With the plane of the ground surface as the datum of reference and ignoring the impact of atmospheric pressure, the seepage boundary condition at the ground surface is ϕ z = 0 = d 1 (3)

With a fixed water head at the outer section of the shield tunnel, its water seepage boundary condition is ϕ x 2 + z + D 2 = R 2 = ϕ 0 (4) where ϕ 0 is the fixed water head at the outer section of the shield tunnel. The method of images is one of the techniques commonly used to solve for the tunnel seepage field, and the key to using this method is to determine the source and the sink. Lei. (1997) disregarded the practice of fixing the centre of the tunnel as a sink, instead setting any point above the centre of the tunnel as the sink with the ground surface serving as the mirror surface. Determining the source according to the principle of the method of images provided a solution for tunnel seepage field analysis. This method avoids the fundamental assumed condition of R/D ≪ 1 and can be used to analyse seepage fields for deep and shallow tunnels. The equation to calculate the seepage field for the shallow marine segment of the shield tunnel ϕ x , z and the water seepage volume per unit width q is ϕ x , z = d 1 − ϕ 0 2 ⁡ ln D R + D R 2 − 1 ⋅ ln x 2 + z + D 2 − R 2 2 x 2 + z − D 2 − R 2 2 + d 1 q = 2 π k d 1 − ϕ 0 ln D R + D R 2 − 1 (5)

The water pressure at any point in the rock surrounding the tunnel is represented by p x , z = γ w ϕ x , z − z (6) where p(x,z) is the pore water pressure at any point in the surrounding rock below the water level, γ w is the weight of the water, z is the location of the water head, which is determined according to the distance between the datum plane and any point above or below the plane.

Because ground water receives long-term replenishment from water at the surface, it is assumed that the groundwater level of the terrestrial segment is essentially equivalent to the water level at the surface and that the groundwater level does not change with tunnel seepage. With the plane of the groundwater level as the datum of reference and ignoring the impact of atmospheric pressure, the seepage boundary condition at the groundwater level is ϕ z = 0 = 0 (7)

The water head at the outer section of the shield tunnel is still assumed to be a fixed water head, and its seepage boundary condition is shown in Eq. 4. Under the special condition when the seepage field of the marine segment of the shield tunnel d ₁ = 0, the seepage field of the terrestrial segment of the shield tunnel is solved using ϕ x , z = − ϕ 0 2 In D − d 2 R + D − d 2 R 2 − 1 ⋅ In x 2 + z + D − d 2 2 − R 2 2 x 2 + z − D − d 2 2 − R 2 2 q = − 2 π k ϕ 0 In D − d 2 R + D − d 2 R 2 − 1 (8) where d ₂ is the thickness of the strata above the groundwater level in the terrestrial segment.

According to the Terzaghi sliding failure mode for the overlying soil body of a shield tunnel, a rock body of thickness dz at a distance z below the ground surface is selected to serve as an object of research for this analysis as shown in Figure 2A. The following assumptions are made for the convenience of using these equations for theoretical analysis: (1) the sliding surfaces within the model are all in their ultimate state of strength, (2) the impact of the vertical seepage force within the sliding section on top of the tunnel is taken into consideration and (3) the vertical seepage gradient is evenly distributed within the rock surrounding the sliding section on top of the tunnel with an average seepage gradient of i ¯ z . A tension-shear coupling effect with the phenomenon of ground surface tensile cracking exists, and its mechanisms are complex. Since this paper emphasizes the soil pressure on top of the tunnel, the destructive impact of the ground surface tensile forces is not discussed any further. The forces experienced by the object of research are separated into self-gravity as − γ e B d z , upper and lower stress as − B σ Z ′ + d σ z ′ and B σ Z ′ , respectively, shear force as 2 τ ′ d z and seepage force as − γ w i ¯ z B d z . When combined with the limit equilibrium analysis method, a differential equation that satisfies the equilibrium conditions can be obtained: d σ z ′ d z = 2 τ ′ B − γ e − γ w i ¯ z (9) where σ z ′ is the effective vertical stress in the rock surrounding the top of the tunnel, γ e is the effective density of the rocky soil body, i ¯ z is the average vertical seepage gradient at the top of the tunnel and τ ′ is the shear force on the sliding surface. The rock and soil body above the sliding surface obeys the Mohr failure criterion with a shear force of τ ′ = K c ′ σ z ′ ⁡ tan ⁡ φ ′ + c ′ (10) where K c ′ is the coefficient of lateral pressure for the sliding surface below the water level, φ ′ is the angle of internal friction within the rocky soil body below the water level and c ′ is the soil cohesion of the rocky soil body below the water level.

