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Advances in Geo-Energy Research 2023, 8(1): 37-44 https://doi.org/10.46690/ager.2023.04.04
Original Article | Open Access | Issue | Published: 10 January 2023

Physics-informed machine learning for solving partial differential equations in porous media

Show Author's Information Liqun Shan1,2,  Chengqian Liu2,  Yanchang Liu2 ( ),  Yazhou Tu1,  Linyu Dong3,  Xiali Hei1
School of Computing and Informatics, University of Louisiana at Lafayette, Lafayette, LA 70503, USA
School of Physical and Electrical Engineering, Northeast Petroleum University, Daqing 163318, P. R. China
Retirement Management Center, Dagang Oilfield, Tianjin 300456, P. R. China
Keywords:
two-phase flow, recurrent neural network, attention mechanism, Porous media, Buckley-Leverett equation, physics-informed neural networks
Cite this article:
Shan L, Liu C, Liu Y, et al. Physics-informed machine learning for solving partial differential equations in porous media. Advances in Geo-Energy Research, 2023, 8(1): 37-44. https://doi.org/10.46690/ager.2023.04.04
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Abstract Full text About this article

Physical phenomenon in nature is generally simulated by partial differential equations. Among different sorts of partial differential equations,the problem of two-phase flow in porous media has been paid intense attention. As a promising direction,physics-informed neural networks shed new light on the solution of partial differential equations. However,current physics-informed neural networks' ability to learn partial differential equations relies on adding artificial diffusion or using prior knowledge to increase the number of training points along the shock trajectory,or adaptive activation functions. To address these issues,this study proposes a physics-informed neural network with long short-term memory and attention mechanism,an ingenious method to solve the Buckley-Leverett partial differential equations representing two-phase flow in porous media. The designed network structure overcomes the dependency on artificial diffusion terms and enhances the importance of shallow features. The experimental results show that the proposed method is in good agreement with analytical solutions. Accurate approximations are shown even when encountering shock points in saturated fields of porous media. Furthermore,experiments show our innovative method outperforms existing traditional physics-informed machine learning approaches.


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About this article

Physics-informed machine learning for solving partial differential equations in porous media

Show Author's information Liqun Shan1,2, Chengqian Liu2, Yanchang Liu2 ( ), Yazhou Tu1, Linyu Dong3, Xiali Hei1
School of Computing and Informatics, University of Louisiana at Lafayette, Lafayette, LA 70503, USA
School of Physical and Electrical Engineering, Northeast Petroleum University, Daqing 163318, P. R. China
Retirement Management Center, Dagang Oilfield, Tianjin 300456, P. R. China

Abstract

Physical phenomenon in nature is generally simulated by partial differential equations. Among different sorts of partial differential equations,the problem of two-phase flow in porous media has been paid intense attention. As a promising direction,physics-informed neural networks shed new light on the solution of partial differential equations. However,current physics-informed neural networks' ability to learn partial differential equations relies on adding artificial diffusion or using prior knowledge to increase the number of training points along the shock trajectory,or adaptive activation functions. To address these issues,this study proposes a physics-informed neural network with long short-term memory and attention mechanism,an ingenious method to solve the Buckley-Leverett partial differential equations representing two-phase flow in porous media. The designed network structure overcomes the dependency on artificial diffusion terms and enhances the importance of shallow features. The experimental results show that the proposed method is in good agreement with analytical solutions. Accurate approximations are shown even when encountering shock points in saturated fields of porous media. Furthermore,experiments show our innovative method outperforms existing traditional physics-informed machine learning approaches.

Keywords: two-phase flow, recurrent neural network, attention mechanism, Porous media, Buckley-Leverett equation, physics-informed neural networks

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Publication history

Received: 09 November 2022
Revised: 05 December 2022
Accepted: 04 January 2023
Published: 10 January 2023
Issue date: April 2023

Copyright

© The Author(s) 2023.

Acknowledgements

Acknowledgements

This research was financially supported by the Natural Science Foundation of Hebei Province (No. E2021107005), Northeast Petroleum University Foundation Founding (No. 2018GP2D-04). We gratefully acknowledge the helpful comments of the editor and anonymous reviewers.

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Open Access This article is distributed under the terms and conditions of the Creative Commons Attribution (CC BY-NC-ND) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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