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

Many 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). For a first test, we do noth parametrize the problem and concentrate on the development of an efficient reduced basis scheme. This allows for separate approximations of the function spaces for the three physical unknowns and the non–linear terms in this numerical scheme. 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. We discuss how the coupling of the unknowns — saturation, velocity and pressure — must be reflected in the generated reduced spaces.