## 1 Introduction  
Interlaid rock is a crucial component of neighborhood tunnels, playing supporting and reinforcing roles between the two tunnels. During tunnel excavation using drilling and blasting methods, the cumulative damage from blasting vibrations on the interlaid rock can lead to a decrease in its mechanical properties [1], posing a threat to the stability of the tunnel structure. In neighborhood tunnels, where the distance between the dual tunnels is relatively small, the constructions of an advance tunnel (AT) and a following tunnel (FT) mutually influence each other. In the research process, it is essential not to consider a single tunnel in isolation. Studies [2; 3; 4; 5] indicate that the excavation of the FT has a greater impact on the AT. [PERSON] et al. [3] monitored the vibration velocity induced by FT blasting and found that the peak particle velocity (PPV) at the monitoring point close to the interlaid rock was 5-to-10-times more than that of away from the interlaid rock in the AT. The results indicate that the negative influence for interlaid rock is mainly from FT blasting. [PERSON] et al. [4] and [PERSON] et al. [5] have also reached this conclusion. Therefore, the key technical aspects [6] of safe construction in neighborhood tunnels mainly include appropriate clearance distance, support parameters, construction methods, and reinforcement techniques for the interlaid rock, among others. For the AT in neighborhood tunnels, its excavation method can be referenced from conventional single-tunnel construction. The selection of an excavation method for the FT, however, leans more towards geological conditions and the surrounding environment. In favorable geological conditions, the full-face method [7] or the benching method [8] are commonly adapted. In challenging geological conditions, options include the center diaphragm method with multiple sections [9; 10], the cross-diaphragm method [11; 12], or the method of circular excavation with reserved core soil [13]. For the study cases where the thickness of the interlaid rock is small, it is necessary to optimize conventional excavation methods. In the excavation of neighborhood tunnels with a 2.7 m thick interlaid rock, [PERSON] et al. [14] have proposed the method of reserving a thin rock layer for segmented excavation to safely excavate the FT. This is an effective solution. However, the thickness of the reserved rock layer is determined based on construction experience. [PERSON] et al. [15] have simplified the cross-section of laterally excavated tunnels into a dual-connected domain problem using the method of equivalent circles. By combining the theory of complex variables and the Schwarz alternating method (SAM), they derived the specific form of the stress function for laterally excavated tunnels and established the relationship between single excavation width, surrounding rock stress, and displacement characteristics. The SAM is a successive approximation analytical method for studying the problems of double circular holes [16; 17] and multiple elliptic holes [18; 19] in solids. [PERSON] et al. [20] and [PERSON] et al. [21] have used the SAM to solve the analytical stress and displacement of twin tunnels. The analytical solutions are verified by the good agreement between the analytical and finite-element method results under the same assumptions. Inspired by those researchers, we similarly simplify the excavation of neighborhood tunnels into a dual-connected domain problem using the SAM. We deduce the impact of the small clearance distance on the surrounding rock stress of the two tunnels and use numerical methods to determine the optimal distance between neighborhood tunnels. This information is then used to calculate the thickness of the reserved rock layer. This paper takes the Shenzhen Liantang tunnel, a typical neighborhood tunnels project, as the engineering background, where the minimum distance between the two tunnels is 0.5 m. The aim is to safely excavate the FT and protect the interlaid rock.