Stochastic analysis of velocity spatial variability in bounded rectangular heterogeneous aquifers

Osnes, Harald

doi:10.1016/s0309-1708(96)00041-3

Cited by 24 publications

(40 citation statements)

References 33 publications

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“…It is probably worthwhile to note that this expression might reasonably be expected to estimate the flux injection mode more accurately than the uniform, since the inverse arithmetic mean velocity appears to be an excellent estimator of the slope of mean mass arrival time curve for all The mass arrival time statistics cannot be considered to be rigorously valid for flow induced by a constant mean gradient across a three-dimensional aquifer of infinite extent. The velocity fields generated by the simulations will certainly exhibit nonstationarity, and arrival time variances should reasonably be expected to be higher than those expected for the infinite aquifer [e.g., Osnes, 1998]. Nevertheless, the qualitative behavior induced by the strong, persistent initial velocity-travel time correlation would certainly manifest itself in a similar fashion.…”

Section: Resultsmentioning

confidence: 99%

Injection mode implications for solute transport in porous media: Analysis in a stochastic Lagrangian Framework

Demmy¹,

Berglund²,

Graham³

1999

Water Resources Research

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Abstract. The effect of solute injection mode is examined in the stochastic-advective' framework. For three-dimensional heterogeneous aquifers, uniform resident injection of solute results in nonlinear propagation of mass arrival time mean and variance with distance. Injection in flux results in a linear propagation of mean mass arrival time and mass arrival time variance that is both lower and appears to reach a linear regime more rapidly than uniform resident injection. Implementation of instantaneous injections for both modes as boundary conditions for mathematical and numerical models is discussed.

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Section: Resultsmentioning

confidence: 99%

Injection mode implications for solute transport in porous media: Analysis in a stochastic Lagrangian Framework

Demmy¹,

Berglund²,

Graham³

1999

Water Resources Research

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“…Some authors simply disregard terms in the solution which contain products of randomly fluctuating quantities on the assumption that such products are relatively small [Bakr et al, 1978;Mizell et al, 1982]; whereas this simplifies the solution, it is generally not justified [Neuman and Orr, 1993a;Neuman et al, 1996;Paleologos et al, 1996;Tartakovsky and Neuman, 1998]. Most available analytical solutions relate statistical moments of head and flux to those of log hydraulic conductivity in unbounded domains under a uniform mean hydraulic gradient [Bakr et al, 1978;Dagan, 1979Dagan, , 1982Dagan, , 1989 [Osnes, 1995[Osnes, , 1998]. …”

Section: We Define the Conditional Ensemble Moments (Q(x))c And (H(x)mentioning

confidence: 99%

Nonlocal and localized analyses of conditional mean steady state flow in bounded, randomly nonuniform domains: 1. Theory and computational approach

Guadagnini¹,

Neuman²

1999

Water Resources Research

152

162

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where the subscript c implies that conditioning is done on the same data as those used to obtain (K(x))c. These conditional moments constitute optimum unbiased predictors of the random, and therefore generally unknown, system states q(x) and h(x), respectively. We are interested here in computing the state predictors (q(X))c and (h(x))c together with the conditional second moments of the associated prediction errors, q'(x) and h'(x), in 12 and on F. Obviously, the same line of reasoning applies to the unconditional counterparts of the above set of statistical quantities. As will soon become obvious, merely replacing the parameters of standard deterministic models by their (conditional or unconditional) ensemble mean values leads to biased and suboptimal results. One way to render deterministic models more suitable for randomly nonuniform media is to use "effective" or "equivalent" parameters traditionally obtained by some method of "upscaling." Among the more rigorous theoretical criteria of equivalence for hydraulic conductivity are those proposed by Indelman and Dagan [1993] which, however, are not easy to implement in practice. A major conceptual difficulty with upscaling is that it postulates a local relationship between (conditional) mean driving force and flux (in the form of Darcy's law) when, in fact, this relationship is generally nonlocal, as we shall soon see. We describe in this paper a formal method of localizing the above relationship (thereby allowing an effective hydraulic conductivity to be defined in an ensemble sense) and explore its performance numerically. Another conceptual difficulty with traditional upscaling is that it requires the a priori definition of a numerical grid even in the absence of firm theoretical guidelines for its selection. Both the nonlocal and localized approaches we describe in this paper are independent of grid specifications; in our approach a grid is introduced only after, not before, problem formulation.Deterministic alternatives to (conditional or unconditional) Monte Carlo simulation seek to predict flow and transport under uncertainty without having to generate random fields or variables. One approach is to write a system of partial differential equations satisfied approximately by the first two (conditional or unconditional) ensemble moments of a quantity such as hydraulic head [Zhang, 1998] In this paper we describe a numerical method derived rigorously from the exact conditional moment equations for steady state flow originally developed by Neuman and Orr [1993a]. The extension of the methodology to unconditional moment equations is straightforward. Taking the conditional ensemble mean (expectation) of (1) where Gc is the deterministic Green's function associated with Exact Second Conditional Moment EquationsThe (54) and I is the identity tensor. Whereas the first term on the right-hand-side of (53) is isotropic, the second term is generally anisotropic. Since Kc(x) is defined on the support scale to, it is a local rather than an upscaled effective para...

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“…Osnes (1998) used semi-analytical methods to obtain velocity covariances in a two-dimensional, bounded domain. James and Graham (1999) developed a numerical method for accurately approximating head and flux covariances and cross-covariances in a finite two-and three-dimensional domain using the mixed finite element method.…”

Section: à2rmentioning

confidence: 99%

Numerical method of moments for solute transport in a nonstationary flow field conditioned on hydraulic conductivity and head measurements

2006

Stoch Environ Res Risk Assess

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Stochastic analysis of velocity spatial variability in bounded rectangular heterogeneous aquifers

Cited by 24 publications

References 33 publications

Injection mode implications for solute transport in porous media: Analysis in a stochastic Lagrangian Framework

Injection mode implications for solute transport in porous media: Analysis in a stochastic Lagrangian Framework

Nonlocal and localized analyses of conditional mean steady state flow in bounded, randomly nonuniform domains: 1. Theory and computational approach

Numerical method of moments for solute transport in a nonstationary flow field conditioned on hydraulic conductivity and head measurements

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