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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.

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
Full Text: DOI
[1] Abadi, Martín; Barham, Paul; Chen, Jianmin; Chen, Zhifeng; Davis, Andy; Dean, Jeffrey; Devin, Matthieu; Ghemawat, Sanjay; Irving, Geoffrey; Isard, Michael, Tensorflow: a system for large-scale machine learning, (OSDI, vol. 16 (2016)), 265-283
[2] Bakhvalov, N. S.; Panasenko, G. P., Homogenization in Periodic Media, Mathematical Problems of the Mechanics of Composite Materials (1984), Nauka: Nauka Moscow · Zbl 0607.73009
[3] Balay, Satish; Buschelman, Kris; Eijkhout, Victor; Gropp, William D.; Kaushik, Dinesh; Knepley, Matthew G.; McInnes, Lois Curfman; Smith, Barry F.; Zhang, Hong, Petsc users manual (2004), Argonne National Laboratory, Technical report, Technical Report ANL-95/11-Revision 2.1. 5
[4] Bosma, Sebastian; Hajibeygi, Hadi; Tene, Matei; Tchelepi, Hamdi A., Multiscale finite volume method for discrete fracture modeling on unstructured grids (ms-dfm), J. Comput. Phys. (2017) · Zbl 1375.76097
[5] Chen, Y.; Durlofsky, Louis J.; Gerritsen, M.; Wen, Xian-Huan, A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations, Adv. Water Resour., 26, 10, 1041-1060 (2003)
[6] Chollet, François, Keras: deep learning library for theano and tensorflow
[7] Chung, Eric T.; Efendiev, Yalchin; Leung, Tat; Vasilyeva, Maria, Coupling of multiscale and multi-continuum approaches, GEM Int. J. Geomath., 8, 1, 9-41 (2017) · Zbl 06853859
[8] Chung, Eric T.; Efendiev, Yalchin; Leung, Wing T.; Wheeler, Mary, Nonlinear nonlocal multicontinua upscaling framework and its applications, Int. J. Multiscale Comput. Eng., 16, 5 (2018)
[9] Chung, Eric T.; Efendiev, Yalchin; Tat Leung, Wing, An adaptive generalized multiscale discontinuous Galerkin method (gmsdgm) for high-contrast flow problems (2014), arXiv preprint · Zbl 1315.65085
[10] Chung, Eric T.; Efendiev, Yalchin; Leung, Wing Tat, Constraint energy minimizing generalized multiscale finite element method, Comput. Methods Appl. Mech. Eng., 339, 298-319 (2018) · Zbl 1440.65195
[11] Chung, Eric T.; Efendiev, Yalchin; Leung, Wing Tat; Vasilyeva, Maria; Wang, Yating, Non-local multi-continua upscaling for flows in heterogeneous fractured media, J. Comput. Phys., 372, 22-34 (2018) · Zbl 1415.76449
[12] D’Angelo, Carlo; Scotti, Anna, A mixed finite element method for Darcy flow in fractured porous media with non-matching grids, ESAIM: Math. Model. Numer. Anal., 46, 2, 465-489 (2012) · Zbl 1271.76322
[13] Durlofsky, L. J.; Efendiev, Y.; Ginting, V., An adaptive local-global multiscale finite volume element method for two-phase flow simulations, Adv. Water Resour., 30, 3, 576-588 (2007)
[14] Efendiev, Yalchin; Galvis, Juan; Li, Guanglian; Presho, Michael, Generalized multiscale finite element methods. Oversampling strategies (2013), arXiv preprint · Zbl 1388.65146
[15] Florez, Horacio; Gildin, Eduardo, Model-order reduction applied to coupled flow and geomechanics, (ECMOR XVI-16th European Conference on the Mathematics of Oil Recovery (2018))
[16] Formaggia, Luca; Fumagalli, Alessio; Scotti, Anna; Ruffo, Paolo, A reduced model for Darcy’s problem in networks of fractures, ESAIM: Math. Model. Numer. Anal., 48, 4, 1089-1116 (2014) · Zbl 1299.76254
[17] Hajibeygi, H.; Kavounis, D.; Jenny, P., A hierarchical fracture model for the iterative multiscale finite volume method, J. Comput. Phys., 230, 24, 8729-8743 (2011) · Zbl 1370.76095
[18] Helmig, Rainer, Multiphase Flow and Transport Processes in the Subsurface: A Contribution to the Modeling of Hydrosystems (1997), Springer-Verlag
