Effect of wettability on two-phase quasi-static displacement: validation of two pore scale modeling approaches

Understanding of pore-scale physics for multiphase flow in porous media is essential for accurate description of various flow phenomena. In particular, capillarity and wettability strongly influence capillary pressure-saturation and relative permeability relationships. Wettability is quantified by the contact angle of the fluid-fluid interface at the pore walls. In this work we focus on the non-trivial interface equilibria in presence of non-neutral wetting and complex geometries. We quantify the accuracy of a volume-of-fluid (VOF) formulation, implemented in a popular open-source computational fluid dynamics code, compared with a new formulation of a level set (LS) method, specifically developed for quasi-static capillarity-dominated displacement. The methods are tested in rhomboidal packings of spheres for a range of contact angles and for different rhomboidal configurations and the accuracy is evaluated against the semi-analytical solutions obtained by Mason and Morrow (1994). While the VOF method is implemented in a general purpose code that solves the full Navier-Stokes (NS) dynamics in a finite volume formulation, with additional terms to model surface tension, the LS method is optimised for the quasi-static case and, therefore, less computationally expensive. To overcome the shortcomings of the finite volume NS-VOF system for low capillary number flows, and its computational cost, we introduce an overdamped dynamics and a local time stepping to speed up the convergence to the steady state, for every given imposed pressure gradient (and therefore saturation condition). Despite these modifications, the methods fundamentally differ in the way they capture the interface, as well as in the number of equations solved and in the way the mean curvature (or equivalently capillary pressure) is computed. This study is intended to provide a rigorous validation study and gives important indications on the errors committed by these methods in solving more complex geometry and dynamics, where usually many sources of errors are interplaying.