First benchmark for the uncertainty analysis based on the example of the French clay site RESULTS

Transcription

1 First benchmark for the uncertainty analysis based on the example of the French clay site RESULTS JRC: S. Prváková, R. Bolado-Lavín, A. Badea and K-F Nilsson ANDRA: G. Pepin, E. Treille EUR EN

2 The Institute for Energy provides scientific and technical support for the conception, development, implementation and monitoring of community policies related to energy. Special emphasis is given to the security of energy supply and to sustainable and safe energy production. European Commission Joint Research Centre Institute for Energy Contact information Address: P.O. Box 1755 ZG, Petten Tel.: Fax: Legal Notice Neither the European Commission nor any person acting on behalf of the Commission is responsible for the use which might be made of this publication. Europe Direct is a service to help you find answers to your questions about the European Union Freephone number (*): (*) Certain mobile telephone operators do not allow access to numbers or these calls may be billed. A great deal of additional information on the European Union is available on the Internet. It can be accessed through the Europa server JRC EUR EN ISSN Luxembourg: Office for Official Publications of the European Communities European Communities, 2008 Reproduction is authorised provided the source is acknowledged Printed in The Netherlands 2

3 Content 1. INTRODUCTION DESCRIPTION OF THE BENCHMARK CONCEPTL MODEL ANALYSIS METHODOLOGY Input parameters Sampling methods OUTPUT Performance Indicators to be computed ADOPTED METHODOLOGY ANDRA JRC-IE RESULTS ANDRA Uncertainty Analysis Results of I Results of Se Results of Nb Sensitivity Analysis Results of I Results of Se Results of Nb JRC-IE RESULTS Uncertainty analysis results Sensitivity analysis results...43 COMPARISON JRC/ANDRA RESULTS CONCLUSIONS AND PERSPECTIVES REFERENCES APPENDICES ANDRA: DETAILED RESULTS Sensitivity analysis I129 - Linear indicators I129 - Rank indicators Se79 - Linear indicators Nb94 - Linear indicators Nb94 - Rank indicators Uncertainty analysis JRC: DETAILED RESULTS Uncertainty analysis Sensitivity analysis

4 List of Figures Figure : geometry and components (i.e. materials) taken into account...8 Figure 2.2-1: Different steps to perform Monte-Carlo calculations...10 Figure : LHS sampling (principles with two parameters)...12 Figure 2.3-1: Indicators to be calculated...12 Figure 3.1-1: Alliances tool used for Monte Carlo simulations...14 Figure : grid used for calculations...15 Figure : a) planar and b) cylindrical option of 2D geometry...16 Figure : Ccdf of Peclet number in undisturbed argillites layer...19 Figure : Simulations of molar flow coming out of an intermediate surface in clay 15 m from the waste package versus time FluxCOX 15m...20 Figure : Quantiles in time of Molar Flow coming out of concrete (waste packages Φsortie colis and EBS Fluxsortie BO)...22 Figure : Quantiles in time of Molar Flow coming in and out of micro-fractures zone (sortie zone micro-fissurée, undisturbed argillites at 15 m FluxCOX 15m and top/bottom Flux sortie COX)...23 Figure : Distribution of Maximal Molar Flow coming out of concrete (waste packages Fluxsortie colis and EBS Flux sortie BO)...24 Figure : Distribution of Maximal Molar Flow coming in and out of undisturbed argillites, Φsortie zone microfissurée, FluxCOX 15m,Fluxsortie COX...24 Figure : Quantiles in time of Molar Flow coming in and out of undisturbed argillites a) waste package Fluxsortie colis, b) disposal cell Fluxsortie BO, c) fractured zone Fluxsortie zone fracturé, d) micro-fissured zone Fluxsortie zone micro-fissurée, e) undisturbed argillites...26 Figure : ccdf Distribution of Maximal Molar Flow coming out of concrete (EBS) Fluxsortie BO and clay layer (top and bottom Fluxsortie COX)...26 Figure : Quantiles in time of Molar Flow coming in and out of undisturbed argillites a) disposal cell Fluxsortie BO, b) fractured zone Fluxsortie zone fracturé, c) micro-fissured zone Fluxsortie zone micro-fissurée, Nb Figure : ccdf of Maximal Molar Flow coming out of concrete (EBS) Fluxsortie BO and clay layer (top and bottom) Fluxsortie COX Nb Figure : Scatter plots maximal molar flow coming out of clay layer Fluxsortie COX I Figure : Rank and linearity indicators between Kd (COX) and molar flow out of clay layer Fluxsortie COX I12929 Figure : Evolution in time of statistical rank indicators for molar flow out of clay layer Fluxsortie COX I Figure : Evolution in time of PRCC rank indicator for all input data Molar Flow coming out of undisturbed argillites Fluxsortie COX I Figure : Sensitivity analysis on maximal molar flow coming out from EBS (concrete) Fluxsortie BO I Figure : Sensitivity analysis on maximal molar flow coming out of clay layer Φsortie COX I Figure : evolution in time of PRCC rank indicator for all input data (molar flow coming out of concrete disposal cell Fluxsortie BO) Se Figure : Sensitivity analysis on maximal molar flow coming out from EBS (concrete) Fluxsortie BO Se Figure : evolution in time of PRCC rank indicator for all input data (molar flow coming out of clay layer top and bottom Fluxsortie COX)...34 Figure : Sensitivity analysis on maximal molar flow coming out of clay layer Fluxsortie COX Se Figure : evolution in time of PRCC rank indicator for all input data (molar flow coming out of EBS) Fluxsortie BO Nb Figure : Sensitivity analysis on maximal molar flow coming out of EBS Fluxsortie BO Nb Figure : Molar flow evolution over time for the three radionuclides getting out of the six layers (1000 curves...39 Figure : Molar flow statistical indicators evolution over time for the 129 I getting out of the six layers; from left...40 Figure : Empirical cumulative distribution function for the peak flows of the 3 radionuclides getting out of the six layers : a) 94 Nb, c) 129 I, e) 79 Se; and for the corresponding time to the peaks: b) 94 Nb, d) 129 I, f) 79 Se. Time in log-scale...42 Figure Evolution over time of 129 I molar flows getting out of the six layers a) waste package Φ sortie colis, b) disposal cell Φ sortie BO, c) fractured zone Φ sortie zone fracturé, d) micro-fissured zone Φ sortie zone micro-fissurée, e) undisturbed argilites (15 m) Φ COX 15m, f) undisturbed argilites,φ sortie COX...46 Figure : Evolution over time of 94 Nb molar flows getting out of the first four layers a) waste package Φ sortie colis, b) disposal cell Φ sortie BO, c) fractured zone Φ sortie zone fracturé, d) micro-fissured zone Φ sortie zone micro-fissurée,...46 Figure : Evolution over time of 79 Se molar flows getting out of the six layers a) waste package Φ sortie colis, b) disposal cell Φ sortie BO, c) fractured zone Φ sortie zone fracturé, d) micro-fissured zone Φ sortie zone micro-fissurée, e) undisturbed argilites (15 m) Φ COX 15m, f) undisturbed argilites,φ sortie COX...48 Figure : I129 - Evolution in time of linear indicators for effective diffusion of argillites...55 Figure : I129 - Evolution in time of linear indicators for effective diffusion of fractured zone...56 Figure : I129 - Evolution in time of linear indicators for effective diffusion of high performance concrete...57 Figure : I129 - Evolution in time of linear indicators for vertical head gradient in argillites

5 Figure : I129 - Evolution in time of linear indicators for permeability of high performance concrete...59 Figure : I129 - Evolution in time of linear indicators for horizontal permeability of argillites...60 Figure : I129 - Evolution in time of linear indicators for permeability of microfissured zone...61 Figure : I129 - Evolution in time of linear indicators for permeability of fractured zone...62 Figure : I129 - Evolution in time of linear indicators for Kd Iodine concrete...63 Figure : I129 - Evolution in time of linear indicators for Kd Iodine argillites...64 Figure : I129 - Evolution in time of linear indicators for vertical permeability in argillites...65 Figure : I129 - Evolution in time of linear indicators for argillites porosity...66 Figure : I129 - Evolution in time of linear indicators for microfissured zone porosity...67 Figure : I129 - Evolution in time of linear indicators for fractured zone porosity...68 Figure : I129 - Evolution in time of rank indicators for effective diffusion of argillites...69 Figure : I129 - Evolution in time of rank indicators for effective diffusion of fractured zone...70 Figure : I129 - Evolution in time of rank indicators for effective diffusion of high performance concrete...71 Figure : I129 - Evolution in time of rank indicators for vertical head gradient in argillites...72 Figure : I129 - Evolution in time of rank indicators for permeability of high performance concrete...73 Figure : I129 - Evolution in time of rank indicators for horizontal permeability of argillites...74 Figure : I129 - Evolution in time of rank indicators for permeability of microfissured zone...75 Figure : I129 - Evolution in time of rank indicators for permeability of fractured zone...76 Figure : I129 - Evolution in time of rank indicators for Kd Iodine concrete...77 Figure : I129 - Evolution in time of rank indicators for Kd Iodine argillites...78 Figure : I129 - Evolution in time of rank indicators for vertical permeability in argillites...79 Figure : I129 - Evolution in time of rank indicators for argillites porosity...80 Figure : I129 - Evolution in time of rank indicators for microfissured zone porosity...81 Figure : I129 - Evolution in time of rank indicators for fractured zone porosity...82 Figure : Se79 - Evolution in time of linear indicators for solubility limit in argillites...83 Figure : Se79 - Evolution in time of linear indicators for Kd in argillites...84 Figure : Se79 - Evolution in time of linear indicators for effective diffusion of argillites...85 Figure : Se79 - Evolution in time of linear indicators for Kd Selenium concrete...86 Figure : Se79 - Evolution in time of linear indicators for Csat Selenium concrete before years...87 Figure : Se79 - Evolution in time of linear indicators for Csat Selenium concrete after years...88 Figure : Se79 - Evolution in time of rank indicators for Kd Selenium concrete...89 Figure : Se79 - Evolution in time of rank indicators for Csat Selenium concrete before years...90 Figure : Se79 - Evolution in time of rank indicators for Csat Selenium concrete after years...91 Figure : Se79 - Evolution in time of rank indicators for vertical permeability in argillites...92 Figure : Se79 - Evolution in time of rank indicators for argillites porosity...93 Figure : Se79 - Evolution in time of rank indicators for microfissured zone porosity...94 Figure : Se79 - Evolution in time of rank indicators for fractured zone porosity...95 Figure : Nb94 - Evolution in time of linear indicators for solubility limit in argillites...96 Figure : Nb94 - Evolution in time of linear indicators for Kd Niobium in argillites...96 Figure : Nb94 - Evolution in time of linear indicators for effective diffusion of argillites...96 Figure : Nb94 - Evolution in time of linear indicators for effective diffusion of high performance concrete...97 Figure : Nb94 - Evolution in time of linear indicators for permeability of high performance concrete...97 Figure : Nb94 - Evolution in time of linear indicators for horizontal permeability of argillites...97 Figure : Nb94 - Evolution in time of linear indicators for argillites porosity...98 Figure : Nb94 - Evolution in time of linear indicators for microfissured zone porosity...98 Figure : Nb94 - Evolution in time of linear indicators for fractured zone porosity...98 Figure : Nb94 - Evolution in time of linear indicators for release rate of hulls of ILW Figure : Nb94 - Evolution in time of linear indicators for release rate of inconel of ILW Figure : Nb94 - Evolution in time of rank indicators for release rate of inconel of ILW Figure : Nb94 - Evolution in time of rank indicators for release rate of stainless steels of ILW Figure : Nb94 - Evolution in time of rank indicators for release rate of stainless steels of ILW Figure : Nb94 - Evolution in time of rank indicators for solubility limit in argillites Figure : Nb94 - Evolution in time of rank indicators for Kd Niobium in argillites Figure : Nb94 - Evolution in time of rank indicators for effective diffusion of argillites Figure : Nb94 - Evolution in time of rank indicators for Kd Niobium concrete Figure : Nb94 - Evolution in time of rank indicators for Csat Niobium concrete before years Figure : Nb94 - Evolution in time of rank indicators for Csat Niobium concrete after years Figure : Nb94 - Evolution in time of rank indicators for argillites porosity Figure : Nb94 - Evolution in time of rank indicators for microfissured zone porosity Figure : Nb94 - Evolution in time of rank indicators for fractured zone porosity Figure : Nb94 - Evolution in time of rank indicators for release rate of hulls of ILW Figure : Nb94 - Evolution in time of rank indicators for release rate of inconel of ILW Figure : Nb94 - Evolution in time of rank indicators for release rate of inconel of ILW

