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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.

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Abstract:We propose an online adaptive local-global POD-DEIM model reduction method for flows in heterogeneous porous media. The main idea of the proposed method is to use local online indicators to decide on the global update, which is performed via reduced cost local multiscale basis functions. This unique local-global online combination allows (1) developing local indicators that are used for both local and global updates (2) computing global online modes via local multiscale basis functions. The multiscale basis functions consist of offline and some online local basis functions. The approach used for constructing a global reduced system is based on Proper Orthogonal Decomposition (POD) Galerkin projection. The nonlinearities are approximated by the Discrete Empirical Interpolation Method (DEIM). The online adaption is performed by incorporating new data, which become available at the online stage. Once the criterion for updates is satisfied, we adapt the reduced system online by changing the POD subspace and the DEIM approximation of the nonlinear functions. The main contribution of the paper is that the criterion for adaption and the construction of the global online modes are based on local error indicators and local multiscale basis function which can be cheaply computed. Since the adaption is performed infrequently, the new methodology does not add significant computational overhead associated with when and how to adapt the reduced basis. Our approach is particularly useful for situations where it is desired to solve the reduced system for inputs or controls that result in a solution outside the span of the snapshots generated in the offline stage. Our method also offers an alternative of constructing a robust reduced system even if a potential initial poor choice of snapshots is used. Applications to single-phase and two-phase flow problems demonstrate the efficiency of our method.

Automating model calibration and production optimization is computationally demanding because of the intensive multiphaseflow-simulation runs that are needed to predict the response of real reservoirs under proposed changes in model inputs. Fast surrogate models have been proposed to speed up reservoir-response predictions without compromising accuracy. Surrogate models either are derived by preserving the physics of the involved processes (e.g., mass balance) to provide reliable long-range predictions or are developed solely on the basis of statistical input/ output relations, in which case they can only provide short-range predictions because of the absence of the physical processes that govern the long-term behavior of the reservoir. We present an alternative approach that combines the advantages of both statistics-based and physics-based methods by reducing the flow predictions in complex 3D models to a 1D flow-network model. The existing injection/production wells in the original model form the nodes or vertices of the flow network. Each pair of wells (nodes) in the flow network is connected by use of a 1D numerical simulation model, resulting in a connected network of 1D gridbased simulation models. The coupling between the individual 1D flow models is enforced at the nodes where network edges intersect. The proposed flow-network model provides a useful and fast tool for characterizing interwell connectivity, estimating drainage volume between each pair of wells, and predicting reservoir production over an extended period of time for optimization purposes. The parameters of the flow-network model are estimated by a robust training approach to ensure that the network model reproduces the response of the full model under a wide range of development strategies. This step helps the network model to preserve its predictive power during optimization iterations when alternative development strategies are proposed and evaluated to find the solution. We demonstrate the effectiveness and applicability of the proposed flow-network model by use of two-phase waterflooding experiments.

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