Within the dewatering radius, the dewatering will cause a funnel-shaped dewatering curve in the soil; as shown in Figure 2, R ₀ is the radius of the dewatering well, H ₀ is the initial water level height of the phreatic aquifer, and H _t is the water level height in the well after dewatering. Based on Dupuit’s assumption, the flow rate at different water levels is equal to the amount of water pumped from the well (Verruijt, 1982): Q = 2 π r h k t d h d r (1) where r is the horizontal distance from the well, h is the water level height at this position, and k _t is the permeability coefficient of soil. The dewatering curve h(r) can be obtained by substituting the water level boundary conditions of the dewatering well h = H 0 , r = R h = H t , r = R 0 and the dewatering radius into Eq. 1: h r = H 0 2 − Q π k ln R r = H 0 2 − H 0 2 − H t 2 ln R r ln R R 0 (2)

According to Sakukin’s formula, the dewatering radius can be calculated by R = 2 s w k H 0 (Verruijt, 1982), s w = H 0 − H t . Within the dewatering radius, the decrease in water level will reduce pore water pressure in the soil, leading to an increase in effective stress, and this increased amplitude can be calculated according to the change in water level after dewatering; It can be divided into two cases according to the relative position of the calculated point.

As shown in Figure 2, point D₁ is above the water level after dewatering, and point D₂ is below the water level after dewatering. Therefore, the effective stress increment σ ₁ and σ ₂ of the two points can be calculated as follows: σ 1 = σ t 1 − σ 01 = h 0 + h 1 γ − h 0 γ + h 1 γ s − h 1 γ w = h 1 γ − γ s + γ w (3a) σ 2 = σ t 2 − σ 02 = h 0 + H 0 − h γ + h 2 − H 0 + h γ s − γ w − h 0 γ + h 2 γ s − h 2 γ w = H 0 − h γ − γ s + γ w (3b) where σ _t and σ ₀ are the effective stresses of D₁ and D₂ points before and after dewatering, respectively; and h ₀ is the buried depth of the initial water level from the ground surface; h ₁ and h ₂ are the elevation differences between D₁ and D₂ points and the initial water level, respectively; γ, γ _s and γ _w are unit weight and saturated unit weight of soil, and unit weight of water, respectively.

Above and below the water level, their effective stress increments after dewatering are different. According to the relative position of the tunnel and water level, the additional stress caused by dewatering is different. Therefore, the additional stress caused by dewatering can be calculated in two cases: all parts of the tunnel are below the water level after dewatering, and part of the tunnel is above the water level, as shown in Figure 3. Based on the coordinate system established in Figure 1, the additional stress on the tunnel under the two conditions is calculated as follows:

When the tunnel is below the water level, the distance between any point on the tunnel and the dewatering well r = x 2 + d 2 , after substituting it into Eq. 2 and Eq.3a can be obtained: σ x = H 0 − H 0 2 − H t 2 ln R x 2 + d 2 ln R R 0 ⋅ γ − γ s + γ w (4)

When the tunnel is partially located above the water level, the additional stress of the tunnel is divided into two parts. The additional stress of the tunnel above the water level is the same and is a fixed value. σ = h 1 γ − γ s + γ w (5) where h ₁ is the height difference between the buried depth of the tunnel axis and the initial water level. The calculation of additional stress of other tunnels located below the water level is similar to Eq. 4. When h(r)= H ₀-h ₁, it is the intersection point between water level and tunnel, so the coordinate of this point can be obtained as x = R H 0 − h 1 2 − H t 2 H 0 2 − H t 2 ⋅ R 0 2 H 0 h 1 − h 1 2 H 0 2 − H t 2 2 − d 2 (6)

In this review, when the tunnel part is located above the water level, the formula for calculating the additional stress of the tunnel is: σ x = H 0 − H 0 2 − H 0 2 − H t 2 ln R x 2 + d 2 ln R R 0 ⋅ γ − γ s + γ w , x 2 > R H 0 − h 1 2 − H t 2 H 0 2 − H t 2 ⋅ R 0 2 H 0 h 1 − h 1 2 H 0 2 − H t 2 2 − d 2 ; h 1 γ − γ s + γ w , x 2 ≤ R H 0 − h 1 2 − H t 2 H 0 2 − H t 2 ⋅ R 0 2 H 0 h 1 − h 1 2 H 0 2 − H t 2 2 − d 2 (7)

The longitudinal deformation of the shield tunnel is composed of bending and shear deformation (Shen et al., 2014; Wu et al., 2015). When the traditional Euler-Bernoulli beam is used to simulate a shield tunnel, the tunnel is regarded as a structure with infinite shear strength, so the shear deformation of the tunnel is ignored. Since shield tunnels are composed of annular segments, unlike continuous structures such as pipelines, Euler-Bernoulli beams will underestimate the longitudinal deformation. The Timoshenko beam can take into account the shear deformation of the tunnel under additional loads (Li et al., 2015). In addition, the Pasternak foundation model can consider the continuity of foundation deformation and simulate the interaction between soil and tunnel well (Xu et al., 2021). Therefore, the tunnel is considered an infinitely long Timoshenko beam on the Pasternak foundation in this research to take into account the shear deformation generated by the tunnel, as shown in Figure 4. The section of the Timoshenko beam is no longer perpendicular to the neutral axis but crosses the normal direction of the neutral axis with an angle due to shear deformation. The stress deformation mode is more complex than the Euler-Bernoulli beam, as shown in Figure 5.

