A Comparison of Closures for Stochastic Advection-Dispersion
Abstract
Perturbation-based moment equations for advection-dispersion equations with random advection have been shown to produce physically unrealistic multimodality. Despite similarities, macrodispersion theory applied to advection-dispersion equations produces moments that do not exhibit such multimodal behavior. This is because macrodispersion approximations, whether explicitly or implicitly, involve renormalized perturbations that remove secularity by including select higher-order terms. We consider basic differences between the two approaches using a low-order macrodispersion approximation to clarify why one produces physically meaningful behavior while the other does not. We demonstrate that using a conventional asymptotic expansion (in the order of velocity fluctuations) leads to equations that cannot produce physically meaningful (macro)dispersion, whether applied to moment equations or macrodispersion theory, proving that the resulting moment equations are in fact hyperbolic in one spatial dimension. We identify higher-order terms that must be added to the conventional expansion to recover second- and fourth-order macrodispersivity approximations. Finally, we propose a closed-form approximation to two-point covariance as a measure of uncertainty, in a manner consistent with the derivation of macrodispersivity. We demonstrate that this and all the macrodispersion-based approximations to moments are more accurate than the alternatives for an example of transport in stratified random media.
- Authors:
-
- Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
- Publication Date:
- Research Org.:
- Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1158509
- Report Number(s):
- PNNL-SA-90925
Journal ID: ISSN 2166-2525; KJ0401000
- DOE Contract Number:
- AC05-76RL01830
- Resource Type:
- Journal Article
- Journal Name:
- SIAM/ASA Journal on Uncertainty Quantification
- Additional Journal Information:
- Journal Volume: 1; Journal Issue: 1; Journal ID: ISSN 2166-2525
- Publisher:
- SIAM
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; stochastic; random fields; macrodispersion; moment equations; solute transport; heterogeneous media
Citation Formats
Jarman, Kenneth D., and Tartakovsky, Alexandre M. A Comparison of Closures for Stochastic Advection-Dispersion. United States: N. p., 2013.
Web. doi:10.1137/120897419.
Jarman, Kenneth D., & Tartakovsky, Alexandre M. A Comparison of Closures for Stochastic Advection-Dispersion. United States. https://doi.org/10.1137/120897419
Jarman, Kenneth D., and Tartakovsky, Alexandre M. 2013.
"A Comparison of Closures for Stochastic Advection-Dispersion". United States. https://doi.org/10.1137/120897419.
@article{osti_1158509,
title = {A Comparison of Closures for Stochastic Advection-Dispersion},
author = {Jarman, Kenneth D. and Tartakovsky, Alexandre M.},
abstractNote = {Perturbation-based moment equations for advection-dispersion equations with random advection have been shown to produce physically unrealistic multimodality. Despite similarities, macrodispersion theory applied to advection-dispersion equations produces moments that do not exhibit such multimodal behavior. This is because macrodispersion approximations, whether explicitly or implicitly, involve renormalized perturbations that remove secularity by including select higher-order terms. We consider basic differences between the two approaches using a low-order macrodispersion approximation to clarify why one produces physically meaningful behavior while the other does not. We demonstrate that using a conventional asymptotic expansion (in the order of velocity fluctuations) leads to equations that cannot produce physically meaningful (macro)dispersion, whether applied to moment equations or macrodispersion theory, proving that the resulting moment equations are in fact hyperbolic in one spatial dimension. We identify higher-order terms that must be added to the conventional expansion to recover second- and fourth-order macrodispersivity approximations. Finally, we propose a closed-form approximation to two-point covariance as a measure of uncertainty, in a manner consistent with the derivation of macrodispersivity. We demonstrate that this and all the macrodispersion-based approximations to moments are more accurate than the alternatives for an example of transport in stratified random media.},
doi = {10.1137/120897419},
url = {https://www.osti.gov/biblio/1158509},
journal = {SIAM/ASA Journal on Uncertainty Quantification},
issn = {2166-2525},
number = 1,
volume = 1,
place = {United States},
year = {Tue Jun 25 00:00:00 EDT 2013},
month = {Tue Jun 25 00:00:00 EDT 2013}
}