Machine-learning of nonlocal kernels for anomalous subsurface transport from breakthrough curves

Xiao Xu; Marta D'Elia; Christian Glusa; John T. Foster

doi:10.3934/naco.2022025

Article Contents

2024, Volume 14, Issue 1: 89-107. Doi: 10.3934/naco.2022025

This issue Previous Article Double stranded smoothing technique in energy minimization problems Next Article On fast convergence rates for generalized conditional gradient methods with backtracking stepsize

Machine-learning of nonlocal kernels for anomalous subsurface transport from breakthrough curves

1.
The Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, USA
2.
Computational Science and Analysis, Sandia National Laboratories, USA
3.
Meta Reality Labs, USA
4.
Computer Science Research Institute, Sandia National Laboratories, USA

^*Corresponding author: John T. Foster
^*Corresponding author: John T. Foster

This paper is handled by Andreas Mang as guest editor.

Received: January 2022

Revised: July 2022

Early access: September 19, 2022

Published online: September 19, 2022

Abstract / Introduction Full Text(HTML) Figure(12) Related Papers Cited by

Abstract

Anomalous behavior is ubiquitous in subsurface solute transport due to the presence of high degrees of heterogeneity at different scales in the media. Although fractional models have been extensively used to describe the anomalous transport in various subsurface applications, their application is hindered by computational challenges. Simpler nonlocal models characterized by integrable kernels and finite interaction length represent a computationally feasible alternative to fractional models; yet, the informed choice of their kernel functions still remains an open problem. We propose a general data-driven framework for the discovery of optimal kernels on the basis of very small and sparse data sets in the context of anomalous subsurface transport. Using spatially sparse breakthrough curves recovered from fine-scale particle-density simulations, we learn the best coarse-scale nonlocal model using a nonlocal operator regression technique. Predictions of the breakthrough curves obtained using the optimal nonlocal model show good agreement with fine-scale simulation results even at locations and time intervals different from the ones used to train the kernel, confirming the excellent generalization properties of the proposed algorithm. A comparison with trained classical models and with black-box deep neural networks confirms the superiority of the predictive capability of the proposed model.

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

Full Text(HTML)

Figure 1. Sketch of a two-dimensional porous medium with periodic heterogeneous conductivity

Download: Full-size image PowerPoint slide

Figure 2. FE mesh of three unit cells

Download: Full-size image PowerPoint slide

Figure 3. Streamlines in the first 3 unit cells

Download: Full-size image PowerPoint slide

Figure 4. Mean square displacement $\left\langle {{{(x - \left\langle x \right\rangle )}^2}} \right\rangle $ of the high-fidelity, particle-tracking solution

Download: Full-size image PowerPoint slide

Figure 5. Coarse-grained density profile $ \bar c_{H\!F} $ at $ t = 72 $

Download: Full-size image PowerPoint slide

Figure 6. The optimal nonlocal spatial kernel

Download: Full-size image PowerPoint slide

Figure 7. BTC predicted by nonlocal model and local models at location 20 (the green dashed line is used to separate training interval and testing interval)

Download: Full-size image PowerPoint slide

Figure 8. $ M\!S\!E $ of predicted BTCs at different locations

Download: Full-size image PowerPoint slide

Figure 9. BTCs predicted by nonlocal dynamic kernel and neural network at location 20 (the green dashed line is used to separate training and testing)

Download: Full-size image PowerPoint slide

Figure 10. $ M\!S\!E $ of predicted BTCs at different location using different size of training data

Download: Full-size image PowerPoint slide

Figure 11. $ M\!S\!E $ of predicted BTCs at different location when $ T_t = 72 $ and $ T_t = 90 $

Download: Full-size image PowerPoint slide

Figure 12. $ M\!S\!E $ of predicted BTCs at different location with different choice of training locations

