# Analytical Study on Surface Settlement Troughs Induced by the Sequential Excavation of Adjacent and Parallel Tunnels in Layered Soils

Authors: Cong Dong; Jian Lin; Guangyong Cao; Hua Cheng; Linlin Shi; Xiaohui Zhang
Journal: Advances in Civil Engineering
 (2022)

---

## Abstract

In this paper, the settlement troughs induced by the excavation of adjacent and parallel tunnels in layered soils were studied. Firstly, a trough width parameter calculation method in layered soils was established based on the propagation model of the plastic zone and unloading disturbance zone. Secondly, the mechanism of the superposition disturbance between parallel tunnels is analyzed, and a method for calculating additional settlement trough is proposed. Finally, the applicability of the proposed model was demonstrated with two case studies of the parallel tunnels. The following conclusions were obtained: the difference of soil properties in layered soils had a significant influence on the width parameter of surface settlement trough; a Gaussian curve can describe the additional settlement caused by superimposed disturbance; and finally, the relationship between the ground loss induced by superposition disturbance and the ground loss induced by the preceding tunnel was approximately linear. The model presented in this paper is highly effective and convenient for use in practice and extends the calculation method of surface settlement trough based on the Gaussian curve.


---

## Body

## 1. Introduction

The settlement of overlying soil above the tunnel is induced by the disturbance of tunnel construction on surrounding soil [1]. The settlement not only affects the stability of surface buildings [2–4], but also jeopardizes the normal use of underground pipelines [5, 6]. The complex structure environment puts forward higher settlement control requirements for tunnel engineering, especially in adjacent and parallel tunnels event. Therefore, the tunneling-induced surface settlement calculation has received extensive attention [7, 8].Peck [9] proposed that the single tunneling-induced surface settlement trough could be distributed by a Gaussian curve as follows:(1)Sx=Smaxexp−x22i2,where Sx is ground settlement; Smax is the maximum surface settlement above the tunnel centerline; x is the horizontal distance from the tunnel centerline; and i is the distance from the tunnel centerline to the curve’s inflection point, which is called trough width parameter, as shown in Figure 1. The empirical methods based on the Gaussian curve have become a generally accepted research model on the surface settlement trough and the interaction of two adjacent and parallel tunnels.Figure 1
Ground settlement trough induced by a single tunnel.Many measured data in engineering practice show that the surface settlement trough induced by the adjacent and parallel tunnels is asymmetrical [10–12]. It is because the following tunnel was excavated in a brown-field site (previously developed site) [13, 14].Some scholars have proposed that the surface settlement trough above the following tunnel, which is affected by the preceding excavation, enlarges and shifts. The settlement trough above the following tunnel could be obtained by modifying and moving according to the settlement date of the preceding tunnel. This method is called the modification factor method [15]. Chapman and Hunt [16] proposed the modified function as follows:(2)S2x=1+k1−B+x2.5i1hS1x,where S1x is the surface settlement trough above the preceding tunnel (same as the single tunnel); S2x is the modified surface settlement trough; k is the value of maximum modification; i1 is the trough width parameter of the preceding tunnel. Dong [15] proposed the deal of maximum modification k could be determined as follows:(3)k=−1.85B2h+D+1.002,where B is the center-to-center spacing of tunnels. Wei and Wei [17] proposed a method of moving settlement trough.Other scholars proposed that calibration ofi and Smax is vital during the settlement trough calculation above the following tunnel. Ma et al. [18] converted the Gaussian curve into a linear form, and two numerical methods were used to estimate the settlement parameters induced by parallel tunnels. Zheng et al. [19] investigated the interaction between parallel tunnels by the physical model test, and the differences between the surface settlement troughs above two tunnels were described in terms of parameters.Many scholars have studied the surface settlement trough induced by adjacent and parallel tunnels with various methods, but the reasons for the changes of surface settlement trough were rarely mentioned from the perspective of interactions. The consistent relationship between model parameters and excavation parameters remains unclear. This paper presents a method for calculating the trough width parameter of surface settlement trough in layered soils. Then, a new calculation method of surface settlement trough above the following tunnel is demonstrated by analyzing the mechanisms of superposition disturbance. Finally, two cases’ study is presented to assess the rationale of the proposed model.


