Novel unstructured grid higher order convection schemes are presented. The schemes are coupled with locally conservative flux continuous control-volume distributed (CVD) finite-volume schemes for the porous medium general tensor pressure equation on structured and unstructured grids in 2-D and 3-D. The schemes are developed for multi-phase flow in porous media. Benefits of the schemes in terms of improved front resolution and medium discontinuity resolution are demonstrated. Comparisons with current methods including the control-volume finite element method highlight the advantages of the new formulation for three dimensional reservoir simulation.