Dynamics of mixing and bimolecular reaction kinetics in aquifers

Abstract

Descriptions of chemical transformation kinetics and hydrologic transport need to be coupled to understand the composition of flowing waters. The coupling of a bimolecular transformation reaction (reactant 1 + reactant 2 → product; rate $r = \kappa c_{1} c_{2}$) with spatially heterogeneous subsurface flows is addressed here. The flow microstructure—that controls the spreading rate of solutes and the mean reactant concentrations (C ₁, C ₂)—creates concentration microstructure whose intensity is characterized by the variances $\sigma_{{c_{1} }}^{2}$, $\sigma_{{c_{2} }}^{2}$, and the cross-covariance $\overline{{c_{1} c_{2} }}$. In addition to the macroscopic overlap of the reactants, that is quantified by the product of the mean concentrations that are routinely modeled using effective dispersion coefficients $D_{ij}$, the concentration microstructure plays an important role in determining reaction macro-kinetics as the mean reaction rate is $\overline{r} = \kappa (C_{1} C_{2} + \overline{{c_{1} c_{2} }} )$. For initially non-overlapping reactants $\overline{{c_{1} c_{2} }} + C_{1} C_{2} = 0$ and $\overline{r} = 0$ under pure advection. It is shown that due to the action of local dispersion, at large time, the $\overline{{c_{1} c_{2} }}$ budget is characterized by a balance between its rate of production and dissipation, which results in $\overline{{c_{1} c_{2} }} \approx 2\tau D_{ij} ({{\partial C_{1} } \mathord{\left/ {\vphantom {{\partial C_{1} } {\partial x_{i} )({{\partial C_{2} } \mathord{\left/ {\vphantom {{\partial C_{2} } {\partial x_{j} )}}} \right. \kern-0pt} {\partial x_{j} )}}}}} \right. \kern-0pt} {\partial x_{i} )({{\partial C_{2} } \mathord{\left/ {\vphantom {{\partial C_{2} } {\partial x_{j} )}}} \right. \kern-0pt} {\partial x_{j} )}}}}$, where τ is the dissipation time-scale characteristic to the destruction of concentration fluctuation by local dispersion. This results in $\overline{r} = \kappa_{\text{eff}} C_{1} C_{2}$, where $\kappa_{\text{eff}} \approx \kappa [1 + 2\tau D_{ij} ({{\partial { \ln }C_{1} } \mathord{\left/ {\vphantom {{\partial { \ln }C_{1} } {\partial x_{i} )({{\partial { \ln }C_{2} } \mathord{\left/ {\vphantom {{\partial { \ln }C_{2} } {\partial x_{j} )}}} \right. \kern-0pt} {\partial x_{j} )}}}}} \right. \kern-0pt} {\partial x_{i} )({{\partial { \ln }C_{2} } \mathord{\left/ {\vphantom {{\partial { \ln }C_{2} } {\partial x_{j} )}}} \right. \kern-0pt} {\partial x_{j} )}}}}]$, which accounts for the influence of concentration microstructure and small-scale mixing on the macroscopic bimolecular kinetics. The effective rate parameter κ_eff is greater than the intrinsic rate constant κ measured under well-mixed conditions if the macroscopic concentration gradients have the same sign (initially overlapping reactants). For the initially non-overlapping reactants which result in macroscopic gradients having opposite signs, κ _eff < κ. The macroscopic reactant concentration gradients, effective dispersion coefficients, and the dissipation time-scale control the reaction macro-kinetics, in addition to the intrinsic rate constant κ and the mean reactant concentrations. The formulation for reaction macro-kinetics developed here helps explain previously reported disparities between laboratory and field-scale transformation rates and also provides a way to represent the influence of reactant concentration microstructure in large-scale descriptions of reactive transport.