# American Institute of Mathematical Sciences

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  July 2021 Early access  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 and 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. [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. [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. [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. [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. [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. [7] D. A. Benson, S. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resources Research, 36 (2000), 1403-1412.  doi: 10.1029/2000WR900031. [8] D. A. Benson, The Fractional Advection-Dispersion Equation: Development and Application, PhD thesis, University of Nevada, Reno, 1998. [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. [10] P. A. Cello, D. D. Walker, A. 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. [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. [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. [13] W. Chen, H. Sun, X. 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. [14] W. Chen, X. 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. [15] R. A. El-Nabulsi, Modifications at large distances from fractional and fractal arguments, Fractals, 18 (2010), 185-190.  doi: 10.1142/S0218348X10004828. [16] W. Fan, X. 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. [17] J. Feng, Fractional fractal geometry for image processing, northwestern university. [18] S. Fomin, V. 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. [19] E. Gerolymatou, I. 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. [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. [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. [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. [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. [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. 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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. [30] S. Javadi, M. Jani and E. Babolian, A numerical scheme for space-time fractional advection-dispersion equation, 7 (2016), 331–343. [31] X. Jiang, M. 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. [32] F. Liu, V. V. Anh, I. Turner and P. Zhuang, Time fractional advection-dispersion equation, J. Appl. Math. Comput., 13 (2003), 233-245.  doi: 10.1007/BF02936089. [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. [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. [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. [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. [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. [38] Y. 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##### 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. [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. [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. [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. [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. [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. [7] D. A. Benson, S. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resources Research, 36 (2000), 1403-1412.  doi: 10.1029/2000WR900031. [8] D. A. Benson, The Fractional Advection-Dispersion Equation: Development and Application, PhD thesis, University of Nevada, Reno, 1998. [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. [10] P. A. Cello, D. D. Walker, A. 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. [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. [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. [13] W. Chen, H. Sun, X. 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. [14] W. Chen, X. 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. [15] R. A. El-Nabulsi, Modifications at large distances from fractional and fractal arguments, Fractals, 18 (2010), 185-190.  doi: 10.1142/S0218348X10004828. [16] W. Fan, X. 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. [17] J. Feng, Fractional fractal geometry for image processing, northwestern university. [18] S. Fomin, V. 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. [19] E. Gerolymatou, I. 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. [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. [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. [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. [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. [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. [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. [26] G. Huang, Q. 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. [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. [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. [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. [30] S. Javadi, M. Jani and E. Babolian, A numerical scheme for space-time fractional advection-dispersion equation, 7 (2016), 331–343. [31] X. Jiang, M. 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. [32] F. Liu, V. V. Anh, I. Turner and P. Zhuang, Time fractional advection-dispersion equation, J. Appl. Math. Comput., 13 (2003), 233-245.  doi: 10.1007/BF02936089. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [44] A. R. Shokri, T. 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. [45] L. Su, W. Wang and Q. Xu, Finite difference methods for fractional dispersion equations, Applied Mathematics and Computation, 216 (2010), 3329-3334.  doi: 10.1016/j.amc.2010.04.060. [46] H. Sun, Z. Li, Y. Zhang and W. Chen, Fractional and fractal derivative models for transient anomalous diffusion: Model comparison, Chaos Solitons Fractals, 102 (2017), 346-353.  doi: 10.1016/j.chaos.2017.03.060. [47] H. Sun, M. M. Meerschaert, Y. Zhang, J. Zhu and W. Chen, A fractal richards' equation to capture the non-boltzmann scaling of water transport in unsaturated media, Advances in Water Resources, 52 (2013), 292-295.  doi: 10.1016/j.advwatres.2012.11.005. [48] H. Sun, Y. Zhang, W. Chen and D. M. Reeves, Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media, Journal of Contaminant Hydrology, 157 (2014), 47-58.  doi: 10.1016/j.jconhyd.2013.11.002. [49] A. A. Tateishi, H. V. Ribeiro and E. K. Lenzi, The role of fractional time-derivative operators on anomalous diffusion, Front. Phys., 5, 2017. doi: 10.3389/fphy.2017.00052. [50] M. Toufik and A. 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Fractal velocity $(V_{F\beta})$ over space for varying fractal dimensions $\beta = 1,0.8,0.5,0.2$
Fractal dispersivity $(V_{F\beta})$ over space for varying fractal dimensions $\beta = 1,0.8,0.5,0.2$
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 ($0.5 \leq \beta \leq 1$)
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 ($0.5 \leq \beta \leq 1$)
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 ($0.6 \leq \beta \leq 1$)
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 ($0.7 \leq \beta \leq 1$)
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 ($0.8 \leq \beta \leq 1$)
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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