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Title: 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:
 [1];  [1]
  1. 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}
}