According to the geological conditions of the project, the strata were simplified as homogeneous and horizontal. Figure 2A is the three-dimensional geological model of the transfer section, and Figure 2B is a model of the cross passage and the main tunnel. The negative direction of the X axis is the tunneling direction of the cross passage, the positive direction of the Y axis is the large mileage direction of the main tunnel, and the positive direction of Z axis is vertical upward.

The distance from the left and right boundary of the model to the left and right side wall of the main tunnel is 69 m (about three times the width of the tunnel). The distance from the bottom boundary to the inverted arch of the main tunnel is 58 m (about three times the height of the tunnel), and the distance from the upper boundary to the tunnel vault is 23 m. The model takes the dimensions of 160 m × 80 m×100 m in the three directions of X, Y, and Z, with a total of 110,515 units and 20,846 nodes. The soil and rock are simulated by the solid element. The primary support, which is applied around the tunnel boundary permanently, is simulated by the plate element with a thickness of 300 mm. The temporary support, which is applied around each heading hole and will be removed after the temporary support is closed into a ring, is simulated by the plate element with a thickness of 200 mm.

The upper surface of the model is the free surface, the vertical displacement constraint is adopted for the bottom boundary, and the horizontal displacement constraint is adopted for the other four boundaries.

According to the survey data of Guanshui Road subway station provided by the exploration department, the physical and mechanical parameters of the soil are shown in Table 1. The constitutive model of soil is simulated through the Mohr-Column model (Lee et al., 2012; Singh and Singh, 2012; Singh and Mandal, 2019; Zhang et al., 2019). The physical and mechanical parameters of the supporting structure are also shown in Table 1, the elastic modulus can be calculated according to Eq. 1. E = E 0 + S g E g S 0 (1) where E is the equivalent elastic modulus of the primary support, E ₀ is the elastic modulus of the sprayed concrete, E _g is the elastic modulus of steel, S _g is the cross-sectional area of the steel bar, and S ₀ is the cross-sectional area of the concrete.

In order to determine the construction scheme for the transfer section between the cross passage and the main tunnel of Guanshui Road Station, three different construction methods, namely, the double-holes interval pillar method, sector expansion method, and gate climbing method were simulated. And the optimal construction method was determined based on the comparisons, including the stress and displacement of the surrounding rock and the supporting system, and the distribution of the plastic zone of the surrounding rock caused by different methods.

In the double-holes interval pillar method, the cross passage and main tunnel are excavated by step method and double side heading method, respectively. The main construction steps of this method are shown in Figure 2C.

1) The upper and middle heading holes of the main tunnel’s right side are excavated towards the large mileage and the support after finishing the excavation of the upper steps of the cross passage is created. As shown in Figure 2C, step ①;

In the sector expansion excavation method, the rock of the transfer section is excavated by multiple sector-shaped excavation bodies, as is shown in Figure 2D. The detailed steps are as follows: 1) When the upper step of the cross passage is excavated to the junction between the cross passage and main tunnel, a portal arch is erected in the cross passage. 2) The transfer section is excavated, and the upper step of the transfer section is divided into three sector-shaped excavation bodies. The excavation width is equal to the width of the cross passage, and the portal arch and primary support are erected immediately after excavation. 3) The lower step of the cross passage and the main tunnel are excavated by the sequence mentioned above. 4) The annular steel support is erected along the main tunnel and the upper arch of the station is closed into a ring through multiple expansion excavation, then construction of the transfer section is completed.

The gate climbing method is a method to excavate the main tunnel at a gradient of 12% along the excavation direction of the cross passage during the construction of the transfer section. Then the reverse excavation method is used to excavate the rock of the transfer section as shown in Figure 2€. The specific steps are as follows: 1) The cross passage section is excavated in full. When it is excavated to the junction part, three portal arches are erected at the junction between the cross passage and the main tunnel. 2) The transfer section is excavated upwards along the excavation direction of the cross passage with the gradient of 12%, the width of excavation part is the same as that of the cross passage and is excavated to the contour line on the other side of the main tunnel. 3) The transfer section is totally divided into eight parts for excavation. Then the portal arch and primary support are erected immediately. 4) The annular steel supports along the contour line of the top of the main tunnel are erected after every step of excavation. Finally, the steel support is closed into a ring and the construction of the transfer section is complete.

In order to compare the influence of different methods on the supporting structure and surrounding rock of the cross passage-main tunnel, the stress, displacement of the surrounding rock-supporting system, and the distribution of the plastic zone of surrounding rock were analyzed by selecting three sections, namely section 1 (the center of the transfer section) and sections 2 and 3 (from the position of 20 and 30 m away from the transfer section, respectively). The positions of the three selected sections are shown in Figure 2B.

