Uncertainty evaluation of mass discharge estimates from a contaminated site using a fully Bayesian framework

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1 M. Troldborg a W. Nowak b N. Tuxen a P. L. Bjerg a R. Helmig b Uncertainty evaluation of mass discharge estimates from a contaminated site using a fully Bayesian framework Stuttgart, Dec 2010 a Department of Environmental Engineering, Technical University of Denmark, Kongens Lyngby, Denmark b Institute of Hydraulic Engineering (LH 2 ) / SimTech, University of Stuttgart, Pfaffenwaldring 61, Stuttgart/ Germany wolfgang.nowak@iws.uni-stuttgart.de Abstract The estimation of mass discharges from contaminated sites is valuable when evaluating the potential risk to downgradient receptors, when assessing the efficiency of a site remediation, or when determining the degree of natural attenuation. Given the many applications of mass discharge estimation, it is important to quantify the associated uncertainties. Here a rigorous approach for quantifying the uncertainty in the mass discharge across a multilevel control plane is presented. The method accounts for (1) conceptual model uncertainty using multiple conceptual models and Bayesian model averaging (BMA), (2) heterogeneity through Bayesian geostatistics with an uncertain geostatistical model, and (3) measurement uncertainty. Through unconditional and conditional Monte Carlo simulation, ensembles of steady state plume realizations are generated. The conditional ensembles honor all measured data at the control plane for each of the conceptual models considered. The probability distribution of mass discharge is obtained by combining all ensembles via BMA. The method was applied to a trichloroethylenecontaminated site located in northern Copenhagen. Four essentially different conceptual models based on two source zone models and two geological models were set up for this site, each providing substantially different prior mass discharge distributions. After conditioning to data, the predicted mass discharge distributions from each of the four conceptual models all approach each other. This indicates that the data set available at the site is strong and that the estimated mass discharge is robust to the assumed conceptual models and their boundary conditions. On the basis of the results, we discuss which of the conceptual models is most likely to reflect the true site conditions and analyze the relative proportions and importance of uncertainties as well as the impact of different data types on mass discharge uncertainty. Keywords Stochastic Hydrogeology Heterogeneous porous media Mass Flux Statistics Conditioning Preprint Series Issue No Stuttgart Research Centre for Simulation Technology (SRC SimTech) SimTech Cluster of Excellence Pfaffenwaldring 7a Stuttgart publications@simtech.uni-stuttgart.de

2 Uncertainty evaluation of mass discharge estimates from a contaminated site using a fully Bayesian framework Mads Troldborg* 1, Wolfgang Nowak 2, Nina Tuxen 3, Poul L. Bjerg 1, Rainer Helmig 2 and Philip J. Binning Department of Environmental Engineering, Technical University of Denmark, Miljoevej, Building 113, 2800 Kgs. Lyngby, Denmark. 2 Institute of Hydraulic Engineering (LH2, Stochastic modelling of hydrosystems), Universität Stuttgart, Pfaffenwaldring 61, Stuttgart, Germany. 3 Orbicon A/S, Ringstedvej 20, 4000 Roskilde, Denmark * Corresponding author. Department of Environmental Engineering, Technical University of Denmark, Miljøvej, Building 113, 2800 Kgs. Lyngby, Denmark; mat@env.dtu.dk; Tel.: ; Fax: Manuscript submitted to Water Resources Research, February

3 Abstract The estimation of mass discharges from contaminated sites has received increased attention over the past decade. Such estimates are valuable when evaluating the potential risk to down-gradient receptors, when assessing the efficiency of a site remediation, or when determining the degree of natural attenuation. Given the many applications of mass discharge estimation, it is important to quantify the associated uncertainties. Here a rigorous approach for quantifying the uncertainty in the mass discharge across a multilevel control plane is presented. The method accounts for i) conceptual model uncertainty using multiple conceptual models and Bayesian model averaging, ii) heterogeneity through Bayesian geostatistics with an uncertain geostatistical model, and iii) measurement uncertainty. Through Monte Carlo simulation, an ensemble of unconditional steadystate plume realizations is generated. By use of the quasi-linear Kalman ensemble generator, the parameter field realizations are conditioned on the site-specific data. Hereby, a posterior ensemble of realizations, all honoring the measured data at the control plane, is generated for each of the conceptual models considered. The probability distribution of mass discharge is obtained by combining all ensembles via Bayesian model averaging. The method is applied to a TCE contaminated site located in northern Copenhagen. Four essentially different conceptual models based on two source zone models and two geological models were set up for this site, each providing substantially different prior mass discharge distributions. After the conditioning to data, the mass discharge distributions for the four conceptual models all approach each other, indicating a strong data set available at the site. The estimated mass discharge was therefore affirmed robust with respect to the assumed conceptual models and their boundary conditions. The uncertainty is evaluated by analyzing the mass discharge distributions, and it is discussed which of the conceptual models is most likely to reflect the true site conditions. Keywords: Mass discharge; Uncertainty evaluation; Risk assessment; Kalman Ensemble Generator, Bayesian inference, Groundwater contamination 2

4 Introduction The estimation of mass discharges (mass/time) from contaminated sites has received increased attention over the past decade. Such estimates are found valuable for assessing natural attenuation rates in plumes [Basu et al., 2006; Bockelmann et al., 2003; Kao and Wang, 2001; King et al., 1999], evaluating the benefits of source zone remediation [Falta et al., 2005; Jawitz et al., 2005; Soga et al., 2004], assessing the potential loading to down-gradient receptors [Feenstra et al., 1996; Kalbus et al., 2007] and prioritizing investigations and clean-up of identified sources on catchment scale [Einarson and Mackay, 2001; Troldborg et al., 2008]. Given the many applications of mass discharge estimates and the growing consensus among regulatory agencies that mass discharge should be used as supplemental performance metrics for risk assessment and remediation design at contaminated sites [Basu et al., 2006], a quantification of the associated uncertainties is important. However, uncertainties related to mass discharge estimates have not yet been given much attention. Mass discharge is defined as the contaminant mass per unit time that migrates across a hypothetical control plane located downstream of the source and perpendicular to the mean groundwater flow. In the field, mass discharge is often determined by measuring the contaminant concentration and the flow rate at a number of locations in a multi-level sampling network (MLS), and then interpolating and aggregating these measurements to cover the entire control plane, usually by assigning a corresponding area to each of the measurements [Kao and Wang, 2001; King et al., 1999; Soga et al., 2004; Tuxen et al., 2003]. Other notable approaches for quantifying the mass discharge are the passive flux meter (PFM) that aims at measuring the mass flux at each monitoring point directly [Annable et al., 2005; Hatfield et al., 2004], and the integral pumping test (IPT) [Bockelmann et al., 2001; Herold et al., 2009]. 3

