Application of a Bayesian approach to stochastic delineation of capture zones

This paper presents a Bayesian Monte Carlo method for evaluating the uncertainty in the delineation of well capture zones and its application to a wellfield in a heterogeneous, multiaquifer system. In the method presented, Bayes' rule is used to update prior distributions for the unknown parameters of the stochastic model for the hydraulic conductivity, and to calculate probability-based weights for parameter realizations using head residuals. These weights are then assigned to the corresponding capture zones obtained using forward particle tracking. Statistical analysis of the set of weighted protection zones results in a probability distribution for the capture zones. The suitability of the Bayesian stochastic method for a multilayered system is investigated, using the wellfield Het Rot at Nieuwrode, Belgium, located in a three-layered aquifer system, as an example. The hydraulic conductivity of the production aquifer is modeled as a spatially correlated random function with uncertain parameters. The aquitard and overlying unconfined aquifer are assigned random, homogeneous conductivities. The stochastic results are compared with deterministic capture zones obtained with a calibrated model for the area. The predictions of the stochastic approach are more conservative and indicate that parameter uncertainty should be taken into account in the delineation of well capture zones.     

二相随机介质有效电导率的广义混合率模型研究(英文)

广义混合率模型目前多应用于岩石的流变、杨氏模量等力学性质的研究,较少应用于多相介质岩石的有效电导率研究。本文利用三维有限元方法计算得到大量二相随机介质模型的有效电导率数据,引入广义混合率的有效电导率模型进行数据拟合,发现广义混合率模型参数J与两相介质电导率比值有关,并首次获得参数 J 与两相介质电导率比值之间的关系式,据此可以快速准确的预测(计算)任意二相介质的有效电导率,其结果较已有的随机介质模型和有效介质理论模型公式更为准确,为精细储集层评价奠定坚实基础。     

Hydrological modelling of temporally-varying catchments: facets of change and the value of information

Abstract

River basins are by definition temporally-varying systems: changes are apparent at every temporal scale, in terms of changing meteorological inputs and catchment characteristics due to inherently uncertain natural processes and anthropogenic interventions. In an operational context, the ultimate goal of hydrological modelling is predicting responses of the basin under conditions that are similar or different to those observed in the past. Since water management studies require that anthropogenic effects are considered known and a long hypothetical period is simulated, the combined use of stochastic models, for generating the inputs, and deterministic models that also represent the human interventions in modified basins, is found to be a powerful approach for providing realistic and statistically consistent simulations (in terms of product moments and correlations, at multiple time scales, and long-term persistence). The proposed framework is investigated on the Ferson Creek basin (USA) that exhibits significantly growing urbanization during the last 30 years. Alternative deterministic modelling options include a lumped water balance model with one time-varying parameter and a semi-distributed scheme based on the concept of hydrological response units. Model inputs and errors are respectively represented through linear and nonlinear stochastic models. The resulting nonlinear stochastic framework maximizes the exploitation of the existing information by taking advantage of the calibration protocol used in this issue.     

On the definition and derivatives of macroscale energy for the description of multiphase systems

Modeling of flow and transport in environmental systems often involves formulation of conservation equations at spatial scales involving tens to hundreds of pore diameters in porous media or the depth of flow in a channel. Quantities such as density, temperature, internal energy, and velocity may not be uniform over these macroscopic length scales. The external gravitational potential causes gradients in density, pressure, and chemical potential even at equilibrium. Despite these complications, it is important to formulate the thermodynamic analysis of environmental systems at the macroscopic scale. Heretofore, this has been accomplished primarily using the approach of rational thermodynamics whereby the thermodynamic dependence of macroscale internal energy on macroscale variables is hypothesized directly without development of any systematic method for transforming microscale energy dependence from the microscale to the macroscale. However when thermodynamic variables are inhomogeneous at the microscale, the functional dependence of macroscale internal energy on macroscale variables is not a simple extension of the microscale case. In the present work, the relation between the definitions of microscale and macroscale intensive thermodynamic variables is established. Expressions for the material derivatives of macroscale internal energy of phases, interfaces, and common lines are derived from and consistent with their microscopic counterparts by integrating to the macroscale. The forms obtained and the consistency required will be important for use in analyses of systems at scales where microscopic heterogeneities cannot be neglected.     

