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Time–space fractional governing equations of transient groundwater flow in confined aquifers: Numerical investigation 
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下载免费PDF全文 In order to model non‐Fickian transport behaviour in groundwater aquifers, various forms of the time–space fractional advection–dispersion equation have been developed and used by several researchers in the last decade. The solute transport in groundwater aquifers in fractional time–space takes place by means of an underlying groundwater flow field. However, the governing equations for such groundwater flow in fractional time–space are yet to be developed in a comprehensive framework. In this study, a finite difference numerical scheme based on Caputo fractional derivative is proposed to investigate the properties of a newly developed time–space fractional governing equations of transient groundwater flow in confined aquifers in terms of the time–space fractional mass conservation equation and the time–space fractional water flux equation. Here, we apply these time–space fractional governing equations numerically to transient groundwater flow in a confined aquifer for different boundary conditions to explore their behaviour in modelling groundwater flow in fractional time–space. The numerical results demonstrate that the proposed time–space fractional governing equation for groundwater flow in confined aquifers may provide a new perspective on modelling groundwater flow and on interpreting the dynamics of groundwater level fluctuations. Additionally, the numerical results may imply that the newly derived fractional groundwater governing equation may help explain the observed heavy‐tailed solute transport behaviour in groundwater flow by incorporating nonlocal or long‐range dependence of the underlying groundwater flow field. 相似文献
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The fractional advection–dispersion equation (FADE) known as its non-local dispersion, has been proven to be a promising tool to simulate anomalous solute transport in groundwater. We present an unconditionally stable finite element (FEM) approach to solve the one-dimensional FADE based on the Caputo definition of the fractional derivative with considering its singularity at the boundaries. The stability and accuracy of the FEM solution is verified against the analytical solution, and the sensitivity of the FEM solution to the fractional order α and the skewness parameter β is analyzed. We find that the proposed numerical approach converge to the numerical solution of the advection–dispersion equation (ADE) as the fractional order α equals 2. The problem caused by using the first- or third-kind boundary with an integral-order derivative at the inlet is remedied by using the third-kind boundary with a fractional-order derivative there. The problems for concentration estimation at boundaries caused by the singularity of the fractional derivative can be solved by using the concept of transition probability conservation. The FEM solution of this study has smaller numerical dispersion than that of the FD solution by Meerschaert and Tadjeran (J Comput Appl Math 2004). For a given α, the spatial distribution of concentration exhibits a symmetric non-Fickian behavior when β = 0. The spatial distribution of concentration shows a Fickian behavior on the left-hand side of the spatial domain and a notable non-Fickian behavior on the right-hand side of the spatial domain when β = 1, whereas when β = −1 the spatial distribution of concentration is the opposite of that of β = 1. Finally, the numerical approach is applied to simulate the atrazine transport in a saturated soil column and the results indicat that the FEM solution of the FADE could better simulate the atrazine transport process than that of the ADE, especially at the tail of the breakthrough curves. 相似文献
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In this paper, the generalised two-dimensional differential transform method (DTM) of solving the time-fractional coupled KdV equations is proposed. The fractional derivative is described in the Caputo sense. The presented method is a numerical method based on the generalised Taylor series expansion which constructs an analytical solution in the form of a polynomial. An illustrative example shows that the generalised two-dimensional DTM is effective for the coupled equations. 相似文献
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Pedro Victor Serra Mascarenhas Ricardo Mendonça de Moraes André Luís Brasil Cavalcante 《国际地质力学数值与分析法杂志》2019,43(11):1956-1977
We propose an extension of the shifted Grünwald-Letnikov method to solve fractional partial differential equations in the Caputo sense with arbitrary fractional order derivative α and with an advective term. The method uses the relation between Caputo and Riemann-Liouville definitions, the shifted Grünwald-Letnikov, and the traditional backward and forward finite difference method. The stability of the method is investigated for the implicit and explicit scheme with homogeneous boundary conditions, and a stability criterion is found for the advective-dispersive equation. An application of the method is used to solve contaminant diffusion and advective-dispersive problems. The numerical solution for the fractional diffusion and fractional advection-dispersion is compared with their respective analytical solutions for different time and space grid refinements. The diffusion simulation exhibited a good fit between the analytical and numerical solutions, with the explicit scheme going from stable to unstable as the time and space refinement changes. The fractional advection-dispersion application produced small deviations from the analytical solution. These deviations, however, are analogous to the numerical dispersions encountered in conventional finite difference solutions of the advection-dispersion equation. The new method is also compared with the traditional L2 method. Notably, an example that involves asymmetrical fractional conditions, a fractional diffusivity that depends on time, and a source term show how the methods compare. Overall, this study assesses the quality and easiness of use of the numerical method. 相似文献