5 Regardless of the method used, the field estimated mass discharge will always be subject to large uncertainties. These uncertainties are highly related to the degree of heterogeneity of the mass flux distribution at the control plane. This heterogeneity is caused by several factors such as spatially and temporally varying flow conditions and complex contaminant distribution in the source zone. It is clear that the more heterogeneous the mass flux distribution is, the finer the monitoring resolution network should be to ensure that the unmeasured areas in the control plane do not influence the estimate significantly. For example, based on extensive field studies at three industrial sites in Ontario, New Hampshire and Florida, Guilbeault et al. [2005] showed that 75 % of the total mass discharge occurred within 5 10 % of the plume cross sectional area, and that a monitoring spacing as small as cm was required to identify high concentration zones. If any such high mass flux zones are missed in the monitoring scheme, a significant underestimation of the mass discharge will be the result. However, at most non-research field sites, the number of monitoring wells is limited, which introduces a higher degree of ignorance and makes it more likely that the unmeasured parts of the control plane are important. So far only a few studies have addressed mass discharge uncertainty and all of these studies have been conducted on synthetic cases. In both Kübert and Finkel [2006] and Rein et al. [2009], the classical field methods for quantifying mass discharge are investigated for estimation errors defined as the deviation of the estimated value from the synthetically generated true value. Kübert and Finkel [2006] analyze the estimation error under different degrees of spatial heterogeneity and find that the spatial sampling intervals at the control plane should be in due proportion to the spatial correlation length scales of the hydraulic conductivity field to reduce the estimation errors. Rein et al. [2009] show that temporally varying flow conditions can lead to significant errors in the mass discharge estimate due to the resulting temporal fluctuations of the concentration measurements and they suggest using time-integrated measurements of concentration to overcome this source of 4

6 uncertainty. Both these studies do not, however, quantify the uncertainty of the mass discharge estimate itself. Li et al. [2007] estimate an empirical distribution of the mass discharge by using a simulation-based approach that generates multiple, equally likely realizations of the joint spatial distribution of concentration and conductivity within the control plane. They use indicator Kriging and probability-field simulation [Srivastava, 1992] to generate the conditional realizations of hydraulic conductivity and concentration. Schwede and Cirpka [2009a] present an inverse Monte- Carlo approach to generate an ensemble of flow and transport realizations that all meet the measurements in the control plane from which they obtain the mass discharge distribution. Their motivation for doing so is that they deem the direct linear geostatistical interpolation (e.g. Kriging) of flow and concentration measurements in a control plane inappropriate, because the flow and transport processes in heterogeneous subsurface environments depend nonlinearly on the hydraulic conductivity field [Schwede and Cirpka, 2009b]. In the study by Schwede and Cirpka [2009a], the estimated flow and concentration field at the control plane were conditioned for a synthetic case, where the boundary conditions and geostatistical parameters were assumed known. However, for field application this will rarely be the case. Often the knowledge about e.g. the contaminant source, the geological and hydrogeological settings is limited, which makes it very difficult to conceptualize these elements and to incorporate them in a model. One way of addressing this lack of knowledge is by using multiple conceptual models to represent the settings at the site. The use of multiple conceptual models has been widely applied in climate change modeling [IPCC, 2007], but has also been recommended for groundwater related problems [Refsgaard et al., 2006]. For example, multiple conceptual models have been used for describing the geological stratification of the same aquifer system [Hojberg and Refsgaard, 2005; Rojas et al., 2008; Troldborg et al., 2007] and also in combination with alternative source zone 5

7 locations [Cypher and Lemke, 2009]. Sohn et al. [2000] perform flow and transport simulations from a contaminated site and use alternative conceptual models describing the geological structure, source locations and the chemicals released at these sources to represent the ignorance in the site characterization. The previous studies of mass discharge uncertainty have not taken the influence of different conceptual models into account. However, in many cases the conceptual model uncertainty has been recognized to be the main source of uncertainty in model prediction [Refsgaard et al., 2006] and is therefore important to account for in an uncertainty evaluation. We present here a holistic and rigorous approach for quantifying the uncertainty in the mass discharge across a multilevel control plane given sparse field measurements. The method is based on a Bayesian inverse modeling approach, where several models representing the site conditions are used to represent our lack of knowledge. The new method is able to account for i) conceptual model uncertainty by use of multiple conceptual models and Bayesian model averaging, ii) heterogeneity through Bayesian geostatistics with an uncertain geostatistical model, and iii) measurement uncertainty. The method shows some similarities to the one presented by Schwede and Cirpka [2009a]. They make use of Monte Carlo simulation to generate an ensemble of unconditional steady-state plume realizations, which subsequently are conditioned on site-specific data. However, where Schwede and Cirpka use the quasi-linear geostatistical method by Kitanidis [1995] to condition each individual realization one by one, we perform the conditioning of the entire ensemble using the quasi-linear Kalman ensemble generator (KEG) developed by Nowak [2009]. Conditioning entire ensembles has been proven computationally much more efficient than the realization-based methods [Franssen and Kinzelbach, 2009; Nowak, 2009]. In this way a posterior ensemble of realizations, all honoring the measured data at the control plane are generated for each of the conceptual models considered. Through Bayesian model averaging, the ensembles are combined and a probability distribution of mass discharge is obtained. 6

8 The novelty of the presented method compared to the work by Schwede and Cirpka [2009a] is i) the use of multiple conceptual models to represent the uncertainty related to the geology and the contaminant source ii), the use of Bayesian geostatistics to describe the uncertain character of heterogeneity of the flow field, and iii) the use of the KEG for the conditioning on data. Furthermore, the method is applied to field data from a contaminated site for which the mass discharge and associated uncertainty is quantified Method for quantifying the uncertainties of mass discharge estimates The procedure used in this paper for quantifying the uncertainty of the mass discharge from a contaminated site is illustrated in Figure 1. The starting point is a set of observations collected from a control plane at the site. The method uses all these measurements to condition hydraulic conductivity fields. That is, a large number of random hydraulic conductivity fields are generated and adjusted in such way that when they are used individually in a forward flow and transport model, all the simulated measurement values meet the observed values within their respective measurement uncertainty bounds. Hereby an ensemble of conditional flow and concentration fields is obtained from which estimates of the mass discharge passing the control plane can be calculated directly. This leads to an empirical mass discharge probability distribution that fully characterizes the uncertainty of the mass discharge estimation. In generating the conditional simulations, the method accounts for uncertain parameter values and also considers different conceptual models represented by different boundary conditions. Thus, an ensemble of conditional simulations is generated for each conceptual model. The methodology for generating the conditional simulations and obtaining the empirical mass discharge distribution is described in detail in the following sections. 7

