Existence and uniqueness of solution to unsteady Darcy-Brinkman problem with Korteweg stress for modelling miscible porous media flow (Q6597305)

The authors study a fluid flow model for a porous medium, based on the Brinkman equation with a force term, a convection-diffusion-reaction equation, and the continuity equation. The fluid is incompressible, newtonian, with constant viscisity. The porous medium is assumed to be heterogeneous. Porosity is a function of the solution (the concentration), thus heterogeneity of the medium and non-linearity of the problem. The authors prove the existence and uniqueness of a solution to the weak formulation of the problem. The model can be applied to high-porosity media, including karst reservoirs. The governing equations are the continuity equation (simply zero divergence of the velocity), reaction-diffusion (concentration changes due to advection, diffusion, and is taken out by a linear reaction term), and a Stokes-type momentum-conservation equation that accounts for acceleration being expressed via not only pressure gradient and viscosity, but also concentration-dependent stress and external force. Integral pressure is assumed to be zero. No equation for pressure is given and it disappears in the weak form. The domain is 2D, time is bounded (a segment). Boundary conditions are no-slip, no-penetration, and no-flux, i.e., zero velocity and zero normal derivative of concentration.