3D inverse modelling of groundwater flow at a fractured site using a stochastic continuum model with multiple statistical populations

 3D groundwater flow at the fractured site of Asp? (Sweden) is simulated. The aim was to characterise the site as adequately as possible and to provide measures on the uncertainty of the estimates. A stochastic continuum model is used to simulate both groundwater flow in the major fracture planes and in the background. However, the positions of the major fracture planes are deterministically incorporated in the model and the statistical distribution of the hydraulic conductivity is modelled by the concept of multiple statistical populations; each fracture plane is an independent statistical population. Multiple equally likely realisations are built that are conditioned to geological information on the positions of the major fracture planes, hydraulic conductivity data, steady state head data and head responses to six different interference tests. The experimental information could be reproduced closely. The results of the conditioning are analysed in terms of ensemble averaged average fracture plane conductivities, the ensemble variance of average fracture plane conductivities and the statistical distribution of the hydraulic conductivity in the fracture planes. These results are evaluated after each conditioning stage. It is found that conditioning to hydraulic head data results in an increase of the hydraulic conductivity variance while the statistical distribution of log hydraulic conductivity, initially Gaussian, becomes more skewed for many of the fracture planes in most of the realisations.     

DYNAMIC RESPONSE VARIABILITY OF STRUCTURES WITH UNCERTAIN PROPERTIES

A modal-based analysis of the dynamic response variability of multiple degree-of-freedom linear structures with uncertain parameters subjected to either deterministic or stochastic excitations is considered. A probabilistic methodology is presented in which random variables with specified probability distributions are used to quantify the parameter uncertainties. The uncertainty in the response due to uncertainties in the structural modelling and loading is quantified by various probabilistic measures such as mean, variance and coefficient of excess. The computation of these probabilistic measures is addressed. A series expansion involving orthogonal polynomials in terms of the system parameters is first used to model the response variability of each contributing mode. Linear equations for the coefficients of each series expansion are derived using the weighted residual method. Mode superposition is then used to derive analytical expressions for the variability and statistics of the uncertain response in terms of the coefficients of the series expansions for all contributing modes. A primary–secondary system and a ten-story building subjected to deterministic and stochastic loads are used to demonstrate the methodology, as well as evaluate its performance by comparing it to existing methods, including the computationally cost-efficient perturbation method.     

Calibration of base flow separation methods with streamflow conductivity

The conductivity mass-balance (CMB) method can be used to calibrate analytical base flow separation methods. The principal CMB assumptions are base flow conductivity is equal to streamflow conductivity at lowest flows, runoff conductivity is equal to streamflow conductivity at highest flows, and base flow and runoff conductivities are assumed to be constants over the period of record. To test the CMB assumptions, fluid conductivities of ground water, surface runoff, and streamflow were measured during wet and dry conditions in a 12-km(2) stream basin. Ground water conductivities at wells varied an average of 6% from dry to wet conditions, while stream conductivities varied 58%. Shallow ground water conductivity varied significantly with distance from the stream, with lowest conductivities of 87 microS/cm near the divide, a maximum of 520 microS/cm 59 m from the stream, and 215 microS/cm 22 m from the stream. Runoff conductivities measured in three rain events remained nearly constant, with lower conductivities of 35 microS/cm near the divide and 50 microS/cm near the stream. The CMB method was applied to the records from 10 USGS stream-gauging stations in Texas, Kentucky, Georgia, and Florida to calibrate the USGS base flow separation technique, HYSEP, by varying the time parameter 2N*. There is a statistically significant relationship between basin areas and calibrated values of 2N*, expressed as N = 0.46A(0.44), with N in days and A in km(2). The widely accepted relationship N = 0.83A(0.2) is not valid for these basins. Other analytic methods can also be calibrated with the CMB method.     

