ACCURATE LOCAL UPSCALING WITH VARIABLE COMPACT MULTI-POINT TRANSMISSIBILITY CALCULATIONS J. V. LAMBERS * , M. G. GERRITSEN † , AND B. T. MALLISON ‡ Abstract. We propose a new single phase local upscaling method that uses spatially varying multi-point transmissibility calculations. The method is demonstrated on two-dimensional Cartesian and adaptive Cartesian grids. For each cell face in the coarse upscaled grid, we create a local fine grid region surrounding the face on which we solve two generic local flow problems. The multi-point stencils used to calculate the fluxes across coarse grid cell faces involve the six neighboring pressure values. They are required to honor the two generic flow problems. The remaining degrees of freedom are used to maximize compactness and to ensure that the flux approximation is as close as possible to being two-point. The resulting multi-point flux approximations are spatially varying (a subset of the six neighbors is adaptively chosen) and reduce to a two-point expressions in cases without full- tensor anisotropy. Numerical tests show that the method significantly improves upscaling accuracy as compared to commonly used local methods and also compares favorably with a local-global upscaling method. Key words. Scale up; Subsurface; Heterogeneity; Flow simulation; Channelized; Permeability; Transmissibility; Multiscale; Adaptivity; 1. Introduction. Subsurface formations typically display high degrees of vari- ability over multiple length scales. The systems may exhibit geometrically complex features with complicated large scale connectivity. The effects of permeability vari- ability, its uncertainty and its potentially complex connectivity must be included in simulations of flow and transport in aquifers or petroleum reservoirs because they can fundamentally impact simulation results. Uncertainty can be taken into account either through stochastic modeling or by simulating a number of deterministic geosta- tistical realizations of the reservoir [8], which is the approach we assume in this work. To reduce computational costs, simulations are generally performed on grids that are coarse compared to the given geocellular grids. These coarsened flow models should adequately represent key behaviors, such as the overall flow rate for given boundary conditions and critical connected flow paths. Multi-scale methods, in our definition, are methods designed to capture multiple scales involved in these fluid flow processes. These include approaches that upscale to a coarse simulation grid and reconstruct the solution on a finer scale, often the geocellular scale, within each coarse grid cell, or approaches based on grid refinement. In the latter case, again appropriate upscaling methods to the grid levels involved must be developed. Hence, any upscaling procedure can be incorporated in a multiscale procedure. In this work we are concerned with transmissibility upscaling, that is, with finding representative coefficients that relate fluxes at faces between coarse cells to pressures of cells neighboring the faces. As demonstrated in [9], transmissibility upscaling is gen- erally found to give substantially improved flow results over permeability upscaling. We limit ourselves to single phase upscaling methods also to find transmissibilities for the upscaled equation 2.1, but these transmissibilities can also be used for multi-phase * Department of Energy Resources Engineering, Stanford University, Green Earth Sciences Build- ing, Stanford, CA 94305-2220, USA † Department of Energy Resources Engineering, Stanford University, Green Earth Sciences Build- ing, Stanford, CA 94305-2220, USA ‡ Chevron Energy Technology Company, San Ramon, CA, USA 1