Runoff contamination by soil chemicals: Time scale approach

Two mass balance equations were used to model the transfer of dissolved chemicals from the soil solution to the surface runoff water and transport of these chemicals to the field outlet end. One mass balance equation is written for the chemicals dissolved in the overland water, and the other for the chemicals within the soil profile. The chemical input into the surface runoff water is by the rate‐limited convective mass transfer process. Two time scales are isolated: the slow time scale represents the diffusion‐based mass transfer process and the fast time scale represents the convective transport of dissolved chemicals by overland flow. Scaling the mass balance equations for the slow time scale yields a small parameter, which multiplies the time derivative of the mass balance equation written for the overland flow, providing a singular perturbation problem. By using the method of matched asymptotic expansion, an inner and outer problem is formulated and solved for each order of approximation. A single composite expansion, uniformly valid over the entire domain, is derived analytically. This approximated solution was compared with an exact analytical solution for the case in which chemicals are initially uniformly distributed throughout a semi‐infinite soil profile. The time scale method was then used to solve a more complicated problem in which chemicals are initially distributed within a certain soil surface layer of a semi‐infinite soil profile.