Dynamic adaptive mesh optimisation for immiscible viscous fingering

Immiscible fingering is challenging to model since it requires a very fine mesh for the numerical method to capture the interaction of the shock front with the capillary pressure. This can result in computationally intensive simulations if a fixed mesh is used. We apply a higher order conservative dynamic adaptive mesh optimisation (DAMO) technique, to model immiscible viscous fingering in porous media. We show that the approach accurately captures the development and growth of the interfacial instability. Convergence is demonstrated under grid refinement with capillary pressure for both a fixed unstructured mesh and with DAMO. Using DAMO leads to significantly reduced computational cost compared to the equivalent fixed mesh simulations. We also present the late-time response of viscous fingers through numerical examples in a 2D rectangular domain and in a 3D cylindrical geometry. Both problems are computationally challenging in the absence of DAMO. The dynamic adaptive problem requires up to 36 times fewer elements than the prohibitively expensive fixed mesh solution, with the computational cost reduced accordingly.