Bimolecular second‐order reactions in spatially varying flows: Segregation induced scale‐dependent transformation rates

Abstract

To quantify chemical transformations in natural hydrologic environments, their coupling with spatially varying flows needs to be understood. Consider the elementary bimolecular second‐order irreversible chemical reaction for which the transformation rate at any point is proportional to the product of the concentrations of two distinct species at that point (r(c1, c2) = κc1c2, κ assumed to be constant). A problem arises in incorporating such chemical reactions in hydrologic transport models. On account of a lack of knowledge of all of the details of flows, or computational limitations, these transport models can only provide estimates of spatial averages of the concentration fields and not the average of the product of the concentration of the different species, . However, that spatial average needs to be assessed to quantify the chemical transformation rate pertinent to the spatially averaged concentration field . For an impulse input of one species in a uniform background of the other species, detailed numerical simulations in laminar shear flow and heterogeneous porous media flow show that the spatial variability of flow causes the small‐scale concentration variations of the different species to be negatively correlated . This small‐scale segregation of solutes gives rise to a scale‐dependent transformation rate, , which can be much smaller than what is found by substituting the mean concentration into the transformation rate expression: . The segregation intensity is strongly related to both the flow variations and the small‐scale mixing mechanisms (diffusion/local dispersion). After an initial increase in the absolute value of the segregation intensity, it slowly decreases with time due to the cumulative smoothing action of diffusion. As in hydrologic problems, the diffusion timescales characteristic to the flow variability scales can be quite large; substituting mean concentrations into rate expressions determined in well‐mixed batch tests is likely to overestimate the chemical transformation rate.