1,721,011 research outputs found
A new look at traditional deterministic flow models and their calibration in the context of randomly heterogeneous media
No full textNonlocal and localized analyses of conditional mean steady state flow in bounded, randomly nonuniform domains, 1, theory and computational approach
No full textNonlocal and localized analyses of conditional mean steady state flow in bounded, randomly nonuniform domains, 2, computational examples
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Type-curve estimation of statistical heterogeneity
No full textThe analysis of pumping tests has traditionally relied on analytical solutions of groundwater flow equations in relatively simple domains, consisting of one or at most a few units having uniform hydraulic properties. Recently, attention has been shifting
toward methods and solutions that would allow one to characterize subsurface heterogeneities in greater detail. On one hand, geostatistical inverse methods are being used to assess the spatial variability of parameters, such as permeability and porosity, on
the basis of multiple cross-hole pressure interference tests. On the other hand, analytical solutions are being developed to describe the mean and variance (first and second statistical moments) of flow to a well in a randomly heterogeneous medium. We explore
numerically the feasibility of using a simple graphical approach (without numerical inversion) to estimate the geometric mean, integral scale, and variance of local log transmissivity on the basis of quasi steady state head data when a randomly heterogeneous
confined aquifer is pumped at a constant rate. By local log transmissivity we mean a function varying randomly over horizontal distances that are small in comparison with a
characteristic spacing between pumping and observation wells during a test. Experimental evidence and hydrogeologic scaling theory suggest that such a function would tend to
exhibit an integral scale well below the maximum well spacing. This is in contrast to equivalent transmissivities derived from pumping tests by treating the aquifer as being locally uniform (on the scale of each test), which tend to exhibit regional-scale spatial
correlations. We show that whereas the mean and integral scale of local log transmissivity can be estimated reasonably well based on theoretical ensemble mean variations of head and drawdown with radial distance from a pumping well, estimating the log
transmissivity variance is more difficult. We obtain reasonable estimates of the latter based on theoretical variation of the standard deviation of circumferentially averaged drawdown
about its mean
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
Extended Power-Law Scaling of Self-Affine Signals Exhibiting Apparent Multifractality
No full textMany earth and environmental variables appear to scale as multiplicative (multifractal) processes with spatial or temporal increments possessing Gaussian or heavy‐tailed distributions. The behavior, characterized by power‐law scaling, is typically limited to intermediate ranges of separation scales (lags) considered, in the case of fully developed turbulence, to be dominated by inertia. It has been established empirically that, in numerous cases (e.g. turbulence, diffusion‐limited aggregates, natural images, kinetic surface roughening, fluvial turbulence, sand wave dynamics, Martian topography, river morphometry, gravel‐bed mobility, barometric pressure, low‐energy cosmic rays, cosmic microwave background radiation, metal‐insulator transition, irregularities in human heartbeat time series, turbulence in edge magnetized plasma of fusion devices and turbulent boundary layers of the Earth’s magnetosphere), this range of lags can be enlarged significantly, at both ends of the spectrum, via a procedure known as Extended Self‐Similarity (ESS). We demonstrate numerically that a similar procedure extends the power‐law scaling range over which additive (self‐affine) signals exhibit apparent multifractality. We conclude that signals appearing to exhibit either standard or extended (such as those listed) multifractal scaling may potentially represent self‐affine processes
Conditioning steady state mean stochastic flow equations on head and hydraulic conductivity measurements
No full textAnisotropic Statistical Scaling of Soil and Sediment Texture in a Stratified Deep Vadose Zone near Maricopa, Arizona.
Get PDFWe analyze the statistical scaling properties of vertical and horizontal increments in soil and sediment texture data measured to a depth of 15 meters over an area of 3600 m2 in a vadose zone near Maricopa, Arizona. The data include sand, silt and clay fractions, their principal components and logit transforms. The statistical scaling properties we uncover are difficult to detect with standard geostatistical methods. They include (a) pronounced intermittency (rough, irregular spatial variability) and antipersistence (tendency for large and small values to alternate rapidly); (b) symmetric, non-Gaussian frequency distributions characterized by heavy tails that often decay with separation distance or lag; (c) nonlinear power-law scaling of sample structure functions (statistical moments of absolute increments) in a midrange of lags, with breakdown in such scaling at small and large lags; (d) extended power-law scaling (linear relations between log structure functions of successive orders) at all lags; (e) nonlinear scaling of power-law exponent with order of sample structure function; and (f) pronounced anisotropy in these behaviors. Similar statistical scaling behaviors are known to be exhibited by a wide variety of earth, environmental and other variables (including ecological, biological, physical, astrophysical and financial). The literature has traditionally interpreted this to imply that the variables are multifractal, which however explains neither the observed breakdown in power-law scaling at small and large lags nor extended power-law scaling. We offer an alternative interpretation that is simpler and consistent with all the above phenomena. Our interpretation views the data as samples from stationary, anisotropic sub-Gaussian random fields subordinated to truncated fractional Brownian motion (tfBm) or truncated fractional Gaussian noise (tfGn). Such sub-Gaussian fields are mixtures of Gaussian fields with random variances. Truncation of monofractal fBm (which is non-stationary) and fGn entails filtering out components below the measurement or resolution scale of the data and above the scale of their sampling domain. Our novel interpretation of the data allows us to obtain maximum likelihood estimates of all parameters characterizing the underlying truncated sub-Gaussian fields. These parameters in turn make it possible to downscale or upscale all statistical moments to situations entailing smaller or larger measurement or resolution and sampling scales, respectively. They also allow one to perform conditional or unconditional Monte Carlo simulations of random field realizations corresponding to these scales
Data assimilation and parameter estimation via ensemble Kalman filter coupled with stochastic moment equations of transient groundwater flow
No full textThe ensemble Kalman filter (EnKF) is a powerful tool for assimilating data in earth system models. The approach allows real time Bayesian updating of system states and parameters as new data become available. This paper focuses on EnKF data assimilation in models of groundwater flow through complex geologic media. It has become common to treat the hydraulic conductivity of such media as correlated random fields conditioned on measured conductivity (medium property) and/or hydraulic head (system state) values. This renders the conductivity nonstationary and the corresponding conditional flow equations stochastic. Solving these equations and coupling them with EnKF generally entails computationally intensive Monte Carlo (MC) simulation. We propose to circumvent the need for MC through a direct solution of approximate nonlocal (integrodifferential) equations that govern the space-time evolution of conditional ensemble means (statistical expectations) and covariances of hydraulic heads and fluxes. We illustrate and explore our approach on synthetic two-dimensional examples in which a well pumps water from a randomly heterogeneous aquifer subject to prescribed head and flux boundary conditions. Embedding the solution in EnKF provides sequential updates of conductivity and head estimates throughout the space-time domain of interest. We demonstrate the computational feasibility and accuracy of our methodology, showing that hydraulic conductivity estimates are more sensitive to early than to later head values and improve with increasing assimilation frequency at early time
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