July  2021, 14(7): 2055-2074. doi: 10.3934/dcdss.2021061

Fractional and fractal advection-dispersion model

1. 

Institute for Groundwater Studies, Faculty of Agricultural and Natural Sciences, University of the Free State, 9301, Bloemfontein, Free State, South Africa

2. 

Department of Mathematics, Faculty of Science and Technology, University of Moulay Ismail, Errachidia, Morocco

* Corresponding author: Amy Allwright

Received  April 2019 Revised  October 2020 Published  May 2021

A fractal advection-dispersion equation and a fractional space-time advection-dispersion equation have been developed to improve the simulation of groundwater transport in fractured aquifers. The space-time fractional advection-dispersion simulation is limited due to complex algorithms and the computational power required; conversely, the fractal advection-dispersion equation can be solved simply, yet only considers the fractal derivative in space. These limitations lead to combining these methods, creating a fractional and fractal advection-dispersion equation to provide an efficient non-local, in both space and time, modeling tool. The fractional and fractal model has two parameters, fractional order ($ \alpha $) and fractal dimension ($ \beta $), where simulations are valid for specific combinations. The range of valid combinations reduces with decreasing fractional order and fractal dimension, and a final recommendation of $ \; 0.7 \leq \alpha, \beta \leq 1 $ is made. The fractional and fractal model provides a flexible tool to model anomalous diffusion, where the fractional order controls the breakthrough curve peak, and the fractal dimension controls the position of the peak and tailing effect. These two controls potentially provide tools to improve the representation of anomalous breakthrough curves that cannot be described by the classical model.

Citation: Amy Allwright, Abdon Atangana, Toufik Mekkaoui. Fractional and fractal advection-dispersion model. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2055-2074. doi: 10.3934/dcdss.2021061
References:
[1]

J. A. Acuna and Y. C. Yortsos, Application of fractal geometry to the study of networks of fractures and their pressure transient, Water Resources Research, 31 (1995), 527-540.  doi: 10.1029/94WR02260.  Google Scholar

[2]

A. Allwright and A. Atangana, Fractal advection-dispersion equation for groundwater transport in fractured aquifers with self-similarities, The European Physical Journal Plus, 133 (2018), Article number: 48. doi: 10.1140/epjp/i2018-11885-3.  Google Scholar

[3]

A. Allwright and A. Atangana, Augmented upwind numerical schemes for a fractional advection-dispersion equation in fractured groundwater systems, Discrete & Continuous Dynamical Systems-S, 13 (2020), 443-466.  doi: 10.3934/dcdss.2020025.  Google Scholar

[4]

A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Chaos, Solitons & Fractals, 102 (2017), 396-406.  doi: 10.1016/j.chaos.2017.04.027.  Google Scholar

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D. Baleanu, S. S. Sajjadi, A. Jajarmi and J. H. Asad, New features of the fractional euler-lagrange equations for a physical system within non-singular derivative operator, The European Physical Journal Plus, 134 (2019), 181. doi: 10.1140/epjp/i2019-12561-x.  Google Scholar

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W. ChenH. SunX. Zhang and D. Korošak, Anomalous diffusion modeling by fractal and fractional derivatives, Comput. Math. Appl., 59 (2010), 1754-1758.  doi: 10.1016/j.camwa.2009.08.020.  Google Scholar

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W. ChenX. Zhang and D. Korošak, Investigation on fractional and fractal derivative relaxation-oscillation models, International Journal of Nonlinear Sciences and Numerical Simulation, 11 (2010), 3-9.  doi: 10.1515/IJNSNS.2010.11.1.3.  Google Scholar

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R. A. El-Nabulsi, Modifications at large distances from fractional and fractal arguments, Fractals, 18 (2010), 185-190.  doi: 10.1142/S0218348X10004828.  Google Scholar

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W. FanX. Jiang and S. Chen, Parameter estimation for the fractional fractal diffusion model based on its numerical solution, Comput. Math. Appl., 71 (2016), 642-651.  doi: 10.1016/j.camwa.2015.12.030.  Google Scholar

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J. Feng, Fractional fractal geometry for image processing, northwestern university. Google Scholar

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S. FominV. Chugunov and T. Hashida, The effect of non-fickian diffusion into surrounding rocks on contaminant transport in a fractured porous aquifer, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 2923-2939.  doi: 10.1098/rspa.2005.1487.  Google Scholar

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[27]

