# Time-Dependent Behavior of Callovo-Oxfordian Claystone for Nuclear Waste Disposal: Uncertainty Quantification from In-Situ Convergence Measurements

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## Abstract

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## 1. Introduction

_{v}≈ σ

_{h}≈ 12.5 MPa), while the major horizontal stress is about 1.3 times higher than the minor stress (σ

_{H}≈ 1.3σ

_{h}) [1]. The excavation of drift in both directions of the major and minor horizontal stresses induces a fractured zone. Indeed, for drifts parallel to σ

_{H}, it was observed that the fractured zone presents a dissymmetrical shape despite the quasi-isotropic stress state in the drift section [2,3]. The extension of fracture is more developed in the horizontal direction of the drift cross section. The observed anisotropy is also confirmed in the convergence measurements that are conducted in the horizontal and vertical directions. The higher convergence, about two times higher, is noted in the horizontal direction, but the evolution in time of both the horizontal and vertical convergence seems similar [2].

## 2. Uncertainty Quantification by Bayesian Inference

#### 2.1. Classical Bayesian Inference

_{1}, y

_{2}… y

_{N}} is a data set of the observations, the random input parameters of the considered model u(θ) are gathered in the vector θ = {θ

_{1}, θ

_{2}, …, θ

_{M}}. The principal idea of BI consists of computing the probabilistic distribution p(θ|y) of the random vector θ conditional on training data y using the Bayes’ theorem [17,18]:

^{2}, the likelihood function p(y|θ) is written as:

#### 2.2. Hierarchical Bayesian Inference

_{j}(j = 1, 2…, N

_{s}), and each data set y

_{j}= {y

_{1}, y

_{2},…,y

_{Nj}} of the jth experiment describes a realization θ

_{j}of the random vector θ of input parameters. Thus, the hierarchical BI consists of quantifying the corresponding unknown statistical moments (i.e., the mean and standard deviation) χ = {μ

_{θ}, σ

_{θ}} of the input parameters θ by considering these so-called hyperparameters as uncertain random variables.

_{j}|χ) for each data set y

_{j}conditional on the hyperparameters χ in Equation (7) presents an important step in the hierarchical BI and different methods have been developed in the literature. For example, Sedehi et al. [21] used the Laplace asymptotic approximation in the integral (Equation (7)), while Nagel and Sudret [22] proposed some advanced MCMC techniques. In their work, Wu et al. [23] approximated the integral by the important sampling method using the proposal distribution p(θ

_{j}|y

_{j}) to reduce the computational cost:

_{j}|y

_{j}) of each data set. Thus, as the main advantage, this hierarchical BI comes as a postprocessing of the results of the classical BI conducted for each data set, through which the likelihood p(y|χ) can be estimated as:

## 3. Numerical Applications Using Synthetic Data

_{0}(Figure 1a). This solution was recently presented in [28], in which the time-dependent behavior of creep rock was characterized by the fractional derivative viscoplastic (FDVP) model. This constitutive model was established from the connection in a series of different components: the fractional-order Maxwell element, the Klevin element, and the Mohr-Coulomb plastic slider. For the sake of simplicity, in this work, we used the simpler FDVP, in which only the fractional-order Maxwell was connected in a series with the Mohr-Coulomb plastic slider (see Figure 1a). In comparison with the contribution in [28], the number of parameters used to characterize the elasto-viscoplastic behavior of rock was lower, being represented by the spring G

_{M}, the fractional-order derivation dashpot η

_{M}and fractional-order coefficient β of Maxwell element, as well as the three well-known parameters of the Mohr-Coulomb model (i.e., the cohesion C, the friction angle φ, and the dilation angle ψ).

_{evp}on the surface tunnel:

_{n}= 1(mm) was generated in the convergence of the tunnel to create the synthetic data for the uncertainty quantification (Figure 1b). Totally, a data set with 50 values of tunnel convergence were artificially generated in the range of 2500 days.

