A diffuse interface model of two-phase flow in porous media

A single-phase, two-component mixture Darcyscale model characterized by a diffuse interface is proposed as a rational alternative to the conventional singular interface Darcyscale empirical model currently employed for analysing two-phase flows of immiscible fluids through porous media. The proposed scheme possesses conceptual as well as computational and analytical advantages. On the conceptual side, we are able to clearly define the macroscale concept of capillary pressure, which in contemporary literature is incorrectly confounded with the well-known microscale, pore-level Laplace boundary condition at the curved interfaces between the immiscible phases. In this same context we offer insights into fundamental issues that have afflicted mixture theories involving 'interpenetrating continua'. This is accomplished in part by clarifying the distinction between whether the continua being referred to constitute phases (consisting of 'immiscible' multiphase continua) or species (present in an inhomogeneous single-phase continuum). In particular, we show that the issue devolves upon the scale at which the phenomenon is viewed. On the computational and analytic sides our scheme offers the advantages of dealing with continuous fields rather than with discontinuous fields, the latter necessitated in contemporary literature by the existence of singular (mobile) phase boundaries, namely interfaces. Also on the analytical side, our scheme permits a formal physicomathematical transition from the diffuse microscale to the coarser singular-surface scale view, by using singular perturbation techniques to achieve the requisite change in scale. Within this framework we formulate, in a rigorous manner, definitions of macroscale quantities, following which we identify the pertinent phase-specific Darcy's laws. Moreover, one can, in principle, calculate the phenomenological coefficients appearing therein in terms of quadratures of the prescribed microscale data. Our macroscale framework treats the Darcyscale continuum as a multicomponent mixture (referred to as the 'diffuse Darcyscale' view) rather than adopting the more commonly employed coarse-grained picture embodied in interpenetrating continua models or singular-surface Darcyscale models. This fine-scale viewpoint is directed towards the foundational aspects of two-phase flows at both the micro- and macroscales-especially with regard to the existence and interpretation of phase-specific Darcy's laws, including capillary pressure as a macroscale field variable. Eventually, we implement this framework with a simplified linear example to substantiate our thesis. Our study embodies several distinct investigations, logically intertwined by the common objective of erecting rational foundations for describing and quantifying two-phase flows through porous media.