An algebraic dynamic multilevel and multiscale method with non-uniform mesh resolution and adaptive algebraic multiscale solver operator for the simulation of two-phase flows in highly heterogeneous petroleum reservoirs

Classical Multiscale Finite Volume (MsFV) methods can produce highly oscillatory pressure solutions (i.e., non-monotonic) for media with high permeability contrasts. This may lead to the production of spurious pressure fields, leading, for instance, to the appearance of gas throughout the reservoir when the pressure erroneously falls below the bubble point pressure, substantially increasing the computational cost to solve the problem due to the necessity of using iterative correction procedures to improve the solution. In this context, in the present paper, we propose an adaptive flow-based dual volume agglomeration strategy for correcting the non-physical terms in the coarse transmissibility matrix by means of a preprocessing local step. These terms can be seen as negative transmissibilities on the coarse scale mesh generated by huge permeability contrasts on the boundaries of the dual mesh. This is done by using a local recalculation of the basis functions in a patch defined by a judicious merging of dual coarse volumes that substantially reduces the spurious oscillations. Classical multilevel and multiscale methods define a uniform level at each coarse control volume, i.e., the same mesh level is used at each coarse control volume. As a result, in multiphase flow problems, there is a necessity to include volumes that do not contain the saturation front in the high-resolution level. To handle this problem, we also present a framework to deal with non-uniform levels at each coarse control volume, which allows the use of fine-scale control volumes only where it is needed, in order to produce smaller coarse scale matrices than those obtained from classical multilevel and multiscale methods. The proposed methodologies were applied to approximate pressure solutions in an Implicit Pressure Explicit Saturation (IMPES) strategy. In the present work, the flow problem is implicitly solved on the multilevel resolution and the transport problem is explicitly solved on the fine-scale. Our strategy was tested with two well-known challenging benchmark problems in order to show its accuracy and efficiency.