The Concept of Block Effective Macrodispersivity and a Unified Approach for Grid-scale and Plume-scale Dependent Transport

We present a new approach for modelling macrodispersivity in spatially variable velocity fields, such as exist in geologically heterogeneous formations. Considering a spectral representation of the velocity, it is recognized that numerical models usually capture low-wavenumber effects, while the large-wavenumber effects, associated with subgrid block variability, are suppressed. While this suppression is avoidable if the heterogeneity is captured at minute detail, that goal is impossible to achieve in all but the most trivial cases. Representing the effects of the suppressed variability in the models is made possible using the proposed concept of block-effective macrodispersivity. A tensor is developed, which we refer to as the block-effective macrodispersivity tensor, whose terms are functions of the characteristic length scales of heterogeneity, as well as the length scales of the model's homogenized areas, or numerical grid blocks. Closed-form expressions are developed for small variability in the log-conductivity and unidirectional mean flow, and are tested numerically. The use of the block-effective macrodispersivities allows conditioning of the velocity field on the measurements on the one hand, while accounting for the effects of unmodelled heterogeneity on the other, in a numerically reasonable set-up. It is shown that the effects of the grid scale are similar to those of the plume scale in terms of filtering out the effects of portions of the velocity spectrum. Hence it is easy to expand the concept of the block-effective dispersivity to account for the scale of the solute body and the pore-scale dispersion.