The study of a coupled flow and transport system in adjacent surface and subsurface regions is of interest for the environmental problem of contaminated aquifers through rivers. The flow in the surface region is characterized by the steady-state Stokes (or Navier-Stokes) equations whereas the flow in the subsurface region is characterized by Darcy's law. The transport equation is coupled to the flow problem via the fluid viscosity and the the flow velocity that appears in both the diffusion and the convection terms of the concentration equation. In other words, the miscible displacement problem is solved in the subsurface. This type of multiphysics couplings is also of importance in the industrial filtration processes

Existence of a weak solution of the coupled problem is first obtained. Discretization algorithms based on the finite element method and the discontinuous Galerkin method are proposed and analyzed. Numerical comparisons between the various numerical solutions are shown.