An analytical solution for the lateral transport of dissolved chemicals in overland flow-varying soil surface concentration

The role of overland flow dynamics on the lateral transport of soil- dissolved chemicals toward an outlet is studied herein using the solutions of the mass conservation equations for overland flow and transport. The transfer of soil chemicals to the overland flow is a rate-limited process proportional to the chemical concentration at the soil surface which is time dependent. Some simplifying conditions are considered, to develop an analytical solution for both equations. The rainfall rate f, the infiltration rate i, and the mass transfer coefficient k are assumed constant. The analytical solution is based on the characteristic method for the two main stages of overland flow, namely, the rising and falling stages. The first stage occurs from pending time to the time when rainfall ceases, at which point the second stage begins. During the second stage, the slope drains and overland flow depth gradually becomes zero. Although the initial water depth for the kinematic wave equation, h₀, could be chosen arbitrarily, its combination with the solute equation requires that h₀ should be greater than zero, since a singularity is obtained for h₀ = 0. Therefore, the solution for overland flow is developed for a general constant initial water depth, but in order to meet real initial conditions, h₀ is chosen to be very small, almost negligible. Use of the characteristic method imposes different solutions for each sub-zone that composes the domain. In one sub-zone that belongs to the falling stage, the characteristic curve is solved by Euler's method. During the rising stage of overland flow the concentration hydrograph decreases rapidly followed by a moderate decrease thereafter. During the falling stage of overland flow, the concentration hydrograph increases gradually. Since here the soil surface concentration is a function of time, the relative influence of the different rate-limited processes determining the shape of the concentration distribution at the slope outlet can be identified and analyzed. At time of pending and during the final period of overland flow recession, the assumption of constant k is not valid and, although mathematically correct, physically erroneous results are obtained. As an example, the analytical solution for c(s)(t), as obtained by Wallach and van Genuchten (1990), is introduced into the solution and the overland flow concentration distributions at the slope outlet were compared and successfully agreed with measured data.