A weak formulation of the coupled problem of flow and transport is discretized and analyzed numerically. The flow problem is characterized by the Navier-Stokes (or Stokes) equations coupled by Darcy equations. The velocity field is obtained by couplings of finite element and discontinuous Galerkin methods. The concentration equation is solved by an improved discontinuous Galerkin method. Convergence of the schemes is obtained. Numerical examples show the robustness of the method for heterogeneous and fractured porous media.