6 Figure : Nb94 - Evolution in time of rank indicators for release rate of stainless steels of ILW Figure : Nb94 - Evolution in time of rank indicators for release rate of stainless steels of ILW Figure : I129 ccdf of maximal molar flow through several surfaces Figure : Se79 ccdf of maximal molar flow through several surfaces Figure : Nb94 ccdf of maximal molar flow through several surfaces Figure 7.2-1: Time evolution for the molar flow of the 3 radionuclides getting out of the following layers (1000 curves for each radionuclide); from left to right and from top to bottom : a) waste package, b) disposal cell, c) fractured zone, d) micro-fissured zone, e) undisturbed argilites (15 m), f) undisturbed argilites Figure 7.2-2: Time evolution for the statistical indicators of the molar flow of 94 Nb getting out of the following layers; from left to right and from top to bottom : a) waste package, b) disposal cell, c) fractured zone, d) micro-fissured zone, e) undisturbed argilites (15 m), f) undisturbed argilites Figure 7.2-3: Time evolution for the statistical indicators of the molar flow of 129 I getting out of the following layers; from left to right and from top to bottom : a) waste package, b) disposal cell, c) fractured zone, d) micro-fissured zone, e) undisturbed argilites (15 m), f) undisturbed argilites Figure 7.2-4: Time evolution for the statistical indicators of the molar flow of 79 Se getting out of the following layers; from left to right and from top to bottom : a) waste package, b) disposal cell, c) fractured zone, d) micro-fissured zone, e) undisturbed argilites (15 m), f) undisturbed argilites Figure 7.2-5: Empirical cumulative distribution function for the maximum release of the 3 radionuclides getting out of all the layers : a) 94 Nb, c) 129 I, e) 79 Se ; and for the corresponding time occurrence : b) 94 Nb, d) 129 I, f) 79 Se Figure 7.2-6: Estimation of the probability density function for the maximum release of the 3 radionuclides getting out of all the layers : a) 94 Nb, c) 129 I, e) 79 Se ; and for the corresponding time occurrence : b) 94 Nb, d) 129 I, f) 79 Se Figure 7.2-7: Release of the 3 radionuclides getting out of all the layers at 5000 years : empirical cumulative distribution function : a) 94 Nb, c) 129 I, e) 79 Se; and estimation of the probability density function: b) 94 Nb, d) 129 I, f) 79 Se Figure : Release of the 3 radionuclides getting out of all the layers at years : empirical cumulative distribution function : a) 94 Nb, c) 129 I, e) 79 Se; and estimation of the probability density function: b) 94 Nb, d) 129 I, f) 79 Se Figure : Release of the 3 radionuclides getting out of all the layers at years : empirical cumulative distribution function : a) 94 Nb, c) 129 I, e) 79 Se; and estimation of the probability density function: b) 94 Nb, d) 129 I, f) 79 Se Figure : Release of the 3 radionuclides getting out of all the layers at years : empirical cumulative distribution function: a) 94 Nb, c) 129 I, e) 79 Se; and estimation of the probability density function : b) 94 Nb, d) 129 I, f) 79 Se. 120 Figure : : Release of the 3 radionuclides getting out of all the layers at years : empirical cumulative distribution function: a) 94 Nb, c) 129 I, e) 79 Se; and estimation of the probability density function : b) 94 Nb, d) 129 I, f) 79 Se. 121 Figure : : Release of the 3 radionuclides getting out of all the layers at years : empirical cumulative distribution function : a) 94 Nb, c) 129 I, e) 79 Se; and estimation of the probability density function : b) 94 Nb, d) 129 I, f) 79 Se Figure : Time evolution of a) SRCs for the molar flows of 94 Nb, b) SRRCs for the molar flows of 94 Nb, c) SRCs for the molar flows of 129 I, d) SRRCs for the molar flows of 129 I, e) SRCs for the molar flows of 79 Se, f) SRRCs for the molar flows of 79 Se; coming out of waste package Figure : Time evolution of a) SRCs for the molar flows of 94 Nb, b) SRRCs for the molar flows of 94 Nb, c) SRCs for the molar flows of 129 I, d) SRRCs for the molar flows of 129 I, e) SRCs for the molar flows of 79 Se, f) SRRCs for the molar flows of 79 Se; coming out of disposal cell Figure : Time evolution of a) SRCs for the molar flows of 94Nb, b) SRRCs for the molar flows of 94Nb, c) SRCs for the molar flows of 129I, d) SRRCs for the molar flows of 129I, e) SRCs for the molar flows of 79Se, f) SRRCs for the molar flows of 79Se; coming out of fractured zone Figure : Time evolution of a) SRCs for the molar flows of 94Nb, b) SRRCs for the molar flows of 94Nb, c) SRCs for the molar flows of 129I, d) SRRCs for the molar flows of 129I, e) SRCs for the molar flows of 79Se, f) SRRCs for the molar flows of 79Se; coming out of micro fissured zone Figure : Time evolution of a) SRCs for the molar flows of 94 Nb, b) SRRCs for the molar flows of 94 Nb, c) SRCs for the molar flows of 129 I, d) SRRCs for the molar flows of 129 I, e) SRCs for the molar flows of 79 Se, f) SRRCs for the molar flows of 79 Se; coming out of undisturbed argillites (15m) Figure : Time evolution of a) SRCs for the molar flows of 94Nb, b) SRRCs for the molar flows of 94Nb, c) SRCs for the molar flows of 129I, d) SRRCs for the molar flows of 129I, e) SRCs for the molar flows of 79Se, f) SRRCs for the molar flows of 79Se; coming out of undisturbed argillites

7 1. Introduction JRC-IE and ANDRA collaborate in the Framework of the Integrated Project PAMINA (Performance Assessment Methodologies in Application to Guide the Development of the Safety Case) to develop and apply advanced methods for performance assessment of a clay repository [1]. The idea is to apply state-of-the-art techniques of different levels of complexity to assess the performance of a French clay repository and to assess the applicability of different methods. This is done by computing i) how uncertainties in input data affect output (the flow of radionuclides in specific parts of the repository system) and ii) to perform sensitivity analysis to rank the processes and input parameters with respect to their relevance for the output. The data used as input to the analyses are based on the large French research programme for a repository in a Cavollo-Oxfordian clay formation, which is the potential host-rock in France [2]. The JRC and ANDRA perform analysis in parallel applying their own methods on two Benchmarks that have been jointly defined. The Benchmarks defines the geometry of the system and its constituents, the processes to be considered, the probabilistic density functions of the data and the coupling between parameters and the output to be computed. This report deals with the first Benchmark for which we have a simplified geometrical description and simplified description of the relevant processes [6]. For the uncertainty analysis the main indicators that are computed include quantiles of the flow for a limited number of radionuclides in different parts of the system and complete probability density functions at specific times. The main indicators for the sensitivity analysis include scatter plots for selected parameters, ranking of parameters with respect to their influence on the output and computation of statistical coefficients such as Pearson to relate input uncertainty to out put uncertainty, Spearman correlation coefficient and standard rank regression coefficient. The First benchmark is also used to calibrate the models, the computational tools and set up the computational methods. The outline of the paper is as follows: In the first section Benchmark is described. This is followed by a description of the methodology adopted by each organization. ANDRA and JRC then present the computed results separately. A more detailed set of results is provided in the Appendices. The results from ANDRA and JRC are then compared and discussed with respect to agreement and differences and the potential sources for differences. The main conclusions are then presented and a roadmap for the second Benchmark is outlined. 7

8 2. Description of the Benchmark 2.1. Conceptual Model The benchmark is based on the study of the release of radionuclides from wastes of an ILW disposal cell embedded in a porous materials for a generic French clay site. The disposal cell, which only contains non-organic waste forms, does not release any hydrogen gas; it includes both activated waste and compacted hulls and end-caps that are supposed to be homogeneously filled. For this Benchmark exercise a simplified representation of the disposal cell is considered (see Figure 2.1.1). It consists of a 2D vertical slice of the middle of the disposal cell. The complete geometrical model includes the thickness (130 meters) of the undisturbed host rock (Callovo-Oxfordien layer) and the disposal cell located in the middle of the clay layer. The waste packages domain is considered as rectangle located in the centre of the circular disposal cell. The materials to be considered in the benchmark model are also indicated in Figure Undisturbed Argilites saines argillites Micro-fissured-zone Zone micro-fissurée «Béton de Béton colisage de colisage» 2D vertical slice simplifications Zone Fractured fracturée zone «Béton de de structure» Figure : geometry and components (i.e. materials) taken into account Colis Waste / Béton Packages/ de remplissage «Béton de remplissage des colis» A detailed description of the Benchmark, the physical model, assumptions and its parameters are given in [6]. The main assumptions to be taken into account in calculations are: 8

9 Materials within disposal cell and its environment are considered as homogeneous and continuous porous media, and are supposed to be water saturated from t=0; hence release of radionuclides from waste packages starts from t=0. Calculations are carried out until 1 million years. The migration of radionuclides in porous media in the aqueous solutions is considered as convective/diffusive/dispersive, taking into account phenomena of sorption (linear and reversible, Kd approach) and precipitation (solubility limit Csat). The convective part is based on steady-state hydraulic results using constant vertical head gradient through the host rock. The waste package is represented by two materials: (i) a specific concrete filling ( béton de remplissage ), in which the source term is distributed in an uniform way, and for the migration there is no sorption and solubility is unlimited from t=0 and; (ii) an over-pack in a specific concrete ( béton de colisage ) with a high level of confining performances (low diffusivity, high Kd and low solubility limit) up to years, and then degrading itself into filling concrete ( béton de structure ). Thus the material properties in the analysis will change at this time. The details for this are given in [6]. The ILW disposal cell out of the wastes domain is filled with a concrete material in which radionuclides are free to migrate with no sorption and with unlimited solubility, and thus, (like béton de remplissage, same deterministic input data), no uncertainty has to be considered. A relevant approach of the geochemical characterization would be a variation in space and time of Kd and Csat taking into account the evolution of different states of concrete (sound, altered, degraded, neutralised conditions). Since the dynamics of propagation of the various alterations of concrete are not known, we consider only one geochemistry which is applied to all the thickness of concrete (all the disposal cell) and whose spectrum of variation (uncertainty) takes into account the various alterations. This is applied by ranges of time. The mechanical disturbance is represented by the Excavated Damaged Zone (EDZ), which consists of a fractured zone with high permeability and of a micro-fissured zone. The geometrical boundaries of the micro-fissured zone does not change with time. The occurrence of a self-sealing phenomenon is integrated into the uncertainty. The chemical disturbance generated by the degradation of the concretes is included in the mechanical disturbance (EDZ) in terms of extension and in migration parameters (hydraulic, sorption and precipitation). The geochemistry considered in the EDZ is the same as in the undisturbed argillites. Table summarizes the various input data whose uncertainty has to be taken into account into benchmark: 9

