Random Space Function Research Articles

Stochastic Inversion of a Tomographic Pumping Test: Identifying Conductivity Horizontal Correlation and Longitudinal Macrodispersivity

AbstractTwo important properties for modeling flow and transport in heterogeneous aquifers are the horizontal integral scale (Ih) of hydraulic logconductivity (Y = ln K) and the longitudinal macrodispersivity (αL). Estimating the former generally requires K measurements in many wells, which are typically not available. In this work, we present a method for estimating Ih and αL from hydraulic tomography (HT) pumping tests. The approach is based on stochastic inversion, assuming head and K are stationary random space functions. Head variance and mean head gradient are calculated from measured head data at large distances from the pumping well and for large times from the start of pumping. These are then used in a theoretical formula for head variance in uniform mean flow to obtain Ih and subsequently αL, requiring only previous knowledge of the vertical integral scale (Iv) and the logconductivity variance . The method is applied to data from an HT pumping test at the Boise Hydrogeophysical Research site and results for Ih and αL are in the range previously reported in the literature. Particularly promising results are found for αL, which is seen to be robust, with relatively little variation for different values of Iv and .

Insight into Spatially Colored Stochastic Heat Equation: Temporal Fractal Nature of the Solution

In this paper, the solution to a spatially colored stochastic heat equation (SHE) is studied. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time has exact, dimension-dependent, global continuity moduli, and laws of the iterated logarithm (LILs). It is obtained that the set of fast points at which LILs fail in this process, and occur infinitely often, is a random fractal, the size of which is evaluated by its Hausdorff dimension. These points of this process are everywhere dense with the power of the continuum almost surely, and their hitting probabilities are determined by the packing dimension dimP(E) of the target set E.

Uncertainty quantification of unsteady source flows in heterogeneous porous media

Unsteady flow generated by a point-like source takes place into a$d$-dimensional porous formation where the spatial variability of the hydraulic conductivity$K$is modelled within a stochastic framework that regards$K$as a stationary, normally distributed random space function (rsf). As a consequence, the hydraulic head$H$becomes also stochastic, and we aim at quantifying its uncertainty. Towards this aim, we have derived the head covariance by means of a perturbation expansion which regards the variance$\unicode[STIX]{x1D70E}^{2}$of the zero meanrsf$\unicode[STIX]{x1D700}=1-K/\langle K\rangle$(hereafter$\langle \rangle$being the ensemble average operator) as a small parameter. The analytical results are expressed in terms of multiple quadratures which are markedly reduced after adopting specific autocorrelation$\unicode[STIX]{x1D70C}$for$\unicode[STIX]{x1D700}$. This enables one to obtain simple results providing straightforward physical insight into the spatial distribution of$H$as a consequence of the heterogeneity of$K$. In view of those applications (pumping tests) aiming at the identification of the hydraulic properties of geological formations, we have focused on a flow generated by a source of instantaneous and constant strength. The attainment of the large time (steady-state) regime is studied in detail.

A scale‐invariant property of the water retention curve in weakly heterogeneous vadose zones

AbstractAbrupt changes of hydraulic properties in a vadose zone are modelled within a stochastic framework, which regards the saturated conductivity and parameters related to the distribution of soil pores as stationary, log‐normally distributed, random space functions. As a consequence, flow variables become random fields, and we aim at deriving an effective Richards equation. To obtain the latter, we adopt a perturbation expansion truncated at the first order (weakly heterogeneous media), which leads to the effective hydraulic conductivity and water retention curves. Overall, the effective properties are scale dependent. However, within the proposed framework, we demonstrate that the inflection point of the laboratory scale retention curve is not affected by the heterogeneity of the vadose zone. Finally, to illustrate the quantitative implications of our results, we consider a monitoring experiment at field scale, and we show how our approach leads to an effective water retention curve, which differs significantly from that which would be obtained without accounting for the above scale‐invariance property.

A local sensitivity analysis for the kinetic Kuramoto equation with random inputs

We present a local sensivity analysis for the kinetic Kuramoto equation with random inputs in a large coupling regime. In our proposed random kinetic Kuramoto equation (in short, RKKE), the random inputs are encoded in the coupling strength. For the deterministic case, it is well known that the kinetic Kuramoto equation exhibits asymptotic phase concentration for well-prepared initial data in the large coupling regime. To see a response of the system to the random inputs, we provide propagation of regularity, local-in-time stability estimates for the variations of the random kinetic density function in random parameter space. For identical oscillators with the same natural frequencies, we introduce a Lyapunov functional measuring the phase concentration, and provide a local sensitivity analysis for the functional.

