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Alberto Guadagnini - One of the best experts on this subject based on the ideXlab platform.
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integration of moment equations in a reduced order modeling strategy for monte carlo simulations of groundwater flow
Journal of Hydrology, 2020Co-Authors: Chuanan Xia, Damiano Pasetto, Mario Putti, Alberto GuadagniniAbstract:Abstract We illustrate and test an approach grounded on embedding moment equations (MEs) of groundwater flow within a Monte Carlo based modeling strategy to yield a Reduced-Order Model (ROM) that enables the efficient and accurate evaluation of probability distributions of Hydraulic Heads in randomly heterogeneous transmissivity fields. The projection space determining the accuracy of the ROM solution is typically computed through the principal component analysis of a selected number of full system model solutions (the so-called snapshots). Computationally expensive sensitivity analyses are then required to assess the independence of the ROM from the snapshots. Here, we propose to compute the projection vectors upon relying on the Hydraulic Head covariance evaluated from the solution of corresponding MEs of groundwater flow. Our workflow to compute Hydraulic Head distributions is organized according to the following steps: (i) approximation of mean Hydraulic Head and Head covariance matrix through (second-order accurate) solutions of MEs; (ii) computation of the leading eigenvectors of the Head covariance matrix to form the basis set for the ROM projection space; and (iii) construction of the ROM. Sample probability density functions of Hydraulic Heads are then efficiently obtained via Monte Carlo simulations relying on the developed ROM. The proposed methodology is compared against snapshot-based ROMs and the full system model in a two- and a three-dimensional steady-state groundwater flow setting where pumping from a point source is superimposed to a mean uniform flow. Our results show that the projection space computed by relying on MEs provides a more accurate ROM solution than the one resulting from reliance on snapshots.
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grid convergence for numerical solutions of stochastic moment equations of groundwater flow
Stochastic Environmental Research and Risk Assessment, 2019Co-Authors: Alberto Guadagnini, Monica Riva, Bill X. Hu, Philippe AckererAbstract:We provide qualitative and quantitative assessment of the results of a grid convergence study in terms of (a) the rate/order of convergence and (b) the grid convergence index, GCI, associated with the numerical solutions of moment equations (MEs) of steady-state groundwater flow. The latter are approximated at second order (in terms of the standard deviation of the natural logarithm, Y, of Hydraulic conductivity). We consider (1) the analytical solutions of Riva et al. (Transp Porous Med 45(1):139–193, 2001) for steady-state radial flow in a randomly heterogeneous conductivity field, which we take as references; and (2) the numerical solutions of the MEs satisfied by the (ensemble) mean and (co)variance of Hydraulic Head and fluxes. Based on 45 numerical grids associated with differing degrees of discretization, we find a supra-linear rate of convergence for the mean and (co)variance of Hydraulic Head and for the variance of the transverse component of fluxes, the variance of radial fluxes being characterized by a sub-linear convergence rate. Our estimated values of GCI suggest that an accurate computation of mean and (co)variance of Head and fluxes requires a space discretization comprising at least 8 grid elements per correlation length of Y, an even finer discretization being required for an accurate representation of the second-order component of mean Heads.
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joint inversion of steady state hydrologic and self potential data for 3d Hydraulic conductivity distribution at the boise hydrogeophysical research site
Journal of Hydrology, 2011Co-Authors: Salvatore Straface, Monica Riva, Francesco Chidichimo, Enzo Rizzo, Warren Barrash, Andre Revil, Michael Cardiff, Alberto GuadagniniAbstract:We combine sedimentological, Hydraulic and geophysical information to characterize the 3D distribution of transport properties of an heterogeneous aquifer. We focus on the joint inversion of Hydraulic Head and self-potential measurements collected during an extensive experimental campaign performed at the Boise Hydrogeophysical Research Site (BHRS), Boise, Idaho, and involving a series of dipole tests. While Hydraulic Head data obtained from piezometric readings in open wells represent a depth-averaged value, self-potential signals provide an estimate of the water table location. The aquifer is conceptualized as a multiple-continuum, where the volumetric fraction of a geo-material within a cell of the numerical flow model is calculated by Multiple Indicator Kriging. The latter is implemented on the basis of available sedimentological information. The functional format of the indicator variograms and associated parameters are estimated on the basis of formal model identification criteria. Self-potential and Hydraulic Head data have been embedded jointly within a three-dimensional inverse model of groundwater flow at the site. Each identified geo-material (category) is assumed to be characterized by a constant Hydraulic conductivity. The latter constitute the set of model parameters. The Hydraulic conductivity associated with a numerical block is then calculated as a weighted average of the conductivities of the geo-materials which are collocated in the block by means of Multiple Indicator Kriging. Model parameters are estimated by a Maximum Likelihood fit between measured and modeled state variables, resulting in a spatially heterogeneous distribution of Hydraulic conductivity. The latter is effectively constrained on the sedimentological data and conditioned on both self-potential and borehole Hydraulic Head readings. Minimization of the Maximum Likelihood objective function allows estimating the relative weight of measurement errors associated with self-potential and borehole-based Head data. The procedure adopted allowed a reconstruction of the heterogeneity of the site with a level of details, which was not obtained in previous studies and with relatively modest computational efforts. Further validation against dipole tests which were not used in the inversion procedure supports the robustness of the results.
