# Local equilibrium and retardation revisited

## TL;DR

This paper critically examines the assumptions behind the retarded advection-dispersion equation in solute transport modeling, revealing that common interpretations of local equilibrium are misleading and that remobilization rate, not equilibrium, governs model validity.

## Contribution

The paper clarifies the mathematical and conceptual foundations of local equilibrium assumptions, demonstrating discrepancies between traditional derivations and rigorous approaches, and provides numerical evidence of differences in transport predictions.

## Key findings

- Retarded ADE embeds a nonphysical constraint on particle immobility.
- Numerical simulations show significant differences between exact and retarded ADE models.
- Remobilization rate, not local equilibrium, controls the validity of the retarded ADE.

## Abstract

In modeling solute transport with mobile-immobile mass transfer (MIMT), it is common to use an advection-dispersion equation (ADE) with a retardation factor, or retarded ADE. This is commonly referred to as making the local equilibrium assumption. Assuming local equilibrium (LE), Eulerian textbook treatments derive the retarded ADE, ostensibly exactly. However, other authors have presented rigorous mathematical derivations of the dispersive effect of mass transfer, applicable even in the case of arbitrarily fast mass transfer. First, we resolve the apparent contradiction between these seemingly exact derivations by adopting a Lagrangian point of view. We show that LE constrains the expected time immobile, whereas the retarded ADE actually embeds a stronger, nonphysical, constraint: that all particles spend the same amount of every time increment immobile. Eulerian derivations of the retarded ADE thus silently commit the gambler's fallacy, leading them to ignore dispersion due to mass transfer that is correctly modeled by other approaches. Second, we present a numerical particle tracking study of transport in a heterogeneous aquifer subject to first-order MIMT. Transport is modeled (a) exactly, and then (b) approximated with the retarded ADE. Strikingly different results are obtained, even though quasi-LE is maintained at all times by the exact MIMT simulation. We thus observe that use of the phrase local equilibrium assumption to refer to ADE validity is not correct. We highlight that solute remobilization rate is the true control on retarded ADE validity, and note that classic "local equilibrium assumption" (i.e., ADE validity) criteria actually test for insignificance of MIMT-driven dispersion relative to hydrodynamic dispersion, rather than for local equilibrium.