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1. Introduction
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approach, macroscopic medium parameters, such as permeability, saturation,
upscaled adsorption rates, and dispersion coefficients need to be introduced.
As the microscopic governing equations contain no information regarding these
macroscopic parameters, they form an undetermined system, insufficient to be
closed unless further equations are supplied. The additional equations, which
are an important part of the continuum theory, are known as constitutive relations.
These relations, such as the relation between permeability and pore
properties of porous media, depend upon the internal constitution of the particular
porous material considered. By upscaling from the pore scale, constitutive
relations can be determined for a specific case. Because the constitutive relations
are ultimately used to model macro-scale problems, understanding of
the pore-scale processes and proper incorporation of their effects in larger-scale
relations must be accomplished.
Although applying the continuum modeling approach is a common practice,
there are some difficulties and drawbacks involved when applying the continuum
approach. Performing experiments to reveal the constitutive relationship
is usually difficult and costly. For example, serious experimental difficulties
are encountered in measuring relative permeabilities or solute dispersivity in
a porous medium. The lack of available constitutive data is frequently cited
by both petroleum and groundwater engineers as a primary barrier to acceptable
predictions (e.g. Abriola and Pinder [1985], Aziz and Settari [1979]).
In addition, although constitutive relations have a crucial bearing on the accuracy
of subsurface flow models, they are approximate solutions and often
uncertain [Miller et al., 1998, Genabeek and Rothman, 1996]. Over the past
two decades, much effort has been expended to develop alternate theories to
the standard approach. For example, Gray and Hassanizadeh have suggested
a more complete approach of modeling multiphase flow based on integration
over a REV to produce mass, momentum, and energy conservation equations
that are formulated based on volume, interfacial area and contact lines [Gray
and Hassanizadeh, 1991b, Hassanizadeh and Gray, 1993, 1979, 1980, Gray and
Hassanizadeh, 1991a].
1.3 Pore-scale modeling approach
Pore-scale modeling provides opportunities to study transport phenomena in
fundamental ways because detailed information is available at the microscopic
pore scale. This offers the best hope for bridging the traditional gap that ex-
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