Multiscale model reduction with local online correction for polymer flooding process in heterogeneous porous media

In this work, we consider a polymer flooding process in heterogeneous media. A system of equations for pressure, water saturation, and polymer concentration describes a mathematical model. For the construction of the fine grid approximation, we use a finite volume method with an explicit time approximation for the transports and implicit time approximation for the flow processes. We employ a loose coupling approach where we first perform an implicit pressure solve using a coarser time step. Subsequently, we execute the transport solution with a minor time step, taking into consideration the constraints imposed by the stability of the explicit approximation. We propose a coupled and splitted multiscale method with an online local correction step to construct a coarse grid approximation of the flow equation. We construct multiscale basis functions on the offline stage for a given heterogeneous field; then, we use it to define the projection/prolongation matrix and construct a coarse grid approximation. For an accurate approximation of the nonlinear pressure equation, we propose an online step with calculations of the local corrections based on the current residual. The splitted multiscale approach is presented to decoupled equations into two parts related to the first basis and all other basis functions. The presented technique provides an accurate solution for the nonlinear velocity field, leading to accurate, explicit calculations of the saturation and concentration equations. Numerical results are presented for two-dimensional model problems with different polymer injection regimes for two heterogeneity fields.