3,615 research outputs found
Recursive 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)
Numerical Investigation of Apparent Multifractality of Samples from Processes Subordinated to Truncated fBm
No full textWe investigate numerically apparent multi-fractal behavior of samples from synthetically generated processes subordinated to truncated fractional Brownian motion (tfBm) on finite domains. We are motivated by the recognition that many earth and environmental (including hydrologic) variables appear to be self-affine (monofractal) or multifractal with Gaussian or heavy-tailed distributions. The literature considers self-affine and multifractal types of scaling to be fundamentally different, the first arising from additive and the second from multiplicative random fields or processes. It has been demonstrated theoretically (Neuman, 2010a, 2011) that square or absolute increments of samples from Gaussian/Lévy processes subordinated to tfBm exhibit apparent/spurious multifractality at intermediate ranges of separation lags, with breakdown in power-law scaling at small and large lags as is commonly exhibited by real data. A preliminary numerical demonstration of apparent multifractality by Neuman (2010b) was limited to Gaussian fields having nearest neighbor autocorrelations and led to rather noisy results. Here we improve upon Neuman's numerical analysis by adopting a much simpler but more complete and accurate generation scheme proposed by Neuman (2011). This allows us to investigate with greater fidelity apparent multifractal behaviors of samples taken from a broader range of processes including Gaussian with and without symmetric Lévy and log-normal (as well as potentially other) subordinators. Our results shed new light on the nature of apparent multifractality which has wide implications vis-a-vis the scaling of many hydrologic as well as other earth and environmental variables
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
Characterizing the spatial variability of transmissivity using stochastic type-curves and numerical inverse analyses of data from a sequence of pumping tests
No full textWe discuss two recent methods of characterizing the spatial variability of a random (natural) log transmissivity field on the basis of observed space-time variations in hydraulic head: a graphical stochastic type-curve method (Neuman et al., 2004, 2007) and a geostatistical method of inverting stochastic mean flow equations (Hernandez et al., 2003, 2006). While both methods allow estimating the unconditional variance and integral (correlation) scale of log transmissivities, geostatistical inversion is computationally more intensive but provides also tomographic images of how log transmissivity estimates and their variance vary in space. We apply the two approaches to synthetic scenarios and to measured late time (quasisteady state) drawdowns from a sequence of transient pumping tests in an unconfined aquifer near Tübingen, Germany
Transport in multiscale log conductivity fields with truncated power variograms
No full textDi Federico and Neuman [this issue] investigated mean uniform steady state groundwater flow in an unbounded domain with log hydraulic conductivity that forms a truncated multiscale hierarchy of statistically homogeneous and isotropic Gaussian fields, each associated with an exponential autocovariance. Here we present leading-order expressions for the displacement covariance, and dispersion coefficient, of an ensemble of solute particles advected through such a flow field in two or three dimensions. Both quantities are functions of the mean travel distance s, the Hurst coefficient H, and the low- and high-frequency cutoff integral scales λ(l) and λ(u). The latter two are related to the length scales of the sampling window (region under investigation) and sample volume (data support), respectively. If one considers transport to be affected by a finite domain much larger than the mean travel distance, so that s << λ(l) < ∞, then an early preasymptotic regime develops during which longitudinal and transverse dispersivities grow linearly with s. If one considers transport to be affected by a domain which increases in proportion to s, then λ(l) and s are of similar order and a preasymptotic regime never develops. Instead, transport occurs under a regime that is perpetually close to asymptotic under the control of an evolving scale λ(l) ~ s. We show that if, additionally, λ(u) << λ(l), then the corresponding longitudinal dispersivity grows in proportion to λ(l)/(1+2H) or, equivalently, s(1+2H). Both these preasymptotic and asymptotic theoretical growth rates are shown to be consistent with the observed variation of apparent longitudinal Fickian dispersivities with scale. We conclude by investigating the effect of variable separations between cutoff scales on dispersion
Effetti di troncamento su campi random multiscala in domini finiti
No full textÈ stato dimostrato da Neuman (1990), Di Federico e Neuman (1997) e Di Federico et al. (1999) che il variogramma di potenza di un campo frattale isotropo o anisotropo può essere costruito sovrapponendo variogrammi esponenziali o Gaussiani di campi (modi) statisticamente omogenei con incrementi mutuamente non correlati e varianza proporzionale a una potenza 2H della scala di correlazione spaziale, do-ve H è il coefficiente di Hurst. Nella sovrapposizione vengono introdotte soglie di troncamento inferiore e superiore, rispettivamente proporzionali alle dimensioni di dominio (finestra campionaria) e scala di suppor-to (volume campionato). Il presente lavoro si propone di valutare numericamente la fondatezza di tale ipote-si. In particolare si esaminano soglie di troncamento di larga scala, attraverso la generazione su dominio fini-to di numerose realizzazioni di campi multivariati Gaussiani bidimensionali con variogrammi di potenza isotropi (che descrivono “fractional Brownian motion”, o fBm). I variogrammi presentano valori di H pari a 0.25 e 0.75, corrispondenti rispettivamente a fBm anti-persistenti e persistenti. Sebbene i variogrammi cam-pionari spaziali omnidirezionali di realizzazioni singole non rispettino il modello di potenza imposto se non in casi isolati, lo riproducono invece fedelmente se considerati e mediati sull’insieme delle realizzazioni ge-nerate. I risultati ottenuti mostrano che il variogramma di potenza si conserva pressoché inalterato in finestre di dimensioni inferiori estratte dal dominio iniziale. Nel presente lavoro si dimostra come i suddetti vario-grammi di potenza, ottenuti da realizzazioni multiple su finestre finite, possano essere rappresentati con suc-cesso per mezzo di variogrammi di potenza troncati con una soglia di larga scala. I risultati hanno notevoli implicazioni nell’interpretazione degli effetti di scala delle variabili idrogeologiche (Neuman e Di Federico, 2003), fondamentali per la comprensione dei fenomeni di flusso e trasporto in formazioni porose naturali
