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

摘要

A stochastic model is developed for the analysis of the traveltime distribution f_τ 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_τ, i.e., the rapid release of solute which preferentially moves through the upper soil. The early branch of f_τ 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_τ, which in many cases resembles the classic linear reservoir model. The resulting shape for f_τ is similar to the Gamma distribution. Analysis of the f_τ 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.