Figure 3 shows the vertical and horizontal displacement nephograms of the supporting structures of the cross passage and the main tunnel after the construction of the double-holes interval pillar method, sector expansion method, and gate climbing method, respectively. In the figure, the vertical displacement is positive in the upward and negative in the downward direction. The horizontal displacement is negative in the heading direction of the cross passage. The maximum vertical displacements of the three methods are all located at the arch bottom of the main tunnel with the small mileage boundary, which are 5.84, 5.66, and 5.78 mm, respectively. The minimum vertical displacements are all located at the vaults of the main tunnel with the small mileage boundary, which are −5.71, −5.82, and −5.86 mm, respectively. The maximum horizontal displacements are located at the arches on both sides of the main tunnel with the small mileage boundary, which are 3.21 and −3.45 mm, 3.14 and −2.86 mm, and 2.92 and −2.75 mm, respectively.

Figure 4A shows that the settlements vary with the construction steps for three sections under the double-holes interval pillar method. As shown in Figure 2B and section 1 is at the right side of the cross passage, section 3 is the cross section of the internal passage, and section 2 is at the middle between section 1 and section 3. In section 3, the construction of upper, medium, and lower heading holes of the internal passage are completed in steps 23, 61, and 85, respectively.

Figure 4A shows that before the excavation of the central core soil, the vault settlements of sections 1, 2 were −0.89 and −1.21 mm, respectively. With the excavation of the left and right heading holes, the vault settlements increase gradually. When the construction steps reach 116 (section 1) and 112 (section 2), respectively, the upper step of the central core soil can be excavated, then the vault settlements increase sharply and finally reach −5.49 and −5.44 mm, respectively. The upper heading hole of the internal passage can be excavated at step 23, causing the vault settlement of the internal passage to increase to −1.11 mm rapidly. With the excavation of the left and right heading holes, the vault settlement of the internal passage gradually increases. The central core soil can be excavated at step 106, resulting in a sharp increase in the vault settlement of the internal passage, which finally stabilizes at −5.13 mm and is less than the vault settlement of section 1 (−5.49 mm).

In Figure 4A, the hance deformation of section 1 slowly increases with the excavation of the left and right upper heading holes and the upper steps of the internal passage. The middle heading hole on the left side is excavated at step 65, causing the hance deformation to rapidly increase to 1.61 mm. When the construction steps reach 100 and 120, the left lower heading hole and the lower step of the central core soil are excavated, respectively, causing the deformation to increase again, and finally stabilize at 2.49 mm. The law of hance horizontal deformation in section 2 is very similar to that in section 1, with three obvious changes, corresponding to the excavation of the middle heading hole on left side, the lower heading hole on left side, and the lower step of the central core soil, and finally stabilizes at 1.86 mm. The hance deformation of section 3 firstly occurs during the excavation of the middle step of the internal passage, then changes significantly during the excavation of the left and right heading holes, and finally stabilizes at 1.66 mm.

Figure 4B shows the settlement curve of the three sections with the construction step after the construction of the sector expansion method is completed. The vault settlement of section 1 reaches −0.91 mm after the construction of the transfer section. When the construction step reaches 46, the central core soil in the middle of the transfer section is excavated, the vault settlement of section 1 increases sharply, and finally stabilizes at −5.09 mm. The vault settlement of sections 2, 3 increases slowly during the construction of the transfer section, which indicates that the construction of the transfer section has little influence on the vault settlement of sections 2, 3. With the excavation of the main tunnel, the vault settlements of sections 2, 3 are similar to those of section 1, and finally stabilizing at −5.01 and −4.81 mm, respectively.

Figure 4B also shows the curve of the hance horizontal displacement of the three sections with the construction step after the completion with the sector expansion method. The upper three sector expanding excavated bodies are excavated at step 10, and the hance deformation of section 1 rapidly increases to 0.36 mm. The middle heading holes on the left and right side of the main tunnel are excavated at step 30, and the value deformation of section 1 increases sharply. Finally, with the completion of the secondary lining construction, the deformation reaches a stable level of about 0.55 mm. The laws of the horizontal hance deformation of sections 2, 3 are similar, which all increase obviously at the excavation stage of left and right heading holes of the main tunnel, and finally stabilize at 0.57 and 0.62 mm, respectively, with the completion of secondary lining construction.

Figure 4C shows the settlement variation of the three sections with construction steps after the completion under the gate climbing method. The variation of the three sections is similar to that of the sector expansion method, and the final vault settlement is stable at −5.21, −4.99, and −4.91 mm, respectively.

Figure 4C also shows the curve of the hance horizontal displacement of the three sections with the construction step after the completion with the gate climbing method. With the excavation of the upper “climbing body,” the hance horizontal deformation of section 1 rapidly increases to 0.58 mm. After completing the construction of the transfer section, it is transferred to the construction of the main tunnel. Affected by this, the deformation of the surrounding rock continues to increase and finally stabilizes at 1.26 mm. The laws of the hance deformation of section 2 and section 3 are basically the same, and both increase obviously when excavating the heading holes and the central core soil of the main tunnel. After the secondary lining, the final deformation reaches 1.37 and 1.46 mm, respectively.