## 2. Calculation of Disturbance Zone and Surface Settlement Trough Induced by Tunnel Excavation in Layered Soils

In order to minimize the detrimental effect of tunneling on surrounding structures, it is necessary to calculate the disturbance zone induced by tunnel excavation accurately. In practical engineering, tunnels are primarily constructed in layered soils. The differences in geotechnical properties significantly affect the plastic zone, unloading disturbance zone, and surface settlement trough.


### 2.1. Basic Assumptions

(1)
The tunnel is excavated in a stratum with a large thickness and covered with multiple soil layers. The surrounding soil is only defined as a standard elastomer.(2)
Instantaneous surface settlement occurs during tunnel excavation, and the ground will not continue to move after the excavation.(3)
The tunnel is excavated in a greenfield, and a Gaussian curve could distribute the settlement trough.


### 2.2. Plastic Zone and Unloading Disturbance Zone in Layered Soils

Assuming that there aren layers of soil overlying the tunnel and the tunnel was excavated in the nth layer of soil, h is the distance from the tunnel centerline to the ground surface. In layered soils, h1 is the thickness of the first soil layer, and so on; hn is the distance from the tunnel centerline to the interface of the n−1th layered soil (Figure 2). The plastic zone and the unloading disturbance zone are calculated by the layered accumulation method.Figure 2
Plastic zone and unloading disturbance zone in layered soils.The tunnel excavation process can be seen as the unloading process of the column hole, and the plastic zone appears in the soil around the tunnel. For the sake of simplicity, the force around the tunnel is assumed to distribute symmetrically. The initial Earth stress and the support force of the tunnel can be calculated as follows:(4)σ0=∑i=1nhiγi,σa=σ0tan245∘−φn2+2cntan45∘−φn2.According to the Fenner formula [20], the radius of the plastic zone caused by tunnel construction can be estimated by(5)a=D21−sinφnσ0+cncotφnσa+cncotφn1−sinφn/2sinφn,where hi is the thickness of the i soil layer; γi is the unit weight of soil of the ith layer; φn is the internal friction angle of the soil layer where the tunnel is located; cn is the cohesion of the soil layer where the tunnel is located.The unloading disturbance zone is produced due to the stress release of the soil above the plastic deformation zone. The shear failure has taken place at the boundary of the unloading disturbance zone. The failure plane of the unloading disturbance zone is tangent to the shearing disturbance zone, which inclines by an angle45°+φ/2 from the horizontal based on Rankine’s Earth pressure theory [21, 22]. Because of the different properties of each soil in layered soils, the failure plane is deflected at the interface between the two soil layers [23, 24]. According to the geometric relationship as shown in Figure 2, the width of the unloading disturbance zone at the interface of the nth soil layer can be calculated as follows:(6)Mn=2hntan45∘+φn/2+asin45∘+φn/2.Hence, the width of the unloading disturbance zone on the ground is obtained as follows:(7)M1=Mn+2∑i=1n−1hitan45∘+φi2.


### 2.3. Surface Settlement Trough and Ground Loss of Single Tunnel

According to the above analysis, it can be seen that the unloading disturbance zone and the settlement trough are equal in width. The trough width parameter in the Gaussian curve determines the width of the settlement trough, so there will be a corresponding relationship between the trough width parameter and the width of the settlement trough. Based on a review of a large number of measured data, Stallebrass and Taylor [25] found that the trough width parameter has a linear correlation with the width of settlement trough, as follows:(8)i1=M15.Another key parameter in the Gaussian curve is the maximum ground settlement just above the tunnel centerline, which determines the depth of the settlement trough and has an inevitable relationship with the ground loss induced by the tunnel excavation. Ground loss is defined as the volume of surface settlement trough per unit length of the tunnel. Although there are some calculation methods for ground loss, it is difficult to select parameters and consider factors that influence the ground loss. Attewell and Farme [21] proposed that the calculation formula of ground loss can be obtained by integrating the Gaussian curve as follows:(9)Vloss=∫−∞+∞Smaxexp−x22i2dx=2πiSmax,where Vloss is the ground loss induced by tunnel excavation.


