# 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  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
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##### References:
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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