A. Jajarmi, S. Arshad and D. Baleanu, A new fractional modelling and control strategy for the outbreak of dengue fever, Phys. A, 535 (2019), 122524. doi: 10.1016/j.physa.2019.122524.  Google Scholar

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show all references

References:
[1]

J. A. Acuna and Y. C. Yortsos, Application of fractal geometry to the study of networks of fractures and their pressure transient, Water Resources Research, 31 (1995), 527-540.  doi: 10.1029/94WR02260.  Google Scholar

[2]

A. Allwright and A. Atangana, Fractal advection-dispersion equation for groundwater transport in fractured aquifers with self-similarities, The European Physical Journal Plus, 133 (2018), Article number: 48. doi: 10.1140/epjp/i2018-11885-3.  Google Scholar

[3]

A. Allwright and A. Atangana, Augmented upwind numerical schemes for a fractional advection-dispersion equation in fractured groundwater systems, Discrete & Continuous Dynamical Systems-S, 13 (2020), 443-466.  doi: 10.3934/dcdss.2020025.  Google Scholar

[4]

A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Chaos, Solitons & Fractals, 102 (2017), 396-406.  doi: 10.1016/j.chaos.2017.04.027.  Google Scholar

[5]

D. Baleanu, A. Jajarmi, S. S. Sajjadi and D. Mozyrska, A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator, \emphChaos, 29 (2019), 083127, 15pp. doi: 10.1063/1.5096159.  Google Scholar

[6]

D. Baleanu, S. S. Sajjadi, A. Jajarmi and J. H. Asad, New features of the fractional euler-lagrange equations for a physical system within non-singular derivative operator, The European Physical Journal Plus, 134 (2019), 181. doi: 10.1140/epjp/i2019-12561-x.  Google Scholar

[7]

D. A. BensonS. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resources Research, 36 (2000), 1403-1412.  doi: 10.1029/2000WR900031.  Google Scholar

[8]

D. A. Benson, The Fractional Advection-Dispersion Equation: Development and Application, PhD thesis, University of Nevada, Reno, 1998. Google Scholar

[9]

M. V. Berry and S. Klein, Integer, fractional and fractal talbot effects, Journal of Modern Optics, 43 (1996), 2139-2164.  doi: 10.1080/09500349608232876.  Google Scholar

[10]

P. A. CelloD. D. WalkerA. J. Valocchi and B. Loftis, Flow dimension and anomalous diffusion of aquifer tests in fracture networks, Vadose Zone Journal, 8 (2009), 258-268.  doi: 10.2136/vzj2008.0040.  Google Scholar

[11]

W. Chen, X. Chen and C. J. R. Sheppard, Optical image encryption based on phase retrieval combined with three-dimensional particle-like distribution, Journal of Optics, 14 (2012), 075402. doi: 10.1088/2040-8978/14/7/075402.  Google Scholar

[12]

W. Chen and Y. Liang, New methodologies in fractional and fractal derivatives modeling, Chaos, Solitons & Fractals, 102 (2017), 72-77.  doi: 10.1016/j.chaos.2017.03.066.  Google Scholar

[13]

W. ChenH. SunX. Zhang and D. Korošak, Anomalous diffusion modeling by fractal and fractional derivatives, Comput. Math. Appl., 59 (2010), 1754-1758.  doi: 10.1016/j.camwa.2009.08.020.  Google Scholar

[14]

W. ChenX. Zhang and D. Korošak, Investigation on fractional and fractal derivative relaxation-oscillation models, International Journal of Nonlinear Sciences and Numerical Simulation, 11 (2010), 3-9.  doi: 10.1515/IJNSNS.2010.11.1.3.  Google Scholar

[15]

R. A. El-Nabulsi, Modifications at large distances from fractional and fractal arguments, Fractals, 18 (2010), 185-190.  doi: 10.1142/S0218348X10004828.  Google Scholar

[16]

W. FanX. Jiang and S. Chen, Parameter estimation for the fractional fractal diffusion model based on its numerical solution, Comput. Math. Appl., 71 (2016), 642-651.  doi: 10.1016/j.camwa.2015.12.030.  Google Scholar

[17]

J. Feng, Fractional fractal geometry for image processing, northwestern university. Google Scholar

[18]

S. FominV. Chugunov and T. Hashida, The effect of non-fickian diffusion into surrounding rocks on contaminant transport in a fractured porous aquifer, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 2923-2939.  doi: 10.1098/rspa.2005.1487.  Google Scholar