10 Undisturbed argillites microfissured zone Fractured zone Concrete High Performance concrete Release rate of metallic components (corrosion rate) Permeability (horizontal/ vertical) up to 1 e 4 years Effective diffusion coefficient up to 1 e 4 years Head ascending vertical gradient in undisturbed argillites Distribution coefficient (time ranges) Solubility limit (time ranges) Diffusion Porosity Kinematic porosity Table : Input data/materials with pdf taken into account for the probabilistic calculations 2.2. Analysis Methodology Monte-Carlo methods will be adopted in this Benchmark. Their main advantages include: i) it is userfriendly and straightforward to understand and implement; ii) the possibility to consider the spectrum of variation of all parameters to investigate; iii) to carry out both uncertainty and sensitivity analysis and to offer a wide range of graphic ouput possibilities. The Monte Carlo methodology is based on several steps with pre-processing, processing and post-processing as outlined in Figure definition of the test-case : geometry, physical processes, definition of probabilistic density function (pdf) definition of correlations and constraints between input data Use Of Alliances SENSITIVITY ANALYSIS Statistic indicators calculations rank and linear correlations {input data/result} UNCERTAINTY ANALYSIS Quantiles, moments, distribution sampling methods numerical set-up of calculations (calibration, ) Results indicateur temps PRE-PROCESSING PROCESSING PROCESSING Figure 2.2-1: Different steps to perform Monte-Carlo calculations POST-PROCESSING PROCESSING Input parameters After having set up the physical and geometrical characteristics of the benchmark, probabilistic density functions (pdfs) need to be defined for the parameters with uncertainty. The pdfs used in this Benchmark are based on a large data set from an intensive characterization programme in Andra in 10

11 combination with expert judgment and/or bibliography. The pdfs include three sources of uncertainty: natural variability (especially for argillites), experimental measurement methods (laboratory + in situ tests) and up-scaling (extrapolation from small samples to repository scale). All pdfs are truncated. The pdfs are given in [6]. The models used for the calculations include many input data that are coupled. In order to get the most physical coherence for each set of input data, correlations and constraints are defined that prescribe natural constraints and couplings between data, and that need to be taken into account when defining the samples. Two basic types of correlations can be distinguished (statistical and static) as well as constraints. A short description is given below. statistical correlation: is a measure of the linear association between two input parameters. It may vary from -1 to +1. The ±1 refer to perfect linear relation with positive or negative slope. Zero means that there is no statistical correlation static correlation: this type of correlation makes it possible to bind two stochastic variables by a function or a scalar. For example, the horizontal permeability of the undisturbed argillites results from the vertical permeability via an "anisotropy" function, itself sampled [6]. constraints between stochastic parameters are identified in order to impose the physical coherence of sets of input data; it consists of applying inequalities between the data, after having used first the statistical correlation. If the constraint is not respected at the end of the sampling (taking into account correlations), then the value of the data is equal to the extreme of the inequality. Consequently some initial pdfs might be affected when using constraints. The constraints are detailed in [6] Sampling methods The sampling method used for the Benchmark is Latin Hypercube Sampling (LHS). The probabilistic density functions (pdf) are cut out into equiprobable layers. The layers are numbered, for instance from 1 to S, and than sampled with a random method using a distribution of the numbers of 1 to S. The LHS method (see Figure 2.2-2) enables to cover the total spectrum of variation of the different parameters. One thousand simulations will be used in the Benchmark. 11

12 Xj Xi Figure : LHS sampling (principles with two parameters) 2.3. Output Performance Indicators to be computed Φ sortie COX Φ sortie zone micro-fissurée Φ COX 15 m 30 m Φ sortie BO Φ sortie colis Φ sortie zone fracturée Figure 2.3-1: Indicators to be calculated The different types of physical indicators to be calculated are the molar flow (Figure 2.3-1) coming out of: the waste packages Φ sortie colis (external envelope surface of waste packages pile), the disposal cell Φ sortie BO (external envelope surface of «béton de structure» material), the fractured zone Φ sortie zone fracturée (external envelope surface of fractured zone material), the micro-fissured zone Φ sortie zone micro-fissurée (external envelope surface of micro-fissured zone material), 12

13 the undisturbed argillites, in the clay Φ COX 15m, at a specific surface located at 15 meters from packages, and from the undisturbed argillites, at the top and bottom. A statistical analysis will be carried for the evolution of the flows above. A more detailed analysis will be done at specific times: 5000 years for Φ sortie colis, Φ sortie BO, Φ sortie zone fracturée, Φ sortie zone micro-fissurée years for Φ sortie colis, Φ sortie BO, Φ sortie zone fracturée, Φ sortie zone micro-fissurée years for Φ sortie colis, Φ sortie BO, Φ sortie zone fracturée, Φ sortie zone micro-fissurée, Φ COX 15m,Φ sortie COX years for Φ sortie zone fracturée, Φ sortie zone micro-fissurée, Φ COX 15m, Φ sortie COX years for Φ sortie zone fracturée, Φ sortie zone micro-fissurée, Φ COX 15m, Φ sortie COX years for Φ sortie zone fracturée, Φ sortie zone micro-fissurée, Φ COX 15m, Φ sortie COX, the maximal release of radionuclides (molar flow) between 0 and years, the occurrence of the maximal release (molar flow) between 0 and years Statistical indicators for uncertainty analysis of the results are: the evolution (in time) of the various (physical) indicators with quantiles at 1%, 5%, 25%, 50% (median), 75%, 95%, 99% plus the mean the probability density function (pdf), the cumulative probability density function (cdf), complementary cumulative probability density function (ccdf) of the various physical indicators; the peak value for each distribution and the time when that happens maximum; and with characterization of the various moments of order 3 and 4 (kurtosis, skewness) of the distributions The objective of the sensitivity analysis is to rank the input parameters with respect to their importance to the radionuclide migration and how they affect the output indicators. The analysis is made by specific statistical coefficients. Coefficients measuring the linearity between the uncertainty of the result and the uncertainty of input data: coefficient of Pearson, PCC, SRC, Coefficients measuring the monotony between the uncertainty of the result and the uncertainty of the input data: Spearman correlation coefficient, partial rank correlation coefficient (PRCC), standard rank regression coefficient (SRRC). 13

14 3. Adopted Methodology 3.1. ANDRA Hydraulic and transfer of radionuclides were performed using the Alliances platform. Alliances platform (Figure 3.1-1), co-developped by ANDRA, CEA, and EDF since 2001, is used to perform calculations and carry out studies on the multi-physic and multi-scale behaviour of a waste repository and its environment. Alliances platform is divided into several modules, in which several codes can be used to compare results. Hydraulic/transfer module has been used; Cast3m and Porflow codes have been used for hydraulic and solute transfer (in advection/diffusion/dispersion, using decay, adsorption and solubility limit). Generator Generator Sampling Evaluation Launcher Launcher Analyzer Analyzer Extraction Calculation Three steps : Creation of the data set generation of stochastic variables evaluation of static correlations Running calculations for each data set Statistical analysis Extract results submitted to analysis Uncertainty and/or sensitivity analysis Post-processing Post-processing Hydraulic code : Cast3M Transport code : Porflow Figure 3.1-1: Alliances tool used for Monte Carlo simulations Calculations have consisted of : - generating the first sets of input data by LHS method, - generating the sets of input data to be performed, (after having rearranged the first sets using correlations - static, statistic, and inequalities), - running calculations using Porflow and Cast3m - extracting results (molar rates in time and maximal molar rate, times of maxima, ), - calculating statistical indicators for : uncertainty analysis : quantiles, pdf and ccdf 14

15 sensitivity analysis (Pearson, PRC, SRC, Spearman, PRCC, SRRC, Kendall, Student, ) All physical and statistical characteristics have been taken into account in calculations. The details are given in [6]. Good time steps are calibrated to cover the whole range of variation and especially for short times, to have a good description of the source terms (especially from Nb94) and a good evaluation of the peak due to the abrupt change of diffusion and permeability of high performance concrete. For the calculation, the mean value of number of time steps are about On an other hand, related to the range of sets of input data, we have to pay attention to ensure a good numerical convergence, finding the best compromise {spatial and time discretization / solver / required accuracy} with physics. One single grid (see Figure 3.1-2), hexahedral mesh, elements) has been used for all calculations, based on the strongest constraints, and based on mesh Peclet Number, skewness, etc. Routines have been developed for post-processing, to check good convergence (mass balance within each material, ) Figure : grid used for calculations 15

16 3.2. JRC-IE The transport of radionuclides was performed using the code GoldSim. GoldSim is a compartment mass transport model tool and it is designed for simulation of contaminants migration within an environmental system. The code can be used for deterministic as well as for probabilistic analysis. The model parameters can be time dependent. Probabilistic parameters are described as probability density functions. One- and two-dimensional options are available for the geometrical description of the system. This first benchmark considers simplified 1D whereas the second benchmark will use 2D geometry with two options of planar and cylindrical geometry (see Figure 3.2-1). a) b) Figure : a) planar and b) cylindrical option of 2D geometry. The transport of radionuclides in the considered concept is diffusion dominant and the diffusion was therefore the only transport process modelled. The probabilistic analysis was performed using the Monte Carlo technique with 1000 simulations using Latin Hypercube Sampling. The Benchmark described in the previous chapter was adopted for the modelling. Thus for the source terms the three radionuclides 129 I, 94 Nb and 79 Se with the specific release rates of the waste matrices were considered. The following three series of parameters defined in the Benchmark description could not be adopted due to the simplifying model assumptions: hydraulic permeabilities - for the barriers: beton de colisage, beton de structure, fractured zone, micro-fissured zone and undisturbed argillites dispersion coefficients - for the barriers: beton de colisage, beton de structure, fractured zone, micro-fissured zone and undisturbed argillites head ascending vertical gradient in the clay barrier. The hydraulic permeabilities and the dispersion coefficients were excluded since only diffusion was modelled. The head gradient was not considered due to the spatial discretisation of the argillite layer. 16

17 The dependence between specific parameters in the Benchmark was defined by the correlation coefficients shown in Table The correlation coefficients related to hydraulic conductivity were not considered in our analysis. Furthermore Goldsim in its present version does not allow a specific parameter to be correlated to more than one parameter. Due to these reasons only 5 correlations were implemented (see Table 1). Undisturbed argillite Micro-fissured zone Fractured zone De ω K De ω K De ω K De ω K De ω K De 0.7 Table : Correlated input parameters considered in the JRC calculations. Micro-fissured zone. The bold number as used by JRC. Those with strike-out were defined in [6] but not used by JRC. De = effective diffusion coefficient, K = permeability, ω = diffusive porosity Several physical constraints between the input variables introduced in [6] were not used in our calculations because the hydraulic conductivities were not applied. The considered constraints were implemented as follows: If De_micro-fissure De_undisturbed argillites then De_micro-fissured zone = De_undisturbed argillites, If De_fracture De_micro-fissure then De_fracture = De_micro-fissure, If porosity_micro-fissure porosity_undisturbed argillites then porosity_micro-fissure = porosity_undisturbed argillites, If porosity_fracture porosity_micro-fissure then porosity_fracture = porosity_micro-fissure. Certainly, implementing only a fraction of the actual correlations, together with this way of implementing the constraints, introduce a bias in the input sample, reshaping the joint distribution and overweighting the line that separates the region of possible values from the region of impossible values. Moreover, due to this way of implementing constraints, some of the properties of a LHS 17

