Transport upscaling in heterogeneous aquifers: What physical parameters control memory functions?

Power law tailing is often observed in the breakthrough curves (BTCs) of tracer tests. Tailing is attributed to heterogeneity of aquifer properties and cannot be properly modeled by means of the homogeneous advection-dispersion equation. Mass transfer models (e. g., continuous time random walk method, multirate mass transfer, or fractional-order advection-dispersion equations) using memory have been widely applied for reproducing observed tails. The relationship between memory parameters obtained from BTC fitting and the parameters characterizing the heterogeneity of hydraulic properties is still unclear. Here we investigate the conditions under which heterogeneity produces the type of tailing observed in the field and how memory functions are influenced by measurable heterogeneity parameters (e. g., variance, variogram, or integral scale of the underlying transmissivity field). We find that the slope of a BTC in a log-log plot is mainly influenced by the connectivity of the underlying permeability field but is insensitive to its variance. The slope BTC reaches asymptotically 2 as connectivity increases. We conclude that an appropriate choice of the memory function allows reproducing the spreading caused by hydraulic heterogeneity but not necessarily the rate of mixing.