Correction to “Prediction of steady state flow in nonuniform geologic media by conditional moments: Exact nonlocal formalism, effective conductivities, and weak approximation” by Shlomo P. Neuman and Shlomo Orr

Neuman, Shlomo P.; Orr, Shlomo

doi:10.1029/93wr00459

Cited by 16 publications

(9 citation statements)

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“…Neuman and coworkers [ Neuman and Orr , 1993; Neuman et al , 1996; Tartakovsky and Neuman , 1998a, 1998b; Guadagnini and Neuman , 1999a, 1999b; Riva et al , 2001; Tartakovsky et al , 2002; Guadagnini et al , 2003] analyzed effective and pseudoeffective hydraulic conductivity by means of moment equations of transient and steady state groundwater flow. Under steady state the predictor of the flux, 〈 q 〉, is written as where it can be easily recognized that 〈 q 〉 is expressed in terms of a Darcian component, which is governed by the hydraulic conductivity predictor, 〈K〉, and a (generally) non‐Darcian one, which is termed residual flux, r .…”

Section: Mean Parallel Flow: Effective and Pseudoeffective Parametersmentioning

confidence: 99%

“…Under steady state the predictor of the flux, 〈 q 〉, is written as where it can be easily recognized that 〈 q 〉 is expressed in terms of a Darcian component, which is governed by the hydraulic conductivity predictor, 〈K〉, and a (generally) non‐Darcian one, which is termed residual flux, r . It has been shown by Neuman and Orr [1993] and Neuman et al [1996] that for a bounded domain, Ω, with impermeable Neumann boundaries, the residual flux is given exactly by the compact explicit expression Here h c is the solution of the following deterministic flow problem where 〈f( x )〉 is the mean source term and the boundary conditions are provided in terms of (ensemble) mean quantities. The kernel is a second‐rank positive semidefinite symmetric tensor, G r ( y , x ) being the random Green's function associated with the groundwater flow problem, i.e., the solution of the (random) groundwater flow equation for the case where the source function is replaced by the Dirac delta function δ ( x − y ), subject to homogeneous boundary conditions [ Greenberg , 1971].…”

Section: Mean Parallel Flow: Effective and Pseudoeffective Parametersmentioning

confidence: 99%

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Representative hydraulic conductivities in saturated groundwater flow

Sánchez-Vila

Guadagnini

Carrera

2006

Reviews of Geophysics

274

209

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Heterogeneity is the single most salient feature of hydrogeology. An enormous amount of work has been devoted during the last 30 years to addressing this issue. Our objective is to synthesize and to offer a critical appraisal of results related to the problem of finding representative hydraulic conductivities. By representative hydraulic conductivity we mean a parameter controlling the average behavior of groundwater flow within an aquifer at a given scale. Three related concepts are defined: effective hydraulic conductivity, which relates the ensemble averages of flux and head gradient; equivalent conductivity, which relates the spatial averages of flux and head gradient within a given volume of an aquifer; and interpreted conductivity, which is the one derived from interpretation of field data. Most theoretical results are related to effective conductivity, and their application to real world scenarios relies on ergodic assumptions. Fortunately, a number of results are available suggesting that conventional hydraulic test interpretations yield (interpreted) hydraulic conductivity values that can be closely linked to equivalent and/or effective hydraulic conductivities. Complex spatial distributions of geologic hydrofacies and flow conditions have a strong impact upon the existence and the actual values of representative parameters. Therefore it is not surprising that a large body of literature provides particular solutions for simplified boundary conditions and geological settings, which are, nevertheless, useful for many practical applications. Still, frequent observations of scale effects imply that efforts should be directed at characterizing well-connected stochastic random fields and at evaluating the corresponding representative hydraulic conductivities

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Section: Mean Parallel Flow: Effective and Pseudoeffective Parametersmentioning

confidence: 99%

Section: Mean Parallel Flow: Effective and Pseudoeffective Parametersmentioning

confidence: 99%

Representative hydraulic conductivities in saturated groundwater flow

Sánchez-Vila

Guadagnini

Carrera

2006

Reviews of Geophysics

274

209

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“…A major conceptual difficulty with upscaling is that it postulates a local relationship between (conditional) mean driving force and flux (Darcy's law) when in fact this relationship is generally nonlocal [Neuman and Orr, 1993;Neuman et al, 1996;Tartakovsky and Neuman, 1998a, b]. Even where localization is possible, the constitutive equations satisfied by conditional mean predictors may be fundamentally different from those satisfied by their random counterparts [Neuman et al, 1999].…”