Maximum and minimum primary stress have an important influence on the stability of the tunnel supporting structure. Figure 5 shows the maximum and minimum primary stress vector diagrams of the supporting structure of the transfer passage-main tunnel after the construction under each method.

Figure 5A shows that after the construction of the double-holes interval pillar method, the direction of the maximum primary stress is distributed along the axis of the main tunnel in most areas, but the direction of the large primary stress becomes disordered near the horsehead gate arch bottom in the transfer section. The maximum primary stress is 4,452.26 kPa, which indicates that the supporting structure at the bottom of the horsehead gate arch bottom is in tension. The small primary stress of the supporting structure is distributed annularly along the tunnel in most areas, stress concentration occurs at the sidewall on both sides of the horsehead gate, and the small primary stress direction near the horsehead gate and the main tunnel becomes disordered. The minimum primary stress is −11,734.92 kPa, which indicates that the supporting structure at the spandrel and sidewall on both sides of the horsehead gate are under pressure.

It can be seen from Figures 5B,C that with the construction of the sector expansion method and gate climbing method, the large primary stress of the supporting structure is distributed along the axial of the main tunnel. Different from the double-holes interval pillar method, the large primary stress of the supporting structures of sector expansion excavation method and gate climbing method are distributed alternately in positive and negative directions at the arch bottom of the transfer section, and the stress concentration occurs at the arch bottom of the main tunnel and horsehead gate. The maximum primary stresses of the two methods are 3,946.15 and 3,786.34 kPa, respectively, indicating that the arch bottom supporting structure of the transfer section is in tension.

In addition, it can be seen from Figures 5B,C that after completing the construction under the sector expansion method and gate climbing method, the small primary stress of the supporting structure is distributed annularly along the tunnel. Stress concentration occurs at the positions of the transfer section between the cross passage and the main tunnel. The minimum primary stresses of the two methods are −10,101.22 and −11,495.45 kPa, respectively, which indicates that the spandrel and sidewall supporting structure on both sides of the horsehead gate are under pressure.

The plastic zone has an important influence over the stability of the tunnel surrounding rock and the supporting system. Figure 6A shows the distribution of the surrounding rock plastic zone after the application of the double-holes interval pillar method in construction. It can be seen that the surrounding rock plastic zone of the main tunnel is mainly distributed at the hance and arch foot on both sides of the main tunnel with a maximum plastic zone radius of 3.11 m. However, the plastic zone of the side wall of the horsehead gate in the transfer section is much wider with a radius of 6.38 m. It also shows that the plastic zone distribution of surrounding rock plastic strain is above 300 με after construction with the double-holes interval pillar method. It can be seen that the maximum plastic strain of the surrounding rock appears at the spandrel on both sides of the horsehead gate where the cross passage connects with the main tunnel (see the circular area in the figure), and the maximum equivalent plastic strain value is 0.001189.

Figures 6B,C give the distribution of the plastic zone of the surrounding rock after the construction under the sector expansion method and the gate climbing method, respectively. It can be seen that the plastic zone distribution of these two methods is similar to the double-holes interval pillar method, but the distribution range is slightly larger. The radii of the maximum plastic zone of the main tunnel are 3.40 and 3.21 m, respectively. The radii of the maximum plastic zone of the side walls of the cross passage are 7.08 and 6.84 m, respectively. The plastic zone distribution of the surrounding rock with plastic strain above 300 με after the construction under the sector expansion method and the gate climbing method are also shown in Figures 6B,C, respectively. It can be seen that the maximum plastic strain of the two methods occurs in the position of the side wall on both sides of the horsehead gate where the cross passage connects with the main tunnel, which is different from the double-holes interval pillar method. The maximum equivalent plastic strain values are 0.001425 and 0.001229, respectively.

Table 5 shows the comparison of the final deformation between the numerical simulation results and the monitoring results among the nine monitoring points in Figures 9, 10. It can be seen that the numerical simulation calculation is smaller than the monitoring deformation at each point. This is mainly due to the following two reasons, on the one hand, for the sake of simplicity, the soil is assumed to be a homogeneous and isotropic continuous medium in the numerical calculation, while the actual soil is a heterogeneous and anisotropic discontinuous medium, so the numerical simulation results are smaller. On the other hand, the surrounding rock of the transfer passage is disturbed by the construction many times during the construction process. The weak structural planes and cracks in the surrounding rock are further cracked after being disturbed, resulting in the gradual deterioration of the surrounding rock mechanical properties. The simulation could not consider the deterioration of the mechanical properties in the surrounding rock after the construction disturbance, which also made the numerical simulation results smaller.

Although there are some differences between the numerical simulation results and the measured results, in general, the monitoring ground settlement is 30 mm less than the control value, and the monitoring horizontal displacement in the tunnel is also 20 mm less than the control value. Therefore, it is safe and feasible to use the double-holes interval pillar method to construct the transfer section of the Guanshui Road Station.