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Research Papers

Discrete Fractional Derivative Based Computational Model to Describe Dynamics of Bed-Load Transport

[+] Author and Article Information
HongGuang Sun

Department of Engineering,
Institute of Soft Matter Mechanics,
Mechanics Hohai University,
1 XiKang Road,
Nanjing 210098, Jiangsu, China
e-mail: shg@hhu.edu.cn

ZhiPeng Li

Department of Engineering,
Institute of Soft Matter Mechanics,
Mechanics Hohai University,
1 XiKang Road,
Nanjing 210098, Jiangsu, China
e-mail: lzp@hhu.edu.cn

Yong Zhang

Department of Geological Sciences,
University of Alabama Tuscaloosa,
Tuscaloosa, AL 35487
e-mail: yzhang264@ua.edu

XiaoTing Liu

Department of Engineering,
Institute of Soft Matter Mechanics,
Mechanics Hohai University,
1 XiKang Road,
Nanjing 210098, Jiangsu, China
e-mail: liuxiaoting@hhu.edu.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 30, 2017; final manuscript received March 25, 2018; published online April 18, 2018. Assoc. Editor: Zaihua Wang.

J. Comput. Nonlinear Dynam 13(6), 061004 (Apr 18, 2018) (9 pages) Paper No: CND-17-1534; doi: 10.1115/1.4039878 History: Received November 30, 2017; Revised March 25, 2018

Bed-load transport in natural rivers exhibits nonlinear dynamics with strong temporal memory (i.e., retention due to burial) and/or spatial memory (i.e., fast displacement driven by turbulence). Nonlinear bed-load transport is discrete in nature due to the discontinuity in the sediment mass density and the intermittent motion of sediment along river beds. To describe the discrete bed-load dynamics, we propose a discrete spatiotemporal fractional advection-dispersion equation (D-FADE) without relying on the debatable assumption of a continuous sediment distribution. The new model is then applied to explore nonlinear dynamics of bed-load transport in flumes. Results show that, first, the D-FADE model can capture the temporal memory and spatial dependency characteristics of bed-load transport for sediment with different sizes. Second, fine sediment particles exhibit stronger super-diffusive features, while coarse particles exhibit significant subdiffusive properties, likely due to the size-selective memory impact. Third, sediment transport with an instantaneous source exhibits stronger history memory and weaker spatial nonlocality, compared to that with a continuous source (since a smaller number of particles might be blocked or buried relatively easier). Hence, the D-FADE provides a strict computational model to quantify discrete bed-load transport, whose nonlinear dynamics can be sensitive to particle sizes and source injection modes, both common in applications.

