Nonlocal and localized analyses of conditional mean steady state flow in bounded, randomly nonuniform domains 2. Computational examples

Alberto Guadagnini, Shlomo P Neuman

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Abstract

In paper 1 [Guadagnini and Neuman, this issue] of this two-part series we presented exact nonlocal equations for first and second conditional ensemble moments of hydraulic head and flux under steady state groundwater flow in bounded, randomly nonuniform domains in the presence of uncertain source and boundary terms. We derived recursive approximations for these equations based on expansion in powers of a small parameter σ(y), which represents the standard estimation error of log hydraulic conductivity; developed a two-dimensional finite element scheme for their solution, under the action of deterministic forcing terms, to first order in σ(y)/2; showed how to localize the conditional mean flow equations so as to yield familiar-looking, Darcian differential equations in which, however, all quantities are conditional on data and therefore nonunique; and proposed that traditional deterministic flow models be reinterpreted in light of these localized equations. We stressed that all quantities which enter into our nonlocal and localized conditional moment equations are defined on a consistent support scale thereby obviating the need for upscaling (from support to grid scale) or the a priori definition of a computational grid; that such a grid can therefore be defined a posteriori based on the degree of smoothness one expects the conditional moments to exhibit; and that conditioning points are often spaced widely enough to render these moments relatively smooth and hence resolvable on a coarser grid than is necessary for Monte Carlo simulations. This notwithstanding, we use in this paper a high-resolution grid to explore superimposed mean-uniform and convergent flows in unconditional and conditional log hydraulic conductivity domains and to compare the relative accuracies of corresponding nonlocal, localized, and Monte Carlo finite element solutions. Whereas our nonlocal solution is nominally restricted to mildly nonuniform media with σ(y)/2 << 1, we find that it actually yields remarkably accurate results for strongly nonuniform media with σ(y)/2 at least as large as 4. Localized solutions deteriorate relative to nonlocal results as one approaches some points of conditioning and singularity or as σ(y)/2 increases. Though they require much less computer time than do nonlocal solutions, localized solutions have the added disadvantage that they yield no information about predictive uncertainty.

Original languageEnglish (US)
Pages (from-to)3019-3039
Number of pages21
JournalWater Resources Research
Volume35
Issue number10
DOIs
StatePublished - 1999

Fingerprint

Hydraulic conductivity
hydraulic conductivity
conditioning
Groundwater flow
Error analysis
Differential equations
upscaling
hydraulic head
Hydraulics
groundwater flow
Fluxes
fluid mechanics
uncertainty
simulation
Monte Carlo simulation
Uncertainty

ASJC Scopus subject areas

  • Aquatic Science
  • Environmental Science(all)
  • Environmental Chemistry
  • Water Science and Technology