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	M. Abbaszadeh-Dehghani and W. Brigham, Analysis of well-to-well tracer flow to determine reservoir layering, Journal of Petroleum Technology, 36 (1984), 1753-1762.
[2]	C. Brgers and C. Greengard, On the mean square displacement in levy walks, SIAM Journal on Applied Mathematics, 80 (2020), 1175-1196. doi: 10.1137/19M1251813.
[3]	W. Chen, H. Sun, X. Zhang and D. Korosak, Anomalous diffusion modeling by fractal and fractional derivatives, Computers & Mathematics with Applications, 59 (2010), 1754-1758. doi: 10.1016/j.camwa.2009.08.020.
[4]	M. D'Elia, M. Gulian, H. Olson and G. E. Karniadakis, Towards a unified theory of fractional and nonlocal vector calculus, Fractional Calculus and Applied Analysis, 24 (2021), 1301-1355. doi: 10.1515/fca-2021-0057.
[5]	M. D'Elia, Q. Du, M. Gunzburger and R. Lehoucq, Nonlocal convection-diffusion problems on bounded domains and finite-range jump processes, Computational Methods in Applied Mathematics, 17 (2017), 707-722. doi: 10.1515/cmam-2017-0029.
[6]	M. D'Elia and C. Glusa, A fractional model for anomalous diffusion with increased variability. analysis, algorithms and applications to interface problems, preprint, arXiv: 2101.11765.
[7]	M. D'Elia and M. Gulian, Analysis of anisotropic nonlocal diffusion models: Well-posedness of fractional problems for anomalous transport, preprint, arXiv: 2101.04289.
[8]	M. D'Elia, M. Gulian, T. Mengesha and J. M. Scott, Connections between nonlocal operators: from vector calculus identities to a fractional helmholtz decomposition, preprint, arXiv: 2112.05317.
[9]	M. D'Elia and M. Gunzburger, The fractional laplacian operator on bounded domains as a special case of the nonlocal diffusion operator, Computers & Mathematics with Applications, 66 (2013), 1245-1260. doi: 10.1016/j.camwa.2013.07.022.
[10]	Q. Du, M. Gunzburger, R. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Review, 54 (2012), 667-696. doi: 10.1137/110833294.
[11]	Q. Du and J. Yang, Fast and accurate implementation of fourier spectral approximations of nonlocal diffusion operators and its applications, Journal of Computational Physics, 332 (2017), 118-134. doi: 10.1016/j.jcp.2016.11.028.
[12]	Y. Edery, A. Guadagnini, H. Scher and B. Berkowitz, Origins of anomalous transport in heterogeneous media: Structural and dynamic controls, Water Resources Research, 50 (2014), 1490-1505.
[13]	L. Holden and O. Lia, A tensor estimator for the homogenization of absolute permeability, Transport in Porous Media, 8 (1992), 37-46.
[14]	J. F. Kelly and M. M. Meerschaert, The Fractional Advection-Dispersion Equation For Contaminant Transport, De Gruyter Reference, De Gruyter, 2019, Chapter 6 of Handbook of Fractional Calculus with Applications in Applications in Physics, Part B.
[15]	M. Levy and B. Berkowitz, Measurement and analysis of non-fickian dispersion in heterogeneous porous media, Journal of Contaminant Hydrology, 64 (2003), 203-226.
[16]	G. Maugin, Nonlocal theories or gradient-type theories-a matter of convenience, Archiv of Mechanics, Archiwum Mechaniki Stosowanej, 31 (1979), 15-26.
[17]	G. A. Maugin, On the thermomechanics of continuous media with diffusion and/or weak nonlocality, Archive of Applied Mechanics, 75 (2006), 723-738.
[18]	S. Mishra, W. E. Brigham and J. Orr Franklin M., Tracer- and pressure-test analysis for characterization of areally heterogeneous reservoirs, SPE Formation Evaluation, 6 (1991), 45-54.
[19]	S. P. Neuman and D. M. Tartakovsky, Perspective on theories of non-fickian transport in heterogeneous media, Advances in Water Resources, 32 (2009), 670-680. doi: 10.12988/ijcms.2007.07043.
