Nonlinear complexity reduction for fast simulation of flow in heterogeneous porous media

Reduced order modeling techniques have been investigated in the context of reservoir simulation and optimization in the past decade in order to mitigate the computational cost associated with the large-scale nature of the reservoir models. Although great progress has been made in basically two fronts, namely, upscaling and model reduction, there has not been a consensus which method (or methods) is preferable in terms of the trade-offs between accuracy and robustness, and if they indeed, result in large computational savings. In the particular case of model reduction, such as the proper orthogonal decomposition (POD), in order to capture the nonlinear behavior of such models, many simulations or experiments are needed prior to the actual online computations and there in not a clear way to deal with the projection of the reduced basis onto the nonlinear terms for fast implementations. This paper presents a step forward to reduced-order modeling in the reservoir simulation framework. In order to overcome the issues with the nonlinear projections, we proposed to use the POD-DEIM algorithm, based on POD combined with the discrete empirical interpolation method (DEIM) proposed for the solution of large-scale partial differential equations. The DEIM is based on the approximation of the nonlinear terms by means of an interpolatory projection of few selected snapshots of the nonlinear terms. In this case, computational savings can be obtained in a forward run of nonlinear models. Also, in order to incorporate information from the multiple length of scales, especially in the case of highly heterogeneous porous media, we suggest the local-global model reduction framework using the multiscale modeling framework. In this case, we will extend the use of the balanced truncation formulation and show how to couple both frameworks.