7,195 research outputs found
Closure to "Hydrodynamic loading on river bridges" by Stefano Malavasi and Alberto Guadagnini
No full textRecursive conditional moment equations for advective transport in randomly heterogeneous velocity fields
No full textFlow and transport parameters such as hydraulic conductivity, seepage velocity, and dispersivity have been traditionally viewed as well-defined local quantities that can be assigned unique values at each point in space-time. Yet in practice these parameters can be deduced from measurements only at selected locations where their values depend on the scale (support volume) and mode (instruments and procedure) of measurement. Quite often, the support of the measurements is uncertain and the data are corrupted by experimental and interpretive errors. Estimating the parameters at points where measurements are not available entails an additional random error. These errors and uncertainties render the parameters random and the corresponding flow and transport equations stochastic. The stochastic flow and transport equations can be solved numerically by conditional Monte Carlo simulation. However, this procedure is computationally demanding and lacks well-established convergence criteria. An alternative to such simulation is provided by conditional moment equations, which yield corresponding predictions of flow and transport deterministically. These equations are typically integro-differential and include nonlocal parameters that depend on more than one point in space-time. The traditional concept of a REV (representative elementary volume) is neither necessary nor relevant for their validity or application. The parameters are nonunique in that they depend not only on local medium properties but also on the information one has about these properties (scale, location, quantity, and quality of data). Darcy’s law and Fick’s analogy are generally not obeyed by the flow and transport predictors except in special cases or as localized approximations. Such approximations yield familiar-looking differential equations which, however, acquire a non-traditional meaning in that their parameters (hydraulic conductivity, seepage velocity, dispersivity) and state variables (hydraulic head, concentration) are information-dependent and therefore, inherently nonunique. Nonlocal equations contain information about predictive uncertainty, localized equations do not. We have shown previously (Guadagnini and Neuman, 1997, 1998, 1999a, b) how to solve conditional moment equations of steady-state flow numerically on the basis of recursive approximations similar to those developed for transient flow by Tartakovsky and Neuman (1998, 1999). Our solution yields conditional moments of velocity, which are required for the numerical computation of conditional moments associated with transport. In this paper, we lay the theoretical groundwork for such computations by developing exact integro-differential expressions for second conditional moments, and recursive approximations for all conditional moments, of advective transport in a manner that complements earlier work along these lines by Neuman (1993)
Statistics of hydraulic head in randomly heterogeneous well fields with truncated multiscale variogram
No full textWe consider two-dimensional steady state flow toward a well that fully penetrates a randomly heterogeneous aquifer. A constant pumping rate is prescribed deterministically at the well while a constant head is maintained at a circular outer boundary of radius L. Flow occurs over an infinite hierarchy of mutually uncorrelated, statistically homogeneous, and isotropic random fields (modes) of natural log transmissivity, Y(r), each of which is associated with a Gaussian variogram. Here we consider only a lower cut-off of the hierarchy. We develop an analytical solution for the mean and variance of hydraulic head based on the nonlocal theory first proposed for steady state flows in bounded, randomly heterogeneous media by Neuman and Orr [1993] and Guadagnini and Neuman [1999a]. In particular, we develop and solve analytically recursive closure approximations of the governing nonlocal moment equations to second order in Y. Analytical solutions are evaluated by Gaussian quadratures. The two-dimensional nature of our solution renders it useful for relatively thin aquifers in which vertical heterogeneity tends to be of minor concern relative to that in the horizontal plane. Potential uses include the analysis of pumping tests and tracer test, the statistical delineation of their respective capture zones, and the analysis of contaminant transport toward fully penetrating wells
