Deep global model reduction learning in porous media flow simulation
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ORIGINAL PAPER
Deep global model reduction learning in porous media flow simulation Siu Wun Cheung1 · Eric T. Chung2 · Yalchin Efendiev3,5 · Eduardo Gildin4
· Yating Wang1 · Jingyan Zhang1
Received: 23 July 2018 / Accepted: 1 November 2019 © Springer Nature Switzerland AG 2019
Abstract In this paper, we combine deep learning concepts and some proper orthogonal decomposition (POD) model reduction methods for predicting flow in heterogeneous porous media. Nonlinear flow dynamics is studied, where the dynamics is regarded as a multi-layer network. The solution at the current time step is regarded as a multi-layer network of the solution at the initial time and input parameters. As for input, we consider various sources, which include source terms (well rates), permeability fields, and initial conditions. We consider the flow dynamics, where the solution is known at some locations and the data is integrated to the flow dynamics by modifying the reduced-order model. This approach allows modifying the reduced-order formulation of the problem. Because of the small problem size, limited observed data can be handled. We consider enriching the observed data using the computational data in deep learning networks. The basis functions of the global reduced-order model are selected such that the degrees of freedom represent the solution at observation points. This way, we can avoid learning basis functions, which can also be done using neural networks. We present numerical results, where we consider channelized permeability fields, where the network is constructed for various channel configurations. Our numerical results show that one can achieve a good approximation using forward feed maps based on multi-layer networks. Keywords Deep learning · Model reduction · POD · Porous media flow · Neural networks
1 Introduction Mathematical models are widely used to describe the underlying physical process in science and engineering disciplines. In many real-life applications, for instance, largescale dynamical systems and control systems, the mathematical models are of high complexity and numerical simulations in high-dimensional systems become challenging. Model order reduction (MOR) is a useful technique in obtaining reasonable approximations with a significantly reduced computational cost of such problems. Through obtaining dominant modes, the dimension of the associated state space is reduced and an approximation of the original model with acceptable accuracy is computed. Proper orthogonal decomposition (POD) has been used in numerical approximations for dynamic systems [20, 23]. Recently, POD has been applied to flow problems in porous media with high contrasts and heterogeneities [3, 4, 12, 15, 16, 22, Eduardo Gildin
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Extended author information available on the last page of the article.
34, 35, 40]. The objective of this work is to develop datadriven POD reduced-order models for such problems using advancements of machine learning. A main difficulty of model order reduction is nonlinear problems. In such
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