A comparison of numerical solutions of partial differential equations with probabilistic and possibilistic parameters for the quantification of uncertainty in subsurface solute transport

Traditionally, uncertainty in parameters are represented as probabilistic distributions and incorporated into groundwater flow and contaminant transport models. With the advent of newer uncertainty theories, it is now understood that stochastic methods cannot properly represent non random uncertainties. In the groundwater flow and contaminant transport equations, uncertainty in some parameters may be random, whereas those of others may be non random. The objective of this paper is to develop a fuzzy-stochastic partial differential equation (FSPDE) model to simulate conditions where both random and non random uncertainties are involved in groundwater flow and solute transport. Three potential solution techniques namely, (a) transforming a probability distribution to a possibility distribution (Method I) then a FSPDE becomes a fuzzy partial differential equation (FPDE), (b) transforming a possibility distribution to a probability distribution (Method II) and then a FSPDE becomes a stochastic partial differential equation (SPDE), and (c) the combination of Monte Carlo methods and FPDE solution techniques (Method III) are proposed and compared. The effects of these three methods on the predictive results are investigated by using two case studies. The results show that the predictions obtained from Method II is a specific case of that got from Method I. When an exact probabilistic result is needed, Method II is suggested. As the loss or gain of information during a probability–possibility (or vice versa) transformation cannot be quantified, their influences on the predictive results is not known. Thus, Method III should probably be preferred for risk assessments.     

Three long-range transport models compared to the ETEX experiment: a performance study

For operational or research purposes (dispersion computations of radioactive effluents during nuclear emergency situations, simulations of chemical pollution in the vicinity of thermal power plants), different models of passive dispersion in the atmosphere have been developed at the Environment Department of EDF’s R and D Division. This report presents the comparison of the performances of three such models: DIFTRA (lagrangian puff model, with operational goal), DIFEUL (three dimensional eulerian) and DIFPAR (Monte Carlo particle model) for the simulation of the first ETEX release, an international tracer campaign during which a passive tracer cloud has been followed over Europe. The results obtained in this study give model vs. experience differences of the same order as the model vs. experience differences observed during an international model comparison experiment using data of the Chernobyl release, the ATMES exercise. In addition to the standard statistical scores used in the evaluation of the performances of the transport models two asymmetric scores (in contradistinction with the Figure of Merit in Space) are proposed: “efficiency” and “power”. Their aim is to separate the two manners in which a model may be wrong: by predicting presence of pollutant while none is measured or conversely predicting absence when pollutant is actually detected.     

Application of first-order reliability method to modelling the fate and transport of benzene in groundwater

Parameter uncertainty plays a significant role in decision making regarding groundwater contamination and remediation, especially for non-conservative chemicals. This paper presents a probabilistic screening model to accommodate parameter uncertainty in the aquifer media, and physical–chemical parameters, using the first-order reliability method (FORM). The application of the developed model is illustrated on transport of benzene in groundwater. The results matched those obtained using the Monte Carlo simulation method, with a smaller number of functional evaluations. Parametric studies were conducted to estimate the effect of various parameters on the results.     

Development of a fuzzy-stochastic nonlinear model to incorporate aleatoric and epistemic uncertainty

Site uncertainties significantly influence groundwater flow and contaminant transport predictions. Aleatoric and epistemic uncertainty are both identified in site characterization and represented using proper uncertainty theories. When one theory best represents one parameter whereas a different theory may be more suitable for another parameter, the hybrid propagation of aleatoric (random) and epistemic (nonrandom) uncertainties will occur. The computational challenges of joint propagation of aleatoric and epistemic uncertainty through groundwater flow and contaminant transport models are significant. A fuzzy-stochastic nonlinear model was developed in this paper to incorporate these two types of uncertain site information and reduce the computational cost. The results show that (1) the computational cost using the nonlinear model is reduced compared with that of using the sparse grid algorithm and Monte Carlo methods; (2) the uncertainty of hydraulic conductivity (K) significantly influences the water head and solute distribution at the observation wells compared to other uncertain parameters, such as the storage coefficient and the distribution coefficient (K_d); and (3) the combination of multiple uncertain parameters substantially affects the simulation results. Neglecting site uncertainties may lead to unrealistic predictions.     

