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1.
The effect of random recharge on uniform steady free-surface flow in heterogeneous porous formations
The effect of parametric uncertainty in recharge rate and spatial variability of hydraulic conductivity upon free-surface flow is investigated in a stochastic framework. We examine the three-dimensional free-surface gravitational flow problem for sloped mean uniform flow in a randomly heterogeneous porous medium under the influence of random recharge. We develop analytic solutions for the variance of free-surface position, head, and specific discharge on the free surface. Additionally, we obtain semi-analytic solutions for the statistical moments of head and specific discharge beneath the free-surface. Statistical moments are derived using a first-order approximation and then compared with their parallel in an unbounded medium. The effect of recharge mean and variability on the statistical moments is analyzed. Results can be applied to more complex flows, slowly varying in the mean. 相似文献
2.
Stochastic analysis is commonly used to address uncertainty in the modeling of flow and transport in porous media. In the stochastic approach, the properties of porous media are treated as random functions with statistics obtained from field measurements. Several studies indicate that hydrological properties depend on the scale of measurements or support scales, but most stochastic analysis does not address the effects of support scale on stochastic predictions of subsurface processes. In this work we propose a new approach to study the scale dependence of stochastic predictions. We present a stochastic analysis of immiscible fluid–fluid displacement in randomly heterogeneous porous media. While existing solutions are applicable only to systems in which the viscosity of one phase is negligible compare with the viscosity of the other (water–air systems for example), our solutions can be applied to the immiscible displacement of fluids having arbitrarily viscosities such as NAPL–water and water–oil. Treating intrinsic permeability as a random field with statistics dependant on the permeability support scale (scale of measurements) we obtained, for one-dimensional systems, analytical solutions for the first moments characterizing unbiased predictions (estimates) of system variables, such as the pressure and fluid–fluid interface position, and we also obtained second moments, which characterize the uncertainties associated with such predictions. Next we obtained empirically scale dependent exponential correlation function of the intrinsic permeability that allowed us to study solutions of stochastic equations as a function of the support scale. We found that the first and second moments converge to asymptotic values as the support scale decreases. In our examples, the statistical moments reached asymptotic values for support scale that were approximately 1/10000 of the flow domain size. We show that analytical moment solutions compare well with the results of Monte Carlo simulations for moderately heterogeneous porous media, and that they can be used to study the effects of heterogeneity on the dynamics and stability of immiscible flow. 相似文献
3.
Thomas M. Bendall 《地球物理与天体物理流体动力学》2019,113(5-6):491-504
ABSTRACTA framework of variational principles for stochastic fluid dynamics was presented by Holm, and these stochastic equations were also derived by Cotter, Gottwald and Holm. We present a conforming finite element discretisation for the stochastic quasi-geostrophic equation that was derived from this framework. The discretisation preserves the first two moments of potential vorticity, i.e. the mean potential vorticity and the enstrophy. Following the work of Dubinkina and Frank, who investigated the statistical mechanics of discretisations of the deterministic quasi-geostrophic equation, we investigate the statistical mechanics of our discretisation of the stochastic quasi-geostrophic equation. We compare the statistical properties of our discretisation with the Gibbs distribution under assumption of these conserved quantities, finding that there is an agreement between the statistics under a wide range of set-ups. 相似文献
4.
《Advances in water resources》1998,21(6):477-485
A numerical approach for approximating statistical moments of hydraulic heads of variably saturated flows in multi-dimensional porous media is developed. The approximation relies on a first-order Taylor series expansion of a finite element flow model and an adjoint state numerical method for variably saturated flows to evaluate sensitivities. This approach can be employed to analyze uncertainties associated with predictions of head of steady-state or transient flows in variably saturated porous media, with any type of boundary and initial conditions. Limitations of stochastic analytical methods such as spectral/perturbation approaches and the time-consuming Monte Carlo simulation technique are thus alleviated. An example is given to demonstrate the utility of the approach and to investigate the temporal evolution of head variances in a variably saturated flow regime. Results show that the fluctuation of the water table can have significant impacts on the propagation of the head variance. 相似文献
5.
