Reliability Estimation for the Joint Waterproof Facilities of Utility Tunnels Based on an Improved Bayesian Weibull Model

Abstract

Safety issues are a major concern for the long-term maintenance and operation of utility tunnels, of which the focal point lies in the reliability of critical facilities. Conventional evaluation methods have failed to reflect the time-dependency and objectivity of the reliability of critical facilities, hence reducing the credibility of the analysis results and posing serious risks to the safety of utility tunnels. Taking joint waterproof facilities as an example, this paper focuses on the scientific problem of how to achieve a dynamic estimation of the reliability of critical facilities throughout the project life cycle of utility tunnels. To this end, an improved Weibull distribution model is proposed to incorporate the actual field conditions that affect the reliability of joint waterproof facilities of utility tunnels. Bayesian methods and Hamiltonian Monte Carlo methods are used to realize the posterior estimation of the model parameters via the observed failure data. The case study shows that the posterior prediction results fit well with the actual observation data. The proposed model can be used to estimate in real time such key reliability indicators as failure rate, failure warning time and expected failure time, which facilitate the safe operation and targeted maintenance of utility tunnels.

1. Introduction

Urban utility tunnels are widely recognized as key infrastructures contributing to sustainable cities [1,2]. The development of the utility tunnel has been promoted nationwide in China since 2015, with a cumulative length of over 8000 km. In such a context, safety issues caused by water leakage, gas leakage, fire and blasting have become a major concern for the long-term operation of utility tunnels [3,4]. One of the key solutions to the safety issues is to implement relevant facilities for the avoidance of such safety incidents [5]. Such facilities that are critical to the safe operation of utility tunnels and the utility cables or pipelines placed therein are referred to as the critical facilities of utility tunnels. The critical facilities of utility tunnels include fire alarm facilities, gas alarm facilities, environment monitor facilities, ventilation facilities, drainage facilities, firefighting facilities, waterproof facilities, power supply and distribution facilities, security facilities and network facilities. The failures of critical facilities can account, either directly or indirectly, for the safety incidents occurred in utility tunnels. Therefore, the reliability of critical facilities is of significant importance for the operation and maintenance of utility tunnels.

Among the critical facilities of utility tunnels, joint waterproof facilities are focal to reliability issues [6]. For one thing, deformation joints enable utility tunnels to adapt to differential land settlements and concentrated structural stresses. For another, the waterproof capabilities of joint facilities tend to be compromised by such factors as deformation, structural stress and hydraulic pressure, thus becoming vulnerable parts of and posing potential risks to the entire waterproof system of a utility tunnel [7,8]. Field investigations by the authors and other practitioners [9] showed that water leakage is quite a pervasive phenomenon for the deformation joints of utility tunnels. As can be seen from the photos in Figure 1, water leakages further induce various safety risks to the operation of utility tunnels, including structural and metal corrosion, electrical short circuit and invasion of toxic and harmful substances. In order to effectively reduce water leakage incidents and control joint failure risks, it is necessary to undertake a safety assessment or reliability estimation for the joint waterproof facilities of utility tunnels.

In recent years, more attention has been paid to the safety issues of urban utility tunnels. Conventional approaches have resorted to various fuzzy set theories for the safety evaluation of utility tunnels. For instance, He et al. (2019) [10] combined such methods as weighted fuzzy Petri nets, even trees and a fire dynamics simulator to assess the cable fire risks in utility tunnels. Sun et al. (2022) [11] adopted the grey clustering method for the safety evaluation of urban utility tunnels. However, these approaches are static and fail to consider the uncertainty characteristics of safety events related with utility tunnel facilities. To better address the safety uncertainties in utility tunnels, the Bayesian network model combined with other models, such as the Dempster–Shafer (D–S) evidence theory [12], bow-tie model composed of a fault tree and event tree [13,14,15] and the Work Breakdown Structure and Risk Breakdown Structure method [16], is widely used to predict the comprehensive risks of all pipelines [5] or of specific pipelines and safety incidents in utility tunnels. The predicted risks derived from the Bayesian network can also be periodically updated using the cumulative observed information recorded in a specific period (e.g., a year). The Bayesian network can also be integrated with numerical simulation methods to quantitatively evaluate the safety event consequences inside and outside utility tunnels. For example, Bai et al., (2022) [17] used the computational fluid dynamics method to quantify the potential gas dispersion and explosion consequences for utility tunnels housing natural gas pipelines. In addition, considering that there are complex types of risks in utility tunnels, a system dynamics simulation was used to achieve a dynamic coupling risk assessment for utility tunnels [18]. Such emerging new methods based on the Bayesian network can to some extent provide references to prevent safety events when applied to urban utility tunnels. However, these methods rely heavily on the observed information, and risk assessment results are likely influenced by the subjectivity of experts’ opinions. Moreover, it is hard for the results of risk assessment to be translated into long-term maintenance guidance for which time is a key indicator since risks are dynamic and time-varying. For instance, if a tunnel facility or a component is assessed with high risks, it is still not clear about when it will fail or how long it will take to reach the lowest required functionality. In this regard, reliability can be a better indicator than risk for the long-term safe operation of urban utility tunnels, because it is generally defined as a function of lifetime and can further derive other key maintenance and operation indicators such as failure rate, failure warning time and expected failure time.

