Theoretical Analysis of Surface Settlement During Parallel Construction of a Double-Track Tunnel with Small Spacing

Abstract

The construction of urban subway tunnels typically induces soil settlement within a specific radius of the surrounding area. However, the accuracy of current methods for predicting ground deformation curves corresponding to the excavation of double-track tunnels with small spacing remains insufficient. Most studies simplify this problem by modeling it as a two-dimensional plane problem; however, the actual ground deformation exhibits pronounced three-dimensional characteristics. Consequently, studying the ground settlement patterns caused by the construction of small-spacing double-track tunnels is crucial. This study is based on the Peck formula, used to calculate surface settlement caused by the excavation of single-track tunnels. By incorporating the maximum settlement offset e and the soil loss rate η(y), the soil displacement calculation formula is derived for small-spacing double-track tunnel excavation. The accuracy of the derived formula is then validated through a case study. The findings provide a theoretical reference for predicting surface settlement induced by the excavation of small-spacing double-track tunnels. Furthermore, the influence of different parameters on ground settlement patterns is explored. The results indicate that appropriately increasing the tunnel spacing, increasing burial depth, and adopting a sequential excavation method for the two tunnels help reduce ground settlement.

1. Introduction

The construction and operation of urban subways have garnered increasing attention across various countries and regions [1,2]. Subway tunnel construction significantly influences the deformation of the surrounding ground. Current research on ground deformation during tunnel construction predominantly relies on empirical methods, analytical approaches, and numerical simulations. Peck [3], through the analysis of extensive surface deformation monitoring data following tunnel excavation, concluded that, under undrained conditions, the volume of ground loss is equal to the volume of the surface settlement trough. Based on these findings, he proposed the Peck formula, which is widely used to predict surface settlement resulting from tunnel construction. Han et al. [4], utilizing field measurement data from various regions in China, modified the original Peck formula and proposed a revised surface settlement prediction formula. The revised formula better accommodates the specific conditions encountered during tunnel excavation in China and is applicable to all soil layer types. Yin et al. [5] provided methods for determining the key parameters of the Peck formula under diverse field engineering conditions, thereby enhancing the applicability of the Peck formula in real-world engineering applications. Based on the Peck formula, the superposition theory was used to predict the surface settlement curve corresponding to double-track tunnel construction. Considering the positional relationship between the two tunnels, Chen et al. [6] derived a formula for estimating deep-soil settlement during double-track tunnel excavation based on the surface settlement predictions for single-track tunnel excavation and incorporating the superposition principle for double-track tunnel scenarios. Wu et al. [7] studied a calculation method for surface settlement caused by the construction of a parallel double-track tunnel. By considering factors such as tunnel spacing and burial depth, they defined the conditions for closely spaced double-track tunnels and derived a formula for calculating the total deep-soil settlement curve under these conditions. Zhang et al. [8] derived a calculation formula for predicting surface settlement during double-track tunnel construction, by considering the spatiotemporal effects of tunnel excavation and the impact of burial depth on the surface settlement. Shi et al. [9] comprehensively studied various factors influencing the surface settlement caused by double-shield tunnel construction, including tunnel spacing, burial depth, cross-sectional diameter, soil parameters, and excavation schemes. Based on the surface settlement data from real-time engineering projects, they proposed a surface settlement distribution curve associated with double-track tunnel construction. Rui et al. [10] investigated the construction process and variations in the construction parameters for double-track parallel tunnels crossing a river. They revealed the deformation patterns of soil induced by double-track parallel tunnel construction under varying construction parameters. Rodríguez et al. [11] presented the application of the finite element method integrated with the Terzaghi principle. They believe that the implementation of the algorithm needs to consider three (3) different stress states. However, existing studies on predicting the ground deformation curves corresponding to the excavation of small-spacing double-track tunnels remain insufficiently accurate. Most studies simplify the problem by analyzing it in two dimensions. Wang [12] improved the classical two-dimensional Peck model by considering the effects of convergence pattern. Wei [13] modified the two-dimensional solution of the unified soil movement model for shield tunneling and proposed a correction formula for the two-dimensional solution of this model. However, they ignore the nonlinear effects of the interaction between small-spacing double-track tunnels, resulting in deviations in the calculation results. Moreover, most studies simplify the problem as a two-dimensional plane analysis, whereas in actual engineering practice, ground deformation in double-track tunnels construction exhibits considerable three-dimensional effects. Various studies have indicated that the maximum ground settlement during double-track tunnel excavation exhibits an offset, and the ground deformation demonstrates significant three-dimensional characteristics [14]. However, traditional methods cannot accurately describe this characteristic. Moreover, construction parameters, such as excavation sequence and burial depth, have a considerable impact on ground settlement; however, many analytical methods have not fully considered these factors, limiting the applicability of prediction results under different construction conditions. Therefore, a more accurate method is needed to predict ground settlement induced by the construction of small-spacing double-track tunnels. It plays a crucial role in optimizing construction plans, reducing settlement risks, and ensuring the safety of surrounding buildings.