9 165 FIGURE Formulation of conceptual models In the first step, a number of conceptual models are formulated, all of which are believed to be adequate representations of the given site. The conceptual models can consider many varying factors, for example the description of the source zone and the geological stratification. The conceptual models are specified through the applied boundary conditions and the boundary values used for each conceptual model can be held uncertain. Each of the conceptual models is assigned a prior belief based on best judgment and the sum of the prior belief should amount to 100 % Setting up the numerical and geostatistical problem The next step involves specifying the numerical grid, implementing the boundary conditions, selecting an appropriate numerical solver and specifying prior values or probability density functions (pdfs) to model parameters and inputs. Parameters that are not expected to influence the simulation results significantly or that are associated with little uncertainty can be kept constant. Measurement errors associated with the obtainment of data will also influence the quality of the mass discharge prediction. The measurement errors ε are here assumed to be Gaussian and independent with zero mean and covariance matrix R, i.e. ε ~ N(0, R). The heterogeneous distribution of hydraulic conductivity (K) is modeled as a random space function defined by a geostatistical model. Here we are assuming that the unknown parameter field vector, s, of n s 1 discretized values of log conductivity lnk(x) follows a multivariate Gaussian distribution with mean vector Xβ and covariance matrix C ss (θ): s β,θ ~ N(Xβ, C ss (θ)) 8

10 where X is an n s p matrix containing p deterministic trend functions, β is the corresponding vector of p 1 trend coefficients and θ are the structural parameters of the covariance function (e.g. Kitanidis, 1986; Nowak, 2009). The above expression holds for known values of β and θ. The values chosen for the structural parameters of the geostatistical model are typically inferred based on data, for example by fitting an empirical semivariogram or covariance model to the observed conductivity data (Pardo-Igúzquiza et al, 2009). However, it generally requires vast data sets to infer legitimate values for the parameters of a geostatistical model. In most cases, the number of hydraulic conductivity measurements is not sufficient to justify fixed parameters. The uncertainty related to the structural parameters is known to influence the uncertainty of the conductivity field and all state variables dependent thereof [Nowak et al., 2009; Pardo-Iguzquiza et al., 2009; Pardo- Iguzquiza, 1999]. The uncertainties related to structural parameters can be acknowledged by the model-based Bayesian approach to geostatistics, where the structural parameters of the geostatistical model themselves are treated as random variables [Diggle and Ribeiro, 2007; Kitanidis, 1986]. This approach is adopted here and the geostatistical model parameters (mean, variance, integral scale etc.) are thus specified through prior pdfs. Furthermore, since the most appropriate type of covariance model is rarely known, the Matérn family of covariance functions will be used. It is given by [Matérn, 1986]: C( l) l = 2 Y = κ 1 2 σ Δx λx Γ( κ) 2 κ ( 2 κl) B ( 2 κl) Δy + λ y 2 κ Δz + λz 2 (1) 9

11 where σ 2 Y is the variance of log-conductivity, l is the anisotropic effective separation distance, Γ( ) is the gamma function, B κ is the modified Bessel of the third kind and order κ, and λ i is the correlation length. As seen from Eq. (1), this family of functions has an additional shape parameter, κ, which determines the shape of the covariance function and therefore also the smoothness of the spatial process s [Diggle and Ribeiro, 2007]. For example, for the specific cases of κ = 0.5, 1,, the Matérn family are the exponential, Whittle and Gaussian covariance model, respectively. By treating the shape parameter as uncertain, the selection of covariance model is also kept uncertain. This resembles Bayesian Model averaging over a continuous spectrum of covariance models [Nowak et al., 2009]. If the trend coefficients β are considered uncertain and described as random variables distributed with expected value β* and covariance C ββ, i.e. β ~ N(β*, C ββ ), the distribution of s is given by s ~ N(Xβ*, G ss (θ)), where G ss = C ss + XC ββ X T is the generalized covariance matrix [Kitanidis, 1993] Updating geostatistical model parameters using direct data As an initial conditioning step, the geostatistical parameter values that are more likely to have produced the observed data are identified. This is done by applying Bayes theorem, where the prior joint distribution of the geostatistical model parameters, p(β,θ), is updated with the measured data, y, to obtain a posterior joint distribution, p(β,θ y): p( β, θ y) p( y β,θ) p( β, θ) (2) where p(y β,θ) is the likelihood of observing the data y given specific sets of structural parameter values, β and θ. To reduce the scope of the work presented here, we limit our approach to consider direct measurements of hydraulic conductivities only, as also done by Woodbury and Ulrych 10

12 [2000]. Since we are assuming that the observed ln(k) data, y, stem from a multivariate Gaussian field, the likelihood function can be expressed as follows: N / 2 1/ 2 1 T 1 p( y β,θ) = (2π ) G yy exp ( y β*) G yy ( y β*) 2 (3) where G yy is the generalized covariance matrix of Y = ln(k) between the measurement locations. Eq. (3) can be evaluated numerically for a large number of random samples taken from the prior joint parameter distribution. Hereby a discrete posterior joint distribution of the structural parameters can be obtained Generating the prior ensemble A prior ensemble of N realizations is now generated for each of the specified conceptual models. This is done via Monte Carlo simulation. First, unconditional random hydraulic conductivity fields s u,i are generated based on the posterior distribution of the structural parameter found in section 2.3. These hydraulic conductivity fields are then used individually as input in a stationary flow and transport model to determine the head and concentration distribution in the domain. For each realization the prior mass discharge passing the given control plane area A can be calculated by integration: MD = n A qcda (4) A where n A is a unit vector normal to the control plane, and q and C are the darcy flux field and concentration field at the control plane, respectively. 11

13 Conceptual model selection At this stage, the performance of the different conceptual models given all the prior assumptions can be tested and used to update the prior belief in each model via Bayes Theorem. Assuming that the measurements errors are Gaussian and independent, the likelihood that the observed data, y, stem from a specific realization y sim,i from the k th conceptual model M k is given by: N / 2 1/ 2 1 T 1 L( y y sim, i, M k ) = (2π ) R exp ( y y sim,i ( M k )) R ( y y sim, i ( M k )) 2 (5) where R is the measurement error covariance and y sim,i (M k ) = f k (s u,i ) + ε i is obtained by extracting the simulated measurement values from the prior ensemble of the k th conceptual model, f k (s u,i ), and adding a random measurement error, ε i, sampled from the specified normal error distribution (c.f. section 2.2). The updated belief in model M k is then, according to Bayesian model averaging: Pr( M k L( y M k ) Pr( M k ) y ) = P (6) L( y M ) Pr( M ) k = 1 k k where P is the total number of conceptual models and L 1 N ( y M k ) = L( y y sim, i, M k ) N i= Updating the ensemble with the Kalman Ensemble Generator The prior ensemble of realizations for each conceptual model is now updated with all the available measurements from the site. Each unconditional hydraulic conductivity field, s u,i, is conditioned using the following updating equation (e.g. Kitanidis, 1995): 12