In this work, we address the problem of characterizing the heterogeneity and uncertainty of hydraulic properties for complex geological settings. Hereby, we distinguish between two scales of heterogeneity, namely the hydrofacies structure and the intrafacies variability of the hydraulic properties. We employ multiple-point geostatistics to characterize the hydrofacies architecture. The multiple-point statistics are borrowed from a training image that is designed to reflect the prior geological conceptualization. The intrafacies variability of the hydraulic properties is represented using conventional two-point correlation methods, more precisely, spatial covariance models under a multi-Gaussian spatial law. We address the different levels and sources of uncertainty in characterizing the subsurface heterogeneity, and explore their effect on groundwater flow and transport predictions. Typically, uncertainty is assessed by way of many images, termed realizations, of a fixed statistical model. However, in many cases, sampling from a fixed stochastic model does not adequately represent the space of uncertainty. It neglects the uncertainty related to the selection of the stochastic model and the estimation of its input parameters. We acknowledge the uncertainty inherent in the definition of the prior conceptual model of aquifer architecture and in the estimation of global statistics, anisotropy, and correlation scales. Spatial bootstrap is used to assess the uncertainty of the unknown statistical parameters. As an illustrative example, we employ a synthetic field that represents a fluvial setting consisting of an interconnected network of channel sands embedded within finer-grained floodplain material. For this highly non-stationary setting we quantify the groundwater flow and transport model prediction uncertainty for various levels of hydrogeological uncertainty. Results indicate the importance of accurately describing the facies geometry, especially for transport predictions. 相似文献
6.
We present a methodology conducive to the application of a Galerkin model order reduction technique, Proper Orthogonal Decomposition (POD), to solve a groundwater flow problem driven by spatially distributed stochastic forcing terms. Typical applications of POD to reducing time-dependent deterministic partial differential equations (PDEs) involve solving the governing PDE at some observation times (termed snapshots), which are then used in the order reduction of the problem. Here, the application of POD to solve the stochastic flow problem relies on selecting the snapshots in the probability space of the random quantity of interest. This allows casting a standard Monte Carlo (MC) solution of the groundwater flow field into a Reduced Order Monte Carlo (ROMC) framework. We explore the robustness of the ROMC methodology by way of a set of numerical examples involving two-dimensional steady-state groundwater flow taking place within an aquifer of uniform hydraulic properties and subject to a randomly distributed recharge. We analyze the impact of (i) the number of snapshots selected from the hydraulic heads probability space, (ii) the associated number of principal components, and (iii) the key geostatistical parameters describing the heterogeneity of the distributed recharge on the performance of the method. We find that our ROMC scheme can improve significantly the computational efficiency of a standard MC framework while keeping the same degree of accuracy in providing the leading statistical moments (i.e. mean and covariance) as well as the sample probability density of the state variable of interest. 相似文献
7.
Valentina Ciriello Vittorio Di Federico Monica Riva Francesco Cadini Jacopo De Sanctis Enrico Zio Alberto Guadagnini 《Stochastic Environmental Research and Risk Assessment (SERRA)》2013,27(4):945-954
We perform global sensitivity analysis (GSA) through polynomial chaos expansion (PCE) on a contaminant transport model for the assessment of radionuclide concentration at a given control location in a heterogeneous aquifer, following a release from a near surface repository of radioactive waste. The aquifer hydraulic conductivity is modeled as a stationary stochastic process in space. We examine the uncertainty in the first two (ensemble) moments of the peak concentration, as a consequence of incomplete knowledge of (a) the parameters characterizing the variogram of hydraulic conductivity, (b) the partition coefficient associated with the migrating radionuclide, and (c) dispersivity parameters at the scale of interest. These quantities are treated as random variables and a variance-based GSA is performed in a numerical Monte Carlo framework. This entails solving groundwater flow and transport processes within an ensemble of hydraulic conductivity realizations generated upon sampling the space of the considered random variables. The Sobol indices are adopted as sensitivity measures to provide an estimate of the role of uncertain parameters on the (ensemble) target moments. Calculation of the indices is performed by employing PCE as a surrogate model of the migration process to reduce the computational burden. We show that the proposed methodology (a) allows identifying the influence of uncertain parameters on key statistical moments of the peak concentration (b) enables extending the number of Monte Carlo iterations to attain convergence of the (ensemble) target moments, and (c) leads to considerable saving of computational time while keeping acceptable accuracy. 相似文献
8.