Building upon Peck’s formula for predicting surface settlement induced by single tunnel excavation, this study addresses the three-dimensional deformation characteristics of the stratum in small-spacing double-track tunnels. To this end, it introduced the horizontal distance e between the position of maximum surface settlement and the centerline of the two tunnels in double-track tunnel excavation, and considered the varying soil loss rate η(y) along the tunnel excavation direction. A three-dimensional solution for soil displacement under small-spacing double-track tunnel excavation was derived and validated through a case study, significantly enhancing the prediction accuracy of surface settlement induced by the excavation of small-spacing double-track tunnels. Furthermore, finite element numerical simulations were conducted to investigate the effects of various factors, including tunnel spacings, burial depths, and excavation sequences, on ground settlement induced by small-spacing double-track tunnel excavation. The findings provide valuable theoretical support and engineering references for predicting ground deformation resulting from the parallel excavation of small-spacing double-track tunnels.

2. Theoretical Analysis of Ground Deformation

2.1. Theoretical Analysis of Ground Deformation Caused by Subsurface Tunnels Construction

Tunnel excavation results in the displacement of the surrounding soil and disrupts the pre-existing stress equilibrium of the strata. Within a specific depth range, the stress within the strata is altered, resulting in a shift toward a new equilibrium state, which is macroscopically observed as ground deformation. Shallow-buried tunnel construction significantly influences ground deformation. During the tunnel construction process, ground deformation primarily occurs as settlement, owing to the following reasons:

• The stress release in the surrounding soil after excavation causes the soil to displace towards the tunnel.
• Gaps form between the support structure and the surrounding soil following excavation.
• As excavation progresses, the support structure undergoes slight shrinkage deformation.
• The tunnel structure undergoes overall downward displacement due to its self-weight.

Because the shrinkage deformation of the support structure is relatively small and overall tunnel subsidence is less common, the first two factors are the primary reasons for ground settlement deformation during tunnel construction.

2.2. Theoretical Analysis of Ground Surface Settlement Induced by Single-Track Tunnel Construction

The lateral ground surface settlement curve corresponding to the construction of shallow-buried tunnels using the New Austrian Tunneling Method is typically predicted using the Peck formula [3]. As shown in Figure 1, Peck proposed that the volume of ground loss caused by tunnel excavation is approximately equal to the volume of the settlement trough. The ground surface settlement curve after excavation followed a Gaussian distribution, and he proposed a formula for predicting the surface settlement under single-track tunnel construction conditions.

The expression of the Peck formula is:

$$S(x)=S_{\max }\exp (-{\displaystyle \frac{x^2}{2i^2}})$$

(1)

$$S_{\max }={\displaystyle \frac{V_{\mathrm{s}}}{i\sqrt{2\pi }}}\approx {\displaystyle \frac{V_{\mathrm{s}}}{2.5i}}$$

(2)

where S(x) represents the ground surface settlement at a distance x from the tunnel axis, S_max is the maximum surface settlement after excavation, vs. is the ground loss per unit length, and i is the width of the settlement trough.

3. Three-Dimensional Theoretical Solution for Ground Settlement Induced by Parallel Double-Track Tunnel Construction

3.1. Current Ground Settlement Prediction Methods and Their Limitations in Double-Track Tunnel Construction

Urban subway tunnels are typically bidirectional, and their construction generally involves two parallel double-track tunnels. When the horizontal spacing between the two tunnels is small, they interact during construction. Consequently, the ground settlement behavior in double-track tunnel construction differs from that observed in single-track tunnel construction. It is essential to further investigate the interaction between the two tunnels and the resulting ground deformation patterns during the construction of a closely spaced double-track tunnel.