14 ( y f (s ) ) s k + (7) 1 c, i = s u,i + G syg yy u,i ε i where s c,i is the conditioned realization of the conductivity field, G yy is the covariance matrix between simulated measurements, and G sy is the cross-covariance between simulated measurements and the hydraulic conductivity field. Typically the covariance matrices in Eq. (7) are approximated through a Jacobian-based linearization around the expected value of s. However, this kind of approximation to the ensemble statistics is found to be biased with non-minimal error. Here, we therefore employ the quasi-linear Kalman ensemble generator (KEG) by Nowak [2009]. The KEG is similar to the ensemble Kalman filter (EnKF) [Evensen, 2007], but where EnKFs condition transient state variables, the KEG conditions geostatistical parameter fields. In the KEG, all the covariances needed in Eq. (7) are extracted directly from the generated prior ensemble, which not only ensures a best unbiased ensemble linearization, but also avoids the computational burden of sensitivity analysis and storage of very large autocovariance matrices. Furthermore, the ensemble based conditioning methods are shown to behave more robustly for nonlinear problems. To improve the robustness and accuracy for non-gaussian and non-linear cases, Nowak [2009] equipped the KEG with a sequential two-step updating scheme, acceptance/rejection sampling, and successive linearization stabilized with a modified Levenberg-Marquardt approach [Nowak and Cirpka, 2004]. The result is a posterior ensemble of model realizations that all meet the observed data at the control plane within the range of measurement error. Since the realizations also resolve natural variability, the issue of dispersion in smoothed or interpolated conductivity fields [Nowak and Cirpka, 2006; Rubin et al., 1999] is avoided. The conditioned mass discharge passing the control plane can now be calculated for each realization and histograms can be created, displaying the posterior mass discharge distribution for each of the conceptual models considered. 13

15 Combining posterior ensembles using Bayesian model averaging As a final step the posterior mass discharge ensembles from each conceptual model are combined through Bayesian model averaging [Hoeting et al., 1999]: P Pr( MD y) = Pr( MD M k, y) Pr( M k y) (8) k = where Pr(MD y) is the final posterior probability distribution of the mass discharge, Pr(MD M k,y) is the posterior mass discharge distribution for conceptual model M k given the observed data y, and Pr(M k y) is the prior probability of model k Case study The above method is applied to a contaminated site located north of Copenhagen, Denmark, to estimate the mass discharge and its associated uncertainty Site description and conceptual models The field site is a former machine factory situated in a paved industrial area (Figure 2). Here, a leaking underground tank containing trichloroethylene (TCE) has resulted in a contamination of the unsaturated zone and an upper sandy aquifer. It is expected that free-phase TCE has been released from the tank directly into the unsaturated zone in the period from 1963 until 1973, where the tank was removed from the site. Five municipal water supply wells are situated within less than 800 meters west and northwest of the site. Since 1987, dissolved TCE has been observed in some of the supply wells, and the study site is believed to be the main source of this contamination. 14

16 326 FIGURE In the period from , several investigations have been carried out at the site to map the contamination, assess the risk of the groundwater contamination associated with the spill and to clarify whether the site was responsible for the contamination at the waterworks. As part of these investigations, a control plane consisting of 28 sampling points was established approximately 160 m downstream the source and perpendicular to groundwater flow to estimate the mass discharge leaving the site (see Figure 2). From measurements of hydraulic conductivity, concentration and the hydraulic gradient at the control plane, the total mass discharge passing the control plane was calculated using the standard estimation technique, where a polygon is designated to each sampling point in the control plane, as shown in Figure 2. The measurements from each point are then assumed to be average values for the entire polygon. The mass discharge passing each polygon can be calculated, and by summing these values a total mass discharge can be estimated. This technique resulted in a mass discharge estimate of about 1 kg TCE/yr [Tuxen et al., 2008]. During the past five years, various hypotheses or conceptual models of the site settings have been formulated to explain the observed contaminant spreading at the site. These conceptual models were set up at different stages during the site characterization as more data were collected. Here we are considering four conceptual models of the site; each assigned a prior belief of 25 %. The conceptual models are formed from different combinations of two geological models of the stratigraphy and two source zone models, as illustrated in Figure 3 and summarized in Table 1. These are briefly described in the following TABLE

17 350 FIGURE Conceptual geological models The overall geology at the site is characterized by 50 meters of quaternary glacial deposits consisting of alternating layers of sand and clay till. The quaternary sediments are dominated by an approximately 40 meter thick sand layer that consists mostly of melt-water sand and gravel. This layer constitutes the generally unconfined upper aquifer in the area around the site. Beneath the quaternary sediments, a limestone formation is encountered at around 30 meters below sea level. The limestone aquifer is the primary groundwater resource in the area. The two aquifers are in the area separated by a clay layer over which hydraulic head differences of 1 to 3 meters are observed, creating a downward vertical gradient from the upper to the lower aquifer. However, geological windows in the separating clay layer have been observed at several locations in the area, and local hydraulic contact between the aquifers is therefore likely to exist. Two conceptual geological models have been formulated based on different interpretations of the stratification of the quaternary deposits in the area around the site. These are shown in Figure 4. Geology A is based on a planar structured geology, where the upper and lower aquifer at the field site is separated by a clay layer. Geology B is based on a so-called kineto-stratigraphical interpretation of the geology [Tuxen et al., 2008], where the layers are inclined and less connected. According to this interpretation, a window could exist in the clay layer just north of the site, resulting in a hydraulic contact between the aquifers FIGURE

18 Conceptual source zone models The first conceptual source zone model (Source I) represents a contaminant source located in the unsaturated zone. Because the site is situated in a paved industrial area, it was assumed that infiltration through the contaminant source was negligible and that the contaminant spreading to groundwater was solely due to gaseous diffusion. This would lead to a radial spreading of contaminants and create a wide, but shallow contaminant plume in the underlying aquifer. This corresponded well with the observed TCE concentrations in both unsaturated zone and in the upper aquifer close to the source. In the area near the TCE tank, the plume was observed to be more than 100 meters wide and only up to 2 meters deep. However, the establishment of the control plane down stream of the source revealed a deep, potentially narrow, plume of high TCE concentrations. As shown in Figure 2, the plume has been observed 20 meters below the groundwater surface at the multi-level sampler F5. It was believed that free-phase leakage could have created a narrow source area that stretched from the soil surface and deep into the upper aquifer. Free-phase transport of TCE was assumed likely because of high soil concentrations measured in the unsaturated zone near the TCE tank. In the unsaturated zone, radial contaminant spreading governed by gas diffusion was still assumed to be the likely cause of the wide contaminant plume in the aquifer. The second conceptual source zone model (Source II) is therefore the same as the first one, but a deep source of residual phase has been included Application of uncertainty method to case study Numerical model and boundary conditions Steady-state groundwater flow and advective-dispersive transport of a conservative tracer is considered. The groundwater flow field at the site has been monitored over a 5 year period without 17