_{M}and β, while the other mechanical properties representing the short-term behavior of rock mass (i.e., the elastoplastic parameters G

_{M}, C, φ, ψ) were assumed known and constants.

_{M}= 3.09(GPa.year) and β = 0.32 obtained from the deterministic calibration as the mean values of the prior distribution. By assuming the Gaussian function of the prior distribution, their chosen standard deviations were calculated from a chosen value of coefficient of variation COV = 30%. The obtained results show that the mean values of the posterior distribution of these two parameters were, respectively, η

_{M}= 3.08(GPa.year) and β = 0.35, with the standard deviations 0.102(GPa.year) and 0.02.

_{M}and β were considered. This effect is highlighted in Figure 3, in which the evolution of mean values of the posterior distribution of η

_{M}and β were plotted as functions of their chosen prior mean values. As expected, the accuracy of the results provided by the classical BI would improve when the prior values approach the exact parameters. We can state from this investigation that the results of the deterministic inversion can provide an appropriate choice for the mean values of the prior distribution of the BI.

_{n}on the results of the identified parameters. As illustrated in Figure 4, the mean values of the posterior distribution of the two parameters η

_{M}and β matched well with their exact values when the additive noise was small. The higher magnitude of noise reduced the correctness of the identified parameters. More precisely, the difference between the posterior mean values and the exact parameters was more pronounced, and their corresponding standard deviations of the posterior functions were also higher when the additive noise magnitude was more important.

_{n}= 1 mm) and aleatoric uncertainty were taken in the synthetic data of tunnel convergence. For the generation of this latter uncertainty, the two parameters (η

_{M}and β) of the FDVP rock were assumed to be random, whose distributions are Gaussian, and the mean values are equal to the ones in Table 1 (i.e., μ

_{η}

_{M}= 3.06(GPa.year) and μ

_{β}= 0.35). The variability of each parameter was characterized by a coefficient of variation (COV). For the sake of simplicity, the same COV = 15% was supposed for these two input parameters (i.e., their corresponding standard deviations were σ

_{η}

_{M}= 0.46(GPa.year) and σ

_{β}= 0.053). As an example, Figure 5 presents ten synthetic data sets of convergence determined at ten sections along the tunnel axis, using both aleatoric uncertainty and additive noise.

_{s}= 5, 10, 20, 50). Following that, the posterior distributions of the hyperparameters (i.e., the mean and standard deviation of the two parameters η

_{M}and β) presented a quite similar tendency. When increasing the number of data sets, the point estimates by hierarchical BI approached the true values of the mean and standard deviation of each parameter of the FDVP rock. Their correspondingly-estimated uncertainty also decreased, as expected, using the higher number sets of convergence.

_{M}and β of FDVP rock can be determined quite well by the classical BI, their corresponding standard deviations were very far from the exact values and from the mean of posterior distribution evaluated by hierarchical BI. Consequently, as summarized in Table 2, the distribution of the two parameters η

_{M}and β ranging from the minimum to the maximum values (which correspond to the lower and upper quantiles of 2.5% and 97.5%) were very different with respect to the exact results and the ones of the hierarchical BI.

## 4. Uncertainty of Time-Dependent Behavior of COx Claystone

#### 4.1. Description of the Numerical Model

_{H}, a dissymmetrical shape of the fractured zone was observed despite the quasi-isotropic stress state (σ

_{v}≈ σ

_{h}≈ 12.5 MPa) in the drift section [2,3]. The fracture zone was more developed in the horizontal direction, with an extension to about 1 time the diameter of the drift (Figure 8a). The convergence measurements also highlighted a higher convergence about two times in the horizontal direction, but the evolution in time of both the horizontal and vertical convergence seemed similar (Figure 8b) [2].

_{i}, φ

_{i}, ψ

_{i}) and of the fractured zone (C

_{f}, φ

_{f}, ψ

_{f}), and three parameters of the viscoplastic Lemaitre model (K, n, m).

