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A review of applications of fractional advection–dispersion equations for anomalous solute transport in surface and subsurface water

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Abstract Fractional advection–dispersion equations (FADEs) have been widely used in hydrological research to simulate the anomalous solute transport in surface and subsurface water. However, a large gap still exists between real‐world application (i.e., being a prediction tool) and theoretical FADEs. To better understand this disparity, the FADEs are firstly reviewed from the perspective of fractional‐in‐time and fractional‐in‐space, as well as the anomalous characteristics described by those functions. Then, challenges for the application of FADEs are summarized, including the theoretical gap of FADEs that needs multidisciplinary efforts to fill, extensive requirements for computation techniques and mathematical knowledge to apply the FADEs, the poor predictability for most parameters in the FADEs, and the limitations for collecting geologic information of flow fields. Then, some suggestions are given for future work, such as developing excellent code sets and mature simulation software with a friendly interface. This kind of work would alleviate the computation workload of hydrologists, especially those without coding expertise. Summarizing and determining the value ranges of the important parameters are needed (e.g., the order of the fractional derivative), through the extensive field, laboratory, and numerical experiments rather than blindly using their mathematical ranges. It is also needed to perform global sensitivity analysis for the FADEs. Meanwhile, comprehensive comparison work is necessary for suggesting a model suitable for a specific problem. Last but not least, further research is clearly needed to establish a link between nonlocal parameters and the heterogeneity property to develop more efficient fractional order partial differential equations. This article is categorized under: Science of Water > Methods Science of Water > Water Quality
Log–log plots of break through curves (BTCs) of non‐Fickian and Fickian transport (blue line) for a pulse injection: (a) BTCs of the early arrival (red line), and (b) BTCs of the heavy tail (red line)
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The multimodal breakthrough curve for solute transport through a subterranean river in a karst area. (Reprinted with permission from Huang et al. (2017). Copyright 2017 Carsologica Sinica. (In Chinese))
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Break through curves (BTCs) and snapshots generated by the space fractional advection–dispersion equation (sFADE) with α ∈ [1.60, 1.99], D = 1, β = 1, and v = 1: (a) Snapshots generated by sFADE with the suppositional observation time at 300 (T) from the injection time. (b) BTCs generated by sFADE with the suppositional observation point at 260 (L) downstream from the injection point
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Breakthrough curves (BTCs) generated by the tempered fractional in time model with different λ. Other parameters were fixed as: v = 1, D = 1, and γ = 0.9
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Break through curves (BTCs) and snapshots generated by the fractional advection–dispersion equation (FTADE) with γ ∈ [0.60, 0.99], D = 1, andv = 0.5: (a) Snapshots generated by FTADE with the suppositional observation time at 500 (T) from the injection time. (b) BTCs generated by FTADE with the suppositional observation point at 260 (L) downstream form the injection point
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Conceptual map of the early arrivals of solutes in a preferential flow in groundwater. Space snapshots are at T0 (injection time), T1, and T2 (T0 < T1 < T2) times, respectively
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Conceptual map of preferential flow in heterogeneous media
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Conceptual map of mass exchange between the main channel and the hyporheic zone (longitudinal section). Space snapshots are at T0 (injection time), T1, and T2 (T0 < T1 < T2) times, respectively
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Conceptual map of a stream (cross‐section) and mass exchange between the main channel and hyporheic zone
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