1,721,270 research outputs found
Effects of evolving scales of heterogeneity on hydraulic head predictions under convergent flow conditions
No full textWe consider two-dimensional steady state flow towards a well that fully penetrates a randomly heterogeneous aquifer, with deterministically prescribed constant head boundary. Flow occurs over an infinite hierarchy of mutually uncorrelated, statistically homogeneous and isotropic random fields (modes) of natural log-transmissivity, Y, each of which is associated with a Gaussian variogram. We consider a lower and upper cut-off of the hierarchy, respectively related to the length scale of the domain and data support (sample volume). This allows directly incorporating the scale dependence of the integral scale of Y into groundwater flow (ensemble) moments and leads to a geostatistical description of the system in terms of a (stationary) Truncated Power Variogram (TPV). We then develop an analytical solution for hydraulic head mean and variance based on recursive approximations of exact nonlocal moment equations. Our solution allows to assess functional dependences of the distribution of the leading (statistical) moments of hydraulic head on parameters of the variogram associated with the hierarchy of log-transmissivity modes. The latter can be determined, for instance, along the lines of Neuman et al. (2008)
Impact of log-transmissivity variogram structure on groundwater flow and transport predictions
No full textWe analyze the impact of the choice of the variogram model adopted to characterize the spatial variability of natural log-transmissivity on the evaluation of leading (statistical) moments of hydraulic heads and contaminant travel times and trajectories within mildly (randomly) heterogeneous two-dimensional porous systems. The study is motivated by the fact that in several practical situations the differences between various variogram types and a typical noisy sample variogram are small enough to suggest that one would often have a hard time deciding which of the tested models provides the best fit. Like-wise, choosing amongst a set of seemingly likely variogram models estimated by means of geostatistical inverse models of flow equations can be difficult due to lack of sensi-tivity of available model discrimination criteria. We tackle the problem within the framework of numerical Monte Carlo simulations for mean uniform and radial flow scenarios. The effect of three commonly used isotropic variogram models, i.e., Gaus-sian, Exponential and Spherical, is analyzed. Our analysis clearly shows that (ensemble) mean values of the quantities of interest are not considerably influenced by the variogram shape for the range of parameters examined. Contrariwise, prediction vari-ances of the quantities examined are significantly affected by the choice of the variogram model of the log-transmissivity field. The spatial distribution of the larg-est/lowest values of the relative differences observed amongst the tested models de-pends on a combination of variogram shape and parameters and relative distance from internal sources and the outer domain boundary. Our findings suggest the need of de-veloping robust techniques to discriminate amongst a set of seemingly equally likely al-ternative variogram models in order to provide reliable uncertainty estimates of state variables
Recharge fronts and stagnation areas for pumping wells
No full textWe consider a single pumping well within a two-dimensional
heterogeneous aquifer, in the presence of a mean uniform background gradient. We analyse the uncertainty associated with the location of the stagnation point which is generated downstream of the well and the maximum lateral extent of the well recharge area. The study is relevant for risk assessment practice, since it allows to estimate the maximum width of the region contributing to the well and to properly locate regions of inversion of
the flow direction. The problem is approached within a numerical Monte Carlo framework and the dependence of the main statistics of the quantities of interest on the moments of the log-conductivity field is studied. An assessment of the impact of one conductivity datum at the pumping location is performed
Radial flow in a bounded randomly heterogeneous aquifer with recharge
No full textWe present analytical expressions for leading statistical moments of vertically averaged hydraulic head under steady state flow to a well that pumps water from a bounded, randomly heterogeneous aquifer, in the presence of a random recharge. The natural logarithm Y = ln T of aquifer transmissivity T is modelled as a statistically homogeneous random field with a Gaussian spatial correlation function. Our solution is based on exact nonlocal moment equations for multidimensional steady state flow in bounded, randomly heterogeneous porous media. In contrast to most stochastic analyses of flow, which require that log transmissivity be multivariate Gaussian, our solution is free of any distributional assumptions. 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. It also applies to thicker aquifers when information about their vertical heterogeneity is lacking, as is commonly the case when measurements of head and flow rate are done in wells that penetrate much of the aquifer thickness
Flussi radiali in formazioni porose eterogenee di estensione finita,
No full textSi presenta una soluzione analitica dei momenti statistici del carico idraulico in condizioni di flusso stazionario verso un pozzo di emungimento in acquifero bi-dimensionale ad eterogeneità aleatoria di estensione finita con assegnate condizioni di carico idraulico al contorno. La media di insieme (stimatore) del carico idraulico soddisfa un’equazione non-locale e non-darciana, dipendendo da un flusso residuo, esprimibile in forma integro-differenziale. La varianza del carico idraulico tende all’infinito al pozzo e decresce in maniera monotona con la distanza, fino ad annullarsi al contorno, per le condizioni deterministiche imposte. Si intruce una trasmissività apparente definita dal rapporto fra il valore atteso del flusso e del gradiente del carico idraulico. Si analizza l’influenza sulle grandezze in esame della dimensione finita del dominio, espressa in termini di scale di correlazione della log-trasmissività e si segnalano possibili ricadute applicative
Perimetrazione di zone di cattura probabilistiche in un acquifero eterogeneo
No full textLa forma ed estensione delle zone di cattura e, conseguentemente, la perimetrazione delle aree di protezione di pozzi emungenti è influenzata dalla variabilità spaziale delle proprietà idrogeologiche dell’acquifero. L’individuazione di tali zone e la valutazione del grado di incertezza a queste associato ha notevole rilevanza ambientale, essendo i pozzi emungenti utilizzati sia per approvviogionamento idrico che per recupero di siti contaminati.