Substituting Eq. 10 into Eq. 9 based on the ground surface effective stress boundary condition σ Z ′ Z = 0 = 0 , an integration calculation can be performed to obtain σ s ′ = B γ e + γ w i ¯ z − 2 c ′ 2 K c ′ ⁡ tan ⁡ φ ′ 1 − exp 2 K c ′ ⁡ tan ⁡ φ ′ B z (11)

Using Eq. 5 to directly solve for the seepage gradient i ¯ z within the sliding section at the top of the marine segment of the tunnel is relatively difficult. The seepage route is shortest at the top of the tunnel, and its seepage gradient is comparatively large. This paper presents calculations based on the most dangerous conditions so that the average seepage gradient i ¯ z within the sliding section is i ¯ z = − 1 D − R ∫ R − D 0 ∂ ϕ 0 , z ∂ z d z = β 1 d 1 − ϕ 0 D − R (12) where β 1 is the water head influence coefficient calculated using the formula β 1 = − ln D 2 − R 2 + R − D D 2 − R 2 + D − R ln D R + D R 2 − 1 (13)

Combining Eqs 11, 12 can solve for the soil pressure at the top of the marine segment of the shield tunnel (z = R - D): σ v ′ = B γ e + γ w i ¯ z − 2 c ′ 2 K c ′ ⁡ tan ⁡ φ ′ 1 − exp 2 K c ′ ⁡ tan ⁡ φ ′ R − D B (14)

For the terrestrial segment, the groundwater level is beneath the ground surface, and for the terrestrial segment of the Linjiang Ocean Shield Tunnel, the groundwater level is generally located above the top of the tunnel. Not taking into consideration the effect of matrix suction and finding separate solutions for the sections above and below the groundwater level, the analytical model shown in Figure 2B is obtained. The differential forces acting on the sliding section of the soil body above the groundwater level are self-gravity, − γ B d z , upper and lower stresses, − B σ Z + d σ z and B σ z , respectively, and shear force, 2 τ d z . The differential forces acting on the sliding section of the soil body below the groundwater level are self-gravity, − γ B d z , upper and lower stresses, − B σ z ′ + d σ z ′ and B σ z ′ , respectively, shear force, 2 τ ′ d z and seepage force, − γ w i ¯ z B d z . The equilibrium conditions for the two parts of the sliding section above and below the groundwater level are: d σ z d z = 2 τ B − γ   Above   the   ground   water   level d σ z ′ d z = 2 τ ′ B − γ e − γ w i z ¯ Below   the   ground   water   level (15) where γ is the natural density of the soil body above the water level.

The shear stress at the face of the sliding section below the groundwater level can be calculated according to Eq. 10 while the shear stress at the face of the sliding section above the groundwater level is τ = K c σ z ⁡ tan ⁡ φ + c (16) where K _c is the coefficient of vertical earth pressure for the shear at the face of the slip surface above the groundwater level and c and φ are cohesion and angle of internal friction of the rock and soil body above the water level, respectively. Considering that the groundwater level in land next to the shore is generally higher and that there exists a capillary water effect, the values c and φ used in the calculations for the rocky soil body above the water level are selected according to their saturated state.