19 showing significant seasonal fluctuations or temporal changes in the groundwater flow directions. The source is modeled as a continuous source term with constant concentration. Sorption is neglected, because retardation is irrelevant at steady-state. Degradation of TCE is not relevant since the upper aquifer is characterized as aerobic and because no degradation products of TCE have been observed at the site. Finally, recharge is not included since the site is located in a paved industrial area [Tuxen et al., 2008]. All flow and transport simulations are carried out using MODFLOW2000 [Harbaugh et al., 2000] and MT3DMS [Zheng and Wang, 1999]. The domain size is 200 x 220 m 2, which is considered to be large enough to avoid undue influence of the boundaries. The depth of the model domain is 20 meters, where the bottom is comprised by the upper clay layer (see Figure 4) and the top is the water table (of the unconfined aquifer). The aquifer is uncontaminated below the upper clay layer. A discretization of [Δx, Δy, Δz] = 2 x 2 x 0.25 meter has been used, resulting in a total of cells. The spectral method by Dietrich and Newsam [1993] for generating random conductivity fields and the Kalman ensemble generator for the conditioning of the random conductivity fields have been implemented in MATLAB. The number of realizations per conceptual model was set to 500. The prior distribution of the geostatistical model parameters is sampled using Latin Hybercube sampling, where each of the five uncertain structural parameters (see section 4.2) is sampled on a 5 % interval giving a total of 20 5 parameter set samples Boundary conditions The selected head boundaries are based on the observed potential map at the site. No-flow conditions (Neumann) are assigned to the western and eastern boundaries, while fixed head boundaries (Dirichlet) are used on the south and north. To obtain the observed north-northeast going flow direction, an eastern head gradient of has been applied along the southern 18

20 boundary. The natural hydraulic gradient in the south-north direction is approximately 0.01, resulting in an average head difference between the southern and northern boundary of 2.1 meters. The bottom hydraulic boundary differs for the two geological models. For the conceptual models without a geological window, no-flow conditions have been applied to the entire bottom of the model domain, while a fixed head boundary has been applied to a rectangular area in the northern part of the domain bottom to conceptualize a geological window (see Figure 2). This window creates a vertical head difference of about 1.2 meters between the top and the bottom of the domain. Different fixed concentration boundaries are applied on the south to represent the two different source zone conceptual models. The deep residual-phase source has an assumed width of 2 meters and extends all the way to the bottom of the domain with a relative concentration of C s,ii = 1. To represent the radial spreading of the gas cloud source, a smooth concentration distribution of about 80 meters width and 2 meters depth has been specified along the southern boundary with a relative centre concentration of C s,i = 0.2. Zero-concentration gradients (Neumann) have been assigned to the western, eastern, northern and bottom boundaries Parameter values Relevant parameter values used in the study site are presented in Table 2. For the steady-state flow and transport simulations, the following parameters are not considered uncertain and thus kept fixed: porosity (n), longitudinal (α L ) and transverse dispersivities (α T ) and the diffusion coefficient in water (D 0 ). Large-scale dispersion is generated by variations in flow caused by the random conductivity fields and so the dispersivities are set to small-scale values. We assume that the model domain is contained within a single geological unit and that a single geostatistical model can be applied to the entire domain. An uncertain geostatistical model for logconductivity is used to describe the variability in the heterogeneous formation, using the Matérn 19

21 family of covariances and a constant global mean. Uncertain geostatistical model parameters are: the global mean value β, the variance σ Y, the correlation lengths in x, y and z directions λ x, λ y and λ z, and the Matérn shape parameter κ. Prior probability density functions (pdfs) for these parameters are given in Table 2. The upper aquifer is assumed to have a global geometric mean conductivity of K g = m/s. The variance of the global mean is assumed a priori to be Q bb = 0.3, which corresponds to a K g 95% confidence interval of approximately one order of magnitude. Uniform prior distributions are assigned to the variance, the correlation lengths and the shape parameter. This reflects a maximum entropy assumption in the presence of two bounds [Jaynes, 1982; Woodbury and Ulrych, 1993]. The upper and lower bounds for variance and scales are based on literature values for comparable aquifers [Bjerg et al., 1992; Christiansen et al., 1998; Gelhar, 1993; Riva et al., 2006]. The lower bounds of the correlation lengths were, however, restricted to four times the numerical grid spacing, i.e. to 8 m and 1 m for the horizontal and vertical correlation lengths, respectively, to resolve the spatial correlation of the conductivity fields. This is considered to be relatively high minimum values, but it was our designated goal to keep the computational costs at a level that is manageable with a standard contemporary Desktop PC, in spite of all the conceptual flexibilities of our method. A uniform prior distribution with range has been specified for the Matern shape parameter κ. When κ is greater than 2, the Matern covariance model generates smooth conductivity fields because their second derivatives are continuous in space [Diggle and Ribeiro, 2007; Handcock and Stein, 1993]. In this way the range from discontinuous to smooth conductivity fields is covered, including the exponential and the Whittle model, whereas the Gaussian covariance model (κ= ) is disregarded, since we do not expect the realizations to be much smoother than twice differentiable [Handcock and Stein, 1993]

22 470 TABLE A data set consisting of 42 hydraulic conductivity measurements (from slug tests and grain-size analysis), 36 hydraulic head measurements and 24 measurements of TCE (all sampled at the control plane) is used to condition the realizations. Most of the measurements are taken from wells and multi-level samplers with 1 meter screen lengths. The measurement error standard deviations are given in Table 2. These measurement errors comprise the likelihood function and include inherent variability, spatial and temporal averaging, and imperfect model representation [Sohn et al., 2000] Results Geostatistical model identification Figure 5 shows histograms of the structural parameter samples drawn from the posterior joint distribution (c.f. section 2.3). Both the variance and the Matérn kappa parameter are fairly well identified within the prior intervals. The variance of the conductivity field is primarily found in the interval between 1.5 and 2.5 indicating a high degree of heterogeneity. The most likely kappa values are in the range of 0.4 to 0.9 pointing to an exponential type of covariance. The correlation lengths are less well identified, especially in the y-direction. A vertical anisotropy is identified and there is also a tendency to horizontal anisotropic effects, where the correlation length in the y- direction generally is observed to be longer than in the x-direction. The correlation length in the z- direction is found to be surprisingly high (between 2 to 4 meter), but this might be explained by the small kappa values, which allows for a lot of variation at scales smaller than the correlation lengths. The sampling density might also not be high enough to capture the spatial variability in the vertical direction. From Figure 5 it is also seen that the prior distributions specified for the correlation 21

23 lengths are impacting the posterior distributions, because many samples are drawn from near the specified bounds. This is especially the case for the correlation length in the x-direction, where most of the samples are near the lower bound. However, the lower bound for this correlation length is limited by the discretization, and the chosen prior distributions are all reasonable compared to values reported in the literature FIGURE Conceptual model selection Next, we updated the prior belief in the four considered conceptual model using equation 5 and 6 (c.f. section 2.5). The results of this updating are shown in Table 3. The results are based on 7500 unconditional simulations with each conceptual model, because 500 realizations were found insufficient to obtain stable average likelihood estimates. When all data are considered, the model likelihood calculation is strongly in favor of conceptual model IIB, which accounts for both a deep spill of residual-phase and presence of a geological window. The belief of model IIB is updated from 25 % to 93 %, while the others are reduced to nearly zero. The reason for this is that model IIB generates a few realizations that match all the available data extremely well. This indicates, however, that the calculated posterior model probabilities are still not robust even after 7500 realizations with each conceptual model. This is also confirmed when running a model selection based on only the available conductivity and head data. Such model selection can determine which conceptual geological model (A or B) performs the best from a hydraulic point of view. Here we would expect that model IA and IIA should obtain approximately the same posterior probability, because they are exactly the same in a hydraulic sense, and that the same is the case for model IB and IIB. However, as shown in Table 3, this is not the case, again implying that more realizations 22