Modeling distributions of air pollutant concentrations—II. Estimation of one and two parameter statistical distributions

In Part I of this series Taylor, Jakeman and Simpson (1986, Atmospheric Environment, 20, 1781–1789) examined the problem of identifying the appropriate distributional form for air pollution concentration data. In this paper we examine the parameter estimation problem. Monte Carlo simulation is used to compare methods for fitting statistical distributions to such data where the distributional form is known. Three methods are investigated for estimating the parameters of the lognormal distribution, two methods for the exponential distribution, three methods for the γ-distribution and four methods for the Weibull distribution. For all distributions and for each method we examine the accuracy with which the upper percentiles of the distribution are evaluated as it is these percentiles which are referred to by air quality standards. For each distribution a simple empirical model, which yields approximations to the relative root mean square error of the percentile estimates against sample size and parameter values, is developed and demonstrated. Thus for each distributional model an estimate of the relative error associated with evaluating high pollutant levels may be readily determined.     

Validation of regression models for nitrate concentrations in the upper groundwater in sandy soils

For Dutch sandy regions, linear regression models have been developed that predict nitrate concentrations in the upper groundwater on the basis of residual nitrate contents in the soil in autumn. The objective of our study was to validate these regression models for one particular sandy region dominated by dairy farming. No data from this area were used for calibrating the regression models. The model was validated by additional probability sampling. This sample was used to estimate errors in 1) the predicted areal fractions where the EU standard of 50 mg l⁻¹ is exceeded for farms with low N surpluses (ALT) and farms with higher N surpluses (REF); 2) predicted cumulative frequency distributions of nitrate concentration for both groups of farms.Both the errors in the predicted areal fractions as well as the errors in the predicted cumulative frequency distributions indicate that the regression models are invalid for the sandy soils of this study area.     

Influence of temporally variable groundwater flow conditions on point measurements and contaminant mass flux estimations

Monitoring of contaminant concentrations, e.g., for the estimation of mass discharge or contaminant degradation rates, often is based on point measurements at observation wells. In addition to the problem, that point measurements may not be spatially representative, a further complication may arise due to the temporal dynamics of groundwater flow, which may cause a concentration measurement to be not temporally representative. This paper presents results from a numerical modeling study focusing on temporal variations of the groundwater flow direction. “Measurements” are obtained from point information representing observation wells installed along control planes using different well frequencies and configurations. Results of the scenario simulations show that temporally variable flow conditions can lead to significant temporal fluctuations of the concentration and thus are a substantial source of uncertainty for point measurements. Temporal variation of point concentration measurements may be as high as the average concentration determined, especially near the plume fringe, even when assuming a homogeneous distribution of the hydraulic conductivity. If a heterogeneous hydraulic conductivity field is present, the concentration variability due to a fluctuating groundwater flow direction varies significantly within the control plane and between the different realizations. Determination of contaminant mass fluxes is also influenced by the temporal variability of the concentration measurement, especially for large spacings of the observation wells. Passive dosimeter sampling is found to be appropriate for evaluating the stationarity of contaminant plumes as well as for estimating average concentrations over time when the plume has fully developed. Representative sampling has to be performed over several periods of groundwater flow fluctuation. For the determination of mass fluxes at heterogeneous sites, however, local fluxes, which may vary considerably along a control plane, have to be accounted for. Here, dosimeter sampling in combination with time integrated local water flux measurements can improve mass flux estimates under dynamic flow conditions.     