If the boundary conditions are σ z z = d 2 = 0 and σ ′ z = 0 = σ z z = 0 and Eqs 15, 16 are combined and then integrated, we can obtain: σ z ′ = B γ e + γ w i z ¯ − 2 c ′ 2 K c ′ ⁡ tan ⁡ φ ′ 1 − exp 2 K c ′ ⁡ tan ⁡ φ ′ B z + γ B − 2 c 2 K c ⁡ tan ⁡ φ 1 − exp − 2 K c ⁡ tan ⁡ φ d 2 B ⋅ exp 2 K c ′ ⁡ tan ⁡ φ ′ B z (17)

According to Eq. 8, the average seepage force gradient i _z within the sliding section of the terrestrial segment of the tunnel is i ¯ z = − 1 D − R − d 2 ∫ R + d 2 − D 0 ∂ ϕ 0 , z ∂ z   d z = β 2 ϕ 0 D − R − d 2 (18) where β 2 is the influence coefficient of the water head, calculated using equation β 2 = ln D − d 2 2 − R 2 + R + d 2 − D D − d 2 2 − R 2 + D − R − d 2 ln D − d 2 R + D − d 2 R 2 − 1 (19)

Combining Eqs 17, 18 can solve for the earth pressure on top of the tunnel for the terrestrial segment of the shield tunnel (z = R + d ₂ - D) as σ z ′ = B γ e + γ w i z ¯ − 2 c ′ 2 K c ′ ⁡ tan ⁡ φ ′ ⋅ 1 − exp 2 K c ′ ⁡ tan ⁡ φ ′ R + d 2 − D B + γ B − 2 c 2 K c ⁡ tan ⁡ φ ⋅ 1 − exp − 2 K c ⁡ tan ⁡ φ d 2 B ⋅ exp 2 K c ′ ⁡ tan ⁡ φ ′ R + d 2 − D B (20)

For a medium load structure model, the water and soil loads P water ,   soil marine and P water ,   soil terrestrial for the top of the tunnel in the marine and terrestrial segments, respectively, are P water ,   soil marine = σ z ′ z = R − D + p w z = R − D (21a) P water ,   soil terrestrial = σ z ′ z = R + d 2 − D + p w z = R + d 2 − D (21b) where p _w is the water pressure at the top of the tunnel, which can be solved for according to Eqs 5–7.

The coefficient of lateral earth pressure is one of the key parameters of the Terzaghi loose earth pressure theory. However, a consensus has still not been reached in the calculation of the coefficient of lateral earth pressure K, with the primary point of controversy being the selection of the principal arch stress to be used in the calculation (Handy, 1985; Wan et al., 2019; Wang et al., 2019; Fan et al., 2022). Xu et al. (2018) and Dong et al. (2022) discovered that a convex principal stress arch is more aligned with actual circumstances and the principal stress arch traces adopt a circular arc form. In addition, the majority of past research and results are based on conditions of sandy strata, and, therefore, the applicability of the coefficient of lateral earth pressure formula to cohesive substrata has not been settled. On the basis of prior research, this paper adopts assumed conditions for the principal stress arch and establishes an analytical model for the convex circular arc form shown in Figure 3. In Figure 3, P _A and P _B are two poles of the Mohr stress circle.

The rocky soil body of the sliding surface is in a limit equilibrium state. Based on the limit equilibrium conditions and the geometric relationship in Figure 3, the horizontal stress σ h and σ v the vertical stress within the sliding rocky soil body can be expressed as σ v = σ 1 cos 2 ⁡ θ + K a sin 2 ⁡ θ σ h = σ 1 K a cos 2 ⁡ θ + sin 2 ⁡ θ (22) where σ 1 is the principal stress at any point within the soil body of the slip surface, θ is the angle of rotation of the principal stress at any point within the soil body of the slip surface, expressing the angle between the direction of principal stress at a certain point and the horizontal while K _a is the coefficient of active earth pressure for the soil body.

