Concentration statistics for transport in heterogeneous media due to stochastic fluctuations of the center of mass velocity

We study concentration statistics for tracer transport in the flow through a heterogeneous porous medium. Spatial heterogeneity and temporal variability cause fluctuations in spatial characteristics of the tracer distribution such as the center of mass velocity v(t) and the spreading rate De(t), which leads to fluctuations in the concentration. These fluctuations can be quantified statistically by the one-point probability density function (PDF) of concentration and by its simplest characteristics, the concentration mean and variance. We focus on the impact of variability in the center of mass velocity and treat De(t) as deterministic. The fluctuating v(t) is modeled as a Gaussian stochastic process. Based on this stochastic description, we derive explicit expressions for the concentration variance and the full one-point concentration PDF. Their behavior is mainly controlled by the ratio of the spatial variance of the tracer distribution κe(t) in single realizations and the variance κm(t) of the plume centroids between realizations. We explore the behavior of concentration variance and PDF for two velocity correlation models, one that leads to a diffusive increase of centroid variance κm(t) and one that yields an asymptotically constant κm(t). The obtained results give insight into the dynamics of concentration statistics in heterogeneous media and the mechanisms by which they are controlled.