Analytical solutions to the convection-dispersion model (CDM) of solute transport require linear reaction terms, strict initial and boundary concentration conditions, and are often complex to evaluate because of the inherent mathematical functions. We present a flexible and mathematically very simple solution to the CDM at steady water flow, labeled a semi-analytical (SA) solution. The SA solution allows for nonlinear reaction terms, variable initial and boundary conditions, and is based on the recently presented moving concentration slope (MCS) model for solute transport. To derive the SA solution, a solute flux approximation at the upper boundary and a small, constant depth increment of 0.5 cm are used, and two features of the MCS model are exploited, i.e., an explicit, depth-integrated flux equation is already inherent in the model and all numerical error and stability equations are unique functions of the solute unit mean travel distance (SUMTD). The SA solution contains seven constants; one is the solute dis-persivity, and the remaining six are functions only of the SUMTD. Excellent agreement between the SA solution and ordinary analytical solutions to the CDM was obtained. For variable boundary conditions, the SA solution was also tested against data for chloride transport in sandy soil columns. Measured and calculated outlet concentrations compared well.
The SA model allows for linear or nonlinear reaction terms without increasing the complexity of the solution. In the case of nonlinear reactions, the SA model offers a simple solution in situations where conventional analytical solutions are not available. This was illustrated by successfully comparing the SA solution, including a Michaelis-Menten reaction term, with measured data for simultaneous transport and reduction of nitrate in porous media columns.
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