Polynomial chaos expansion for global sensitivity analysis applied to a model of radionuclide migration in a randomly heterogeneous aquifer

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.     

Numerical approach to Cryptosporidium risk assessment using reliability method

A previously developed Cryptosporidium transport model is solved numerically to investigate the transport and interactions between Cryptosporidium, water and surface sediment and to estimate the risk of surface water contamination by Cryptosporidium. The primary objective of this study is to expand the work of Yeghiazarian (Ph.D. dissertation, Cornell University 2001)where the analytical solution of the Cryptosporidium transport model was obtained for a simple case of specific attachment of Cryptosporidium oocysts to fine soil particles wherein some parameters have zero values. However, some studies have shown several cases where these parameters are not zero. This necessitated further study to generate a solution to the complete Cryptosporidium transport model. Utilizing the finite difference method, the Cryptosporidium transport model is solved numerically for the general case of a system with any parameter values. Previously, first- and second-order reliability methods (FORM and SORM) were employed for risk assessment using analytical transport results (Yeghiazarian, Ph.D. dissertation, Cornell University, 2001), but in this work, FORM and SORM are applied to the numerical solution of the Cryptosporidium transport model to estimate the risk of Cryptosporidium contamination in surface water. The risk of surface water contamination is estimated by the probability that the Cryptosporidium concentration in surface water at a given time and location exceeds a safety threshold. The numerical solution is interfaced with the general-purpose reliability code, CALREL, to estimate the probability of failure on one hillslope. The sensitivity of system reliability to process parameters is reported.     

Stochastic modelling and risk analysis of groundwater pollution using FORM coupled with automatic differentiation

The first order reliability method (FORM) has been widely used in probabilistic modelling of groundwater problems. The FORM approach requires an iterative optimization procedure to find out the system failure point (the most probable point).The advantages of this approach are that it does not require many computations in comparison with other methods when applied to simple problems, and it produces reasonably accurate results. However, it has been found that the computations of FORM can equal or exceed that of other methods in case of large number of variables.In this paper, a new implementation of FORM was proposed with more efficiency and accuracy than the traditional FORM method. In the proposed approach, automatic differentiation is used to obtain the gradient vector of the limit state function, which is required by FORM, instead of using finite difference estimation. This way, the first order derivative was obtained with a very good accuracy, and with less computational effort. Based on the obtained results, it is found that the proposed implementation of FORM is a very good tool for probabilistic risk assessment and uncertainty analysis in groundwater problems.     

Designing a monitoring network for detecting groundwater pollution with stochastic simulation and a cost model

A method is presented to design monitoring networks for detecting groundwater pollution at industrial sites. The goal is to detect the pollution at some distance from the site’s boundary so that it can be cleaned up or hydrologically contained before contaminating groundwater outside the site. It is assumed that pollution may occur anywhere on the site, that transport is by advection only and that no retardation and chemical reactions take place. However, the approach can be easily extended to include designated (and uncertain) source areas, dispersion and reactive transport. The method starts from the premise that it is impossible to detect 100% of all the contaminant plumes with reasonable costs and therefore seeks a balance between the risk of pollution and network density. The design approach takes account of uncertainty in the flow field by simulating realisations of conductivity, groundwater head and associated flow fields, using geostatistical simulation and a groundwater flow model. The realisations are conditioned to conductivity and head observations that may already be present on the site. The result is an ensemble of flow fields that is further analysed using a particle track program. From this the probability of missing a contaminant plume originating anywhere on the terrain can be estimated for a given network. From this probability follows the risk, i.e. the expected costs of an undetected pollution. The total costs of the monitoring strategy are calculated by adding the risk of pollution to the costs of installing and maintaining the monitoring wells and the routinely performed chemical analyses. By repeating this procedure for networks of varying well numbers, the best network is chosen as the one that minimises total cost. The method is illustrated with a simulated example showing the added worth of exploratory wells for characterising hydraulic conductivity of a site.     

Dissolution of nonaqueous phase liquid pools in anisotropic aquifers

A two-dimensional numerical transport model is developed to determine the effect of aquifer anisotropy and heterogeneity on mass transfer from a dense nonaqueous phase liquid (DNAPL) pool. The appropriate steady state groundwater flow equation is solved implicitly whereas the equation describing the transport of a sorbing contaminant in a confined aquifer is solved by the alternating direction implicit method. Statistical anisotropy in the aquifer is introduced by two-dimensional, random log-normal hydraulic conductivity field realizations with different directional correlation lengths. Model simulations indicate that DNAPL pool dissolution is enhanced by increasing the mean log-transformed hydraulic conductivity, groundwater flow velocity, and/or anisotropy ratio. The variance of the log-transformed hydraulic conductivity distribution is shown to be inversely proportional to the average mass transfer coefficient.     

