Online conservative generalized multiscale finite element method for highly heterogeneous flow models

In this work, we consider an online enrichment procedure in the context of the Generalized Multiscale Finite Element Method (GMsFEM) for the two-phase flow model in highly heterogeneous porous media. The coefficient of the pressure equation is referred to as the permeability and is the main source of heterogeneity within the model. The elliptic pressure equation is solved using online GMsFEM, which is coupled with a hyperbolic transport equation where local conservation of mass is necessary. To satisfy the conservation property, we aim at constructing conservative fluxes within the space of multiscale basis functions through the use of a postprocessing technique. In order to improve the accuracy of the pressure and velocity solutions in the online GMsFEM, we apply a systematic online enrichment procedure. The increase in pressure accuracy due to the online construction is inherited by the conservative flux fields and the desired saturation solutions from the coupled transport equation. Despite the fact that the coefficient of the pressure equation is dependent on the saturation which may vary in time, we construct an approximation space using the absolute permeability field (λ(S) = 1) and no further basis updates follow. Numerical results corresponding to three different types of heterogeneous permeability coefficients are exhibited to show the performance of the proposed methodology.