[19]

E. GerolymatouI. Vardoulakis and R. Hilfer, Modelling infiltration by means of a nonlinear fractional diffusion model, Journal of Physics D: Applied Physics, 39 (2006), 4104-4110.  doi: 10.1088/0022-3727/39/18/022.  Google Scholar

[20]

J. Gomez-Aquilar, L. Torres, H. Yepez-Martinez, D. Baleanu, J. Reyes and I. Sosa, Fractional liénard type model of a pipeline within the fractional derivative without singular kernel, Adv. Difference Equ., 2016 (2016), Paper No. 173, 13 pp. doi: 10.1186/s13662-016-0908-1.  Google Scholar

[21]

D. J. Goode, C. Tiedeman, P. J. Lacombe, T. E. Imbrigiotta, A. M. Shapiro and F. H. Chapelle, Contamination in Fractured-Rock Aquifers: Research at the Former Naval Air Warfare Center, West Trenton, New Jersey, , Fact Sheet, 2007. doi: 10.3133/fs20073074.  Google Scholar

[22]

C. Hall, Anomalous diffusion in unsaturated flow: Fact or fiction?, Cement and Concrete Research, 37 (2007), 378-385.  doi: 10.1016/j.cemconres.2006.10.004.  Google Scholar

[23]

R. Hilfer and L. Anton, Fractional master equations and fractal time random walks, Phys. Rev. E, 51 (1995), R848–R851. doi: 10.1103/PhysRevE.51.R848.  Google Scholar

[24]

J. Hristov, Derivatives with non-singular kernels from the caputo-fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models, Frontiers in Fractional Calculus, Curr. Dev. Math. Sci., 1 (2017), 269-341.  doi: 10.2174/9781681085999118010013.  Google Scholar

[25]

F. Huang and F. Liu, The fundamental solution of the space-time fractional advection-dispersion equation, J. Appl. Math. Comput., 18 (2005), 339-350.  doi: 10.1007/BF02936577.  Google Scholar

[26]

G. HuangQ. Huang and H. Zhan, Evidence of one-dimensional scale-dependent fractional advection-dispersion, Journal of Contaminant Hydrology, 85 (2006), 53-71.  doi: 10.1016/j.jconhyd.2005.12.007.  Google Scholar

[27]

A. Jajarmi, S. Arshad and D. Baleanu, A new fractional modelling and control strategy for the outbreak of dengue fever, Phys. A, 535 (2019), 122524. doi: 10.1016/j.physa.2019.122524.  Google Scholar

[28]

A. Jajarmi, D. Baleanu, S. S. Sajjadi and J. H. A. and, A new feature of the fractional euler-lagrange equations for a coupled oscillator using a nonsingular operator approach, Front. Phys., 7 (2019), 196. doi: 10.3389/fphy.2019.00196.  Google Scholar

[29]

A. Jajarmi, B. Ghanbari and D. Baleanu, A new and efficient numerical method for the fractional modeling and optimal control of diabetes and tuberculosis co-existence, Chaos, 29 (2019), 093111, 15pp. doi: 10.1063/1.5112177.  Google Scholar

[30]

S. Javadi, M. Jani and E. Babolian, A numerical scheme for space-time fractional advection-dispersion equation, 7 (2016), 331–343. Google Scholar

[31]

X. JiangM. Xu and H. Qi, The fractional diffusion model with an absorption term and modified fick's law for non-local transport processes, Nonlinear Anal. Real World Appl., 11 (2010), 262-269.  doi: 10.1016/j.nonrwa.2008.10.057.  Google Scholar

[32]

F. LiuV. V. AnhI. Turner and P. Zhuang, Time fractional advection-dispersion equation, J. Appl. Math. Comput., 13 (2003), 233-245.  doi: 10.1007/BF02936089.  Google Scholar

[33]

S. Lu, F. J. Molz and G. J. Fix, Possible problems of scale dependency in applications of the three-dimensional fractional advection-dispersion equation to natural porous media, Water Resources Research, 38 (2002), 4–1–4–7. doi: 10.1029/2001WR000624.  Google Scholar

[34]

C. Masciopinto and D. Palmiotta, Flow and transport in fractured aquifers: New conceptual models based on field measurements, Transport in Porous Media, 96 (2012), 117-133.  doi: 10.1007/s11242-012-0077-y.  Google Scholar

[35]

