Parametrized Model Order Reduction Applied to Reservoir Simulation

Abstract

The development of efficient numerical reservoir simulation is an essential step in devising advanced production optimization strategies and uncertainty quantification methods applied to porous media flow. In this case, a highly accurate and detailed description of the underlying models lead to a solution of a set of partial differential equations, which after discretization, induce dynamical systems of very large dimensions. In order to overcome the computational costs associated with these large-scale models, several forms of model-order reduction have been proposed in the literature. In porous media flow, two different approaches are used: (1) a "coarsening" of the discretization grid in a process called upscaling; and (2) a reduction in the number of state variables (i.e., pressure and saturation) directly in a process called approximation of dynamical systems. Recently, the the idea of combining both approaches have been proposed using the control-relevant upscaling (CRU) methodology. In this paper, we investigate the use of the so-called parametric model order reduction (PMOR) techniques applied to porous media flow simulation in a system-theoretical framework. PMOR entails the generation of reduced-order models which retains the functional dependency on specific parameters of the original large-scale system. In particular, this work focuses on the the application of PMOR to the case of single-phase flow, in which the dependencies of the porous media properties, such as, permeability, and the discretization parameter, such as, grid sizes, is investigated. The the main ideas behind model order reduction will be reviewed, including the general framework of interpolatory projection techniques and applications to single-phase flow test cases will be developed.