Reliability refers to the probability of a system operation with intended functions in the context of set time limit and specific environment [19]. It varies with operation time and system status. However, similar to the aforementioned risk assessment endeavors, conventional methods, such as set pair analysis [20] and the technique for order preference by similarity to an ideal solution (TOPSIS) [21], cannot reflect the dynamic and time-varying characteristics of utility tunnel reliability [22,23]. In addition, these methods can only offer subjective and descriptive estimations, reducing the credibility of the results. In comparison, machine learning incorporated with Bayesian methods and Markov chain Monte Carlo (MCMC) makes it possible to achieve a dynamic and credible reliability estimation [24]. Hitherto, however, there has been an absence of research attempts made to apply these methods to the reliability estimation for the joint waterproof facilities of utility tunnels. In this sense, this study will fill such a research gap by demonstrating how these methods can be used to reveal the time-dependent reliability of critical facilities, in particular joint waterproof facilities, throughout the project life cycle of utility tunnels.

The observed failure regularities of joint waterproof facilities are consistent with the classical failure mode characterized by the bathtub curve, which is widely practiced in reliability estimation and deterioration modeling for many other products. The bathtub curve is comprised of three project life stages [19], including an early failure stage with a decreasing failure rate due to improper designs and implementations, a useful life stage characterized by random failures with a relatively constant failure rate and a wear-out stage with a sharply increasing failure rate before breakdown. The three life stages of the bathtub curve can be modeled by the Weibull distribution with flexible shape parameters [19]. For this reason, Weibull distribution is chosen to model the failure and reliability of joint waterproof facilities of utility tunnels. Nevertheless, the failure rate of joint waterproof facilities is not just determined by lifetime, but also subject to the actual field conditions, such as geological and hydrological conditions and ground treatment, against which the facilities are implemented. Therefore, it is necessary to incorporate such field condition covariates into the Weibull distribution [25]. Consequently, an improved Weibull distribution model will be proposed in this study. With the aid of Bayesian statistical estimation methods and MCMC methods, the key model parameters of the improved Weibull model at any momentary positions can be derived, enabling a dynamic reliability estimation for the joint waterproof facilities of utility tunnels.

Toward a dynamic reliability estimation incorporating actual field conditions for critical facilities of utility tunnels, this paper will first introduce the methodology of the improved Bayesian Weibull model in Section 2, then demonstrate its applicability with a case study in Section 3. The findings of this study will be concluded in Section 4. The proposed model and the findings are expected to be utilized by the utility tunnel operators or the entrusted technical companies to facilitate the safe operation and targeted maintenance of utility tunnels.

2. Methodology

The proposed methodological framework of reliability estimation for critical utility tunnel facilities is comprised of three parts as shown in Figure 2, including the development of an improved Weibull model incorporated with Cox proportional hazard model, Bayesian estimation of model parameters and approximation of model parameters. In order to take field conditions into account, the improved model in the first part has multiple parameters, based on which the key indicators for the long-term operation and maintenance of utility tunnel facilities can be derived. However, these key indicators cannot be quantified until the model parameters are calculated for a specific case scenario. Toward the estimation of model parameters, the Bayesian inference theorem is adopted in the second part. The Bayesian estimation incorporates both prior expert knowledge and observed data samples to generate a joint posterior distribution of model parameters, based on which the integral computation equations of model parameters can be derived. Since these integral equations are multi-dimensional and difficult to solve, the third part uses the Hamiltonian Monte Carlo (HMC) sampling method to approximate the values of model parameters. These values will then be substituted into the proposed model to obtain key indicator values at any momentary position, which will provide useful guidance for the future long-term operation and maintenance of critical utility tunnel facilities.