### 2.4. Surface Settlement Trough above Parallel Tunnels

Through the statistical analysis of the surface settlement in published case history, it is found that the settlement trough above the parallel tunnels is difficult to describe by a single Gaussian curve. Therefore, Suwansawat and Einstein [12] proposed a superposition calculation method to describe the surface settlement trough above a parallel tunnel. According to the method, the surface settlement trough caused by the excavation of adjacent and parallel tunnels would consist of two components: the surface settlement troughs of the preceding tunnel and the following tunnel. Both troughs caused can be described by Gaussian curve. The first component equals the surface settlement trough induced by the single tunnel. However, the calculation of the second component (i.e., surface settlement trough above the following tunnel) would be the critical point of this method.


## 2.1. Basic Assumptions

(1)
The tunnel is excavated in a stratum with a large thickness and covered with multiple soil layers. The surrounding soil is only defined as a standard elastomer.(2)
Instantaneous surface settlement occurs during tunnel excavation, and the ground will not continue to move after the excavation.(3)
The tunnel is excavated in a greenfield, and a Gaussian curve could distribute the settlement trough.


## 2.2. Plastic Zone and Unloading Disturbance Zone in Layered Soils

Assuming that there aren layers of soil overlying the tunnel and the tunnel was excavated in the nth layer of soil, h is the distance from the tunnel centerline to the ground surface. In layered soils, h1 is the thickness of the first soil layer, and so on; hn is the distance from the tunnel centerline to the interface of the n−1th layered soil (Figure 2). The plastic zone and the unloading disturbance zone are calculated by the layered accumulation method.Figure 2
Plastic zone and unloading disturbance zone in layered soils.The tunnel excavation process can be seen as the unloading process of the column hole, and the plastic zone appears in the soil around the tunnel. For the sake of simplicity, the force around the tunnel is assumed to distribute symmetrically. The initial Earth stress and the support force of the tunnel can be calculated as follows:(4)σ0=∑i=1nhiγi,σa=σ0tan245∘−φn2+2cntan45∘−φn2.According to the Fenner formula [20], the radius of the plastic zone caused by tunnel construction can be estimated by(5)a=D21−sinφnσ0+cncotφnσa+cncotφn1−sinφn/2sinφn,where hi is the thickness of the i soil layer; γi is the unit weight of soil of the ith layer; φn is the internal friction angle of the soil layer where the tunnel is located; cn is the cohesion of the soil layer where the tunnel is located.The unloading disturbance zone is produced due to the stress release of the soil above the plastic deformation zone. The shear failure has taken place at the boundary of the unloading disturbance zone. The failure plane of the unloading disturbance zone is tangent to the shearing disturbance zone, which inclines by an angle45°+φ/2 from the horizontal based on Rankine’s Earth pressure theory [21, 22]. Because of the different properties of each soil in layered soils, the failure plane is deflected at the interface between the two soil layers [23, 24]. According to the geometric relationship as shown in Figure 2, the width of the unloading disturbance zone at the interface of the nth soil layer can be calculated as follows:(6)Mn=2hntan45∘+φn/2+asin45∘+φn/2.Hence, the width of the unloading disturbance zone on the ground is obtained as follows:(7)M1=Mn+2∑i=1n−1hitan45∘+φi2.


## 2.3. Surface Settlement Trough and Ground Loss of Single Tunnel

According to the above analysis, it can be seen that the unloading disturbance zone and the settlement trough are equal in width. The trough width parameter in the Gaussian curve determines the width of the settlement trough, so there will be a corresponding relationship between the trough width parameter and the width of the settlement trough. Based on a review of a large number of measured data, Stallebrass and Taylor [25] found that the trough width parameter has a linear correlation with the width of settlement trough, as follows:(8)i1=M15.Another key parameter in the Gaussian curve is the maximum ground settlement just above the tunnel centerline, which determines the depth of the settlement trough and has an inevitable relationship with the ground loss induced by the tunnel excavation. Ground loss is defined as the volume of surface settlement trough per unit length of the tunnel. Although there are some calculation methods for ground loss, it is difficult to select parameters and consider factors that influence the ground loss. Attewell and Farme [21] proposed that the calculation formula of ground loss can be obtained by integrating the Gaussian curve as follows:(9)Vloss=∫−∞+∞Smaxexp−x22i2dx=2πiSmax,where Vloss is the ground loss induced by tunnel excavation.