Nonlocal and localized analyses of nonreactive solute transport in bounded randomly heterogeneous porous media: Computational Analysis
No full textIn a companion paper we presented exact (though implicit and not closed) nonlocal conditional first and second moment equations for nonreactive advective–dispersive transport under both steady state and transient flow regimes in bounded, randomly heterogeneous porous domains. To allow solving our nonlocal equations we developed recursive moment equations in Laplace space for the special case of steady state flow to second order in the standard deviation of natural log hydraulic conductivity, Y, which is generally nonhomogeneous, and proposed a higher-order iterative closure scheme. We also showed that, under a limited set of conditions, the mean transport equation can be localized to yield a familiar-looking advection–dispersion equation with a conditional macrodispersion tensor that generally varies in space–time. The purpose of this paper is to explore the behavior and assess the accuracy and computational efficiency of our moment solutions in comparison to conditional and unconditional Monte Carlo simulations. To do so, we present a high-accuracy computational algorithm for our iterative nonlocal and recursive localized moment equations and corresponding computational results in two spatial dimensions conditional on measurements of Y. Our algorithm solves the moment equations by finite elements in Laplace-transformed space and inverts the solution numerically back into the time domain. Conditional results obtained with our iterative algorithm compare well with Monte Carlo simulations for log-conductivity variance of 0.3 and Peclet number Pe = 100 defined in terms of the integral scale of Y, and for Pe = 10 in the unconditional case. As log-conductivity variance, Pe and time increase the quality of our iterative moment solution deteriorates. We show that this is due to our disregarding velocity moments of order higher than two and propose that including such moments should render our iterative solution workable over a wider range of these parameters. Second-order recursive nonlocal and space-localized results are considerably less accurate than those obtained with our iterative nonlocal algorithm. Even though our moment solution does not require computing (space–time localized) macrodispersion coefficients, we nevertheless do so to examine the influence of boundaries and conditioning on their behavior. Our results support an earlier observation by the second author [Morales Casique E, Neuman SP, Guadagnini A. Nonlocal and localized analyses of nonreactive solute transport in bounded randomly heterogeneous porous media: theoretical framework. Adv Water Resour, in press., Neuman SP. Universal scaling of hydraulic conductivities and dispersivities in geologic media. Water Resour Res 1990;26(8):1749–58], based on world-wide tracer test results, that the rate at which apparent longitudinal dispersivity increases with scale diminishes with conditioning. In preliminary runs conducted on a relatively small grid without optimizing our algorithms and without parallelization, the moment solutions required considerably less computer time than did the Monte Carlo simulations
Mean travel time of conservative solutes in randomly heterogeneous unbounded domains under mean uniform flow
No full textWe derive a closed-form expression for mean travel time of a conservative solute migrating under uniform in the mean flow conditions within an infinite stationary field with simple exponential correlation of the natural logarithm of hydraulic conductivity. Our
expression is developed from a consistent second-order expansion in sY (standard deviation of the log hydraulic conductivity) of the equations for moments of travel time and trajectories of conservative solutes in two-dimensional randomly nonuniform flows of Guadagnini et al. [2001]. As such, it is nominally valid for moderately heterogeneous fields, with sY < 1. Its validity for larger heterogeneity degrees is tested against numerical
Monte Carlo simulations. Our results clarify the nonlinear effect in the mean travel time with respect to distance that has been observed numerically (and modeled empirically) in
the literature
Effects of uncertainty of lithofacies, conductivity and porosity distributions on stochastic interpretations of a field scale tracer test.
