Eulerian derivation of the fractional advection–dispersion equation

Schumer, R.; Benson, David A.; Meerschaert, Mark M.; Wheatcraft, Stephen W.

doi:10.1016/s0169-7722(00)00170-4

Cited by 339 publications

(204 citation statements)

References 33 publications

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“…(3) On taking ν = 1, η = ν, λ = 2 and g(x) = C o (x) in Theorem 3.1, we obtain following result which was considered by Liu et al [12] ∂ 20) subject to the initial condition…”

Section: )mentioning

confidence: 54%

“…There are some other authors who considered the fractional ADE. A space fractional ADE with Eulerian derivation was derived by Schumer et al [20] which is used to describe solute plume evolution with a large probability of particles moving significantly ahead of and behind the mean solute velocity. The physical interpretation of space-time fractional advection-dispersion equation (FADE) is given by Schumer et al [20].…”

Section: Introductionmentioning

confidence: 99%

“…A space fractional ADE with Eulerian derivation was derived by Schumer et al [20] which is used to describe solute plume evolution with a large probability of particles moving significantly ahead of and behind the mean solute velocity. The physical interpretation of space-time fractional advection-dispersion equation (FADE) is given by Schumer et al [20]. A space-time fractional advection-dispersion equation (FADE) is a generalization of the classical ADE in which the first-order space derivative is replaced with Hilfer composite fractional derivatives (see Hilfer [8]) of order 0 < µ < 1 and 0 ≤ ν ≤ 1 and the second-order space derivative is replaced with the space fractional Laplacian operator of order 0 < α ≤ 2.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Analytic solution of generalized space time advection-dispersion equation with fractional Laplace operator

Jain¹,

Agarwal²

2016

J. Nonlinear Sci. Appl.

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“…(3) On taking ν = 1, η = ν, λ = 2 and g(x) = C o (x) in Theorem 3.1, we obtain following result which was considered by Liu et al [12] ∂ 20) subject to the initial condition…”

Section: )mentioning

confidence: 54%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Analytic solution of generalized space time advection-dispersion equation with fractional Laplace operator

Jain¹,

Agarwal²

2016

J. Nonlinear Sci. Appl.

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“…[9] A generalized Fick's law [Paradisi et al, 2001;Schumer et al, 2001;Kim and Kavvas, 2006] indicates that the dispersive flux F is proportional to a fractional derivative of solute concentration:…”

Section: Model 1: the Ff-adementioning

confidence: 99%

Space‐fractional advection‐dispersion equations with variable parameters: Diverse formulas, numerical solutions, and application to the Macrodispersion Experiment site data

Zhang

Benson

Meerschaert

et al. 2007

Water Resources Research

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126

123

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[1] To model the observed local variation of transport speed, an extension of the homogeneous space-fractional advection-dispersion equation (fADE) to more general cases with space-dependent coefficients (drift velocity V and dispersion coefficient D) has been suggested. To provide a rigorous evaluation of this extension, we explore the underlying physical meanings of two proposed, and one other possible form, of the fADE by using the generalized mass balance law proposed by Meerschaert et al. (2006). When the classical Fick's law is replaced by its generalized form in the first-order mass conversation law, the original fADE with constant parameters extends to the advection-dispersion equation (ADE) with fractional flux (denoted as FF-ADE). When the net inflow of dispersive flux is from nonlocal concentration gradients following a fractional divergence, we get the ADE with fractional divergence (FD-ADE). When the total net inflow from both nonlocal advection and nonlocal concentration gradients follows a fractional form, the fADE contains fully fractional divergence (FFD-ADE). These three fADEs with constant parameters can also be obtained by proper choice of the two memory kernels in the nonlocal dispersive constitutive theory proposed by Cushman et al. (1994), while the space-variable fADEs correspond to the conditional nonlocal theory proposed by Neuman (1993) after specifying the general (Lévy-type) functional form of the random distribution. The corresponding Langevin Markov models can be found in many cases, where the Lagrangian stochastic processes can be conditioned directly on local aquifer properties at any practical, measurable level and resolution. The resulting Lagrangian random-walk particle tracking methods, along with previous numerical solutions using implicit Euler finite differences, distinguish and elucidate the plume behavior described by these fADEs. The fractional models are applied to fit the tritium plumes measured at the Macrodispersion Experiment test site. When the local parameters gleaned from the hydraulic conductivity (K) distribution are varied even slightly (i.e., a two-zone model), allowing the use of observed hydraulic conditions at the site, the model fits are well within the variability of the data. The extended fADEs describe the fast and space-dependent leading edges of measured plumes in the regional-scale alluvial system, which was underestimated and could not be fully captured by the original fADE with constant parameters. Applications also favor the FD-ADE model because of the ease of implementation and consistency with previous analysis of the K statistics.

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“…Recently, scientists found that the classical Fick's law (1.3) is not adequate to describe anomalous diffusion which occurred in field and laboratory studies, such as transport of a solute in heterogeneous porous media. Instead, a fractional Fick's law is proposed [22] (1.4)…”

Section: Introductionmentioning

confidence: 99%

Efficient spectral–Galerkin methods for fractional partial differential equations with variable coefficients

Mao

Shen

2016

Journal of Computational Physics

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Eulerian derivation of the fractional advection–dispersion equation

Cited by 339 publications

References 33 publications

Analytic solution of generalized space time advection-dispersion equation with fractional Laplace operator

Analytic solution of generalized space time advection-dispersion equation with fractional Laplace operator

Space‐fractional advection‐dispersion equations with variable parameters: Diverse formulas, numerical solutions, and application to the Macrodispersion Experiment site data

Efficient spectral–Galerkin methods for fractional partial differential equations with variable coefficients

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