The upper and lower adjacent overlapping tunnels in this study are shown in Figure 1. The maximum buried depths of the vault of the upper and lower tunnels are 130 and 149 m, respectively, and the longitudinal position relationship is shown in Figure 2. The tunnel area is mainly woodland with an elevation of 217.9–302.3 m. The minimum height difference between the upper and lower tunnel track surfaces in the tunnel mileage section of this study is about 14.3 m, and the minimum net rock layer is about 2.5 m. The entire section is located on a straight line, and a single-sided slope with a longitudinal slope of 3% is designed.

The tunnel is excavated by the mining method and supported in advance by the large pipe shed. In general, the construction is carried out from the bottom to the top, and after the construction of the lower tunnel is completed, the construction of the upper tunnel will be carried out. Figure 3 shows the construction diagram of the upper and lower tunnels, which are constructed by the two-step temporary inverted arch method, with temporary cross bracing. The initial support of the upper and lower tunnels is C25 shotcrete with a thickness of 30 and 27 cm, respectively, and the lining is C35 reinforced concrete with a thickness of 55 and 70 cm, respectively.

The stratum in the tunnel area is a monoclinic structure with a complex geological structure and poor geological development, mainly including accumulations, coal seam gas, soft rock, and trench soft soil. The overall surrounding rock of the tunnel is relatively soft, the joint fissures are relatively developed, and it is covered with artificially filled soil, silty clay, and silty clay. The surface water in the site area is mainly gully water, which is recharged by atmospheric rainfall, while the groundwater is mainly poured water and fissure water, which is mainly recharged by infiltration such as atmospheric rainfall. It has little impact on concrete and tunnel construction.

The vertical spacing between the upper and lower tunnels is about 2∼5 m, the horizontal spacing is about 0–22 m, and the buried depth of the upper tunnels gradually increases from 3 to 75 m. Five typical sections in the study section are selected for research. The relative position relationship between the upper and lower tunnels is shown in Figure 4, and the corresponding buried depth and spacing are shown in Table 1.

The interaction between the stratum and the structure is analyzed and studied by finite element numerical simulation. Taking into account the range of interaction between the research object and the surrounding soil, select the influence range of tunnel excavation (3–5 times the tunnel radius). According to the relevant design data, the mechanical parameters of the rock mass around the tunnel section are selected. In the calculation model, the surrounding rock is simulated by the two-dimensional elements of the Mohr-Coulomb constitutive model. The secondary lining is simulated by linear elastic constitutive two-dimensional elements. One-dimensional beam elements with an elastic constitutive model are adopted for concrete and temporary cross braces. In some cases, bolt support can be equivalent to soil reinforcement in numerical simulation. The detailed values of the material parameters of the model are shown in Table 2.

Set the normal displacement constraint boundary on the side and bottom of the model, and the boundary on the top of the model is a free surface. As shown in Figure 5, take Section 4 as an example to illustrate the numerical simulation grid division of the model. The model is 110 m long and 118 m high. The upper and lower tunnel excavation zones and the secondary lining model are shown in Figure 6.

According to the actual working conditions, the lower tunnel is first excavated, and then the upper tunnel is excavated. The construction sequence of the two tunnels is shown in Table 3. According to the overlap, the tunnel construction process is divided into 12 construction stages according to the simulation of key steps.

Take Section 4 as an example to present the simulation results and analyze them. Figure 7 shows the vertical displacement cloud image caused by the completion of Step12. During the excavation of tunnels, arch crown settlement and arch bottom uplift will occur. After the construction of the upper tunnel, the vault settlement of the lower tunnel increased from 8.9 to 9.7 mm, an increase of 9%, compared with the results of step 7. And the distribution of stratum settlement is asymmetric, and the settlement trend is inclined below the joint action of the upper and lower tunnels.

Figure 8 shows the horizontal displacement cloud diagram caused by the completion of Step12. During the excavation of the tunnel, the arch waist and the area above it will produce horizontal deformation, which can increase the potential factors of the secondary lining tension cracking. After the construction of the upper tunnel, the horizontal deformation of the arch waist and above the lower tunnel increased from 10.7 to 11.3 mm, an increase of 6%, and the horizontal convergence increased from 23.2 to 25.6 mm, an increase of 10%, compared with the results of step 7. It can be seen that the horizontal convergence deformation of the arch waist and the area above increased due to the construction of the upper tunnel.

Figure 9 shows the maximum stress cloud image caused by the completion of Step12. During the excavation of the tunnel, there is a concentration of shear stress around the tunnel, which is mainly concentrated in the range of 3–8 m around the tunnel, that is, around the end of the anchor rod. Therefore, in actual construction, the quality of the anchor rod should be ensured, especially the depth of the anchor rod and the stability of the soil near the tunnel. At the same time, the shear stress concentration area at the stratum connection part is significantly higher than that of the surrounding stratum, which means that the stratum soil in this area will be damaged first, and it should be properly reinforced during construction. In addition, the shear stress concentration of the lower tunnel is mainly concentrated above the arch waist, which is close to the secondary lining crack found on site.