The futures of regional design
No full textThe climate crisis has grown worse, with impacts more severe, widespread, unpredictable, and what’s worse, not dealt with globally in a meaningful way. The devastating wildfires in Australia in late 2019 and the western United States of America (USA) in 2020 seemed to underscore that consensus, with tens of millions of acres, thousands of homes burned, many lives lost, including an estimated one billion animals. As humans and our constructions - roads, infrastructures, buildings - destroy and invade formerly intact habitats across the globe, species of all kinds interact in new ways. As cities have grown into metropolises, megacities, and city regions, people witness the increasing urgency to plan and manage these behemoths so that their residents can lead healthy&prosperous lives, sustainably. The contribution that regional design makes to resolving these conundrums is to highlight the relatively new arena of governance that comports with the actual spatial scale of urban phenomena now and into the future - the region.Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.Spatial Planning and Strateg
On the geostatistical characterization of hierarchical media
No full textThe subsurface consists of porous and fractured materials exhibiting a hierarchical
geologic structure, which gives rise to systematic and random spatial and directional
variations in hydraulic and transport properties on a multiplicity of scales. Traditional
geostatistical moment analysis allows one to infer the spatial covariance structure of such
hierarchical, multiscale geologic materials on the basis of numerous measurements on
a given support scale across a domain or ‘‘window’’ of a given length scale. The resultant
sample variogram often appears to fit a stationary variogram model with constant variance
(sill) and integral (spatial correlation) scale. In fact, some authors, who recognize that
hierarchical sedimentary architecture and associated log hydraulic conductivity fields tend
to be nonstationary, nevertheless associate them with stationary ‘‘exponential-like’’
transition probabilities and variograms, respectively, the latter being a consequence of the
former. We propose that (1) the apparent ability of stationary spatial statistics to
characterize the covariance structure of nonstationary hierarchical media is an artifact
stemming from the finite size of the windows within which geologic and hydrologic
variables are ubiquitously sampled, and (2) the artifact is eliminated upon characterizing
the covariance structure of such media with the aid of truncated power variograms, which
represent stationary random fields obtained upon sampling a nonstationary fractal over
finite windows. To support our opinion, we note that truncated power variograms arise
formally when a hierarchical medium is sampled jointly across all geologic categories and
scales within a window; cite direct evidence that geostatistical parameters (variance and
integral scale) inferred on the basis of traditional variograms vary systematically with
support and window scales; demonstrate the ability of truncated power models to capture
these variations in terms of a few scaling parameters; show that exponential and truncated
power variograms are often difficult to distinguish from each other, which helps
explain why hierarchical data may appear to fit the former; note that truncated power
models are unique in their ability to represent multiscale random fields having either
Gaussian or heavy-tailed symmetric Levy stable probability distributions; detail the way in
which these models allow conditioning the spatial statistics of such fields on multiscale
measurements via cokriging; and illustrate these capabilities on multiscale hydraulic
data from an unconfined aquifer near Tu ̈bingen, Germany
Type curve interpretation of late-time pumping test data in randomly heterogeneous aquifers
No full textThe properties of heterogeneous media vary spatially in a manner that can seldom be described with certainty. It may, however, be possible to describe the spatial variability of these properties in terms of geostatistical parameters such as mean, integral (spatial correlation) scale, and variance. Neuman et al. (2004) proposed a graphical method to estimate the geostatistical parameters of (natural) log transmissivity on the basis of quasi–steady state head data when a randomly heterogeneous confined aquifer is pumped at a constant rate from a fully penetrating well. They conjectured that a quasi–steady state, during which heads vary in space-time while gradients vary only in space, develops in a statistically homogeneous and horizontally isotropic aquifer as it does in a uniform aquifer. We confirm their conjecture numerically for Gaussian log transmissivities, show that time-drawdown data from randomly heterogeneous aquifers are difficult to interpret graphically, and demonstrate that quasi–steady state distance-drawdown data are amenable to such interpretation by the type curve method of Neuman et al. The method yields acceptable estimates of statistical log transmissivity parameters for fields having either an exponential or a Gaussian spatial correlation function. These estimates are more robust than those obtained using the graphical time-drawdown method of Copty and Findikakis (2003, 2004a). We apply the method of Neuman et al. (2004) simultaneously to data from a sequence of pumping tests conducted in four wells in an aquifer near Tuebingen, Germany, and compare our transmissivity estimate with estimates obtained from 312 flowmeter measurements of hydraulic conductivity in these and eight additional wells at the site. We find that (1) four wells are enough to provide reasonable estimates of lead log transmissivity statistics for the Tu ̈bingen site using this method, and (2) the time-drawdown method of Cooper and Jacob (1946) underestimates the geometric mean transmissivity at the site by 30–40%
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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