An Analytical Solution for the Lateral Transport of Dissolved Chemicals in Overland Flow

Judith Rivlin (Byk)*, Rony Wallach

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


The role of overland flow dynamics on the dissolved chemical transport toward an outlet is studied herein using the solutions of the mass conservation equations for overland flow and transport. Some simplifying conditions are considered, to develop an analytical solution for both equations. The rainfall rate ƒ, the infiltration rate i ,the mass transfer coefficient k, and the soil surface concentration cs are assumed constant. The solution, which is analytical, is based on the characteristics method for the two main stages of overland flow, namely, the rising and falling stages. The first stage occurs from ponding 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 induces that h0 should be greater than zero, since a singularity is obtained for h0 = 0. Therefore the solution for the 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. As an initial condition for the solute equation,c(x, 0) = cs is taken. This is based on the postulation that some mixing occurs with the impact of the raindrops during the first stage of water depth buildup. The initial condition also implies that k → ∞ which, according to the convective mass transfer theory, is obtained for h → 0. An analytical solution is obtained by the characteristics method. As a result, different solutions are obtained for each subzone that composes the domain. In one subzone that belongs to the falling stage, the characteristic curve is solved by Euler's method. The concentration hydrograph decreases rapidly during the rising stage of overland flow to a constant value. During the falling stage of overland flow the concentration hydrograph increases gradually to a value larger than cs, which depends on the ratio k/(k − i). Therefore at the time of ponding 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.

Original languageAmerican English
Pages (from-to)1031-1040
Number of pages10
JournalWater Resources Research
Issue number4
StatePublished - Apr 1995


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