Cite this

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title = "Nonlocal and localized analyses of conditional mean steady state flow in bounded, randomly nonuniform domains 2. Computational examples",
abstract = "In paper 1 [Guadagnini and Neuman, this issue] of this two-part series we presented exact nonlocal equations for first and second conditional ensemble moments of hydraulic head and flux under steady state groundwater flow in bounded, randomly nonuniform domains in the presence of uncertain source and boundary terms. We derived recursive approximations for these equations based on expansion in powers of a small parameter σ(y), which represents the standard estimation error of log hydraulic conductivity; developed a two-dimensional finite element scheme for their solution, under the action of deterministic forcing terms, to first order in σ(y)/2; showed how to localize the conditional mean flow equations so as to yield familiar-looking, Darcian differential equations in which, however, all quantities are conditional on data and therefore nonunique; and proposed that traditional deterministic flow models be reinterpreted in light of these localized equations. We stressed that all quantities which enter into our nonlocal and localized conditional moment equations are defined on a consistent support scale thereby obviating the need for upscaling (from support to grid scale) or the a priori definition of a computational grid; that such a grid can therefore be defined a posteriori based on the degree of smoothness one expects the conditional moments to exhibit; and that conditioning points are often spaced widely enough to render these moments relatively smooth and hence resolvable on a coarser grid than is necessary for Monte Carlo simulations. This notwithstanding, we use in this paper a high-resolution grid to explore superimposed mean-uniform and convergent flows in unconditional and conditional log hydraulic conductivity domains and to compare the relative accuracies of corresponding nonlocal, localized, and Monte Carlo finite element solutions. Whereas our nonlocal solution is nominally restricted to mildly nonuniform media with σ(y)/2 << 1, we find that it actually yields remarkably accurate results for strongly nonuniform media with σ(y)/2 at least as large as 4. Localized solutions deteriorate relative to nonlocal results as one approaches some points of conditioning and singularity or as σ(y)/2 increases. Though they require much less computer time than do nonlocal solutions, localized solutions have the added disadvantage that they yield no information about predictive uncertainty.",
author = "Alberto Guadagnini and Neuman, {Shlomo P}",
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N2 - In paper 1 [Guadagnini and Neuman, this issue] of this two-part series we presented exact nonlocal equations for first and second conditional ensemble moments of hydraulic head and flux under steady state groundwater flow in bounded, randomly nonuniform domains in the presence of uncertain source and boundary terms. We derived recursive approximations for these equations based on expansion in powers of a small parameter σ(y), which represents the standard estimation error of log hydraulic conductivity; developed a two-dimensional finite element scheme for their solution, under the action of deterministic forcing terms, to first order in σ(y)/2; showed how to localize the conditional mean flow equations so as to yield familiar-looking, Darcian differential equations in which, however, all quantities are conditional on data and therefore nonunique; and proposed that traditional deterministic flow models be reinterpreted in light of these localized equations. We stressed that all quantities which enter into our nonlocal and localized conditional moment equations are defined on a consistent support scale thereby obviating the need for upscaling (from support to grid scale) or the a priori definition of a computational grid; that such a grid can therefore be defined a posteriori based on the degree of smoothness one expects the conditional moments to exhibit; and that conditioning points are often spaced widely enough to render these moments relatively smooth and hence resolvable on a coarser grid than is necessary for Monte Carlo simulations. This notwithstanding, we use in this paper a high-resolution grid to explore superimposed mean-uniform and convergent flows in unconditional and conditional log hydraulic conductivity domains and to compare the relative accuracies of corresponding nonlocal, localized, and Monte Carlo finite element solutions. Whereas our nonlocal solution is nominally restricted to mildly nonuniform media with σ(y)/2 << 1, we find that it actually yields remarkably accurate results for strongly nonuniform media with σ(y)/2 at least as large as 4. Localized solutions deteriorate relative to nonlocal results as one approaches some points of conditioning and singularity or as σ(y)/2 increases. Though they require much less computer time than do nonlocal solutions, localized solutions have the added disadvantage that they yield no information about predictive uncertainty.

AB - In paper 1 [Guadagnini and Neuman, this issue] of this two-part series we presented exact nonlocal equations for first and second conditional ensemble moments of hydraulic head and flux under steady state groundwater flow in bounded, randomly nonuniform domains in the presence of uncertain source and boundary terms. We derived recursive approximations for these equations based on expansion in powers of a small parameter σ(y), which represents the standard estimation error of log hydraulic conductivity; developed a two-dimensional finite element scheme for their solution, under the action of deterministic forcing terms, to first order in σ(y)/2; showed how to localize the conditional mean flow equations so as to yield familiar-looking, Darcian differential equations in which, however, all quantities are conditional on data and therefore nonunique; and proposed that traditional deterministic flow models be reinterpreted in light of these localized equations. We stressed that all quantities which enter into our nonlocal and localized conditional moment equations are defined on a consistent support scale thereby obviating the need for upscaling (from support to grid scale) or the a priori definition of a computational grid; that such a grid can therefore be defined a posteriori based on the degree of smoothness one expects the conditional moments to exhibit; and that conditioning points are often spaced widely enough to render these moments relatively smooth and hence resolvable on a coarser grid than is necessary for Monte Carlo simulations. This notwithstanding, we use in this paper a high-resolution grid to explore superimposed mean-uniform and convergent flows in unconditional and conditional log hydraulic conductivity domains and to compare the relative accuracies of corresponding nonlocal, localized, and Monte Carlo finite element solutions. Whereas our nonlocal solution is nominally restricted to mildly nonuniform media with σ(y)/2 << 1, we find that it actually yields remarkably accurate results for strongly nonuniform media with σ(y)/2 at least as large as 4. Localized solutions deteriorate relative to nonlocal results as one approaches some points of conditioning and singularity or as σ(y)/2 increases. Though they require much less computer time than do nonlocal solutions, localized solutions have the added disadvantage that they yield no information about predictive uncertainty.

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