Effective conductivity in steady well-type flows through porous formations

A steady flow generated by a well of given strength takes place in a two-dimensional heterogeneous porous formation where the conductivity K is modeled as a random space function (RSF). As a consequence, the flow-variables become RSF s, and we wish to compute the effective conductivity (EC) by means of the self-consistent approximation. Toward this aim, the porous formation is sought as a collection of circular, non overlapping, inclusions with different (and statistically independent) conductivities. We compute the EC by adapting a procedure which was originally developed for mean uniform flows. Overall, the EC is found to be position-dependent, and therefore it can not be regarded as a medium’s property (unless one is dealing with large distances from the well). Then, it is shown how results of the present study can be used for practical applications.

Integrating transient behavior as a new dimension to WHPA delineation

Abstract The protection of groundwater abstraction is primarily based on the delineation of “time-of-travel capture zones” known as Well Head Protection Areas (WHPAs). Their delineation is subject to several uncertainties: the scarcity of hydrogeological data due to limited budgets for exploration and additional uncertainty from time-varying flow conditions. While the former has lead to the use of probabilistic strategies, no existing approaches for WHPA delineation consider jointly geological uncertainties, transient flow conditions and the associated additional uncertainties. We propose to extend existing steady-state probabilistic approaches for WHPA delineation to account for transient flow conditions and their related uncertainties. In our synthetic scenario, transiency is represented through four selected transient drivers: (I) regional groundwater flow direction, (II) regional strength of the hydraulic gradient, (III) natural groundwater recharge and (IV) pumping rate. As a result of our proposed probabilistic framework, we obtain probabilistic catchment maps that consider the joint effect of (uncertain) transient flow conditions and geological uncertainty. To represent hydraulic conductivity we use a random space function using a Matern covariance function. In order to reduce computational costs, so that we can conduct Monte Carlo analysis over both sources of uncertainty, we represent transiency as a dynamic superposition of steady-state flow solutions. Overall, we show that (1) each transient driver defines a distinctive pattern of temporal catchment membership, with the ambient flow direction driving most of the changes. (2) The transient analysis enhances decision support for probabilistic WHPA delineation by providing additional information in terms of time reliability levels. Now, a WHPA can be defined from this analysis by selecting a reliability level for time and one for geological uncertainty. (3) Time reliability addresses a different kind of risk when compared to geological reliability: While a lack of time reliability implies a known risk to the well for a fraction of time, a lack of geological reliability means that one does not know whether the well is at risk. (4) In the presence of uncertain transient and geological conditions, high time reliability levels are cheap compared to high reliability levels against geological uncertainties. (5) The combined use of time-geological reliability information with land use information leads towards more targeted risk reduction measures that align with the two different types of risk addressed by time reliability and geological reliability.

Stochastic analysis of unsaturated steady flows above the water table

AbstractSteady flow takes place into a three‐dimensional partially saturated porous medium where, due to their spatial variability, the saturated conductivity Ks, and the relative conductivity Kr are modeled as random space functions (RSF)s. As a consequence, the flow variables (FVs), i.e., pressure‐head and specific flux, are also RSFs. The focus of the present paper consists into quantifying the uncertainty of the FVs above the water table. The simple expressions (most of which in closed form) of the second‐order moments pertaining to the FVs allow one to follow the transitional behavior from the zone close to the water table (where the FVs are nonstationary), till to their far‐field limit (where the FVs become stationary RSFs). In particular, it is shown how the stationary limits (and the distance from the water table at which stationarity is attained) depend upon the statistical structure of the RSFs Ks, Kr, and the infiltrating rate. The mean pressure head has been also computed, and it is expressed as , being ψ a characteristic heterogeneity function which modifies the zero‐order approximation of the pressure head (valid for a vadose zone of uniform soil properties) to account for the spatial variability of Ks and Kr. Two asymptotic limits, i.e., close (near field) and away (far field) from the water table, are derived into a very general manner, whereas the transitional behavior of ψ between the near/far field can be determined after specifying the shape of the various input soil properties. Besides the theoretical interest, results of the present paper are useful for practical purposes, as well. Indeed, the model is tested against to real data, and in particular it is shown how it is possible for the specific case study to grasp the behavior of the FVs within an environment (i.e., the vadose zone close to the water table) which is generally very difficult to access by direct inspection.

