# Stephen W. Wheatcraft

The traditional 2ndorder advectiondispersion equation (ADE) does not adequately describe the movement of solute tracers in aquifers. This study examines and rederives the governing equation. The analysis starts with a generalized notion of particle movements, since the secondorder equation is trying to impart Brownian motion on a mathematical plume at any(More)
The macrodispersion experiments (MADE) at the Columbus Air Force Base in Mississippi 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 transport equation that uses fractional-order dispersion derivatives has fundamental solutions that are Lévy’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.(More)
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 Lévy’s a-stable densities that resemble(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)
The traditional conservation of mass equation is derived using a first-order Taylor series to represent flux change in a control volume, which is valid strictly for cases of linear changes in flux through the control volume. We show that using higher-order Taylor series approximations for the mass flux results in mass conservation equations that are(More)
• Journal of contaminant hydrology
• 2001
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)
We develop the basic tools of fractional vector calculus including a fractional derivative version of the gradient, divergence, and curl, and a fractional divergence theorem and Stokes theorem. These basic tools are then applied to provide a physical explanation for the fractional advection–dispersion equation for flow in heterogeneous porous media. r 2005(More)
• Ground water
• 2005
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)
• 1