A small perturbations solution for nonequilibrium chemical transport through soils with relatively high desorption rate

Rony Wallach*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


An analytical solution is presented for reactive solute transport or nonuniform flow when the reaction rate or the diffusive transfer rate to stagnant water is small compared to the flow velocity or large compared to he contact time. For these cases many transport models assume local chemical or physical equilibrium for conceptual and mathematical simplification. In this paper the commonly used nonequilibrium formulation for one-dimensional, steady-flow transport was used as the starting point. Dimensionless variables were defined and substituted into the mass balance equations forming dimensionless equations and boundary and initial conditions. There are two characteristic timescales associated with the two rate-limited processes: The fast characteristic timescale is related to the chemical desorption and the slow timescale is related to the convective-dispersive transport. By scaling the first-order rate equation that accounts for linear sorption kinetics (or the dissolved chemical exchange between mobile and stagnant porosities in structured soils), a singular perturbations problem was' obtained in which a small parameter, ε multiplies the time derivative. This small parameter is the ratio between the slow and fast timescales. Inner, τ = O(ε), and outer; τ = O(1), solutions were developed, and the method of matched asymptotic expansion was used to form a uniform solution over the entire time domain. The leading-order approximation, obtained for s → 0 and valid for τ = O(1), yields equations that are usually obtained by the local equilibrium assumption (LEA). A initial layer, τ = O(ε), near τ = 0 is formed within which the initial concentrations equilibrate. The condition under which the local equilibrium assumption is valid, ε = v2/k,D << 1, which depends on the system parameters such as flow velocity, dispersion coefficient, and the variables. This condition was also obtained in other studies by solving the problem in either kinetic or LEA form and comparing the solutions to determine under what conditions the local equilibrium approximation is close to the kinetic solution. As ε decreases, the LEA is valid, and the initial layer, in which the initial concentrations reach local equilibrium, is shorter.

Original languageAmerican English
Pages (from-to)149-154
Number of pages6
JournalWater Resources Research
Issue number1
StatePublished - Jan 1998


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