Fractional dispersion, Lévy motion, and the MADE tracer tests

Abstract. 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 plume development at the MADE site. The new governing equation is based on a fractional, rather than integer, order of differentiation. This order (), based on MADE plume measurements, is approximately 1.1. The hydraulic conductivity (K) increments also follow a power law of order D 1:1: We conjecture that the heavy-tailed K distribution gives rise to a heavy-tailed velocity field that directly implies the fractional-order governing equation derived herein. Simple arguments lead to accurate estimates of the velocity and dispersion constants based only on the aquifer hydraulic properties. This supports the idea that the correct governing equation can be accurately determined before, or after, a contamin-ation event. While the traditional ADE fails to model a conservative tracer in the MADE aquifer, the fractional equation predicts tritium concentration profiles with remarkable accuracy over all spatial and temporal scales. Key words: fractional derivative, fractional Laplacian, anomalous dispersion, Lévy motion, -stable, heavy tails, Fokker–Planck equation, MADE site