Solute dispersion by 1D stepped velocity fluctuations

Abstract

The effect of fluctuations in the drift velocity on dispersion by a porous medium is investigated. An analytical model is developed which represents the effect of a single discrete step in the velocity of a 1 dimensional flow as a multiplicative factor that modifies the underlying linear growth in solute variance predicted by the standard advection-dispersion equation. The algebraic structure of the model identifies two variable combinations ∆ and α that characterize the step and the rate of stochastic dispersion respectively, in terms of which a simple formula for the downstream effect of the step on dispersion is obtained. This formalism is next applied to a sequence of 3 steps representing a velocity fluctuation, and it is shown that while kinetic compression effects cancel out across such a fluctuation, the stochastic dispersion increases for any plausible combination of ∆ and α. This implies that a dispersion enhancement factor ƒ is associated with a fluctuation, and a simple formula is obtained for this in terms of variables that describe the fluctuation length and amplitude. Moreover, the algebra leads to the definition of a natural length scale Λ related to the Peelet number of the flow. Repeated application of this formula is used to find the cumulative dispersion enhancement by a sequence of identical fluctuations, leading to an expression for dispersivity as a function of the distance traversed by a solute plume. Key features of the model are that the dispersivity behaves differently for traversal lengths above and below Λ, and that above this transition it is proportional to a fractional power of the traversal length. These features are in agreement with experimental observation of scale-dependent dispersivity, but quantitatively the observed growth in dispersivity over several orders of magnitude is not obtained for any reasonable choice of parameter values.