Groundwater,salt and contaminant transport in a coastal aquifer system with a low-permeability layer in the intertidal zone

Low-permeability layer (LPL), formed by natural deposit or artificial reclamation and commonly found below the intertidal zone of coastal groundwater system, can retard the ingress of seawater and contaminants, and shorten the travel time of the land-sourced contaminant to the marine environment compared with a homogenous sandy coastal aquifer. However, there is limited understanding on how an intertidal LPL, a condition occurred in a coastal aquifer at Moreton Bay, Australia, influences the groundwater and contaminant transport across the shallow beach aquifer system. We characterized the aquifer hydrological parameters, monitored the in situ groundwater heads, and constructed a 2-D numerical model to analyses the cross-shore hydrological processes in this stratified system. The calibrated model suggests that in the lower aquifer, the inland-source fresh groundwater flowed horizontally towards the sea, upwelled along the freshwater–saltwater interface, and exited the aquifer at the shore below the LPL. Whereas in the upper aquifer, the tidally driven seawater circulation formed a barrier that prevented fresh groundwater from horizontal transport and discharge to the beach above the LPL, thereby directing its leakage to the lower aquifer. A contaminant represented by a conservative tracer was ‘released’ the upper aquifer in the model and results showed that the spreading extent of the contaminant plume, the maximum rate of contaminant discharge to the ocean, and its plume length decreased compared with a simulation case in a homogenous sandy aquifer. Sensitivity analysis was also conducted to investigate the characteristics of the LPL, including its continuity and hydraulic conductivity, which were found to vary along the beach at Moreton Bay. The result shows that with a lower hydraulic conductivity and continuous layer of LPL reduced the groundwater exchange and contaminant transport between upper and lower aquifer. The findings from the combined field and modelling investigations on the impact of an intertidal LPL on coastal aquifer systems highlight its significant implications to alter the groundwater and mass transport across the land–ocean interface.     

Probabilistic sensitivity and modeling of two-dimensional transport in porous media

A reliability approach is used to develop a probabilistic model of two-dimensional non-reactive and reactive contaminant transport in porous media. The reliability approach provides two important quantitative results: an estimate of the probability that contaminant concentration is exceeded at some location and time, and measures of the sensitivity of the probabilistic outcome to likely changes in the uncertain variables. The method requires that each uncertain variable be assigned at least a mean and variance; in this work we also incorporate and investigate the influence of marginal probability distributions. Uncertain variables includex andy components of average groundwater flow velocity,x andy components of dispersivity, diffusion coefficient, distribution coefficient, porosity and bulk density. The objective is to examine the relative importance of each uncertain variable, the marginal distribution assigned to each variable, and possible correlation between the variables. Results utilizing a two-dimensional analytical solution indicate that the probabilistic outcome is generally very sensitive to likely changes in the uncertain flow velocity. Uncertainty associated with dispersivity and diffusion coefficient is often not a significant issue with respect to the probabilistic analysis; therefore, dispersivity and diffusion coefficient can often be treated for practical analysis as deterministic constants. The probabilistic outcome is sensitive to the uncertainty of the reaction terms for early times in the flow event. At later times, when source contaminants are released at constant rate throughout the study period, the probabilistic outcome may not be sensitive to changes in the reaction terms. These results, although limited at present by assumptions and conceptual restrictions inherent to the closed-form analytical solution, provide insight into the critical issues to consider in a probabilistic analysis of contaminant transport. Such information concerning the most important uncertain parameters can be used to guide field and laboratory investigations.     

