A New Flow-Kinematics-Based Model for Time-Dependent Effective Dispersion in Mixing-Limited Reactions

Abstract

A new upscaling procedure that provides 1D representations of 2D mixing-limited reactive transport systems is developed and applied. A key complication with upscaled models in this setting is that the procedure must differentiate between interface spreading, driven by the spatially variable velocity field, and mixing, through which components contact one another and react. Our model captures the enhanced mixing caused by spreading through use of a time-dependent effective dispersion term. The early-time behavior of this dispersion is driven by flow kinematics, while at late times it reaches a Taylor-dispersion-like limit. The early-time behavior is modeled here using a very fast (purely advective) particle tracking procedure. The only free parameter in the model is the late-time asymptotic effective dispersion. This quantity is estimated using a fit involving a dimensionless grouping of system variables and a few reference results, or by calibrating with the corresponding conservative (non-reacting) case. Numerical results for bimolecular reaction systems are generated using a pseudo-spectral approach capable of resolving fronts at high Peclet numbers. Results are presented for three different types of 2D velocity fields over a wide range of parameters. The upscaled model is shown to provide highly accurate results for the conversion factor, along with reasonable approximations of the spatial distribution of reaction occurrence. The model is also shown to be valid to upscale mixing in non-reacting systems.