Local reaction and diffusion in porous media transport models

Abstract

The problem of when advection‐dispersion models apply for reactive transport in porous media is addressed. Assuming local mass balances, including arbitrary homogeneous and interfacial chemical reactions, are known, volume averaging is applied to obtain a set of equations for the average concentrations. It is shown that timescale constraints must be satisfied in addition to the well‐known length‐scale constraint needed for volume averaging. The timescale for simulation must be longer than a diffusion timescale in the representative elementary volume, t/TD ≫ 1. In addition, interfacial reaction timescales must be larger than meaningful diffusion timescales, Tr/TD ≫ 1. When these constraints are satisfied, the usual dispersion coefficient exists and is time‐invariant and independent of reactions. Reaction rate expressions and all mass transfer fluxes can be expressed in terms of the average concentrations of the macroscopic model. Even when surface reactions are fast, it is shown that the fluid volume can be subdivided into small enough regions such that the appropriate time constraint Tr/TD ≫ 1 is satisfied. An average model can be obtained that includes mass transfer resistance expressed in terms of a mass transfer coefficient. The mass transfer coefficient is defined and is shown to depend only on the geometry of the porous medium and the flow field. This work provides a theoretical basis for the commonly used advection‐dispersion models for reactive transport at the Darcy scale and provides both length‐scale and timescale constraints for when they apply.