Abstract: We present a locally conservative enriched Galerkin finite element method that can be applied for solving parabolic equation as well as Stokes equation in a unified fashion. We present the reason why local conservation is important by establishing continuous and discrete maximum principle of coupled flow and transports. We also present that the resulting system can be solved effectively using a fast solver based on algebraic Multigrid method. Finally, a number of application areas will be presented, which include modeling of Tornado and enhanced Oil recovery.