In this paper, we present a global-local model reduction for fast multiscale reservoir simulations in highly heterogeneous porous media with applications to optimization and history matching. Our proposed approach identifies a low dimensional structure of the solution space. We introduce an auxiliary variable (the velocity field) in our model reduction that allows achieving a high degree of model reduction. The latter is due to the fact that the velocity field is conservative for any low-order reduced model in our framework. Because a typical global model reduction based on POD is a Galerkin finite element method, and thus it can not guarantee local mass conservation. This can be observed in numerical simulations that use finite volume based approaches. Discrete Empirical Interpolation Method (DEIM) is used to approximate the nonlinear functions of fine-grid functions in Newton iterations. This approach allows achieving the computational cost that is independent of the fine grid dimension. POD snapshots are inexpensively computed using local model reduction techniques based on Generalized Multiscale Finite Element Method (GMsFEM) which provides (1) a hierarchical approximation of snapshot vectors (2) adaptive computations by using coarse grids (3) inexpensive global POD operations in a small dimensional spaces on a coarse grid. By balancing the errors of the global and local reduced-order models, our new methodology can provide an error bound in simulations. Our numerical results, utilizing a two-phase immiscible flow, show a substantial speed-up and we compare our results to the standard POD-DEIM in finite volume setup.

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 em- pirical 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.

Summary We present a global/local model reduction for fast multiscale reservoir simulations in highly heterogeneous porous media. Our approach identifies a low-dimensional structure in the solution space. We introduce an auxiliary variable (the velocity field) in our model reduction that achieves a high compression of the model. This compression is achieved because the velocity field is conservative for any low-order reduced model in our framework, whereas a typical global model reduction that is based on proper-orthogonal-decomposition (POD) Galerkin projection cannot guarantee local mass conservation. The lack of mass conservation can be observed in numerical simulations that use finite-volume-based approaches. The discrete empirical interpolation method (DEIM) approximates fine-grid nonlinear functions in Newton iterations. This approach delivers an online computational cost that is independent of the fine-grid dimension. POD snapshots are inexpensively computed with local model-reduction techniques that are based on the generalized multiscale finite-element method (GMsFEM) that provides (1) a hierarchical approximation of the snapshot vectors, (2) adaptive computations with coarse grids, and (3) inexpensive global POD operations in small dimensional spaces on a coarse grid. By balancing the errors of the global and local reduced-order models, our new methodology provides an error bound in simulations. Our numerical results, by use of a two-phase immiscible flow, show a substantial speedup, and we compare our results with the standard POD-DEIM in a finite-volume setup.

Despite great advances in reservoir simulation capabilities with the introduction of high-performance computing (HPC) platforms and enhanced solvers, high fidelity grid-based simulation still remains a challenging task. This task is especially demanding for fine-resolved geological reservoirs with multiphase and multicomponents and in production optimization and uncertainty quantification frameworks where several calls of the large scale simulation model need to be performed. Model order reduction techniques have been applied in porous media flow modeling in order to obtain fast simulation surrogates to alleviate the high computational cost associated with these simulations. The difficulty of applying model reduction techniques to porous media flow arises from the nonlinearity of the system. In order to overcome the issue of nonlinearity, we introduce the bilinear form of the dynamical system which in many cases produces a satisfying approximation of the system. The bilinear approximation is a simple form of the parent system and it is linear in the input and linear in the state but it not linear in both jointly. The bilinear form is computed using truncated multidimensional Taylor's expansion of the system terms. Examples are presented to illustrate this recent approach for the case of single phase flow modeling, and comparisons are made with the case of linearized models and the full nonlinear models. In addition, we present methods to reduce the complexity of the bilinear system of equations and results are presented that compare linear model reduction (balanced truncation) and nonlinear model reductions, such as POD techniques.