Global-local model reduction for heterogeneous Forchheimer flow

In this paper, we propose a mixed Generalized Multiscale Finite Element Method (GMsFEM) for solving nonlinear Forchheimer flow in highly heterogeneous porous media. We consider the two term law form of the Forchheimer equation in the case of slightly-compressible single-phase flows. We write the resulting system in terms of a degenerate nonlinear flow equation for pressure when the nonlinearity depends on the pressure gradient. The proposed approach constructs multiscale basis functions for the velocity field following Mixed-GMsFEM as developed in Chung et al. (2015). To reduce the computational cost resulting from solving nonlinear system, we combine the GMsFEM with Discrete Empirical Interpolation Method (DEIM) to compute the nonlinear coefficients in some selected degrees of freedom at each coarse domain. In addition, a global reduction method such as Proper Orthogonal Decomposition (POD) is used to construct the online space to be used for different inputs or initial conditions. We present numerical and theoretical results to show that in addition to speeding up the simulation we can achieve good accuracy with a few basis functions per coarse edge. Moreover, we present an adaptive method for basis enrichment of the offline space based on an error indicator depending on the local residual norm. We use this enrichment method for the multiscale basis functions at some fixed time levels. Our numerical experiments show that these additional multiscale basis functions will reduce the current error if we start with a sufficient number of initial offline basis functions.