A Stefan Problem Arising from Two-Species Reaction-Diffusion with Applications to Chemical Oxidation of Contaminants in Fractured Rock

A Stefan Problem Arising from Two-Species Reaction-Diffusion with Applications to Chemical Oxidation of Contaminants in Fractured Rock

Harihar Rajaram
Department of Civil, Environmental and Architectural Engineering
University of Colorado at Boulder

This study was motivated the problem of chemical oxidation of dissolved solvents (often referred to as DNAPLs for "dense nonaqueous phase liquids") in a rock matrix by delivering oxidants such as permanganate through fractures. Under continuous flushing/recirculation, the concentrations of permanganate are maintained at a constant level in fractures, while it diffuses into the rock matrix and reacts with the DNAPL. The permanganate-DNAPL reaction is typically described as a bimolecular reaction, based on experimental kinetic data. Due to the relatively rapid rate of the oxidation reaction, an appropriately defined Damkohler number is large. Under these conditions, a thin reaction front develops and propagates into the rock matrix at a rate controlled by diffusion. The reaction front can be described as a moving boundary by analogy with the classical Stefan problem in heat transfer with phase change. The propagation of the reaction front can be quantified using a reaction front diffusivity, which can be calculated explicitly. The reaction front diffusivity is shown to depend on the initial concentrations of DNAPL and oxidant, and their effective diffusivities. Scaling arguments are proposed to quantify the temporal dynamics of the DNAPL consumption rate and the width of the reaction zone. The results of the analysis for (i) reaction front propagation, (ii) oxidant and DNAPL consumption rate and (iii) reaction zone width, are all confirmed by numerical simulations. An early time regime was identified, wherein the reaction front has yet to form. The duration of this early-time regime and a perturbation analysis to quantify the consumption rate in this regime are also presented. The consumption rate behaves as sqrt(t)) in this early time regime, transitioning to the diffusion-controlled 1/sqrt(t) behavior after the reaction front forms. At late time in a rock matrix of finite width, boundary effects are incorporated via an approximate integral balance approach, also drawn from previous work on heat transfer. Numerical simulations in heterogeneous porous media indicate that the front is stable with respect to perturbations arising from natural heterogeneity, consistent with the few available field observations.