Do Hydrodynamic Dispersion Coefficients Exist?

Abstract

Tracer dispersion is the simplest example of the flow of an inhomogeneous two-component mixture. We assume that the two components, the “tracer” and the “solvent,” have identical behavior but are distinguishable in the sense that they are assigned different “colors.” The model may be simple, but it is neither trivial nor irrelevant. For instance, the solvent can represent a pure fluid and the tracer a contaminant. If the solvent occupies the space within some porous medium and is stationary, the tracer has a diffusion coefficient somewhat less than its value in the pure solvent (the latter we denote as D0). If, however, the solvent flows through the porous medium with some mean flow velocity ¯ v, then, depending on the magnitude of ¯ v, the dispersion coefficient D (defined as the tracer diffusion coefficient in a frame of reference moving with the mean flow velocity) can be very much greater than D0. Understanding this phenomenon of “hydrodynamic dispersion” [1] is obviously important for modeling pollutant transport in ground water. Hydrodynamic dispersion arises when the variations in the local fluid velocity spread the tracer over a larger volume than one would expect by diffusion alone (this is nicely illustrated in Ref. [2]). The Peclet number Pe is a measure of the relative importance of diffusion and convection. It is defined as Pe ; U p l p yD0, where U p and l p are, respectively, a characteristic velocity and a characteristic length scale. At high values of Pe, tracer transport over distances larger than l p is dominated by convection, and dispersion is dominated by the variations