Pictures of Blockscale Transport: Effective versus Ensemble Dispersion and Its Uncertainty

Using a Lagrangian framework, we discuss different conceptualizations of the blockscale dispersion tensor. We distinguish effective and ensemble blockscale dispersion, which measure the impact of subscale velocity fluctuations on solute dispersion. Ensemble dispersion quantifies subscale velocity fluctuations between realizations, which overestimates the actual velocity variability. Effective dispersion on the other hand quantifies the actual blockscale velocity variability and thus reflects the impact of subscale velocity fluctuations on mixing and spreading. Based on these concepts, we quantify the impact of subscale velocity fluctuations on solute particle spreading and determine the governing equations for the coarse-grained concentration distributions. We develop analytical and semi-analytical expressions for the average and variance of the blockscale dispersion tensor in 3D flow fields as a function of the structural parameters characterizing the subsurface. Our results illustrate the relevance of the blockscale, the initial scale of the solute body and local-scale dispersion in controlling the uncertainty of the plume's dispersive behavior. The analysis performed in this work has implications in numerical modeling (i.e. grid design) and allows to quantify the uncertainty of the blockscale dispersion tensor.