## 2.4. Surface Settlement Trough above Parallel Tunnels

Through the statistical analysis of the surface settlement in published case history, it is found that the settlement trough above the parallel tunnels is difficult to describe by a single Gaussian curve. Therefore, Suwansawat and Einstein [12] proposed a superposition calculation method to describe the surface settlement trough above a parallel tunnel. According to the method, the surface settlement trough caused by the excavation of adjacent and parallel tunnels would consist of two components: the surface settlement troughs of the preceding tunnel and the following tunnel. Both troughs caused can be described by Gaussian curve. The first component equals the surface settlement trough induced by the single tunnel. However, the calculation of the second component (i.e., surface settlement trough above the following tunnel) would be the critical point of this method.


## 3. Calculation of Surface Settlement Trough above the following Tunnel Based on Superposition Technique

The existing calculation models have complex calculation processes and narrow application scope. Therefore, a calculation model of surface settlement trough above the following tunnel with broad applicability is established by analyzing the interaction mechanism between adjacent and parallel tunnels.


### 3.1. Overlapping Disturbance of Adjacent and Parallel Tunnel

The soil stability changes in the unloading disturbance area above the preceding tunnel due to the excavation disturbance. The soil in the overlapping zone has been disturbed so that the surface settlement trough above the following tunnel will be larger than the preceding tunnel. In the overlapping zone, the increase of settlement is called additional settlement.Considering the influence of the overlapping disturbance, it is reasonable that the surface settlement trough above the following tunnel could divide into two components for calculation. The first component is the surface settlement trough caused by tunnel excavation without considering the interaction and would be equal to the surface settlement trough of a single tunnel. The second component is the asymmetric settlement resulting from the overlapping disturbance in the overlapping zone, as shown in Figure3. The calculation of asymmetric settlement is the critical point of this method.Figure 3
Calculation diagram of calculation method based on superposition theory.


### 3.2. Additional Surface Settlement Trough Induced by Overlapping Disturbance

The mechanism of soil disturbance in the process of overlapping disturbance is complex, and the additional settlement is the result of various disturbances in the excavation process. It is hard to calculate the surface settlement accurately. As shown in Figure4, through the analysis of multiple groups of measured data, it could be found that the additional settlement trough caused by the overlapping disturbance also appears as a Gaussian curve, and the settlement characteristics can be described by a Gaussian curve as follows:(10)S′x=Smax′exp−x′22i′2,where x′ is the horizontal distance to the surface; S′x is additional surface settlement; i′ is the width parameter of additional settlement trough, Smax′ is the maximum additional settlement. The surface additional settlement trough and soil loss caused by overlapping disturbance are two key points of research.Figure 4
Comparison between the measured data of additional settlement and the fitting curves. (a) [26]. (b) [27]. (c) [28]. (d) [29].
(a)(b)(c)(d)In order to describe the surface additional settlement trough caused by superposition disturbance, it is necessary to determine the width parameteri′ and the maximum additional settlement Smax′. As shown in Figure 3, according to the definition and model of overlapping disturbance, the width of surface overlap disturbance is obtained as follows:(11)L=M1−B.And the width parameter of additional settlement trough is obtained based on (8), as follows:(12)i′=L5.The maximum additional settlement is closely related to the ground loss in the overlapped disturbance zone.


### 3.3. Ground Loss Induced by Overlapping Disturbance

The analysis of additional settlement found that it is difficult to establish a simple corresponding relationship between two adjacent and parallel tunnels in terms of the maximum surface settlement. Considering that the stratum conditions and the construction parameters of the primary and overlapping disturbance zone are similar, a simple correlation between the two adjacent and parallel tunnels can be established through the ground loss. However, the mechanism of soil loss caused by overlapping disturbance is complex, and it is difficult to monitor accurately. Therefore, the ground loss induced by overlapping disturbance can only be explored based on the existing measured settlement data. The additional settlement data can be obtained according to the following procedure:(1)
Translate the settlement trough of the postconstructed tunnel to the proconstructed tunnel so that the positions of the two tunnels are overlapped.(2)
Subtract the preceding tunnel’s settlement from the following tunnel’s settlement to obtain an additional surface settlement caused by superposition disturbance.(3)
The additional settlement value caused by the superposition disturbance is fitted and integrated by the gauss formula. And then, the soil loss caused by superposition disturbance is calculated.


