The main factors affecting the adjacent tunnel lining are tunnel heading pressure and tail-grouting pressure. This section investigates the influence of different grouting pressure and supporting pressure on the adjacent EB tunnel lining during WB TBM-driven process (see

Figure 8). Grouting pressure and supporting pressure are controlled by the coefficients

${K}_{g}$ and

${K}_{s}$.

Table 3 shows the values of grouting pressure G and supporting pressure S in different working conditions, where

$G={K}_{g}\times {G}_{0}$,

$S={K}_{s}\times {S}_{0}$ (

${G}_{0}$ and

${S}_{0}$ are the values of Case 0).

#### 4.1. Ground Surface Settlement

Figure 9 shows the influence of the coefficients

${K}_{g}$ and

${K}_{s}$ on ground surface settlement. As shown in

Figure 9a, the TBM tail of the second tunnel was parallel to ER60 of the first tunnel in a different case, the grouting pressure range from 0.6 to 1.4

${G}_{0}$ with an increment of 0.2

${G}_{0}$. The first tunnel construction results in a typical settlement trough which is consistent with the Gaussian distribution assumption [

32]. Lower grouting pressures result in larger surface settlement above the second tunnel. Double-bottom settlement type can be observed during the tunneling process of the second tunnel. While the grouting pressure is 0.6

${G}_{0}$ (Case 1), the maximum settlement of 7.5 mm occurs above the center line of the second tunnel.

Figure 9b shows the settlement curves under different face supporting pressure when the second tunnel face arrived at ER 60. The heading pressures of different cases range from 0.6 to 1.4

${S}_{0}$ with the gradient of 0.2

${S}_{0}$. Comparing

Figure 9b with

Figure 9a, we can observe that the surface settlement of the first tunnel induced by the second tunnel advance is more sensitive to the grouting pressure than face supporting pressure.

#### 4.2. Tunnel Displacement and Convergence

(1) Tunnel displacement

The displacements of ER60 caused by the grouting pressure during the excavation of WR38 to WR50 (see

Figure 8) are presented in

Figure 10. The increasing C/D ratio in

Figure 10 indicates the TBM advancing at the WB. A positive value of the vertical axis indicates lateral displacement towards the second tunnel. As illustrated in

Figure 10, the lateral displacement towards the second tunnel decreases with increasing grouting pressure, indicating a notable ground supporting under sufficient grouting pressures. The inward displacement increases with the TBM advance in Case 2, Case 0, Case 3, and Case 4. However, the opposite phenomenon can be observed in Case 1, indicating that the effect of soil movement towards the second tunnel exceeds that of the second tunnel ovalization.

As shown in

Figure 11, settlement can be observed at the top of the first tunnel during the TBM-driven process of the second tunnel. The first tunnel settlement shows negative correlation with the face supporting pressure. With the TBM advance, the maximum settlement of each case occurs at a C/D ratio within a range of 0.56–0.59. As the TBM was driven away from the monitoring section (i.e., ER60), the vertical displacement increases gradually, ascribing to the influence of the second tunnel upheaval. It can be observed from

Figure 11 that the cases with a

${K}_{s}>1$ cannot prevent the first tunnel from settlement, although the settlement is small enough (less than 1 mm which is not easy to be measured by ordinary monitoring approaches) to be ignored.

(2) Tunnel convergence

Figure 12 illustrates the influence of grouting pressure and face supporting pressure on the convergence of ER60 in the first tunnel. As shown in

Figure 12, a horizontally oval-shaped lining deformation can be observed. However, the lining ovalization is not symmetric, ascribing to the squeezing effect caused by the TBM-driven process of the second tunnel. As can be seen from

Figure 12, with the increase of grouting pressure, the convergence of ER60 decreases at the top and increases at the side wall adjacent to the second tunnel, demonstrating that lower grouting pressure results in a smaller squeezing effect on the first tunnel. In

Figure 12b, the development of lining convergence under different supporting pressure (i.e., Cases 5 to 8) shows the similar trend as that under different grouting pressures (i.e., Cases 1 to 4). However, different face supporting pressures have minor influence on the convergence of the first tunnel (ER60), ascribing to the reason that the acting direction of the face supporting pressure is perpendicular to the first tunnel convergence.

