A new approach to modeling flow in porous media via mixed finite element methods on non-matching multiblock grids is presented. The method is somewhat simpler and less expensive than the mortar method, since flux continuity is incorporated in the velocity space. Theoretical a priori estimates and numerical studies for single-phase flow show almost optimal convergence for pressure and interior velocity. Computational results for two-phase flow indicate that the enhanced velocity method provides accuracy comparable to the mortar method at substantially reduced cost. This work is joint with John Wheeler and Ivan Yotov.