Currently, the prediction of ground settlement induced by parallel double-track tunnel construction typically employs the direct superposition method [15]. In this method, the ground settlement deformation caused by the construction of each tunnel is calculated separately, and the results are subsequently superimposed to obtain a formula for the ground settlement deformation during parallel double-track tunnel construction, as follows:

$$S\left(x\right)=S_{\max ,1}\exp \left[-{\displaystyle \frac{{\left(x-L/2\right)}^2}{2i_1^2}}\right]+S_{\max ,r}\exp \left[-{\displaystyle \frac{{\left(x-L/2\right)}^2}{2i_{\mathrm{r}}^2}}\right]$$

(3)

where S(x) represents the total ground surface settlement, i_l and ir are the width parameters of the settlement trough for the left and right tunnels, respectively, L is the distance between the axes of the two tunnels, and S_max,l and S_max,r are the maximum ground surface settlements for the left and right tunnels under separate excavation conditions, respectively.

Equation (3) treats the construction of the double-track tunnel as two independent events, neglecting the interaction between the two tunnels during construction. When the distance between the double-track tunnels is large, the sequence of tunnel construction has minimal impact on the surrounding soil displacement, and the interaction between the tunnels can be neglected. In such cases, Equation (3) provides a reasonable approximation for predicting the ground settlement caused by double-track tunnel excavation. However, when the distance between the double-track tunnels is small, the sequence of tunnel construction significantly affects the changes in the settlement curve, and considerable interaction occurs between the tunnels. The application of Equation (3) may lead to significant discrepancies in the results. Therefore, a more refined calculation method must be developed to achieve accurate prediction.

Suwansawat S. et al. [16] proposed that the influence of the preceding tunnel must be considered when predicting the ground settlement induced by subsequent tunnel construction. The values of i and η for the preceding tunnel differ from those of the subsequent tunnel. However, the settlement profile of the subsequent tunnel follows a normal distribution. Based on this premise, a more accurate formula for the ground settlement deformation under the influence of the preceding tunnel in a twin parallel tunnel construction was derived:

$$S\left(x\right)={\displaystyle \frac{\pi R^2{\eta }_{\mathrm{first}}}{i_{\mathrm{first}}\sqrt{2\pi }}}\exp \left[-{\displaystyle \frac{{\left(x-L/2\right)}^2}{2i_{\mathrm{first}}^2}}\right]+{\displaystyle \frac{\pi R^2{\eta }_{\mathrm{second}}}{i_{\mathrm{second}}\sqrt{2\pi }}}\exp \left[-{\displaystyle \frac{{\left(x-L/2\right)}^2}{2i_{\mathrm{second}}^2}}\right]$$

(4)

where i_first and i_second represent the settlement trough width coefficients of the preceding and subsequent tunnels, respectively, and η_first and η_second represent the ground loss ratios of the preceding and subsequent tunnels, respectively.

This method accounts for the influence of the tunnel spacing on the settlement and assumes that the total ground settlement is not symmetric. However, the determination of the four unknown parameters (i_first, i_second, η_first, and η_second) remains unresolved, and no specific methods are provided for determining i_second and η_second.

Chapman [17] extended the Peck formula and proposed a correction for the ground settlement of a subsequent tunnel, accounting for the disturbance caused by the preceding tunnel. It was proposed that, in clayey soils, ground settlement due to the subsequent tunnel must be corrected by approximately 60%. The corrected formula is as follows:

$$S\left(x\right)=S{\left(x\right)}_1(1+(m(1-{\displaystyle \frac{\left|0.5L+x\right|}{2.5-3kh}})))$$

(5)

where S(x)₁ is the uncorrected ground settlement value, m is the ground settlement correction coefficient with a value of 0.6 in clay, k is the width coefficient of the settlement trough, and h is the burial depth along the tunnel axis.

Although this method incorporates the effect of the preceding tunnel on the subsequent tunnel, the results indicate that the maximum ground settlement caused by the subsequent tunnel excavation exceeds that of the preceding tunnel. However, some field measurements have indicated the opposite phenomenon. Moreover, the correction coefficient m is a constant, without considering the influence of L.

3.2. Definition of Closely Spaced Double-Track Tunnels

Numerous related studies [18,19] have shown that the ground surface settlement curve induced by the construction of parallel double-track tunnels is related to the distance L between the two tunnels and the burial depth h. Based on the relationship between L and h, double-track tunnels can be categorized into non-closely spaced and closely spaced cases.

Cording et al. [20] indicated that, for various types of clay, the horizontal distance from the edge of the ground settlement trough to the tunnel axis is approximately h + R.

Wu et al. [7] proposed a relative horizontal distance coefficient C = L/(h + R) for the axes of a double-track tunnel. He suggested that this coefficient C can be used to define the closely spaced condition for a double-track tunnel, indicating that when C < 0.6, the distance between the centerlines of the double-track tunnel is considered closely spaced.