Radiation Transport in Random Media With Large Fluctuations

Neutral particle transport in media exhibiting large and complex material property spatial variation is modeled by representing cross sections as lognormal random functions of space and generated through a nonlinear memory-less transformation of a Gaussian process with covariance uniquely determined by the covariance of the cross section. A Karhunen-Loeve decomposition of the Gaussian process is implemented to effciently generate realizations of the random cross sections and Woodcock Monte Carlo used to transport particles on each realization and generate benchmark solutions for the mean and variance of the particle flux as well as probability densities of the particle reflectance and transmittance. A computationally effcient stochastic collocation method is implemented to directly compute the statistical moments such as the mean and variance, while a polynomial chaos expansion in conjunction with stochastic collocation provides a convenient surrogate model that also produces probability densities of output quantities of interest. Extensive numerical testing demonstrates that use of stochastic reduced-order modeling provides an accurate and cost-effective alternative to random sampling for particle transport in random media.

Efficient geostatistical inversion of transient groundwater flow using preconditioned nonlinear conjugate gradients

In the geostatistical inverse problem of subsurface hydrology, continuous hydraulic parameter fields, in most cases hydraulic conductivity, are estimated from measurements of dependent variables, such as hydraulic heads, under the assumption that the parameter fields are autocorrelated random space functions. Upon discretization, the continuous fields become large parameter vectors with O(104−107) elements. While cokriging-like inversion methods have been shown to be efficient for highly resolved parameter fields when the number of measurements is small, they require the calculation of the sensitivity of each measurement with respect to all parameters, which may become prohibitive with large sets of measured data such as those arising from transient groundwater flow.We present a Preconditioned Conjugate Gradient method for the geostatistical inverse problem, in which a single adjoint equation needs to be solved to obtain the gradient of the objective function. Using the autocovariance matrix of the parameters as preconditioning matrix, expensive multiplications with its inverse can be avoided, and the number of iterations is significantly reduced. We use a randomized spectral decomposition of the posterior covariance matrix of the parameters to perform a linearized uncertainty quantification of the parameter estimate.The feasibility of the method is tested by virtual examples of head observations in steady-state and transient groundwater flow. These synthetic tests demonstrate that transient data can reduce both parameter uncertainty and time spent conducting experiments, while the presented methods are able to handle the resulting large number of measurements.

An analytical model for carrier-facilitated solute transport in weakly heterogeneous porous media

Carrier-facilitated solute transport in heterogeneous aquifers is studied within a Lagrangian framework. Dissolved solutes and carriers are advected by steady random groundwater flow, which is modeled by Darcy's law with uncertain hydraulic conductivity that is treated as a stationary random space function. We derive general expressions for the spatial moments of the dissolved concentration and the concentration associated with the carrier phase. In order to reduce the computational effort, we use previously derived solutions for the flow field. This enables us to obtain closed-form solutions for the spatial moments of the two concentration fields. The mass and center of gravity of the two propagating plumes depend only on the mean velocity field and chemical/degradation processes. The higher (second and third) moments are affected by the coupling between reactions (sorption/desorption and degradation) among the three phases (i.e., dissolved, carrier and sorbed concentrations) and the aquifer’s heterogeneity. We investigate the potentially enhancing effect of carriers by comparing spatial moments of the two propagating plumes. The forward/backward mass transfer rates between the liquid and carrier phases, and the degradation coefficients are identified as critical parameters. The carrier's role is most prominent when detachment from carrier sites is slow, provided that degradation on the carriers is smaller than that in the liquid phase.

Analysis and Upscaling of Unsaturated Flow through Randomly Heterogeneous Soil

The spatial variability of soil hydraulic properties has to be considered to provide realistic predictions of unsaturated flow and transport at field scale. Steady unsaturated flow through randomly heterogeneous soils is analyzed and upscaled in this study. Numerical experiments of unsaturated flow have been performed on single realizations of unsaturated soil parameters generated by using threedimensional turning band random generator. In the first part of the study a linear relation for unsaturated hydraulic conductivity as a function of soil suction (.) given by K = Ksea. is used, with constant beta and saturated hydraulic conductivity (K-s) as perfectly correlated random space functions. In the first part of the study, linear conductivity relation given by Gardner is used, with Gardner's constant (beta) and saturated conductivity (Ks) as perfectly correlated random space functions. The steady flow fields generated are analyzed to explore the effect of variability of the product (a): the pore-size distribution parameter whose reciprocal represents the characteristic capillary length (h(cap)) and the correlation length (lambda Z) of the generated soil properties. A set of one-dimensional simulations are performed, with various values of the product beta lambda(Z) ratio of correlation length to capillary dispersion length (lambda(z)/ hcap = 1, >> 1 and e 1) used to compare the moisture distribution tendency. The resulting steady pressure fields are compared and analyzed. In some cases, the soil behaves as homogeneous, and in other cases as stratified. The second part of the study seeks to identify simple upscaling laws for block-scale nonlinear constitutive relationships under high tension, with the parameters of their measurement scale counterparts represented by the van Genuchten model. Random fields are generated to represent layering of perfect and imperfectly stratified soils, and numerical simulations are performed in flows that are perpendicular and parallel to stratification. Upscaling of hydraulic conductivity curve Kd.and moisture retention curve theta(Psi) is performed for perfectly stratified and imperfectly stratified soils, and anisotropy is also studied for both types of layering considered. It is demonstrated that the upscaled nonlinear curves can be represented by simple exponential relations and/ or by linear relations obtained by piecewise linearization. The linear relations obtained can be expressed in equivalent terms that can be used in analytical solutions to directly obtain effective hydraulic conductivity values. (C) 2016 American Society of Civil Engineers.

