# Modeling Long-Distance Forward and Backward Diffusion Processes in Tracer Transport Using the Fractional Laplacian on Bounded Domains

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model Development

#### 2.2. Finite Difference Method Scheme for the Fractional Laplacian-Based Model

## 3. Results and Discussion

#### 3.1. Diffusion Regimes

#### 3.2. Sensitivity Analysis of Model Parameters

## 4. Applications

#### 4.1. Case 1: Solute Transport in Groundwater Flow

#### 4.2. Case 2: Intermediate-Scale Flume Experiments of Bedload Sediments

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Bouchaud, J.P.; Georges, A. Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Phys. Rep.
**1990**, 195, 127–293. [Google Scholar] [CrossRef] - Berkowitz, B.; Cortis, A.; Dentz, M.; Scher, H. Modeling non-Fickian transport in geological formations as a continuous time random walk. Rev. Geophys.
**2006**, 44, RG2003. [Google Scholar] [CrossRef] - Wang, Y.; Sun, H.; Fan, S.; Gu, Y.; Yu, X. A nonlocal fractional peridynamic diffusion model. Fractal Fract.
**2021**, 5, 76. [Google Scholar] [CrossRef] - Li, Z.; Yuan, S.; Tang, H.; Zhu, Y.; Sun, H. Quantifying nonlocal bedload transport: A regional-based nonlocal model for bedload transport from local to global scales. Adv. Water Resour.
**2023**, 177, 104444. [Google Scholar] [CrossRef] - Hao, X.; Sun, H.; Zhang, Y.; Li, S.; Yu, Z. Co-transport of arsenic and micro/nano-plastics in saturated soil. Environ. Res.
**2023**, 228, 115871. [Google Scholar] [CrossRef] [PubMed] - Hatano, Y.; Hatano, N. Dispersive transport of ions in column experiments: An explanation of long-tailed profiles. Water Resour. Res.
**1998**, 34, 1027–1033. [Google Scholar] [CrossRef] - Liu, L.; Zhang, S.; Chen, S.; Liu, F.; Feng, L.; Turner, I.; Zheng, L.; Zhu, J. An Application of the Distributed-Order Time-and Space-Fractional Diffusion-Wave Equation for Studying Anomalous Transport in Comb Structures. Fractal Fract.
**2023**, 7, 239. [Google Scholar] [CrossRef] - Li, Z.; Kiani Oshtorjani, M.; Chen, D.; Zhang, Y.; Sun, H. Dynamics of Dual-Mode Bedload Transport With Three-Dimensional Alternate Bars Migration in Subcritical Flow: Experiments and Model Analysis. J. Geophys. Res. Earth Surf.
**2023**, 128, e2022JF006882. [Google Scholar] [CrossRef] - Adams, E.E.; Gelhar, L.W. Field study of dispersion in a heterogeneous aquifer: 2. Spatial moments analysis. Water Resour. Res.
**1992**, 28, 3293–3307. [Google Scholar] [CrossRef] - Zhang, Y.; Benson, D.A.; Reeves, D.M. Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications. Adv. Water Resour.
**2009**, 32, 561–581. [Google Scholar] [CrossRef] - Metzler, R.; Rajyaguru, A.; Berkowitz, B. Modelling anomalous diffusion in semi-infinite disordered systems and porous media. New J. Phys.
**2022**, 24, 123004. [Google Scholar] [CrossRef] - Sun, H.; Zhang, Y.; Baleanu, D.; Chen, W.; Chen, Y. A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul.
**2018**, 64, 213–231. [Google Scholar] [CrossRef] - ten Hagen, B.; van Teeffelen, S.; Löwen, H. Brownian motion of a self-propelled particle. J. Phys. Condens. Matter
**2011**, 23, 194119. [Google Scholar] [CrossRef] - Ghosh, P.K.; Misko, V.R.; Marchesoni, F.; Nori, F. Self-propelled Janus particles in a ratchet: Numerical simulations. Phys. Rev. Lett.
**2013**, 110, 268301. [Google Scholar] [CrossRef] [PubMed] - Nelissen, K.; Misko, V.; Peeters, F. Single-file diffusion of interacting particles in a one-dimensional channel. Europhys. Lett.
**2007**, 80, 56004. [Google Scholar] [CrossRef] - Taloni, A.; Marchesoni, F. Single-file diffusion on a periodic substrate. Phys. Rev. Lett.
**2006**, 96, 020601. [Google Scholar] [CrossRef] - Boffetta, G.; De Lillo, F.; Musacchio, S. Anomalous diffusion in confined turbulent convection. Phys. Rev. E
**2012**, 85, 066322. [Google Scholar] [CrossRef] - Metzler, R.; Klafter, J. The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep.
**2000**, 339, 1–77. [Google Scholar] [CrossRef] - Sun, L.; Qiu, H.; Wu, C.; Niu, J.; Hu, B.X. A review of applications of fractional advection–dispersion equations for anomalous solute transport in surface and subsurface water. Wiley Interdiscip. Rev. Water
**2020**, 7, e1448. [Google Scholar] [CrossRef] - Cushman, J.H. The Physics of Fluids in Hierarchical Porous Media: Angstroms to Miles; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2013; Volume 10. [Google Scholar]
- Foufoula-Georgiou, E.; Ganti, V.; Dietrich, W. A nonlocal theory of sediment transport on hillslopes. J. Geophys. Res. Earth Surf.