No full textWe investigate the importance of selecting two different methodologies for the determination of hydraulic conductivity from available grain-size distributions on the stochastic modeling of the depth-averaged breakthrough curve observed during a forced-gradient tracer test experiment. The latter was performed in the Lauswiesen alluvial aquifer, located near the city of Tuebingen, Germany, by injecting NaBr into a well at a distance of about 50 m from a pumping well. We also examine the joint effect of the choice of the transport model adopted to describe solute transport at the site and the way the spatial distribution of porosity is assessed. In the absence of direct measurements of porosity, we consider: (a) the model used by Riva et al. (2006, 2008), which relates the natural logarithms of effective porosity and conductivity through an empirical, experimentally-based, linear relationship derived for a nearby experimental site; and (b) a model based on a commonly used relationship linking the total porosity to the coefficient of uniformity of grain size distributions. Transport is described in terms of a purely advective process and/or by including mass exchange processes between mobile and immobile regions. Modeling of flow and transport is performed within a Monte Carlo framework, upon conceptualizing the aquifer as a random composite medium. Our results indicate that the model adopted to describe the correlation between conductivity and porosity and the way grain-sieve information are incorporated to depict the heterogeneous distribution of hydraulic conductivity can have relevant effects in the interpretation of the data at the site. All the conceptual models employed to describe the structural heterogeneity of the system and transport features can reasonably reproduce the global characteristics of the experimental depth-averaged breakthrough curve. Specific details, such as the peak concentration and the time of first arrival, can be better reproduced by a double porosity transport model when a correlation between conductivity and porosity based on grain size information at the site is considered. The best prediction of the late-time behavior of the measured breakthrough curves, in terms of the observed heavy tailing, is offered by directly linking porosity distribution to the spatial variability of particle size information
Trasporto di soluti reattivi in acquiferi
No full textSi presentano alcuni recenti sviluppi relativi alla modellazione di processi di trasporto di soluti reattivi in acquiferi sotterranei. Molte reazioni di interesse applicativo avvengono in presenza di sistemi geochimicamente complessi. La formulazione generale di un problema di trasporto reattivo si articola su molteplici aspetti e richiede di specificare un numero elevato di specie acquose e non acquose, nonché di identificare la distribuzione spazio-temporale dei tassi delle reazioni. La metodologia proposta da De Simoni et al. (2005, 2007) consente il calcolo diretto dei tassi di reazione associati a scenari di trasporto multi-specie mediante il disaccoppiamento del sistema di equazioni che descrive l’evoluzione dei soluti nel dominio. Uno degli obiettivi del presente lavoro consiste nel presentare l’applicazione di tale metodologia disaccoppiata per descrivere le caratteristiche salienti associate al mescolamento di acque a diversa composizione chimica in un mezzo poroso. Si discutono a tal fine i risultati della modellazione numerica di esperimenti di dissoluzione condotti in un sistema carbonatico omogeneo alla scala di laboratorio, in cui si induce il mescolamento di acqua dolce e salata. Si illustrano, quindi, possibilità e prospettive di estensione della metodologia investigata. In particolare, si presentano espressioni per la funzione di densità di probabilità e gli associati momenti (di insieme) di primo e secondo ordine dei tassi delle reazioni che hanno luogo in un acquifero stratificato ad eterogeneità aleatoria
Nonlocal and localized analyses of nonreactive solute transport in bounded randomly heterogeneous porous media: Theoretical framework
No full textSolute transport in randomly heterogeneous media is described by stochastic transport equations that are typically solved via Monte Carlo simulation. A promising alternative is to solve a corresponding system of statistical moment equations directly. We present exact (though not closed) implicit conditional first and second moment equations for advective–dispersive transport in finite domains. The velocity and concentration are generally non-stationary due to possible trends in heterogeneity, conditioning on data, temporal variations in velocity, fluid and/or solute sources, initial and boundary conditions. Our equations are integro-differential and include non-local parameters depending on more than one point in space-time. To allow solving these equations, we close them by perturbation and develop recursive moment equations in Laplace space for the special case of steady state flow, to second order in the standard deviation of (natural) log hydraulic conductivity. We also propose a higher-order iterative closure. Our recursive equations and iterative closure are formally valid for mildly heterogeneous media, or well-conditioned strongly heterogeneous media in which the random component of heterogeneity is relatively small. The non-local moment equations suggest (and a companion paper [Morales Casique E, Neuman SP, Guadagnini A. Nonlocal and localized analyses of nonreactive solute transport in bounded randomly heterogeneous porous media: computational analysis. Adv Water Resour, submitted for publication] demonstrates numerically) that, in general, transport cannot be validly described by means of Fick_s law with a (constant or variable) macrodispersion coefficient. We show nevertheless that, under a limited set of conditions, the mean transport equation can be localized to yield a familiar-looking advection–dispersion equation with a conditional macrodispersion tensor that varies generally in spacetime. In a companion paper [Morales Casique E, Neuman SP, Guadagnini A. Nonlocal and localized analyses of nonreactive solute transport in bounded randomly heterogeneous porous media: computational analysis. Adv Water Resour, submitted for publication] we present a high-accuracy computational algorithm for our iterative non-local and recursive localized moment equations, assessing their accuracy and computational efficiency in comparison to unconditional and conditional Monte Carlo simulations
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