L. Nyikos and T. Pajkossy, Fractal dimension and fractional power frequency-dependent impedance of blocking electrodes, Electrochimica Acta, 30 (1985), 1533-1540.  doi: 10.1016/0013-4686(85)80016-5.  Google Scholar

[36]

K. Owolabi and A. Atangana, Robustness of fractional difference schemes via the caputo subdiffusion-reaction equations, Chaos Solitons Fractals, 111 (2018), 119-127.  doi: 10.1016/j.chaos.2018.04.019.  Google Scholar

[37]

Y. Z. Povstenko, Fundamental solutions to time-fractional advection diffusion equation in a case of two space variables, Math. Probl. Eng., 2014 (2014), 1-7.  doi: 10.1155/2014/705364.  Google Scholar

[38]

Y. Povstenko, Space-time-fractional advection diffusion equation in a plane, in Lecture Notes in Electrical Engineering, Springer International Publishing, 320 (2015), 275–284. doi: 10.1007/978-3-319-09900-2_26.  Google Scholar

[39]

M. Rieu and G. Sposito, Fractal fragmentation, soil porosity, and soil water properties: Ⅰ. theory, Soil Science Society of America Journal, 55 (1991), 1231-1238.  doi: 10.2136/sssaj1991.03615995005500050006x.  Google Scholar

[40]

Q. Rubbab, I. A. Mirza and M. Z. A. Qureshi, Analytical solutions to the fractional advection-diffusion equation with time-dependent pulses on the boundary, AIP Advances, 6 (2016), 075318. doi: 10.1063/1.4960108.  Google Scholar

[41]

M. Santos and I. Gomez, A fractional fokker-planck equation for non-singular kernel operators, J. Stat. Mech. Theory Exp., 2018 (2018), 123205. doi: 10.1088/1742-5468/aae5a2.  Google Scholar

[42]

S. G. Schmelling and R. R. Ross, Contaminant transport in fractured media: Models for decision makers, Groundwater, 28 (1990), 272-279.  doi: 10.1111/j.1745-6584.1990.tb02259.x.  Google Scholar

[43]

A. M. Shapiro, The challenge of interpreting environmental tracer concentrations in fractured rock and carbonate aquifers, Hydrogeology Journal, 19 (2010), 9-12.  doi: 10.1007/s10040-010-0678-x.  Google Scholar

[44]

A. R. ShokriT. Babadagli and A. Jafari, A critical analysis of the relationship between statistical- and fractal-fracture-network characteristics and effective fracture-network permeability, SPE Res Eval & Eng, 19 (2016), 494-510.  doi: 10.2118/181743-pa.  Google Scholar

[45]

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Figure 1.  Fractal velocity $ (V_{F\beta}) $ over space for varying fractal dimensions $ \beta = 1,0.8,0.5,0.2 $
Figure 2.  Fractal dispersivity $ (V_{F\beta}) $ over space for varying fractal dimensions $ \beta = 1,0.8,0.5,0.2 $
Figure 3.  Simulation results illustrated for distance (m) in the x-direction along a line over time (d) for the fractional order $ \alpha = 1 $ (simplifies to a local order), and varying fractal dimensions ($ 0.5 \leq \beta \leq 1 $)
Figure 4.  Simulation results illustrated for distance (m) in the x-direction along a line over time (d) for the fractional order $ \alpha = 0.9 $ (simplifies to a local order), and varying fractal dimensions ($ 0.5 \leq \beta \leq 1 $)
Figure 5.  Simulation results illustrated for distance (m) in the x-direction along a line over time (d) for the fractional order $ \alpha = 0.8$ (simplifies to a local order), and varying fractal dimensions ($ 0.6 \leq \beta \leq 1 $)
Figure 6.  Simulation results illustrated for distance (m) in the x-direction along a line over time (d) for the fractional order $ \alpha = 0.7$ (simplifies to a local order), and varying fractal dimensions ($ 0.7 \leq \beta \leq 1 $)
Figure 7.  Simulation results illustrated for distance (m) in the x-direction along a line over time (d) for the fractional order $ \alpha = 0.6, 0.5 $ (simplifies to a local order), and varying fractal dimensions ($ 0.8 \leq \beta \leq 1 $)
Table 1.  Simulation results illustrated for distance (m) in the x-direction along a line over time (d) for the fractional order α = 0:7 (simplifies to a local order), and varying fractal dimensions (0:7 ≤ β ≤ 1)
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