Assessing the natural attenuation of organic contaminants in aquifers using plume-scale electron and carbon balances: model development with analysis of uncertainty and parameter sensitivity

A quantitative methodology is described for the field-scale performance assessment of natural attenuation using plume-scale electron and carbon balances. This provides a practical framework for the calculation of global mass balances for contaminant plumes, using mass inputs from the plume source, background groundwater and plume residuals in a simplified box model. Biodegradation processes and reactions included in the analysis are identified from electron acceptors, electron donors and degradation products present in these inputs. Parameter values used in the model are obtained from data acquired during typical site investigation and groundwater monitoring studies for natural attenuation schemes. The approach is evaluated for a UK Permo-Triassic Sandstone aquifer contaminated with a plume of phenolic compounds. Uncertainty in the model predictions and sensitivity to parameter values was assessed by probabilistic modelling using Monte Carlo methods. Sensitivity analyses were compared for different input parameter probability distributions and a base case using fixed parameter values, using an identical conceptual model and data set. Results show that consumption of oxidants by biodegradation is approximately balanced by the production of CH4 and total dissolved inorganic carbon (TDIC) which is conserved in the plume. Under this condition, either the plume electron or carbon balance can be used to determine contaminant mass loss, which is equivalent to only 4% of the estimated source term. This corresponds to a first order, plume-averaged, half-life of > 800 years. The electron balance is particularly sensitive to uncertainty in the source term and dispersive inputs. Reliable historical information on contaminant spillages and detailed site investigation are necessary to accurately characterise the source term. The dispersive influx is sensitive to variability in the plume mixing zone width. Consumption of aqueous oxidants greatly exceeds that of mineral oxidants in the plume, but electron acceptor supply is insufficient to meet the electron donor demand and the plume will grow. The aquifer potential for degradation of these contaminants is limited by high contaminant concentrations and the supply of bioavailable electron acceptors. Natural attenuation will increase only after increased transport and dilution.     

Using stochastic risk assessment in setting information priorities for managing dioxin impact from a municipal waste incinerator

The objectives of this study were to assess site-specific carcinogenic risk of incinerator-emitted dioxins in a manner reflecting pollutant transfer across multimedia and multi-pathways. The study used site-specific environmental and exposure information and combined the Monte Carlo method with multimedia modeling to produce probability distributions of risk estimates. The risk estimates were further categorized by contaminated environmental media and exposure pathways that are experienced by human receptors in order to pinpoint significant sources of risk. Rank correlation coefficients were also calculated along with the Monte Carlo sampling to identify key factors that influenced estimation of risk. The results showed that ingestion accounted for more than 90% of the total risk and that risk control on ingestion of eggs, aboveground vegetables, and poultry should receive priority. It was also found that variation of parameters with variability accounted for around 35% of the total risk variance, while uncertainty contributed to the remaining 65%. Intake rates of aboveground vegetables, eggs, and poultry were the key parameters with the largest contribution to variance. In addition, sufficient sampling and analysis of dioxin contents in eggs, aboveground vegetables, poultry, soil, and fruit should be performed to improve risk estimation because the variation in concentrations in these media accounted for the largest overall risk variance. Finally, focus should be placed on reduction of uncertainty associated with the risk estimation through ingestion of aboveground vegetables, eggs, poultry, fruit, and soil because the risk estimates associated with these exposure pathways had the largest variance.     

A method for estimating distributions of mass transfer rate coefficients with application to purging and batch experiments

Mass transfer between aquifer material and groundwater is often modeled as first-order rate-limited sorption or diffusive exchange between mobile zones and immobile zones with idealized geometries. Recent improvements in experimental techniques and advances in our understanding of pore-scale heterogeneity demonstrate that two (or even a few) rate coefficients are insufficient in many cases. Here, we investigate a piece-wise linear model for a continuous distribution of rate coefficients, that has several advantages over previously used ‘statistical' distribution models (with functional form from gamma or lognormal PDF's): (1) distributions of arbitrary, even bimodal, shapes can be represented; (2) linear estimation methods can be applied to determine the distribution from experimental data; (3) the uncertainty in the distribution can be determined for each of its sections; and (4) the relationship between the time scales of available data and those of estimatable mass transfer processes can be investigated. A statistical model refinement algorithm is presented that reduces the number of parameters (sections of the piece-wise linear model) to the admissible minimum. We show that purging experiments allow estimation of a wider zone of the rate distribution than do batch experiments, and hence will provide predictions that are accurate over a wider range of time scales. Finally, in an application to TCE gas-purging desorption data, the piece-wise linear rate-distribution model has a higher probability of being adequate than those using a gamma or lognormal distribution or a single rate coefficient.     

Stochastic modeling of chemical oxygen demand transformations in gravity sewers.

