Temporal moment analysis for stochastic-advective vertical solute transport in heterogeneous unsaturated soils

Temporal moment analysis of solute transport in unsaturated soils subjected to rainfall events is typically achieved by the numerical solution of the flow field from the Richards equation followed by a numerical solution of the advection-dispersion equation before computing moments. These numerical solutions are computationally very intensive, and may not provide the insights that are possible from simpler analytical representations. In this study, temporal moments of solute transport for unsteady unsaturated flows under rainfall conditions at the soil surface are presented for the first time. A local-scale model for water movement is derived from a sharp front approximation and is combined with a model for transport of solute particles along the main characteristics of the flow field. Expressions for travel times from the local-scale model are first presented for pre- and post-ponding conditions. These local solutions are upscaled to field-scale solute transport by adopting a log-normally distributed spatial hydraulic conductivity field. Semi-analytical expressions are developed for temporal moments of travel times. These expressions are compared to 1-D Monte Carlo simulation results, and to 3-D numerical results for model corroboration. The model is used to investigate the behavior of macroscopic Eulerian effective velocities and dispersion at the field-scale. Expressions for asymptotic effective properties show that effective velocity achieves a constant value while effective dispersion increases linearly with depth. The roles of pre-ponding and post-ponding conditions in determining field-scale dispersion are described.