24 are needed for a stable model selection. Still, it provides us with an idea of which conceptual model is more likely to represent the observed data best TABLE Conditioning of conceptual models to data For all the conceptual models considered, except for model IA, the ensemble of hydraulic conductivity fields was successfully conditioned on the available measurements of conductivity, head and concentration. This is illustrated in Figure 6 for model IA and IIA, where the observed measurement values are plotted against the simulated ensemble mean values before and after the conditioning. The 90 % confidence interval of the simulated posterior value (error bars) and of the assumed measurement error distribution (dotted lines) are also displayed. The quality of the conditioning for model IIA is judged to be highly satisfactory, especially for the log-conductivity and head data, where the simulated ensemble mean values and the 90 % confidence intervals accurately match the observations. The simulated posterior concentration values also match the observations at a satisfactory level, but the fit appears to be less good. This is because of the non- Gaussian distribution of concentration due to the larger non-linear relationship between concentration and conductivity, but also because of the larger measurement uncertainty specified for concentration (Table 2). Though model IA is conditioned nicely to the conductivity and head data, it fails to simulate the high concentrations measured in deeper part of the control plane. This can be seen as strong evidence against model IA and suggests that this conceptual should be discarded FIGURE 6 23

25 Figure 7 shows the prior and posterior ensemble mean concentration distribution at the control plane for the four conceptual models. The simulated contaminant plumes at the control plane for model IIA and IIB become narrower after the conditioning to data. For model IA and IB, the observed deep plume pattern is obtained after the conditioning, even though the contaminants are released via a shallow source term in these models. The presence of the geological window is observed to facilitate the downward migration of the contamination, which is especially pronounced for model IB FIGURE To obtain the observed contamination pattern at the control plane requires tweaking of the conductivity field. This is illustrated in Table 4, where the ensemble statistics of the hydraulic conductivity fields before and after the conditioning to data are presented. The conditioning does not greatly affect the global mean value β itself, but the variance of the global mean value Q ββ is reduced for all the conceptual models. Though the variance of ln(k) is reduced to the specified measurement uncertainty at the measurement locations (and also in an area around the measurement), the average variance of the conductivity field realizations, σ 2 Y, is increased for all the conceptual models. This is especially the case for model IA, where large corrections are needed to obtain the deep contamination pattern at the control plane, resulting in several conditioned realizations for which the variance of the conductivity field is inconsistent with the assumed prior. For model IB, less tweaking of the conductivity field is required compared to model IA, which is attributed to the presence of the geological window. As shown in Table 4, the coefficient of variation of σ 2 Y has also been determined to evaluate the uncertainty of the conditioned conductivity field variances. While the coefficient of variation 24

26 remains unchanged after the conditioning for model IB, IIA and IIB, it is almost doubled for model IA. This again implies that the conditioning of the conductivity fields is less good for model IA than for the other three models. The results from model IA are not surprising. Since there is nothing in the prior setup of this model that facilitates a downward movement of the contamination, the only way the contamination can be forced deeper into the aquifer is by introducing more heterogeneity in the system. However, the results from model IA show that no statistically consistent tweaking of the conductivity fields can produce a plume as deep as the one observed at the field site. The conceptual model without either a geological window or a deep residual-phase spill is thus considered highly unlikely. This conclusion is dependent on the assumption that the heterogeneity follows a multivariate Gaussian distribution. It might be possible to obtain the observed contaminant plume at the control plane with model IA by applying a different model of the heterogeneity, for example by using transition probability geostatistics [e.g. Lee et al., 2007]. However, since the model domain considers a single geological facies, the log-normal distribution of conductivity is here considered justifiable TABLE The results in Table 4 agree well with the model selection discussed in the previous section, where the most likely model (model IIB) is found to be the model that requires the least modification of the conductivity fields to fit the data. The other models required more iterations with the KEG and thus more tweaking of the conductivity fields in order to meet the data. It could be claimed that this is an indication of less appropriate boundary conditions and that these models should therefore be discarded. However, even though these results might be in favor of model IIB, both model IB and IIA are able to satisfactorily match data without relying on unreasonable 25

27 parameter values. Both model IB and IIA are therefore considered to be possible representations of the site settings, while model IA is regarded to be highly improbable Mass discharge uncertainty Figure 8 and Figure 9 show, respectively, the estimated mass discharge probability density function and the mass discharge cumulative density function for the four conceptual models before and after conditioning. Table 5 summarizes some key statistics for the estimated mass discharge distributions. The prior distributions reflect the mass discharge uncertainty without including any data at the control plane. There is a significant difference between the prior mass discharge distributions for model I and model II owing to the difference in the conceptual source models. Not surprisingly, the models accounting for the presence of a deep residual-phase source result in the highest estimated mass discharges. The prior ensemble mean mass discharge for model IIA and IIB are about four times larger than for model IA and IB. The chosen geological model (A or B) does not appear to have a significant effect on the estimated prior mass discharge distributions. After conditioning, the estimated mass discharge distributions for the four conceptual models approach each other. For model IIA a moderate decrease of the mass discharge estimates are observed, while the mass discharge estimates from model IIB are relatively unaffected. On the other hand, the mass discharge estimates from conceptual model IA and IB are increased after conditioning (seen as a shift of the distributions to the right). The standard deviation of the estimated mass discharge for model IB, IIA and IIB is reduced after taking data into account, as expected. This is not the case for model IA, where an increase is observed in the predicted mass discharge standard deviation. Though the relative uncertainty of the mass discharge estimate (coefficient of variation) for model IA is still reduced after conditioning, the increased mass discharge standard deviation could again imply that model IA is an inappropriate conceptual model. 26

28 The data set at the site is quite strong and the estimated mass discharge distribution is relatively robust with respect to the choice of conceptual models and their boundary conditions. This can be seen by noting that model conditioning results in no significant difference between the outputs of the four conceptual models FIGURE FIGURE TABLE Figure 8 also shows that the obtained mass discharge of 1 kg TCE/yr based on the classical estimation method lies nicely within the predicted posterior distributions for all four models. This implies that the available data set at the field site can provide a reliable mass discharge estimate. With the method presented here, we can now add legitimate uncertainty bounds to this estimate. It should be noted that the control plane considered here is located 160 meters downstream of the source, which means that the spatial concentration distribution at the control plane is smoothed by local hydrodynamic dispersion and that the spatial variability of contaminant mass flux can therefore be expected to be smaller. This could explain the agreement between the mass discharge value from the classical approach and the predicted ensemble estimates. If the control plane is located at shorter distances from the source much higher concentration gradients are likely to be observed, making the classical estimation method less adequate. Schwede and Cirpka (2009a) found that the empirical posterior mass discharge distribution can be approximated by a log-normal distribution, log-n(μ, σ), without introducing significant errors, 27