Section: Introductionmentioning

confidence: 99%

“…We explore the extent to which our assumptions regarding the properties of a are justified by providing a brief review of published studies concerning its spatial variability. Treating it as a random constant allows us to develop exact conditional first-and second-moment equations for steady state unsaturated flow which have integrodifferential forms similar to those developed for steady state saturated flow by Neuman and Orr [1993], Neuman et al [1996], and Guadagnini and Neuman [1997,1998]. Upon introducing the additional assumption that a has a relatively small variance, the Kirchhoff transformation allows us to solve these stochastic moment equations analytically by perturbation in terms of a small parameter representing the variance of the natural logarithm of saturated hydraulic conductivity.…”

Section: Introductionmentioning

confidence: 99%

Conditional stochastic averaging of steady state unsaturated flow by means of Kirchhoff Transformation

Tartakovsky¹,

Neuman²,

Lü³

1999

Water Resources Research

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Abstract. We consider the effect of measuring randomly varying soil hydraulic properties on one's ability to predict steady state unsaturated flow subject to random sources and/or initial and boundary conditions. Our aim is to allow optimum unbiased prediction of system states (pressure head, water content) and fluxes deterministically, without upscaling and without linearizing the constitutive characteristics of the soil. It has been shown by Neuman et al. [1999] that such prediction is possible by means of first ensemble moments of system states and fluxes, conditioned on measured values of soil properties, when the latter scale in a linearly separable fashion as proposed by Vogel et al. [1991]; the uncertainty associated with such predictions can be quantified by means of the corresponding conditional second moments. The derivation of moment equations for soils whose properties do not scale in the above manner requires linearizing the corresponding constitutive relations, which may lead to major inaccuracies when these relations are highly nonlinear, as is often the case in nature. When the scaling parameter of pressure head is a random variable independent of location, the steady state unsaturated flow equation can be linearized by means of the Kirchhoff transformation for gravity-free flow. Linearization is also possible in the presence of gravity when hydraulic conductivity varies exponentially with pressure head. For the latter case we develop exact conditional firstand second-moment equations which are nonlocal and therefore non-Darcian. We solve these equations analytically by perturbation for unconditional vertical infiltration and compare our solution with the results of numerical Monte Carlo simulations. Our analytical solution demonstrates in a rigorous manner that the concept of effective hydraulic conductivity does not apply to ensemble-averaged unsaturated flow except when gravity is the sole driving force.

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“…Neuman and Orr [1993] and Neuman et al [1996] developed an exact nonlocal formalism for the prediction of steady state flow in randomly heterogeneous geologic media by conditional moments under the action of uncertain forcing terms (sources and boundary conditions). They started from the premise that Darcy's law applies locally, at some support scale 60, which need not constitute a representative elementary volume (REV) in any traditional sense of the term.…”

Section: Introductionmentioning

confidence: 99%

Transient flow in bounded randomly heterogeneous domains: 1. Exact conditional moment equations and recursive approximations

Tartakovsky¹,

Neuman²

1998

Water Resources Research

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102

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Abstract. We consider the effect of measuring randomly varying local hydraulic conductivities K(x) on one's ability to predict transient flow within bounded domains, driven by random sources, initial head, and boundary conditions. Our aim is to allow optimum unbiased prediction of local hydraulic heads h(x, t) and Darcy fluxes q(x, t) by means of their ensemble moments, (h(x, t))c and (q(x, t))c, conditioned on measurements of K(x). We show that these predictors satisfy a compact deterministic flow equation which contains a space-time integrodifferential "residual flux" term. This term renders (q(x, t))c nonlocal and non-Darcian so that the concept of effective hydraulic conductivity looses meaning in all but a few special cases. Instead, the residual flux contains kernels that constitute nonlocal parameters in space-time that are additionally conditional on hydraulic conductivity data and thus nonunique. The kernels include symmetric and nonsymmetric second-rank tensors as well as vectors. We also develop nonlocal equations for second conditional moments of head and flux which constitute measures of predictive uncertainty. The nonlocal expressions cannot be evaluated directly without either a closure approximation or high-resolution conditional Monte Carlo simulation. To render our theory workable, we develop recursive closure approximations for the moment equations through expansion in powers of a small parameter which represents the standard estimation error of natural log K(x). These approximations are valid to arbitrary order for either mildly heterogeneous or well-conditioned strongly heterogeneous media. They allow, in principle, evaluating the conditional moments numerically on relatively coarse grids, without upscaling, by standard methods such as finite elements.

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Correction to “Prediction of steady state flow in nonuniform geologic media by conditional moments: Exact nonlocal formalism, effective conductivities, and weak approximation” by Shlomo P. Neuman and Shlomo Orr

Cited by 16 publications

References 0 publications

Representative hydraulic conductivities in saturated groundwater flow

Representative hydraulic conductivities in saturated groundwater flow

Conditional stochastic averaging of steady state unsaturated flow by means of Kirchhoff Transformation

Transient flow in bounded randomly heterogeneous domains: 1. Exact conditional moment equations and recursive approximations

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