Complex variable solution for ground deformation responses considering the grouting pressure of shallow shield

During shield tunneling in shallow strata, ground deformation is highly sensitive to grouting pressure response. Excessive grouting pressure can cause ground uplift, and an excessively low grouting pressure can cause excessive ground settlement, affecting the surrounding environment. Current studies seldom analyze the influence of shield excavation on ground deformation when the grouting pressure is considered. To ensure micro-disturbance of the surrounding environment during shield tunneling, this study presents a complex variable analysis method for analyzing the influence of grouting pressure on surface settlement. This solution maps the semi-infinite strata outside the segment into circular domains by introducing a complex function conformal mapping method. The nonuniform convergence deformation pattern BC-4 was introduced as the boundary condition for the displacement around the tunnel. The Laurent series expansion of the analytical function considering the grouting pressure in the circular domain was substituted into each boundary condition to solve the analytical function. Subsequently, the accuracy of the analytical method was verified by combining it with a finite element numerical method. Finally, the influence of grouting pressure on ground deformation under different working conditions was analyzed through parameterization. This analytical method can provide a theoretical basis for controlling grouting pressure parameters in shield tunnels.

For the empirical method, Peck [1] initially fitted surface settlement using a Gaussian curve.Subsequent research revised and supplemented Peck's formula [2][3].The parameters of the empirical method are simple and practical.However, they cannot explain the deformation mechanism of the stratum, and most of the parameters are subjective.A stratum often corresponds to one or multiple modified empirical formulas; therefore, the empirical method has limitations.In numerical analysis methods, the finite element, finite difference, discrete element, and other methods [4][5][6] are primarily used to study the response of the stratum to tunnel excavation.Numerical methods have a high

Assumption and geometry of the problem
Analysis of the surface deformation caused by shallow shield tunnel excavation is a common, simply connected domain problem in a semi-infinite space, as shown in Figure 1.To solve the semi-infinite domain problem with a hole, this study used the conformal mapping method in complex variable analysis to map the half-infinite domain (R-plane) to a circular ring domain on a new complex plane (Rplane).The mapping equation is shown in Equation (1) [10].The following assumptions were made: 1) The tunnel is infinitely long in the longitudinal direction and satisfies the plane strain condition; 2) The formation is isotropic, homogeneous, and elastic; 3) The lining is in close contact with the surrounding rock without friction and satisfies the deformation coordination equation.
where z is any point in the R plane, z=x+iy, ζ is any point in the R plane, ζ=ε+iη, a=h(1-α 2 )/(1+α 2 ), α=(h-(h 2 -r 2 ) 0.5 )/r, h is the distance between the tunnel center and ground surface, and r is the outer diameter of the lining.The original boundaries Lh and Ls in the R plane were mapped and transformed into Ωh and Ωs boundaries on the R plane, respectively.

Basic equation
According to the basic theory of elastic mechanics, there must be two analytical functions, φ(z) and ψ(z), under the condition of constant physical force.φ(z) and ψ(z) can be used to represent the displacement and stress at any point in the entire elastic body, as shown in Equation (2).
where μ is the shear modulus of the elastic body, and g h (z) is the displacement function.According to the assumption of plane strain, G=3-4v [12].The solution to Equation ( 2) is closed.Therefore, when the analytical functions φ(z) and ψ(z) are obtained, the displacement and stress fields of the entire stratum can be calculated.The analytical functions are typically solved using boundary conditions.The displacement of any boundary is given by Equation ( 2), and the stress boundary conditions of any boundary are given by Equation (3).
where z0 and z are any two points on the integration boundary, tx and ty are the stresses along the x and y directions on the boundary, respectively, and the integration constant Cz0 is the result of any starting value z0 of the integration, usually considered as 0.
To consider the influence of grouting pressure on ground deformation caused by tunnel excavation, this study simplified the grouting pressure as a surface force distributed along the tunnel circumference, which does not consider the attenuation mechanism of the grouting pressure caused by the permeation of grout in the stratum.According to the literature [11], the analytical function of a semi-infinite domain with a single hole and free surface can be expressed as follows: where zc is the singular point in the entire semi-infinite domain; φ(z) and ψ(z) are nonanalytical.It is evident that zc=-ih in the R plane.Fx h +iFx h is the resultant force applied at the boundary of the tunnel and represents the grouting pressure acting on the tunnel circumference.Grouting pipes are typically uniformly arranged along the segment to apply a uniform grouting pressure and maintain uniform hole deformation and segment stability, as shown in Figure 2. The upward buoyancy force generated by the slurry along the y-axis is obtained by integrating along the hole circumference, as shown in Equation (5).When the grouting pressure is applied unevenly, a bias pressure is generated, as shown in Figure 2. In this study, the uneven grouting pressure in the half zone is categorized into Modes 1, 2, 3, and 4, with the influence of these grouting modes on surface settlement discussed in subsequent sections.
To solve the expression of the analytical function, the analytical functions φ(z) and ψ(z) were mapped into the R-plane.Subsequently, φ0(z) and ψ0(z) in in Equations ( 6) and (7) were determined to satisfy the following equation: Because φ(z) and ψ(z) is analytical in the R-plane, φ0(z) and ψ0(z) is also analytical in the R-plane.Thus, φ0(z) and ψ0(z) satisfy the Laurent expansion, as shown in Equation (9).Therefore, solving φ (z) and ψ (z) can be simplified to solving the Laurent expansion coefficient of φ0(ζ) and ψ0(ζ).

