Adaptive Strategies in the Multilevel Multiscale Mimetic (M3) Method for Two-Phase Flows in Porous Media
Konstantin Lipnikov, D. Moulton, Daniil Svyatskiy · Multiscale Modeling and Simulation · 2011
The multilevel multiscale mimetic ([Formula: see text]) method was proposed in Lipnikov, Moulton, and Svyatskiy [J. Comput. Phys., 227 (2008), pp. 6727–6753] for simulating two-phase flows (water and oil) in a heterogeneous reservoir. The governing equations are an elliptic equation for the reservoir pressure and a hyperbolic equation for the water saturation. The challenge lies in the influence that fine-scale features of the porous medium may have on coarse-scale properties of the solution. This challenge is accentuated in highly heterogeneous media with large correlation lengths that defy the use of simple parameter averaging techniques. The [Formula: see text] method builds a structure preserving multilevel hierarchy of models that are locally conservative at each level. The mimetic finite difference discretization handles full tensor permeabilities and general unstructured meshes. We introduced a unified model upscaling technology that highlights the impact of a well model on the upscaled pressure. We developed and analyzed numerically two new adaptive strategies. First, an adaptive mesh coarsening strategy based on the equidistribution of fluxes is developed for accelerating the solution of the transport equation. Second, the multilevel hierarchy of models is only regenerated when a change in the velocity field exceeds some threshold. The effectiveness of these adaptive strategies is verified with numerical experiments of a well-driven flow.