101,888 research outputs found
Reply to 'Comment on Nonstationary flow and nonergodic transport in random porous media by N. Suciu and C. Vamos [#2007WR005946]'
No full textThe paper reports the replay to comment by Suciu and Vamoş [2007] on a published paper [Darvini and Salandin, 2006] and demonstrates the correctness of the previously obtained results. New insights about the fulfillment of the ergodic hypothesis in numerical experiments are also added
Sistemi di distribuzione con rotture delle condotte e richiesta della portata aleatorie: una tecnica di valutazione dell'affidabilità
No full textLe tecniche di valutazione dell’affidabilità applicate ai sistemi di distribuzione acquedottistici, mirano a quantificare la capacità di soddisfare la richiesta di portata nelle diverse condizioni in cui il sistema stesso può venirsi a trovare [e.g. Mays, 2000]. Le condizioni del sistema dipendono, oltre che dall’andamento planoaltimetrico e dal diametro delle condotte, dalla loro scabrezza, dalla eventualità del fuori servizio di alcuni elementi (condotte, valvole, pompe, ecc.), dalla accidentale insufficienza delle sorgenti. Anche la stessa richiesta di portata, variabile sia nel tempo che nello spazio, risulta di incerta definizione, e concorre, assieme alle altre variabili aleatorie coinvolte, a limitare la validità delle classiche soluzioni deterministiche. Queste ultime, usualmente adottate nella progettazione e nella gestione dei sistemi acquedottistici, risultano al più rappresentative di un particolare stato – molte volte ipotetico – del sistema, senza tener conto di quella che è l’effettiva frequenza probabile del suo verificarsi nel corso della vita dell’opera: condizione, questa, di fondamentale importanza nella progettazione e nella gestione di un’opera di utilizzazione quale un sistema acquedottistico. Per contro con l’approccio probabilistico non risulta sempre possibile tenere in conto in modo efficiente le diverse cause d’incertezza, ma, in relazione al particolare problema che si deve trattare – progettazione, gestione o manutenzione del sistema – vengono analizzati solo alcuni degli aspetti che contribuiscono all’affidabilità, assumendo come deterministiche l’una o l’altra quantità sulla base di ipotesi che, seppur ragionevoli, risultano sempre in qualche misura limitative. Per cercare di ovviare a tale problema è descritta nel seguito una tecnica di valutazione dell’affidabilità di un sistema acquedottistico capace di tener contemporaneamente conto della fallanza meccanica delle componenti, dell’aleatorietà spazio – temporale della domanda e della distribuzione incerta della scabrezza delle condotte. A tale scopo, ad una tecnica (numerica) di simulazione completa che permette di valutare l’affidabilità del sistema soggetto a guasti delle sue componenti considerando la variabilità temporale a grande scala della portata immessa [Salandin & Bertola, 1996; Darvini & Salandin, 2004; Salandin & Darvini, 2004], è stata accoppiata una soluzione in forma chiusa, limitata al primo ordine nei termini fluttuanti, per tener conto sia della variabilità spaziale a minor scala temporale della domanda, sia della scabrezza incerta delle condotte [Xu & Goulter, 1998]. Definito in base alla conoscenza delle portate immesse in rete dall’azienda l’istogramma di frequenza di queste ultime, è possibile - attorno al valor medio di ciascuna classe -, considerare le fluttuazioni di portata a scala inferiore con tecniche che prevedono la linearizzazione delle equazioni. I risultati del modello semianalitico così ottenuto permettono la costruzione di un indice affidabilistico locale utile nella valutazione delle prestazioni della rete in funzione dell’articolato insieme delle grandezze aleatorie contemporaneamente considerate. Il modello proposto è stato applicato al caso della rete sintetica Anytown [Walsky, 1987] ben noto in letteratura. I risultati ottenuti mettono in luce come l’incertezza nella distribuzione spaziale dei valori di scabrezza e la variabilità a piccola scala della richiesta di portata possano modificare sostanzialmente le stime di affidabilità basate sulla sola probabilità di fallanza meccanica delle singole componenti del sistema
Handling subsurface transport in nonstationary velocity fields
No full textIn the stochastic subsurface analysis most of transport solutions available in literature are obtained under the assumption of statistical homogeneity of velocity field even though in real-world applications this hypothesis is not always verified. Several causes may induce statistical inhomogeneity of flow. This may derive from the influence of boundaries in a limited domain, complex flow configurations (related to pumping and/or injecting wells), spatial nonstationarity of the properties of formations (distinct geological layers and zones) as well as from conditioning on measurements. While several contributes discusses the effects on the flow statistic, only few attempts to find a general solution of transport problems suitable in absence of statistical homogeneity of velocity were carried out. A method is here proposed to handle different and concurrent causes of the flow field nonstationarity and to develop a plume evolution in a domain of finite size. The goal is reached by expanding the steady state flow equation in Taylor series limited to first-order and by the recursive application of finite element method. The unknowns are the piezometric head mean values and its derivatives respect to the fluctuating porous media hydraulic conductivity. Spatial moments of a solute plume are a posteriori obtained by a consistent Lagrangian analysis starting from the knowledge of the velocity field covariance matrices without any restriction regarding the statistical homogeneity of flow and/or ergodicity conditions. By this approach any combination of finite complex boundary, internal sources or sinks, fully inhomogeneous hydraulic conductivity characteristics can be handled. The application of the proposed method to some test cases in domains of finite extension gives results in agreement with literature findings. To give a measure of the capability of the method the examples here developed take into account the spatial velocity nonstationarity coming from inhomogeneous hydraulic conductivity also. A Monte Carlo analysis expressly developed gives a term of comparison