Boundary condition
On the R-plane, any point ζ can be represented by Euler's formula (i.e., ζ=ρe iθ =ρσ).After simple symbolic operations, this can be expressed as follows: Considering the free boundary Lh, mapped to be the Ωh boundary on the R-plane, and substituting Equation (10) and ζ=ρσ (on the Ωh boundary, ρ=1)into Equation ( 6) yield the following: Considering the boundary Ls, mapped to be the Ωs boundary on the R-plane, and substituting Equation (10) and ζ=ρσ (on the Ωh boundary, ρ=α) into Equation ( 7) yield the following: . Gonzalez et al. [14] reported that ground deformation is caused by ground loss and nonuniform convergence deformation.Ground loss can be attributed to the radial shrinkage of the rock surrounding the tunnel, leading to changes in the tunnel volume.Nonuniform convergence deformation results in tunnel ovalization and overall settlement, and the tunnel volume does not change, as shown in Figure 3.To comprehensively consider the aboveground deformation, this study refers to the BC-4 displacement condition proposed by Park [15], representing the ground deformation caused by shield excavation, as shown in Equation (13).
where r1 is the outer diameter of the lining, θ ' is the angle between the center of the tunnel and horizontal direction, and g is the gap parameter.Equation ( 13) expresses the displacement boundary condition g h (z) given by Equation ( 2).During shield tunneling, the gap at the tail of the shield is filled by grouting, and the gap parameters change owing to the different grouting filling rates.When the consistency of the slurry is low, and the permeability coefficient of the stratum is large, most of the slurry is lost.The grouting filling rate is 0, and the gap parameters do not decrease.However, when the slurry properties are good, the grouting filling rate is almost 100%, and the gap parameter is 0. Therefore, considering the grouting filling rate, Equation ( 13) must consider a reduction.For convenience, the gap parameter and grouting filling rate are assumed to satisfy a linear relationship: g ' =(1-m)g, where g ' is the gap parameter considering the grouting filling rate, and m is the grouting filling rate.

Solution
To solve the Laurent expansion coefficient of the analytical functions φ0(ζ) and ψ0(ζ), the sum of the coefficients of ζ k with the same power in boundary conditions (11) and ( 12) must be 0.For the consistency and simplicity of notation, functions fo s (σ) and go s (ασ) in Equations ( 11) and ( 12) are represented as expansions.
In Equation (12), solving the expansion is challenging because the denominator of each coefficient contains 1-ασ.Therefore, Equation ( 12) is multiplied by 1-ασ.Accordingly, go s (ασ) can be represented by Equation (15).The displacement boundary condition of Equation ( 13) is represented by the overall coordinates in Figure 1, and its expression in the R-plane is obtained through conformal mapping.Finally, the expansion of 2μ(1-ασ)g h (ασ) is shown in Equation ( 16).The logarithmic term in go s (ασ) is expanded using a Taylor series.Finally, The expansion formula of (1-ασ)go s (ασ) is shown in Equation (17).
IOP Publishing doi:10.1088/1755-1315/1333/1/012037 By substituting Equations ( 9), ( 16), and (17) into Equations ( 11) and ( 12) and comparing the coefficient of σ k with same power, the recursive function Equations ( 18) and ( 19) of the final solution can be obtained.When a0 is obtained, all coefficients can be solved based on the recursive function.The solution method for a0 is referred to in reference [10], and solution method is established using Python.

Verification of solution
To verify the reliability of the complex variable analysis method, Plaxis was selected to establish a finite element model for shield tunneling, as shown in Figure 4.The distance between the center of the tunnel and surface was 12 m, and the tunnel radius was 3 m.The width and height of the model were both 40 m.Horizontal constraints were applied to the left and right sides, longitudinal constraints to the front and rear sides, and fixed constraints at the bottom.The stratum adopted the Mohr-Coulomb constitutive model with the following constitutive parameters: soil weight of18 kN/m 3 , elastic modulus of 30 MPa, Poisson's ratio of 0.3, cohesive force of 22 kPa, internal friction angle of 20 o , and slurry weight of18 kN/m 3 .The entire shield tunneling process was completed in three steps: Step 1: Soil excavation and shell application.Soil excavation was simulated by killing the soil units within the excavation range.The shell plate and interface elements between the shield shell and soil were activated.
Step 2: Surface element analysis.The grouting behind the segment was simulated by applying a line load radially to the tunnel.
Step 3: Lining application.The plate element representing the lining and the interface element between the lining and soil were activated to simulate the lining construction.The boundary conditions for the displacement around the hole were determined according to Equation (13), and the gap parameters were 100, 140, and 200 mm.The grouting mode adopted was Mode 1, and the grouting pressure and filling rate were 200 kPa and 80%, respectively.Figure 5 shows the calculation results for the analytical and numerical solutions.

