Flow through porous media may be described at either of two length scales. At the scale of a single pore, fluids flow according to the Navier-Stokes equations and the appropriate boundary conditions. At a larger, volume-averaged scale, the flow is usually thought to obey a linear Darcy law relating flow rates to pressure gradients and body forces via phenomenological permeability coefficients. Aside from the value of the permeability coefficient, the slow flow of a single fluid in a porous medium is well-understood within this framework. The situation is considerably different, however, for the simultaneous flow of two or more fluids: not only are the phenomenological coefficients poorly understood, but the form of the macroscopic laws themselves is subject to question. I describe a numerical study of immiscible two-phase flow in an idealized two-dimensional porous medium constructed at the pore scale. Results show that the macroscopic flow is a nonlinear function of the applied forces for sufficiently low levels of forcing, but linear thereafter. The crossover, which is not predicted by conventional models, occurs when viscous forces begin to dominate capillary forces; i.e., at a sufficiently high capillary number. In the linear regime, the flow may be described by the linear phenomenological law ui=&Sgr;jLijfj, where the flow rate ui of the ith fluid is related to the force fj applied to the jth fluid by the matrix of phenomenological coefficients Lij, which depends on the relative concentrations of the two fluids.

The diagonal terms are proportional to quantities commonly referred to as ''relative permeabilities.'' The cross terms represent viscous coupling between the two fluids; they are conventionally assumed to be negligible and require special experimental procedures to observe in a laboratory. In contrast, in this numerical study the cross terms are straightforward to measure and are found to be of significant size. The cross terms are additionally observed to be approximately equal, which is the behavior predicted by Onsager's reciprocity theorem. However, persistent transient effects can render the reciprocity unobservable. The numerical study is performed with a discrete numerical model of the molecular dynamics of immiscible mixtures called the immiscible lattice gas. The immiscible lattice gas models both the Navier-Stokes equations and surface tension. Numerical tests presented here additionally provide quantitative validation of the method's ability to simulate wetting phenomena and the effects of capillary pressure. Whereas the numerical study of the linear phenomenological laws utilizes a highly simplified porous medium with one pore and two throats, numerical examples of wetting and nonwetting invasion experiments in a geometrically complex 2-D porous medium are also provided. ¿ American Geophysical Union 1990