Model Discrimination in Water Distribution Systems
No full textAbstractModel discrimination procedures are useful tools for selection of the best models to be used to represent a specific process. In the paper a sequential discrimination procedure is adopted in the water distribution system analysis for selection of the most suitable model to perform hydraulic simulations and for estimation of precise model parameters, that are usually treated as independent techniques. The results indicate that the procedure allows to discriminate among different models, based on the collected experimental data. This results is achieved by selecting the best additional experiments in the design of the sensor location for pressure and/or flow measurements
From the local to the regional scale. What is the effect of missing vertical heterogeneity moving from fully 3-D to 2-D depth averaged dispersion models?
No full textThe plume evolution in natural porous formations is strongly affected by the erratic variability of the hydraulic conductivity K that exhibits a three-dimensional correlation structure. In regional domains, the effect of the vertical heterogeneity combines itself with that one due to the horizontal variability of K, and when the plume has travelled a large number of (horizontal) integral scales, under the hypothesis that the transmissivity spatial distribution prevails, its evolution can be analyzed by two-dimensional models. Until this limit is reached, the vertical and horizontal variability of K are combined to give a fully three-dimensional dispersion process and the application of depth averaged models may gives erroneous results. In order to analyze the effects on trasport deriving from this simplification, we present the results of some numerical experiments that compare the three-dimensional plume evolution with two-dimensional simulations developed by tacking into account different hydraulic conductivity spatial distributions. The comparison between results of numerical simulations and theoretical considerations based on first order solution suggests a possible way to take into account the vertical variability of K in a depth averaged model
Simulations of non ergodic transport induced by inhomogeneous flow fields in heterogeneous porous formations of hierarchical sedimentary architecture
No full textDealing with subsurface flow and transport in naturally heterogeneous formations neither statistical homogeneity nor ergodicity are usually met in real world applications. If the stationarity is obeyed by Lagrangian velocity field the ergodicity of transport can be ascribed to the only size of the injection area, but this is not the case in all situations where flow field doesn't meet statistical homogeneity. In this case not only the size of the area where the concentration is injected, but also its location as well as the spatial variability of mean flux play a relevant role in the expected plume evolution. Among other causes, the hierarchical sedimentary architecture of porous formations combined with the finite size of the domain may be considered as one of the main causes of the flow field inhomogeneity and of the subsequent evidences of the non ergodic transport in real cases. To give a better understanding of this phenomenon synthetic 2-D cases of isotropic log conductivity field with integral scale λ1 whose expected value is assumed as a periodic function of the space coordinates of prescribed amplitude are investigated. The wave length of the mean log conductivity fluctuation is λ2, properly chosen to satisfy the relationship λ1 < λ2 < L, being L the characteristic dimension of the finite domain. The time evolution of the spatial moments of the plume driven by a statistically inhomogeneous steady state random velocity field is analyzed by varying the amplitude of the periodic mean flow and its wave length λ2 and by taking into account different sizes of injection area. These moments are achieved by space- time integration of the velocity field covariance structure derived according to the first-order Taylor series expansion by the stochastic finite element method. The discussion of the results leads to a better understanding into the ergodicity lack that affects solute transport in heterogeneous aquifers, by giving a comparative measure of the relevance of two possible causes, that is the flow field inhomogeneity due to the hierarchical sedimentary architecture of porous formations combined with a limited domain and the injection area of finite size
Examples of subsurface solute spreading driven by inhomogeneous velocity fields
No full textMost of solutions of flow and transport problems available in literature are obtained under assumptions of stationarity of flow field and/or ergodicity of transport, even though in real-world analyses these hypotheses cannot be always verified. For instance, into problems that are met in practical applications, the statistics of flow are often location dependent and the stationarity of flow is violated. The nonstationarity of velocity field may originate from finite domain boundaries, complex flow configurations (pumping and injecting), nonstationarity of medium properties or conditioning of the log conductivity field to measurements of head or conductivity. Moreover the lack of ergodicity related to the finite size of solute sources makes difficult transport analyses that are often carried out by rough schematizations. Some examples of solute dispersion of nonergodic passive solute plume in heterogeneous formations with nonstationary flow conditions are here considered and solved by a new approach. By this method the spatial moments of finite initial volume of solute are obtained from the statistics of velocity field evaluated by first-order perturbation expansion of steady state flow equation in Taylor series combined with a finite element discretization. The approach allows to handle nonstationarity due to several causes and is here applied to some test cases in bounded domains. A comparison of numerical results in terms of particle displacements moments with known solutions available in literature and with Monte Carlo simulations gives a measure of the effect related to the lack of plume ergodicity and of flow spatial stationarity in real-world transport analyses