Probabilistic sensitivity and modeling of two-dimensional transport in porous media

A reliability approach is used to develop a probabilistic model of two-dimensional non-reactive and reactive contaminant transport in porous media. The reliability approach provides two important quantitative results: an estimate of the probability that contaminant concentration is exceeded at some location and time, and measures of the sensitivity of the probabilistic outcome to likely changes in the uncertain variables. The method requires that each uncertain variable be assigned at least a mean and variance; in this work we also incorporate and investigate the influence of marginal probability distributions. Uncertain variables includex andy components of average groundwater flow velocity,x andy components of dispersivity, diffusion coefficient, distribution coefficient, porosity and bulk density. The objective is to examine the relative importance of each uncertain variable, the marginal distribution assigned to each variable, and possible correlation between the variables. Results utilizing a two-dimensional analytical solution indicate that the probabilistic outcome is generally very sensitive to likely changes in the uncertain flow velocity. Uncertainty associated with dispersivity and diffusion coefficient is often not a significant issue with respect to the probabilistic analysis; therefore, dispersivity and diffusion coefficient can often be treated for practical analysis as deterministic constants. The probabilistic outcome is sensitive to the uncertainty of the reaction terms for early times in the flow event. At later times, when source contaminants are released at constant rate throughout the study period, the probabilistic outcome may not be sensitive to changes in the reaction terms. These results, although limited at present by assumptions and conceptual restrictions inherent to the closed-form analytical solution, provide insight into the critical issues to consider in a probabilistic analysis of contaminant transport. Such information concerning the most important uncertain parameters can be used to guide field and laboratory investigations.     

钢框架结构的非线性静力抗震可靠性分析 II:可靠度指标及其灵敏度

将有限元反应及其灵敏度分析与结构可靠度分析的近似解析方法结合起来,可以进行具有隐式功能函数的大型复杂结构的可靠性分析。在基于位移的非线性纤维梁柱单元及其灵敏度直接微分表达式的基础上,通过力学变换、概率变换和反应灵敏度,将结构可靠度计算方法FORM和SORM与有限元方法有机地集成在一起。依据现行抗震设计规范,建立了钢框架结构典型构件承载能力和结构层间变形能力的抗震极限状态方程,利用地震作用的等效随机静力模型,采用非线性有限元静力可靠度方法,对一实际工程结构的抗震可靠度及其灵敏度进行了概率分析和评价,结果表明:尽管在大震作用下该结构的层间弹塑性变形可靠度较高,但是构件极限承载能力的可靠度指标较低,仍然存在失效的可能性。因此,仅验算"小震"作用下结构的承载能力可靠度和"大震"作用下结构的变形能力可靠度是不够的,还需要验算在"中震"和"大震"作用下结构的极限承载能力可靠度。     

钢框架结构的非线性静力抗震可靠性分析Ⅱ:可靠度指标及其灵敏度

将有限元反应及其灵敏度分析与结构可靠度分析的近似解析方法结合起来,可以进行具有隐式功能函数的大型复杂结构的可靠性分析。在基于位移的非线性纤维梁柱单元及其灵敏度直接微分表达式的基础上,通过力学变换、概率变换和反应灵敏度,将结构可靠度计算方法FORM和SORM与有限元方法有机地集成在一起。依据现行抗震设计规范,建立了钢框架结构典型构件承载能力和结构层间变形能力的抗震极限状态方程,利用地震作用的等效随机静力模型,采用非线性有限元静力可靠度方法,对一实际工程结构的抗震可靠度及其灵敏度进行了概率分析和评价,结果表明：尽管在大震作用下该结构的层间弹塑性变形可靠度较高,但是构件极限承载能力的可靠度指标较低,仍然存在失效的可能性。因此,仅验算“小震”作用下结构的承载能力可靠度和“大震”作用下结构的变形能力可靠度是不够的,还需要验算在“中震”和“大震”作用下结构的极限承载能力可靠度。     

Optimization of groundwater-monitoring networks for identification of the distribution of a contaminant plume

A new methodology is proposed to optimize monitoring networks for identification of the extent of contaminant plumes. The optimal locations for monitoring wells are determined as the points where maximal decreases are expected in the quantified uncertainty about contaminant existence after well installation. In this study, hydraulic conductivity is considered to be the factor that causes uncertainty. The successive random addition (SRA) method is used to generate random fields of hydraulic conductivity. The expected value of information criterion for the existence of a contaminant plume is evaluated based on how much the uncertainty of plume distribution reduces with increases in the size of the monitoring network. The minimum array of monitoring wells that yields the maximum information is selected as the optimal monitoring network. In order to quantify the uncertainty of the plume distribution, the probability map of contaminant existence is made for all generated contaminant plume realizations on the domain field. The uncertainty is defined as the sum of the areas where the probability of contaminant existence or nonexistence is uncertain. Results of numerical experiments for determination of optimal monitoring networks in heterogeneous conductivity fields are presented.     

