Algebraic Dynamic Multilevel (ADM) Method For Simulations Of Multiphase Flow With An Adaptive Saturation Interpolator

An Algebraic Dynamic Multilevel (ADM) method for simulations of multiphase flow in heterogeneous porous media with an adaptive enriched multiscale formulation for saturation unknowns is presented. ADM maps the fine-scale fully-implicit (FIM) discrete system of equations to a dynamic multilevel system, the resolution of which is defined based on the location of the fluid fronts. The map between the dynamic multilevel resolutions is performed algebraically by sequences of restriction and prolongation operators. While finite-volume restriction operators are necessary to ensure mass conservation at all levels, different interpolation strategies can be considered for each main unknown (e.g., pressure and saturation). For pressure, the multiscale basis functions are used to accurately capture the effect of fine-scale heterogeneities at all levels. In previous works, all other unknowns (e.g., saturation) were interpolated with piece-wise constant functions. Hence, the multiscale nature of saturation equation was not fully exploited. Here, an adaptive interpolation strategy, thus a multiscale transport formulation, is employed for the saturation unknowns that allows to preserve most details of the fine-scale saturation distribution even in regions where a coarser resolution is employed. In regions where the ratio between the coarse and the fine-scale saturation updates is detected to be constant throughout the time-dependent simulation, such ratio is stored and employed as interpolator for subsequent time-steps in which a coarser resolution is employed. Numerical results are presented to study the accuracy and efficiency of the method and the advantages of such interpolation strategy for test cases including challenging non-linear physics, i.e. gravitational and capillary effects.