Mining Geostatistics to Quantify the Spatial Variability of Certain Soil Flow Properties

The functional dependence of the relative unsaturated hydraulic conductivity (UHC) Kr (ψ) exp(αψ) upon the matric potential ψ, [L], via the soil-dependent parameter α, [L−1], has been traditionally regarded as a deterministic process (i.e. α ∼ constant). However, in the practical applications one is concerned with flow domains of large extents where α undergoes to significant spatial variations as consequence of the disordered soil's structure. To account for such a variability (hereafter also termed as “heterogeneity”) we adopt the mining geostatistical approach, which regards α as a random space function (RSF). To quantify the heterogeneity of α, estimates of local-values were obtained from ∼ 100 locations along a trench where an internal drainage test was conducted. The analysis of the statistical moments of α demonstrates (in line with the current literature on the matter) that the log-transform ζ ln α can be regarded as a structureless, normally distributed, RSF. An novel implementation of the present study in the context of the “Internet of Things” (IoT) is outlined.

Sound source localization in a randomly inhomogeneous medium using matched statistical moment method.

This paper investigates the problem of sound source localization from acoustical measurements obtained by an array of microphones. The sound propagation medium is assumed to be randomly inhomogeneous, being modelled by a random function of space. In this case, classical source localization methods (e.g., beamforming, near-field acoustical holography, and time reversal) cannot be used anymore. Therefore, an approach based on the statistical moments of acoustical measurement is proposed to solve the aforementioned problem. In this work, a Karhunen-Loève expansion is used so that the random medium can be represented by a small number of uncorrelated and identically distributed random variables. The statistical characteristics of the measurements in terms of probability density function and statistical moments are also studied. Then, the sound source is localized by minimizing the error of statistical moments between the real measurements obtained from the microphone array and the measurements simulated from an assumed source. Finally, a numerical example is introduced to justify the proposed method. This experiment shows that the random field can be replicated by a very small number of random variables, the statistical moments of measurements guarantee the convergence, and the source location can be accurately estimated using the proposed source localization method.

Solute transport in aquifers with evolving scale heterogeneity

Abstract Transport processes in groundwater systems with spatially heterogeneous properties often exhibit anomalous behavior. Using first-order approximations in velocity fluctuations we show that anomalous superdiffusive behavior may result if velocity fields are modeled as superpositions of random space functions with correlation structures consisting of linear combinations of short-range correlations. In particular, this corresponds to the superposition of independent random velocity fields with increasing integral scales proposed as model for evolving scale heterogeneity of natural porous media [Gelhar, L. W. Water Resour. Res. 22 (1986), 135S-145S]. Monte Carlo simulations of transport in such multi-scale fields support the theoretical results and demonstrate the approach to superdiffusive behavior as the number of superposed scales increases.

Stochastic analysis of steady seepage underneath a water-retaining wall through highly anisotropic porous media

Steady seepage is determined by a head drop upstream/downstream of a water-retaining wall. Due to its erratic variations, hydraulic log-conductivity$Y=\ln K$is modelled as a stationary random space function (RSF). We deal with a highly anisotropic porous formation, i.e. an axisymmetric medium where the horizontal correlation integral scale of$Y$is much larger than the vertical one. The goal of computing the resulting flow field within a stochastic framework is complicated by non-uniformity of the mean flow. Simple (closed-form) expressions for the correlation functions of the flow variables as well as the mean head are derived. We use these results to quantify the impact of spatial variability of$Y$upon the probability that the exit volumetric flow rate downstream of the wall is greater than that obtained by regarding the formation as homogeneous (with constant hydraulic conductivity). In particular, we show that the spatial variability of$Y$may lead to predictions (and consequently to design choices) which significantly differ from those achieved by regarding the porous formation as homogeneous.