K, the ratio of the horizontal earth pressure to the vertical earth pressure at any point, is expressed as K = K a cos 2 ⁡ θ + sin 2 ⁡ θ cos 2 ⁡ θ + K a sin 2 ⁡ θ (23)

Because the rocky soil body within the sliding section is in an active fault state, the coefficient of active earth pressure in the equation is K a = σ 3 / σ 1 , K a = tan 2 π / 4 − φ / 2 .

The average vertical stress is σ ¯ v = 1 B ∫ − B / 2 B / 2 σ V = σ 1 2 sin θ 0 1 − K a 2 sin 2 θ 0 + 1 + K a θ 0 (24) where θ 0 is the angle between the principal stress σ 1 at point A in the shear-slip surface and the horizontal.

The distribution coefficient m for the vertical earth pressure is m = σ v σ ¯ v = 4 sin θ 0 cos 2 ⁡ θ + K a sin 2 ⁡ θ 1 − K a sin ⁡ 2 θ 0 + 2 1 + K a θ 0 (25)

According to Figure 3, the shear stress τ at point A of the shear-slip surface satisfies τ = K A σ v A ⁡ tan ⁡ φ + c (26) where K _A is the ratio of the horizontal earth pressure and σ v A is the vertical earth pressure, respectively, at point A of the shear-slip surface. Substituting Eq. 25 into Eq. 26 simplifies the formula to τ = m A K A σ ¯ v ⁡ tan ⁡ φ + c (27)

The coefficient of lateral earth pressure for the shear-slip surface is K c = m A K A = 4 K a cos 2 θ 0 + sin 2 θ 0 sin θ 0 1 − K a sin ⁡ 2 θ 0 + 2 1 + K a θ 0 (28)

According to the geometrical relationship between the Mohr stress circle and the extreme failure envelope of the τ − σ plane, it is known that tan ⁡ θ 0 = σ v ⁡ tan ⁡ φ + c σ v − σ 3 σ 3 = c sin ⁡ φ − 1 cos ⁡ φ − σ v sin ⁡ φ − 1 cos 2 ⁡ φ (29)

Equation 29 can be simplified to obtain tan ⁡ θ 0 = cos φ 2 + sin φ 2 cos φ 2 − sin φ 2 (30)

Because of φ, the angle of internal friction within the soil body is typically less than 90⁰ and cos φ 2 > sin φ 2 . Eq. 30 then becomes tan ⁡ θ 0 = cos φ 2 + sin φ 2 cos φ 2 − sin φ 2 = tan π 4 + φ 2 (31)

from which can be obtained θ 0 = π / 4 + φ / 2 (32)

It is clear from Eq. 32 that the formula for the angle of rotation of the principal stress obtained in this paper is consistent with the formula for the angle of rotation of the principal stress in sandy strata obtained by Handy (1985). Further, it can be seen that the angle of rotation of the principal stress in the soil body is only associated with the angle of internal friction within the soil body and that the coefficient of lateral earth pressure is a function of the angle of rotation of principal stress. This shows that this paper’s model for calculating the coefficient of lateral earth pressure can be applied to both sandy and cohesive strata. By substituting Eq. 32 into Eq. 28 and simplifying, the coefficient of lateral earth pressure of the shear-slip surface can be obtained: K c = 2 1 + K a + 1 − K a sin ⁡ φ sin π 4 + φ 2 1 − K a cos ⁡ φ + 2 1 + K a π 4 + φ 2 (33)

A comparative analysis has been performed using this paper’s model for calculating the coefficient of lateral earth pressure and other existing formulas as shown in Figure 4. Through an analysis of the figure, when the results from this paper’s model are placed alongside those from existing theoretical calculations, the validity of this paper’s model for calculating the coefficient of lateral earth pressure is clear. Furthermore, it can also be seen from the figure that the coefficient of lateral earth pressure changes with increases in the angle of internal friction within the rocky soil body by first slightly decreasing and then increasing. This could be primarily due to the impact of the soil arching effect during the soil deformation process, showing that there are certain conditions where the soil arching effect is a driving influence on the coefficient of lateral earth pressure. Before the soil arching effect begins, the coefficient of lateral earth pressure decreases slightly with an increase in the angle of internal friction. After the soil arching effect forms, soil arching is a driving influence on the changes in the coefficient of lateral earth pressure, and as the angle of internal friction increases, the coefficient of lateral earth pressure increases and the soil arching effect is more evident, which aligns with established facts.

