In the last decade, multi‐phase flow in porous media has become a prominent topic in hydrologic research. This has been motivated by the widespread occurrence of subsurface contamination problems involving sparingly soluble liquids, often referred to as non‐aqueous phase liquids (NAPLs). NAPL contamination problems require analysis of the simultaneous movement of at least two fluid phases, NAPL and water. This is somewhat different than the traditional multiphase flow problem in hydrology, namely water movement in unsaturated soils. In the traditional treatment of unsaturated systems, the movement of air, the non‐aqueous phase, is not of interest, and its movement is ignored. Both NAPL‐water and air‐water systems are similar in that capillary forces, acting at the pore scale, usually dictate the pore‐scale distribution of each fluid phase. Pore‐scale fluid distributions then dictate the continuum‐scale properties such as relative saturation and relative permeability.

Abstract. Physical description of multiphase flow in porous media ideally should be based on conservation principles. In practice, however, Darcy's law is employed as the foundation of multiphase flow studies. Darcy's law is an empirical surrogate for momentum conservation based on data obtained from experimental study of one-dimensional single-phase flow. In its original form [Darcy, 1856], Darcy's law contained a single, constant coefficient that depended on the properties of the medium. Since 1856, Darcy's relation has been heuristically and progressively altered by allowing this coefficient to be a spatially dependent, nonlinear function of fluid and solid phase properties, particularly of the quantities of these phases within the flow system. The shortcoming of this approach is that the governing flow equation is obtained by enhancing a simple empirical coefficient with complex functional dependencies rather than by simplifying general conservation principles. As a result, some of the important physical phenomena are not properly accounted for. Also, some assumptions intrinsic to the equations are overlooked, making accurate simulation more of an art than an entirely scientific exercise. A more general and more theoretically appealing approach to the derivation of conservation principles for multiphase flow has been evolving over the last 30 years. This approach employs a mathematical procedure for deriving conservation principles at the length scale of interest, followed by imposition of thermodynamic constraints to restrict the generality of these expressions. The product of this approach is a set of balance equations that provides a framework in which the assumptions inherent in a hypothesized model of multiphase flow are clearly stated. Requirements for more comprehensive and physically complete models can then be specified. INTRODUCTIONThis paper is organized as follows: The introduction includes discussion of heuristic extensions of Darcy's [1856] observations to more general problems of singlephase and multiphase flow. Next, some shortcomings of these extensions are described. A theoretical approach to deterministic modeling of multiphase flow that relies entirely on fundamental conservation laws and thermodynamics is then outlined. Finally, computational and experimental network models are discussed as an avenue for critical assessment of the theoretical approach. The appendix is a tutorial that examines hysteresis in capillary pressure within the context of capillary tubes. A glossary of technical terms may be found at the end of the paper; these terms are italicized the first time they appear in text. Darcy's Law for Single-Phase FlowEquations that describe flow in porous media are important for modeling many processes of practical interest. Most likely, the first porous media problem for which a theoretical description was sought was related to movement of water in geologic formations for purposes of water supply. Although many formations may contain water, only those formations that also have th...

Proper mathematical description of multiphase fluid displacement in porous media requires specification of appropriate constitutive relations, including the relationship between capillary pressure and saturation. This pressure-saturation relationship may be measured at the laboratory core scale, but extension to larger scales is difficult in heterogeneous materials. To date, stochastic and volume averaging methods have been used to estimate the "effective" pressure-saturation relation in heterogeneous media. A third method, based on network percolation models, may be developed and used to define effective relationships. This last method is appealing in that pore scale physics are directly incorporated into the model, assumptions are clear, and numerical tests are most similar to those performed in the laboratory. Numerical results based on network models indicate that heterogeneities have a significant impact on the effective relationship. In addition, these results indicate that application of any sort of linear averaging of the individual small-scale relationships to define an effective relationship may provide incorrect results.

Abstract. Pore-scale network models of two-phase capillary displacement in porous media may be used to predict constitutive relationships between the capillary pressure, saturation, and relative permeabilities at the continuum scale. Since these constitutive relationships reflect ensembles of pore-scale events, spatial correlation in the pore structure is expected to be an important consideration. Correlations in the pore structure are also evident in recent experimental studies. Pore-scale network models are used here in a quantitative investigation of the influence of correlations on the capillary pressuresaturation-relative permeability relationships. The pore space is modeled using a cubic lattice larger than 12 times the spatial correlation scale in each direction. The following two types of correlations are incorporated: spatial correlation between the radii of bonds oriented along the same direction and cross correlation between the radii of bonds emanating from the same site in all directions. Results demonstrate that the saturationrelative permeability relationship is not very sensitive to unidirectional spatial correlation but is sensitive to cross correlation. Pore-scale model parameters are fit to capillary pressure-saturation data from several real soils, followed by a prediction of the relative permeability curves for the same softs. The predicted relative permeabilities are compared to the measured values and predictions using the traditional van Genuchten relationships. The pore-scale model is shown to fit the capillary-pressure saturation curves and predict the saturation-relative permeability curves with a degree of accuracy comparable to the van Genuchten relationships. While these computations have been restricted to unconsolidated soils, they demonstrate the potential of pore-scale models for predicting other transport properties such as dispersivities and mass transfer coefficients.

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