ure Fig
Figure 5 shows that the numerical solution was consistent with the analytical solution obtained in this study.Owing to the effect of grouting pressure, the surface did not settle after excavation but rather uplifted.The maximum uplift occurred on the central axis of the tunnel, and the uplift decreased along both sides of the central axis.In the numerical solution, under three different gap parameters, the maximum surface uplift values were 6.66, 3.31, and -1.58 mm.The analytical results were 7.01, 3.68, and -1.32 mm, yielding errors of 4.9, 10, and 16.5% between the two methods.These results demonstrate the reliability of the complex variable analytical solution presented in this study.

Parametric analyses
To study the influence of various factors on surface deformation, this study designed 15 working conditions based on tunnel burial depth, gap parameters, grouting pressure, grouting filling rate, and grouting mode, as listed in Table 1.The elastic modulus of the soil was 30 MPa, and the slurry weight was 18 kN/m 3 under all conditions.Based on these 15 working conditions, the calculated results of the surface deformation are shown in Figure 6, and the partial displacement cloud map in the y-axis is shown in Figure 7.  Figure 6 (a) shows that as the burial depth of the tunnel became shallower, the disturbance to the surface after the tunnel excavation was more significant, and the surface settlement was larger.Under the working conditions of Case 1, the maximum surface settlement was 25.64 mm.However, as the burial depth decreased, the surface settlement trough became narrower, consistent with actual engineering.Figure 6 (b) shows that as the gap parameter increased, the surface settlement increased.This is because the gap parameter characterizes the formation loss and nonuniform convergence deformation.After the gap parameter increased, the ground deformation increased.As shown in Figure 6 (c), as the grouting pressure increased, the overall surface settlement decreased.If the grouting type is Mode 1, the grouting pressure is integrated along the tunnel circumference to obtain a large positive force along the y-axis, supporting the downward deformation of the surrounding rock.Moreover, as the grouting pressure increases, further surface uplift may occur.Figure 6 (d) shows the influence of the grouting filling rate on the surface displacement as the grouting filling rate increased; the surface settlement decreased as the grouting becamesfuller.Therefore, in engineering, strictly controlling the grouting pressure and grouting filling rate is necessary to ensure full grouting.If the slurry properties are poor, and full grouting cannot be ensured, the grouting pressure should be increased appropriately. Figure 6 (e) shows that the different grouting modes have different effects on the surface settlement and settlement trough.When the grouting pressure was uneven, a bias pressure was generated on the surrounding rock.When the bias pressure direction was along the positive y-axis, the surface settlement caused by the tunnel excavation could be reduced.However, when the bias direction was along the negative y-axis, the surface settlement increased further.A bias direction along the positive or negative x-axis direction caused a deviation in the entire shield posture, as shown in the cloud diagram of Figures 7 (c) and (d); that is, the tunnel moved toward the positive or negative x-axis direction, causing the settlement trough to move.In addition, the maximum ground settlement position was not on the original tunnel axis.In summary, the surface displacement was found to be highly sensitive to grouting pressure, particularly in shallow tunnels.To ensure minimum disturbance on the surface, ensuring full grouting is necessary, and applying a relatively uniform grouting pressure is necessary to ensure that the surface does not generate excessive settlement, uplift, and tunnel offset.
IOP Publishing doi:10.1088/1755-1315/1333/1/0120379 6.Conclusion Based on the complex variable analysis method, this study analyzed the influence of factors, such as grouting pressure and grouting filling rate, on the surface displacement caused by shield tunneling.The main conclusions are as follows: (1) The complex analysis method considering the grouting pressure proposed in this study can effectively simulate the surface displacement caused by grouting pressure during shield tunneling, and its reliability was verified through comparison with the finite element numerical method.
(2) As the burial depth of the shield tunnel increased, the surface settlement decreased and the settlement trough widened.As the gap parameters increased, surface settlement also increased.
(3) The surface displacement response was sensitive to the grouting pressure.When the grouting pressure generated a resultant force along the positive y-axis, the surface settlement decreased as the grouting pressure increased, and an uplift occurred when a large grouting pressure was applied to the surrounding rock.The surface settlement increased when the grouting pressure generated a resultant force along the negative y-axis.When the grouting pressure generated a resultant force along the x-axis, the posture of the shield tunnel generated an offset, and the surface settlement trough moved accordingly.The surface settlement decreased with an increase in the grouting filling rate.

Figure 1 .
Figure 1.Schematic diagram of a semi-infinite domain mapping with a hole.

Figure 4 .
Figure 4. Finite element model.Comparison of analytical and numerical solutions for surface settlement

Table 1 .
Parameter values under 15 working conditions.