The Multiscale Finite-Volume (MSFV) method is designed to reduce the computational cost by splitting the original global problem into a set of localized problems (with approximate local boundary conditions) coupled by a coarse global problem. The MSFV error can be reduced iteratively (i-MSFV) using an improved (smoothed) multiscale solution to enhance the localization conditions or by using a Krylov subspace method (e.g. GMRES) preconditioned by the MSFV system. In a multiphase flow system, a balance between accuracy and computational efficiency should be achieved by finding a minimum number of i-MSFV iterations (on pressure) that are necessary to achieve the desired accuracy of the saturation solution.

In this work, we extend the i-MSFV method to sequential implicit simulations of time dependent problems. To control the error of the coupled saturation-pressure system, we analyze the transport error due to approximate flux fields. We then propose a-priori error estimate based on the residual of the pressure equation. At the beginning of the simulation we iterate until the specified pressure accuracy is achieved. To minimize the required iterations in time-dependent multiphase flow problems, this initial solution is also utilized to improve the localization assumption at later time steps. Additional iterations are employed only when the residual becomes larger than a specified threshold value. The saturation equation error in presence of an approximate conservative MSFV velocity field is anlayzed. Finally, a saturation error expression is derived, and a residual based strategy is introduced to control numerical errors. Numerical results show that the proposed strategy results in efficient and accurate simulation of multiphase flow in heterogeneous porous media.

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