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 vari able velocity fields, such as exist in geologically heterogeneous formation s. Considering a spectral representation of the velocity, it is recognized that numerical models usually capture low-wavenumber effects, while the lar ge-wavenumber effects, associated with subgrid block variability, are suppr essed. While this suppression is avoidable if the heterogeneity is captured at minute detail, that goal is impossible to achieve in all but the most t rivial cases. Representing the effects of the suppressed variability in the models is made possible using the proposed concept of block-effective macr odispersivity. A tensor is developed, which we refer to as the block-effect ive macrodispersivity tensor, whose terms are functions of the characterist ic length scales of heterogeneity, as well as the length scales of the mode l's homogenized areas, or numerical grid blocks. Closed-form expressions ar e developed for small variability in the log-conductivity and unidirectiona l mean flow, and are tested numerically. The use of the block-effective mac rodispersivities allows conditioning of the velocity field on the measureme nts on the one hand, while accounting for the effects of unmodelled heterog eneity on the other, in a numerically reasonable set-up. It is shown that t he effects of the grid scale are similar to those of the plume scale in ter ms 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 ac count for the scale of the solute body and the pore-scale dispersion.