Numerical approximations for a miscible displacement process in porous media

A reservoir simulation problem describing a miscible displacement process is stated and analyzed, and a semidiscrete numerical scheme based on a Galerkin method with mesh refinements is presented for its solution. The underlying physical problem involves two miscible, incompressible fluids of equal viscosity flowing in a porous medium. Injection and production wells are rigorously modelled, which leads to a formulation containing nearly singular source and sink terms. The singularities arising in the solution are described in some detail, and it is shown how knowledge of their forms motivate the choice of an appropriate mesh refinement for the numerical scheme. Optimal order quadratic convergence is shown for the resulting semidiscrete approximation using piecewise linear elements.