Predicting DNAPL mass discharge and contaminated site longevity probabilities: Conceptual model and high‐resolution stochastic simulation

AbstractImproper storage and disposal of nonaqueous‐phase liquids (NAPLs) has resulted in widespread contamination of the subsurface, threatening the quality of groundwater as a freshwater resource. The high frequency of contaminated sites and the difficulties of remediation efforts demand rational decisions based on a sound risk assessment. Due to sparse data and natural heterogeneities, this risk assessment needs to be supported by appropriate predictive models with quantified uncertainty. This study proposes a physically and stochastically coherent model concept to simulate and predict crucial impact metrics for DNAPL contaminated sites, such as contaminant mass discharge and DNAPL source longevity. To this end, aquifer parameters and the contaminant source architecture are conceptualized as random space functions. The governing processes are simulated in a three‐dimensional, highly resolved, stochastic, and coupled model that can predict probability density functions of mass discharge and source depletion times. While it is not possible to determine whether the presented model framework is sufficiently complex or not, we can investigate whether and to which degree the desired model predictions are sensitive to simplifications often found in the literature. By testing four commonly made simplifications, we identified aquifer heterogeneity, groundwater flow irregularity, uncertain and physically based contaminant source zones, and their mutual interlinkages as indispensable components of a sound model framework.

Scattering of internal tides by irregular bathymetry of large extent

AbstractWe present an analytical theory of scattering of tide-generated internal gravity waves in a continuously stratified ocean with a randomly rough seabed. Based on a linearized approximation, the idealized case of constant mean sea depth and Brunt–Väisälä frequency is considered. The depth fluctuation is assumed to be a stationary random function of space, characterized by small amplitude and a correlation length comparable to the typical wavelength. For both one- and two-dimensional topographies the effects of scattering on the wave phase over long distances are derived explicitly by the method of multiple scales. For one-dimensional topography, numerical results are compared with Bühler & Holmes-Cerfon (J. Fluid Mech., vol. 678, 2011, pp. 271–293), computed by the method of characteristics. For two-dimensional topography, new results are presented for both statistically isotropic and anisotropic cases.

Bayesian approach to quantify parameter uncertainty and impacts on predictive flow and mass transport in heterogeneous aquifer

Groundwater flow and mass transport predictions are subjected to uncertainty due to heterogeneity of hydraulic conductivity, whose variability in space is considerably higher than that of other hydraulic properties relevant to groundwater flow. To characterize the distribution of hydraulic conductivity, random space function (RSF) is often used. The Bayesian approach was applied to quantitatively study the effect of parameter uncertainty in RSF on a hypothetical two-dimensional uniform groundwater flow and mass transport. Specifically, the parameter uncertainty transmitted to macrodispersion in mass transport model was also inferred. The results showed that the posterior probability distributions of parameters were updated after Bayesian inference. The numerical experiments indicated that the overall predictive uncertainty was increased with simulating time along the flow direction. As to the relative contribution of the two types of uncertainty, it indicated that parametric uncertainty was a little more important than stochastic uncertainty for the predictive uncertainty of hydraulic head. When the uncertainty of hydraulic head as well as macrodispersion was transported to mass transport model, a much bigger contribution of stochastic uncertainty was observed. Therefore, parametric uncertainty should not be neglected during the process of subsurface simulation.

Travel time approach to kinetically sorbing solute by diverging radial flows through heterogeneous porous formations

Diverging radial flow takes place in a heterogeneous porous medium where the log conductivity Y = ln K is modeled as a stationary random space function (RSF). The flow is steady, and is generated by a fully penetrating well. A linearly sorbing solute is injected through the well envelope, and we aim at computing the average flux concentration (breakthrough curve). A relatively simple solution for this difficult problem is achieved by adopting, similar to Indelman and Dagan (1999), a few simplifying assumptions: (i) a thick aquifer of large horizontal extent, (ii) mildly heterogeneous medium, (iii) strongly anisotropic formation, and (iv) large Peclet number. By introducing an appropriate Lagrangian framework, three‐dimensional transport is mapped onto a one‐dimensional domain (τ, t) where τ and t represent the fluid travel and current time, respectively. Central for this approach is the probability density function of the RSF τthat is derived consistently with the adopted assumptions stated above. Based on this, it is shown that the travel time can be regarded as a Gaussian random variable only in the far field. The breakthrough curves are analyzed to assess the impact of the hydraulic as well as reactive parameters. Finally, the travel time approach is tested against a forced‐gradient transport experiment and shows good agreement.