C. -M. Chang M. W. Kemblowski 《Stochastic Environmental Research and Risk Assessment (SERRA)》1994,8(4):281-300
In this article, we are concerned with the statistics of steady unsaturated flow in soils with a fractal hydraulic conductivity distribution. It is assumed that the spatial distribution of log hydraulic conductivity can be described as an isotropic stochastic fractal process. The impact of the fractal dimension of this process, the soil pore-size distribution parameter, and the characteristic length scale on the variances of tension head and the effective conductivity is investigated. Results are obtained for one-dimensional and three-dimensional flows. Our results indicate that the tension head variance is scale-dependent for fractal distribution of hydraulic conductivity. Both tension head variance and effective hydraulic conductivity depend strongly on the fractal dimension. The soil pore-size distribution parameter is important in reducing the variability of the unsaturated hydraulic conductivity and of the fluxes. 相似文献
9.
Non-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. 相似文献
10.
An extension of the symmetric-moving-average (SMA) scheme is presented for stochastic synthesis of a stationary process for approximating any dependence structure and marginal distribution. The extended SMA model can exactly preserve an arbitrary second-order structure as well as the high order moments of a process, thus enabling a better approximation of any type of dependence (through the second-order statistics) and marginal distribution function (through statistical moments), respectively. Interestingly, by explicitly preserving the coefficient of kurtosis, it can also simulate certain aspects of intermittency, often characterizing the geophysical processes. Several applications with alternative hypothetical marginal distributions, as well as with real world processes, such as precipitation, wind speed and grid-turbulence, highlight the scheme’s wide range of applicability in stochastic generation and Monte-Carlo analysis. Particular emphasis is given on turbulence, in an attempt to simulate in a simple way several of its characteristics regarded as puzzles. 相似文献
11.
Groundwater is a primary source of drinking water worldwide, but excess nutrients and emerging contaminants could compromise groundwater quality and limit its usage as a drinking water source. As such contaminants become increasingly prevalent in the biosphere, a fundamental understanding of their fate and transport in groundwater systems is necessary to implement successful remediation strategies. The dynamics of surface water-groundwater (hyporheic) exchange within a glacial, buried-valley aquifer system are examined in the context of their implications for the transport of nutrients and contaminants in riparian sediments. High conductivity facies act as preferential flow pathways which enhance nutrient and contaminant delivery, especially during storm events, but transport throughout the aquifer also depends on subsurface sedimentary architecture (e.g. interbedded high and low conductivity facies). Temperature and specific conductance measurements indicate extensive hyporheic mixing close to the river channel, but surface water influence was also observed far from the stream-aquifer interface. Measurements of river stage and hydraulic head indicate that significant flows during storms (i.e., hot moments) alter groundwater flow patterns, even between consecutive storm events, as riverbed conductivity and, more importantly, the hydraulic connectivity between the river and aquifer change. Given the similar mass transport characteristics among buried-valley aquifers, these findings are likely representative of glacial aquifer systems worldwide. Our results suggest that water resources management decisions based on average (base) flow conditions may inaccurately represent the system being evaluated, and could reduce the effectiveness of remediation strategies for nutrients and emerging contaminants. 相似文献
12.
We present a diagrammatic method for solving stochastic 1-D and 2-D steady-state flow equations in bounded domains. The diagrammatic method results in explicit solutions for the moments of the hydraulic head. This avoids certain numerical constraints encountered in realization-based methods. The diagrammatic technique also allows for the consideration of finite domains or large fluctuations, and is not restricted by distributional assumptions. The results of the method for 1-D and 2-D finite domains are compared with those obtained through a realization-based approach. Mean and variance of head are well reproduced for all log-conductivity variances inputted, including those larger than one. The diagrammatic results also compare favorably to hydraulic head moments derived by standard analytic methods requiring a linearized form of the flow equation. 相似文献
13.