Andrés Sahuquillo - One of the best experts on this subject based on the ideXlab platform.
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solving the steady state groundwater flow equation for finite linear aquifers using a generalized fourier series approach in two dimensional domains
International Journal for Numerical Methods in Engineering, 2009Co-Authors: Jose E Capilla, Andrés Sahuquillo, David Pulidovelazquez, Joaquín AndreuAbstract:A method to solve steady linear groundwater flow problems using generalized Fourier Series is developed and particularized for multiple Fourier series in two-dimensional domains. It leads to a linear vector equation whose solution provides a finite number of generalized Fourier coefficients approximating the Hydraulic Head field. Its implementation is shown and two relevant properties are found for the system matrix. It is always symmetric and, once computed, if additional Fourier terms are needed for a better approximation of the Hydraulic Head field, previously computed matrix elements remain invariant, i.e. only new rows and columns are added to the system matrix. The method is demonstrated in three simple cases with different geometries and transmissivity fields, where solutions are compared with analytical and finite element method results. Thus, the method is verified as an alternative to other flow solvers. Additionally, it provides a direct way to obtain the spectral form of the flow equation solution, given a spectral representation of transmissivity, and can be easily extended to obtain continuous velocity fields and their approximated spectral expressions. Copyright © 2009 John Wiley & Sons, Ltd.
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joint simulation of transmissivity and storativity fields conditional to steady state and transient Hydraulic Head data
Advances in Water Resources, 1999Co-Authors: Harriejan Hendricks Franssen, Jose E Capilla, Jaime J Gomezhernandez, Andrés SahuquilloAbstract:Abstract The self-calibrated method has been extended for the generation of equally likely realizations of transmissivity and storativity conditional to transmissivity and storativity data and to steady-state and transient Hydraulic Head data. Conditioning to transmissivity and storativity data is achieved by means of standard geostatistical co-simulation algorithms, whereas conditioning to Hydraulic Head data, given its non-linear relation to transmissivity and storativity, is achieved through non-linear optimization, similar to standard inverse algorithms. The algorithm is demonstrated in a synthetic study based on data from the WIPP site in New Mexico. Seven alternative scenarios are investigated, generating 100 realizations for each of them. The differences among the scenarios range from the number of conditioning data, to their spatial configuration, to the pumping strategies at the pumping wells. In all scenarios, the self-calibrated algorithm is able to generate transmissivity–storativity realization couples conditional to all the sample data. For the specific case studied here the results are not surprising. Of the piezometric Head data, the steady-state values are the most consequential for transmissivity characterization. Conditioning to transient Head data only introduces local adjustments on the transmissivity fields and serves to improve the characterization of the storativity fields.
Allan D Woodbury - One of the best experts on this subject based on the ideXlab platform.
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a full bayesian approach to the inverse problem for steady state groundwater flow and heat transport
Geophysical Journal International, 2006Co-Authors: Yefang Jiang, Allan D WoodburyAbstract:SUMMARY The full (hierarchal) Bayesian approach proposed by Woodbury & Ulrych and Jiang et al. is extended to the inverse problem for 2-D steady-state groundwater flow and heat transport. A stochastic conceptual framework for the heat flow and groundwater flow is adopted. A perturbation of both the groundwater flow and the advection-conduction heat transport equations leads to a linear formulation between Heads, temperature and logarithm transmissivity [denoted as ln (T)]. A Bayesian updating procedure similar to that of Woodbury & Ulrych can then be performed. This new algorithm is examined against a generic example through simulations. The prior mean, variance and integral scales of ln (T) (hyperparameters) are treated as random variables and their pdfs are determined from maximum entropy considerations. It is also assumed that the statistical properties of the noise in the Hydraulic Head and temperature measurements are also uncertain. Uncertainties in all pertinent hyperparameters are removed by marginalization. It is found that the use of temperature measurements is showed to further improve the ln (T) estimates for the test case in comparison to the updated ln (T) field conditioned on ln (T) and Head data; the addition of temperature data without Hydraulic Head data to the update also aids refinement of the ln (T) field compared to simply interpolating ln (T) data alone these results suggest that temperature measurements are a promising data source for site characterization for heterogeneous aquifer, which can be accomplished through the full-Bayesian methodology.