### 3.4. Back Analysis of Ground Loss Induced by Overlapping Disturbance

A total of 14 tunnel excavation cases are collected in Tables1 and 2 to study the distribution of ground loss caused by overlapping disturbance. All cases are parallel tunnels excavated by the EPB shield machine. Table 1 summarizes the details of each tunnel project, and Table 2 summarizes the surface settlement data from the measured and calculated data. In Table 2, Vloss′ is the ground loss caused by superposition disturbance and Vloss1 is the ground loss caused by the preceding tunnel.Table 1
Engineering details.
Case NO.Project nameGround conditionsD (m)Reference1Bangkok MRTA projectSoft clay, hard clay6.45Suwansawat and Einstein [12]2The new Milan metro line 5Gravelly sand6.7Fargnoli et al. [7]3Wuhan Yangtze river tunnelMiscellaneous fill, silty clay, Mucky soil, silty clay, silt, silty clay, fine sand12.6Ma [30]4Nanjing metro line 1Silt, mucky silt clay6.4Li [31]Table 2
Data obtained from measurement and fitting.
Case NO.Monitoring sectionh (m)B (m)L (m)i′ (m)Smax′ (mm)Vloss′ (m3/m)i1 (m)Smax1 (mm)Vloss1 (m3/m)1CS-8D20.11519.263.853.40.03396.50.1472SS-5T-52E-S22.22016.453.294.30.0351315.40.5013S19151510.422.081.90.01078.80.1544S351516.58.721.741.80.00768.80.1315HS-217.42017.463.4927.00.236843.70.8736HS-318.42020.984.1919.10.201729.90.5237HS-535.62039.917.987.70.1551522.60.7338HS-637.62042.158.437.00.1481512.80.4829HS-7442049.269.853.30.083226.40.35210HS-846.52051.910.384.10.116236.00.34711H1111312.092.424.40.026619.00.28512H616.41320.084.026.60.067815.80.31613H7171321.204.244.30.046917.80.40114H15141316.773.355.70.048713.90.243Note: Cases 1-2 are from Project 1, Cases 3-4 are from Project 2, Cases 3–10 are from Project 3, and Cases 11–14 are from Project 4.For convenience, the least-squares principle is used to carry out linear fitting ofVloss′ in the overlapping disturbance area. All calculated Vloss′ for cases 1–14 from the back analysis are plotted as a function of Vloss1 in Figure 5. The approximate relationship of Vloss′∼Vloss1 can be expressed as(13)Vloss′=0.0249+0.2845Vloss.Figure 5
Fitting results.The coefficient of determination (R2) was used to compare the goodness of fitting. 72% of the data show a good fitting result (R2=0.7146), which indicates that Vloss′ has a significant linear positive correlation with Vloss1. It can be concluded that the ground loss caused by the overlapping disturbance can be obtained through the ground loss of the following tunnel.


### 3.5. Calculation Process of Settlement Trough for the following Tunnel

The settlement trough calculation procedure is shown in Figure6. Firstly, S1x and Vloss1 are monitored by the measured data after the preceding tunnel excavated. Secondly, S′x is determined by i′ and Vloss′, which are calculated by (12) and (13). Finally, based on superposition technique, S2x is predicted by superimposing S′x on S1x.Figure 6
Flowchart of the calculation procedure.


## 3.1. Overlapping Disturbance of Adjacent and Parallel Tunnel

The soil stability changes in the unloading disturbance area above the preceding tunnel due to the excavation disturbance. The soil in the overlapping zone has been disturbed so that the surface settlement trough above the following tunnel will be larger than the preceding tunnel. In the overlapping zone, the increase of settlement is called additional settlement.Considering the influence of the overlapping disturbance, it is reasonable that the surface settlement trough above the following tunnel could divide into two components for calculation. The first component is the surface settlement trough caused by tunnel excavation without considering the interaction and would be equal to the surface settlement trough of a single tunnel. The second component is the asymmetric settlement resulting from the overlapping disturbance in the overlapping zone, as shown in Figure3. The calculation of asymmetric settlement is the critical point of this method.Figure 3
Calculation diagram of calculation method based on superposition theory.