Wei et al. [21] analyzed the problem using random medium theory, considering the relationship between the edge-to-edge distance within a double-track tunnel (L − D) and the ground surface settlement curve. He modified the horizontal distance to be the distance to the edge of the tunnel (h + D) and proposed the following formula for defining closely spaced double-track tunnels:

$$P={\displaystyle \frac{L^2}{\left(L-D\right)\left(h+D\right)}}$$

(6)

where D represents the tunnel excavation width, and p is the closely spaced distance defining coefficient (with the range for closely spaced distance being P ≤ 1.1). A smaller P value indicates a shorter distance between the axes of a double-track tunnel.

3.3. Derivation of the Three-Dimensional Solution for Ground Displacement Induced by Excavation of Closely Spaced Parallel Double-Track Tunnels

Extensive field measurements indicate that the ground settlement curve for a closely spaced parallel double-track tunnel follows a normal distribution. During the construction of a parallel double-track tunnel, the excavation of the tunnel first excavated affects the subsequent tunnel, resulting in the maximum value of the total ground settlement curve often deviating from the centerline between the two tunnels. Additionally, the calculation parameters differ from those used for single-track tunnels. The positional relationship and directional coordinates of the parallel double-track tunnel are shown in Figure 2, where the y-direction represents the advancement direction of the tunnel.

As shown in Figure 3, New et al. [22] proposed that the maximum ground settlement induced by the construction of a parallel double-track tunnel could be offset. They introduced an offset parameter e to modify the Peck formula, and proposed the following:

$$S\left(x\right)={\displaystyle \frac{\pi R^2{\eta }_{\mathrm{all}}}{i_{\mathrm{all}}\sqrt{2\pi }}}\exp \left[-{\displaystyle \frac{{\left(x-e\right)}^2}{2i_{\mathrm{all}}^2}}\right]$$

(7)

$${\eta }_{\mathrm{all}}={\displaystyle \frac{{V_{\mathrm{s}}}_{\mathrm{all}}}{\pi R^2}}$$

(8)

where e is the horizontal distance of the maximum settlement value deviating from the centerline, with positive values indicating a deviation towards the direction of the tunnel first excavated. Furthermore, V_sall represents the total soil loss per unit length, η_all denotes the total soil loss rate, and i_all is the coefficient for the total settlement trough width.

An analysis of extensive field data showed that when the axes of the twin parallel tunnels were close together, the total ground settlement caused by construction followed a normal distribution curve. The aforementioned formula can be used directly to calculate the ground settlement. However, this formula only applies to ground settlement and is not suitable for calculating deep-soil layer settlement. Therefore, it is necessary to modify the settlement trough width coefficient i.

Mair et al. [23] demonstrated via centrifuge tests that the width coefficient i for cohesive soils is given by:

$$i\left(z\right)=0.5\mathrm{h}-0.325\mathrm{z}$$

(9)

Han et al. [24] supplemented and extended Mair’s formula to address its limitation of being applicable only to cohesive soils. They proposed the following:

$$i\left(z\right)=i(0)\left(1-az/h\right),i(0)=kh$$

(10)

where a is a parameter that considers the soil conditions with values ranging from 0 to 1 and k is the settlement trough coefficient. In the absence of regional experience, a is recommended to be 0.65 for cohesive soils and 0.50 for sandy soils.

Jiang et al. [25] assumed that the volume of the settlement trough formed by tunnel excavation in deep soil layers was equal to the volume of soil loss. The settlement trough curves for each soil layer were represented by a normal distribution function. Employing regression analysis, they proposed empirical calculation formulas for the soil layer settlement at different depths caused by single-line tunnel excavation:

$$i\left(z\right)=i{\left(1-z/h\right)}^{0.3}$$

(11)

where z represents the depth of the soil layer, with a value range of 0 ≤ z < h − R.

Sun et al. [26] used numerical simulations to obtain transverse settlement trough curves at different soil depths and proposed the following formula:

$$i\left(z\right)=i{\left(1-z/h\right)}^{0.5}$$

(12)

Wei [27] modified Equations (11) and (12) to broaden their applicability and proposed the following:

$$i\left(z\right)=i{\left(1-z/h\right)}^n$$

(13)

where n is an influencing factor related to the tunnel radius and soil conditions. A smaller n value results in a larger i(z) value. For cohesive soils, n ranges from 0.35 to 0.85; for sandy soils, it ranges from 0.85 to 1.00.

