Localized Model Reduction in Porous Media Flow∗

This paper introduces a new localized approach to construct an efficient reduced order model for fluid flow simulation and optimization in porous media flow. For nonlinear systems, one of the most common methodology used is the proper orthogonal decomposition (POD) combined with discrete empirical interpolation method (DEIM) due to its computational efficiency and good approximation. Whereas regular POD-DEIM approximates the fine scale model with just one single reduced subspace, the localized POD (LPOD) and localized DEIM (LDEIM) that are introduced in this work compute several local subspaces. Each subspace characterize a region of solutions and all together they not only can approximate the high fidelity model better, but also reduced the computational cost of simulation. LPOD and LDEIM use classification approach to find these regions in the offline computational phase. After obtaining each class, POD and DEIM is applied to construct the basis of the reduced space. In the online phase, at each time step, the reduced states and functions will be used to find the most representative basis for POD and DEIM without requiring fine scale information. The advantages of LPOD and LDEIM are shown in a numerical example of two phase flow in porous media.