Local modeling of instability onset for global finger evolution

Abstract

Simulation of density-driven instabilities requires flexible methods to deal with the different spatial and temporal scales involved. Downscaling approaches based on standard adaptive grid refinement aim at resolving the fine-scale details only in the region of interest, but they may become computationally expensive in presence of very corrugated unstable fronts because the problem to be solved approaches the size of the fully refined system. The Downscaling Multiscale Finite-Volume (DMsFV) method overcomes this issue by splitting the problems into a set of localized subproblems that interact only through a global problem. However, in presence of convective instabilities (e.g. density-driven fingers) the diffusion scale has to be resolved only at early times to capture the evolution of infinitesimal random perturbations, whereas at later times fingers have developed and merged, allowing the use of a coarser numerical description. Based on this observation, we present an adaptive algorithm which splits the simulation into three stages: an onset stage in which a set of localized problems is solved independently to capture the initial growth of the instabilities; a transition stage in which the DMsFV method is used to couple local and global scales; and a global stage in which only a fully coarsened description of the problem is employed. The dissolution–diffusion–convection problem (which is typically studied in the context of CO2 sequestration) is chosen as an example to evaluate the accuracy of the adaptive algorithm. For this problem, the use of a coarse grid that does not resolve the fine-scale details at earlier times leads to a dramatic underestimation of mass influx and penetration depth. On the contrary, the solutions obtained with the adaptive algorithm are in good agreement with the reference solution (obtained with a fully refined discretization) and are able to capture total mass influx and penetration depth with excellent accuracy. This demonstrates the need and the effectiveness of modeling local details during the instability onset to capture large-scale features of the concentration patterns at later times.