Reduced Basis Model Reduction of Parametrized Two—Phase Flow in Porous Media

Abstract

AbstractMany applications, e.g. in control theory and optimization depend on time-consuming parameter studies of parametrized evolution equations. Reduced basis methods are an approach to reduce the computation time of numerical simulations for these problems. The methods have gained popularity for model reduction of different numerical schemes with remarkable results preferably for scalar and linear problems with affine dependence on the parameter as in Patera and Rozza (2007). Over the last few years, the framework for the reduced basis methods has been continuously extended for non–linear discretizations, coupled problems and arbitrary dependence on the parameter, e.g. Grepl et al. (2007); Drohmann et al. (2010); Carlberg et al. (2011).In this presentation, we apply the framework developed in Drohmann et al. (2010) on a problem that combines all these difficulties. The considered problem models two–phase flow in a porous medium discretized by the finite volume method like in Michel (2004). The two–phase flow equation is of interest, e.g. in the context of oil recovery.For a first test, we concentrate on the development of an efficient reduced basis scheme without any parametrization. This reduced basis scheme is derived by two model reduction steps from the high dimensional finite volume scheme. Firstly, a Galerkin projection on the so–called reduced basis space — a function space spanned by snapshots of the high dimensional solution — is performed. Secondly, the non–linear operators are approximated by an efficiently computable empirical interpolant. We shortly introduce the main aspects of the reduced basis method including the concept of offline/online decomposition, empirical operator interpolation method and reduced basis generation by greedy algorithms.The generalized formulation of the presented reduced basis scheme allows for a separation of the reduced basis space into function spaces for the three physical unknowns — saturation, velocity and pressure — and the approximation of the non–linear terms. It is discussed, how the coupling of the unknowns must be reflected in the generated reduced spaces. Furthermore, we compare the computational complexities of the high–dimensional to the low–dimensional computations theoretically an by numerical experiments. All the presented experiments are implemented with our reduced basis software package RBmatlab.