#### 4.2. Results of the Bayesian Inversion and Discussions

_{s}= 36) can be generated for the hierarchical BI instead of six initial data sets (N

_{s}= 6). As observed in Figure 12, the increase of the number of data sets reduced the uncertainty of the hyperparameters of the viscoplastic COx claystone, notably their mean values represented by a narrower posterior distribution. Finally, in Table 5, we summarized the ranging values evaluated at the 2.5% and 97.5% quantiles of the viscoplastic properties of host rock that were calculated from the two BI methods. The range of each parameter was reduced as expected by increasing the number of data sets.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- IPCC Special Report “Global Warming 1.5 °C”. 2018. Available online: https://www.ipcc.ch/sr15/ (accessed on 20 April 2022).
- Armand, G.; Noiret, A.; Zghondi, J.; Seyedi, D.M. Short-and long-term behaviors of drifts in the Callovo Oxfordian claystone at the Meuse/Haute-Marne Underground Research Laboratory. J. Rock Mech. Geotech. Eng.
**2013**, 5, 221–230. [Google Scholar] [CrossRef] [Green Version] - Armand, G.; Leveau, F.; Nussbaum, C.; de la Vaissiere, R.; Noiret, A.; Jaeggi, D.; Landrein, P.; Righini, C. Geometry and properties of the excavation induced fractures at the Meuse/Haute-Marne URL drifts. Rock Mech. Rock. Eng.
**2014**, 47, 21–41. [Google Scholar] [CrossRef] - Armand, G.; Conil, N.; Talandier, J.; Seyedi, D.M. Fundamental aspects of the hydromechanical behavior of Callovo-Oxfordian claystone: From experimental studies to model calibration and validation. Comput. Geotech.
**2017**, 85, 277–286. [Google Scholar] [CrossRef] - Alonso, M.; Vu, M.N.; Vaunat, J.; Armand, G.; Gens, A.; Plua, C. Effect of thermohydro-mechanical coupling on the evolution of stress in the concrete liner of an underground drift in the Cigéo project. IOP Conf. Ser. Earth Environ. Sci.
**2021**, 833, 012200. [Google Scholar] [CrossRef] - Souley, M.; Vu, M.N.; Armand, G. 3D Modelling of Excavation-Induced Anisotropic Responses of Deep Drifts at the Meuse/Haute-Marne URL. Rock Mech. Rock Eng.
**2022**, 55, 4183–4207. [Google Scholar] [CrossRef] - Mánica, M.A.; Gens, A.; Vaunat, J.; Armand, G.; Vu, M.N. Numerical simulation of underground excavations in an indurated clay using non-local regularisation. Part 1: Formulation and base case. Géotechnique
**2021**, 1–21. [Google Scholar] [CrossRef] - Yu, Z.; Shao, J.; Duveau, G.; Vu, M.N.; Armand, G. Numerical modeling of deformation and damage around underground excavation by phase-field method with hydromechanical coupling. Comput. Geotech.
**2021**, 138, 104369. [Google Scholar] [CrossRef] - Zhao, J.J.; Shen, W.Q.; Shao, J.F.; Liu, Z.B.; Vu, M.N. A constitutive model for anisotropic clay-rich rocks considering micro-structural composition. Int. J. Rock Mech. Min. Sci.
**2022**, 151, 105029. [Google Scholar] [CrossRef] - Saitta, A.; Lopard, G.; Petizon, T.; Armand, G. Projet Cigéo (France)—Modélisation du comportement des argilites de la galerie GRD du laboratoire souterrain de Meuse/Haute-Marne. In Proceedings of the Congrès AFTES 2017, Paris, France, 13–15 November 2017. [Google Scholar]
- Vu, M.N.; Guayacan-Carrillo, L.M.; Armand, G. Excavation induced over pore pressure around drifts in the Callovo-Oxfordian claystone. Eur. J. Environ. Civil Eng.
