On the concept of macroscopic capillary pressure in two-phase porous media flow

Although two-phase fluid flow in porous media has been an established research field for decades, its theoretical background is still incomplete. In particular, while a universal definition of capillary pressure exists at the microscale, its upscaling to the macro-scale is still rather vague. In this work, a clear and rigorous definition of the macroscopic capillary pressure is proposed, which follows naturally from application of the method of volume averaging to interface properties in multiphase systems. The relationship between the macroscopic capillary pressure and the average properties of the medium is given by the macroscopic momentum balance for the fluid-fluid interfaces, in a form which can be interpreted as a generalized Young-Laplace equation at the macro-scale. We then present simulation results of drainage in a porous region extracted from a three-dimensional micro-CT image of a real carbonate rock, and show how our formulation differs from the standard one which is commonly employed in field-scale computational codes.