We present a novel microscopic model of sorption and convection of ions in heterogeneous media. Our model is based on an analogy to electron transport in a semiconductor. A new feature of our model is a power law random distribution of the adsorption time of ions. Diverging standard deviation of the distribution function yields anomalous ion transport. We show that this anomalous transport explains a concentration profile with a long tail that has been observed in column experiments. We successfully fit recent experimental data. Finally, we propose new experiments by which we can check the validity of our model. ¿ 1998 American Geophysical Union