18 sample do not hold any more. Imposing correctly constraints on joint distributions and multiple correlations, or in general dependencies, are issues to be dealt with in the next step of this benchmark. The probabilistic transport calculations provide the molar flow of the three radionuclides that come out from each one of the main barriers versus time for up to one million years. The uncertainty and sensitivity analysis has been carried out with the help of in-house developed Matlab programs. Graphic and numeric techniques have been used. All output variables mentioned in the benchmark specifications [6] have been studied in the uncertainty analysis (). Quantiles have been estimated using the corresponding order statistics; i.e.: y (950) the 950 th smallest observation of output variable Y has been used as the estimator of quantile 0.95 of Y, y (990) as the estimator of quantile 0.99 o Y and so on; (x (500) +x (501) )/2 has been used to estimate the median. Though confidence intervals could have been provided for all quantiles, they haven t been plotted in order to keep figures understandable and not overloaded with information. In addition to the statistics proposed, the standard deviation has also been computed as the main measure of spread. The evolution of all numeric statistics over time has been represented. Regarding graphic techniques, probability density functions (pdfs) have been estimated through the use of smoothing techniques, while probability distribution functions have been estimated using the empirical cumulative distribution function (ecdf). Since complementary cumulative distribution functions (ccdfs) and empirical cumulative distribution functions provide information that is 100% redundant, only cdfs are plotted. The sensitivity analysis (SA) developed in this work has been restricted to the use of multiple regression-based techniques. The coefficient of determination (R 2 ) has been used as the main reference to check the quality of the regressions computed. Low values of R 2 (close to 0) mean that the linear sensitivity analysis model used is too simple (and not appropriate) to analyse the model under study, high R 2 values (close to 1) mean that the SA technique is adequate to perform the SA study. Regressions have been computed on the raw values and also on their ranks. The sensitivity indices used have been the standardised regression coefficients (SRCs) and the Standardised Rank Regression Coefficients (SRRCs). Neither Partial Correlation Coefficients (PCCs) nor Partial Rank Correlation coefficients (PRCCs) have been used due to the overlapping information that both sets deliver. The evolution of SRCs, SRRCs and R 2 s over time is represented in different figures for dynamic variables in order to determine the input parameters that have the strongest impact at different times. The information has been given in tables for the peak flows and the time to the peak. 18

19 4. Results 4.1. ANDRA Uncertainty Analysis Results of I129 Preliminary analysis of main physical process in radionuclide transfer Figure shows the complementary cumulative density function (ccdf) of the simulations versus the Peclet number in the clay layer. The lower the Peclet number is the more diffusion dominant is the migration. If Pe < 2 we consider the migration to be diffusion controlled. From Figure it follows that 98% of the simulations have a Peclet number below 2 and hence confirms that the main radionuclide transport phenomenon in the clay is diffusion. An additional analysis of diffusion travel times shows that transfer of aqueous solutes is very slow and limited, with very long travel times that range from several hundreds of thousands of years to many tens of million of years. Hence advection has a very minor role. L. Kv( COX ). gradh Pe = De( COX ) L Diffusion Convection + Diffusion Figure : Ccdf of Peclet number in undisturbed argillites layer 19

20 Quantiles in time For each surface, the results for all the 1000 simulations are plotted versus time. Figure plots the individual simulations for the molar rate through a surface located at 15 meters from the waste packages. This heavy graph enables: to focus on some calculations whose result might appear strange from a physical point of view (bad convergence, ), in more of systematic numerical checking, to evaluate the large variability of the results both in time and on the levels of releases, to have a first evaluation of the relevant parameters on the results and to identify representative runs and associated curves (extreme, medium, minimal). PARAMETER De (COX) VALUE 4,5e-12 m²/s De(COX) Kd (COX) Kv (COX) Kd (concrete) 0 (retard.coeffic = 1) 1,3e-13 m/s 4e-5 (retard.coeffic ) Kd(COX) ω acc (COX) 0,03 Molar flux [mol/yr yr] Run 175 Great variability in time PARAMETER VALUE De (COX) 1,8e-13 m²/s Kd (COX) 1 e -3 (retard.coeffic = 25) Kv (COX) 1,3e-13 m/s Kd (concrete) 3e-4 (retard.coeffic = 3 ) ω acc (COX) 0,05 Run 247 Time [yrs[ yrs] Figure : Simulations of molar flow coming out of an intermediate surface in clay 15 m from the waste package versus time FluxCOX 15m From Figure we note that there is a large variation in the time at which the peak molar flow is attained as well as in the peak value itself. It can also be noted that for many simulations the maximum of release has not been reached at one million of years. Focusing on some sets of input data it can be shown that high molar rates are associated with high values of diffusion and low retardation coefficients, and the opposite for the attenuated molar rate (example of runs 175 and 247). On the other hand, it appears that the variability of the results is not very sensitive to other parameters. Specific sensitivity carried out should confirm this particular topic. 20

21 Figure and Figure give the evolution of quantiles of molar rates coming out of waste packages, the EBS (concrete disposal cell), the micro-fissured zone (entering undisturbed argillites), at the intermediate surface at 15m from waste packages, and at the clay layer (top and bottom). Some direct observations are: The abrupt increase in molar rate is due to model of degradation of the concretes, very simplified for these calculations (switch of model at years increase 4 orders of magnitude on the diffusion coefficient). The concrete degradation would be a long-term process rather than a sudden event and the peak in the flow would be drastically smaller. The uncertainty of the input parameters (hydraulic/migration and adsorption) of concretes generates an important dispersion of the results; for example, at 1000 years, between 1st and 99th centile, there is a difference of 3 and 5 orders of magnitude on the results respectively out of the packages and the disposal cell. In both cases, however, the median and mean are close. The jump of release at years, due to the change in characteristics of the concrete, disappears when moving away from the disposal cell, due to long travel times of argillites, The dispersion of the results in the geological medium (clay layer) is even more important because it combines both variations of the parameters of concrete, EDZ and undisturbed argillites. Results in the clay layer indicate a strong capacity of argillites to attenuate the release of radionuclides; almost 50% of the curves have either significant release after one million years or a total attenuation before one million years. For the clay, 15 meters of migration is sufficient to get a significant dispersion of the results. Compared to the flow out of the micro-fractured zone, the median is attenuated by an order of magnitude and the peak value is attained 10 5 years later. 21

22 Waste packages EBS Figure : Quantiles in time of Molar Flow coming out of concrete (waste packages sortie colis and EBS Fluxsortie BO) Coming out of microfissured zone = Into undist. argillites 22

23 Coming out of intermediate surface (15 m from packages) Coming out of undisturbed argillites (top and bottom) Figure : Quantiles in time of Molar Flow coming in and out of micro-fractures zone (sortie zone microfissurée, undisturbed argillites at 15 m FluxCOX 15m and top/bottom Flux sortie COX) Distribution Moments The cumulative density function gives the probability that the random variable is less or equal to a specific value. The complementary density function gives the probability that the random variable is larger or equal to a specific value. For the molar flux in a repository one is more interested in the probability of exceeding a value and it is therefore more natural to use the ccdf. Figure and Figure plot the ccdf for the peak molar flux at the different parts of the repository. It is seen in Figure that the peak value for waste package and the EBS occurs clearly at 10,000 years. As mentioned above the peak value for maximal molar rate coming out clay layer has in many simulations not been reached at 1 million years. In this case the maximal value adopted in the analysis is the value at one million years. 23

24 X = maximal molar flux coming out of EBS P(X> 10-5 = 90%) probability of an «event» Moments (Log) Mean SD Skewness Kurtosis Value -2,78 0,37-0,4 2,13 Figure : Distribution of Maximal Molar Flow coming out of concrete (waste packages Fluxsortie colis and EBS Flux sortie BO) X = maximal molar flux coming out of argillites P(X < 10-8 = 50%) probability of an «event» Moments (Log) Mean Value -6,88 SD Skewness Kurtosis 0,95-3, m Figure : Distribution of Maximal Molar Flow coming in and out of undisturbed argillites, sortie zone microfissurée, FluxCOX 15m,Fluxsortie COX As indicated on the graph, ccdf curve show for example that there is a 50% probability that the maximum molar rate coming out of clay layer is larger than 10-8 mol/year. Moreover, calculations of the various moments provide information on the shape of the distribution of the result. For example, the negative skewness for the maximum molar at 15 m from the waste packages, indicates that it is skewed to the right. The kurtosis provides information about the flatness of the distribution. The greater the value, the more peaked is the distribution. For a normal distribution it is 3. The value 24 indicates a "pointed" distribution and borders of distribution that are not significant. 24

25 Results of Se79 Figure shows the evolution of the molar flow through several surfaces from waste packages to top and bottom of clay layer. As for I129, an abrupt increase in the molar rate occurs at years due to the model of concrete degradation. Thanks to the high level of retardation in concrete, the release is completely smoothed from the EBS, with maxima of release after years for more than 75% of numerical simulations. At the top and bottom of the clay layer, maximal molar rate exceeds mol/yr only for 20% of the simulations. Looking at the shapes of curves, we expect solubility limit to have less influence than Kd factor. A sensitivity analysis will be carried on relevant ranges of time, from 10 4 to 10 6 years for molar rate coming out of EBS, and from 10 5 to 10 6 years for molar rate coming out of clay layer. Figure shows the ccdf for the peak molar flux coming out of the EBS and the undisturbed clay. The peak molar flux for Se79 at these locations is several orders of magnitude than for I129. a) b) b) c) 25

26 d) e) Figure : Quantiles in time of Molar Flow coming in and out of undisturbed argillites a) waste package Fluxsortie colis, b) disposal cell Fluxsortie BO, c) fractured zone Fluxsortie zone fracturé, d) micro-fissured zone Fluxsortie zone micro-fissurée, e) undisturbed argillites argillites concrete Figure : ccdf Distribution of Maximal Molar Flow coming out of concrete (EBS) Fluxsortie BO and clay layer (top and bottom Fluxsortie COX) Results of Nb94 Figure gives the evolution in time of molar flow through several surfaces. Results show that release of Nb94 is totally attenuated in the micro-fissured zone. Given the short half-life (about years) and the long travel times (including retardation, more than several hundreds of thousands years) in EDZ, retardation, associated with diffusion, seems to play a major influence on the attenuation. Hence we don t consider any release from micro-fissured zone and thus, sensitivity analysis is carried out only in near field (see Figure ). We also should expect release rate of source term to be have a relatively large influence as durations and travel times are quite equivalent. 26

27 a) b) c) Figure : Quantiles in time of Molar Flow coming in and out of undisturbed argillites a) disposal cell Fluxsortie BO, b) fractured zone Fluxsortie zone fracturé, c) micro-fissured zone Fluxsortie zone micro-fissurée, Nb94 Figure : ccdf of Maximal Molar Flow coming out of concrete (EBS) Fluxsortie BO and clay layer (top and bottom) Fluxsortie COX Nb94 27