2.1. An Improved Weibull Model Incorporating Field Conditions

In the classical Weibull model, supposing we have independently identically distributed lifetime t, the failure probability density function of the two-parameter Weibull distribution for joint waterproof facilities is:

$$f(t)=\frac{m}{\eta }{(\frac{t}{\eta })}^{m-1}\exp \left[-{(\frac{t}{\eta })}^m\right]$$

(1)

where m is a unitless shape parameter and η is a scale parameter. The Weibull distribution cumulative distribution function F(t) is often written as:

$$F(t)={\displaystyle {\mathrm{\int }}_0^{t}f(t)dt=1-\exp \left[-{(\frac{t}{\eta })}^m\right]}$$

(2)

as well as the reliability function R(t) is:

$$R(t)=1-F(t)=\exp \left[-{(\frac{t}{\eta })}^m\right]$$

(3)

The failure rate function (or hazard function) λ(t) can then be derived as:

$$\mathit{\lambda }(t)=\frac{f(t)}{R(t)}=\frac{m}{\eta }{(\frac{t}{\eta })}^{m-1}$$

(4)

From Equation (4), it can be seen that the shape parameter m enables the Weibull distribution to be applied to the aforementioned three life stages of the bathtub curve. That is, the shape parameter m models the three stages of early failure, useful life and wear-out with the corresponding values less than one, equal to one and greater than one.

The proposed method that incorporates the explanatory variables of field conditions into the Weibull model follows the Cox proportional hazard model [26], which allows a simultaneous exploration of the effects of several covariates on reliability [27].

With the Cox proportional hazard model, the failure rate of the joint waterproof facilities of utility tunnels can be extended to incorporate field condition factors as follows:

$$\mathit{\lambda }(t,X)=\mathit{\lambda }(t)\exp ({\beta }_0+{\displaystyle \sum _{i=1}^n{\beta }_i{X}_i})$$

(5)

where λ(t) is the basic failure rate of the joint waterproof facilities of utility tunnels that only depends on lifetime t, X_i denotes the covariates of the ith principal field condition factor that influences the reliability distribution and the regression coefficient β_i represents the degree of influence of the ith covariate.

Hence the key indicators for the Weibull model, i.e., the failure rate function λ(t), the reliability function R(t) and the failure probability density function f(t), can be expressed as follows:

$$\mathit{\lambda }(t,X)=\frac{m}{\eta }{(\frac{t}{\eta })}^{m-1}\exp ({\beta }_0+{\displaystyle \sum _{i=1}^n{\beta }_i{X}_i})$$

(6)

$$R(t,X)=\exp \left[-{(\frac{t}{\eta })}^m\exp ({\beta }_0+{\displaystyle \sum _{i=1}^n{\beta }_i{X}_i})\right]$$

(7)

$$f(t,X)=\frac{m}{\eta }{(\frac{t}{\eta })}^{m-1}\exp (Z)\exp \left[-{(\frac{t}{\eta })}^m\exp (Z)\right]$$

(8)

where $Z={\beta }_0+{\displaystyle \sum _{i=1}^n{\beta }_i{X}_i}$.

From the improved Weibull model, two key indicators in the reliability analysis for joint waterproof facilities can be derived. The first key indicator is the failure warning time. It is defined as the time (unit: days) when the reliability decreases to 80% of the original state, which is deduced from Equation (7) as follows:

$$T_{0.8}=\eta {\left[-\frac{\ln 0.8}{\exp ({\beta }_0+{\displaystyle \sum _{i=1}^n{\beta }_i{X}_i})}\right]}^{1/m}$$

(9)

The other key indicator is expected failure time (unit: days) which is calculated as:

$$E(T)={\displaystyle {\mathrm{\int }}_0^{\infty }tf(t)dt=\eta {\left[\exp ({\beta }_0+{\displaystyle \sum _{i=1}^n{\beta }_i{X}_i})\right]}^{-1/m}}\mathrm{\Gamma }(\frac{m+1}{m})$$

(10)

From the aforementioned analysis process, it is evident that the dynamic reliability estimation and subsequent early warning of failure risks can be delivered if the model parameters of m, η and β_i can be obtained at any momentary position. However, as the improved Weibull model increases the number of model parameters, the derivation of these parameters remains a challenge owing to the deficiencies in empirical research and the complexities in the varying bathtub stages. Therefore, the Bayesian parameter estimation methods are required to tackle the challenge.

