Recent advances in multiscale methods have shown great promise in modeling multiphase flow in highly detailed heterogeneous domains. Existing multiscale methods, however, solve for the flow field (pressure and total-velocity) only. Once the fine-scale flow field is reconstructed, the saturation equations are solved on the fine scale. With the efficiency in dealing with the flow equations greatly improved by multiscale formulations, solving the saturation equations on the fine scale becomes the relatively more expensive part. In this paper, we describe an adaptive multiscale finite-volume (MSFV) formulation for the nonlinear transport (saturation) equations. A general algebraic multiscale formulation consistent with the operator based framework proposed by Zhou and Tchelepi (SPEJ 13:267-173) is presented. Thus, the flow and transport equations are solved in a unified multiscale framework. Two types of multiscale operators, namely restriction and prolongation, are used to construct the multiscale saturation solution. The restriction operator is defined according to the local sum of the fine-scale transport equations in a coarse gridblock. Three adaptive prolongation operators are defined according to the local saturation history at a particular coarse block. The three operators have different computational complexity, and they are used adaptively in the course of a simulation run. When properly used, they yield excellent computational efficiency while preserving accuracy. This adaptive multiscale formulation has been tested using several challenging problems with strong heterogeneity, large buoyancy effects, and changes in the well operating conditions (e.g., switching injectors and producers during simulation). The results demonstrate that adaptive multiscale transport calculations are in excellent agreement with fine-scale reference solutions, but with a much lower computational cost.
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