## 1 Introduction  
In water network infrastructure construction, the utilization of water conveyance tunnels is widespread for achieving the interconnection of rivers, lakes, and reservoirs, facilitating the optimal allocation of water resources. The lining support structure plays a crucial role in ensuring the safe and stable operation of water conveyance tunnels. Currently, there is extensive research focused on understanding the bearing mechanisms of lining support. Scholars have employed various methods to analyze the factors influencing the effectiveness of lining support from diverse perspectives. The depth of the tunnel plays a significant role in determining the effectiveness of lining support. Analysis typically revolves around two scenarios: deep burial and shallow burial. The distinction between them hinges on whether the impact of the ground stress gradient around the tunnel can be disregarded. [PERSON] and [PERSON] et al. [1; 2] first solved the stress results for a circular tunnel with lining by using the complex function power series method proposed by [PERSON] [3] and considered the misfit and interaction between surrounding rock and lining. [PERSON] et al. [4; 5] also conducted research on lining support mechanisms by using the complex function power series method and conformal transformation method to obtain the stress components at any point in the lining or surrounding rock, and they considered the influence of different contact modes between surrounding rock and lining. [PERSON] et al. [6] presented complex variable solutions for soil and lining deformation based on a deterministic nonuniform convergence boundary model for displacement-controlled conditions at the tunnel opening, indicating the key parameters affecting the tunneling-induced soil deformation including tunneling gap, soil and lining physical characteristics, and the buried depth, as well as lining thickness. For the analysis of the lining support mechanism of a shallow tunnel, [PERSON] et al. [7] used complex potential functions combined with a conformal mapping method to determine stress components within the concrete lining and the surrounding rock mass but with the assumption of uniform radial displacement of the surrounding rock inner edge. [PERSON] et al. [8] proposed a semi-analytical solution for stress and displacement of a lined circular tunnel at shallow depth, based on full-contact interaction between surrounding rock and lining. The current study assumes the tunnel to be deeply buried; thus, the original loading conditions of the tunnel are depicted in Figure 1.  
According to the obtained research results, the contact mode between surrounding rock and lining has a significant impact on the bearing effect of the lining. The main forms of contact include complete contact [4; 9; 10], pure slip contact [5; 11; 12; 13], and frictional slip [14; 15; 16; 17]. Among these, complete contact mode denotes continuous stress and displacement coordination at the contact surface, and pure slip contact mode denotes that this occurs only in the radial direction of the contact surface, stress continuity and displacement continuity are satisfied, and there is no shear force transfer. Frictional slip mode is a state between the above two contact modes. The study in this paper is based on the assumption of pure slip contact between the tunnel and the lining. Figure 1: Schematic diagram of circular hydraulic tunnel with lining. The interaction between the surrounding rock and lining structures of tunnels with different hole patterns is also an important research direction. [PERSON] et al. [18] proposed a semi-analytical elastic solution for the stress field of a lined non-circular tunnel using a complex variable method, and then, [PERSON] et al. [9; 10; 11] put forward an analytical solution and took the support delay and contact modes into consideration. According to the design specifications for hydraulic tunnels in China, the hole type used in pressure tunnels should be circular.