When the distance between the water level and the ground surface is d ₁ = d ₂ = 10 m, the cohesive strength of the rocky soil body is c = 20 kPa, the angle of internal friction φ ¯ = 30⁰, the tunnel is fully drained, and the saturated density of the rocky soil body is 19 kN/m². The water and soil loading on the top of the shield tunnel changes with different tunnel depth-diameter ratios D/R as shown in Figure 5. Through an analysis of the figure, it is clear that as D/R increases, the water and soil loading on the top of the shield tunnel consistently increases; and its rate of increase gradually lessens primarily due to the fact that as D/R increases, the impact of the soil arching effect becomes more evident. Under the same calculation conditions, the water and soil load on the marine segment is greater than that on the terrestrial segment primarily because the impact of the water level is higher on the marine segment, which aligns with known facts and reflects the rationality of this paper’s method.

When the distance between the water level and the ground surface is d ₁ = d ₂ = 10 m, the cohesive strength of the rocky soil body is c = 20 kPa, the angle of internal friction φ = 30⁰, the tunnel is fully drained, the radius of the tunnel is R = 5 m, and the tunnel depth is D = 30 m. The water and soil loading on the top of the shield tunnel changes with different values of the saturated density γ of the rocky soil body, as shown in Figure 6. Through an analysis of the figure, it is clear that as the volumetric weight of the soil body increases, the water and soil load on the top of the tunnel experiences a gradual linear increase.

When the distance between the water level and the ground surface is d ₁ = d ₂ = 10 m, the cohesive strength of the rocky soil body is c = 20 kPa, the angle of internal friction φ = 30°, the radius of the tunnel is R = 5 m, the tunnel depth is D = 30 m, the saturated density of the rocky soil body is 19 kN/m² and the coefficient of seepage for the rocky soil body is k = 0.001 m/d. The water and soil loading on the top of the shield tunnel changes with different values of the water head at the inner boundary of the tunnel as shown in Figures 7, 8. In the figures, d is the distance between the water level and the ground surface. Through an analysis of the figure, it is clear that as the water head at the outer edge of the section increases, the ability of the tunnel to seep water gradually decreases and the water and soil load on the top of the tunnel presents a consistent linear increase. As the volume of seepage water per unit width increases, the water and soil load on the top of the tunnel reveals a gradual linear decrease.

The Maliuzhou segment of the Hengqin Tunnel is the object of our research. This tunnel is located in the section between Working Well No. Three and Financial Island. The Maliuzhou Waterway is in the YDK8 + 100∼ + 700 segment with a waterway width of 600 m and water depth of 5–8 m. The geological profile of the Maliuzhou Waterway shield construction segment of the Hengqin tunnel is shown in Figure 9. The strata surrounding the tunnel are primarily silt, fine clay, medium-grade, coarse and fine sands and weathered granite. According to the engineering geological conditions and engineering characteristics of Hengqin Tunnel, static penetration test, standard penetration test and dynamic penetration test, pressure meter test and indoor soil test were carried out for the main soil layers. Through the statistical analysis of these test results, the paper obtains the values of cohesion and angle of internal friction of geotechnical masses. The primary physical and mechanical parameters are shown in Table 1. The shield construction section of the tunnel uses a total of four composite-style earth pressure equilibrium shields with linings of prefabricated reinforced concrete. The shield tunnel’s inner diameter is 7.7 m, its outer diameter is 8.5 m, its segment width is 1.6 m and its segment liner thickness is 400 mm.