28 Sensitivity Analysis Scatter plots Results of I E E-07 Kd (clay) Scatter plots 1.0E E E E E E-13 Pearson coefficient = -0,3 Strong negative correlation {Maximal Molar Flux coming out of undisturbed clay} / {Input Data} 1.0E E E-06 Effective diffusion De (clay) E-06 Head gradient (clay) 1.0E E E E E E E E E E E E-14 Strong positive correlation Pearson coefficient = 0,4 1.00E-13 Pearson coefficient = E E E-14 No correlation 1.00E E E E E E E E E E E E-12 Figure : Scatter plots maximal molar flow coming out of clay layer Fluxsortie COX I129 Scatter plots are very useful to focus on local sensitivity (around scalar values within spectrum of variation of each input parameter, correlated with results). Figure gives an example of correlations between the maximal molar rate at top and bottom of clay layer and 3 input parameters. When the data in scatter plots fall within a narrow band, the correlation is strong (for instance between the diffusion and Kd); when it is scattered, the correlation is weak (example of the head gradient). According to the slope of scatter plots, we distinguish positive correlation (the indicator increases when the diffusion coefficient increases) from negative correlation (the indicator decreases when Kd sorption increases). This correlation (input/ouput) is also given by Pearson coefficient, calculated on a "global" way (taking into account all the spectrum of variation) 28

29 Sensitivity analysis in time Various statistical indicators of correlation and regression were plotted in order to get a good understanding how the uncertainty of the input parameters affect the uncertainty of the results over the studied time range. These indicators were plotted in the time ranges for which flow is significant. An analysis on extremely low values of flow (lower than mol/yr) has no practical relevance. Also for example, looking at quantiles Figure 4.1-4, sensitivity analysis carried out for molar rate coming out of clay layer is interesting only in time range from years to 1 million years. Figure represents rank and linearity sensitivity indicators between the Kd (adsorption) data and the molar flow rate out of clay layer: they all give the same tendencies to a few hundreds of thousands of years. Moreover, we can notice a good agreement for indicators from each group (rank or linearity): Spearman, PRCC, and SRRC for the rank indicators, Pearson, PCC and SRC for the linearity indicators. However, taking into account the large variability of most input data (several orders of magnitude of variation of uncertainty), sensitivity analysis is carried with rank indicators. Kd (clay) Rank indicators Kd (clay) Linear indicators Time (years) Time (years) Figure : Rank and linearity indicators between Kd (COX) and molar flow out of clay layer Fluxsortie COX I129 Figure and Figure check and confirm the first results observed on scatter plots: strong correlation between the uncertainty of molar flow and Kd (adsorption) and De (diffusion of clay layer), respectively positive and negative. The head gradient has no influence on the results. 29

30 De (clay) Head Gradient Rank indicators Rank indicators Time (years) Time (years) Figure : Evolution in time of statistical rank indicators for molar flow out of clay layer Fluxsortie COX I129 PRCC sensitivity analysis carried out on the whole set of input data in time shows that between years and 1 million years only diffusion De and Kd sorption are important. The contribution in term of variance of the other parameters is negligible. Evolution in time of PRCC rank indicator {Molar Molar Flux coming out of undisturbed argillites / ALL input data} De (COX ) Kd (COX) Time (years) Figure : Evolution in time of PRCC rank indicator for all input data Molar Flow coming out of undisturbed argillites Fluxsortie COX I129 Sensitivity analysis coefficients applied to maximal values Main results In order to evaluate the influence of the various parameters in near and far field, sensitivity analysis is carried out on the maximal molar rate through various surfaces of the system (global timeless analysis). According to previous results on sensitivity indicators the values of the different coefficients 30

31 are very similar. We therefore only consider the average of the rank coefficients, Spearman PRCC, and SRRC as main sensitivity indicator. Molar rate out of concrete Figure indicates a ranking of the key input parameters on the maximum molar rate coming out of disposal cell (concrete). It highlights the influence of parameters of the medium located just after the concrete; indeed, diffusion and Kd of fractured zone have a major influence on the gradient of concentration at the interface between the two zones, and thus molar rate out of concrete (the higher the Kd fractured zone is, the higher the molar rate is). The diffusion process is very dominant, so results are quite independent of the permeability. It is a bit surprising that the effective diffusion coefficient should be the most important parameter. This is linked to the model assumption that the diffusion properties change by several orders of magnitude at 10,000 years when the peak flow is attained. Finally, input parameters for which the uncertainty has a major influence on the uncertainty of the result are, by decreasing order of importance: diffusion of the high performance concrete, Kd and diffusion of the fractured zone, and finally Kd of the concrete. These results can also be useful in case of safety altered scenarios, such as borehole scenario (no geological barrier involved in migration). Sensitivity analysis on Maximal molar flux coming out from EBS (concrete( concrete) No influence of other input data Uncertainties of of :: --High Highperformance concrete concrete (diffusion, retardation) --Fractured zone zone (diffusion, retardation) --Concrete (retardation) are are the themost mostinfluent on on the theuncertainty of ofthe themaximal molar molarflux flux Kd (concrete( concrete) De (fractured( zone) Kd (fractured( zone, COX) De (high perf. 0,00 0,10 0,20 0,30 0,40 0,50 Mean (abs(rank statistical indicators)) 0,60 concrete) 0,70 Spearman/PRCC/SRRC Figure : Sensitivity analysis on maximal molar flow coming out from EBS (concrete) Fluxsortie BO I129 31

32 Molar rate out of clay layer Figure indicates a ranking of the most influent input parameters on the maximum molar rate coming out of clay layer. The influence of parameters of the materials and media in the near field (concrete and EDZ) is very small due to the very long migration times in the clay layer. The only input parameters whose uncertainty has a major influence on the uncertainty of the result are, by decreasing order of importance: Kd (sorption) and De (effective diffusion) of the argillites. No influence of other input data Uncertainties of ofundisturbed argillites (diffusion, retardation) are are the the most most influent on on the the uncertainty of of the the maximal molar molar flux flux De (undisturbed COX) Kd (COX) 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 statistical indicators)) Spearman/PRCC/SRRC Mean (abs(rank( statistical indicators)) Figure : Sensitivity analysis on maximal molar flow coming out of clay layer sortie COX I Results of Se79 Molar rate out of concrete Figure gives PRCC rank sensitivity indicator between each input data with uncertainty for Se79 and maximal molar rate coming out disposal cell (EBS). Figure gives sensitivity analysis on maximal molar rate. The main observations are: the uncertainties of the geochemical input data (concrete and clay) have a major influence in time on the uncertainty of the molar rate. Levels of Selenium Kd (and its uncertainties) in concrete are sufficient to increase travel time within the concrete and to keep a level of concentration always under concrete s solubility limit. Thus, concrete solubility limit has no influence on the result. On the other hand, the geochemical parameters Csat, Kd and diffusion 32

33 in the fractured zone have a major influence on the gradient of concentration at the interface between the two zones, and thus molar rate out of concrete. The reason is that the higher the Kd fractured zone is, the higher the more molar rate is; and the higher the Csat fractured zone is, the lower the molar rate is. The influence of concrete input data is more important at earlier times than at later ones. Input parameters whose uncertainty has a major influence on the uncertainty of the molar flux are, by decreasing order of importance: Kd of concrete, Kd and Csat of fractured zone, diffusion of argillites. De(high perf. concrete) Kd(fractured zone, COX) De(fractured zone) De(COX) Csat(fractrured zone, COX) Kd(concrete) Time (years) Figure : evolution in time of PRCC rank indicator for all input data (molar flow coming out of concrete disposal cell Fluxsortie BO) Se79 Sensitivity analysis on Maximal molar flux coming out from EBS (concrete) No influence of other input data Uncertainties of of :: --Concrete Concrete (retardation) --Fractured zone zone (retardation, solubility) --Undisturbed argillites argillites (Diffusion) are are the the most most influent influent on on the the uncertainty of of the the maximal maximal molar molar flux flux S De (COX) Csat (fractured zone, COX) Kd (fractured zone, COX) Kd (concrete) 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 Mean (abs(rank( statistical indicators)) Spearman/PRCC/SRRC Figure : Sensitivity analysis on maximal molar flow coming out from EBS (concrete) Fluxsortie BO Se79 33

34 Molar rate out of clay layer Figure gives the PRCC rank sensitivity indicator between each input data related to Se79 and the molar rate coming out of clay layer (top and bottom). Figure gives sensitivity analysis on the maximal molar rate. The influence of parameters of the materials and media in the near field (concrete and EDZ) is not visible any more, except for Kd concrete, for which in may simulations the travel time in the concrete is similar to that of the clay. This means that Kd concrete plays an important role even far from waste packages. Input parameters whose uncertainty has a major influence on the uncertainty of the result are, by decreasing order of importance: Kd (sorption) and De (effective diffusion) of argillites, Kd of concrete and solubility limit of Se79 in argillites. Csat(COX) De(COX) Kd(concrete) Kd(COX) Figure : evolution in time of PRCC rank indicator for all input data (molar flow coming out of clay layer top and bottom Fluxsortie COX) 34

35 Sensitivity analysis on maximal molar flux coming out of clay layer No influence of other input data Uncertainties of of -undisturbed argillites (diffusion, retardation, solubility limit) limit) --concrete (retardation) are are the the most most influent on on the the uncertainty of of the the maximal molar molar flux flux Csat (COX) Kd (concrete) De (COX) Kd (COX) 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 Spearman/PRCC/SRRC Mean (abs(rank( statistical indicators)) Figure : Sensitivity analysis on maximal molar flow coming out of clay layer Fluxsortie COX Se Results of Nb94 Figure gives PRCC rank sensitivity indicator between each input data related to Nb94 and molar rate coming out disposal cell (EBS). Figure presents a sensitivity analysis on maximal molar rate. Main results are as follows: Kd of the concrete is the input data whose uncertainty has the largest influence in the studied time range on the uncertainty of the molar rate. In fact PRCC nearly equals to one from 8000 years to years. The values of Kd (and its uncertainties) in concrete, associated with diffusion, limit the release and delay the maximum of release. Attenuation is increased thanks to the short half-life. In many cases, release rate from waste components also affect the level of release when the duration is similair the travel time. This topic especially concerns ILW2, whose initial activity (inventory) and range of uncertainty is much more important than for ILW1. As it is the case with Se79, the Nb94 solubility limit has no influence on the result because of high Kd values in concrete and clay. In fact concentration of Nb94 solute never exceeds solubility limit in this area. Parameters of the medium located just after the interface (Kd and diffusion of fractured zone) have also a major influence on the results (see comments in Se79 section). 35

36 Finally, input parameters whose uncertainty has a major influence on the uncertainty of the maximal molar rate are, by decreasing order of importance: Kd of concrete, Kd of fractured zone, diffusion of fractured zone, release rate of stainless steel from ILW2 waste packages. De (high perf. concrete) De (fractured zone) Kd (COX) Release rate stainless steel (ILW2) Release rate inconel steel (ILW2) Kd (concrete) Time [years] Figure : evolution in time of PRCC rank indicator for all input data (molar flow coming out of EBS) Fluxsortie BO Nb94 Sensitivity analysis on Maximal molar flux coming out from EBS (concrete) No influence of other input data Uncertainties of of :: --Concrete (retardation) --Fractured zone zone (retardation, diffusion) --release release rate rate (stainless steel) steel) are are the the most most influent on on the the uncertainty of of the the maximal molar molar flux flux Release rate (ILW2, stainless steel) De (fractured zone) Kd(fractured zone, COX) Kd (concrete) 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Mean (abs(rank( statistical indicators)) Spearman/PRCC/SRRC Figure : Sensitivity analysis on maximal molar flow coming out of EBS Fluxsortie BO Nb94 36

