Ensemble phase averaged equations for multiphase flows in porous media. Part 2: A general theory

Most models for multiphase flows in a porous medium are based on a straightforward extension of Darcy’s law, in which each fluid phase is driven by its own pressure gradient. The pressure difference between the phases is thought to be an effect of surface tension and is called capillary pressure. Independent of Darcy’s law, for liquid imbibition processes in a porous material, diffusion models are sometime used. In this paper, an ensemble phase averaging technique for continuous multiphase flows is applied to derive averaged equations and to examine the validity of the commonly used models. Closure for the averaged equations is quite complicated for general multiphase flows in a porous material. For flows with a small ratio of the characteristic length of the phase interfaces to the macroscopic length, the closure relations can be simplified significantly by an approximation with a second order error in this length ratio. This approximation reveals the information of the length scale separation obscured during an averaging process and leads to an equation system similar to Darcy’s law, but with additional terms. Based on interactions on phase interfaces, relations among closure quantities are studied.