## 4 Conclusions  
In order to achieve a dynamic reliability estimation for joint waterproof facilities of utility tunnels, this paper proposed an improved Weibull distribution model to incorporate the actual field conditions, such as geological and hydrological conditions and ground treatment. Parameters of the proposed model were estimated with the aid of Bayesian methods and Hamiltonian Monte Carlo (HMC) methods via the observed failure data collected from the case study. Through the results of this study, the following conclusions can be drawn:  
1. Compared with conventional risk assessment or safety evaluation, reliability estimation is a better approach for the long-term safe operation of critical utility tunnel facilities, as it can achieve a dynamic and time-varying prediction of future risks that can be translated into long-term maintenance guidance;
2. The improved Weibull distribution model incorporating the field condition covariates can effectively simulate the failure rate distribution of joint waterproof facilities of utilities tunnels. For cast-in-situ utility tunnels in this case study, the specific covariates are identified as groundwater type, repair, quality defect, geological condition and foundation treatment;
3. The HMC sampling method is capable of sampling a complex failure rate model such as the improved Weibull model proposed in this study. In the case of limited observation sample size, the posterior estimates of model parameters obtained by HMC sampling, with Gamma priors for Weibull model parameters and standard normal distribution priors for field condition covariates, can achieve adequate convergence and fit well with the observed data and the actual engineering experiences.
4. The posteriors of model parameters can be used to estimate in real time the reliability-related indicators such as the failure rate, the failure warning time and expected failure time, providing references for risk investigation and targeted maintenance of joint waterproof facilities as well as other critical facilities of utility tunnels.  
To implement the proposed model into utility tunnel operation and maintenance practices, it is better to have a large observation sample from a specific utility tunnel project. With more failure data of utility tunnel facilities being accumulated, this proposed model is expected to have better reliability estimation performances. Meanwhile, tentative research can be carried out in the future using other models, such as a three-parameter Weibull distribution model which appears more complex but might provide better fitting for specific cases. Moreover, it should be noted that the currently proposed model can be rather difficult for the potential users, such as the utility tunnel operators or the technical companies entrusted for the regular maintenance of a utility tunnel, to understand and employ. Therefore, it is best to encapsulate the methodologies and algorithms into a software package in the future that is capable to display and export a straightforward reliability analysis report after data import and simple clicks by the users.  
Conceptualization, F.-L.P., Y.-K.Q. and C.Y.; Methodology, Y.-K.Q. and C.Y.; Software, C.Y.; Validation, Y.-K.Q.; Formal analysis, C.Y.; Data curation, C.Y.; Writing--original draft, Y.-K.Q.; Writing--review & editing, F.-L.P.; Visualization, Y.-K.Q.; Supervision, F.-L.P.; Funding acquisition, F.-L.P. and Y.-K.Q. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Natural Science Foundation of China (NSFC) (grant number 42201284); the China Postdoctoral Science Foundation (grant number BX2021220); the National Natural Science Foundation of China (NSFC) (grant number 52090083).  
Not applicable.  
Data Availability Statement: The data presented in this study are available on request from the corresponding author.