A transport equation that uses fractional-order dispersion derivatives has fundamental solutions that are L6vy's a-stable densities. These densities represent plumes that spread proportional to time •/•, have heavy tails, and incorporate any degree of skewness. The equation is parsimonious since the dispersion parameter is not a function of time or distance. The scaling behavior of plumes that undergo L6vy motion is accounted for by the fractional derivative. A laboratory tracer test is described by a dispersion term of order 1.55, while the Cape Cod bromide plume is modeled by an equation of order 1.65 to 1.8. tions that use fractional-order derivatives. The motions can be heavy-tailed, implying extremely long-term correlation and fractional derivatives in time [Giona and Roman, 1992; Compte, 1996] and/or space [e.g., Gorenrio and Mainardi, 1998; Saichev and Zaslavsky, 1997; Benson, 1998; Chaves, 1998; Meerschaert et al., 1999]. Second-order dispersion arises when tails of this distribution are sufficiently "thin" that relative to an observation or measurement event, time-consuming or very large motions are effectively ruled out. Typically, a sufficient number of thintailed motions must also be integrated before DeMoive's central limit theorem is a good assumption and before ergodicity and Fickian transport ensue. During this pre-Fickian phase of transport, scale-dependent dispersion coefficients can be used in a local ADE, although a nonlocal formulation is a more accurate model of these motions [Neuman, 1993; Cushman et al., 1994]. Solutions of certain nonlocal equations are gained via fast Fourier and Laplace transforms for relatively homogeneous media [e.g., Deng et al., 1993]. dispersion tensor is a very handy predictive equation, since solutions are easily gained. The fractional-order forms of the ADE are similarly useful. General nonlocal forms of the second-order ADE are less so, since solutions must be gained by numerical convolution. Similarly, explicit numerical modeling by generation of random K fields and subsequent Monte Carlo simulation is a very time-consuming process that is commonly used to gain information about a plume's pre-Fickian behavior. The fractional ADE used herein is spatially nonlocal and models particles that experience very large transitions. The large transitions may arise from high heterogeneity [Benson et al., 2000] and very long spatial autocorrelation [ Benson, 1998]. If the transition probabilities follow a power law, then analytic solutions of the fractional ADE may serve as good stand-alone models of transport throughout a plume's development. In this study we examine whether the fractional-order ADE is also a useful model for transport in relatively homogeneous material. Evidence of Fractional, or a-Stable, Behavior in Relatively Homogeneous MaterialThe fractional ADE predicts concentration versus time and distance in closed form, once the scaled a-stable density (fundamental solution) is known. In this spirit, two experiments are analyzed in the simplest way possible to ...

[1] A fractal mobile/immobile model for solute transport assumes power law waiting times in the immobile zone, leading to a fractional time derivative in the model equations. The equations are equivalent to previous models of mobile/immobile transport with power law memory functions and are the limiting equations that govern continuous time random walks with heavy tailed random waiting times. The solution is gained by performing an integral transform on the solution of any boundary value problem for transport in the absence of an immobile phase. In this regard, the output from a multidimensional numerical model can be transformed to include the effect of a fractal immobile phase. The solutions capture the anomalous behavior of tracer plumes in heterogeneous aquifers, including power law breakthrough curves at late time, and power law decline in the measured mobile mass. The MADE site mobile tritium mass decline is consistent with a fractional time derivative of order g = 0.33, while Haggerty et al. 's [2002] stream tracer test is well modeled by a fractional time derivative of order g = 0.3.

Abstract. A governing equation of stable random walks is developed in one dimension.This Fokker-Planck equation is similar to, and contains as a subset, the second-order advection dispersion equation (ADE) except that the order (a) of the highest derivative is fractional (e.g., the 1.65th derivative). Fundamental solutions are Ldvy's a-stable densities that resemble the Gaussian except that they spread proportional to time •/", have heavier tails, and incorporate any degree of skewness. The measured variance of a plume undergoing Ldvy motion would grow faster than Fickian plume, at a rate of time 2/", where 0 < a <_ 2. The equation is parsimonious since the parameters are not functions of time or distance. The scaling behavior of plumes that undergo Ldvy motion is accounted for by the fractional derivatives, which are appropriate measures of fractal functions. In real space the fractional derivatives are integrodifferential operators, so the fractional ADE describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space. Solute transport in subsurface material may be viewed as a purely probabilistic problem [e.g., Cushman, 1990; Bhattacharya and Gupta, 1990]. This approach is intimately tied to the classical divergence (Eulerian) point of view through a string of mathematical equivalences. Einstein [1956] first explored this method by assuming that a single microscopic particle was continuously bombarded by other particles, resulting in Brownian random walk. By assuming a finite variance process and taking appropriate limits, he found that the resulting Green's function of the probability of finding a particle somewhere in space was a Gaussian (Normal) probability density. If the motions of a large number of particles are assumed independent, then the particle probability and the concentration of a diffusing tracer are interchangeable, leading to the equivalence of the process of Brownian motion and the governing equation OC/Ot = •V2C. The most important assumption tied to Brownian motion and the second-order, Fickian diffusion equation is that a particle's motion has little or no spatial correlation. Since long walks in the same direction are rare, the variance of a particle's excursion distance is finite.The classical descriptions of a local dispersion tensor based on the second-order diffusion equation [Gelhat and Axness, 1983;Dagan, 1984Dagan, , 1988 Neumann and Zhang, 1990] carry similar assumptions: The aquifer velocity contrasts must be small and the mean travel distance must be large compared to a typical velocity correlation length. These assumptions arise 1413

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