According to the preliminary geological exploration results of Hengqin Tunnel, two sections of the shield tunnel were selected to measure the changes in water and soil pressure on the outside of the shield tunnel in the direction of Well No. Three on the northern shore beneath the waterway and in the direction of Financial Island to the south. A basic overview of the measured sections is as follows:

Section I: YDK8 + 339.5 at a buried depth of 18.7 m and water depth of 5–8 m in the bottom section of the Maliuzhou Waterway. Overlying strata is composed of fluid plastic silt, fine plastic clay and a thin layer of silty clay. The tunnel passes through a stratum of moderate-density coarse sand.

Section II: YDK8 + 765.1 at a buried depth of 22.5 m and water depth of 5–8 m in the northern shore of the Maliuzhou Waterway. Overlying strata is composed of loose fine sand, fluid plastic silt, a thin layer of slightly dense fine sand, fine plastic clay and clay. The tunnel passes through an arched stratum of plastic clay, a middle stratum of moderately dense sand and a bottom stratum of weatherized granite.

According to Figure 10, it is clear that the two measured sections located in B₂ are essentially located at the top of the tunnel and field measurements can be used to obtain the distributed changes in the pressure of water and soil in the outer B₂ section at locations I and II as shown in Figures 11, 12. It can be seen that the initial water pressure of section II is higher than that of section I, which is mainly due to the different testing time of the two initial water pressures. Due to the limitation of on-site installation and testing conditions, the initial water pressure of section II is measured after the shield tail passes through about two ring segments. At this time, the underground water is affected by grouting and surrounding rock deformation, which will produce a certain water pressure on the outside of the segment. The initial water pressure of section I is measured when the shield tail almost passed section I, and the water pressure outside the segment is almost zero. After water and soil loading had essentially stabilized, the total soil pressure at the top of the tunnel in Section I was 0.309 MPa, the water pressure was 0.144 MPa and the effective soil pressure was 0.165 MPa. The total soil pressure at the top of the tunnel in Section II was 0.243 MPa, the water pressure was 0.174 MPa and the effective soil pressure was 0.068 MPa.

It is rather difficult to make a direct comparison between the water and soil pressures at the top of the tunnel in Sections I and II. The following formula can be used to find a weighted average of the physical parameters of the soil strata above the tunnel to achieve an effective simplification of the conditions of the strata where these sections of the tunnel lie: x ¯ = ∑ 1 4 x i h i ∑ 1 4 h i (34) where x is the weighted average of the physical parameters of the soil body, x _i is the value of the physical parameter at strata i and h _i is the thickness of strata i.

The strata above the top of the tunnel was selected to perform a weighted calculation. Segment I is located in strata completely below the water level with separate strata of coarse sand, silt and fine clay. Some of the strata above the tunnel in Segment II are located above the water level and require separate calculations. The strata located below the water level are comprised of fine clay, silt and fine sand. The strata above the water level are comprised of fine sand. The results of a weighted calculation of the physical parameters of the strata of segments I and II are shown in Table 2.

Based on the aforementioned parameters for the strata, different methods, including the Terzaghi method which does not take into consideration the impact of groundwater seepage, were used to perform a predictive analysis of the value of the water and soil pressure at the top of the tunnel. Calculations were performed according to the total stress method. The values for the water and soil pressure at the top of the tunnel obtained through these methods are shown in Table 3 and Figure 13. From an analysis of Figure 13, it is clear that the results from actual field measurements fall between the results of the fully-drained and non-drained conditions proposed in this paper while the results from calculations using the Dimitrios (2007) and Terzaghi methods have a small trend. For this project, the largest margin of error for the results obtained through this paper’s calculation method is 5.17% while those of the DK and Terzaghi methods are nearly 80% and 34.63%, respectively. This shows that the calculation method proposed by this paper has good precision and validity and is able to effectively predict the water and soil load on the top of the tunnel.