Computational investigation of porous media phase field formulations: Microscopic, effective macroscopic, and Langevin equations

We consider upscaled/homogenized Cahn-Hilliard/Ginzburg-Landau phase field equations as mesoscopic formulations for interfacial dynamics in strongly heterogeneous domains such as porous media. A recently derived effective macroscopic formulation, which takes systematically the pore geometry into account, is computationally validated. To this end, we compare numerical solutions obtained by fully resolving the microscopic pore-scale with solutions of the upscaled/homogenized porous media formulation. The theoretically derived convergence rate ${\cal O}(\epsilon^{1/4})$ is confirmed for circular pore-walls. An even better convergence of order ${\cal O}(\epsilon^{1})$ holds for square shaped pore-walls. We also compute the homogenization error over time for different pore geometries. We find that the quality of the time evolution shows a complex interplay between pore geometry and heterogeneity. Finally, we study the coarsening of interfaces in porous media with computations of the homogenized equation and the microscopic formulation fully resolving the pore space. We recover the experimentally validated and theoretically rigorously derived coarsening rate of ${\cal O}(t^{1/3})$ in the periodic porous media setting. In the case of \emph{critical quenching} and after adding thermal noise to the microscopic porous media formulation, we observe that the influence of thermal fluctuations on the coarsening rate shows after a short, expected phase of universal coarsening, a sharp transition towards a different regime.