Quasi-incompressible models for binary fluid flows in porous media

We derive a quasi-incompressible hydrodynamic phase field model and consistent physical boundary conditions for flows of binary incompressible fluids of distinct intrinsic densities in porous media subject to an external force using the generalized Onsager principle (GOP). When the external force is conservative, the model not only conserves mass and volume fraction but also dissipates the total mechanical energy with respect to the consistent boundary conditions in a fixed domain. In the case of gravity, an extended family of hydrodynamical phase field models parametrized by a specific volume fraction parameter, ϕˆ, is derived via the GOP in the context of the buoyancy force, which reduces to the previous model at ϕˆ=0. In the case of a constant mobility and equal intrinsic densities in the binary fluid system, the extended model is identical to the original one for any ϕˆ. While ϕˆ is chosen as the spatially averaged volume fraction in the general case, the extended model may not be dissipative and thus different from the model at ϕˆ=0. Numerical examples are presented to highlight the difference between the two models and interfacial instability with respect to distinct densities when the two models are distinct.