Second order linear thermodynamically consistent approximations to nonlocal phase field porous media models

Abstract

We present a weakly compressible, thermodynamically consistent, hydrodynamical model for the binary fluid flow in porous media using the one-fluid multiple component formulation, consisting of a force balance equation with a weak inertia, weakly perturbed continuity equation, and a volume-preserving, nonlocal Allen–Cahn equation for the phase field variable. This model is called the nonlocal Allen–Cahn-Extended-Darcy (NACED) model. In the incompressible and/or the inertialess limit, it reduces to two limiting models that we focus on in this study: the incompressible, nonlocal Allen–Cahn-Extended-Darcy (NACED) model and nonlocal Allen–Cahn–Darcy (NACD) model, respectively. The weakly compressible model provides a pathway for us to develop thermodynamically consistent numerical algorithms for the incompressible models. Guided by the energy variational formulation of the models, we use the energy quadratization method in time and finite difference method in space on staggered grids to derive second-order, linear, coupled and decoupled, energy dissipation rate preserving schemes for the incompressible models. The decoupled schemes for NACED model are obtained by exploiting the intrinsic relation between the incompressible and weakly compressible model. We then prove that the linear systems resulted from the linear schemes are uniquely solvable. Mesh refinement tests are conducted to confirm the convergence rates and benchmark examples of binary fluid flow motion in porous media with and without gravity are presented to showcase the schemes’ accuracy and usefulness.