By combining Equations (7) and (13), the deep-soil layer settlement caused by the construction of parallel twin tunnels can be obtained. Additionally, this formula accounts for the asymmetry of the settlement curve. For instance, consider the excavation on the left-hand side.

$$S_z\left(x\right)={\displaystyle \frac{S_{\max \mathrm{all}}}{{\left(1-z/h\right)}^n}}\exp \left[-{\displaystyle \frac{{\left(x-e\right)}^2}{2i_{\mathrm{all}}^2{\left(1-z/h\right)}^{2n}}}\right]$$

(14)

$${S_{\max }}_{\mathrm{all}}={\displaystyle \frac{\pi R^2{\eta }_{\mathrm{all}}}{i_{\mathrm{all}}\sqrt{2\pi }}}$$

(15)

Relevant studies [14] have indicated that soil deformation resulting from tunnel excavation and soil loss exhibits significant three-dimensional characteristics. The deformation at any given point (x, y, z) in relation to the ground surface, extending above the tunnel crown, is not solely related to the distance x from the axis and depth z, but also to the excavation direction y. The soil deformation in front of the excavation face decreased progressively in the positive y-direction, whereas the deformation behind the excavation face increased progressively in the negative y-direction and eventually stabilized after a certain distance. Therefore, the soil loss rate η is not a constant, but varies continuously along the excavation y-direction.

By introducing the expression for soil layer loss rate along the excavation direction η(y) proposed by Wei [28], the two-dimensional solution of soil settlement caused by twin tunnel excavation is extended to a three-dimensional solution.

$$\eta \left(y\right)={\displaystyle \frac{\eta }{2}}\left[1-{\displaystyle \frac{y}{\sqrt{y^2+h^2}}}\right]$$

(16)

By combining Equations (14)–(16), the three-dimensional solution for the deep-soil displacements caused by the excavation of closely spaced parallel twin tunnels is obtained as follows:

$$S_z\left(x,y,z\right)={\displaystyle \frac{\pi R^2{\eta }_{\mathrm{all}}}{2i_{\mathrm{all}}\sqrt{2\pi }{\left(1-z/h\right)}^n}}\exp \left[-{\displaystyle \frac{{\left(x-e\right)}^2}{2i_{\mathrm{all}}^2{\left(1-z/h\right)}^{2n}}}\right]\times \left[1-{\displaystyle \frac{y}{\sqrt{y^2+h^2}}}\right]$$

(17)

where i_all, η_all, e, and n are parameters that can be selected with reference to the previous study [7].

Table 1. Results of the back-analysis results on the measured values.

Serial Number	Section Name	h/m	L/m	ifirst/m	iall/m	e/m	ηfirst/m	ηall/m
1	23-AR-001	22	10.3	15	17	2	4.86	7.22
2	26-AR-001	18.5	20	13	18	4	4.42	7.71
3	CS-8B	19	17.5	12	14	3	0.74	1.03
4	CS-8D	20.1	14.5	10	13	1	0.69	1.10
5	SS-5T-52e-s	22.2	20	13	17	0	1.71	3.94
6	SS-5T-52e-o	26	15	14	17	0	0.92	1.44
7	23-G3-007-019	19	20	9	11	8	2.78	3.78
8	HS1	12.6	20	9	11	8	2.78	3.78
9	HS2	17.4	20	6	9	0	1.14	1.83
10	HS3	18.4	20	8	11	0	1.07	1.94
11	HS4	22.9	20	11	15	10	1.43	2.81
12	HS5	35.6	20	15	18	0	0.72	1.71
13	HS6	37.6	20	15	18	0	0.72	1.71
14	HS7	44	20	22	25	0	0.39	1.08
15	HS8	46.5	20	23	26	0	0.29	0.91

is sourced from study [7] and presents the fitting of ground settlement induced by the preceding tunnel, as well as the total ground settlement, using the Peck formula. As observed in Table 1, the average value of parameter e is 1.87 m. However, determining the exact value of e is challenging, and its specific selection should be made by referring to the actual engineering conditions.

3.4. Case Verification

To validate Equation (17) derived in this study, we apply it to a real-world engineering case involving the excavation of a double-track tunnel in Section III of the Shenyang Metro Line 3. The project spanned the section between Sanhao Street Station and the Industrial Exhibition Hall Station, where the total length of the left tunnel was 570.62 m, while that of the right tunnel was 594.54 m. A top-down bench method with a composite lining structure was adopted for the construction.