**2010**, 115, F00A16. [Google Scholar] [CrossRef] - Union, J.I.G. Advection diffusion equation models in near-surface geophysical and environmental sciences. J. Indian Geophys. Union
**2013**, 17, 117–127. [Google Scholar] - Hilfer, R. Applications of Fractional Calculus in Physics; World Scientific: Singapore, 2000. [Google Scholar]
- Klafter, J.; Lim, S.; Metzler, R. Fractional Dynamics: Recent Advances; World Scientific: Singapore, 2012. [Google Scholar]
- Tawfik, A.M.; Hefny, M.M. Subdiffusive Reaction Model of Molecular Species in Liquid Layers: Fractional Reaction-Telegraph Approach. Fractal Fract.
**2021**, 5, 51. [Google Scholar] [CrossRef] - Kim, S.; Kavvas, M.L. Generalized Fick’s law and fractional ADE for pollution transport in a river: Detailed derivation. J. Hydrol. Eng.
**2006**, 11, 80–83. [Google Scholar] [CrossRef] - Baeumer, B.; Kovács, M.; Meerschaert, M.M.; Sankaranarayanan, H. Reprint of: Boundary conditions for fractional diffusion. J. Comput. Appl. Math.
**2018**, 339, 414–430. [Google Scholar] [CrossRef] - Zhang, Y.; Yu, X.; Li, X.; Kelly, J.F.; Sun, H.; Zheng, C. Impact of absorbing and reflective boundaries on fractional derivative models: Quantification, evaluation and application. Adv. Water Resour.
**2019**, 128, 129–144. [Google Scholar] [CrossRef] - Jannelli, A.; Ruggieri, M.; Speciale, M.P. Analytical and numerical solutions of time and space fractional advection–diffusion–reaction equation. Commun. Nonlinear Sci. Numer. Simul.
**2019**, 70, 89–101. [Google Scholar] [CrossRef] - Yin, M.; Ma, R.; Zhang, Y.; Chen, K.; Guo, Z.; Zheng, C. A Dual Heterogeneous Domain Model for Upscaling Anomalous Transport With Multi-Peaks in Heterogeneous Aquifers. Water Resour. Res.
**2022**, 58, e2021WR031128. [Google Scholar] [CrossRef] - Angstmann, C.N.; Henry, B.I.; Jacobs, B.A.; McGann, A.V. An explicit numerical scheme for solving fractional order compartment models from the master equations of a stochastic process. Commun. Nonlinear Sci. Numer. Simul.
**2019**, 68, 188–202. [Google Scholar] [CrossRef] - Furbish, D.J.; Roering, J.J. Sediment disentrainment and the concept of local versus nonlocal transport on hillslopes. J. Geophys. Res. Earth Surf.
**2013**, 118, 937–952. [Google Scholar] [CrossRef] - Zhang, Y. Backward Particle Tracking of Anomalous Transport in Multi-Dimensional Aquifers. Water Resour. Res.
**2022**, 58, e2022WR032396. [Google Scholar] [CrossRef] - Lischke, A.; Pang, G.; Gulian, M.; Song, F.; Glusa, C.; Zheng, X.; Mao, Z.; Cai, W.; Meerschaert, M.M.; Ainsworth, M.; et al. What is the fractional Laplacian? A comparative review with new results. J. Comput. Phys.
**2020**, 404, 109009. [Google Scholar] [CrossRef] - Scherer, R.; Kalla, S.L.; Tang, Y.; Huang, J. The Grünwald–Letnikov method for fractional differential equations. Comput. Math. Appl.
**2011**, 62, 902–917. [Google Scholar] [CrossRef] - Li, C.; Qian, D.; Chen, Y. On Riemann-Liouville and caputo derivatives. Discret Dyn. Nat. Soc.
**2011**, 2011, 562494. [Google Scholar] [CrossRef] - De Oliveira, E.C.; Tenreiro Machado, J.A. A review of definitions for fractional derivatives and integral. Math. Probl. Eng.
**2014**, 2014, 238459. [Google Scholar] [CrossRef] - Zoia, A.; Rosso, A.; Kardar, M. Fractional Laplacian in bounded domains. Phys. Rev. E
**2007**, 76, 021116. [Google Scholar] [CrossRef] - D’Elia, M.; Gunzburger, M. The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator. Comput. Math. Appl.
**2013**, 66, 1245–1260. [Google Scholar] [CrossRef] - Zhang, Y.; Zhou, D.; Yin, M.; Sun, H.; Wei, W.; Li, S.; Zheng, C. Nonlocal transport models for capturing solute transport in one-dimensional sand columns: Model review, applicability, limitations and improvement. Hydrol. Process.
**2020**, 34, 5104–5122. [Google Scholar] [CrossRef] - Duo, S.; van Wyk, H.W.; Zhang, Y. A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem. J. Comput. Phys.
**2018**, 355, 233–252. [Google Scholar] [CrossRef] - Gao, T.; Duan, J.; Li, X.; Song, R. Mean exit time and escape probability for dynamical systems driven by Lévy noises. SIAM J. Sci. Comput.
**2014**, 36, A887–A906. [Google Scholar] [CrossRef] - Huang, Y.; Oberman, A. Numerical methods for the fractional Laplacian: A finite difference-quadrature approach. SIAM J. Numer. Anal.
**2014**, 52, 3056–3084. [Google Scholar] [CrossRef] - Sun, H.; Li, Z.; Zhang, Y.; Chen, W. Fractional and fractal derivative models for transient anomalous diffusion: Model comparison. Chaos Solitons Fractals
**2017**, 102, 346–353. [Google Scholar] [CrossRef] - Yin, M.; Zhang, Y.; Ma, R.; Tick, G.R.; Bianchi, M.; Zheng, C.; Wei, W.; Wei, S.; Liu, X. Super-diffusion affected by hydrofacies mean length and source geometry in alluvial settings. J. Hydrol.
**2020**, 582, 124515. [Google Scholar] [CrossRef]

