### Local Equilibrium and Retardation Revisited

We present that 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 (LEA). Assuming local equilibrium, Eulerian textbook treatments derive the retarded ADE, ostensibly exactly. However, other authors have presented rigorous mathematical derivations of the dispersive effect of MIMT, applicable even in the case of arbitrarily fast mass transfer. We resolve the apparent contradiction between these seemingly exact derivations by adopting a Lagrangian point of view. We show that local equilibrium 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. We then present a particle tracking simulation illustrating how poor an approximation the retarded ADE may be, even when mobile and immobile plumes are continually near local equilibrium. Finally, we note that classic “LEA” (actually, retarded ADE validity) criteria test for insignificance of MIMT-driven dispersion relative tomore »

- Publication Date:

- Report Number(s):
- LA-UR-16-22097

Journal ID: ISSN 0017-467X

- Grant/Contract Number:
- AC52-06NA25396; 11145687

- Type:
- Accepted Manuscript

- Journal Name:
- Ground Water

- Additional Journal Information:
- Journal Volume: 56; Journal Issue: 1; Journal ID: ISSN 0017-467X

- Publisher:
- Wiley - NGWA

- Research Org:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org:
- USDOE Office of Science (SC)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 58 GEOSCIENCES; 97 MATHEMATICS AND COMPUTING

- OSTI Identifier:
- 1477643

- Alternate Identifier(s):
- OSTI ID: 1418694

```
Hansen, Scott K., and Vesselinov, Velimir Valentinov.
```*Local Equilibrium and Retardation Revisited*. United States: N. p.,
Web. doi:10.1111/gwat.12566.

```
Hansen, Scott K., & Vesselinov, Velimir Valentinov.
```*Local Equilibrium and Retardation Revisited*. United States. doi:10.1111/gwat.12566.

```
Hansen, Scott K., and Vesselinov, Velimir Valentinov. 2017.
"Local Equilibrium and Retardation Revisited". United States.
doi:10.1111/gwat.12566. https://www.osti.gov/servlets/purl/1477643.
```

```
@article{osti_1477643,
```

title = {Local Equilibrium and Retardation Revisited},

author = {Hansen, Scott K. and Vesselinov, Velimir Valentinov},

abstractNote = {We present that 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 (LEA). Assuming local equilibrium, Eulerian textbook treatments derive the retarded ADE, ostensibly exactly. However, other authors have presented rigorous mathematical derivations of the dispersive effect of MIMT, applicable even in the case of arbitrarily fast mass transfer. We resolve the apparent contradiction between these seemingly exact derivations by adopting a Lagrangian point of view. We show that local equilibrium 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. We then present a particle tracking simulation illustrating how poor an approximation the retarded ADE may be, even when mobile and immobile plumes are continually near local equilibrium. Finally, we note that classic “LEA” (actually, retarded ADE validity) criteria test for insignificance of MIMT-driven dispersion relative to hydrodynamic dispersion, rather than for local equilibrium.},

doi = {10.1111/gwat.12566},

journal = {Ground Water},

number = 1,

volume = 56,

place = {United States},

year = {2017},

month = {7}

}