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References

Rosgen, D. L. , 1994, “ A Classification of Natural Rivers,” Catena, 22(3), pp. 169–199. [CrossRef]
Sear, D. , 1994, “ River Restoration and Geomorphology,” Aquat. Conserv., 4(2), pp. 169–177. [CrossRef]
Bravard, J. , Goichot, M. , and Tronchère, H. , 2014, “ An Assessment of Sediment-Transport Processes in the Lower Mekong River Based on Deposit Grain Sizes, the Cm Technique and Flow-Energy Data,” Geomorphology, 207, pp. 174–189. [CrossRef]
Einstein, H. , 1937, “Bedload Transport as a Probability Problem,” Water Resource Publication, Littleton, CO, pp. 105–108.
Meyer-Peter, E. , and Müller, R. , 1948, “ Formulas for Bed-Load Transport,” IAHSR 2nd meeting, Stockholm, Sweden, June 7–9.
Valyrakis, M. , Diplas, P. , and Dancey, C. L. , 2011, “ Prediction of Coarse Particle Movement With Adaptive Neuro-Fuzzy Inference Systems,” Hydrol. Process., 25(22), pp. 3513–3524. [CrossRef]
Barry, J. J. , Buffington, J. M. , and King, J. G. , 2004, “ A General Power Equation for Predicting Bed Load Transport Rates in Gravel Bed Rivers,” Water. Resour. Res., 43(10), pp. 2709–2710.
Bathurst, J. C. , 2007, “ Effect of Coarse Surface Layer on Bed-Load Transport,” J. Hydraul. Eng., 133(11), pp. 1192–1205. [CrossRef]
Chien, N. , and Wan, Z. H. , 1999, “ The Mechanics of Sediment Transport,” Am. Soc. Civ. Eng., 105(2), pp. 268–268.
Gorenflo, R. , Mainardi, F. , Moretti, D. , Pagnini, G. , and Paradisi, P. , 2002, “ Discrete Random Walk Models for Space–Time Fractional Diffusion,” Chem. Phys., 284(1–2), pp. 521–541. [CrossRef]
Wilcock, P. R. , and Crowe, J. C. , 2003, “ Surface-Based Transport Model for Mixed-Size Sediment,” J. Hydraul. Eng., 129(2), pp. 120–128. [CrossRef]
Rossikhin, Y. A. , and Shitikova, M. V. , 1997, “ Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids,” ASME Appl. Mech. Rev., 50(1), pp. 15–67.
Cushman, J. H. , 1991, “ On Diffusion in Fractal Porous Media,” Water. Resour. Res., 27(4), pp. 643–644. [CrossRef]
Cushman, J. H. , Hu, X. L. , and Ginn, T. R. , 1994, “ Nonequilibrium Statistical Mechanics of Preasymptotic Dispersion,” J. Stat. Phys., 75(5–6), pp. 859–878. [CrossRef]
Cushman, J. H. , and Ginn, T. R. , 2000, “ Fractional Advection-Dispersion Equation: A Classical Mass Balance With Convolution-Fickian Flux,” Water. Resour. Res., 36(12), pp. 3763–3766. [CrossRef]
Dentz, M. , and Berkowitz, B. , 2003, “ Transport Behavior of a Passive Solute in Continuous Time Random Walks and Multirate Mass Transfer,” Water. Resour. Res., 39(5), pp. 285–285. [CrossRef]
Haggerty, R. , McKenna, S. A. , and Meigs, L. C. , 2000, “ On the Late-Time Behavior of Tracer Test Breakthrough Curves,” Water. Resour. Res., 36(12), pp. 3467–3479. [CrossRef]
Grabasnjak, M. , 2003, “ Random Particle Motion and Fractional-Order Dispersion in Highly Heterogeneous Aquifers,” Ph.D. thesis, University of Nevada, Reno, NV.
Herrick, M. G. , Benson, D. A. , Meerschaert, M. M. , and McCall, K. R. , 2002, “ Hydraulic Conductivity, Velocity, and the Order of the Fractional Dispersion Derivative in a Highly Heterogeneous System,” Water. Resour. Res., 38(11), pp. 1227–1239. [CrossRef]
Singh, A. K. , Yadav, V. K. , and Das, S. , 2017, “ Dual Combination Synchronization of the Fractional Order Complex Chaotic Systems,” ASME J. Comput. Nonlinear Dyn., 12(1), p. 011017. [CrossRef]
Sun, H. G. , Chen, D. , Zhang, Y. , and Chen, L. , 2015, “ Understanding Partial Bed-Load Transport: Experiments and Stochastic Model Analysis,” J. Hydrol., 521, pp. 196–204. [CrossRef]
Zhang, Y. , Chen, D. , Garrard, R. , Sun, H. G. , and Lu, Y. H. , 2016, “ Influence of Bed Clusters and Size Gradation on Operational Time Distribution for Non-Uniform Bed-Load Transport,” Hydrol. Process., 30(17), pp. 3030–3045. [CrossRef]
Bradley, D. N. , Tucker, G. E. , and Benson, D. A. , 2010, “ Fractional Dispersion in a Sand Bed River,” J. Geophys. Res. Earth Surf., 115(F1), p. F00A09.
Zhang, Y. , Meerschaert, M. M. , and Packman, A. I. , 2012, “ Linking Fluvial Bed Sediment Transport Across Scales,” Geophys. Res. Lett., 39(20), p. 20404.
Martin, R. L. , Jerolmack, D. J. , and Schumer, R. , 2012, “ The Physical Basis for Anomalous Diffusion in Bed Load Transport,” J. Geophys. Res. Earth, 117(F1), p. F01018.
Tarasov, V. , 2016, “ Leibniz Rule and Fractional Derivatives of Power Functions,” ASME J. Comput. Nonlinear Dyn., 11(3), p. 031014. [CrossRef]
Holm, M. T. , 2011, “ The Theory of Discrete Fractional Calculus: Development and Application,” Ph.D. dissertation, University of Nebraska, Lincoln, NE, pp. 1120–1127.
Atici, F. M. , and Eloe, P. W. , 2007, “ A Transform Method in Discrete Fractional Calculus,” IJDE, 2(2), pp. 165–176.
Wu, G. C. , Baleanu, D. , and Xie, H. P. , 2016, “ Riesz Riemann-Liouville Difference on Discrete Domains,” Chaos, 26(8), p. 084308. [CrossRef] [PubMed]
Wu, G. C. , Baleanu, D. , Deng, Z. G. , and Zeng, S. D. , 2015, “ Lattice Fractional Diffusion Equation in Terms of a Riesz-Caputo Difference,” Phys. A, 438, pp. 335–339. [CrossRef]
Armanini, A. , and Di Silvio, G. , 1988, “ A One-Dimensional Model for the Transport of a Sediment Mixture in Non-Equilibrium Conditions,” J. Hydraul. Res., 26(3), pp. 275–292. [CrossRef]
Nikora, V. , Habersack, H. , Huber, T. , and McEwan, I. , 2002, “ On Bed Particle Diffusion in Gravel Bed Flows Under Weak Bed Load Transport,” Water. Resour. Res., 38(6), pp. 11–17. [CrossRef]
Agarwal, R. , Belmekki, M. , and Benchohra, M. , 2009, “ A Survey on Semilinear Differential Equations and Inclusions Involving Riemann-Liouville Fractional Derivative,” Adv. Differ. Equ., 2009(1), p. 981728.
Podlubny, I. , 1998, Fractional Differential Equations, Academic Press, San Diego, CA, pp. 43–48.
Abdeljawad, T. , 2011, “ On Riemann and Caputo Fractional Differences,” Comput. Math. Appl., 62(3), pp. 1602–1611. [CrossRef]
Mozyrska, D. , and Girejko, E. , 2013, “ Overview of Fractional h-Difference Operators,” Advances in Harmonic Analysis and Operator Theory, Springer, Cham, Switzerland, pp. 253–268. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Numerical results of the normalized concentration for bed-load sediment, calculated using the D-FADE model with different time fractional derivative α and space fractional derivative β. (a) Time evolution curves of bed-load with different α. (b) Space evolution curves of bed-load with different β. The other model parameters are as follows: v = 0.05763 m/minα and D = 0.04 mβ/minα.