[20]	A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai and S. Chintala, Pytorch: An imperative style, high-performance deep learning library, in Advances in Neural Information Processing Systems 32, Curran Associates, Inc., (2019), 8024-8035, http://papers.neurips.cc/paper/9015-pytorch-an-imperative-style-high-performance-deep-learning-library.pdf.
[21]	D. W. Pollock, Semianalytical computation of path lines for finite-difference models, Groundwater, 26 (1988), 743-750.
[22]	J. Revels, M. Lubin and T. Papamarkou, Forward-mode automatic differentiation in Julia, preprint, arXiv: 1607.07892.
[23]	R. Schumer, D. Benson, M. Meerschaert and S. Wheatcraft, Eulerian derivation of the fractional advection-dispersion equation, Journal of Contaminant Hydrology, 48 (2001), 69-88.
[24]	P. Seleson and M. Parks, On the role of the influence function in the peridynamic theory, International Journal of Multiscale Computational Engineering, 9 (2011), 689-706.
[25]	T. Shen, R. G. Moghanloo and W. Tian, Decoupling of channeling and dispersion effects by use of multiwell tracer test, SPE Reservoir Evaluation & Engineering, 21 (2017), 54-72. doi: 10.2118/187960-PA.
[26]	S. A. Silling, Origin and effect of nonlocality in a composite, Journal of Mechanics of Materials and Structures, 9 (2014), 245-258.
[27]	S. A. Silling and E. Askari, A meshfree method based on the peridynamic model of solid mechanics, Computers & structures, 83 (2005), 1526-1535.
[28]	E. Stalgorova and T. Babadagli, Field-scale modeling of tracer injection in naturally fractured reservoirs using the random-walk particle-tracking simulation, SPE Journal, 17 (2012), 580-592.
[29]	L. Sun, H. Qiu, C. Wu, J. Niu and B. X. Hu, A review of applications of fractional advection ispersion equations for anomalous solute transport in surface and subsurface water, WIREs Water, 7.
[30]	J. Suzuki, M. Gulian, M. Zayernouri and M. D'Elia, Fractional modeling in action: A survey of nonlocal models for subsurface transport, turbulent flows, and anomalous materials, preprint, arXiv: 2110.11531.
[31]	A. Tyukhova, M. Dentz, W. Kinzelbach and M. Willmann, Mechanisms of anomalous dispersion in flow through heterogeneous porous media, Physical Review Fluids, 1 (2016), 074002.
[32]	O. Weckner and R. Abeyaratne, The effect of long-range forces on the dynamics of a bar, Journal of the Mechanics and Physics of Solids, 53 (2005), 705-728. doi: 10.1016/j.jmps.2004.08.006.
[33]	X. Xu, M. D'Elia and J. T. Foster, A machine-learning framework for peridynamic material models with physical constraints, preprint, arXiv: 2101.01095. doi: 10.1016/j.cma.2021.114062.
[34]	X. Xu and J. T. Foster, Deriving peridynamic influence functions for one-dimensional elastic materials with periodic microstructure, preprint, arXiv: 2003.05520. doi: 10.1007/s.
[35]	H. You, Y. Yu, S. Silling and M. D'Elia, A data-driven peridynamic continuum model for upscaling molecular dynamics, Computer Methods in Applied Mechanics and Engineering, 389 (2022), 114400. doi: 10.1016/j.cma.2021.114400.
[36]	H. You, Y. Yu, S. Silling and M. D'Elia, Data-driven learning of nonlocal models: from high-fidelity simulations to constitutive laws, Accepted in AAAI Spring Symposium: MLPS, 2021.
[37]	H. You, Y. Yu, N. Trask, M. Gulian and M. D'Elia, Data-driven learning of robust nonlocal physics from high-fidelity synthetic data, Computer Methods in Applied Mechanics and Engineering, 374 (2021), 113553. doi: 10.1016/j.cma.2020.113553.