2.2. Bayes Estimation of Model Parameters

According to the Bayesian inference theorem, the joint posterior distribution of the parameters of m, η and β_i to be determined using the observed censoring samples D for the improved Weibull model can be obtained as:

$$f(m,\eta ,{\beta }_i|D)=\frac{L(D|m,\eta ,{\beta }_i)f(m,\eta ,{\beta }_i)}{{\displaystyle \int \cdot \cdot \cdot {\displaystyle \underset{(m,\eta ,{\beta }_i)}{\int }L(D|m,\eta ,{\beta }_i)f(m,\eta ,{\beta }_i)d(m,\eta ,{\beta }_i)}}}$$

(11)

where:

$$L(D|m,\eta ,{\beta }_i)={\displaystyle \prod _{j=1}^{n}f(t_j,X)}$$

(12)

where L(D|m, η, β_i) is the likelihood of Weibull model parameters and f(m, η, β_i) is the joint prior distribution of the parameters.

Given that there are no revealed function form references for the multi-dimensional probability density function f(m, η, β_i), we need to designate the initial distributions the parameters of m, η and β_i assuming they are independently distributed. Considering the ease of computation resulting from conjugate prior distributions and the experiences and knowledges, we assume that the unknown parameters m and η follow independent Gamma priors such that m ∼ Gamma (a₁, b₁), η ∼ Gamma (a₂, b₂), and the prior of parameter β_i follows normal distribution as β_i ∼ N (μ_i, σ_i²), where hyper parameters a₁, b₁, a₂, b₂, μ_i and σ_i are assumed as non-negative and known. In this manner, the prior distributions of the unknown Weibull parameters can be expressed as:

$$f(m)=\frac{b^{a_1}}{\mathrm{\Gamma }(a_1)}{m}^{a_1-1}\exp (-b_{1}m),m\in [0,+\infty ]$$

(13)

$$f(\eta )=\frac{b^{a_2}}{\mathrm{\Gamma }(a_2)}{\eta }^{a_2-1}\exp (-b_2\eta ),\eta \in [0,+\infty ]$$

(14)

$$f({\beta }_i)=\frac{1}{\sqrt{2\mathit{\pi }}{\sigma }_i}\exp \left[-\frac{{({\beta }_i-{\mu }_i)}^2}{2{\sigma }_i{}^2}\right],\beta \in [-\infty ,+\infty ]$$

(15)

Consequently, the joint prior distribution f(m, η, β_i) is:

$$f(m,\eta ,{\beta }_i)=f(m)f(\eta ){\displaystyle \prod _{i=1}^{k}f({\beta }_i)}$$

(16)

By substituting Equations (13) and (16) into Equation (11), the joint posterior distribution function of Weibull parameters on observed data samples can be derived. Subsequently, the Bayes estimate of each individual parameter can be obtained as the mean of the posterior function by the following equation, taking the parameter m as an example:

$$\widehat{m}={\displaystyle \mathrm{\int }\cdot \cdot \cdot {\displaystyle \underset{(\eta ,{\beta }_i)}{\mathrm{\int }}m\cdot f(m,\eta ,{\beta }_i|D)d(m,\eta ,{\beta }_i)}}$$

(17)

However, it is rather challenging to solve the multi-dimensional integral equations such as Equations (11) and (17) with an explicit expression for Weibull parameters. To tackle this challenge, the Hamiltonian Monte Carlo (HMC) method will be employed to approximate the Bayes estimation.

2.3. Hamiltonian Monte Carlo (HMC) Sampling Method

The choice of approximation methods and samplers for of Bayesian parameter estimation depends on the characteristics of posterior distribution. For instance, the commonly used MCMC algorithms, such as random walk Metropolis and Gibbs sampling, are more challenging to tune and converge [28] and less efficient in generating large effective sample sizes [29] for high-dimensional distribution functions, e.g., the improved Weibull model in this study, in comparison with the HMC method and the No-U-turn Sampler (NUTS) that extends HMC. HMC is a gradient sampling method established from the Hamiltonian Dynamics theory, rather than the random-walk mechanism, to predict sample locations, thus achieving a higher sampling rate.

