The presence of solute concentration fluctuations at spatial scales much below the scale of resolution is a major challenge for modeling reactive transport in porous media. Overlooking small-scale fluctuations, which is the usual procedure, often results in strong disagreements between field observations and model predictions, including, but not limited to, the overestimation of e˙ective reaction rates. Existing innovative approaches that account for local reactant segregation do not provide a general mathematical formulation for the generation, transport and decay of these fluctuations and their impact on chemical reactions. We propose a Lagrangian formulation based on the random motion of fluid particles carrying solute concentrations whose departure from the local mean is relaxed through multi-rate interaction by exchange with the mean (MRIEM). We derive and analyze the macroscopic description of the local concentration covariance that emerges from the model, showing its potential to simulate the dynamics of mixing-limited processes. The action of hydrodynamic dispersion on coarse-scale concentration gradients is responsible for the production of local concentration covariance, whereas covariance destruction stems from the local mixing process represented by the MRIEM formulation. The temporal evolution of integrated mixing metrics in two simple scenarios shows the trends that characterize fully-resolved physical systems, such as a late-time power-law decay of the relative importance of incomplete mixing with respect to the total mixing. Experimental observations of mixing-limited reactive transport are successfully reproduced by the model.

The growth of bioflm in porous media causes disruption in the groundwater flow patterns. In conservative tracer tests performed in columns, this translates in significant changes in the observed breakthrough curves (BTCs) in bio-ammended as compared to biofilm-free porous media, then translated to bioclogging-dependent interpreted hydraulic parameters. While the impact upon reduction in saturated hydraulic conductivity values has been widely explained and modeled, this has not been the case for the reported significant increase in apparent dispersivity values. We present here simple, yet practical, expressions for the evaluation of enhanced effective dispersivity coefficients in bio-ammended saturated porous media, based on the modification of the BTCs (in terms of temporal moments) with respect to the biofilm-free porous media and for a number of proposed models with slightly different underlying hypotheses. The advantage of the expressions provided is that they are written in terms of observables that are relatively easy to measure in the lab or the field, contrarily to existing expressions that relate the effect to channelization caused by pore constriction assuming a simple geometry for the biofilm. One model is then used to interpret nine column experiments from the literature, allowing to use data of measured dispersivity, porosities, and flow rate to provide a lumped parameter that incorporates accessibility of solutes to biofilm by a mass transfer term between the biofilm and the flowing water.