A comparative review of upscaling methods for solute transport in heterogeneous porous media

The classical Fickian model for solute transport in porous media cannot correctly predict the spreading (the dispersion) of contaminant plumes in a heterogeneous subsurface unless its structure is completely characterized. Although the required precision is outside the reach of current field characterization methods, the advection–dispersion model remains the most widely used model among practitioners. Two approaches can be adopted to solve the effect of physical heterogeneity on transport. First, based on a given characterization of the spatial structure of the subsurface, upscaling methods allow the computation of apparent scale-dependent parameters (especially longitudinal dispersivity) to be used in the classical Fickian model. In the second approach, upscaled (non-Fickian) transport equations with scale-independent parameters are used. In this paper, efforts are made to classify and review upscaling methods for Fickian transport parameters and non-Fickian upscaled transport equations for solute transport, with an emphasis on their mathematical properties and their (one-dimensional) analytical formulations. In particular, their capacity to model scale effects in apparent longitudinal dispersion is investigated. Upscaling methods and upscaled models are illustrated in the case of two three-dimensional synthetic aquifers, with lognormal hydraulic conductivity distributions characterized by variance values of 2 and 8.