The New Potential for Understanding Groundwater Contaminant Transport

The groundwater remediation field has been changing constantly since it first emerged in the 1970s. The remediation field has evolved from a dissolved‐phase centric conceptual model to a DNAPL‐dominated one, which is now being questioned due to a renewed appreciation of matrix diffusion effects on remediation. Detailed observations about contaminant transport have emerged from the remediation field, and challenge the validity of one of the mainstays of the groundwater solute transport modeling world: the concept of mechanical dispersion (Payne et al. 2008). We review and discuss how a new conceptual model of contaminant transport based on diffusion (the usurper) may topple the well‐established position of mechanical dispersion (the status quo) that is commonly used in almost every groundwater contaminant transport model, and evaluate the status of existing models and modeling studies that were conducted using advection‐dispersion models.     

Assessing environmental risks through fuzzy parameterized probabilistic analysis

A fuzzy parameterized probabilistic analysis (FPPA) method was developed in this study to assess risks associated with environmental pollution-control problems. FPPA integrated environmental transport modeling, fuzzy transformation, probabilistic risk assessment, fuzzy risk quantification into a general risk assessment framework, and was capable of handling uncertainties expressed as fuzzy-parameterized stochastic distributions. The proposed method was applied to two environmental pollution problems, with one being about the point-source pollution in a river system with uncertain water quality parameters and the other being concerned with groundwater contaminant plume from waste landfill site with poorly known contaminant physical properties. The study results indicated that the complex uncertain features had significant impacts on modeling and risk-assessment outputs; the degree of impacts of modeling parameters were highly dependent on the level of imprecision of these parameters. The results also implied that FPPA was capable of addressing vagueness or imprecision associated with probabilistic risk evaluation, and help generate risk outputs that could be elucidated under different possibilistic levels. The proposed method could be used by environmental managers to evaluate trade-offs involving risks and costs, as well as identify management solutions that sufficiently hedge against dual uncertainties.     

Groundwater contaminant transport including adsorption and first order decay

This study deals with the transport of a contaminant in groundwater. The contaminant is subject to first order decay or linear adsorption. Its displacement can be modeled by a random walk process in which particles are killed at exponentially distributed times. Dirichlet problems are derived for the rate and mean time at which contaminated particles reach a particular part of the boundary of a certain domain. These Dirichlet problems are solved asymptotically for two types of 2D-flow patterns: flow parallel to the boundary of a domain and arbitrary flow towards a well in an aquifer.     

Groundwater contaminant transport including adsorption and first order decay

This study deals with the transport of a contaminant in groundwater. The contaminant is subject to first order decay or linear adsorption. Its displacement can be modeled by a random walk process in which particles are killed at exponentially distributed times. Dirichlet problems are derived for the rate and mean time at which contaminated particles reach a particular part of the boundary of a certain domain. These Dirichlet problems are solved asymptotically for two types of 2D-flow patterns: flow parallel to the boundary of a domain and arbitrary flow towards a well in an aquifer.     

Characterization of groundwater contaminant source using Bayesian method

Contaminant source identification in groundwater system is critical for remediation strategy implementation, including gathering further samples and analysis, as well as implementing and evaluating different remediation plans. Such problem is usually solved with the aid of groundwater modeling with lots of uncertainty, e.g. existing uncertainty in hydraulic conductivity, measurement variance and the model structure error. Monte Carlo simulation of flow model allows the input uncertainty onto the model predictions of concentration measurements at monitoring sites. Bayesian approach provides the advantage to update estimation. This paper presents an application of a dynamic framework coupling with a three dimensional groundwater modeling scheme in contamination source identification of groundwater. Markov Chain Monte Carlo (MCMC) is being applied to infer the possible location and magnitude of contamination source. Uncertainty existing in heterogonous hydraulic conductivity field is explicitly considered in evaluating the likelihood function. Unlike other inverse-problem approaches to provide single but maybe untrue solution, the MCMC algorithm provides probability distributions over estimated parameters. Results from this algorithm offer a probabilistic inference of the location and concentration of released contamination. The convergence analysis of MCMC reveals the effectiveness of the proposed algorithm. Further investigation to extend this study is also discussed.     