Wastewater quality characteristics in terms of biomass, its substrates, and the corresponding kinetic and stoichiometric parameters were determined based on 109 wastewater samples originating from five different campaigns in four different sewer networks. Quality parameters were determined by model calibration of measured wastewater oxygen uptake rates applying a model that describes the aerobic breakdown of wastewater organic matter. Thereafter, the distributions of the parameters were analyzed. Two of the five datasets were obtained at the upstream end of a five-km-long, intercepting gravity sewer. For each of these upstream wastewater samples, downstream samples were collected with a delay corresponding to the residence time. The upstream distributions of the wastewater composition were used as boundary conditions for a Monte Carlo simulation. The calculated downstream distributions were compared to the measured downstream distributions and good agreement was observed.     

Evaluation of air quality zone classification methods based on ambient air concentration exposure

Air quality zones are used by regulatory authorities to implement ambient air standards in order to protect human health. Air quality measurements at discrete air monitoring stations are critical tools to determine whether an air quality zone complies with local air quality standards or is noncompliant. This study presents a novel approach for evaluation of air quality zone classification methods by breaking the concentration distribution of a pollutant measured at an air monitoring station into compliance and exceedance probability density functions (PDFs) and then using Monte Carlo analysis with the Central Limit Theorem to estimate long-term exposure. The purpose of this paper is to compare the risk associated with selecting one ambient air classification approach over another by testing the possible exposure an individual living within a zone may face. The chronic daily intake (CDI) is utilized to compare different pollutant exposures over the classification duration of 3 years between two classification methods. Historical data collected from air monitoring stations in Kuwait are used to build representative models of 1-hr NO₂ and 8-hr O₃ within a zone that meets the compliance requirements of each method. The first method, the “3 Strike” method, is a conservative approach based on a winner-take-all approach common with most compliance classification methods, while the second, the 99% Rule method, allows for more robust analyses and incorporates long-term trends. A Monte Carlo analysis is used to model the CDI for each pollutant and each method with the zone at a single station and with multiple stations. The model assumes that the zone is already in compliance with air quality standards over the 3 years under the different classification methodologies. The model shows that while the CDI of the two methods differs by 2.7% over the exposure period for the single station case, the large number of samples taken over the duration period impacts the sensitivity of the statistical tests, causing the null hypothesis to fail. Local air quality managers can use either methodology to classify the compliance of an air zone, but must accept that the 99% Rule method may cause exposures that are statistically more significant than the 3 Strike method.

Implications: A novel method using the Central Limit Theorem and Monte Carlo analysis is used to directly compare different air standard compliance classification methods by estimating the chronic daily intake of pollutants. This method allows air quality managers to rapidly see how individual classification methods may impact individual population groups, as well as to evaluate different pollutants based on dosage and exposure when complete health impacts are not known.     

Methods for Quantifying Variability and Uncertainty in AP-42 Emission Factors: Case Studies for Natural Gas-Fueled Engines

Abstract

Quantitative methods for characterizing variability and uncertainty were applied to case studies of oxides of nitrogen and total organic carbon emission factors for lean-burn natural gas-fueled internal combustion engines. Parametric probability distributions were fit to represent inter-engine variability in specific emission factors. Bootstrap simulation was used to quantify uncertainty in the fitted cumulative distribution function and in the mean emission factor. Some methodological challenges were encountered in analyzing the data. For example, in one instance, five data points were available, with each data point representing a different market share. Therefore, an approach was developed in which parametric distributions were fitted to population-weighted data. The uncertainty in mean emission factors ranges from as little as ~±10% to as much as -90 to 21+180%. The wide range of uncertainty in some emission factors emphasizes the importance of recognizing and accounting for uncertainty in emissions estimates. The skewness in some uncertainty estimates illustrates the importance of using numerical simulation approaches that do not impose restrictive symmetry assumptions on the confidence interval for the mean. In this paper, the quantitative method, the analysis results, and key findings are presented.     

