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Learning macroscopic parameters in nonlinear multiscale simulations using nonlocal multicontinua upscaling techniques. (English) Zbl 1436.76041
Summary: In this work, we present a novel nonlocal nonlinear coarse grid approximation using a machine learning algorithm. We consider unsaturated and two-phase flow problems in heterogeneous and fractured porous media, where mathematical models are formulated as general multicontinuum models. We construct a fine grid approximation using the finite volume method and embedded discrete fracture model. Macroscopic models for these complex nonlinear systems require nonlocal multicontinua approaches, which are developed in our earlier work [“Nonlinear nonlocal multicontinua upscaling framework and its applications”, Int. J. Multiscale Comput. Eng. 16, No. 5, 487–507 (2018; doi:10.1615/IntJMultCompEng.2018027832)]. These rigorous techniques require complex local computations, which involve solving local problems in oversampled regions subject to constraints. The solutions of these local problems can be replaced by solving original problem on a coarse (oversampled) region for many input parameters (boundary and source terms) and computing effective properties derived by nonlinear nonlocal multicontinua approaches. The effective properties depend on many variables (oversampled region and the number of continua), thus their calculations require some type of machine learning techniques. In this paper, our contribution is two fold. First, we present macroscopic models and discuss how to effectively compute macroscopic parameters using deep learning algorithms. The proposed method can be regarded as local machine learning and complements our earlier approaches on global machine learning [“Deep multiscale model learning”, J. Comput. Phys. 406, Article ID 109071, 19 p. (2020; doi:10.1016/j.jcp.2019.109071); “Reduced-order deep learning for flow dynamics”, Preprint, arXiv:1901.10343]. We consider a coarse grid approximation using two upscaling techniques with single phase upscaled transmissibilities and nonlocal nonlinear upscaled transmissibilities using a machine learning algorithm. We present results for two model problems in heterogeneous and fractured porous media and show that the presented method is highly accurate and provides fast coarse grid calculations.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
68T07 Artificial neural networks and deep learning
76S05 Flows in porous media; filtration; seepage
76T06 Liquid-liquid two component flows
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