Impact of uncertainty on the porous media description in the subsurface transport analysis
No full textIn the modelling of flow and transport phenomena in naturally heterogeneous media, the spatial variability of hydraulic properties, typically the hydraulic conductivity, is generally described by use of a variogram of constant sill and spatial correlation. While some analyses reported in the literature discuss of spatial inhomogeneity related to a trend in the mean hydraulic conductivity, the effect in the flow and transport due to an inexact definition of spatial statistical properties of media as far as we know had never taken into account. The relevance of this topic is manifest, and it is related to the uncertainty in the definition of spatial moments of hydraulic log-conductivity from an (usually) little number of data, as well as to the modelling of flow and transport processes by the Monte Carlo technique, whose numerical fields have poor ergodic properties and are not strictly statistically homogeneous. In this work we investigate the effects related to mean log-conductivity (logK) field behaviours different from the constant one due to different sources of inhomogeneity as: i) a deterministic trend; ii) a deterministic sinusoidal pattern and iii) a random behaviour deriving from the hierarchical sedimentary architecture of porous formations and iv) conditioning procedure on available measurements of the hydraulic conductivity. These mean log-conductivity behaviours are superimposed to a correlated weakly fluctuating logK field. The time evolution of the spatial moments of the plume driven by a statistically inhomogeneous steady state random velocity field is analyzed in a 2-D finite domain by taking into account different sizes of injection area. The problem is approached by both a classical Monte Carlo procedure and SFEM (stochastic finite element method). By the latter the moments are achieved by space-time integration of the velocity field covariance structure derived according to the first-order Taylor series expansion. Two different goals are foreseen: 1) from the results it will be possible to distinguish the contribute in the plume dispersion of the uncertainty in the statistics of the medium hydraulic properties in all the cases considered, and 2) we will try to highlight the loss of performances that seems to affect the first-order approaches in the transport phenomena that take place in hierarchical architecture of porous formations
Effective dispersion in conditioned transmissivity fields
No full textWe analyze the impact of conditioning to measurements of hydraulic transmissivity on the transport of a conservative solute. The effects of conditioning on solute transport are widely discussed in the literature, but most of the published works focuses on the reduction of the uncertainty in the prediction of the plume dispersion. In this study both ensemble and effective plume moments are considered for an instantaneous release of a solute through a linear source normal to the mean flow direction, by taking into account different sizes of the source. The analysis, involving a steady and spatially inhomogeneous velocity field, is developed by using the stochastic finite element method. Results show that conditioning reduces the ensemble moment in comparison with the unconditioned case, whereas the effective dispersion may increase because of the contribution of the spatial moments related to the lack of stationarity in the flow field. As the number of conditioning points increases, this effect increases and it is significant in both the longitudinal and transverse directions. Furthermore, we conclude that the moment derived from data collected in the field can be assessed by the conditioned second-order spatial moment only with a dense grid of measured data, and it is manifest for larger initial lengths of the plume. Nevertheless, it seems very likely that the actual dispersion of the plume may be underestimated in practical applications
Flow and transport in heterogeneous porous formations subject to uncertainty in statistical description of the hydraulic log-conductivity spatial behaviour
No full textThe effect of a periodic spatial behaviour of the mean log-conductivity (logK) in random heterogeneous aquifers was investigated by numerical simulations. The adopted schematization reflects the spatial inhomogeneity due to an inexact definition of spatial statistical properties of media and can be considered a rough description of the hierarchical sedimentary architecture of porous formations. The solution of investigated cases was achieved by use of both the stochastic finite element method (SFEM) and the Monte Carlo simulations (MC). From the results it was possible to distinguish the contribute of the fluctuating mean log-conductivity field in the plume evolution. Comparison between SFEM and MC outcome reveals strong nonlinearity effects related to the spatial variability of the mean of the logK field also when its variance is negligible
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