Efficient Numerical Simulation of Density‐Driven Flows: Application to the 2‐ and 3‐D Elder Problem

Abstract

AbstractModeling density‐driven flow in porous media is challenging due to the nonlinear coupling between flow and transport equations, the large domains of interest and the wide range of time and space scales involved. Solving this type of problem numerically using a fixed mesh can be prohibitively expensive. Here, we apply a dynamic mesh optimization (DMO) technique along with a control‐volume‐finite element method to simulate density‐driven flows. DMO allows the mesh resolution and geometry to vary during a simulation to minimize an error metric for one or more solution fields of interest, refining where needed and coarsening elsewhere. We apply DMO to the Elder problem for several Rayleigh numbers. It is demonstrated that DMO accurately reproduces the unique two‐dimensional (2D) solutions for low Rayleigh number cases at significantly lower computational cost compared to an equivalent fixed mesh, with speedup of order ×16. For unstable, high Rayleigh number 2D cases, multiple steady‐state fingering solutions exist and are all captured by our approach with high accuracy and significantly reduced computational cost, with speedup of order ×6. Velocity‐dependent dispersion is shown to have a small impact on the 2D numerical solutions. The lower computational cost of simulations using DMO allows extension of the high Rayleigh number case to a three dimensional (3D) configuration. We demonstrate new 3D fingering patterns that have not been observed previously. Early time, transient 3D patterns represent combinations of the previously observed, steady‐state 2D solutions, but all evolve to a single, steady‐state finger in the late time limit.