**2020**, 1–16. [Google Scholar] [CrossRef] - Tran, N.T.; Do, D.P.; Hoxha, D.; Vu, M.N.; Armand, G. Kriging-based reliability analysis of the long-term stability of a deep drift constructed in the Callovo-Oxfordian claystone. J. Rock Mech. Geotech. Eng.
**2021**, 13, 1033–1046. [Google Scholar] [CrossRef] - Tran, N.T.; Do, D.P.; Hoxha, D.; Vu, M.N.; Armand, G. Modified AK-MCS method and its application on the reliability analysis of underground structures in the rock mass. J. Sci. Tech. Civil Eng. (STCE)-HUCE
**2022**, 16, 38–54. [Google Scholar] [CrossRef] - Fortsakis, P.; Kavvadas, M. Estimation of time dependent ground parameters in tunnelling using back analyses of convergence data. In Proceedings of the Euro:Tun 2009, Bochum, Germany, 9–11 September 2009. [Google Scholar]
- Lecampion, B.; Constantinescu, A.; Nguyen, D.M. Parameter identification for lined tunnels in a viscoplastic medium. Int. J. Numer. Anal. Methods Geomech.
**2002**, 26, 1191–1211. [Google Scholar] [CrossRef] [Green Version] - Do, D.P.; Vu, M.N.; Tran, N.T.; Armand, G. Closed-form solution and reliability analysis of deep tunnel supported by a concrete liner and a covered compressible layer within the viscoelastic Burger rock. Rock Mech. Rock Eng.
**2021**, 54, 2311–2334. [Google Scholar] [CrossRef] - Do, D.P.; Tran, N.T.; Hoxha, D.; Vu, M.N.; Armand, G. Kriging-based optimization design of deep drift in the rheological Burger rock. IOP Conf. Ser. Earth Environ. Sci.
**2021**, 833, 012155. [Google Scholar] [CrossRef] - Zhang, J.; Yin, J.; Wang, R. Basic framework and main methods of uncertainty quantification. Math. Probl. Eng.
**2020**, 2020, 6068203. [Google Scholar] [CrossRef] - Rappel, H.; Bee, L.A.A.; Hale, J.S.; Bordas, S.P.A. Beyasian inference for the stochastic identification of elastoplastic material parameters: Introduction, misconceptions, and additional insight. arXiv
**2016**, arXiv:1606.02422. [Google Scholar] - Wagner, P.R.; Nagel, J.; Marelli, S.; Sudret, B. UQLab User Manuel—Bayesian Inference for Model Calibration and Inverse Problem; Report # UQLab-V1.4-113; Risk, Safety and Uncertainty Quantification: Zurich, Switzerland, 2021. [Google Scholar]
- Sedehi, O.; Papadimitriou, C.; Katafygiotis, L.S. Probabilistic Hierarchical Bayesian framework for time-domain model updating and robust predictions. Mech. Sys. Sig. Proc.
**2019**, 123, 648–673. [Google Scholar] [CrossRef] - Nagel, J.B.; Sudret, B. A unified framework for multilevel uncertainty quantification in Bayesian inverse problems. Probabilistic Eng. Mech.
**2016**, 43, 68–84. [Google Scholar] [CrossRef] [Green Version] - Wu, S.; Angelikopoulos, P.; Beck, J.L.; Koumoutsakos, P. Hierarchical stochastic model in Bayesian inference for engineering applications: Theoretical implications and efficient approximation. ASCE-ASME J. Risk Uncert. Eng. Sys. Part B. Mech. Eng.
**2019**, 5, 011006. [Google Scholar] [CrossRef] [Green Version] - Marelli, S.; Sudret, B. UQLab: A framework for uncertainty quantification in Matlab. In Proceedings of the 2nd International Conference on Vulnerability and Risk Analysis and Management (ICVRAM2014), Liverpool, UK, 13–16 July 2014. [Google Scholar]
- Li, C.; Jiang, S.H.; Li, J.; Huang, J. Bayesian approach for sequential probabilistic back analysis of uncertain geomechanical parameters and reliability updating of tunneling-induced ground settlements. Adv. Civil Eng.
**2020**, 2020, 8528304. [Google Scholar] [CrossRef] - Miro, S.; König, M.; Hartmann, D.; Schanz, R. A probabilistic analysis of subsoil parameters uncertainty impacts on tunnel-induced ground movements with a back-analysis study. Comput. Geotech.
**2015**, 68, 38–53. [Google Scholar] [CrossRef] - Rappel, H.; Bee, L.A.A.; Bordas, S.P.A. Beyasian inference to identify parameters in viscoelasticity. Mech. Time Depend. Mater.
**2018**, 22, 221–258. [Google Scholar] [CrossRef] - Kabwe, E.; Karakus, M.; Chanda, E.K. Time-dependent solution for non-circular tunnels considering the elasto-viscoplastic rockmass. Int. J. Rock Mech. Min. Sci.
**2020**, 133, 104395. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Circular tunnel in the FDVP rock; (

