Upscaling the impact of transport limitations in fluid-solid reactions using a chemical continuous time random walk

Fluid-solid reactions play a key role in a wide range of biogeochemical processes. Transport limitations at the pore scale limit the amount of solute available for reaction, so that reaction rates measured under well-mixed conditions tend to strongly overestimate rates occurring in natural and engineered systems. Although different models have been proposed to capture this phenomenon, linking pore-scale structure, flow heterogeneity, and local reaction kinetics to upscaled effective kinetics remains a challenging problem.

We present a new theoretical framework to upscale these dynamics based on the chemical continuous time random walk framework. The approach is based on the concept of inter-reaction times, which incur delays compared to well-mixed conditions due to the times between contacts of transported reactants with the solid phase. We consider a simple chemical reaction in order to focus on the effects of transport limitations and medium structure, namely a second-order degradation reaction between a fluid-phase reactant and a solid-phase reactant distributed uniformly over the fluid-solid interface, where only the fluid reactant is consumed. Our formulation quantifies the global kinetics of fluid-reactant mass as it undergoes advection, diffusion, and reaction. Predictions are in agreement with numerical simulations of transport in stratified channel flows and Stokes flow through a beadpack. The theory captures the decrease of effective reaction rates compared to the well-mixed prediction with increasing Damköhler number due to transport limitations.