Si analizza l’effetto delle eterogeneità locali sulla demarcazione di zone di protezione di pozzi localizzati nel sito di “Lauswiesen” (Germania). Si utilizzano a tal fine informazioni fornite da indagini sperimentali condotte sia alla scala di laboratorio che di campo.
Le analisi granulometriche eseguite su campioni indisturbati evidenziano la presenza di depositi alluvionali caratterizzati da eterogeneità elevata, da elevata conducibilità idraulica . Tre diversi litotipi (facies) che descrivono l’eterogeneità dell’acquifero dal punto di vista litologico sono identificati mediante analisi multivariata. La variabilità spaziale di ciascuna facies viene analizzata separatamente tramite procedura geostatistica a variabili indicizzate. Utilizzando i dati indicatori e il semivariogramma sperimentale si generano, in uno schema Monte Carlo, realizzazioni condizionate della distribuzione tridimensionale delle facies a diversa conducibilità idraulica all’interno dell’acquifero. Essendo ogni facies completamente caratterizzata dai suoi parametri idraulici, dalla sua variabilità interna e dalle sue proprietà strutturali, il risultato fornisce campi tri-dimensionali dei parametri dell’acquifero eliminando difficoltà legate ad upscaling. Tramite tale procedura Monte Carlo si ottiene una stima delle zone di cattura dinamiche tridimensionali, a cui viene associata una quantificazione della corrispondente incertezza. I risultati mostrano che la delineazione delle zone di cattura è fortemente influenzata dall’aleatorietà intrinseca sia della conducibilità idraulica che della porosità, evidenziando importanti implicazioni nella progettazione di misure di protezione di pozzi e/o nella pianificazione di interventi di decontaminazione di acquiferi mediante pompaggio
Assessment of uncertainty associated with the estimation of well catchments by moment equations
No full textNon-local stochastic moment equations are used successfully to analyze groundwater flow in randomly heterogeneous media.
Here we present a moment equations-based approach to quantify the uncertainty associated with the estimation of well catchments.
Our approach is based on the development of a complete second order formalism which allows obtaining the first statistical moments of the trajectories of conservative solute particles advected in a generally non-uniform groundwater flow. Approximate equations of moments of particles trajectories are then derived on the basis of a second order expansion in terms of the standard deviation of the aquifer log hydraulic conductivity. Analytical expressions are then obtained for the predictors of locations of mean stagnation points, together with their associated uncertainties. We implement our approach on heterogeneous media in bounded two-dimensional domains, with and without including the effect of conditioning on hydraulic conductivity information. The impact of domain size, boundary conditions, heterogeneity and non-stationarity of hydraulic conductivity on the prediction of a well catchment is explored. The results are compared against Monte Carlo simulations and semi-analytical solutions available in the literature. The methodology is applicable to both infinite and bounded domains and is free of distributional assumptions (and so applies to both Gaussian and non-Gaussian log hydraulic conductivity fields) and formally includes the effect of conditioning on available information
Characterization of Spatially Heterogeneous Environmental Variables Through Multi‐Modal Generalized Sub‐Gaussian Distributions
Full text linkWe provide a sound theoretical framework for the characterization of randomly heterogeneous spatial fields exhibiting multi-modal, long-tailed probability densities. Multi-modal distributions are at the core of conceptual models employed to represent heterogeneity of hydrogeological or geochemical systems across which one can otherwise distinguish diverse regions whose location is uncertain. Within each region, the quantity of interest shows a distinct heterogeneous pattern that can be described through a generally non-Gaussian distribution. Our analytical model embeds the joint formulation of the probability density of the target variable and its spatial increments. The distributions of the latter scale with separation distance between locations at which increments are evaluated. This feature is in line with documented experimental observations of a variety of Earth system quantities. Our stochastic modeling framework integrates approaches based on unimodal non-Gaussian fields described through a Generalized Sub-Gaussian model and (multi-modal) distributions resulting from mixtures of Gaussian fields. These are recovered as specific instances within our comprehensive formulation. We apply this framework to an experimental data set consisting of a collection of dissolution rate fields obtained from high-resolution nanoscale measurements acquired through Atomic Force Microscopy and documenting the dissolution behavior of a calcite sample under continuous flow conditions. Our findings demonstrate the capability of our stochastic approach to elucidate key statistical traits and scaling features inherent in the heterogeneous distributions of these types of environmental variables
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