G. Christakos C. T. Miller D. Oliver 《Stochastic Environmental Research and Risk Assessment (SERRA)》1993,7(3):213-239
As is well known, a complete stochastic solution of the stochastic differential equation governing saturated groundwater flow leads to an infinite hierarchy of equations in terms of higher-order moments. Perturbation techniques are commonly used to close this hierarchy, using power-series expansions. These methods are applied by truncating the series after a finite number of terms, and products of random gradients of conductivity and head potential are neglected. Uncertainty regarding the number or terms required to yield a sufficiently accurate result is a significant drawback with the application of power series-based perturbation methods for such problems. Low-order series truncation may be incapable of representing fundamental characteristics of flow and can lead to physically unreasonable and inaccurate solutions of the stochastic flow equation. To support this argument, one-dimensional, steady-state, saturated groundwater flow is examined, for the case of a spatially distributed hydraulic conductivity field. An ordinary power-series perturbation method is used to approximate the mean head, using second-order statistics to characterize the conductivity field. Then an interactive perturbation approach is introduced, which yields improved results compared to low-order, power-series perturbation methods for situations where strong interactions exist between terms in such approximations. The interactive perturbation concept is further developed using Feynman-type diagrams and graph theory, which reduce the original stochastic flow problem to a closed set of equations for the mean and the covariance functions. Both theoretical and practical advantages of diagrammatic solutions are discussed; these include the study of bounded domains and large fluctuations. 相似文献
14.
L. D. Oliver G. Christakos 《Stochastic Environmental Research and Risk Assessment (SERRA)》1995,9(4):269-296
We present a diagrammatic method for solving stochastic 1-D and 2-D steady-state flow equations in bounded domains. The diagrammatic method results in explicit solutions for the moments of the hydraulic head. This avoids certain numerical constraints encountered in realization-based methods. The diagrammatic technique also allows for the consideration of finite domains or large fluctuations, and is not restricted by distributional assumptions. The results of the method for 1-D and 2-D finite domains are compared with those obtained through a realization-based approach. Mean and variance of head are well reproduced for all log-conductivity variances inputted, including those larger than one. The diagrammatic results also compare favorably to hydraulic head moments derived by standard analytic methods requiring a linearized form of the flow equation. 相似文献
15.
In this work we develop a new multiscale procedure to compute numerically the statistical moments of the stochastic variables which govern single phase flow in heterogeneous porous media. The technique explores the properties of the log-normally distributed hydraulic conductivity, characterized by power-law or exponential covariances, which shows invariance in its statistical structure upon a simultaneous change of the scale of observation and strength of heterogeneity. We construct a family of equivalent stochastic hydrodynamic variables satisfying the same flow equations at different scales and strengths of heterogeneity or correlation lengths. Within the new procedure the governing equations are solved in a scaled geology and the numerical results are mapped onto the original medium at coarser scales by a straightforward rescaling. The new procedure is implemented numerically within the Monte Carlo algorithm and also in conjunction with the discretization of the low-order effective equations derived from perturbation analysis. Numerical results obtained by the finite element method show the accuracy of the new procedure to approximated the two first moments of the pressure and velocity along with its potential in reducing drastically the computational cost involved in the numerical modeling of both power-law and exponential covariance functions. 相似文献
16.
Greg Holloway 《Surveys in Geophysics》2004,25(3-4):203-219
Traditional ocean modeling treats fields resolved on the model grid according to the classical dynamics of continua. Variability on smaller scales is included through sundry eddy viscosities, mixing coefficients and other schemes. In this paper we develop an alternative approach based on statistical dynamics. First, we recognize that we treat probabilities of flows, not the flows themselves. Modeled dependent variables are the moments (expectations) of the probabilities of possible flows. Second, we address the challenge to obtain the equations of motion for the moments of probable flows rather than the (traditional) equations for explicit flows. For linear terms and on larger resolved scales, the statistical equations agree with classical dynamics where those of traditional modeling works well. Differences arise where traditional modeling would relegate unresolved motion to eddy viscosity, etc.. Instead, changes of entropy (<-log P> over the probability distribution of possible flows) with respect to the modeled moments act as forcings upon those moments. In this way we obtain a consistent framework for specifying the terms which, traditionally, represent subgridscale effects. Although these statistical equations are close to the classical equations in many ways, important differences are also evident; here, two phenomena are described where the results differ. We consider eddies interacting with bottom topography. It is seen that traditional eddy viscosity and/or topographic drag, which would reduce large scale flows toward rest, are wrong. The second law of thermodynamics is violated; the arrow of time is running backwards! From statistical dynamics, approximate corrections are obtained, yielding a practical improvement to the fidelity of ocean models. Another phenomenon occurs at much smaller scales in the turbulent mixing of heat and salt. Even when both heat and salt are stably stratifying, their rates of turbulent transfer should differ. This suggests a further model improvement. 相似文献
17.