On the geostatistical approach to the inverse problem

The geostatistical approach to the inverse problem is discussed with emphasis on the importance of structural analysis. Although the geostatistical approach is occasionally misconstrued as mere cokriging, in fact it consists of two steps: estimation of statistical parameters (“structural analysis”) followed by estimation of the distributed parameter conditional on the observations (“cokriging” or “weighted least squares”). It is argued that in inverse problems, which are algebraically undetermined, the challenge is not so much to reproduce the data as to select an algorithm with the prospect of giving good estimates where there are no observations. The essence of the geostatistical approach is that instead of adjusting a grid-dependent and potentially large number of block conductivities (or other distributed parameters), a small number of structural parameters are fitted to the data. Once this fitting is accomplished, the estimation of block conductivities ensues in a predetermined fashion without fitting of additional parameters. Also, the methodology is compared with a straightforward maximum a posteriori probability estimation method. It is shown that the fundamental differences between the two approaches are: (a) they use different principles to separate the estimation of covariance parameters from the estimation of the spatial variable; (b) the method for covariance parameter estimation in the geostatistical approach produces statistically unbiased estimates of the parameters that are not strongly dependent on the discretization, while the other method is biased and its bias becomes worse by refining the discretization into zones with different conductivity.     

Convergence of deterministic and stochastic approaches in optimal remediation design of a contaminated aquifer

The comparison between two series of optimal remediation designs using deterministic and stochastic approaches showed a number of converging features. Limited sampling measurements in a supposed contaminated aquifer formed the hydraulic conductivity field and the initial concentration distribution used in the optimization process. The deterministic and stochastic approaches employed a single simulation–optimization method and a multiple realization approach, respectively. For both approaches, the optimization model made use of a genetic algorithm. In the deterministic approach, the total cost, extraction rate, and the number of wells used increase when the design must satisfy the intensified concentration constraint. Growing the stack size in the stochastic approach also brings about same effects. In particular, the change in the selection frequency of the used extraction wells, with increasing stack size, for the stochastic approach can indicate the locations of required additional wells in the deterministic approach due to the intensified constraints. These converging features between the two approaches reveal that a deterministic optimization approach with controlled constraints is achievable enough to design reliable remediation strategies, and the results of a stochastic optimization approach are readily available to real contaminated sites.     

Comparison of deterministic ensemble Kalman filters for assimilating hydrogeological data

Groundwater models are critical decision support tools for water resources management and environmental remediation. However, limitations in site characterization data and conceptual models can adversely affect the reliability of groundwater models. Therefore, there is a strong need for continuous model uncertainty reduction. Ensemble filters have recently emerged as promising high-dimensional data assimilation techniques. Two general categories of ensemble filters exist in the literature: perturbation-based and deterministic. Deterministic ensemble filters have been extensively studied for their better performance and robustness in assimilating oceanographic and atmospheric data. In hydrogeology, while a number of previous studies demonstrated the usefulness of the perturbation-based ensemble Kalman filter (EnKF) for joint parameter and state estimation, there have been few systematic studies investigating the performance of deterministic ensemble filters. This paper presents a comparative study of four commonly used deterministic ensemble filters for sequentially estimating the hydraulic conductivity parameter in low- and moderately high-dimensional groundwater models. The performance of the filters is assessed on the basis of twin experiments in which the true hydraulic conductivity field is assumed known. The test results indicate that the deterministic ensemble Kalman filter (DEnKF) is the most robust filter and achieves the best performance at relatively small ensemble sizes. Deterministic ensemble filters often make use of covariance inflation and localization to stabilize filter performance. Sensitivity studies demonstrate the effects of covariance inflation, localization, observation density, and conditioning on filter performance.     

