## 5 Conclusions  
In order to understand the underlying process of shield joint leakage and the influential law of factors on it, this research describes an analytical method for shield lining joint leakage calculation, which is built on the Persson contact mechanics and percolation theory. Various influencing factors, including the joint gap, the joint offset, the roughness of the contact surface, the service time, and groundwater pressure on the shield joint leakage, can be reflected by the parameters in the model. Based on this method, the critical leakage state and the self-sealing effect were defined, and their calculations are derived using the law of [PERSON]'s inner friction. Then, taking the Dapu Road Tunnel in Shanghai (China) as a case study, the calculation results of self-sealing stress was compared with results in the literature, and the influential law of factors on shield joint leakage was analyzed, and the calculation results were verified through the leakage data observed at the tunnel. The following conclusions were drawn:  
1. Any sealing interface is permeable; they are mainly sealed by the interfacial fluid, which can form a plug, preventing further penetration. We called this phenomenon the self-sealing effect. The greater the squeezing pressure, the narrower the size of the critical percolation channel, which further provides stronger self-sealing stress. 2. The joint gap is often accompanied by joint offset. For the case where the joint offset is independent of the joint gap, the offset with considerable contact stress will enhance the joint waterproof performance, while, for the gap-dependent offset, the offset will increase the leakage. The leakage mainly depends on the increasing gap, and the effect of the offset change on leakage is slight. 3. The leakage increases with the aging of the sealing material. The aging model of Shi CH is utilized to calculate the leakage trend for one year, and the results show good agreement with the monitoring value of the second Dapu Road Tunnel. 4. The orthogonal array testing is designed with four factors including the joint gap, the surface roughness, the service time, and groundwater pressure in five levels. The range analysis showed that the roughness of the contact surface is the most influential, followed then by the service time, groundwater pressure and the joint gap. 5. The applicability of this model was verified by comparing the calculation results with experimental results in the literature. However, due to the differences between the hypothesis and the actual situation of engineering, this model aims to help more clearly understand the underlying processes of shield lining joint leakage and the influential laws of various factors rather than accurately predict it. Conceptualization, Y.W. ; Funding acquisition, Y.Y. and L.W. ; Methodology, Y.W. ; Software, Y.W. ; Supervision, Y.Y. and L.W. ; Validation, Y.W. ; Visualization, Y.W. ; Writing--original draft, Y.W. ; Writing--review & editing, F.S. and L.W. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Key Research and Development Program of China: 2017 YFC0805008. The authors declare no conflict of interest. Definitions are given here in the following sequence: Bold quantities denote vectors or tensors.