## 3.2. Additional Surface Settlement Trough Induced by Overlapping Disturbance

The mechanism of soil disturbance in the process of overlapping disturbance is complex, and the additional settlement is the result of various disturbances in the excavation process. It is hard to calculate the surface settlement accurately. As shown in Figure4, through the analysis of multiple groups of measured data, it could be found that the additional settlement trough caused by the overlapping disturbance also appears as a Gaussian curve, and the settlement characteristics can be described by a Gaussian curve as follows:(10)S′x=Smax′exp−x′22i′2,where x′ is the horizontal distance to the surface; S′x is additional surface settlement; i′ is the width parameter of additional settlement trough, Smax′ is the maximum additional settlement. The surface additional settlement trough and soil loss caused by overlapping disturbance are two key points of research.Figure 4
Comparison between the measured data of additional settlement and the fitting curves. (a) [26]. (b) [27]. (c) [28]. (d) [29].
(a)(b)(c)(d)In order to describe the surface additional settlement trough caused by superposition disturbance, it is necessary to determine the width parameteri′ and the maximum additional settlement Smax′. As shown in Figure 3, according to the definition and model of overlapping disturbance, the width of surface overlap disturbance is obtained as follows:(11)L=M1−B.And the width parameter of additional settlement trough is obtained based on (8), as follows:(12)i′=L5.The maximum additional settlement is closely related to the ground loss in the overlapped disturbance zone.


## 3.3. Ground Loss Induced by Overlapping Disturbance

The analysis of additional settlement found that it is difficult to establish a simple corresponding relationship between two adjacent and parallel tunnels in terms of the maximum surface settlement. Considering that the stratum conditions and the construction parameters of the primary and overlapping disturbance zone are similar, a simple correlation between the two adjacent and parallel tunnels can be established through the ground loss. However, the mechanism of soil loss caused by overlapping disturbance is complex, and it is difficult to monitor accurately. Therefore, the ground loss induced by overlapping disturbance can only be explored based on the existing measured settlement data. The additional settlement data can be obtained according to the following procedure:(1)
Translate the settlement trough of the postconstructed tunnel to the proconstructed tunnel so that the positions of the two tunnels are overlapped.(2)
Subtract the preceding tunnel’s settlement from the following tunnel’s settlement to obtain an additional surface settlement caused by superposition disturbance.(3)
The additional settlement value caused by the superposition disturbance is fitted and integrated by the gauss formula. And then, the soil loss caused by superposition disturbance is calculated.


## 3.4. Back Analysis of Ground Loss Induced by Overlapping Disturbance

A total of 14 tunnel excavation cases are collected in Tables1 and 2 to study the distribution of ground loss caused by overlapping disturbance. All cases are parallel tunnels excavated by the EPB shield machine. Table 1 summarizes the details of each tunnel project, and Table 2 summarizes the surface settlement data from the measured and calculated data. In Table 2, Vloss′ is the ground loss caused by superposition disturbance and Vloss1 is the ground loss caused by the preceding tunnel.Table 1
Engineering details.
Case NO.Project nameGround conditionsD (m)Reference1Bangkok MRTA projectSoft clay, hard clay6.45Suwansawat and Einstein [12]2The new Milan metro line 5Gravelly sand6.7Fargnoli et al. [7]3Wuhan Yangtze river tunnelMiscellaneous fill, silty clay, Mucky soil, silty clay, silt, silty clay, fine sand12.6Ma [30]4Nanjing metro line 1Silt, mucky silt clay6.4Li [31]Table 2
Data obtained from measurement and fitting.
Case NO.Monitoring sectionh (m)B (m)L (m)i′ (m)Smax′ (mm)Vloss′ (m3/m)i1 (m)Smax1 (mm)Vloss1 (m3/m)1CS-8D20.11519.263.853.40.03396.50.1472SS-5T-52E-S22.22016.453.294.30.0351315.40.5013S19151510.422.081.90.01078.80.1544S351516.58.721.741.80.00768.80.1315HS-217.42017.463.4927.00.236843.70.8736HS-318.42020.984.1919.10.201729.90.5237HS-535.62039.917.987.70.1551522.60.7338HS-637.62042.158.437.00.1481512.80.4829HS-7442049.269.853.30.083226.40.35210HS-846.52051.910.384.10.116236.00.34711H1111312.092.424.40.026619.00.28512H616.41320.084.026.60.067815.80.31613H7171321.204.244.30.046917.80.40114H15141316.773.355.70.048713.90.243Note: Cases 1-2 are from Project 1, Cases 3-4 are from Project 2, Cases 3–10 are from Project 3, and Cases 11–14 are from Project 4.For convenience, the least-squares principle is used to carry out linear fitting ofVloss′ in the overlapping disturbance area. All calculated Vloss′ for cases 1–14 from the back analysis are plotted as a function of Vloss1 in Figure 5. The approximate relationship of Vloss′∼Vloss1 can be expressed as(13)Vloss′=0.0249+0.2845Vloss.Figure 5
Fitting results.The coefficient of determination (R2) was used to compare the goodness of fitting. 72% of the data show a good fitting result (R2=0.7146), which indicates that Vloss′ has a significant linear positive correlation with Vloss1. It can be concluded that the ground loss caused by the overlapping disturbance can be obtained through the ground loss of the following tunnel.