Spatial scale effect on the upper soil effective parameters of a distributed hydrological model

Nonlinear dynamics and spatial variability in hydrological systems make the formulation of scaling theories difficult. Therefore, the development of knowledge related to scale effects, scaling techniques, parameterization and linkages of parameters across scales is highly relevant. The main purpose of this work is to analyse the spatial effect of the static storage capacity parameter H_u and the saturated hydraulic conductivity parameter k_s from microscale (sub‐grid level) to mesoscale (grid level) and its implication to the definition of an optimum cell size. These two parameters describe the upper soil water characteristics in the infiltration process conceptualization of the TETIS hydrological model. At microscale, the spatial heterogeneity of H_u and k_s was obtained generating random parameter fields through probability distribution functions and a spatial dependence model with pre‐established correlation lengths. The effective parameters at mesoscale were calculated by solving the inverse problem for each parameter field. Results indicate that the adopted inverse formulation allows transferring the nonlinearity of the system from microscale to the mesoscale via non‐stationary effective parameters. Their values at each cell and time step are in the range of zero to the mean value of the parameter at microscale. The stochastic simulations showed that the variance of the estimated effective parameters decreases when the ratio between mesoscale cell size and correlation length at microscale increases. For a ratio greater than 1, we found cell sizes having the characteristics of a representative elementary area (REA); in such case, the microscale variability pattern did not affect the system response at mesoscale. Copyright © 2011 John Wiley & Sons, Ltd.     

Validity limits for the van Genuchten–Mualem model and implications for parameter estimation and numerical simulation

The calculation of the relative hydraulic conductivity function based on water retention data is an attractive and widely used approach, since direct measurements of unsaturated conductivities are difficult. We show theoretically under which conditions an air-entry value for water retention data is definitely required when using the statistical approach of Mualem. Moreover we rigorously specify the conditions for which the classical van Genuchten–Mualem model leads to wrong predictions of relative hydraulic conductivity and, hence, an alternative formulation including an air-entry value should be used. Significant consequences are demonstrated for the inverse parameter estimation based on multistep outflow experiments. Furthermore it is shown that the use of a physically correct formulation of the water retention curve including an air-entry value and the derived hydraulic conductivity function influences not only the stability of numerical simulations but also their final results. This is especially grave as simulations with van Genuchten–Mualem parameters are frequently used to compare experiments and simulations and to draw conclusions on the correctness of Richards’ equation.     

Mapping optimization based on sampling size in earth related and environmental phenomena

A critical sampling grid can be defined for an earth related natural variable distributed in space, according to established theoretical results and under certain mathematical conditions. Sampling above this critical limit does not substantially improve mapping results, while based on this limit the ideal process of reproducing the original phenomenon is theoretically defined. The aim of the present paper is, by using an innovative approach; to investigate the validity of commonly used interpolation algorithms, both stochastic and deterministic, below and above this critical sampling limit. When sampling is dense, application to a simulated spatial random field shows that the results are equally accurate with those derived with more sophisticated stochastic methods. On the other hand, when the sampling grid is sparse, deterministic methods produce less accurate results, therefore stochastic algorithms with minimum estimation error are a much better option. To further demonstrate these points, the interpolation algorithms were applied in three different sampling grid densities in a contaminated waste disposal site in Russia.     

Truncated multiGaussian fields and effective conductance of binary media

Truncated Gaussian fields provide a flexible model for defining binary media with dispersed (as opposed to layered) inclusions. General properties of excursion sets on these truncated fields are coupled with a distance-based upscaling algorithm and approximations of point process theory to develop an estimation approach for effective conductivity in two-dimensions. Estimation of effective conductivity is derived directly from knowledge of the kernel size used to create the multiGaussian field, defined as the full-width at half maximum (FWHM), the truncation threshold and conductance values of the two modes. Therefore, instantiation of the multiGaussian field is not necessary for estimation of the effective conductance. The critical component of the effective medium approximation developed here is the mean distance between high conductivity inclusions. This mean distance is characterized as a function of the FWHM, the truncation threshold and the ratio of the two modal conductivities. Sensitivity of the resulting effective conductivity to this mean distance is examined for two levels of contrast in the modal conductances and different FWHM sizes. Results demonstrate that the FWHM is a robust measure of mean travel distance in the background medium. The resulting effective conductivities are accurate when compared to numerical results and results obtained from effective media theory, distance-based upscaling and numerical simulation.     

