing takes place in the direction of the macroscopic
flow. Since pore space heterogeneities strongly affect D L and D T , the implication is
that the heterogeneities also affect miscible displacements. Unlike the effective diffu-
sivities, the dispersion coefficients D L and D T depend on the mean flow velocity. In
principle, the relation between the dispersion coefficients and the mean flow velocity
depends on the value of the Peclet number Pe, defined as, Pe = λυ/D m where λ is a
characteristic length scale of the porous medium, and D m is the molecular diffusivity.
The relation between the dispersion coefficients and Pe can be non-linear (see, e.g.,
Sahimi, 1993a, 1995, for comprehensive discussions), but in numerical simulations
one usually assumes that

D L = α L υ, D T = α T υ (8.4)

where α L and α T are, respectively, the longitudinal and transverse dispersivities.
Roughly speaking, if the length scale of the observations or measurements is larger
than the dispersivities, then the CD equation is applicable to describing the dispersion
process (and, thus, miscible displacements). Dispersion is called anomalous if it
cannot be described by the CD equation (Sahimi, 1993b).
Use of the dispersivities permits a phenomenological description of transport
through porous media. However, a fundamental understanding of the phenomena
is obtained only if the dispersivities can be related to the basic physical properties of
porous media, such as their porosity, hydraulic conductivity and/or pore size distribu-
tion. To develop such relations, two approaches have been developed in the past. One
approach constructs empirical correlations between the measured dispersivities and
important morphological parameters of the porous medium. For example, Harleman
and Rumer (1963) found, based on the analysis of their experimental data, that the
longitudinal dispersivity is proportional to the square root of the mean hydraulic con-
ductivity of the porous medium. However, attempts to verify this relationship under