Stochastic analysis of transport in hillslopes: Travel time distribution and source zone dispersion

A stochastic model is developed for the analysis of the traveltime distribution f(tau) in a hillslope. The latter is described as made up from a surficial soil underlain by a less permeable subsoil or bedrock. The heterogeneous hydraulic conductivity K is described as a stationary random space function, and the model is based on the Lagrangian representation of transport. A first-order approach in the log conductivity variance is adopted in order to get closed form solutions for the principal statistical moments of the traveltime. Our analysis indicates that the soil is mainly responsible for the early branch of f(tau), i.e., the rapid release of solute which preferentially moves through the upper soil. The early branch of f(tau) is a power law, with exponent variable between -1 and -0.5; the behavior is mainly determined by unsaturated transport. The subsoil response is slower than that of the soil. The subsoil is mainly responsible for the tail of f(tau), which in many cases resembles the classic linear reservoir model. The resulting shape for f(tau) is similar to the Gamma distribution. Analysis of the f(tau) moments indicates that the mean traveltime is weakly dependent on the hillslope size. The traveltime variance is ruled by the distribution of distances of the injected solute from the river; the effect is coined as source zone dispersion. The spreading due to the K heterogeneity is less important and obscured by source zone dispersion. The model is tested against the numerical simulation of Fiori and Russo (2008) with reasonably good agreement, with no fitting procedure.