The standard cross-section of the interval tunnel had a width and height of 6.40 and 6.63 m, respectively. The initial support structure was 0.25 m thick and constructed using C25 shotcrete, while the secondary lining structure was 0.35 m thick and was composed of C40 concrete. The specific dimensions of the tunnel cross section are shown in Figure 4. The burial depth of the tunnel axis was h = 18.00 m, and the horizontal distance between the axes of the two tunnels was L = 12.84 m. The tunnels primarily passed through gravelly sand and rounded gravel layers, with the left tunnel excavated before the right tunnel.

According to Equation (6), the small-spacing double-track tunnel criterion coefficient for this project is P = 1.05, which is less than 1.10, thus satisfying the conditions for applying Equation (17) derived in this study. The Peck formula was used to fit the measured surface settlement curve corresponding to the excavation of the tunnel first excavated. The following values were obtained via back analysis: η_first = 0.85% and i_first = 13.09 m. As shown in Figure 5, the surface settlement curve corresponding to the excavation of the tunnel first excavated follows a normal distribution.

Based on the method in reference [7], the following values are calculated: i_all = 16.81 m, η_all = 1.28%, n = 0.90, e = 2.00 m (towards the tunnel first excavated). Taking y = 30.00 m as the calculation condition for Equation (17) and fitting the measured data through back analysis, we obtain i_all = 17.19 m and η_all = 1.87%. These values differed from the theoretical calculation results by 2.21% and 0.59%, respectively, with the measured values being slightly larger than theoretical values. As shown in Figure 6, the surface settlement curve corresponding to the excavation of the small-spacing double-track tunnel follows a normal distribution. The maximum settlement deviated towards the tunnel first excavated, and the results calculated using the method proposed in this study were consistent with the measured data, verifying the accuracy of the formula.

4. Influence of Different Parameters on Ground Settlement Caused by Excavation of Double-Track Tunnel

Studies have indicated that the maximum ground settlement during the excavation of double-track tunnels does not always occur directly above the tunnel centerline but could likely shift by a certain offset. The magnitude of this offset is influenced by several factors, including tunnel spacing, excavation sequence, and tunnel depth. In this section, based on an engineering case, a numerical calculation model is established using MIDAS/GTS2024 finite element analysis software. By controlling the variables, a numerical simulation was conducted to study the impact of different parameters on the ground settlement caused by double-track tunnel excavation.

4.1. Numerical Calculation Model Establishment

4.1.1. Soil Model Constitutive Relationships

A modified Mohr–Coulomb constitutive model [29] was employed to simulate the surrounding strata of the tunnel. This model is an optimized version of the Mohr–Coulomb model, which provides as accurate representation of the soil behavior under tunnel excavation conditions. The modified Mohr–Coulomb model is particularly suitable to account for the variations in loading and unloading stiffness moduli, which improves the alignment of the simulation results obtained under tunnel excavation conditions with the actual engineering data. The stress–strain curve for the modified Mohr–Coulomb model under the triaxial compression tests is shown in Figure 7.

4.1.2. Computational Parameters for Simulating the Soil Layers

Based on the geotechnical investigation data from the excavation project of Shenyang Metro Line 3, and with reference to design documents and relevant standards, this tunnel segment primarily passes through gravelly sand and rounded gravel layers. The calculated parameters selected for each soil layer are listed in

Table 2. Computational parameters for simulating soil layers.

Name	Unit Weight (KN/m3)	Secant Modulus (MPa)	Unloading Stiffness (MPa)	Poisson’s Ratio	Cohesion (kPa)	Internal Friction Angle (°)
Miscellaneous fill	17.0	10	30	0.28	10	6.0
3-4 Grit	19.0	42	126	0.28	3	34.0
3-5-0 Boulder	20.1	80	240	0.29	0	37.0
4-4-0 Grit	20.1	59	177	0.29	0	37.0

.

4.1.3. Computational Parameters for the Tunnel Structure

The tunnel structure was simulated using an isotropic elastic material. The initial support structure of the tunnel was composed of C25 shotcrete, and the secondary lining was composed of C40 reinforced concrete. The specific material parameters for each structure are listed in

Table 3. Computational parameters for tunnel structure structures.

Name	Unit Weight (KN/m3)	Elastic Modulus (MPa)	Poisson’s Ratio
Initial support	22	23,000	0.2
Secondary lining	25	32,500	0.2

.

