Abstract
A wide variety of multiscale methods have been proposed in the literature to improve simulation runtimes and provide better scaling to large models. With a few notable exceptions, the methods proposed so far are mostly limited to structured grids. We present a new multiscale restricted-smoothed basis (MsRSB) method that is designed to be applicable to stratigraphic and fully unstructured grids. Like many other multiscale methods, it is based on a coarse partition of an underlying fine grid with a set of prolongation operators (also called multiscale basis functions) that map from unknowns associated with the fine grid cells to unknowns associated with the coarse grid blocks. These mappings are constructed by restricted
smoothing: starting from a constant, a localized iterative scheme is applied directly to the fine-scale discretization to give prolongation operators that are consistent with the local properties of the differential operators. The resulting method has three main advantages: First, there are almost no requirements on the geometry and topology of the fine and the coarse grids. Coarse partitions and good prolongation operators can therefore easily be constructed on complex models involving high media contrasts and unstructured cell connections introduced by faults, pinch-outs, erosion, local grid refinement, etc. Moreover, the coarse grid can easily be adapted to features in the geo-cellular model or any precomputed flow field to improve
accuracy. Secondly, the method is accurate and robust when compared to existing multiscale methods. In particular, the method does not need to recompute local basis functions to account for transient behavior: dynamic mobility changes are incorporated by continuing previous iterations with a few extra steps. This way, the cost of updating the prolongation operators will be proportional to the amount of change in fluid mobility and one avoids tolerance-based updates. Finally, since the MsRSB method is formulated on top of a cell-centered, conservative, finite-volume method, it is applicable to any flow model in which one can isolate a pressure equation; in the paper, we discuss incompressible two-phase flow and compressible, three-phase, black-oil type models. For our fine-scale discretization, we use the standard two-point flux-approximation scheme, but the method could equally well have been formulated using a multipoint discretization.
Several numerical examples are presented to highlight features of the method. First, we compare the MsRSB method with the multiscale finite-volume (MsFV) method for single-phase flow problems with
petrophysical parameters from the SPE 10 benchmark. Then we perform several validation studies using
two-phase flow geometry and petrophysical properties from simulation models of two real fields (Gullfaks
and Norne), as well as a compressible gas-injection case described by the black-oil equations.
smoothing: starting from a constant, a localized iterative scheme is applied directly to the fine-scale discretization to give prolongation operators that are consistent with the local properties of the differential operators. The resulting method has three main advantages: First, there are almost no requirements on the geometry and topology of the fine and the coarse grids. Coarse partitions and good prolongation operators can therefore easily be constructed on complex models involving high media contrasts and unstructured cell connections introduced by faults, pinch-outs, erosion, local grid refinement, etc. Moreover, the coarse grid can easily be adapted to features in the geo-cellular model or any precomputed flow field to improve
accuracy. Secondly, the method is accurate and robust when compared to existing multiscale methods. In particular, the method does not need to recompute local basis functions to account for transient behavior: dynamic mobility changes are incorporated by continuing previous iterations with a few extra steps. This way, the cost of updating the prolongation operators will be proportional to the amount of change in fluid mobility and one avoids tolerance-based updates. Finally, since the MsRSB method is formulated on top of a cell-centered, conservative, finite-volume method, it is applicable to any flow model in which one can isolate a pressure equation; in the paper, we discuss incompressible two-phase flow and compressible, three-phase, black-oil type models. For our fine-scale discretization, we use the standard two-point flux-approximation scheme, but the method could equally well have been formulated using a multipoint discretization.
Several numerical examples are presented to highlight features of the method. First, we compare the MsRSB method with the multiscale finite-volume (MsFV) method for single-phase flow problems with
petrophysical parameters from the SPE 10 benchmark. Then we perform several validation studies using
two-phase flow geometry and petrophysical properties from simulation models of two real fields (Gullfaks
and Norne), as well as a compressible gas-injection case described by the black-oil equations.