Stochastic‐Convective Transport with Nonlinear Reaction: Mathematical Framework

Abstract

A stochastic‐convective reactive (SCR) transport method is developed for one‐dimensional steady transport in physically heterogeneous media with nonlinear degradation. The method is free of perturbation amplitude limitations and circumvents the difficulty of scale dependence of phenomenological parameters by avoiding volume‐averaged specifications of diffusive/dispersive fluxes. The transport system is conceptualized as an ensemble of independent convective‐reactive streamlines, each characterized by a randomized convective velocity (or travel time). Dispersive effects are treated as a component of the randomness in the streamline velocity ensemble, so no explicit expression for hydrodynamic dispersive flux is written in the streamline transport equation. The expected value of the transport over the stream tube ensemble is obtained as an average of solutions to the reactive convection equation according to the stream tube (travel time) probability distribution function. In this way, transport with reaction can be expressed in terms of global‐scale random variables, such as solute travel time and travel distance, which are integrals of the stochastic variables such as velocity. Derivations support the hypothesis that via the SCR the decay process can be factored out of the mechanical transport behavior (as reflected by movement of a passive tracer) and scaled independently. Solution strategies are presented for general linear and nonlinear kinetic reactions. Demonstration simulations show that for Fickian transport with nonlinear reactions the SCR and convection dispersion equation can give different results. Ginn et al. (this issue) extend the SCR solution to coupled nonlinear equations, to accommodate Michaelis‐Menten biodegradation of solute with an accounting of microbial growth.