4.1.4. Model Mesh Generation

A numerical calculation model was developed using MIDAS/GTS2024 finite element analysis software to simulate and analyze the construction process of a shallow-buried tunnel close to the pile foundations of an overpass. The structural dimensions in the finite element model were set based on the actual measurements, and the boundary dimensions were determined based on the Saint-Venant equivalent principle [30]. The influence range was considered to extend three to five times the width of the tunnel excavation section. Consequently, the three-dimensional dimensions of the finite element model were set as 80 m in length, 50 m in width, and 40 m in height. The tunnel section was constructed employing the upper and lower bench methods for safety. The initial support structure was 0.25 m thick and constructed using C25 shotcrete, whereas the secondary lining structure was 0.35 m thick and constructed using C40 concrete.

In the simulation, it was assumed that each soil layer was uniformly and horizontally distributed, and that the stress and deformation of each soil layer and structural component remained within the elastoplastic range. The soil layers were modeled using a modified Mohr–Coulomb constitutive model. Furthermore, all soil layers and secondary lining structures were represented with three-dimensional solid elements, whereas the initial support structure was modeled with two-dimensional plate elements.

4.1.5. Load Boundary Conditions

• Model Load Conditions:

The entire model was subjected to gravitational loading. After achieving geostress equilibrium in the initial stage, the nodal displacements were reset to zero.

2.

Model Boundary Constraint Conditions:

The bottom of the entire model was fixed, while the top was free.

The displacements in the x and y directions were constrained on both sides.

The rotational degree of freedom Rz was constrained for one-dimensional pile elements.

The subsoil cap and secondary lining structures were initially modeled as part of the strata prior to excavation. During the bridge construction and excavation stages, the attributes of the cap and secondary lining structures must be updated.

4.1.6. Construction Phase Simulation

The excavation sequence in the project adopted the benching tunnelling method, which is suitable for stratified soils where short-term stability is achievable after excavation. This approach, which is an improvement over full-face excavation, divides the excavation cross section into upper and lower sections to be sequentially excavated. As shown in Figure 8, in MIDAS/GTS, the benching tunnelling method simulation proceeded as follows:

• Excavate the upper bench, applying initial support to the upper section.
• Excavate the lower bench and apply the initial support to the lower section, while simultaneously proceeding with the next step for the upper bench.
• Once the initial support structure stabilizes, apply the secondary lining structure.
• Repeat steps 1–3, alternating between the upper and lower bench excavations until completion of the entire section.

Figure 8. Construction process simulation of the benching tunnelling method. (a) Excavate the upper bench. (b) Excavate the lower bench. (c) Completion of the initial support. (d) Application of the secondary lining structure.

Figure 8. Construction process simulation of the benching tunnelling method. (a) Excavate the upper bench. (b) Excavate the lower bench. (c) Completion of the initial support. (d) Application of the secondary lining structure.

Figure 9a shows a schematic diagram of the numerical calculation model. The purple soil layer corresponds to “Miscellaneous fill” in Table 2, the green soil layer corresponds to “3-4 Grit” in Table 2, the cyan soil layer corresponds to “3-5-0 Boulder” in Table 2, and the bottommost loess-colored soil layer corresponds to “4-4-0 Grit” in Table 2. The specific calculation parameters for each soil layer are detailed in Table 2. Figure 9b is a schematic diagram of the loads and boundary conditions.

4.2. Influence of Tunnel Spacing on Ground Settlement

In the real-world engineering project, the excavation diameter of the tunnel was D = 6.42 m, and the tunnel burial depth was h = 18.00 m. According to Equation (6) for defining small-spacing double-track tunnels, the range of L values that satisfy the small-spacing double-track tunnel is between 10.6136 m and 16.2484 m. To further study and summarize the variation patterns of ground settlement during the excavation of double-track tunnels at different spacing intervals, four representative spacing values were selected for simulation: L = 10.6 m, L = 12.6 m, L = 14.6 m, and L = 16.6 m.

Figure 10 depicts a comparison of the surface settlement curves caused by the double-track excavation of tunnels associated with different spacings. The results indicated that the ground settlement curve followed a normal distribution regardless of the spacing, but its shape varied with the tunnel spacing. When the spacing was small, the settlement curve was V-shaped, indicating a concentrated settlement in the central area between the two tunnels. As the tunnel spacing increased, the curve gradually evolves into a U-shape, indicating a wider range of settlements with a more uniform distribution. At a spacing of L = 16.6 m, the curve becomes W-shaped, indicating a reduced interaction between the two tunnels, resulting in a multi-peaked distribution.