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The success of transient storage (TS) modeling for natural streams depends, in part, on the ability to describe the dispersion process accurately. Evidence based on stream tracer data shows that solute transport processes often do not follow the classical second-order dispersion model (e.g., early breakthrough and faster than Fickian travel times were observed). While models based on space-fractional dispersion are a promising alternative, different definitions of fractional derivatives exist in the literature. Unlike integer-order derivatives, fractional derivatives represent convolutions of concentration with long-range spatial correlation and numerical approximations can produce dense matrices. Therefore issues of both accuracy and computational efficiency need to be examined to successfully identify model parameters for natural streams. In this paper, we first compare the performance of several numerical approaches for solving the space-fractional dispersion equation. We examine three different numerical approaches to approximate the space-fractional derivatives including: (a) a fully-implicit scheme based on the shifted Grünwald–Letnikov (GL) approximation (b) a three-point implicit representation based on the GL formula and (c) a three-point implicit scheme based on mass conservation and the Caputo definition of the fractional derivative. We then use an operator-splitting technique to evaluate a TS model based on space-fractional dispersion (the FSTS model) and test the model against analytical solutions and stream tracer data. A sequence acceleration method (Richardson extrapolation) significantly improves the performance of all schemes examined. Results indicate that the fully-implicit GL method with Richardson extrapolation produces the most accurate solutions while the three-point implicit GL scheme has a stringent time-step restriction to produce acceptable solutions. The three-point implicit scheme based on the Caputo derivative produces accurate solutions in a fraction of the time taken by the fully-implicit GL method and represents the best trade-off between accuracy and computational efficiency for practical applications. The scheme is suitable for parameter estimation and is used to successfully describe tracer data in a natural stream. 相似文献
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The inherent heterogeneity of geological media often results in anomalous dispersion for solute transport through them, and how to model it has been an interest over the past few decades. One promising approach that has been increasingly used to simulate the anomalous transport in surface and subsurface water is the fractional advection–dispersion equation (FADE), derived as a special case of the more general continuous time random walk or the stochastic continuum model. In FADE, the dispersion is not local and the solutes have appreciable probability to move long distances, and thus reach the boundary faster than predicted by the classical advection–dispersion equation (ADE). How to deal with different boundaries associated with FADE and their consequent impact is an issue that has not been thoroughly explored. In this paper we address this by taking one-dimensional solute movement in soil columns as an example. We show that the commonly used FADE with its fractional derivatives defined by the Riemann–Liouville definition is problematic and could result in unphysical results for solute transport in bounded domains; a modified method with the fractional dispersive flux defined by the Caputo derivatives is presented to overcome this problem. A finite volume approach is given to numerically solve the modified FADE and its associated boundaries. With the numerical model, we analyse the inlet-boundary treatment in displacement experiments in soil columns, and find that, as in ADE, treating the inlet as a prescribed concentration boundary gives rise to mass-balance errors and such errors could be more significant in FADE because of its non-local dispersion. We also discuss a less-documented but important issue in hydrology: how to treat the upstream boundary in analysing the lateral movement of tracer in an aquifer when the tracer is injected as a pulse. It is shown that the use of an infinite domain, as commonly assumed in literature, leads to unphysical backward dispersion, which has a significant impact on data interpretation. To avoid this, the upstream boundary should be flux-prescribed and located at the upstream edge of the injecting point. We apply the model to simulate the movement of Cl− in a tracer experiment conducted in a saturated hillslope, and analyse in details the significance of upstream-boundary treatments in parameter estimation. 相似文献
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