Time–space fractional governing equations of transient groundwater flow in confined aquifers: Numerical investigation

Tu, Tongbi; Ercan, Ali; Kavvas, M. L.

doi:10.1002/hyp.11500

Cited by 14 publications

(4 citation statements)

References 53 publications

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“…On the other hand, those of Well 2 represented mainly Gaussian characteristics, which sometimes failed to capture the peaks of the skewedprobability distributions of Well 2 groundwater levels. Time series analysis of groundwater level fluctuations of the two wells demonstrated that the observed fractal behavior is site specific, and there is a need for generalized governing equations of groundwater flow processes that can model both the long-memory behavior and the Brownian finite-memory behavior Tu et al, 2017).…”

Section: Discussionmentioning

confidence: 99%

Fractal scaling analysis of groundwater dynamics in confined aquifers

Ercan

Kavvas

2017

Earth Syst. Dynam.

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Abstract. Groundwater closely interacts with surface water and even climate systems in most hydroclimatic settings. Fractal scaling analysis of groundwater dynamics is of significance in modeling hydrological processes by considering potential temporal long-range dependence and scaling crossovers in the groundwater level fluctuations. In this study, it is demonstrated that the groundwater level fluctuations in confined aquifer wells with long observations exhibit site-specific fractal scaling behavior. Detrended fluctuation analysis (DFA) was utilized to quantify the monofractality, and multifractal detrended fluctuation analysis (MF-DFA) and multiscale multifractal analysis (MMA) were employed to examine the multifractal behavior. The DFA results indicated that fractals exist in groundwater level time series, and it was shown that the estimated Hurst exponent is closely dependent on the length and specific time interval of the time series. The MF-DFA and MMA analyses showed that different levels of multifractality exist, which may be partially due to a broad probability density distribution with infinite moments. Furthermore, it is demonstrated that the underlying distribution of groundwater level fluctuations exhibits either non-Gaussian characteristics, which may be fitted by the Lévy stable distribution, or Gaussian characteristics depending on the site characteristics. However, fractional Brownian motion (fBm), which has been identified as an appropriate model to characterize groundwater level fluctuation, is Gaussian with finite moments. Therefore, fBm may be inadequate for the description of physical processes with infinite moments, such as the groundwater level fluctuations in this study. It is concluded that there is a need for generalized governing equations of groundwater flow processes that can model both the long-memory behavior and the Brownian finite-memory behavior.

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Section: Discussionmentioning

confidence: 99%

Fractal scaling analysis of groundwater dynamics in confined aquifers

Ercan

Kavvas

2017

Earth Syst. Dynam.

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“…It should also be noticed that the applications of FADEs in hydrology are mainly focused on the simulation of tracer experiments. Previous studies have shown that FADEs can also be used to simulate the change of subsurface water tables (Tu, Ercan, & Levent, 2018) and the transport of suspended sediment in an unsteady flow (Nie et al, 2018), to describe the vertical distribution of suspended sediment in a steady flow (Nie et al, 2017), and to forecast pollutant drift on coastal water surfaces (Qin et al, 2017). Yu et al (2018) also proved that the fractional derivate model could be used to simulate colloid transport in the dense vegetation and soil systems.…”

Section: Challenges and Suggestions For Future Workmentioning

confidence: 99%

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

Sun

Qiu

et al. 2020

WIREs Water

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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

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“…The elementary structure of time-fractional convection-dispersion equation (TFCDE) uses the derivative term with fractional index instead of derivative term with integer order in time where the range of fractional order derivative is generally considered to be 0 to 1 but it can also be further extended to the range 1-2 [17][18][19][20]. The advantage of using fractional convectiondispersion equation (FCDE) over classical form of CDE is that fractional one is much more applicable to simulate the variation of the subsurface water table and also for the simulation of colloid migration in the soil system and dense vegetation [21,22]. Due to nonlinearity arises from the fractional order terms, it is quite complicated to present analytical or numerical solutions for FCDE as compare to classical one.…”

Section: Introductionmentioning

confidence: 99%

Anomalous transport for multispecies reactive system with first order decay: time-fractional model

Chaudhary¹,

Singh²

2022

Phys. Scr.

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The prediction of pollutant migration and its concentration variation in the subsurface hydrology is vitally requisite for the assessment and restorative treatment of polluted groundwater system. Pollutant migration for the multispecies reactive system cannot be reliably investigated by classical form of convection-dispersion equation (CDE), due to the presence of more than one reactive species. This paper establishes a time-fractional model for multispecies reactive system under the first order sequential reaction network to understand the anomalous or non-Fickian migration phenomenon for reactive pollutants. At present, most of the fractional models are presented for the classical CDE to investigate migration phenomenon for single species system, not for the multispecies reactive system due to the complexity of the modelled framework. The impact of fractional derivative model is analysed for variable dependent migration parameters and constant parameters, both for the multispecies reactive migration phenomenon. The fractional derivative is expressed in the Caputo sense and to handle the non-linearity of problem, Homotopy perturbation method (HPM) is adopted. The advantage of this method, to get the solutions, is that the HPM is independent of small parameters required for the deformation process as used in other existing perturbation techniques, which make it much more convenient to use for non-linear systems. The impact of the fractional derivative index and other migration parameters is graphically depicted for the reactive species and significant influence of fractional term is observed. The derived solutions are then validated by using the corresponding solutions obtained by other existing well-established methods to ensure the convergence of the HPM solutions. As there do not exist any solutions for such fractional models for multispecies reactive system, this study may be advantageous to convey better understanding for the anomalous or non-Fickian migration for reactive pollutants and their remediation planning in the groundwater resources.

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Time–space fractional governing equations of transient groundwater flow in confined aquifers: Numerical investigation

Cited by 14 publications

References 53 publications

Fractal scaling analysis of groundwater dynamics in confined aquifers

Fractal scaling analysis of groundwater dynamics in confined aquifers

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

Anomalous transport for multispecies reactive system with first order decay: time-fractional model

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