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Abstract

Summary

There is growing interest in employing Machine Learning (ML) strategies to solve forward and inverse computational physics problems. The physics-informed machine learning (PIML) frameworks developed by Raissi et al.[ ] and Zhu et al.[ ] are prominent examples. The basic idea is to encode the partial differential equations (PDE) that govern the flow physics into the neural network. This encoding is achieved by enriching the loss function with the governing conservation equation. Using the initial and boundary conditions, the network is then able to learn the solution of the forward problem without any labeled data. The scarcity of site-specific “labeled” data presents serious challenges to modeling of Enhanced Oil Recovery (EOR) processes. Thus, if PIML approaches can be used to model the nonlinear flow and transport that govern EOR processes, then they could change the practice of reservoir simulation.

In this work, we explore the application of a particular PIML approach to solve the nonlinear hyperbolic equation that describes nonlinear immiscible two-phase flow in porous media. Specifically, we are concerned with the forward solution of a Riemann problem - a nonlinear conservation law together with piecewise constant data having a single discontinuity. It is well known that it is hard to solve this nonlinear transport problem, especially with a non-convex flux function, due to emergence of saturation shocks in the domain. The focus is on the pure forward problem, i.e., the absence of previously simulated (so-called labeled) data in the interior of the domain. The PIML framework breaks down for this nonlinear hyperbolic problem with non-convex flux function. We have found that it is essential to add a diffusion term to the underlying nonlinear PDE. That is, we used the parabolic form of the equation with a finite Peclet number. When the loss function includes a finite amount of diffusion, the neural network can actually produce reasonable approximations of the forward solution when shocks and mixed waves (shocks and rarefactions) are present.

For the obtained neural networks we also analyze the training process and provide 2-D visualizations of the loss landscape, then we discuss possible reasons for the observed behavior.

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/content/papers/10.3997/2214-4609.202035147
2020-09-14
2024-03-29
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References

  1. Aziz, K. and Settari, A.
    [1979] Petroleum reservoir simulation. Applied Science Publishers, 476.
    [Google Scholar]
  2. Cybenko, G.
    [1989] Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals and Systems, 2(4), 303–314.
    [Google Scholar]
  3. Glorot, X. and Bengio, Y.
    [2010] Understanding the difficulty of training deep feedforward neural networks. Proceedings of the 13th International Conference on Artificial Intelligence and Statistics, 249–256.
    [Google Scholar]
  4. Krizhevsky, A., Sutskever, I. and G. E.Hinton
    . [2012] Imagenet classification with deep convolutional neural networks. Proceedings of the 25th International Conference on Neural Information Processing Systems, 1, 1097–1105.
    [Google Scholar]
  5. Lax, P. D.
    [1973] Hyperbolic System of Conservation Laws and the Mathematical Theory of Shock Waves. Society for Industrial and Applied Mathematics, 1–48.
    [Google Scholar]
  6. [2006] Hyperbolic partial differential equations. American Mathematical Soc., 14.
    [Google Scholar]
  7. Mnih, V., Kavukcuoglu, K., Silver, D., Graves, A., Antonoglou, I., Wierstra, D. and Riedmiller, M.
    [2013] Playing Atari with deep reinforcement learning. arXiv preprint arXiv:1312.5602.
    [Google Scholar]
  8. Nocedal, J. and Wright, S. J.
    [2006] Numerical Optimization. SpringerNew York.
    [Google Scholar]
  9. Peaceman, D. W.
    [1991] Fundamentals of Numerical Reservoir Simulation. Elsevier Science Inc.
    [Google Scholar]
  10. Raissi, M., Perdikaris, P., Karniadakis, G. E.
    [2019] Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686–707.
    [Google Scholar]
  11. Silver, D.
    [2016] Mastering the game of Go with deep neural networks and tree search. Nature529.7587, p. 484.
    [Google Scholar]
  12. Stewart, R. and Ermon, S.
    [2017] Label-free supervision of neural networks with physics and domain knowledge, Thirty-First AAAI Conference on Artificial Intelligence, 2576–2582.
    [Google Scholar]
  13. Sutskever, I., Vinyals, O. and Le, Q. V.
    [2014] Sequence to sequence learning with neural networks. Proceedings of the 27th International Conference on Neural Information Processing Systems, 2, 3104–3112.
    [Google Scholar]
  14. Zhu, Y., Zabaras, N., Koutsourelakis, P.-S. and Perdikaris, P.
    [2019] Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data. Journal of Computational Physics, 394, 56–81.
    [Google Scholar]
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