Stephen W Wheatcraft

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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 (␣) of the highest derivative is fractional (e.g., the 1.65th derivative). Fundamental solutions are Lévy's ␣-stable densities that resemble(More)
A fractional advection-dispersion equation (ADE) is a generalization of the classical ADE in which the second-order derivative is replaced with a fractional-order derivative. In contrast to the classical ADE, the fractional ADE has solutions that resemble the highly skewed and heavy-tailed breakthrough curves observed in field and laboratory studies. These(More)
A transport equation that uses fractional-order dispersion derivatives has fundamental solutions that are Lévy's ␣-stable densities. These densities represent plumes that spread proportional to time 1/␣ , have heavy tails, and incorporate any degree of skewness. The equation is parsimonious since the dispersion parameter is not a function of time or(More)
The macrodispersion experiments (MADE) at the Columbus Air Force Base in Missis-sippi were conducted in a highly heterogeneous aquifer that violates the basic assumptions of local second-order theories. A governing equation that describes particles that undergo Lévy motion, rather than Brownian motion, readily describes the highly skewed and heavy-tailed(More)
A mathematical method called subordination broadens the applicability of the classical advection-dispersion equation for contaminant transport. In this method, the time variable is randomized to represent the operational time experienced by different particles. In a highly heterogeneous aquifer, the operational time captures the fractal properties of the(More)
Alternative fractional models of contaminant transport lead to a new travel time formula for arbitrary concentration levels. For an evolving contaminant plume in a highly heterogeneous aquifer, the new formula predicts much earlier arrival at low concentrations. Travel times of contaminant fronts and plumes are often obtained from Darcy's law calculations(More)
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