Nonlinear reactions and nonuniform flows

Analytic solutions are developed for the advection‐reaction problem with nonuniform flows and nonlinear reactions. We consider a stratified flow field and a flow field due to the operation of an injection‐extraction pair of wells in a homogeneous medium. In each case we derive the transfer function for transport between a pair of surfaces in the flow field. We consider linear/nonlinear aqueous kinetics and the nonlinear equilibrium adsorption of a Langmuir radionuclide with decay. Using a stream tube concept in conjunction with these transfer functions, we develop analytic solutions to the flux‐averaged concentration response of these reactive solutes. Resident concentrations are obtained by solving a simpler batch reaction system in place of coupled partial differential equations. We use these solutions to evaluate two numerical approaches. One of them is based on an operator‐split approach that employs a conventional finite difference scheme with total variation diminishing fluxes for the advection operator. The other is based on the use of advected particles to compute a numerical approximation of the transfer function. We consider a boundary‐value problem and establish that the particle‐tracking approach yields near‐analytic solutions for both the flux‐averaged and resident concentrations. We consider an initial‐value problem and evaluate the “dilution index” [ Kitanidis , 1994] associated with the advective flow field and the numerical scheme. A flow field that highly distorts a plume and increases its surface area exhibits a larger dilution index, and most numerical schemes for transport in such a flow field suffer larger numerical dispersion. For linear reactions the predicted amount of plume degradation is unaffected by plume deformation, while the errors in predictions for nonlinear reactions increase as the plume deformation increases.