**b**) calculated convergence on the surface of tunnel without and with additive noise.

**Figure 2.**Results of the classical BI using the synthetic data of tunnel convergence in creep rock: (

**a**) prior distribution; (

**b**) posterior distribution of the time-dependent behavior parameters (η

_{M}, β) of the FDVP model; (

**c**) the posterior predictions.

**Figure 3.**Mean values of the posterior distribution obtained from classical BI using different prior values: (

**a**) parameter η

_{M}; (

**b**) parameter β of the FDVP rock.

**Figure 4.**Posterior distribution of: (

**a**) parameter η

_{M}; (

**b**) parameter β of the FDVP rock as a function of additive noise magnitude σ

_{n}.

**Figure 5.**Synthetic data of tunnel convergence using: (

**a**) aleatoric uncertainty; (

**b**) both aleatoric and epistemic uncertainty.

**Figure 6.**Posterior distribution of: (

**a**) mean value; (

**b**) standard deviation value of the two parameters η

_{M}and β of FDVP rock determined by hierarchical BI using different numbers of synthetic data sets.

**Figure 7.**Posterior distribution of: (

**a**) mean value; (

**b**) standard deviation value of the two parameters η

_{M}and β of FDVP rock using the classical and hierarchical BI.

**Figure 8.**(

**a**) Induced fracture network; (

**b**) horizontal and vertical convergences observed in the drifts excavated following the direction of major horizontal stress [1].

**Figure 9.**(

**a**) Geometrical model with the elliptical fractured zone around the drift; (

**b**) elastic-perfectly plastic Mohr Coulomb model of intact and fractured rocks.

**Figure 10.**Evolution in time of horizontal and vertical convergences at a section of the drift: comparison of experimental and numerical simulation of the deterministic problem.

**Figure 11.**(

**a**) Prior distribution; (

**b**) posterior samples of the Lemaitre parameters (n, 1/m, 1/K) of the COx claystone determined by classical BI using convergence data of one section of drift.

**Figure 12.**Posterior distribution of: (

**a**) mean value; (

**b**) standard deviation value of Lemaitre parameters of COx claystone determined by classical and hierarchical BI using the convergence data of six sections of drift.

G_{M}(GPa) | η_{M}(GPa.year) | C (MPa) | φ (°) | ψ (°) | β | P_{0}(MPa) | R (m) | λ |
---|---|---|---|---|---|---|---|---|

1.73 | 3.06 | 6 | 20 | 0 | 0.35 | 12.5 | 2.6 | 1 |

**Table 2.**Minimum and maximum values (corresponding to lower quantile 2.5% and upper quantile 97.5%) of FDVP parameters using classical and hierarchical BI.