37 4.2. JRC-IE results Uncertainty analysis results The target of the uncertainty analysis () is to get as much information as possible about the output variables under study. In total eighteen dynamic output variables and thirty-six non-dynamic output variables are considered. The dynamic eighteen output variables are the molar flows for each radionuclide ( 129 I, 94 Nb and 79 Se) at any time before 10 6 years getting out through the external surface of each engineered and geological barriers considered in the repository design. Six surfaces are considered: the external boundary of the waste packages (WP), the external boundary of the disposal cell (DC), the external boundary of the fractured zone (FZ), the external boundary of the micro-fissured zone (MZ), two horizontal layers (considered as only one concerning the flow getting out through it) within the undisturbed argilites 15 m. above and below the centre of the disposal cell (15m) and the two horizontal upper and lower layers bounding the undisturbed argilites geological formation (). The non-dynamic output variables are the peak flows and the time to the peak flows for each radionuclide and each barrier. Special attention will be paid to the results obtained for 129 I, given the relevance of this radionuclide. Main graphic and numeric results will be included in this section, while supporting additional material is included in Appendix 7.2 and will be referred to as needed (Tables 7.2-x and Figures 7.2-x). A lower threshold of mol y -1 has been set for all graphic representations of uncertainty (not for numeric - tables). Figure shows the molar flow evolution over time for the three radionuclides getting out of the six layers (1000 curves per radionuclide). The first remarkable fact is the effect of radioactive decay on 94 Nb. This radionuclide disappears completely in all layers for times beyond y; in fact no 94 Nb contamination gets out of the in the whole period and only negligible quantities reach 15m, see Figure and Table and Table in addition to Figure The second remarkable fact is the effect of changing the effective diffusion coefficient at 10 4 y. A sudden increase in the molar flow, which produces sharp peaks, can be seen for the three radionuclides at that time in the first three barriers. Such effect can also be seen in MZ and 15m but only for 79 Se, see Figure and Figure The effect of the distribution coefficient of 129 I in the beton de structure can also be seen in figures Figure b and Figure c. The split of the 10 3 runs in two sets, corresponding to the 37

38 two parts of the probability distribution function used to define the mentioned input parameter, is obvious. For each simulation 209 time steps have been saved. The time discretization is coarse than for ANDRA. This can be clearly seen at the 10,000 when the concrete properties change abruptly resulting in a peak in the flow rate. At this time step JRC used 9800, and years as opposed to X years by ANDRA, the peak flow for the waste package and the disposal cell are therefore smaller in the JRC computation. Figure shows Molar flow statistical indicators evolution over time for the 129 I getting out of the six layers. A general trend observable in this figure is the shift of the molar flows to lower values and later times as the contaminant arrives at more external layers, so it can be seen that at in WP the 99% quantile line reaches a maximum close to 10-4 mol y -1 at circa y, while in FZ the same line reaches a maximum close to 10-5 mol y -1 at circa y, and in it is close to 10-7 mol y -1 at circa y. Moreover, flows go across the mol y -1 threshold in only after 10 4 y, while they widely exceed that threshold in WP since the very beginning of the simulations. This has to do with the fact that the longer the contaminant takes to get though the layer the more the contaminant is spread over time. The flow distributions of 129 I are not very skewed and are quite symmetric, except for and 15m at early times (before 10 5 y); most of the time the standard deviation is smaller that the mean and the median is also close to it. Another important fact that deserves to be remarked is the existence of a time for each barrier, after 10 4 y, when the uncertainty in orders of magnitude reaches a minimum. This happens at approximately y in FZ, at approx y in MZ, and later than 10 6 y in 15m and. Note that there is a bottle neck, or narrowest point, in the uncertainty bands defined in each plot. This is related to the existence of a crossing point of realisations, when runs producing early flows go across runs producing late flows, see Figure (specially c, d, e and f). The flow distributions of 94 Nb and of 79 Se do not show so good characteristics (see Figure 7.2-2, Figure and Table 7.2-3, Table 7.2-5). In the case of these two radionuclides, with the partial exception of the two internal layers, the distributions are quite skewed towards high values (large positive values of the skewness coefficient). The standard deviations are larger than the means most of the time, sometimes up to one order of magnitude larger, and the means are systematically larger than the medians. Maximum values are most of the time one order of magnitude larger than the means and in many cases more than two orders of magnitude larger. Moreover, kurtosis are in many cases huge. Figure Figure and Figure Figure and Table Table provide all the information concerning the uncertainty analysis for the dynamic output variables. 38

39 a) b) c) d) e) f) Figure : Molar flow evolution over time for the three radionuclides getting out of the six layers (1000 curves per radionuclide); from left to right and from top to bottom : a) waste package Φ sortie colis, b) disposal cell Φ sortie BO, c) fractured zone Φ sortie zone fracturé, d) micro-fissured zone Φ sortie zone micro-fissurée, e) undisturbed argilites (15 m) Φ COX 15m, f) undisturbed argilites,φ sortie COX. 39

40 a) b) c) d) e) f) Figure : Molar flow statistical indicators evolution over time for the 129 I getting out of the six layers; from left to right and from top to bottom : a) waste package Φ sortie colis, b) disposal cell Φ sortie BO, c) fractured zone Φ sortie zone fracturé, d) micro-fissured zone Φ sortie zone micro-fissurée, e) undisturbed argilites (15 m) Φ COX 15m, f) undisturbed argilites, Φ sortie COX. The last part of the uncertainty analysis is dedicated to the non-dynamic output variables. contains the empirical cumulative distribution functions (ecdfs) for the 18 peak flows under study and their associated time to the peak flows. More extensive information is provided in Figure 7.2-5, Figure and Table and Table A first remark should be done. Computations with Goldsim have 40

41 been obtained for 208 fixed time steps, the same in all runs. When estimating peaks, the maximum value among the 208 available values per run has been chosen as the peak flow; no interpolation has been computed. Furthermore, the time step corresponding to the maximum value selected has been considered as the time to the peak flow for that run. Proceeding in such a way has certainly introduced a small bias in the estimation of the peak flows, in particular for the 10,000 change in properties. It has also forced the time to the peak flow output variables to take only a finite number of different values (always much fewer than 208), which will affect the uncertainty analysis results and the sensitivity analysis results. The effect of taking only a limited number of time values on the time to the peak flows may also be seen in Figure b), d) and f), where the corresponding ecdfs have a stair-like shape (see also Figure b) to realize the effect on the smoothed empirical probability density functions). This has also been the cause of getting all the peaks for 79 Se in WP at y, so that the time to the peak had null standard deviation in this case and the coefficient of skewness and kurtosis were not computable. The distributions of peak flow shift towards smaller values as we move to outer layers, while the distributions of time to the peaks shift to larger values. This is a general trend observed in the three radionuclides. Peak values for 94 Nb sharply decrease as we move to outer layers, becoming negligible beyond the MZ. The decrease is also sharp for 79 Se when passing from the WP to the DC, but later on it is smooth. The decrease is smooth always in the case of 129 I. The sharp decrease for 94 Nb has much to do with its small half-life. Table B.1 also shows that in a few runs (less than 1%), neither 129 I nor 79 Se produce any flow at all in. Figure shows that, for all radionuclides, the dispersion in terms of orders of magnitude of the peak flow distributions increases as we move to outer layers. The opposite is true when the dispersion is measured in absolute terms, through the standard deviation. The dispersion of the distributions of the time to the peak flows grows in absolute terms as we move to outer layers. It is also important to remark that in, the flow was still growing (true peak not reached) when the simulation stopping time was reached in more than 60% of the runs for 129 I and in more than 70% of the runs for 79 Se. This is also true in a few percent of the cases, between 1% and 5%, for both radionuclides in 15m. 41

42 ECDF waste package disposal cell fractured zone micro fissured zone undisturbed argilites (15 m) undisturbed argilites log (max(release)) a) b) ECDF time [yrs] waste package disposal cell fractured zone micro fissured zone undisturbed argilites (15 m) undisturbed argilites ECDF waste package disposal cell fractured zone micro fissured zone undisturbed argilites (15 m) undisturbed argilites ECDF waste package disposal cell fractured zone micro fissured zone undisturbed argilites (15 m) undisturbed argilites e+00 2e+05 4e+05 6e+05 8e+05 1e+06 log(max(release)) c) d) time [yrs] ECDF waste package disposal cell fractured zone micro fissured zone undisturbed argilites (15 m) undisturbed argilites ECDF waste package disposal cell fractured zone micro fissured zone undisturbed argilites (15 m) undisturbed argilites log(max(release)) 0e+00 2e+05 4e+05 6e+05 8e+05 1e+06 e) f) Figure : Empirical cumulative distribution function for the peak flows of the 3 radionuclides getting out of the six layers : a) 94 Nb, c) 129 I, e) 79 Se; and for the corresponding time to the peaks: b) 94 Nb, d) 129 I, f) 79 Se. Time in logscale. time [yrs] As in the case of the flows over time, it can be seen in Figure and Figure and in Table that the peak flow distributions for 129 I are more symmetric and much less spread in terms of orders of magnitude than the peak flow distributions for the other two radionuclides. The standard deviations of 42

43 all the peak flow distributions are larger than the corresponding means for 94 Nb and 79 Se, while they are smaller in the case of 129 I. Skewness coefficients are also quite larger in the case of 94 Nb and 79 Se than in the case of 129 I, and the same happens with the kurtosis. In general, the peak flow distributions for 129 I show better properties from the point of view of the estimability of its characteristic parameters Sensitivity analysis results The target of the sensitivity analysis is to get as much information as possible about the impact of the different input uncertainty parameters on each output variable. Eighteen dynamic output variables and 36 non-dynamic output variables are studied in this work. The dynamic variables are the flows getting out of the six layers over time for the three radionuclides considered. The thirty six non-dynamic variables are the peak flows for each layer and each radionuclide and the corresponding times to the peak flows. Twenty four input uncertainty parameters are considered in the analysis (all the uncertain input parameters described in reference 6, excluding permeabilities). All the results of the sensitivity analysis are collected in appendix 7.2.2; Figure to Figure report results for dynamic output variables and Table to Table report results about the non-dynamic ones. In order to keep the text of this chapter simple, the following coded names have been used for the input parameters: Name Radionuclide release rate in stainless steel (B1) Radionuclide release rate in inconel (B1) Radionuclide release rate in coques (B5) Radionuclide release rate in stainless steel (B5) Radionuclide release rate in inconel (B5) Effective diffusion coefficient in béton de colisage Iodine distribution coefficient Kd(I) in béton de structure Niobium distribution coefficient Kd (Nb) in béton de structure Selenium distribution coefficient Kd (Se) in béton de structure Niobium solubility limit in béton de structure before y Niobium solubility limit in béton de structure after y Selenium solubility limit in béton de structure before y Selenium solubility limit in béton de structure after y Effective diffusion coefficient in the fractured zone Diffusive porosity in the fractured zone Effective diffusion coefficient in the micro fissured zone Diffusive porosity in the micro fissured zone Effective diffusion coefficient in the undisturbed argilites Diffusive porosity in the undisturbed argilites Iodine distribution coefficient Kd(I) in the undisturbed argilites Niobium distribution coefficient Kd (Nb) in the undisturbed argilites Selenium distribution coefficient Kd (Se) in the undisturbed argilites Niobium solubility limit in the undisturbed argilites Coded name rb1ac rb1in rb5co rb5ac rb5in DeWp Kd I DC Kd Nb DC Kd Se DC Cs1 Nb DC Cs2 Nb DC Cs1 Se DC Cs2 Se DC DeFZ PorFZ DeMZ PorMZ De Por Kd I DC Kd Nb DC Kd Se DC Cs Nb 43

