Upscaling retardation factors in 2-D porous media

Even under the simple linear isotherm adsorption model, the parameters controlling adsorption under field conditions are frequently approximated by the values derived from batch experiments. First, the measurement scale and conditions are very different from those at the model scale. Second, the parameters are heterogeneous in space and, at most, there is some information about them at a few locations within an aquifer. For these two reasons, there is a need to consider how to treat the heterogeneity of the parameters that control adsorption, i.e. the retardation factor or distribution coefficient, and a need to establish upscaling rules to transfer the information about parameters at the measurement scale to those at the scale of model grid blocks. This paper presents some best upscaling rules for the distribution coefficient accounting for different heterogeneous structures for both the hydraulic conductivity and the distribution coefficient. Exhaustive numerical simulations are carried out by combining different heterogeneity patterns of the hydraulic conductivity and the distribution coefficient, the cross correlation between them, overall degree of variability, and time dependence. It is demonstrated that under certain conditions, e.g. large variances and small correlation lengths of hydraulic conductivity InK(x) and distribution coefficient InKd(x), the geometric mean is a good approximation for the upscaled retardation factor.