Approximate solutions for cation transport during unsteady, unsaturated soil water flow

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Abstract
Approximate analytical solutions are derived for the transport of reactive solutes during unsteady flow of water in unsaturated porous media. The solutions apply to solutes that undergo adsorption reactions including ion exchange, specific adsorption, or negative adsorption (anion exclusion). No particular form of adsorption or exchange equation is assumed, and solutions are developed for both unfavorable (concave upward) and favorable (convex upward) isotherms, as well as for multisite isotherms. The main approximations that are assumed in deriving the solutions are (1) that local chemical and physical equilibrium is attained rapidly relative to the time scale for transport, (2) that the reaction takes place in a region of constant total charge concentration in both the solution and solid phases, (3) that the effects of dispersion and the shape of the adsorption isotherm may be treated separately and then combined, and (4) that as has been assumed previously for unsteady flow of nonreactive solutes, the water content and pore water velocity may be treated as constant over the region of dispersion. The solutions were tested using experimental data for the transport of exchanging cations during horizontal infiltration in laboratory columns. The solution for unfavorable isotherms was compared with data for sodium invading a calcium‐dominated soil; that for multisite isotherms, which combines the solutions for both favorable and unfavorable isotherms, was compared with data for potassium invading a calcium‐dominated soil. In both cases, good agreement was found between the measured distributions of solution and sorbed cations and those predicted using these solutions together with independently measured cation exchange isotherms.