## ABSTRACT

Relative permeability is a concept proposed in the 1930’s and 1940’s to describe multiphase flow using equations constructed in analogy with Darcy’s equation for single phase flow. Although this description was treated with some skepticism at first it eventually became the standard description of multiphase flow in the engineering literature. This occurred because there were no credible alternatives at the time. However it is now possible to rigorously construct the equations of multiphase flow and examine the role of relative permeability in some detail. The concept of relative permeability, or as referred to by Rose (2000) “extensions of Darcy’s Law” has been discussed by numerous authors Muscat (1937, 1949), Hubbert (1940 1956), Buckley and Leverett (1942), Rose (1949), Yuster (1951), Slattery (1970), Bear (1972), Scheidegger (1974), Whittaker (1980), de la Cruz and Spanos (1983), Dullien (1992), Bentsen (1994), Spanos (2002). As well a rather outrageous concept has also crept in to the debate, that relative permeability is associated with Onsager’s relations. The papers that discuss this have mistakenly associated the thermodynamics of molecular mixtures with porous media. In the case of porous media the phases are mixed at various scales. As a result Onsager’s relations place very strong restrictions on the dynamic interactions of the various phases such that the laws of physics are obeyed at all scales. These relations are straightforward to construct and have nothing to do with permeability. In the present paper it is shown that relative permeability is not a parameter at all but rather a function of the dynamic variables and that relative permeability may be determined by both generalizing the equations of motion and introducing an additional degree of freedom into the motions. The associated additional dynamic variable, saturation, introduces an additional equation of motion associated with the large scale pressure difference between the fluid phases. This additional large scale equation of motion allows for a resolution of both the Buckley-Leverett paradox and the closure problem encountered in many engineering analyses using volume averaging to construct large scale flow equations and relative permeability. The construction of Onsager’s relation in the case of thermostatic compression is presented in the Appendix as an illustration of the constraints they impose on the dynamical motions of porous media and the associated parameters. In the present discussion the following scales will be considered:

• Microscale (molecular scale) Physical interactions described in terms of molecular dynamics. Mixing of the phases is described by molecular diffusion.