Simulating the fate and transport of TCE from groundwater to indoor air

This work provides an exploratory analysis on the relative importance of various factors controlling the fate and transport of volatile organic contaminants (in this case, TCE) from a DNAPL source zone located below the water table and into the indoor air. The analysis is conducted using the multi-phase compositional model CompFlow Bio, with the base scenario problem geometry reminiscent of a field experiment conducted by Rivett [Rivett, M.O., (1995), Soil–gas signatures from volatile chlorinated solvents: Borden field experiments. Groundwater, 33(1), 84–98.] at the Borden aquifer where groundwater was observed to transport a contaminant plume a substantial distance without vertical mass transport of the contaminant across the capillary fringe and into the vadose zone. Results for the base scenario model indicate that the structure of the permeability field was largely responsible for deflecting the groundwater plume upward towards the capillary fringe, permitting aqueous phase diffusion to transport the TCE into the vadose zone. Alternative permeability realizations, generated as part of a Monte Carlo simulation process, at times deflected the groundwater plume downwards causing the extended thickness of the saturated zone to insulate the vadose zone from exposure to the TCE by upward diffusive transport. Comparison of attenuation coefficients calculated using the CompFlow Bio and Johnson and Ettinger [Johnson, P.C. and Ettinger, R.A., (1991), Heuristic model for predicting the intrusion rate of contaminant vapors into buildings. Environmental Science and Technology, 25, 1445–1452.] heuristic model exhibited fortuitous agreement for the base scenario problem geometry, with this agreement diverging for the alternative permeability realizations as well as when parameters such as the foundation slab fracture aperture, the indoor air pressure drop, the capillary fringe thickness, and the infiltration rate were varied over typical ranges.     

Rigorous uncertainty assessment in contaminant transport inverse modelling: a case study of fluoride diffusion through clay liners

Inverse methods used in assessing landfill liner design have not yet taken advantage of current developments in inverse procedures. Here, a method for inverting contaminant transport models is presented including a general error model and procedures for differentially weighted multiple response regression. General error models are employed in cases where the residuals are heteroscedastic and correlated, and lead to valid inference on model parameter and predictive uncertainty. The Shuffled Complex Evolution algorithm is used to optimise model parameters. Model parameter uncertainty is assessed by exploring the posterior probability distribution with the Metropolis algorithm, a Markov chain Monte Carlo sampling method. The inverse method is applied to simultaneously determine the sorption and diffusion parameters from laboratory diffusion cell experiments. In these experiments, fluoride migration through kaolin clays was measured by sampling the source and collector cells over time. To uniquely determine the transport model parameters, it was necessary to simultaneously fit the observed data from two independent diffusion cell experiments with different initial concentrations. The jointly fitted transport model parameters compared well with those fitted to independent batch experiments.     

Coupling geochemistry with a particle tracking transport model

Coupling geochemistry and transport appears unavoidable since it is rare that either of these two phenomena alone can account for the movement of solutes in groundwater. The chemical model is based on thermodynamic equilibrium. The method used is a Gibbs free energy minimization constrained by mass balances. The model calculates the aqueous speciation, the precipitation and the dissolution of pure minerals or solid solutions. The transport equation is solved by the random walk technique which avoids the problem of numerical dispersion for transport, but may be more time consuming than finite differences or elements if a large number of particles are necessary in order to get a sufficiently “smooth” solution. However, when the chemistry deals with a realistic number of elements (e.g., > 10), the cost of the chemistry computation largely dominates that of transport. Special techniques had to be developed in order to solve problems linked to the conditions present in some of the CEC CHEMVAL tests (boundary with fixed concentrations and very low Péclet numbers). The coupling consists of calculating the exchanges of chemical elements between two populations. The first population is sedentary, constituted by a mesh of fixed cells representing the composition of the solid phase. The other population is nomadic, represented by a set of particles which are advected by groundwater flow. A vector of real numbers is associated with each mobile particle. This vector accounts for the mass of each element dissolved in the moving liquid phase. For this reason, the transport equation is only solved once for the whole set of elements. The main assumptions that were necessary to perform the coupling in a simple way are discussed. Two applications are presented: (1) a verification compared to an analytical solution; and (2) the simulation of a percolation experiment through a sandstone core.     