## 3.5. Calculation Process of Settlement Trough for the following Tunnel

The settlement trough calculation procedure is shown in Figure6. Firstly, S1x and Vloss1 are monitored by the measured data after the preceding tunnel excavated. Secondly, S′x is determined by i′ and Vloss′, which are calculated by (12) and (13). Finally, based on superposition technique, S2x is predicted by superimposing S′x on S1x.Figure 6
Flowchart of the calculation procedure.


## 4. Field Project Cases Validation

A total of 2 sets of data are investigated from 2 field tunnel projects to validate the calculation model for the trough width parameter and the settlement troughs above the parallel tunnels. Parameters and known quantities are listed in Table3.Table 3
Information of the investigated tunnel cases.
Case no.Soil profileh (m)γ (kN/m3)c (kPa)φ (°)References1Miscellaneous fill3.119.416.250Huang [32]Silty clay4.319.85915.3Sand3.119Argillaceous siltstone10.520.5125322Miscellaneous fill4181510Li [33]Round gravel619.825.931Medium-coarse sand4.520.32729.5Gravelly sand9.420.22930


### 4.1. Changsha Metro Line 2 Project

Changsha Metro Line 2 project included two tunnels constructed using an EBP shield with a diameter of 6.00 m. Field settlements were obtained from section DK16 + 265. The depth of each tunnel axis is approximately 18 m, and the spacing between the center of two adjacent tunnels is 13 m. The soil profile at the instrumented site comprises miscellaneous fill, silty clay, sand, and argillaceous siltstone. The tunnel is located in argillaceous siltstone.


### 4.2. Shenyang Utility Tunnel Project

Shenyang utility tunnel project, which involved two tunnels, was built in Shenhe District of Shenyang City. Two tunnels were excavated using an EBP shield with a diameter of 6.00 m. Field settlements were obtained from section K8 + 577. The depth of each tunnel axis is approximately 18 m, and the spacing between the center of two adjacent tunnels is 12 m. The soil profile at the instrumented site comprises miscellaneous fill, round gravel, medium-coarse sand, and gravelly sand. The tunnel is located in gravelly sand.


### 4.3. The Trough Width Parameter

The proposed formula fori (i.e., (8)) is validated in two cases. Comparison between measured data and the i calculated results is shown in Table 4. In the first case, the relative error of i calculated by the proposed formula is 9.1%, and in the second case, the relative error is 1.4%.Table 4
Calculation results of trough width parameters.
Case NO.Monitoring sectionField data (m)Calculation result (m)1DK16 + 2655.56.02K8+5776.86.7It can be seen that different soil properties in the layered soils affect the trough width parameters induced by the tunnel, and it must be considered in the calculation. The proposed formulas fori could accurately calculate the trough width parameters in the field project.


### 4.4. The Settlement Troughs above the Parallel Tunnels

In two cases, the Gaussian curve is used to describe the surface settlement troughs above the parallel tunnels. The calculation results of the modification factor method are shown in Figures7(a) and 8(a), and the calculation results of the proposed method based on the superposition technique are also presented in Figures 7(b) and 8(b), respectively.Figure 7
Comparison between the measured data and the calculated curves in case 1.
(a)(b)Figure 8
Comparison between the measured data and the calculated curves in case 2.
(a)(b)A comparison between the calculated settlement troughs and the data points is shown in Figures7 and 8. In the first case, only 50% of the data using the modification factor method show a good fitting result (R2=0.645), while 80% of the data using the calculation method based on the superposition technique show a good fitting result (R2=0.852). But in the second case, 75% of the data using the modification factor method show a good fitting result (R2=0.928), and 83% of the data using the calculation method based on superposition technique show a good fitting result (R2=0.952). The comparison demonstrates that the proposed method based on the superposition technique reasonably describes the feature of surface settlement trough above the following tunnel.


