Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media

Sun, HongGuang; Zhang, Yong; Chen, Wen; Reeves, Donald M.

doi:10.1016/j.jconhyd.2013.11.002

Cited by 134 publications

(63 citation statements)

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“…faster movement found in preferential flow pathways), and subdiffusion is characterised by a growth rate slower than linear growth (i.e. slower movement in low-permeability zones) [20][21][22][23]. These shortcomings of traditional advection-dispersion equation have led to alternative non-local conceptualisations of flow and transport, and numerous methods for addressing scale and space-time dependencies.…”

Section: New Groundwater Transport Model: Fractal Advection-dispersiomentioning

confidence: 99%

“…These alternative methods include stochastic averaging of the classical advection-dispersion equation, the multiple-rate mass transfer method, the continuous time random walk method, the time fractional advection-dispersion equation method, the space fractional advection-dispersion equation method, and others [20]. In these alternative methods, the dispersive state is allowed to vary between superdiffusion, subdiffusion and normal diffusion, referred to as transient dispersion [22]. However, each method might be formulated for a specific transition and thus not appropriate for all types of transient dispersion.…”

Section: New Groundwater Transport Model: Fractal Advection-dispersiomentioning

confidence: 99%

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Fractal advection-dispersion equation for groundwater transport in fractured aquifers with self-similarities

Allwright

Atangana

2018

Eur. Phys. J. Plus

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Abstract. Groundwater transport within a fractured aquifer with a fractal nature exhibiting self-similarity cannot be accurately simulated by the classical Fickian advection-dispersion transport equation without a detailed characterisation of the fracture network and heterogeneity of the system. Because the information to characterise such a system to an appropriate level of detail is often not available, most applications fail to accurately simulate the observed contaminant transport. In response to the current limitations of transport modelling using the advection-dispersion equation, especially in fractured media, a fractal advectiondispersion groundwater transport equation is developed. Fractal differentiation is discussed in terms of the fractal derivative and the fractal integral. The fractal derivative is commonly applied and known, yet the fractal integral is developed in this paper along with the appropriate theorem and proof. The numerical approximation of the fractal derivative and integral are given, where Simpson's 3/8 Rule and Boole's Rule for numerical integration are applied for the fractal integral, and the forward and Crank-Nicolson finite difference schemes are applied for the fractal derivative. Upon the given foundation, the fractal advectiondispersion transport equation is formulated to develop a new groundwater transport model. The qualitative properties of the fractal advection-dispersion equation are investigated to determine boundedness, existence and uniqueness of the solution. To validate the developed fractal advection-dispersion equation, a numerical simulation is performed with different fractal dimensions to demonstrate the applicability of the model to fractal groundwater systems. The numerical simulations found that anomalous diffusion could be better modelled by incorporating a fractal dimension, especially where limited information is available on preferential pathways.

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Section: New Groundwater Transport Model: Fractal Advection-dispersiomentioning

confidence: 99%

Section: New Groundwater Transport Model: Fractal Advection-dispersiomentioning

confidence: 99%

Fractal advection-dispersion equation for groundwater transport in fractured aquifers with self-similarities

Allwright

Atangana

2018

Eur. Phys. J. Plus

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“…From Equation (41) c on the coordinate z or the operator method can also be used (see [58,59]). As a result we obtain the generalized boundary conditions at the median surface Σ (see Figure 2): (46) and (47) we get the conditions of perfect diffusive contact (32) and (33) whereas from (50) and (51) we obtain the conditions of perfect contact (9) and (11).…”

Section: Generalized Conditions Of Nonperfect Contactmentioning

confidence: 99%

“…The entropy approach to anomalous diffusion was used to analyze the magnetic resonance images in biological issues [16][17][18]. Fractional reaction-diffusion and advection-diffusion equations were studied by many authors (see [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34], among many others). Several numerical methods were used to solve the problem: the implicit and explicit difference schemes [19][20][21][22], the Adomian decomposition method [23], the homotopy perturbation method [24] and homotopy analysis method [25], the collocation methods [26,27], the finite element method [28].…”

Section: Introductionmentioning

confidence: 99%

“…Next they employ the boundary particle method to solve the transformed problem and implement the Stehfest numerical inverse Laplace transform. The space-time-fractional advection-diffusion equation with derivatives of the variable order was studied in [32]; the implicit Eulerian scheme and a Lagrangian solver were applied to solve this equation. The fractal and fractional derivatives were used in [33] to model anomalous diffusion; the Crank-Nicholson finite difference discretization scheme was utilized.…”

Section: Introductionmentioning

confidence: 99%

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Generalized Boundary Conditions for the Time-Fractional Advection Diffusion Equation

Povstenko

2015

Entropy

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Abstract:The different kinds of boundary conditions for standard and fractional diffusion and advection diffusion equations are analyzed. Near the interface between two phases there arises a transition region which state differs from the state of contacting media owing to the different material particle interaction conditions. Particular emphasis has been placed on the conditions of nonperfect diffusive contact for the time-fractional advection diffusion equation. When the reduced characteristics of the interfacial region are equal to zero, the conditions of perfect contact are obtained as a particular case.

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A subordinated advection model for uniform bed load transport from local to regional scales

Zhang

Martin

Chen

et al. 2014

J. Geophys. Res. Earth Surf.

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Sediment tracers moving as bed load can exhibit anomalous dispersion behavior deviating from Fickian diffusion. The presence of heavy-tailed resting time distributions and thin-tailed step length distributions motivate adoption of fractional-derivative models (FDMs) to describe sediment dispersion, but these models require many parameters that are difficult to quantify. Here we propose a considerably simplified FDM for anomalous transport of uniformly sized grains along straight channels, the subordinated advection equation (SAE), which is based on the concept of time subordination. Unlike previous FDM models with time index between 0 and 1, our SAE model adopts a value of between 1 and 2. This describes random velocities deviating significantly from the mean velocity and models both long resting periods and relatively fast displacements. We show that the model quantifies the dynamics of four bed load transport experiments recorded in the literature. In addition to , SAE model parameters-velocity and capacity coefficient-are related to the mean and variance of particle velocities, respectively. Successful application of the SAE model also implies a universal probability density for the heavy-tailed waiting time distribution (with finite mean) and a relatively lighter tailed step length distribution for uniform bed load transport from local to regional scales.

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Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media

Cited by 134 publications

References 50 publications

Fractal advection-dispersion equation for groundwater transport in fractured aquifers with self-similarities

Fractal advection-dispersion equation for groundwater transport in fractured aquifers with self-similarities

Generalized Boundary Conditions for the Time-Fractional Advection Diffusion Equation

A subordinated advection model for uniform bed load transport from local to regional scales

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