Time behavior of solute transport in heterogeneous media: transition from anomalous to normal transport

We study the time behavior of solute transport in a heterogeneous medium. We consider a spatially biased continuous time random walk (CTRW) governed by ψ(s,t), the joint probability density for an event–displacement s with an event–time t. In this effective transport framework the concentration distribution of a solute is given by a generalized master equation (GME). We present results detailing the time dependence for the resident and flux concentrations, the center of mass velocity and the macroscopic dispersion coefficients of the solute plume. The Laplace transform of the GME is converted to a spatial differential equation resembling the advection dispersion equation (ADE), with Laplace space dependent coefficients though, and can then be solved explicitly in Laplace space. The transport behavior of the solute is then determined by accurate numerical inverse Laplace transforms. The confirmation of the accuracy of our methods is demonstrated by the excellent agreement with efficient random walk simulations based on the same ψ(s,t). The ψ(s,t) is given by the product of a Gaussian distribution for s and a truncated power-law distribution for t. This particular choice allows for a systematic study of the time regimes of anomalous and normal transport behavior and the transition from normal to anomalous behavior. The presented results show new aspects for the modeling of solute transport in heterogeneous media, in particular the effect of the system “memory” on plume patterns at asymptotically long times. In a specific application we solve for the contaminant flux entering a stream from a point injection of tracer in a catchment. The results are discussed as an independent test of a model of fractal stream chemistry in catchments.