Theoretical advances in the upscaling of multiphase-flow models were
reported using several fundamentally different approaches. Gray and
Hassanizadeh [1991a,b] and Hassanizadeh and Gray [1993a,b]
derived an improved macroscopic physical description of two-phase flow
from conservation of mass, momentum, and energy, and the second law of
thermodynamics. Among the aims were to avoid the usual questionable
extension of the single-phase Darcy law via relative permeability, and to
obtain equations defined entirely in terms of macroscopic state variables,
thereby averting confusion between scales. Theoretical and experimental
paradoxes arising from traditional assumptions of capillary-gravity
equilibrium were discussed. The complete two-phase model was quite
complicated, but with quasi-equilibrium assumptions its variables
were reduced to the standard ones along with macroscopic capillary
pressure, 
, and the specific interfacial area, 
, between
the phases. 
was a function of 
as well as wetting-phase
saturation and was not in general equal to the difference between
macroscopic nonwetting- and wetting-phase pressures; this held only
at equilibrium, and the difference yielded a dynamic capillary
equation. The model also added an interfacial-area equation to
the usual set and a saturation-gradient term to the Darcy potential
gradient in the momentum equation, and a nonempirical flow-coefficient
tensor was derived to replace relative hydraulic conductivity. The
inclusion of interfacial-area dependence of capillary pressure was
expected to eliminate the need for hysteresis. The model required
more data than a standard one, but its physical basis made the data
potentially easier to describe and measure without having to rely on
complex empirical functional dependencies.
Wang and Beckermann [1993] applied a two-step process, dual-scale volume averaging, to multiphase flow. They proved general dual-scale averaging theorems applicable to a variety of transport processes, avoiding the need to rederive the averaging for each equation set. Representative elementary volumes were allowed to change with time. If one physical phase flowed on both characteristic length scales, it was represented by two phases, one at each scale, with a time-varying imaginary interface between the scales.
King et al. [1993] extended real-space renormalization, based on an analogy with a network of resistors, to two-phase flow. In a numerical implementation, they found it to be 100 times faster than traditional petroleum-industry methods, which generate pseudo-relative permeabilities by successively moving from finer to coarser grids.
For upscaling of capillary pressure, Kueper and McWhorter [1992] used a macroscopic percolation theory. Each point of the percolation lattice, instead of being a pore, was a local-scale porous medium with given porosity, permeability, and saturation-capillary pressure relation. The result was a hysteretic large-scale capillary pressure as a function of large-scale saturation. This assumed capillary-dominated flow in which a static capillary-pressure curve could model dynamic conditions and gravity could be ignored. Macroscopic trapping could be modeled by the order in which lattice points were invaded, and in contrast to microscopic percolation, both phases could be continuous via local-scale relations.
Research on upscaling, in its infancy for multiphase flow, is of considerable future significance because it could determine the impact of other fundamental research on the practice of modeling. For example, with regard to phase partitioning, the real issue is the form that a model should have after averaging up to simulation grid scale. This may be quite different from the models being derived at laboratory scale from experiments. It is evident that much important fundamental work remains.