Hamiltonian dynamics describe the energy transfer between kinetic energy and potential energy of particles moving in a system. Assuming a particle’s position vector is s and its velocity vector is ϕ, the state of the particle can be described as $\chi =(s,\varphi )$, and the joint canonical distribution of the two vectors is:

$$\mathit{\pi }(s,\varphi )=\mathit{\pi }(\varphi |s)\mathit{\pi }(s)$$

(18)

Assume that the Hamiltonian, H(s,ϕ), satisfies:

$$\mathit{\pi }(s,\varphi )\propto \exp \left[-H(s,\varphi )\right]$$

(19)

where:

$$H(s,\varphi )=E(s)+K(\varphi )=E(s)+\frac{1}{2}{\displaystyle \sum _i{\varphi }_i{}^2}$$

(20)

where E(s) and K(ϕ) denote potential energy and kinetic energy, and are defined as follows:

$$\begin{array}{l}K(\varphi )=-\log \mathit{\pi }(\varphi )\\ E(s)=-\log \mathit{\pi }(s)\end{array}$$

(21)

If π(s) is the target distribution to sample, the HMC method does not directly generate a sample of π(s). Rather, it simulates a sample from the joint canonical distribution π(s,ϕ), which can be specified by putting Equations (20) and (21) into Equation (19) as:

$$\mathit{\pi }(s,\varphi )=\frac{1}{Z}\exp \left[-H(s,\varphi )\right]=\mathit{\pi }(s)\mathit{\pi }(\varphi )$$

(22)

where Z is a normalizing constant, and π(s) and π(ϕ) are independent from each other. The independency between π(s) and π(ϕ) means that the sampling of s via the sample of π(s,ϕ) with any predetermined π(ϕ) can well fit the distribution of π(s). In this sampling process, π(ϕ) acts as an auxiliary variable, generally being designated as normal distribution N (0,1).

According to the Hamiltonian equations of the Hamiltonian Dynamics:

$$\begin{array}{l}\frac{d{s}_i}{dt}=\frac{\partial H}{\partial {\varphi }_i}={\varphi }_i\\ \frac{d{\varphi }_i}{dt}=-\frac{\partial H}{\partial s_i}=-\frac{\partial E}{\partial s_i}\end{array}$$

(23)

the values of s can be derived just by knowing $\partial E/\partial s_i$.

In practice, the aforementioned sampling process needs to be simulated by a leap-frog iterative algorithm, which includes three steps:

$$\begin{array}{l}{\varphi }_i(t+\epsilon /2)={\varphi }_i(t)-\frac{\epsilon }{2}\frac{\partial }{\partial s_i}E\left[s(t)\right]\\ s_i(t+\epsilon )=s_i(t)+\epsilon {\varphi }_i(t+\epsilon /2)\\ {\varphi }_i(t+\epsilon )={\varphi }_i(t+\epsilon /2)-\frac{\epsilon }{2}\frac{\partial }{\partial s_i}E\left[s(t+\epsilon /2)\right]\end{array}$$

(24)

Equation (24) accomplishes the state transition via a change in the velocity variable ϕ with half a time step, while ensuring that the state transition process is reversible and the Hamiltonian H(s,ϕ) remains unchanged in this process. Subsequently, integrated with the aid of NUTS, the HMC sampling process can achieve a much higher sampling rate than other purely random sampling methods.

3. Case Study

3.1. Data Description and Preparation

The raw data of this case study were mainly sourced from the site inspection and maintenance records of the deformation joints, 1104 in total, of a cast-in-situ double cabin utility tunnel project located in a city in Jiangsu Province, China. The raw data included the serial numbers of deformation joints, state descriptions (such as leakage, exudation, corrosion, crack, deformation, etc.), waterproof failure judgments, the environment conditions, maintenance records and the reasons for failure of the deformation joints. Specifically, waterproof failures are identified if there are leakage phenomena of water dripping, outflow or inrush in deformation joints, or the exudation area exceeds the standard. The time interval from the start of operation to the occurrence of waterproof failure is defined as the initial failure time. Once the waterproof facilities of deformation joints are repaired, the start of the failure time refers to the completion time of repair.

The data preparation process involved data cleansing, classification and standardization. As analyzed in Section 2, the required data for the case study of joint waterproof facilities included the failure time (unit: days) and the covariates of the field environment conditions. From the start of operation of the utility tunnel in June 2018 to June 2019, a total of 90 deformation joint failure data were recorded along with the required field condition information. A sample of the recorded data (translated from Chinese) after the preparation processes is showcased in

Table 1. A sample of the recorded deformation joint failure data.

