The effects of heterogeneity on solute transport in porous media: anomalous dispersion

Mixing of solute in sub-surface flows, for example in the leakage of contaminants into groundwater, is quantified by a dispersion coefficient that depends on the dispersivity of the medium and the transport velocity. Many previous models of solute transport data have assumed Fickian behaviour with constant dispersivity, but found that the inferred dispersivity increases with the distance from the point of injection. This approach assumes that the dispersion on either side of the advective front is symmetric and that the concentration there is close to half of the injected value. However, field data show consistent asymmetry about the advective front. Here, motivated by experimental data and a fractal interpretation of porous media, we consider a simplified heterogeneous medium described by a dispersivity with a power-law dependence on the downstream distance from the source and explore the nature of the asymmetry obtained in the solute transport. In a heterogeneous medium of this type, we show that asymmetry in solute transport gradually increases with the increase in heterogeneity or fractal dimension, and the concentration at the advective front becomes increasingly different from 50% of that at the inlet. In particular, for a sufficiently heterogeneous medium, the concentration profiles at late-time approach a non-trivial steady solution and, as a result, the concentration at a downstream location will never reach that at the inlet. By fitting the Fickian solution to our results, we are able to connect the parameters from our model to those found from experimental data, providing a more physically grounded approach for interpreting them.