**Figure 1.**Conceptual map of the local and the nonlocal diffusion in both forward and backward directions.

**Figure 2.**Simulated tracer concentration of the fractional Laplacian-based model (6) using the implicit FDM scheme. (

**a**) The snapshots of the fractional Laplacian-based model at different times. (

**b**) The semilog plot of snapshots at different times. All parameters are set as $\beta =1.4$, $v=0.1$ m/min, $D=0.4$ m

^{β}/min, $R=1$, and $\gamma =1.7$.

**Figure 3.**The snapshots and BTCs with different values of fractional order $\beta $ for the fractional Laplacian-based model (6). The fractional order $\beta $ varies from 1.3 to 1.9. (

**a**) The spatial distribution of tracer particles at $T=60$ min. (

**b**) The temporal evolution of tracer particles at $X=100$ m. Other parameters are set as $v=0.6$ m/min, $D=0.4$ m

^{β}/min, $R=1$, and $\gamma =1+\beta /2$.

**Figure 4.**Comparison between the documented snapshots (symbols) and the best-fit results using the classical ADE and fractional Laplacian-based models at four times (t = 27, 132, 224, and 328 days) along the 300 m long heterogeneous media. All parameters are list in Table 1. (

**a**) 27 days, (

**b**) 132 days, (

**c**) 224 days, and (

**d**) 328 days.

**Figure 5.**(

**a**) The fitting results for the fractional order $\beta $ and its variation with time. (

**b**) The evolution of the tracer particle mean squared displacement (MSD) with time (red solid line), and the MSD for normal diffusion (gray solid line) is included for comparison. The gray and blue dashed lines represent the evolution trend of MSD at early and later times, respectively.

**Figure 6.**Modeled (lines) versus observed (symbols) snapshots for bedload sediments with a diameter of 1 mm; three models were compared: the classical ADE model (black lines), the PD model (a circular influence domain with radius $\delta =80\Delta x$) (blue lines), and the fractional Laplacian-based model (fractional order $\beta =1.1$) (red lines) at two times (t = 155 min and 355 min) along the 20 m long flume. (

**a**) t = 155 min, (

**b**) t = 355 min.

Time (Days) | v (m/Day) | D (m/Day)^{β} | R | $\mathit{\beta}$ |
---|---|---|---|---|

27 | 0.018 | 3 | 10 | 1.40 |

132 | 1.30 | |||

224 | 1.25 | |||

328 | 1.16 |

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**MDPI and ACS Style**

Li, Z.; Tang, H.; Yuan, S.; Zhang, H.; Kong, L.; Sun, H.
Modeling Long-Distance Forward and Backward Diffusion Processes in Tracer Transport Using the Fractional Laplacian on Bounded Domains. *Fractal Fract.* **2023**, *7*, 823.
https://doi.org/10.3390/fractalfract7110823

**AMA Style**

Li Z, Tang H, Yuan S, Zhang H, Kong L, Sun H.
Modeling Long-Distance Forward and Backward Diffusion Processes in Tracer Transport Using the Fractional Laplacian on Bounded Domains. *Fractal and Fractional*. 2023; 7(11):823.
https://doi.org/10.3390/fractalfract7110823

**Chicago/Turabian Style**

Li, Zhipeng, Hongwu Tang, Saiyu Yuan, Huiming Zhang, Lingzhong Kong, and HongGuang Sun.
2023. "Modeling Long-Distance Forward and Backward Diffusion Processes in Tracer Transport Using the Fractional Laplacian on Bounded Domains" *Fractal and Fractional* 7, no. 11: 823.
https://doi.org/10.3390/fractalfract7110823