Method | η_{M} (GPa.year) | β | ||
---|---|---|---|---|

Min | Max | Min | Max | |

Exact | 2.16 | 3.96 | 0.25 | 0.45 |

Classical BI (N_{s} = 10) | 3.34 | 3.50 | 0.35 | 0.39 |

Classical BI (N_{s} = 50) | 3.28 | 3.37 | 0.38 | 0.41 |

Hierarchical BI (N_{s} = 10) | 0.95 | 5.56 | 0.25 | 0.46 |

Hierarchical BI (N_{s} = 50) | 1.45 | 4.80 | 0.16 | 0.53 |

E (GPa) | υ | C_{i}(MPa) | φ_{i}(°) | ψ_{i}(°) | C_{f}(MPa) | φ_{f}(°) | ψ_{f}(°) |
---|---|---|---|---|---|---|---|

6.6 | 0.3 | 6 | 20 | 0 | 1 | 25 | 5 |

**Table 4.**Mean and standard deviation (Std) of viscoplastic properties of COx claystone at different sections using the classical BI.

Section | n | 1/m | 1/K (GPa^{−1}) | |||
---|---|---|---|---|---|---|

Mean | Std | Mean | Std | Mean | Std | |

OHZ170B | 9.22 | 1.78 | 0.26 | 0.040 | 8.25 | 1.95 |

OHZ170C | 8.97 | 1.72 | 0.25 | 0.039 | 7.19 | 1.79 |

OHZ170D | 8.94 | 1.69 | 0.24 | 0.038 | 7.15 | 1.78 |

OHZ170E | 8.92 | 1.70 | 0.25 | 0.038 | 7.35 | 1.83 |

OHZ170F | 8.48 | 1.55 | 0.23 | 0.035 | 7.07 | 1.78 |

OHZ170G | 8.95 | 1.63 | 0.25 | 0.037 | 7.49 | 1.88 |

Six sections | 8.66 | 1.82 | 0.23 | 0.041 | 8.36 | 2.16 |

**Table 5.**Minimum and maximum values (corresponding to the lower quantile 2.5% and the upper quantile 97.5%) of viscoplastic Lemaitre parameters using classical and hierarchical BI.

Method | n | 1/m | 1/K (GPa^{−1}) | |||
---|---|---|---|---|---|---|

Min | Max | Min | Max | Min | Max | |

Classical BI | 5.09 | 12.24 | 0.15 | 0.31 | 4.12 | 12.60 |

Hierarchical BI (N_{s} = 6) | 3.91 | 13.74 | 0.10 | 0.40 | 2.68 | 12.17 |

Hierarchical BI (N_{s} = 36) | 8.07 | 10.42 | 0.18 | 0.26 | 5.63 | 9.39 |

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**MDPI and ACS Style**

Do, D.-P.; Tran, N.-T.; Hoxha, D.; Vu, M.-N.; Armand, G.
Time-Dependent Behavior of Callovo-Oxfordian Claystone for Nuclear Waste Disposal: Uncertainty Quantification from In-Situ Convergence Measurements. *Sustainability* **2022**, *14*, 8465.
https://doi.org/10.3390/su14148465

**AMA Style**

Do D-P, Tran N-T, Hoxha D, Vu M-N, Armand G.
Time-Dependent Behavior of Callovo-Oxfordian Claystone for Nuclear Waste Disposal: Uncertainty Quantification from In-Situ Convergence Measurements. *Sustainability*. 2022; 14(14):8465.
https://doi.org/10.3390/su14148465

**Chicago/Turabian Style**

Do, Duc-Phi, Ngoc-Tuyen Tran, Dashnor Hoxha, Minh-Ngoc Vu, and Gilles Armand.
2022. "Time-Dependent Behavior of Callovo-Oxfordian Claystone for Nuclear Waste Disposal: Uncertainty Quantification from In-Situ Convergence Measurements" *Sustainability* 14, no. 14: 8465.
https://doi.org/10.3390/su14148465