A new transmissibility / relative permeability system for upscaling subgrid-scale reservoir heterogeneity to coarser finite-difference simulation cells has been developed. Material balance and entropy considerations require coarse-grid transmissibilities to be calculated from the fine scale full-field Laplace solutions and to be non-negative. These requirements are satisfied by the deformation of originally regular coarse-grid boundaries in a transformed space with axes defined by flow potentials and streamfunctions. The resulting coarse grids are bundles of streamtube segments with exact transmissibilities that reproduce the input Laplace solution as the material balance solutions. The Buckley-Leverett displacement profiles are conceptually mapped onto each streamtube and are superposed to build rigorous effective relative permeabilities through a new formula that directly relates the saturation profiles between streamtubes. Test upscaling problems with two orders of magnitude reduction in the number of simulation cells for an extremely heterogeneous system demonstrate the applicability of the proposed method.