Grahic Jump Location
Fig. 2

The simulated snapshots of bed-load spatial distribution for particle size of 1 mm at time (a) t = 155 min and (b) t = 355 min, α = 0.82, β = 1.65, U = 0.05763 m/minα, D = 0.025 mβ/minα

Grahic Jump Location
Fig. 3

The simulated snapshots of bed-load spatial distribution for particle size of 3 mm at time (a) t = 270 min and (b) t = 430 min, α = 0.82, β = 1.75, U = 0.04071 m/minα, D = 0.045 mβ/minα

Grahic Jump Location
Fig. 4

The simulated snapshots of bed-load spatial distribution for particle size of 7 mm at time (a) t = 65 min and (b) t = 345 min, α = 0.65, β = 1.82, U = 0.11983 m/minα, D = 0.050 mβ/minα

Grahic Jump Location
Fig. 5

The simulated snapshots of bed-load spatial distribution for particle size (a) dm = 1 mm, at time t = 117 min, U = 0.0700 m/minα, D = 0.025 mβ/minα, α = 0.65, and β = 1.75; (b) dm = 3 mm, at time t = 86 min, U = 0.08260 m/minα, D = 0.045 mβ/minα, α = 0.43, and β = 1.80; and (c) dm = 7 mm, at time t = 33 min, U = 0.17950 m/minα, D = 0.050 mβ/minα, α = 0.31, and β = 1.84

Grahic Jump Location
Fig. 6

Sensitivity analysis of space fractional derivative α and β in the D-FADE model for particle sizes 1 mm and 7 mm. (a) Numerical result for the particle of size 1 mm at t = 155 min with α = 0.72, 0.82, and 0.92, respectively. (b) Particle size 1 mm at t = 155 min with β = 1.45, 1.65, and 1.85, respectively. (c) Numerical result of particle size 7 mm at t = 65 min with α = 0.55, 0.65, and 0.75, respectively. (d) Particle size 7 mm at t = 65 min with β = 1.62, 1.82, and 1.98, respectively.

Grahic Jump Location
Fig. 7

Time fractional derivative order sensitivity analysis in the D-FADE model for 1 mm particles: (a) β = 1.65, U = 0.0700 m/minα, D = 0.025 mβ/minα, and t = 117 min; and (b) α = 0.65, U = 0.0700 m/minα, D = 0.025 mβ/minα, and t = 117 min

Grahic Jump Location
Fig. 8

Dispersion coefficient sensitivity analysis in the D-FADE model: (a) for 1 mm particles, α = 0.82, β = 1.65, U = 0.05763 m/minα, D = 0.025 mβ/minα, and t = 155 min; and (b) for 1 mm particles, α = 0.65, β = 1.75, U = 0.0700 m/minα, D = 0.025 mβ/minα, and t = 117 min

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