According to Timoshenko beam deformation theory, the relationship between tunnel bending moment M, shear force, and Q and displacement w is: Q = κ G A d w x d x − θ (8) M = − E t I t d θ d x (9) where κ is the equivalent section coefficient, the tunnel is the annular section, κ= 0.5; G is the tunnel shear modulus, G=E _t/2(1+v _t); v _t is Poisson’s ratio of tunnel; A is the annular section area of the tunnel; E _t is the elastic modulus of the tunnel; I _t is the moment of inertia of the tunnel. According to the equilibrium differential equation: p x D d x − q x D d x − d Q = 0 (10) p x D 2 d x 2 + Q d x − q x D 2 d x 2 − d M = 0 (11) where q(x) is the additional load generated by dewatering, q(x)= σ(x); D is tunnel diameter. p(x) is the ground reaction force. According to the Pasternak foundation model, p(x) is obtained as p x = k w x − g s d 2 w x d x 2 (12) where k and g _s are the elastic coefficient and shear coefficient of the foundation, respectively, which can be calculated as follows (AttewellB et al., 1986; Tanahashi, 2004): k = 1.3 E s D 1 − v 2 E s D 4 E t I t 1 / 12 (13) g s = E s t 6 1 + v (14) where E _s soil elastic modulus; v is Poisson’s ratio of soil; t is the soil shear layer thickness, and t=6D is used for calculation (Xu, 2005).

Through Eqs. 8–12, the governing equation of tunnel displacement w(x) can be obtained as: E t I t g s D κ G A + 1 d 4 w x d x 4 − k D E t I t κ G A + g s D d 2 w x d x 2 + k D w x = q x − E t I t κ G A d 2 q x d x 2 D (15)

Five groups of tests are designed to study the influence of the distance d between the tunnel and the dewatering well on the tunnel deformation while the other parameters remain unchanged, whose distances are 6, 8, 10, 12, and 14 m, respectively. It can be seen from Figure 8 that when the spacing d increases from 6 to 14 m, the maximum vertical displacement of the tunnel decreases from 5.72 to 3.89 mm. This is because when the distance between the tunnel and the dewatering well gradually increases, the influence of dewatering on the tunnel is weakened. Therefore, the dewatering well should be as far away from the tunnel as possible to avoid excessive tunnel deformation in the project.

Five groups of tests with different tunnel shear modulus G_p were taken to study its influence on tunnel deformation while the other parameters remained unchanged, including 1/125, 1/25, 1/5, and 5 times the original shear modulus. Figure 9 shows the tunnel displacement curves caused by pre-dewatering under different tunnel shear modulus G_p. As can be seen from Figure 9, when the tunnel shear modulus G_p decreases from 5 G_p to 1/125 G_p, the maximum vertical displacement of the tunnel rapidly and significantly increases from 4.38 to 14.93 mm. This is because when the shear modulus G_p decreases, the ability of the tunnel to resist the influence of dewatering is weakened. Therefore, strengthening the soil around the tunnel can reduce tunnel deformation.

Five groups of tests with different soil elastic modulus E _s were designed to study its influence on tunnel deformation while other parameters remained unchanged, which were 10, 20, 30, 40, and 50 MPa. It can be seen from Figure 10 that the maximum vertical displacement of the tunnel rapidly decreases from 12.36 to 2.75 mm as the soil elastic modulus E _s increases from 10 to 50 MPa. That is, because when the elastic modulus of soil E _s increases, the foundation is less likely to deform. On the other hand, when the tunnel bends, the foundation can provide a more significant reaction force to prevent tunnel deformation. Therefore, strengthening the soil around the tunnel can reduce tunnel deformation.

Five groups of tests with different H _t were designed to study its influence on tunnel deformation while other parameters remained unchanged, which were 30, 25, 20, 15, and 10 m, and the corresponding s_w was 10, 15, 20, 25, and 30 m respectively. Figure 12 shows the water level curve in the strata near the dewatering well under different water level dropping depths s _w. It can be seen that with the increase of the dropping depth of water level s _w in the well, the water level in the surrounding stratum also decreases, and the decreasing range gradually decreases. When the dewatering s _w reaches 20 m, the groundwater level is still above the buried depth of the tunnel axis, and the additional stress on the tunnel can be calculated by Eq. 4. However, when s _w of water level drawdown reaches 25 m, the part, that is, the closest to the dewatering well on the tunnel is already below the water level; In this case, the additional stress on this part is calculated by Eq. 5. Figure 13 shows the additional stress sustained by the tunnel under different water level drop depth s _w. It can be seen that the additional stress on the tunnel increases with the increase of s _w of the water level. This is because the more the water level drops, the less water buoyancy the soil above the tunnel receives. As shown in Figure 13, when s _w reaches 25 and 30 m, the maximum value of the additional stress on the tunnel will not increase, and the range of the whole deal of the additional stress will expand. This is because part of the tunnel is already above the water level as s _w=25 m. At the same time, as the s _w continues to increase, more of the tunnel is located above the water level. However, the additional stress on the tunnel will not increase when the water level is below the tunnel. Therefore, the curves of s _w =25 and s _w =30 show inconsistent change trends compared with several other curves.

Figure 14 shows the tunnel displacement curve caused by pre-dewatering under different water level drop depth s _w. It can be seen from Figure 14 that the maximum vertical displacement of the tunnel increases from 4.49 to −8.99 mm when the water level drawdown s _w increases from 10 to 30 m, but the increased amplitude gradually weakens. That is, because when the water level drops depth s _w gradually increases, the influence of dewatering on the surrounding water level also increases, which leads to the increase of the effect of dewatering on the tunnel. Furthermore, the drop in water level causes the increase of additional loads on the tunnel. Therefore, with the increase of s _w, the tunnel displacement due to dewatering increases overall.