44 Selenium solubility limit in the undisturbed argilites Cs Se Regression analyses based on the raw values and on the ranks have been performed for all dynamic and non-dynamic output variables. The first generic result shows that the quality of the regressions, measured in terms of their coefficients of determination (R 2 ), is much better when the ranks of the values are used instead of the raw values in most of the cases (compare R 2 values in left and right columns in Figure to Figure and Table to Table 7.2-9). This is the reason to base all our analyses on the study of the SRRCs rather than the SRCs. This means that, in most of the cases, it is convenient to analysis the dependence between outputs and inputs from the point of view of monotonic relations instead of from the point of view of linear relations. Results for dynamic outputs are always reported for the period 10 to 10 6 y. Figure shows the results of the regressions based on the ranks of the values for the molar flow of 129 I getting out of the six layers. It can be noted that parameters associated with the other radionuclides, such as Kd Se show up in I129 plots. There should of course be no physical relation between the Kds for different radionuclides.; this is just an artifact from including all input parameters in the regression analysis. The effect of the degradation of the concrete properties at 10 4 y can be seen in Figure a) to d). A discontinuity in R 2 and in the most relevant parameters may be seen in the first three figures, while in the fourth one slope changes happen. No specific effect can be seen in the results for 15m and for. The second general comment about the results for 129 I over time is about the existence of a minimum in the value of R 2. In the six plots, at a given time, R 2 reaches a minimum. This happens at approx y in WP, at approx y in DC and so on. Only in this minimum does not happen. It is obvious the shift to later times of this minimum as we look at outer layers. This minimum is always located at a time when SRRCs associated to the most relevant input parameters switch their sign. Figure does also show that, regarding 129 I, the most important parameters are the distribution coefficients of this radionuclide in the béton de structure (Kd I DC) and in the undisturbed argilites (Kd I ), and the effective diffusion coefficient in the béton de colisage (DeWP) and in the undisturbed argilites (De), though their importances change over time and from one to another layer. The 129 I flow getting out of the waste packages (Figure a)) is most affected at early times (before 10 4 y) by Kd I DC and DeWP. After that time Kd I becomes the most important parameter, followed by Kd I DC and De. It is also remarkable the change of the sign of the SRRCs for these input 44

45 parameters over time. A similar behaviour can be seen in Figure b) and Figure c) regarding the second and the third layers. Regarding the 129 I flow getting out of the micro fissured zone (figure Figure d), Kd I becomes the most important input parameter at all times, followed by Kd I DC and De. A change in the sign of the three SRRCs happens at approx y. In the next layer, Kd I becomes even more predominant. In the last layer, Kd I becomes almost the only important parameter, followed by De. All flows getting out of this layer were null before 500 y, that is why SRRCs are not represented for earlier times (Figure f)) De WP R R 2 SRRCs and R De De MZ -0.4 De FZ Kd I DC Kd I Time (years) a) b) SRRCs and R De WP De FZ Kd I DC De De MZ Kd I Kd Se Time (years) R R 2 SRRCs and R De MZ Kd I De WP De FZ De SRRCs and R Kd I DC Kd Se De De MZ -0.8 Kd I DC -0.8 Kd I Time (years) c) d) Time (years) 45

46 1 0.8 R R Kd Se 0.6 SRRCs and R Kd I DC De De MZ SRRCs and R Kd Se De Kd I Time (years) Time (years) e) f) Figure Evolution over time of 129 I molar flows getting out of the six layers a) waste package Φ sortie colis, b) disposal cell Φ sortie BO, c) fractured zone Φ sortie zone fracturé, d) micro-fissured zone Φ sortie zone micro-fissurée, e) undisturbed argilites (15 m) Φ COX 15m, f) undisturbed argilites,φ sortie COX.. Kd I R R SRRCs and R De WP Kd Nb SRRCs and R De FZ De Wp Cs1 Nb DC Kd Nb -0.4 Cs1 Nb DC Kd Nb DC Time (years) a) b) -0.8 Kd Nb DC Time (years) R R 2 SRRCs and R De MZ De FZ De WP Cs1 Nb DC Kd Nb SRRCs and R Cs1 Nb DC Kd Nb DC -0.8 Kd Nb DC Time (years) -0.8 Kd Nb Time (years) c) d) Figure : Evolution over time of 94 Nb molar flows getting out of the first four layers a) waste package Φ sortie colis, b) disposal cell Φ sortie BO, c) fractured zone Φ sortie zone fracturé, d) micro-fissured zone Φ sortie zone micro-fissurée,. 46

47 As in the case of 129 I, the effect of the degradation of the concrete properties at 10 4 y on the 94 Nb flows can be seen in Figure a) to Figure c). The effect does not affect 94 Nb flows getting out of the micro fissured zone. It is also remarkable the very high quality of the regressions obtained for the second third and fourth layers (R 2 close to 1) for most of the times. This is mostly due to the fact that SRRCs associated to most important input parameters do not change their signs over time and they have very high absolute values. In the first layer, the two most important input parameters change their SRRCs signs at approx y. Results for 94 Nb in 15m and for are not reported in here because in the former case results are very similar to those obtained in MZ (see Figure b)) while in the latter case all flows are null and no SA results could be obtained (see Figure b)). The most important parameters regarding the flow of 94 Nb are the distribution coefficients of this radionuclide in the béton de structure (Kd Nb DC) and in the undisturbed argilites (Kd Nb ), the effective diffusion coefficient in the béton de colisage (DeWP) and the Nb solubility limit (Cs1 Nb DC), though DeWP is moderately relevant only for the results in the first layer at early times. Cs1 Nb DC is moderately important at intermediate times in the first layer and its importance is dramatically reduced as we look at outer layers. Kd Nb DC is the most important input parameter in the first three layers, and it losses its primacy in favour of Kd Nb only in the fourth and fifth layers. As in the case of 129 I and 94 Nb, the effect of the degradation of the concrete properties at 10 4 y on the 79 se flows can be seen in Figure The effect does not affect 79 Se flows getting out of the sixth layer (external boundary of the undisturbed argilites). It is also remarkable the very high quality of the regressions obtained for the first five layers (R 2 close to 1) at early times (before 10 4 y). There is a general decrease in the quality of the regressions after that time in all layers except in the sixth one and, partially, in the first one. This is mostly due to the fact that SRRCs associated to most important input parameters decrease at late times. Only for the flow getting out of WP and of after 10 4 y R 2 remains high. 47

48 De WP R R 2 SRRCs and R Kd Se De De FZ Cs Se SRRCs and R De WP De FZ Kd Se De Cs Se Kd Se DC -0.8 Kd Se DC Time (years) a) b) Time (years) SRRCs and R De WP Kd Se De FZ Cs Se R 2 De SRRCs and R Kd Se DC R 2 Cs Se De -0.8 Kd Se DC -0.8 Kd Se Time (years) c) d) Time (years) R R 2 SRRCs and R Kd Se DC De Cs Se SRRCs and R De Cs Se Kd I Kd Se Time (years) Time (years) Kd Se e) f) Figure : Evolution over time of 79 Se molar flows getting out of the six layers a) waste package Φ sortie colis, b) disposal cell Φ sortie BO, c) fractured zone Φ sortie zone fracturé, d) micro-fissured zone Φ sortie zone micro-fissurée, e) undisturbed argilites (15 m) Φ COX 15m, f) undisturbed argilites,φ sortie COX. Figure shows that, regarding 79 Se, the most important parameters are the distribution coefficients of this radionuclide in the béton de structure (Kd Se DC) and in the undisturbed argilites (Kd Se ), the 48

49 effective diffusion coefficient in the béton de colisage (DeWP) and in the undisturbed argilites (De), and the solubility limit of Se in the undisturbed argilites (Cs Se ). Kd Se DC and Kd Se are the two most important input parameters affecting 79 Se results. Kd Se DC is the most important one regarding the results for the first three layers while it switches to the second position in the three outer layers, in favour of Kd Se. DeWP is moderately important only for the flow of 79 Se getting out of the first three layers until 10 4 y. It is completely irrelevant after that time and has no impact on the results for the flow getting out of the three outer layers. Kd Se DC and Kd Se. Cs Se becomes moderately important in all layers only after 10 4 y. Tables Table Table show the results of the regressions performed for the peak flows getting out of each layer and the times to the peaks. The first two tables report results based on the raw values while the last to report results based on the ranks of the values. As it was in the case in the previous analyses, the R 2 values obtained in the case of analysing raw values are quite smaller than the ones obtained for the ranks. Again, this analysis will be based on the study of the monotonic relations instead of linear relations (tables Table and Table will be referred to in the next paragraphs). Regarding 129 I peak flow, Kd I DC is the most important parameter in the inner layers, while Kd I followed by De are the most important parameters in the outer layers, see Table Table DeWP is the most important parameter in WP. Regarding the times to the peak for the same radionuclide, Kd I DC and DeWP are the most important parameters in the internal layers while Kd I is the only parameters that really matters in the outer layers. Regarding 94 Nb peak flows, only the Kds have a relevant impact, Kd Nb DC in the inner layers and Kd Nb in the outer layers. The parameter that affects most the times to the peak flows of 94 Nb in the four intermediate layers is Cs1 Nb DC, followed by Kd Nb DC. For the first layer results are neither reliable nor significant and in the last layer peak flows were all null. The parameter that affects most 79 Se peak flow in the first five layers is Kd Se DC. In intermediate layers, this parameter is followed in order of importance by Cs Se, De and Ks Se. Ks Se and De are the most important parameters in the last layer, see Table Table The times to the peak flow is controlled by Cs Se in WP, by Kd Se DC in the next two layers and by Kd Se in the three outer layers. 49

50 Comparison JRC/ANDRA Results Before comparing the results from ANDRA and JRC we would just summarize the main differences in the modelling. ANDRA models all aspects described in the Benchmark; it is a 2D-model with all transport modes included and all constraints implemented. JRC uses a one-dimensional geometrical description; diffusion is the only considered transport mode and only a limited set of the constraints were actually implemented. Even though there is quite a large difference in the level of complexity to radionuclide migration the computed molar flows are quite similar. For I129 this can be seen by comparing Figure and Figure from ANDRA with Figure for JRC. The peak values and the time when it occurs are quite similar. The exception is that the peak in molar flow from waste package and the disposal cell are higher for ANDRA than JRC. This difference is caused by the difference in time discretization around years as discussed above. There is also a good correlation between ANDRA and JRC for Se79 and Nb94 (compare Figure and Figure for Se79 and Figure and Figure for Nb94). The cumulative distribution functions are a bit more difficult to compare since JRC plots the cumulative probability density function (ecdf) whereas ANDRA plots its complement (ccdf = 1-ecdf). The 50% probability is however the same and can be directly compared. The peak flow from the waste package and to a lesser extent from the disposal cell is caused by the years transition, which is higher for ANDRA than for JRC. This is clearly seen by comparing Figure from ANDRA (about moles/year) and Figure c) from JRC (about moles/year). As we move away from the waste package and the years peak gets attenuated, the JRC and ANDRA analyses converge and at top and bottom of the undisturbed argillite the 50% probability for the molar flow is 10-8 moles/year in both cases. The rank of the key input parameters is similar for molar flow of I129 getting out of the ; the Kd of the argillite is the most important parameter followed by the argillite s diffusion coefficient. The agreement is also quite good for Se79 with Kd Se, De, Cs Se being the most important parameters. For the ranking of the parameters for the peak flow the difference between JRC and ANDRA is much larger for the waste package and the disposal cell; compare for instance Figure and Table 7.2-8). This is also expected since the peak value is affected by the discretization of the transition at years. There is good agreement for the ranking of the input parameters that affect the molar flow in the clay and their indicator values. It can be noted though that JRC gives systematically higher values for Kd. The reason for that could not be clarified. 50