Figure 10 shows the distribution cloud map of the formation plastic zone caused by the completion of Step12. Part of the plastic zone appears around the tunnel excavation area, and it is easy to form a penetrating plastic zone between the upper and lower tunnels. Moreover, the plastic zone of the lower tunnel is mainly located above the arch waist on both sides, which is also consistent with the location of the shear stress concentration. The plastic zone reflects the distribution of rock mass instability and failure, especially at the stratum connection. The distribution of the plastic zone is relatively obvious, and there is serious penetration. Therefore, in the actual construction, one needs to pay attention to the proper reinforcement of the rock mass at the connection between the upper and lower tunnels to avoid the penetration of the plastic zone.

Figure 11 shows the vertical stress distribution cloud diagram of the secondary lining caused by the completion of Step 12. It can be seen that the tensile stress of the secondary lining of the lower tunnel appears to stress concentration near the left and right arch waists, and the maximum tensile stress in the Y direction is 4.15 and 4.46 mpa, respectively. The tensile stress distribution of the left and right arch waist is significant.

According to the actual measurement on site, after the construction of the upper tunnel (post-excavation tunnel) was completed, longitudinal cracks appeared in some areas of the lower tunnel (existing tunnel). As shown in Figure 12, the cracks occurred mainly at the right waist of the lower tunnel, which was adjacent to the upper tunnel. As shown in Figure 13, cracks appeared in both the left and right arches of the lower tunnel. Table 4 has given the specific conditions of the cracks. It can be found that the crack length is 12 m, the crack width is 0.2∼0.4 mm, and the crack depth is 7.7∼12.5 cm.

When comparing the tensile stress calculation result of the obvious tensile area of the secondary lining of the lower tunnel caused by the construction of the upper tunnel with the standard value of the tensile strength of C35 concrete of 2.2 MPa, it can be found that the maximum tensile stress of the secondary lining of the lower tunnel obtained by the cross-section simulation calculation is greater than 2.2 MPa, so cracks extending along the Z direction in the XY plane will occur, which appear as longitudinal cracks.

Although the above is a quantitative judgment, to further analyze the development law of tunnel cracks caused by construction mechanical properties, it is necessary to analyze the cause and law of the secondary lining cracks of the lower tunnel based on the characteristics of stratum deformation and the distribution law of stratum shear stress and plastic zone.

Figure 14 shows the tunnel displacement curve of each section after the completion of step 12, and Figure 15 shows the absolute change rate of the influence of the upper tunnel on the lower tunnel, in which the tunnel buried depth, tunnel vertical clear distance, and tunnel horizontal distance from Section 1 to Section 5 gradually increase. According to Figures 14, 15, in terms of vault settlement, when the buried depth is shallow and the tunnel distance is close, the construction of the upper tunnel will rebound the vault settlement of the lower tunnel to a certain extent, in which Section 1 rebounds by 12%. With the increase in the distance between the two tunnels, it will gradually increase the vault settlement of the lower tunnel, in which Section 5 settlement increases by 7%. In terms of invert displacement, the influence of the upper tunnel construction on the invert of the lower tunnel uplift is relatively small, of which Section 1 has the largest change rate, and the uplift is only increased by 5%. In terms of horizontal convergence, the construction of the upper tunnel will lead to an increase in the horizontal displacement of the lower tunnel. The greater the distance between the two tunnels, the greater the increase and the maximum increase is 11%. It can be seen that the vault settlement and horizontal convergence of the lower tunnel at different sections are mainly affected, while the inverted arch has little effect. Considering the vertical deformation of the arch crown and the horizontal deformation of the arch waist, greater oblique deformation above the arch waist will be obtained. The deformation law of the upper part of the arch waist reflects the causes of the cracks in the lower tunnel.

Figure 16 shows the maximum shear stress distribution of different sections after step 12. It can be seen that the shear stress is mainly concentrated in the range of 3∼8 m around the tunnel. In this range, the overlapped tunnel arrangement has the greatest influence, and the most significant part is still in the upper part of the lower arch. As the upper and lower tunnels are gradually staggered on the plane, this influence gradually weakens, and as the distance increases and staggers, the maximum shear stress is gradually distributed in the area between the two tunnels, and the influence of the right arch of the lower tunnel is more obvious than that of the left arch.

1) The characteristics of tunnel displacement and the distribution of shear stress indicate that there is a greater risk of cracks near the left and right arch waists of the lower tunnel, which not only reflects the location of the cracks, but also explains the reasons for the changes in the cracks.