Multiscaling fractional advection‐dispersion equations and their solutions
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The multiscaling fractional advection-dispersion equation (ADE) as discussed by the authors is a multidimensional model of solute transport that encompasses linear, Fickian, and super-Fickian dispersion.Abstract:
[1] The multiscaling fractional advection-dispersion equation (ADE) is a multidimensional model of solute transport that encompasses linear advection, Fickian dispersion, and super-Fickian dispersion. The super-Fickian term in these equations has a fractional derivative of matrix order that describes unique plume scaling rates in different directions. The directions need not be orthogonal, so the model can be applied to irregular, noncontinuum fracture networks. The statistical model underlying multiscaling fractional dispersion is a continuous time random walk (CTRW) in which particles have arbitrary jump length distributions and finite mean waiting time distributions. The meaning of the parameters in a compound Poisson process, a subset of CTRWs, is used to develop a physical interpretation of the equation variables. The Green's function solutions are the densities of operator stable probability distributions, the limit distributions of normalized sums of independent, and identically distributed random vectors. These densities can be skewed, heavy-tailed, and scale nonlinearly, resembling solute plumes in granular aquifers. They can also have fingers in any direction, resembling transport along discrete pathways such as fractures.read more
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Finite difference approximations for fractional advection-dispersion flow equations
TL;DR: In this paper, the authors developed practical numerical methods to solve one dimensional fractional advection-dispersion equations with variable coefficients on a finite domain and demonstrated the practical application of these results is illustrated by modeling a radial flow problem.
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Fractal mobile/immobile solute transport
TL;DR: In this paper, a fractal mobile/immobile model for solute transport with power law waiting times in the immobile zone was proposed, leading to a fractional time derivative in the model equations, which captures the anomalous behavior of tracer plumes in heterogeneous aquifers.
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Finite difference methods for two-dimensional fractional dispersion equation
TL;DR: In this article, a practical alternating directions implicit method to solve a class of two-dimensional initial-boundary value fractional partial differential equations with variable coefficients on a finite domain is discussed.
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Perspective on theories of non-Fickian transport in heterogeneous media
TL;DR: In this paper, the authors focus on four approaches that give rise to nonlocal representations of advective and dispersive transport of nonreactive tracers in randomly heterogeneous porous or fractured continua.
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A second-order accurate numerical method for the two-dimensional fractional diffusion equation
TL;DR: This numerical method combines the alternating directions implicit (ADI) approach with a Crank-Nicolson discretization and a Richardson extrapolation to obtain an unconditionally stable second-order accurate finite difference method.
References
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Book
Convergence of Probability Measures
TL;DR: Weak Convergence in Metric Spaces as discussed by the authors is one of the most common modes of convergence in metric spaces, and it can be seen as a form of weak convergence in metric space.
Book
Dynamics of fluids in porous media
TL;DR: In this paper, the Milieux poreux Reference Record was created on 2004-09-07, modified on 2016-08-08 and the reference record was updated in 2016.
An Introduction To Probability Theory And Its Applications
TL;DR: A First Course in Probability (8th ed.) by S. Ross is a lively text that covers the basic ideas of probability theory including those needed in statistics.
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