Number of Data Records	Failure Time	Groundwater Type	Repair	Quality Defect	Geological Condition	Foundation Treatment
0	100	Unconfined	Without	Without	Medium	Without
1	34	Unconfined	Without	Without	Medium	Without
2	28	Unconfined	Without	Without	Soft soil	With
3	28	Unconfined	With	With	Medium	Without
4	62	Unconfined	Without	Without	Medium	Without
…	…	…	…	…	…	…
85	73	Unconfined	Without	Without	Medium	Without
86	15	Unconfined	With	With	Soft soil	With
87	243	Unconfined	Without	Without	Medium	Without
88	312	Unconfined	Without	Without	Medium	Without
89	19	Unconfined	Without	Without	Medium	Without

, which will be further explained by

Table 2. Possible reasons for the failure of the joint waterproof facilities of cast-in-situ utility tunnels.

Factors	Failure Reasons	Influence Modes
Component wear-out	Waterproof components such as membranes and waterstops reach their service life and lose their waterproofing functions.	Long-term influence with certain regularities
Construction disturbance	External construction activities damage waterproof components or cause excessive deformations that lead to the failure of waterproof facilities.	Contingent influence that can be avoided by setting safeguarding zones
Differential land settlement	Dislocations and rotations of deformation joints caused by differential land settlements lead to the failure of waterproof facilities.	Long-term influence that is strong in the early stage due to accelerating land settlement but weakens with time
Construction defect	Improper construction methods lead to component installation defects. For example, the internal waterstops fall off during concrete casting, or do not bond well to the side structures due to insufficient vibration.	Early-stage influence
Component defect	Inferior waterproof products reduce the service life of joint waterproof facilities.	Early-stage influence
Natural disaster	Fire or earthquake disasters damage the waterproof system.	Contingent influence

and

Table 3. Field condition covariates and corresponding values.

Covariates	Field Conditions	Values
X1	Groundwater type	Unconfined: 0; Confined: 1
X2	Repair	With: 0; Without: 1
X3	Quality defect	With: 0; Without: 1
X4	Geological condition	Rock or hard soil: 0; Soft soil: 1; Medium: 0.5
X5	Foundation treatment	With: 0; Without: 1

in the following Section 3.2.

3.2. Model Building and Sampling

(1)

Identification of field condition covariates

Waterproofing designs for the deformation joints of utility tunnels are specific to construction methods. In China, utility tunnels are generally constructed by cast-in-situ concrete, which is also the case in this case study. For this reason, this paper will focus on the deformation joints of cast-in-situ utility tunnels, for which the waterproof facilities are typically designed as shown in Figure 3.

It can be seen from Figure 3 that such a waterproofing system consists, from exterior to interior, of a waterproof membrane, a sealant (for tunnel top or sidewalls) or an external waterstop (for tunnel bottom), a foam board filler, an internal waterstop, a foam board filler and a sealant. The joint leakages in cast-in-situ utility tunnels indicate that the entire waterproofing system or some of its components are out of service. Based on the authors’ field investigation, there are multiple factors as listed in Table 2 that can lead to the failure of joint waterproof facilities.

As contingent influence factors, such as construction disturbances and natural disasters, are characterized with low occurrence probability, short effect time and less regularity in the long-term operation and maintenance process, their impacts on the reliability estimation for the joint waterproof facilities of utility tunnels depends more on the scenario-specific analysis. Therefore, this case study only selected the predicable long-term and early-stage failure factors to identify the field condition covariates of joint waterproof facilities. In this manner, the field condition covariates that are related to the long-term and early-stage failure factors of joint waterproof facilities were identified as groundwater type, repair, quality defects, geological conditions and foundation treatment, as listed in Table 3.

(2)

Specification of Prior Distributions for Model Parameters

Theoretically, as long as there are enough observation data, the Bayesian estimation can derive accurate posteriors from any prior distributions. However, in the case of limited samples of this case study, the specification prior distributions will have a great influence on the parameter estimation results. According to the improved Weibull model incorporating field conditions listed in Table 3, it was required to specify the priors of shape parameter m, scale parameter η and covariates β₀, β₁, β₂, β₃, β₄ and β₅.