Additionally, analysis of the maximum total settlement indicates that as the spacing increased from 10.6 m to 16.6 m, the maximum settlement decreased from 8.74 mm to 6.37 mm. This trend indicates that appropriately increasing the tunnel spacing helps reduce the maximum ground settlement. Furthermore, the location of the maximum settlement shifted towards the tunnel first excavated, with the offset values being 0.87 m, 1.13 m, 4.01 m, and 5.97 m, corresponding to increasing spacing.

4.3. Influence of Excavation Sequence on Ground Settlement

To investigate the influence of different excavation sequences on the ground settlement, three typical working conditions were simulated based on real-world engineering cases:

• Sequential excavation: left tunnel excavated first, followed by the right tunnel after completion.
• The left tunnel excavated halfway before starting the right tunnel.
• Simultaneous Excavation of both Tunnels.

Figure 11 depicts a comparison of the surface settlements caused by the double-track excavation of tunnels associated with different excavation sequences. As illustrated, sequential excavation (Condition 1) resulted in the smallest ground settlement, with a maximum total settlement of 5.91 mm, which was smaller than that in Conditions 2 (6.11 mm) and 3 (6.35 mm). This is likely because sequential excavation allows the soil to deform and adjust, thereby reducing the impact of subsequent excavations. Furthermore, under Condition 3, the maximum settlement occurred at the centerline of the two tunnels, whereas under Conditions 1 and 2, the maximum settlement shifted towards the tunnel first excavated. The offset under Condition 1 was greater than that under Condition 2, with offsets of 5.04 m and 2.13 m, respectively.

4.4. Influence of Burial Depth on Ground Settlement

To study the impact of tunnel depth on ground settlement, three depth scenarios were simulated based on real-time project conditions: h = 9.62 m, h = 12.66 m, and h = 15.70 m. The dimensions of these three working condition models were kept consistent, with only the tunnel burial depth being changed.

Figure 12 depicts the surface settlement caused by the double-track excavation of the tunnels associated with different burial depths. As the depth increases, the maximum total settlement decreases from 6.52 mm to 5.69 mm, and the maximum settlement position gradually moves towards the tunnel centerline. The offset values for the different depths were 6.11, 4.02, and 1.01 m. These results indicate that increasing the tunnel depth reduces the effect of the first tunnel excavation on the second tunnel, resulting in a more evenly distributed settlement. Additionally, the shape of the settlement curve changes with depth, transitioning from a W-shape to a U-shape. These trends indicate that increasing the tunnel depth can effectively reduce the surface settlement and promote a more uniform distribution, which is critical for underground construction design and implementation. In summary, to reduce ground deformation, the tunnel spacing should be appropriately increased during the construction design phase. Based on ground conditions, excavation should be carried out at a deeper position whenever possible. Moreover, the excavation progress of the two tunnels should be maximally staggered to allow sufficient time for soil deformation adjustment. For shorter tunnels, a sequential excavation method should be adopted whenever possible.

5. Conclusions

Based on a study of the Shenyang Metro Line 3 project, this study proposed a displacement calculation formula for soil deformation resulting from the excavation of a small-spacing double-track tunnel. This study investigated the patterns of ground settlement deformation induced by these tunnel excavations under the influence of various parameters. The conclusions drawn are as follows.

• Based on the Peck formula for predicting the surface settlement induced by single-track tunnel excavation, this study proposed a definition for small-spacing double-track tunnels. The horizontal offset e and soil loss rate η (y) were introduced to derive a displacement calculation formula for soil deformation during small-spacing double-track tunnel excavation. The accuracy of the proposed formula was validated.
• The deformation patterns of ground settlement induced by small-spacing double-track tunnel excavation influenced by various parameters were characterized. As the tunnel spacing increased, the ground settlement curve transitioned from a V-shape to a U-shape, and eventually to a W-shape, and the maximum settlement gradually decreased. The location of the maximum settlement shifted towards the tunnel first excavated. Additionally, as the burial depth increased, the ground settlement curve evolved from a W-shape to a U-shape, and the maximum settlement decreased as the location of the maximum settlement shifted towards the tunnel centerline. Sequential excavation induced the smallest ground settlement; however, it was significantly influenced by the tunnel first excavated. In contrast, simultaneous excavation resulted in the largest settlement, and the maximum settlement occurred at the centerline. As the burial depth increased, the influence of the tunnel excavation on the ground settlement decreased, and the settlement distribution became increasingly uniform.