Dynamic mesh optimisation for unstable density-driven flow modelling: Application to the Elder problem in 2- and 3D

Density-driven flows in porous media are frequently encountered in natural systems and arise from the gravitational instabilities introduced by fluid density gradients. They have a significant economic and environmental impact, and numerical modelling is often used to predict the behaviour of these flows for risk assessment, reservoir characterisation or management. However, modelling density-driven flow in porous media is very 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. We propose a dynamic mesh optimisation (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 minimise an error metric for one or more solution fields of interest, refining where needed and coarsening elsewhere. First, we apply DMO to the original two-dimensional Elder problem for several Rayleigh numbers. We demonstrate that DMO very accurately reproduces the unique solutions for low Rayleigh number cases at significantly lower computational cost compared to an equivalent fixed mesh, with speedup of order x13. For unstable high Rayleigh number cases, multiple steady-state solutions exist, and we show that they are all captured by our approach with high accuracy and significantly reduced computational cost, with speedup of order x4. We then extend the high Rayleigh number case to a three-dimensional configuration and demonstrate new steady-state solutions that have not been observed previously. The use of DMO allows efficient investigation of the Ra>400 Elder problem solutions both in 2D and 3D. Its high efficiency also paves the way for real-scale density-driven flow simulations within more realistic, complex geometries, such as heterogeneous aquifers or sedimentary basins.