Since the data records were ranged in the first year of the utility tunnel project in this case study, it was concluded that the joint waterproof facilities were in the early failure stage of operation and maintenance. In this stage, the failure probability of joint waterproof facilities is highly subject to such field condition factors such as land subsidence and construction defects. The corresponding value of the shape parameter m in the early failure stage ranged from 0 to 1, i.e., 0 < m < 1. Assuming that the initial expectations of parameters m and η were 0.8 and 1.5, the Gamma priors that the unknown parameters m and η follow could consequently be specified as m ∼ Gamma (0.8, 1) and η ∼ Gamma (1.5, 1). For field condition covariates, due to the uncertainties of the covariate influence extent in different scenarios, they were temporarily assumed to follow the standard normal distribution, that is, β_i ∼ N (0,1). The positive and negative values of β_i indicate the positive and negative effects of covariables, respectively.

(3)

Posterior Estimation and Sampling

Considering there were few relevant empirical references for the setting of the continuous variable t, this case study used the self-adjusting NUTS (No-U-turn Sampler) for posterior sampling. Key indicators of the sampling process are listed as follows: the sample size was 6000, the fine-tuned sample size was 6000, the target acceptance rate was 0.8, and the number of parallel Markov chains was two.

Figure 4 shows the sampling results of the eight model parameters. The left side of Figure 4 displays the distribution curve of sampling results of each parameter, with the dotted line and solid line representing the sampling results of two parallel Markov chains, and the right side plots the specific values of each sampling process. As can be seen from Figure 4, the sampling process showed the basic characteristics of uniform sampling, which also reflected the randomness of sampling. The sampling results of the two parallel Markov chains were close, indicating that the estimation results of model parameters were reliable.

3.3. Results and Discussion

(1)

Convergence monitoring

This study chose the Gelman–Rubin method [30] to monitor convergence for model parameters on Markov chains. For an iterative simulation with m sequences (m equal 2 in the case study) of length n, the convergence can be monitored via the indicator of potential scale reduction estimated by the following equations:

$$B=\frac{n}{m-1}{\displaystyle \sum _{i=1}^m{{\displaystyle ({\overline{x}}_{i.}-{\overline{x}}_{..})}}^2}$$

(25)

$$W=\frac{1}{m}{\displaystyle \sum _{i=1}^m\left[\frac{1}{n-1}{\displaystyle \sum _{j=1}^n{{\displaystyle ({\overline{x}}_{ij}-{\overline{x}}_{i.})}}^2}\right]}$$

(26)

$$\widehat{R}=\sqrt{\frac{V\widehat{a}r(x\left|y\right.)}{W}}=\sqrt{\frac{\frac{n-1}{n}W+\frac{1}{n}B}{W}}$$

(27)

where B denotes the variance between the m sequence means; W denotes the average of the m within-sequence variances; x is the estimated model parameter; Var(x|y) represents the current variance estimate of the marginal posterior distribution of x; $\widehat{R}$ denotes the potential scale reduction. When n approaches infinity and Markov chains converge, the variance estimate Var(x|y) and within-sequence variance will approach the true variance of parameter x. Therefore, the convergence of estimated parameters can be judged to be adequate when $\widehat{R}$ nears 1. Figure 5 shows the 94% high density intervals (HDI) of all model parameters on the left, and the estimated values of potential scale reduction on the right, from which it allows concluding that all model parameters have adequate convergence on Markov chains.

(2)

Model and sampling evaluation

In this case study, Bayesian Fraction of Missing Information (BFMI) was used to measure the efficiency of sampling from a posterior distribution of parameters in each iteration, which was computed by the following diagnostic [31].

$$\mathrm{BFMI}\approx \frac{{\displaystyle \sum _{i=1}^N{(E_n-E_{n-1})}^2}}{{\displaystyle \sum _{i=0}^N{(E_n-\overline{E})}^2}}$$

(28)

In the HMC algorithm, the essence of BFMI computed by Equation (28) is the ratio of energy transition and marginal energy. If the energy transfer is too small, the sampling efficiency is low, indicating that the sampling parameters need to be adjusted. When BFMI approaches 1.0, it indicates that the sampling efficiency of parameters is high, and approximately independent samples can be collected from the posterior distribution.

Figure 6 shows the BFMI analysis results of this case study. BFMI values of the two parallel Markov chains were 1.15 and 1.10, respectively. Moreover, it is evident that the energy transition almost shared the same distribution with the marginal energy, indicating that the sampling efficiency of the sampling algorithm was relatively high under the current parameter setting.

