Lagrangian modeling of mixing-limited reactive transport in porous
media: multi-rate interaction by exchange with the mean
Abstract
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.