29 where the mean μ and the standard deviation σ of the log-variable are determined by fitting to the first two moments of the posterior mass discharge ensemble: μ and σ 1 S ln( M ) ln M = 2 S ln 1 + M 2 = 2 2 where M and S 2 are the posterior ensemble mean and variance, respectively. In Figure 8, the lognormal distribution based on the ensemble mean and variance has been plotted for each conceptual model. It is seen that the log-normal distribution is in good agreement with the estimated posterior mass discharge distributions. This is interesting from a computational point of view, because it requires much fewer realizations to obtain the ensemble mean and variance than to derive the entire distribution. The number of realizations should, however, still be large enough to give a reasonable approximation of the covariance matrices needed for the updating in equation (7). By applying the presented approach to conceptual model IIB with 100, 250, 500 and 1000 realizations, respectively, we have here found that 500 realizations are sufficient to obtain stable first two moments of the mass discharge. The four posterior mass discharge distributions are combined using Bayesian Model Averaging (BMA) as shown in Figure 10. The ensembles have been combined using three different set of model beliefs. In the first BMA the ensembles are considered equally likely, while in the second BMA the ensembles from the four models have been weighted according to the previously conducted model selection (Table 3, all data). However, as this model selection did not result in stable model weights and as model IA is considered to be highly unlikely based on the unsuccessful conditioning to concentration data, a third BMA has been included, where model IA, IB, IIA and 28

30 IIB have been given weights of 0%, 25%, 25% and 50%, respectively. By averaging over all the conceptual models, the uncertainty of the mass discharge estimate is increased depending on how the different models are weighted FIGURE Though the use of multiple conceptual models seems to be a more robust way of handling surprises related to conceptual model uncertainty, we do not claim that conceptual model uncertainty is now fully accounted for. The approach presented allows only for an incorporation of the conceptual models we can perceive. New data could therefore still result in surprises leading to rejection of some of the conceptual models and/or formulation of new conceptual models. It can be questioned whether the possible site settings are fairly represented by the chosen conceptual models. Other source and geological models could have been used and the geostatistical model describing the heterogeneity could have been more sophisticated by including trends or other factors. All these things are, however, relatively straightforward to include in the method. The conceptual models presented for the case study can be considered quite general. Of course, the modeled residual-phase source zone does not resemble the complexity of a true NAPL spill with many high concentration blobs in a complex pattern. However, since the control plane is located 160 meters downstream of the source zone, the uniform concentration boundary used to represent the residual-phase spill is considered adequate. If the control plane was located closer to the source zone this might not be the case. In the case presented, the conceptual models do not greatly affect assessment of mass discharge uncertainty. The choice of conceptual model is, however, very important for the risk posed by the site. The existence of a geological window will lead to a much higher risk for the down-stream 29

31 water supply wells, because these abstract water from the lower aquifer. If a window is not present the contamination at the site is likely to pose little threat to the water supply wells Summary and conclusions A rigorous method for quantifying the uncertainty of mass discharges from contaminated sites has been developed. The method is based on Bayesian inverse modeling, where multiple conceptual models representing the site conditions are used to reflect the uncertainty related to the choice of boundary conditions. The new method can account for unknown heterogeneity through Bayesian geostatistics with an uncertain geostatistical model. An ensemble of conditional realizations of the flow and concentration distribution at the downstream control plane is generated by use of the quasi-linear Kalman Ensemble Generator from which a posterior mass discharge distribution for each of the conceptual models considered is obtained. Through Bayesian model averaging these posterior distributions are combined by weighting them according to the belief associated with each of the conceptual models. This method has several advantages over the standard mass discharge estimation approach. First of all, it provides legitimate uncertainty bounds for the mass discharge estimate, which is essential for any risk assessment or decision support purposes. Because the method simulates flow and transport on conditioned conductivity fields, each mass discharge realization honors both the available data and natural variability. This approach ensures that the nonlinearity of the underlying processes producing the observations in the control plane is accounted for and that unphysical results of interpolation are avoided. Secondly, the method allows for an inclusion of any data that can be related to the hydraulic conductivity field, such as direct mass flux measurements from passive flux meters, hydraulic head measurements, concentration data and measurements of hydraulic conductivity from e.g. grain-size 30

32 analysis, slug tests, and pumping tests. Furthermore, the method also allows the use of data obtained up- or downstream of the control plane for conditioning. The main outcome of the method is the mass discharge distributions, but the method also provides uncertainty estimates of the spatial distribution of conductivity, head and concentration, not only at the control plane, but for the entire field site. These kinds of results can be used for optimizing the design of further site investigations. For example, areas with high variance (uncertainty) in the estimated concentration or flow field can be targeted for additional data collection, or the investigations can be targeted at supporting or invalidating different conceptual models, thereby reducing the uncertainty. The method also allows for a simple form of conceptual model testing. This is carried out by use of Bayes Theorem, where the ability of each conceptual model to simulate the data a priori is evaluated. Furthermore, if the KEG fails to update the ensemble related to a specific conceptual model, the belief in this conceptual model should be strongly reduced or might even be discarded. The method has been applied to a TCE contaminated site north of Copenhagen for which four conceptual models were set up reflecting the uncertainty involved when specifying boundary conditions for a real field site. The 500 realizations generated for each conceptual model were all conditioned on a data set consisting of 42 conductivity measurements, 36 head measurements and 24 concentration measurements. In this case study the following is concluded: 1. The conceptual model accounting for both a deep residual-phase source and a downward hydraulic gradient due to a geological window (model IIB) is the most likely of the four models. It is highly unlikely that the conceptual model that does not include a geological window or a deep residual-phase spill (model IA) can reflect the true site settings. 2. Though the prior mass discharge distributions for the four conceptual models are substantially different, the mass discharge distributions approach each other after 31

33 conditioning to the data. It is therefore concluded that the current data set at the site is strong and the mass discharge estimate is affirmed robust to the chosen conceptual models. 3. The deterministic mass discharge estimate obtained from the classical estimation method agrees well with the estimated posterior mass discharge distribution. This is ascribed to the strong available data set and to the fact that the control plane was located 160 meters downstream of the source, leading to a smaller spatial variability of the contaminant mass flux at the control plane. 4. The method is computationally costly, primarily due to the relatively large domain and the correspondingly large number of grid cells used in this study. At the same time, a relatively large number of realizations are required to obtain accurate approximations of the ensemble statistics needed for model conditioning. We found that 500 realizations appear to give a good approximation of the mass discharge distribution passing the control plane. 5. However, Bayesian Model Averaging (BMA) of the posterior mass discharge distributions requires much larger ensembles. We found that 7500 unconditional realizations with each conceptual model were insufficient to obtain stable average likelihood estimates necessary for the BMA. The convergence of BMA may therefore be an issue for further research. 6. Though computationally demanding, the approach is considered to of practical interest. It is very flexible and can easily be modified or expanded to account for other uncertain elements. The method is valuable, because it allows the relaxation of deterministic assumptions. Furthermore, the method is easy to implement and requires only a suitable numerical groundwater and transport model for the given site

