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

Abstract

We derive incompressible and weakly compressible hydrodynamical models for binary fluid flows in porous media using one fluid multiple component formulations, guided by the generalized Onsager principle. These thermodynamically consistent models are named as the incompressible and weakly compressible nonlocal Allen-Cahn-Brinkman (NACB) and nonlocal Allen-Cahn-Darcy (NACD) model, respectively. These models are energy dissipative, consisting of a simplified force balance equation, continuity equation, and a volume-preserving, nonlocal Allen-Cahn equation for the phase field variable. Guided by the energy variational formulation of the models, we use the energy quadratization method to derive second-order, linear, coupled and decoupled, energy dissipation rate preserving, semi-discrete schemes in time for the models. We then apply second-order finite difference methods on staggered-grids in space to discretize the semi-discrete schemes to obtain fully discrete schemes. We explore the intrinsic relation between the decoupled schemes and the weakly compressible NACB model and prove that the linear systems resulted from the linear schemes are uniquely solvable. Mesh refinement tests are conducted to confirm the convergence rates. Then, benchmark examples of binary fluid flow motion in porous media with and without gravity are presented to showcase the schemes' accuracy and usefulness.