Dilution of Reactive Plumes: Evolution of Concentration Statistics Under Diffusion and Nonlinear Reaction

Abstract

Concentration fields of solutes in porous media often exhibit large fluctuations, driven by physical and chemical heterogeneity from the pore to the Darcy scale. For many applications, ranging from reactive transport modeling to toxicology, the knowledge of mean concentrations is not sufficient, and quantifying concentration variability is necessary. The probability density function (PDF) of concentration quantifies the frequency of occurrence of concentration values throughout a spatial domain. While evolution equations and analytical solutions for the concentration PDF exist for conservative solutes, less is known about its evolution under the joint action of transport and reaction. In this work, we investigate how dilution of a reactive plume by diffusion affects the statistics of concentrations. While mixing has no effect on first-order reactions, its coupling with nonlinear reactions leads to non-trivial effective kinetics relevant for a broad range of reactive transport problems. We study the evolution of the concentration PDF under diffusion and nonlinear reaction in one spatial dimension, which represents a critical step toward further coupling with heterogeneous advection. We show that the dependence of the scalar dissipation rate on concentration encodes the impact of diffusive transport on the concentration PDF and derive a dynamical equation for its time evolution. Using a weak-coupling approximation for the reaction and diffusion dynamics, we derive analytical predictions for the concentration PDF and its moments. Our results provide new insights into how diffusion and reaction control concentration variability and open new opportunities for coupling mixing models with chemical reactions.