To evaluate the efficacy of the improved Weibull model, we used the obtained sampling results to make a posterior prediction of the independent variables and compare them with the actual observed values. Figure 7 shows the comparison between the posterior prediction and the observed failure time of joint waterproof facilities, from which it can be seen that the posterior prediction results fitted well with the actual observation data. Meanwhile, it is evident that both the posterior prediction and the observed data showed an early failure tendency, which further demonstrates the validity of the model building.

(3)

Estimation of Model Parameters

Figure 8 shows the estimated posterior distributions of the model parameters at the current time, from which the mean values and credible intervals of the posterior distributions can be obtained (

Table 4. Posterior estimates of the model parameters.

Parameters	m	η	β0	β1	β2	β3	β4	β5
Average values	1.0	5.2	−3.6	0.66	0.48	1.3	1.1	−0.83
94% credible intervals	[0.88, 1.2]	[1.5, 9.2]	[−5.0, −2.3]	[0.22, 1.1]	[−0.35, 1.4]	[0.4, 2.1]	[−0.27, 2.3]	[−1.5, −0.12]

). It can be seen from Table 4 that m = 1 indicates that the joint waterproof facilities of utility tunnels were currently in the useful life stage with a relatively constant failure rate. The high values of β₃ and β₄ reflect the great influence of quality defects and geological conditions on failure rate, while the negative value of β₅ reflects the positive effect of foundation treatment on reducing failure rate. The results were consistent with the solid engineering experiences, indicating that the improved Weibull model is capable of deriving the influence mechanism of field conditions on the failure rate from observed data.

(4)

Reliability estimation results and managerial implications

Based on the estimated model parameters listed in Table 4, this case study estimated the key reliability indicators of joint waterproof facilities for the utility tunnel project when t = 365 days (i.e., one more year of operation from the current time), and the results are analyzed with figures as follows.

Figure 9 shows the waterproof failure rates of deformation joints in the case study utility tunnel project after one-year’s operation. Among them, the failure rates of deformation joints numbered #56 to #72 exceeded the 10% warning limit, for which special diagnosis and risk investigation are expected.

Figure 10 shows the estimation results for joint waterproof facilities of the case study utility results in the accelerated wear-out stage where sharply increasing failure rates are expected after a relatively stable stage. The upper and lower bounds of the black line represent the warning time T_0.8 (when the reliability drops to 80%) and the expected failure time E(T) respectively. The deformation joints estimated at level 1 and level 2 were characterized by a short stable working time and should be treated with strengthened inspections.

According to the estimated reliability indicator, various maintenance strategies can be tailored for joint waterproof facilities predicted with different reliability performances. Figure 11 shows the reliability changing tendencies of deformation joints numbered #12 (colored in blue) and #156 (colored in orange), respectively. The reliability and failure behaviors of these two deformation joints showed different forms. In this case study, 80% and 40% were assumed as critical reliability thresholds, based on which the service lifetime of a joint waterproof facility can be divided into three stages, i.e., stable period, unstable period and unreliable period. With reliability ranging from the initial to 80%, joint waterproof facilities were considered to work in a stable period. When reliability drops lower than 80% but higher than 40%, joint waterproof facilities will enter an unstable working period. During the unstable period, the failure rate of joint waterproof facilities will increase significantly, and inspection efforts should be strengthened to avoid unexpected waterproof failures. When reliability is estimated lower than 40%, it is required to evaluate the waterproof performance of the deformation joint and take necessary measures, such as specified testing and maintenance. As can be seen from Figure 11, the decreasing rate of reliability of joint #12 was significantly higher than that of #156. According to the estimation results in Figure 11, the #12 deformation joint reached the early warning position (80%) in about 50 days and reached the failure critical point (40%) in about 235 days. In contrast, the stable period of the #156 deformation joint was predicted to be close to 200 days, and the unstable period lasted more than two years. According to the failure characteristics of different periods, plans can be made, respectively, to optimize the inspection and maintenance strategies while ensuring waterproof quality.

It should be noted, however, that it is not easy to validate the predictions of the proposed model in a real case scenario. In general, it is appropriate to compare future observed data with predicted results to verify the model accuracy. However, for critical utility tunnel facilities such as joint waterproof facilities, it is unbearable to let the facilities fail without any interventional maintenances considering that they are already predicted with high failure risks. A plausible solution is to select an experimental sample of deformation joints that have failed before but did not cause severe damages to the utility tunnel. Then we can use the comparison results between the observed failure time of these joints from the current time on and their predicted failure time for the validation of prediction accuracy.

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.