#### 4.3. Bending Moment of Lining Segment

Figure 13 illustrates the changes in the lining bending moment of the segments in different cases. As can be seen in

Figure 13, the grouting pressure of the second tunnel advance has significant effect on the bending moment of ER60 in the first tunnel. However, similar to the influence on the first tunnel convergence, the face supporting pressure has little influence on the bending moment of the first tunnel, since the face supporting pressure mainly affects the stress in the axial direction of the tunnel. As the grouting pressure increases, the positive bending moment decreases at the lining sidewall (see

Figure 13a). This can be attributed to the unloading effect caused by insufficient grouting pressure in the second tunnel. The maximum positive bending moment in all cases is 113.4 kN·m in Case 1, with the minimum magnitude of 85.0 kN·m in Case 4. In all cases, the maximum magnitude of negative bending moment is −91.82 (Case1), and the minimum is −63.03 kN·m (Case 4).

In

Figure 14, the relationship between the bending moment

$M$ and the grouting pressure index

${K}_{g}$ is fitted with linear functions (see Equation (19)). The similarity

${R}^{2}$ in different fitting functions are over 0.99.

where

$a$ and

$b$ are the linear fitting parameters.

In

Figure 14a, the max negative bending moment

${M}_{n}$ on the top of lining decrease with increasing

${K}_{g}$. In addition, with the decrease of C/D, the slop

${b}_{j}$ and the magnitude of intercept

${a}_{j}$ decrease. In

Figure 14b, the max positive bending moment on the EB sidewall lining decreases with the increase of

${K}_{g}$. With the decrease of C/D, the magnitude of slop

${b}_{i}$ and the intercept

${a}_{i}$ of different fitting lines decrease. Notably, all fitting lines intersect at the point adjacent to

${K}_{g}=1$ (i.e., Case 0) in which the grouting pressure during the second tunnel construction is equal to

${G}_{0}$, revealing that Case 0 is the most suitable case since the variation of the bending moment in the first tunnel is stable with the TBM advance of the second tunnel.

Figure 15 shows that the relationship between face supporting pressure and the bending moment under different tunnel depth can be fitted by Lognormal function:

where

${M}_{0},A,w,{K}_{c}$ are the fitting parameters in the Lognormal function with the details demonstrated in

Table 4.

As shown in

Figure 15a, when C/D exceeds 0.56, as the grouting pressure increases, the bending moment on the first tunnel side-wall firstly increases and then continuously decreases with the maximum value at

${K}_{s}=0.7$. This can be explained as that the first tunnel bending moment can be affected by the loosened zone of the second tunnel, which formed by the soil arching effect. When the supporting pressure is greatly reduced, it not only affects the soil within the tunnel, but also the soil above the tunnel, resulting in the change of the loosened zone [

47]. When C/D is lower than 0.56, the bending moment of the first tunnel continuously decreases as the supporting pressure increases. In

Figure 15b, the bending moment can also be fitted by a lognormal curve, as expressed in Equation (20). With the increase of K

_{s}, the bending moment continuously increases. When C/D is greater than 0.58, the bending moment value increases as the C/D decreases. However, the bending moment of ER55 (C/D = 0.6084) is greater than ER60 (C/D = 0.5821). This is due to the fact that when the overburden is thick enough, soil arching can be formed, which effectively reduces the bending moment of the first tunnel lining.

Figure 15a and

Table 4 show that the peak phenomenon in bending moment of the sidewall gradually disappears with the C/D decreased to 0.55. Comparing with

Figure 15b, a conclusion can be drawn that the critical C/D, under which the horizontal (see

Figure 15a) and vertical (see

Figure 15b) soil arching effect disappears, locates between 0.55–0.60. This is in consistent with previous research [

33].