Global variable compact multipoint methods for accurate upscaling with full-tensor effects

New transmissibility upscaling procedures designed to accurately capture full-tensor effects are developed and applied. These techniques are based on variable compact multipoint (VCMP) flux approximations. VCMP is extended to irregular grids. Two approaches for including global flow information within the VCMP upscaling procedure are considered—one in which the upscaled model is determined directly and one in which iteration of the coarse-scale model is used to minimize the mismatch between coarsescale fluxes and integrated fine-scale fluxes. To guarantee monotonicity, the VCMP stencils are adapted to assure the coefficient matrix is an M-matrix whenever nonmonotone solutions are encountered. The new VCMP procedures are applied to multiple realizations of two-dimensional fine-scale permeability descriptions for coarse models defined on both Cartesian and irregular quadrilateral grids. Both log-normally distributed permeability fields with oriented layers and channelized models are considered. Six different upscaling techniques (extended local, direct global, and iterated global, each using both two-point and VCMP T. Chen · M. G. Gerritsen (B) · J. V. Lambers · L. J. Durlofsky Department of Energy Resources Engineering, Stanford University, Stanford, CA, 94305-2220, USA e-mail: [email protected] T. Chen e-mail: [email protected] J. V. Lambers e-mail: [email protected] L. J. Durlofsky e-mail: [email protected] flux approximations) are assessed for four different sets of global boundary conditions. The global VCMP techniques consistently display high degrees of accuracy for total flow rate, L2 flux error, and L2 pressure error. For the oriented-layer cases, where full-tensor effects are important, the global VCMP methods are shown to provide clearly better overall accuracy than analogous methods based on two-point flux approximations. For channelized cases in which full-tensor effects are not significant, both types of methods provide high levels of accuracy. The selective M-fix procedure is also shown to lead to improved accuracy, which can be significant in some cases. In total, for the systems considered here, the new global VCMP upscaling techniques are observed to provide the best overall accuracy of any of the upscaling methods investigated.