[19] Hou, Thomas Y.; Wu, Xiao-Hui, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134, 1, 169-189 (1997) · Zbl 0880.73065
[20] Kingma, Diederik P.; Ba, Jimmy, Adam: a method for stochastic optimization (2014), arXiv preprint
[21] Krizhevsky, Alex; Sutskever, Ilya; Hinton, Geoffrey E., Imagenet classification with deep convolutional neural networks, (Advances in Neural Information Processing Systems (2012)), 1097-1105
[22] LeCun, Yann; Bengio, Yoshua; Hinton, Geoffrey, Deep learning, Nature, 521, 7553, 436 (2015)
[23] Lee, S. H.; Lough, M. F.; Jensen, C. L., Hierarchical modeling of flow in naturally fractured formations with multiple length scales, Water Resour. Res., 37, 3, 443-455 (2001)
[24] Logg, Anders; Mardal, Kent-Andre; Wells, Garth, Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, vol. 84 (2012), Springer Science & Business Media · Zbl 1247.65105
[25] Martin, Vincent; Jaffré, Jérôme; Roberts, Jean E., Modeling fractures and barriers as interfaces for flow in porous media, SIAM J. Sci. Comput., 26, 5, 1667-1691 (2005) · Zbl 1083.76058
[26] Richards, Lorenzo Adolph, Capillary conduction of liquids through porous mediums, Physics, 1, 5, 318-333 (1931) · Zbl 0003.28403
[27] Sánchez-Palencia, Enrique, Non-homogeneous media and vibration theory, (Non-homogeneous Media and Vibration Theory, vol. 127 (1980)) · Zbl 0432.70002
[28] Ţene, Matei; Al Kobaisi, Mohammed Saad; Hajibeygi, Hadi, Algebraic multiscale method for flow in heterogeneous porous media with embedded discrete fractures (f-ams), J. Comput. Phys., 321, 819-845 (2016) · Zbl 1349.76394
[29] Vasilyeva, M.; Tyrylgin, A., Convolutional neural network for fast prediction of the effective properties of domains with random inclusions, J. Phys. Conf. Ser., 1158, Article 042034 pp. (2019)
[30] Vasilyeva, Maria; Chung, Eric T.; Cheung, Siu Wun; Wang, Yating; Prokopev, Georgy, Nonlocal multicontinua upscaling for multicontinua flow problems in fractured porous media, J. Comput. Appl. Math., 355, 258-267 (2019) · Zbl 1432.76181
[31] Vasilyeva, Maria; Chung, Eric T.; Efendiev, Yalchin; Kim, Jihoon, Constrained energy minimization based upscaling for coupled flow and mechanics, J. Comput. Phys., 376, 660-674 (2019) · Zbl 1416.76163
[32] Vasilyeva, Maria; Chung, Eric T.; Leung, Wing Tat; Alekseev, Valentin, Nonlocal multicontinuum (nlmc) upscaling of mixed dimensional coupled flow problem for embedded and discrete fracture models (2018), arXiv preprint · Zbl 1431.76130
[33] Vasilyeva, Maria; Chung, Eric T.; Leung, Wing Tat; Wang, Yating; Spiridonov, Denis, Upscaling method for problems in perforated domains with non-homogeneous boundary conditions on perforations using non-local multi-continuum method (nlmc) (2018), arXiv preprint · Zbl 07073326
[34] Vasilyeva, Maria; Tyrylgin, Aleksey, Machine learning for accelerating effective property prediction for poroelasticity problem in stochastic media (2018), arXiv preprint
[35] Wang, Min; Cheung, Siu Wun; Leung, Wing Tat; Chung, Eric T.; Efendiev, Yalchin; Wheeler, Mary, Reduced-order deep learning for flow dynamics (2019), arXiv preprint
[36] Wang, Yating; Cheung, Siu Wun; Chung, Eric T.; Efendiev, Yalchin; Wang, Min, Deep multiscale model learning (2018), arXiv preprint
[37] Yang, Yanfang; Ghasemi, Mohammadreza; Gildin, Eduardo; Efendiev, Yalchin; Calo, Victor, Fast multiscale reservoir simulations with pod-deim model reduction, SPE J., 21, 06, 2-141 (2016)
[38] Zhao, Lina; Chung, Eric T., An analysis of the nlmc upscaling method for high contrast problems (2019), arXiv preprint · Zbl 1437.65207
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