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a full bayesian approach to the groundwater inverse problem for steady state flow
Water Resources Research, 2000Co-Authors: Allan D Woodbury, Tadeusz J UlrychAbstract:A full-Bayesian approach to the estimation of transmissivity from Hydraulic Head and transmissivity measurements is developed for two-dimensional steady state groundwater flow. The approach combines both Bayesian and maximum entropy viewpoints of probability. In the first phase, log transmissivity measurements are incorporated into Bayes' theorem, and the prior probability density function is updated, yielding posterior estimates of the mean value of the log transmissivity field and covariance. The two central moments are generated assuming that the prior mean, variance, and integral scales are “hyperparameters”; that is, they are treated as random variables in themselves which is contrary to classical statistical approaches. The probability density functions (pdfs) of these hyperparameters are, in turn, determined from maximum entropy considerations. In other words, pdfs are chosen for each of the hyperparameters that are maximally uncommitted with respect to unknown information. This methodology is quite general and provides an alternative to kriging for spatial interpolation. The final step consists of updating the conditioned natural logarithm transmissivity (ln(T)) field with Hydraulic Head measurements, utilizing a linearized aquifer equation. It is assumed that the statistical properties of the noise in the Hydraulic Head measurements are also uncertain. At each step, uncertainties in all pertinent hyperparameters are removed by marginalization. Finally, what is produced is a ln(T) field conditioned on measurements of both Hydraulic Heads and log transmissivity and covariances of the ln(T) field. In addition, we can also produce resolution matrices, confidence (credibility) limits, and the like for the ln(T) field. It is shown that the application of the methodology yields good estimates of transmissivities, even when Hydraulic Head measurements are noisy and little or no information is specified on mean values of ln(T), variance of ln(T), and integral scales.
Jose E Capilla - One of the best experts on this subject based on the ideXlab platform.
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solving the steady state groundwater flow equation for finite linear aquifers using a generalized fourier series approach in two dimensional domains
International Journal for Numerical Methods in Engineering, 2009Co-Authors: Jose E Capilla, Andrés Sahuquillo, David Pulidovelazquez, Joaquín AndreuAbstract:A method to solve steady linear groundwater flow problems using generalized Fourier Series is developed and particularized for multiple Fourier series in two-dimensional domains. It leads to a linear vector equation whose solution provides a finite number of generalized Fourier coefficients approximating the Hydraulic Head field. Its implementation is shown and two relevant properties are found for the system matrix. It is always symmetric and, once computed, if additional Fourier terms are needed for a better approximation of the Hydraulic Head field, previously computed matrix elements remain invariant, i.e. only new rows and columns are added to the system matrix. The method is demonstrated in three simple cases with different geometries and transmissivity fields, where solutions are compared with analytical and finite element method results. Thus, the method is verified as an alternative to other flow solvers. Additionally, it provides a direct way to obtain the spectral form of the flow equation solution, given a spectral representation of transmissivity, and can be easily extended to obtain continuous velocity fields and their approximated spectral expressions. Copyright © 2009 John Wiley & Sons, Ltd.