Examining the Use of USEPA's Generic Attenuation Factor in Determining Groundwater Screening Levels for Vapor Intrusion

A value of 0.001 is recommended by the United States Environmental Protection Agency (USEPA) for its groundwater‐to‐indoor air Generic Attenuation Factor (GAFG), used in assessing potential vapor intrusion (VI) impacts to indoor air, given measured groundwater concentrations of volatile chemicals of concern (e.g., chlorinated solvents). The GAFG can, in turn, be used for developing groundwater screening levels for VI given target indoor air quality screening levels. In this study, we examine the validity and applicability of the GAFG both for predicting indoor air impacts and for determining groundwater screening levels. This is done using both analysis of published data and screening model calculations. Among the 774 total paired groundwater‐indoor air measurements in the USEPA's VI database (which were used by that agency to generate the GAFG) we found that there are 427 pairs for which a single groundwater measurement or interpolated value was applied to multiple buildings. In one case, up to 73 buildings were associated with a single interpolated groundwater value and in another case up to 15 buildings were associated with a single groundwater measurement (i.e., that the indoor air contaminant concentrations in all of the associated buildings were influenced by the concentration determined at a single point). In more than 70% of the cases (390 of 536 paired measurements in which horizontal building‐monitoring well distance was recorded) the monitoring wells were located more than 30 m (and one up to over 200 m) from the associated buildings. In a few cases, the measurements in the database even improbably implied that soil gas contaminant concentrations increased, rather than decreased, in an upward direction from a contaminant source to a foundation slab. Such observations indicate problematic source characterization within the data set used to generate the GAFG, and some indicate the possibility of a significant influence of a preferential contaminant pathway. While the inherent value of the USEPA database itself is not being questioned here, the above facts raise the very real possibility that the recommended groundwater attenuation factors are being influenced by variables or conditions that have not thus far been fully accounted for. In addition, the predicted groundwater attenuation factors often fall far beyond the upper limits of predictions from mathematical models of VI, ranging from screening models to detailed computational fluid dynamic models. All these models are based on the same fundamental conceptual site model, involving a vadose zone vapor transport pathway starting at an underlying uniform groundwater source and leading to the foundation of a building of concern. According to the analysis presented here, we believe that for scenarios for which such a “traditional” VI pathway is appropriate, 10^?4 is a more appropriately conservative generic groundwater to indoor air attenuation factor than is the EPA‐recommended 10^?3. This is based both on the statistical analysis of USEPA's VI database, as well as the traditional mathematical models of VI. This result has been validated by comparison with results from some well‐documented field studies.     

Probabilistic risk assessment of contaminant transport in groundwater and vapour intrusion following remediation of a contaminant source

Community-scale simulations were performed to investigate the risk to groundwater and indoor air receptors downgradient of a contaminated site following the remediation of a long-term source. Six suites of Monte Carlo simulations were performed using a numerical model that accounted for groundwater flow, reactive solute transport, soil gas flow, and vapour intrusion in buildings. The model was applied to a three-dimensional, community-scale (250 m × 1000 m × 14 m) domain containing heterogeneous, spatially correlated distributions of the hydraulic conductivity, fraction of organic carbon, and biodegradation rate constant, which were varied between realizations. Analysis considered results from both individual realizations as well as the suite of Monte Carlo simulations expressed through several novel, integrated parameters, such as the probability of exceeding a regulatory standard in either groundwater or indoor air. Results showed that exceedance probabilities varied considerably with the consideration of biodegradation in the saturated zone, and were less sensitive to changes in the variance of hydraulic conductivity or the incorporation of heterogeneous distributions of organic carbon at this spatial scale. A sharp gradient in exceedance probability existed at the lateral edges of the plumes due to variability in lateral dispersion, which defined a narrow region of exceedance uncertainty. Differences in exceedance probability between realizations (i.e., due to heterogeneity uncertainty) were similar to differences attributed to changes in the variance of hydraulic conductivity or fraction of organic carbon. Simulated clean-up times, defined by reaching an acceptable exceedance probability, were found to be on the order of decades to centuries in these community-scale domains. Results also showed that the choice of the acceptable exceedance probability level (e.g., 1 vs. 5 %) would likely affect clean up times on the order of decades. Moreover, in the scenarios examined here, the risk of exceeding indoor air standards was greater than that of exceeding groundwater standards at all times and places. Overall, simulations of coupled transport processes combined with novel spatial and temporal quantification metrics for Monte Carlo analyses, provide practical tools for assessing risk in wider communities when considering site remediation.