On the local concentration probability density function of solutes reacting upon mixing

Abstract

The probability density function (pdf) of solute concentration is a useful tool for modeling transport of contaminants in heterogeneous aquifers that is increasingly used in risk assessment and, more generally, as a mean to quantify uncertainty in transport modeling. In order to be effective the pdfs should be linked in a simple manner to the spatial variability model of hydraulic properties, pore‐scale (local) dispersion, and a suitable parametrization of the geochemical processes. We analyze the pdf and concentration moments of two aqueous species in equilibrium with their precipitate reacting upon mixing in two‐ and three‐dimensional geological formations. The speciation equations, resulting from application of the chromatographic theory, provide the link between concentration pdfs (and moments) of the aqueous species and that of a passive tracer. Within this framework, we investigate the role of pore‐scale dispersion and macrodispersion in enhancing mixing, and thus reaction between the aqueous species, in the case of an instantaneous injection of a water with contrasting concentrations with respect to the ambient water under the constraint that in both waters the two aqueous species are in equilibrium with their precipitate. The main conclusion of our analysis is that for the pore‐scale dispersion typically observed in natural formations, the local concentration pdfs of both species are far from being Gaussian, and therefore the first two concentration moments provide very limited information of the underlying transport dynamics. Instead, the pdfs provide crucial information for applications, such as the probability of exceeding a given concentration, for example, the regulatory limit, at a particular location within the domain of interest. Furthermore, by analyzing the second‐order moments of the concentration, we show that mixing is strongly affected by space dimensionality and that the two‐dimensional approach, often used for computational convenience, may severely underestimate reaction rates in real settings.