51 5. Conclusions and Perspectives This report describes the first Benchmark of a joint study between ANDRA and JRC to apply advanced methods for uncertainty and sensitivity analysis of a clay repository. The molar flow rates in different parts of the repository system for three radionuclides (I129, Se79 and Nb94) have been singled out as the output. The analysis is based on the Monte Carlo simulation technique. The input parameters that control the radionuclide migration in the analysis have been derived as part of the French research programme for radioactive waste management. This first Benchmark will be followed by a second Benchmark. In fact it should be more seen as a pre-study and the synthesis of the two analyses have not been very detailed in this report. Nevertheless a number of conclusions can be drawn. The molar flow rate in the different parts of the system computed by JRC and ANDRA are very similar except for the peak value in the waste package and disposal cell. This peak value is caused by the stepwise change in the concrete properties at years, which gives a very high peak but with a short duration. ANDRA has a higher peak value because it has a much finer time discretization around this event. The computed sensitivity standard rank regression coefficients with respect to the molar flow for key input parameters are similar when plotted versus time. There is large difference though when looking at the peak flow rate out of the waste package and the disposal cell because of the difference in the time discretization at years. ANDRA s model is much more complex than JRC s in terms of geometrical description, transport processes and parameter constraints. The good agreement in results (except for the peak) between the more complex ANDRA and the simpler JRC analysis is because the molar flux is diffusion controlled and the distribution coefficient (Kd) and the effective diffusion coefficient (De) control the molar flow out of the clay layer. The Benchmark 1 is the basis for the second Benchmark. Additional aspects that we would like to address in the second Benchmark include: A more realistic description of the repository s geometry (Figure 2.1-1). In Benchmark 1 only the pathway into the host rock was modelled, but the the pathway along the disposal cell is also important. This requires a three-dimensional model. A three-dimensional analysis would of course require significantly more computational time and the number of simulations would therefore need to be restricted. The Monte Carlo simulation technique with 1000 runs or more is therefore not feasible. The idea is to 51

52 perform a limited set of 3D analyses to build response surface which can be used for a very large number of simulations. A comparison between the Monte Carlo technique and the response surface technique should be conducted. In addition to the regression techniques, local sensitivity techniques will also be explored. More advanced techniques for sensitivity analysis when scatter plot relations are not monotonic will also be tested. These include Sobol indices and Fourier Amplitude Sensitivity Test (FAST); these techniques generally require a very large number of data points. The parameter constraints are not compatible with the Latin Hypercube Sampling technique. Other sampling techniques for the Monte Carlo simulation such as stratified sampling will therefore also be investigated. The degradation of the concrete was modelled as an instantaneous event at years. This gives rise to a peak in the maximum molar flow from the waste package and disposal cell. In reality the degradation takes place over a long time and the associated peak in not physical. A more realistic description of the degradation, for instance linear over a specific time range, should be adopted in the second Benchmark. The objective of the two Benchmarks is to provide recommendations for the use of different methods for uncertainty and sensitivity analysis of a real clay repository. Hence much emphasis in the second Benchmark will be given to the practical guidelines. 52

53 6. References [1] PAMINA: Performance Assessment Methodologies in Application of to Guide the Development of the Safety Case, EURATOM Integrated project Contract No. FP , Annex 1 Description of Work. [2] Andra, Dossier 2005 Argile Synthesis report : Evaluation of the feasibility of a geological repository (2005) [3] Andra, Dossier 2005 Argile Safety evaluation of a geological repository (2005) [4] Andra, Dossier 2005 Argile Phenomenological evolution of a geological repository (2005) [5] Andra, Dossier 2005 Argile Architecture and management of a geological repository (2005) [6] G. Pepin, F. Plas, S. Prváková and K-F Nilsson, Milestone First Benchmark specification for the uncertainty analysis based on the example of the French clay site, April

54 7. Appendices 7.1. ANDRA: Detailed results Sensitivity analysis I129 - Linear indicators 54

55 Figure : I129 - Evolution in time of linear indicators for effective diffusion of argillites 55

56 Figure : I129 - Evolution in time of linear indicators for effective diffusion of fractured zone 56

57 Figure : I129 - Evolution in time of linear indicators for effective diffusion of high performance concrete 57

58 Figure : I129 - Evolution in time of linear indicators for vertical head gradient in argillites 58

59 Figure : I129 - Evolution in time of linear indicators for permeability of high performance concrete 59

60 Figure : I129 - Evolution in time of linear indicators for horizontal permeability of argillites 60

61 Figure : I129 - Evolution in time of linear indicators for permeability of microfissured zone 61

62 Figure : I129 - Evolution in time of linear indicators for permeability of fractured zone 62

63 Figure : I129 - Evolution in time of linear indicators for Kd Iodine concrete 63

64 Figure : I129 - Evolution in time of linear indicators for Kd Iodine argillites 64

65 Figure : I129 - Evolution in time of linear indicators for vertical permeability in argillites 65

66 Figure : I129 - Evolution in time of linear indicators for argillites porosity 66

67 Figure : I129 - Evolution in time of linear indicators for microfissured zone porosity 67

68 Figure : I129 - Evolution in time of linear indicators for fractured zone porosity 68

69 I129 - Rank indicators Figure : I129 - Evolution in time of rank indicators for effective diffusion of argillites 69

70 Figure : I129 - Evolution in time of rank indicators for effective diffusion of fractured zone 70

71 Figure : I129 - Evolution in time of rank indicators for effective diffusion of high performance concrete 71

72 Figure : I129 - Evolution in time of rank indicators for vertical head gradient in argillites 72

73 Figure : I129 - Evolution in time of rank indicators for permeability of high performance concrete 73

74 Figure : I129 - Evolution in time of rank indicators for horizontal permeability of argillites 74

75 Figure : I129 - Evolution in time of rank indicators for permeability of microfissured zone 75

76 Figure : I129 - Evolution in time of rank indicators for permeability of fractured zone 76

77 Figure : I129 - Evolution in time of rank indicators for Kd Iodine concrete 77

78 Figure : I129 - Evolution in time of rank indicators for Kd Iodine argillites 78

79 Figure : I129 - Evolution in time of rank indicators for vertical permeability in argillites 79

80 Figure : I129 - Evolution in time of rank indicators for argillites porosity 80

81 Figure : I129 - Evolution in time of rank indicators for microfissured zone porosity 81

82 Figure : I129 - Evolution in time of rank indicators for fractured zone porosity 82

83 Se79 - Linear indicators Figure : Se79 - Evolution in time of linear indicators for solubility limit in argillites 83

84 Figure : Se79 - Evolution in time of linear indicators for Kd in argillites 84

85 Figure : Se79 - Evolution in time of linear indicators for effective diffusion of argillites 85

86 Figure : Se79 - Evolution in time of linear indicators for Kd Selenium concrete 86

87 Figure : Se79 - Evolution in time of linear indicators for Csat Selenium concrete before years 87

88 Figure : Se79 - Evolution in time of linear indicators for Csat Selenium concrete after years 88

89 Figure : Se79 - Evolution in time of rank indicators for Kd Selenium concrete 89

90 Figure : Se79 - Evolution in time of rank indicators for Csat Selenium concrete before years 90

91 Figure : Se79 - Evolution in time of rank indicators for Csat Selenium concrete after years 91

92 Figure : Se79 - Evolution in time of rank indicators for vertical permeability in argillites 92

93 Figure : Se79 - Evolution in time of rank indicators for argillites porosity 93

94 Figure : Se79 - Evolution in time of rank indicators for microfissured zone porosity 94

95 Figure : Se79 - Evolution in time of rank indicators for fractured zone porosity 95

96 Nb94 - Linear indicators Figure : Nb94 - Evolution in time of linear indicators for solubility limit in argillites Figure : Nb94 - Evolution in time of linear indicators for Kd Niobium in argillites Figure : Nb94 - Evolution in time of linear indicators for effective diffusion of argillites 96

97 Figure : Nb94 - Evolution in time of linear indicators for effective diffusion of high performance concrete Figure : Nb94 - Evolution in time of linear indicators for permeability of high performance concrete Figure : Nb94 - Evolution in time of linear indicators for horizontal permeability of argillites 97

98 Figure : Nb94 - Evolution in time of linear indicators for argillites porosity Figure : Nb94 - Evolution in time of linear indicators for microfissured zone porosity Figure : Nb94 - Evolution in time of linear indicators for fractured zone porosity 98

99 Figure : Nb94 - Evolution in time of linear indicators for release rate of hulls of ILW2 Figure : Nb94 - Evolution in time of linear indicators for release rate of inconel of ILW1 Figure : Nb94 - Evolution in time of rank indicators for release rate of inconel of ILW2 99

100 Figure : Nb94 - Evolution in time of rank indicators for release rate of stainless steels of ILW1 Figure : Nb94 - Evolution in time of rank indicators for release rate of stainless steels of ILW Nb94 - Rank indicators Figure : Nb94 - Evolution in time of rank indicators for solubility limit in argillites 100

101 Figure : Nb94 - Evolution in time of rank indicators for Kd Niobium in argillites Figure : Nb94 - Evolution in time of rank indicators for effective diffusion of argillites Figure : Nb94 - Evolution in time of rank indicators for Kd Niobium concrete 101

102 Figure : Nb94 - Evolution in time of rank indicators for Csat Niobium concrete before years Figure : Nb94 - Evolution in time of rank indicators for Csat Niobium concrete after years Figure : Nb94 - Evolution in time of rank indicators for argillites porosity 102

103 Figure : Nb94 - Evolution in time of rank indicators for microfissured zone porosity Figure : Nb94 - Evolution in time of rank indicators for fractured zone porosity Figure : Nb94 - Evolution in time of rank indicators for release rate of hulls of ILW2 103

104 Figure : Nb94 - Evolution in time of rank indicators for release rate of inconel of ILW1 Figure : Nb94 - Evolution in time of rank indicators for release rate of inconel of ILW2 Figure : Nb94 - Evolution in time of rank indicators for release rate of stainless steels of ILW1 104

105 Figure : Nb94 - Evolution in time of rank indicators for release rate of stainless steels of ILW2 105

106 ccdf I Uncertainty analysis ccdf Se79 Figure : I129 ccdf of maximal molar flow through several surfaces Figure : Se79 ccdf of maximal molar flow through several surfaces 106

107 ccdf Nb94 Figure : Nb94 ccdf of maximal molar flow through several surfaces 107

108 7.2. JRC: Detailed results Uncertainty analysis 108

109 Evolution in time of the 3 radionuclides, for all the realizations Figure 7.2-1: Time evolution for the molar flow of the 3 radionuclides getting out of the following layers (1000 curves for each radionuclide); from left to right and from top to bottom : a) waste package, b) disposal cell, c) fractured zone, d) micro-fissured zone, e) undisturbed argilites (15 m), f) undisturbed argilites. 109

110 Evolution in time of the main statistical indicators of the molar flow of 94 Nb Figure 7.2-2: Time evolution for the statistical indicators of the molar flow of 94 Nb getting out of the following layers; from left to right and from top to bottom : a) waste package, b) disposal cell, c) fractured zone, d) microfissured zone, e) undisturbed argilites (15 m), f) undisturbed argilites. 110

111 Evolution in time of the main statistical indicators of the molar flow of 129 I Figure 7.2-3: Time evolution for the statistical indicators of the molar flow of 129 I getting out of the following layers; from left to right and from top to bottom : a) waste package, b) disposal cell, c) fractured zone, d) micro-fissured zone, e) undisturbed argilites (15 m), f) undisturbed argilites. 111

112 Evolution in time of the main statistical indicators of the molar flow of 79 Se Figure 7.2-4: Time evolution for the statistical indicators of the molar flow of 79 Se getting out of the following layers; from left to right and from top to bottom : a) waste package, b) disposal cell, c) fractured zone, d) microfissured zone, e) undisturbed argilites (15 m), f) undisturbed argilites. 112