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joint simulation of transmissivity and storativity fields conditional to steady state and transient Hydraulic Head data
Advances in Water Resources, 1999Co-Authors: Harriejan Hendricks Franssen, Jose E Capilla, Jaime J Gomezhernandez, Andrés SahuquilloAbstract:Abstract The self-calibrated method has been extended for the generation of equally likely realizations of transmissivity and storativity conditional to transmissivity and storativity data and to steady-state and transient Hydraulic Head data. Conditioning to transmissivity and storativity data is achieved by means of standard geostatistical co-simulation algorithms, whereas conditioning to Hydraulic Head data, given its non-linear relation to transmissivity and storativity, is achieved through non-linear optimization, similar to standard inverse algorithms. The algorithm is demonstrated in a synthetic study based on data from the WIPP site in New Mexico. Seven alternative scenarios are investigated, generating 100 realizations for each of them. The differences among the scenarios range from the number of conditioning data, to their spatial configuration, to the pumping strategies at the pumping wells. In all scenarios, the self-calibrated algorithm is able to generate transmissivity–storativity realization couples conditional to all the sample data. For the specific case studied here the results are not surprising. Of the piezometric Head data, the steady-state values are the most consequential for transmissivity characterization. Conditioning to transient Head data only introduces local adjustments on the transmissivity fields and serves to improve the characterization of the storativity fields.
John A Cherry - One of the best experts on this subject based on the ideXlab platform.
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structural and statistical characterization of joints and multi scale faults in an alternating sandstone and shale turbidite sequence at the santa susana field laboratory implications for their effects on groundwater flow and contaminant transport
Journal of Structural Geology, 2016Co-Authors: Antonino Cilona, Atilla Aydin, Jeremias Likerman, Beth L Parker, John A CherryAbstract:Abstract This paper describes the properties of faults and fractures in the Upper Cretaceous Chatsworth Formation exposed at Santa Susana Field Laboratory and its surroundings (Simi Hills, California), where groundwater flow and contamination have been studied for over three decades. The complex depositional architecture of this turbidite consisting of alternating sandstones and shales, interacting with formative stress conditions are responsible for multi-scale fault hierarchies and permeable fractures in which nearly all groundwater flow occurs. Intensity and distribution of background fractures and their relation to bedding thickness are established for sandstones, the dominant lithology. The architecture of faults with increasing displacement is described, and relationships among fault dimensional parameters captured. Data from ∼400 boreholes and piezometers reveal the effect of faults and fractures on groundwater flow. Large Hydraulic Head differences, observed across fault zones with shale-rich cores, indicate these structures as cross-flow barriers. Moreover, Hydraulic Head profiles under ambient conditions, and pumping tests suggest strong Hydraulic connectivity in all directions to depth of hundreds of meters. This outcrop-based structural characterization relates the horizontal Hydraulic conductivity to the observed well-connected fracture network, and explains the strong vertical connectivity across low-Hydraulic conductivity shales as faults and sheared fractures provide flow pathways.
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characteristics of high resolution Hydraulic Head profiles and vertical gradients in fractured sedimentary rocks
Journal of Hydrology, 2014Co-Authors: Jessica R Meyer, Beth L Parker, John A CherryAbstract:Summary Accurately identifying the position of vertical Hydraulic conductivity (Kv) contrasts is critical to the delineation of hydrogeologic units that serve as the basis for conceptual and numerical models of groundwater flow. High resolution Head profiles have identified the position and thickness of Kv contrasts in clayey aquitards but this potential has not yet been thoroughly evaluated in sedimentary rocks. This paper describes an experiment in which Head profiles with the highest, technically feasible resolution were obtained using Westbay® multilevel systems (MLS) installed in 15 cored holes at three sedimentary rock research sites with contrasting geologic and flow system conditions. MLSs were installed to maximum depths between 90 and 260 m with 2–5 monitoring zones per 10 m. Head profiles were measured over multiyear periods. The vertical component of Hydraulic gradient (i.e., vertical gradient) was calculated for each pair of adjacent monitoring intervals in every MLS and then categorized based on its repeatability to facilitate interpretation of Kv contrasts and comparisons within boreholes, between boreholes at the same site, and between sites. The Head and vertical gradient profiles from all three sites display systematic (i.e., simple, geometric) shapes defined by repeatable intervals of no to minimal vertical gradient, indicating relatively high Kv units, bounded by shorter depth intervals with large (up to −50 m/m) vertical gradients, indicating relatively low Kv units. The systematic nature of the profiles suggests flow in regular and interconnected fracture networks rather than dominated by a few key fractures with irregular orientations. The low Kv units were typically thin, with their positions and thicknesses not predicted by lithostratigraphy or detailed lithologic, geophysical, and horizontal Hydraulic conductivity data. Hence, the position and thickness of units with contrasting Kv would not be evident if MLSs with the conventional number of monitoring zones had been used. Furthermore, the detailed profiles can be strongly diagnostic of hydrogeologic unit boundaries or layers and can be used to improve the quantitative assessment of flow system conditions that is foundational to understanding contaminant plume migration.