《Advances in water resources》1999,23(1):49-57
Travel-time statistics for non-reacting tracers in fractal and multifractal media are addressed through numerical simulations. The logarithm of hydraulic conductivity is modeled using fractional Brownian motion (fBm) and more recently developed multifractal model based on bounded fractional Levy motion (bfLm). These models have been shown previously to accurately reproduce statistical properties of large conductivity datasets. The ensemble-mean travel time increases nearly linearly with travel distance and the variance in the travel time increases nearly parabolically with travel distance. This is consistent with near-field analytical approximations developed for non-fractal media and suggests that these analytical results may have some degree of robustness to non-ideal features in the random-field models. The magnitudes of the travel-time moments are dependent on the system size. For fBm media, this size dependence can be explained using an effective variance that increases with increasing size of the flow system. However, the magnitudes of the travel-time moments are also sensitive to other non-ideal effects such as deviations from Gaussian behavior. This sensitivity illustrates the need for careful aquifer characterization and conditional numerical simulation in practical situations requiring accurate estimates of uncertainty in the plume position. 相似文献
18.
19.
This paper deals with the lower order (first four) nonstationary statistical moments of the response of linear systems with random stiffness and random damping properties subject to random nonstationary excitation modeled as white noise multiplied by an envelope function. The method of analysis is based on a Markov approach using stochastic differential equations (SDE). The linear SDE with random coefficients subject to random excitation with deterministic initial conditions are transformed to an equivalent nonlinear SDE with deterministic coefficients and random initial conditions subject to random excitation. In this procedure, new SDE with random initial conditions, deterministic coefficients and zero forcing functions are introduced to represent the random variables. The joint statistical moments of the response are determined by considering an augmented dynamic system with state variables made up of the displacement and velocity vectors and the random variables of the structural system. The zero time-lag joint statistical moment equations for the augmented state vector are derived from the Itô differential formula. The statistical moment equations are ordinary nonlinear differential equations where hierarchy of moments appear. The hierarchy is closed by the cumulant neglect closure method applied at the fourth order statistical moment level. General formulation is given for multi-degree-of-freedom (MDOF) systems and the performance of the method in problems with nonstationary excitations and large variabilities is illustrated for a single-degree-of-freedom (SDOF) oscillator. 相似文献
20.
An Evaluation of Interpolation Methodologies for Generating Three-Dimensional Hydraulic Property Distributions from Measured Data 总被引：2，自引：0，他引：2
The process of attempting to model ground-water systems requires a good understanding of the spatial variation of aquifer hydraulic properties. The capabilities of the more recent innovative flowmeters such as the electromagnetic and heat pulse flowmeters provide the sensitivity to measure ambient flows and pump-induced flows. These flowmeters provide the measurements of pump-induced vertical flows which are analyzed to obtain vertical variations in horizontal hydraulic conductivity, K(z). With discrete areal K-values, K(x, y), and vertical profiles of K, provided by multiwell testing, the essential elements are present to produce a three-dimensional hydraulic conductivity field. The advent of these new flow measuring devices has contributed much to the motivation behind this paper. This paper presents the results of applying deterministic and stochastic methodology to the three-dimensional interpolation of hydraulic properties, specifically, hydraulic conductivity, K. Three of the approaches applied in this paper are deterministic in nature, inverse-distance weighting, inverse-distance-squared weighting, and ordinary kriging, while the fourth is a stochastic approach based on self-affine fractals. All of the methods are applied to measured data collected from 14 wells at a site in the United States near Mobile, Alabama. The three-dimensional K-distributions generated by each of the methods are used as inputs to an advective based transport model with the resulting model output compared to a two-well tracer study run previously at the same site. 相似文献