[1] Space-fractional advection-dispersion models provide attractive alternatives to the classical advection-dispersion equation for model applications that exhibit early arrivals and plume skewness. This paper develops a flexible method for estimating the parameters of the fractional transport model on the basis of spatial plume snapshots or temporal breakthrough curve data. A particle-tracking approach provides error bars for the parameter estimates and a general method for model fitting and comparison via optimal weighted least squares. A simple model of concentration variance, based on the particletracking approach, identifies the optimal weights.

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A fractional advection-dispersion equation (fADE) has been advocated for heavytailed flows where the usual Brownian diffusion models fail. A stochastic differential equation (SDE) driven by a stable Lévy process gives a forward equation that matches the spacefractional advection-dispersion equation and thus gives the stochastic framework of particle tracking for heavy-tailed flows. For constant advection and dispersion coefficient functions, the solution to such SDE itself is a stable process and can be derived easily by least square parameter fitting from the observed flow concentration data. However, in a more generalized scenario, a closed form for the solution to a stable SDE may not exist. We propose a numerical method for solving/generating a stable SDE in a general set-up. The method incorporates a discretized finite volume scheme with the characteristic line to solve the fADE or the forward equation for the Markov process that solves the stable SDE. Then we use a numerical scheme to generate the solution to the governing SDE using the fADE solution. Also, often the functional form of the advection or dispersion coefficients are not known for a given plume concentration data to start with. We use a Levenberg-Marquardt (L-M) regularization method to estimate advection and dispersion coefficient function from the observed data (we present the case for a linear advection) and proceed with the SDE solution construction described above.