Methods for quantifying variability and uncertainty in AP-42 emission factors: case studies for natural gas-fueled engines

Quantitative methods for characterizing variability and uncertainty were applied to case studies of oxides of nitrogen and total organic carbon emission factors for lean-burn natural gas-fueled internal combustion engines. Parametric probability distributions were fit to represent inter-engine variability in specific emission factors. Bootstrap simulation was used to quantify uncertainty in the fitted cumulative distribution function and in the mean emission factor. Some methodological challenges were encountered in analyzing the data. For example, in one instance, five data points were available, with each data point representing a different market share. Therefore, an approach was developed in which parametric distributions were fitted to population-weighted data. The uncertainty in mean emission factors ranges from as little as approximately +/-10% to as much as -90 to +180%. The wide range of uncertainty in some emission factors emphasizes the importance of recognizing and accounting for uncertainty in emissions estimates. The skewness in some uncertainty estimates illustrates the importance of using numerical simulation approaches that do not impose restrictive symmetry assumptions on the confidence interval for the mean. In this paper, the quantitative method, the analysis results, and key findings are presented.     

Analysis of the number of flux chamber samples and study area size on the accuracy of emission rate measurements

Monte Carlo simulations were conducted on a set of flux chamber measurements at a landfill to estimate the relationship between the number of flux chamber samples and study area size on the emission rate measurement accuracy. The spatial variability of flux was addressed in the study by utilizing an existing flux chamber measurement data set that is one of the most dense flux chamber sampling arrays published to date for a landfill. At a probability of 95%, the Monte Carlo simulations indicated that achieving an accuracy within 10% with the flux chamber method is highly unlikely. An accuracy within 20% was achieved for small areas of less than about 0.2 hectares using 220 flux chamber measurements, but achieving this level of accuracy for area emission sources, of similar or greater variability, that are larger than this is highly unlikely. An accuracy within 30% was achieved up to the Full Area of about 0.4 hectares if more than approximately 120 samples were obtained. Even for an accuracy within 50%, at least 40 flux chamber measurements were needed for the Full Area of about 0.4 hectares. Available methods of estimating the number of samples required were compared to the Monte Carlo simulation results. The Monte Carlo simulations indicate that, in general, more samples are required than determined from an existing statistical method, which is a function of the mean and standard deviation of the population. Specifying the number of samples based on a regulatory method results in very poor accuracy. A modification to the statistical method for estimating the number of samples, or for estimating an accuracy for a given probability and number of samples, is proposed.

Implications: The flux chamber method is the most widely used method of measuring fugitive emission rates from area sources. However, extrapolation of a set of individual flux chamber samples to a larger area results in area flux measurement values of unknown accuracy. Quantification of the accuracy of the extrapolation of a set of flux chamber measurements would be beneficial for understanding the confidence that can be placed on the measurement results. Guidance as to the appropriate number of flux chamber measurements to achieve a desired level of accuracy would benefit flux chamber method practitioners.     

Quantifying the effects of fumarate on in situ reductive dechlorination rates

In situ methods are needed to evaluate the effectiveness of chemical amendments at enhancing reductive dechlorination rates in groundwater that is contaminated with the priority pollutant, trichloroethene (TCE). In this communication, a method that utilizes single-well, “push–pull” tests to quantify the effects of chemical amendments on in situ reductive dechlorination rates is presented and demonstrated. Five push–pull tests were conducted in each of five monitoring wells located in a TCE-contaminated aquifer at the site of a former chemical manufacturing facility. Rates for the reductive dechlorination of the fluorinated TCE-surrogate, trichlorofluoroethene (TCFE), were measured before (test 1) and after (test 5) three successive additions (tests 2–4) of fumarate. Fumarate was selected to stimulate the growth and activity of indigenous microorganisms with the metabolic capability to reduce TCFE and TCE. In three wells, first-order rate constants for the reductive dechlorination of TCFE increased by 8.2–92 times following fumarate additions. In two wells, reductive dechlorination of TCFE was observed after fumarate additions but not before. The transformation behavior of fumarate was also monitored following each fumarate addition. Correlations between the reductive dechlorination of TCFE and the reduction of fumarate to succinate were observed, indicating that these reactions were supported by similar biogeochemical conditions at this site.