## 4.1. Changsha Metro Line 2 Project

Changsha Metro Line 2 project included two tunnels constructed using an EBP shield with a diameter of 6.00 m. Field settlements were obtained from section DK16 + 265. The depth of each tunnel axis is approximately 18 m, and the spacing between the center of two adjacent tunnels is 13 m. The soil profile at the instrumented site comprises miscellaneous fill, silty clay, sand, and argillaceous siltstone. The tunnel is located in argillaceous siltstone.


## 4.2. Shenyang Utility Tunnel Project

Shenyang utility tunnel project, which involved two tunnels, was built in Shenhe District of Shenyang City. Two tunnels were excavated using an EBP shield with a diameter of 6.00 m. Field settlements were obtained from section K8 + 577. The depth of each tunnel axis is approximately 18 m, and the spacing between the center of two adjacent tunnels is 12 m. The soil profile at the instrumented site comprises miscellaneous fill, round gravel, medium-coarse sand, and gravelly sand. The tunnel is located in gravelly sand.


## 4.3. The Trough Width Parameter

The proposed formula fori (i.e., (8)) is validated in two cases. Comparison between measured data and the i calculated results is shown in Table 4. In the first case, the relative error of i calculated by the proposed formula is 9.1%, and in the second case, the relative error is 1.4%.Table 4
Calculation results of trough width parameters.
Case NO.Monitoring sectionField data (m)Calculation result (m)1DK16 + 2655.56.02K8+5776.86.7It can be seen that different soil properties in the layered soils affect the trough width parameters induced by the tunnel, and it must be considered in the calculation. The proposed formulas fori could accurately calculate the trough width parameters in the field project.


## 4.4. The Settlement Troughs above the Parallel Tunnels

In two cases, the Gaussian curve is used to describe the surface settlement troughs above the parallel tunnels. The calculation results of the modification factor method are shown in Figures7(a) and 8(a), and the calculation results of the proposed method based on the superposition technique are also presented in Figures 7(b) and 8(b), respectively.Figure 7
Comparison between the measured data and the calculated curves in case 1.
(a)(b)Figure 8
Comparison between the measured data and the calculated curves in case 2.
(a)(b)A comparison between the calculated settlement troughs and the data points is shown in Figures7 and 8. In the first case, only 50% of the data using the modification factor method show a good fitting result (R2=0.645), while 80% of the data using the calculation method based on the superposition technique show a good fitting result (R2=0.852). But in the second case, 75% of the data using the modification factor method show a good fitting result (R2=0.928), and 83% of the data using the calculation method based on superposition technique show a good fitting result (R2=0.952). The comparison demonstrates that the proposed method based on the superposition technique reasonably describes the feature of surface settlement trough above the following tunnel.


## 5. Conclusion

This paper establishes a new calculation method of surface settlement trough induced by the parallel tunnels in layered soils. The primary aim was to address the problem that settlement trough above the following tunnel was affected by the preceding tunnel in adjacent and parallel tunnels event. The following conclusions can be obtained based on the results of the work:(1)
The difference of soil properties in layered soils significantly affects the width parameter of surface settlement trough. In the calculation, the plastic zone and unloading disturbance zone are firstly determined in layered soils. And then, according to the linear relationship between the width parameter and the width of the unloading disturbance zone, the width parameter is calculated by the linear formula.(2)
Based on the superposition technique of tunnel settlement, the surface settlement above the following tunnel can be divided into the single tunneling-induced surface settlement and additional settlement induced by overlapping disturbance. The Gaussian curve can approximate the additional settlement trough according to the fitting results.(3)
Through extensive data exploration, the method for calculating the ground lossVloss′ induced by overlapping disturbance is proposed. With the increase of Vloss1, the value of Vloss′ is gradually increased. The approximate relationship of Vloss′∼Vloss1 can be expressed as(14)Vloss′=0.0249+0.2845Vloss,(4)
The calculated and measured settlement troughs above the parallel tunnels are in good agreement for the case study. Generally, the proposed model is effective and convenient for use in practice.



---

*Source: 2489711-2022-02-15.xml*