Upscaling mass transport with heterogeneous reaction, adsorptionand accumulation in porous media

Modeling reactive mass transport in porous media systems is often performed using effective-medium approaches due to the difficulties of solving the microscale equations throughout the entire system. Under an effective-medium framework, transport processes at the solid-fluid interface are usually assumed to be quasi-steady with respect to the transport in the bulk phase that saturates the pores of the system. This has led to effective-medium models in which the reaction term is present in the macroscopic mass balance equation, which cannot be used to predict transport at the early stages of the process but rather under steady conditions. To address this issue, this work presents an alternative modeling approach in which transport at the interface is assumed to be unsteady. This leads to a macroscopic model consisting of a set of two coupled partial differential equations that can be used to predict the average concentration in the phase and at the interface under unsteady conditions. These equations are derived using the volume averaging method, which also allows for predicting the associated effective-medium coefficients by solving the corresponding closure problems in periodic unit cells. The model is validated through comparisons with direct numerical simulations under several transport and reaction conditions. From this analysis, and by comparison with a previously derived interface-steady model, specific situations in which the two-equation model excels compared to previous approaches are identified. The results of this work are relevant in many water resources, chemical, and biological systems involving the transport and adsorption of reactive species under unsteady conditions.