Transient flowmeter test: semi-analytic crossflow model

The transient flowmeter test (TFMT) provides more information about the well–aquifer system than the traditional quasi-steady-state flowmeter test (QFMT). The TFMT duration may be much shorter than that of a QFMT, which is desirable at highly contaminated sites where the extracted water has to be treated as hazardous waste. Here we present the TFMT model that accounts for inter-layer crossflow, a thick skin surrounding the well, and wellbore storage. The model is derived under the simplifying assumptions of the pseudo-steady-state inter-layer crossflow and the uniform wellface flux within each layer. The semi-analytic solution is inverted numerically from the Laplace domain to the time domain. Layer and skin parameters are estimated from the TFMT data via the modified Levenberg–Marquardt algorithm. The estimation is robust when the initial parameter guesses are close to their true values. Otherwise, a computationally expensive search among the local minima of the objective function is necessary to find the parameter estimates. The modeling errors and the associated parameter estimation errors are evaluated in a number of synthetic TFMTs and compared to the corresponding results obtained with a general numerical model that relaxes the two simplifying assumptions. The TFMT provides reasonably accurate estimates of hydraulic conductivities for the aquifer layers and the damaged skins and order-of-magnitude estimates of layer specific storativities and hydraulic conductivities for the normal skin. The skin specific storativities should not be estimated from a TFMT. Multi-rate TFMTs with a step-variable pumping rate yield significantly more accurate parameters than constant-pumping-rate TFMTs. The calculated modeling errors may be useful in estimating the magnitude of parameter estimation errors from the TFMT. Our field tests in a coastal aquifer at the Lizzie Site in North Carolina (USA) demonstrate the feasibility of a TFMT for aquifer characterization. The downhole hydraulic conductivity profiles from our field and synthetic TFMTs are consistent with the corresponding profiles from QFMTs.     

Estimation of reservoir flood control operation risks with considering inflow forecasting errors

A method for quantifying inflow forecasting errors and their impact on reservoir flood control operations is proposed. This approach requires the identification of the probability distributions and uncertainty transfer scheme for the inflow forecasting errors. Accordingly, the probability distributions of the errors are inferred through deducing the relationship between its standard deviation and the forecasting accuracy quantified by the Nash–Sutcliffe efficiency coefficient. The traditional deterministic flood routing process is treated as a diffusion stochastic process. The diffusion coefficient is related to the forecasting accuracy, through which the forecasting errors are indirectly related to the sources of reservoir operation risks. The associated risks are derived by solving the stochastic differential equation of reservoir flood routing via the forward Euler method. The Geheyan reservoir in China is selected as a case study. The hydrological forecasting model for this basin is established and verified. The flood control operation risks in the forecast-based pre-release operation mode for different forecasting accuracies are estimated by the proposed approach. Application results show that the proposed method can provide a useful tool for reservoir operation risk estimation and management.     

A comparative study of closed-form hydraulic conductivity prediction models

The Mualem and the Burdine hydraulic conductivity prediction models are considered in combination with the van Genuchten analytical retention curve, as well as the Brooks and Corey prediction model. An equivalence is presented between the retention curves of these models. A comparative study follows between hydraulic conductivities that are based on equivalent retention curves. A unified presentation of prediction models provides a framework for the whole analysis. The treatment of the equivalence problem consists in a minimization procedure characterized by uncoupling of the parameters and analytical evaluation of the objective function. Exact analytical equivalence relations are given for significant parts of the parameter ranges, and, for the remaining parts, analytical approximations are proposed. The comparisons between hydraulic conductivities are carried out via an inequality analysis. It is shown that the hydraulic conductivity of the Burdine model is less than that of the other models for extended ranges of equivalent parameters.