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Finite Difference Approximations for Fractional Advection-Dispersion Flow Equations
, 2004
"... Fractional advection-dispersion equations are used in groundwater hydrology tomqU- the transport of passive tracers carried by fluid flow in a porous mrousq In this paper we develop practical numtical mumti to solve one dimUEBDqyU fractional advection-dispersion equations with variable coefficients ..."
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Cited by 96 (10 self)
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Fractional advection-dispersion equations are used in groundwater hydrology tomqU- the transport of passive tracers carried by fluid flow in a porous mrousq In this paper we develop practical numtical mumti to solve one dimUEBDqyU fractional advection-dispersion equations with variable coefficients on a finite domeqV The practical application of these results is illustrated by mqUEIB# a radial flow problem Use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site.
Radial fractionalorder dispersion through fractured rock
- Water Resources Research
, 2004
"... [1] A solute transport equation with a fractional-order dispersion term is a model of solute movement in aquifers with very wide distributions of velocity. The equation is typically formulated in Cartesian coordinates with constant coefficients. In situations where wells may act as either sources or ..."
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Cited by 5 (1 self)
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[1] A solute transport equation with a fractional-order dispersion term is a model of solute movement in aquifers with very wide distributions of velocity. The equation is typically formulated in Cartesian coordinates with constant coefficients. In situations where wells may act as either sources or sinks in these models, a radial coordinate system provides a more natural framework for deriving the resulting differential equations and the associated initial and boundary conditions. We provide the fractional radial flow advection-dispersion equation with nonconstant coefficients and develop a stable numerical solution using finite differences. The hallmark of a spatially fractional-order dispersion term is the rapid transport of the leading edge of a plume compared to the classical Fickian model. The numerical solution of the fractional radial transport equation is able to reproduce the early breakthrough of bromide observed in a radial tracer test conducted in a fractured granite aquifer. The early breakthrough of bromide is underpredicted by the classical radial transport model. Another conservative, yet nonnaturally occurring solute (pentaflourobenzoate), also shows early breakthrough but does not conclusively support the bromide model due to poor detection at very low concentrations. The solution method includes, through a procedure called subordination, the effects of solute partitioning on immobile water.
Acknowledgments
, 2015
"... Fractional derivatives were invented by Leibniz soon after their integer-order cousins. Recently, they have found practical appli-cations in many areas of science and engineering. Fractional dif-fusion equations replace the integer derivatives in the traditional diffusion equation with their fractio ..."
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Fractional derivatives were invented by Leibniz soon after their integer-order cousins. Recently, they have found practical appli-cations in many areas of science and engineering. Fractional dif-fusion equations replace the integer derivatives in the traditional diffusion equation with their fractional counterparts. These mod-els for anomalous diffusion govern limits of continuous time ran-dom walks, where a random waiting time separates random parti-cle jumps. A power law probability distribution for particle jumps leads to fractional derivatives in space. Power law waiting times correspond to time-fractional derivatives. Particle traces are ran-dom fractals, whose dimension relates to the orders of the frac-tional derivatives. Parameter estimation requires novel statisti-cal techniques, since power law data contains many outliers, and these tail values are the most important feature of the data. Nu-merical methods apply nonlocal variants of standard Euler finite
Article Diffusion in Relatively Homogeneous Sand Columns: A Scale-Dependent or Scale-Independent Process?
, 2013
"... entropy ..."
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"... A stochastic fractal model of the Universe related to the fractional Laplacian ..."
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A stochastic fractal model of the Universe related to the fractional Laplacian
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"... The effect of non-Fickian diffusion into surrounding rocks on contaminant transport in a fractured porous aquifer ..."
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The effect of non-Fickian diffusion into surrounding rocks on contaminant transport in a fractured porous aquifer