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38 929 Figure captions Figure 1. Flow chart for the Bayesian inverse modeling approach of estimating the mass discharge probability distribution from a contaminated site Figure 2. Map of the study site area with the location of the leaking TCE tank and the control plane of multilevel samplers. The model domain and boundary conditions are indicated. The manually interpolated measurements of concentration at the control plane downstream the source zone together with the grid applied to estimate the mass discharge deterministically have also been shown Figure 3. The four conceptual models of the study site, see description in Table Figure 4. Map of the area around the study site with the location of the geological transect and the chosen model domain. Two interpretations of the geological stratification along the transect (Geology A and B) are illustrated Figure 5. Histograms of structural parameter samples drawn from the posterior joint distribution Figure 6. Plot of the observed and simulated ensemble mean values of hydraulic conductivity, head and concentration before and after the conditioning. Left column: model IA. Right column: model IIA. Open circles: prior mean values. Black diamonds: posterior mean values. Error bars: 90 % confidence interval of the posterior simulated values. Dotted lines: assumed 90 % measurement error interval. 37

39 Figure 7: Ensemble mean concentration distribution (in g TCE/l) at the control plane for the four conceptual models before (left column) and after conditioning (right column). Note that different concentration legends are used for model I and model II Figure 8. Estimated mass discharge distributions for the four conceptual models. Gray line with dots: prior mass discharge distribution. Black line: posterior mass discharge distribution. Gray line: log-normal pdf obtained from sample mean and variance. Black arrow: field estimated mass discharge based on classical method. Cross: mean of posterior mass discharge ensemble. Vertical lines: 16% and 84% quantiles of posterior mass discharge ensemble, respectively Figure 9. Estimated cumulative mass discharge distributions for the four conceptual models before (A) and after conditioning (B) Figure 10. Final posterior mass discharge distributions determined by combining the mass discharge distributions from the four conceptual models using Bayesian Model Averaging. Gray line with dots: models have been assigned equal weights. Black line with dots: models are weighted according to table 3 (all data). Black line: model IA, IB, IIA and IIB have been given weights of 0%, 25%, 25% and 50%, respectively

40 Table 1: Overview of conceptual models for contaminant source and geology at the study site Source Geology Model IA I: Gas cloud A: Geological window not present Model IB I: Gas cloud B: Geological window present Model IIA II: Gas cloud + free-phase spill A: Geological window not present Model IIB II: Gas cloud + free-phase spill B: Geological window present

41 Table 2: Parameter values used for the case study Unit Distribution Value Numerical grid Domain size [L x, L y, L z ] m [220, 200, 20] Grid spacing [Δx, Δy, Δz] m [2, 2, 0.25] Transport parameters Dispersivities [α l, α th, α tv ] cm Fixed [10, 0.5, 0.1] Effective porosity, n e - Fixed 0.25 Diffusion in water, D 0 m 2 /s Fixed 1e-9 Geostatistical model fory = lnk Global mean, β = ln(k g ) - Normal Mean: ln(2e-5) Variance Q ββ : Variance σ Y - Uniform Integral scale [λ x, λ y, λ z ] m Uniform X: 8-50 Y: 8-50 Z: 1-4 Matérn shape parameter κ - Uniform Measurement error and their standard deviations ln(k), σ R,Y - Normal 0.3 Head, σ R,h m Normal Concentration, σ R,c μg/l Normal Relative error: 0.2 Absolute error: 0.02

42 Table 3: Prior and posterior probabilities of the four considered conceptual models. The posterior probabilities have been calculated based on all the available data, but also based on hydraulic data only. Model IA Model IB Model IIA Model IIB Prior probability 25 % 25 % 25 % 25 % Posterior probability (lnk + head data) 19 % 30 % 34 % 17 % Posterior probability (all data) 4 % 3 % 0 % 93 %

43 Table 4: Hydraulic conductivity (K) field statistics before and after conditioning. β: global mean of ln(k). Q ββ : variance of β, σ Y 2 : variance of ln(k). CV[σ Y 2 ]: coefficient of variation of σ Y 2. Conceptual model β Q ββ 2 σ Y CV[σ 2 Y ] Prior Posterior Prior Posterior Prior Posterior Prior Posterior Model IA ln(2.0e-5) ln(2.5e-5) Model IB ln(2.0e-5) ln(1.2e-5) Model IIA ln(2.0e-5) ln(2.7e-5) Model IIB ln(2.0e-5) ln(1.7e-5)

44 Table 5: Mean, standard deviation, coefficient of variation and 16% to 84% quantiles of the estimated mass discharges (kg/yr) for the 4 conceptual models before and after conditioning. Conceptual model Mean Standard deviation CV 16% to 84% quantiles Prior Posterior Prior Posterior Prior Posterior Prior Posterior Model IA Model IB Model IIA Model IIB

45 Formulate P conceptual site models. Assign prior probability to each model. Specify numerical grid and solver. Define boundary conditions. Define prior parameter pdfs Update geostatistical model parameters with direct data using Bayes Theorem. N realizations Sample updated parameter pdf. Generate random field. Run flow and transport. Extract simulated measurements. Determine mass discharge Evaluate prior ensemble statistics or all conceptual mode els Repeat f Check for ensemble convergence Accepted? Go to next realization Accepted? Rejected? Update realization with data. Run flow and transport Evaluate goodness of fit Rejected? All realizations accepted? Calculate posterior mass discharge. Assess posterior ensemble statistics Obtain mass discharge probability distribution by combining all ensembles via Bayesian Model Averaging.

46 Denmark A A RAP10 F1 B1 F2 F2A F3 F4A F4 F5 RAP9 F6 ma asl > 100 μg/l μg/l μg/l 1-25 μg/l

47 30 m S Model IA N S Model IIA N Source Control plane Source Control plane 15 m Sand GWT Sand GWT 0 m Flow Flow?? -15 m 30 m Clay till Model IB Clay till Model IIB Source Control plane Source Control plane 15 m Sand GWT Sand GWT 0 m Flow Flow??? -15 m Clay till?? Clay till?? 0 m 100 m 200 m 0 m 100 m 200 m

48 30 m A Geology A A B3 B F m Clay till -30 m Sand Limestone 0 m 500 m 1000 m 30 m A Geology B A B F5 B1 0 m Clay till -30 m Sand Limestone 0 m 500 m 1000 m

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