Assessment of uncertainty associated with the estimation of well catchments by moment equations

Riva, Mònica; Guadagnini, Alberto; Simoni, Michela

doi:10.1016/j.advwatres.2005.07.005

Cited by 18 publications

(10 citation statements)

References 32 publications

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“…Recent applications of first-order methods have been extended to the determinations of stochastic well capture zones (e.g. Kunstmann and Kinzelbach, 2000;Stauffer et al, 2002Stauffer et al, , 2004Lu and Zhang, 2003;Zhang and Lu, 2004;Lessoff and Indelman, 2004;Bakr and Butler, 2005;Riva et al, 2006;Kunstmann and Kastens, 2006;Indelman et al, 2006). Most studies on the subject dealt with depth-averaged two-dimensional problems (Kunstmann and Kinzelbach, 2000;Stauffer et al, 2002Stauffer et al, , 2004Lu and Zhang, 2003;Zhang and Lu, 2004;Bakr and Butler, 2005;Riva et al, 2006;Kunstmann and Kastens, 2006).…”

Section: Introductionmentioning

confidence: 99%

“…Varljen and Schafer, 1991;Franzetti and Guadagnini, 1996;2292 C.-F. Ni et al: Quantifying flow and remediation zone uncertainties Lessoff and Indelman, 2004;Indelman et al, 2006;Riva et al, 2006;Kunstmann and Kastens, 2006;Guha, 2008). The MCS is conceptually straightforward for determining stochastic remediation zones in heterogeneous aquifers.…”

Section: Introductionmentioning

confidence: 99%

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Quantifying flow and remediation zone uncertainties for partially opened wells in heterogeneous aquifers

Lin

et al. 2011

Hydrol. Earth Syst. Sci.

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Abstract. This study presents a numerical first-order spectral model to quantify transient flow and remediation zone uncertainties for partially opened wells in heterogeneous aquifers. Taking advantages of spectral theories in solving unmodeled small-scale variability in hydraulic conductivity (K), the presented nonstationary spectral method (NSM) can efficiently estimate flow uncertainties, including hydraulic heads and Darcy velocities in r-and z-directions in a cylindrical coordinate system. The velocity uncertainties associated with the particle backward tracking algorithm are then used to estimate stochastic remediation zones for scenarios with partially opened well screens. In this study the flow and remediation zone uncertainties obtained by NSM were first compared with those obtained by Monte Carlo simulations (MCS). A layered aquifer with different geometric mean of K and screen locations was then illustrated with the developed NSM. To compare NSM flow and remediation zone uncertainties with those of MCS, three different small-scale K variances and correlation lengths were considered for illustration purpose. The MCS remediation zones for different degrees of heterogeneity were presented with the uncertainty clouds obtained by 200 equally likely MCS realizations. Results of simulations reveal that the first-order NSM solutions agree well with those of MCS for partially opened wells. The flow uncertainties obtained by using NSM and MCS show identically for aquifers with small ln K variances and correlation lengths. Based on the test examples, the remediation zone uncertainties (bandwidths) are not sensitive to the changes of small-scale ln K correlation lengths. However, the increases of remediation zone uncertainties (i.e. the Correspondence to: C.-F. Ni (nichuenfa@geo.ncu.edu.tw) uncertainty bandwidths) are significant with the increases of small-scale ln K variances. The largest displacement uncertainties may have several meters of differences when the ln K variances increase from 0.1 to 1.0. Such conclusions are also valid for the estimations of remediation zones in layered aquifers.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quantifying flow and remediation zone uncertainties for partially opened wells in heterogeneous aquifers

Lin

et al. 2011

Hydrol. Earth Syst. Sci.

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“…This approach is known in the literature as moment equations (see, e.g., [41,46,4,30,39,42,28]). We do not detail here the derivation and algorithmic implementation of the moment equations, which can be found in [10].…”

Section: Taylor Expansion Let 0 < σ < 1 Be the Standard Deviation Ofmentioning

confidence: 99%

Perturbation Analysis for the Darcy Problem with Log-Normal Permeability

Bonizzoni¹,

Nobile²

2014

SIAM/ASA J. Uncertainty Quantification

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Abstract. We study the single-phase flow in a saturated, bounded heterogeneous porous medium. We model the permeability as a log-normal random field. We perform a perturbation analysis, expanding the solution in Taylor series. The approximation properties of the Taylor polynomial are studied, and the local convergence of the Taylor series is proved. With a counterexample we show that, in general, the Taylor series is not globally convergent to the stochastic solution as the polynomial degree goes to infinity. Nevertheless, for small variability of the permeability field and low degree of the Taylor polynomial, the perturbation approach is feasible and provides a good approximation of both the stochastic solution and the statistical moments of the stochastic solution. We derive an upper bound on the norm of the residual of the Taylor series, which predicts the optimal degree of the Taylor polynomial to consider. The upper bound is quite pessimistic. In the simple case of a permeability field described by only one random variable, we show numerically that a simple "tuning" of the upper bound, which uses estimates of the growth of the derivatives, provides sharp bounds.

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“…For example, if we model single or multi-phase flow in a porous medium, randomness arises in the permeability tensor, due to the impossibility of a full characterization of conductivity properties of subsurface media, but also in the source term, typically pressure gradients or impervious boundaries. See for example Tartakovsky & Neuman (1998), Guadagnini & Neuman (1999a,b), Zhang (2002), Riva et al (2006), Babuška et al (2007) and Franssen et al (2009). Similar situations appear in many other applications, such as combustion flows, earthquake engineering, biomedical engineering and finance.…”

Section: Introductionmentioning

confidence: 99%

“…This case can be treated for small randomness by a perturbation approach (Taylor or Neumann expansions, see e.g. Tartakovsky & Neuman, 1998;Guadagnini & Neuman, 1999a,b;Riva et al, 2006 from the hydrology literature, and Babuška & Chatzipantelidis, 2002;Cohen et al, 2011;Bonizzoni & Nobile, 2013;Bonizzoni, 2013) and is currently under investigation. The outline of the paper is the following: in Section 2, we recall the Sobolev spaces of differential forms and the main results on the mixed formulation of the Hodge-Laplace problem in the deterministic setting, stating the well-posedness of the problem and translating it into the language of PDEs using proxy fields.…”

Section: Introductionmentioning

confidence: 99%

Moment equations for the mixed formulation of the Hodge Laplacian with stochastic loading term

Bonizzoni

Buffa²,

Nobile³

2013

IMA Journal of Numerical Analysis

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We study the mixed formulation of the stochastic Hodge-Laplace problem defined on an n-dimensional domain D (n 1), with random forcing term. In particular, we focus on the magnetostatic problem and on the Darcy problem in the three-dimensional case. We derive and analyse the moment equations, that is, the deterministic equations solved by the mth moment (m 1) of the unique stochastic solution of the stochastic problem. We find stable tensor product finite element discretizations, both full and sparse, and provide optimal order-of-convergence estimates. In particular, we prove the inf-sup condition for sparse tensor product finite element spaces.Keywords: finite element exterior calculus; Hodge Laplacian; mixed finite elements; uncertainty quantification; stochastic partial differential equations; moment equations; sparse tensor product approximation.

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Assessment of uncertainty associated with the estimation of well catchments by moment equations

Cited by 18 publications

References 32 publications

Quantifying flow and remediation zone uncertainties for partially opened wells in heterogeneous aquifers

Quantifying flow and remediation zone uncertainties for partially opened wells in heterogeneous aquifers

Perturbation Analysis for the Darcy Problem with Log-Normal Permeability

Moment equations for the mixed formulation of the Hodge Laplacian with stochastic loading term

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