Research on the Fine Control of the Influence of Pipe-Jacking Parameter Deviation on Surrounding Stratum Deformation

Abstract

Based on the Zhuyuan–Bailonggang sewage interconnection pipe project in Shanghai, the ABAQUS finite element software was used in numerical simulations to study the fine control of stratum disturbances caused by pipe jacking parameter deviation in soft soil areas. Combining the simulation results with onsite measured data, the Peck formula was used to predict surface settlement. The results indicate the following: (1) The jacking speed and face pressure are negatively correlated with surface settlement. Under the maximum positive deviation and negative deviations in the jacking speed, after the tail passes through the monitoring section D₀ 16 ring, the maximum value of settlement at point B8 increases by 21.6% and decreases by 12.8%, respectively. Increasing the jacking speed increases the area with stress change ratio R < 0 at monitoring section D₀, and the arch foot at the tail of the pipe jacking machine decreases the surface settlement. In contrast, when the face pressure deviates from its average value, the variation range is less than 1%. (2) The pipe slurry coefficient and surface subsidence are positively correlated. Under the maximum positive deviation and the maximum negative deviation, the tail passes through the monitoring section D₀ 16 ring, and the maximum settlement value at B8 decreases by 4.9% and increases by 16.5%, respectively. The increase in the coefficient reduces the area with R < 0 at D₀ and increases the surface settlement. (3) In the order of descending strength, surface settlement is affected by the jacking speed, slurry friction coefficient, and face pressure. (4) To predict the maximum surface settlement value due to deviations in the jacking parameters, the Peck formula was modified using a correction factor $\alpha$ ranging from 0.6 to 3.0 and a settlement trough width correction factor $\beta$ ranging from 1.6 to 4.0. The modified prediction curve is in closer agreement with the actual conditions.

1. Introduction

In underground municipal engineering, the pipe jacking method is widely used for underground utility corridors and small-diameter tunnels because of its advantages of high construction speed and low overall cost. However, the impact of construction on the surrounding strata remains a challenging issue that urgently needs to be addressed.

To address the issue of stratum response caused by pipe jacking construction, multifaceted and multilevel research studies have been conducted. It has been reported that ground loss has the greatest impact on stratum deformation, followed by the side friction of the machine head and, finally, the grouting pressure [1,2]. In addition, Niu et al. [3] and Ren et al. [4] considered the effects of additional thrust at the front, grouting pressure, and pipe–soil friction, and used the Mindlin displacement solution to develop a surface settlement prediction model based on the stochastic medium theory. Wang et al. [5] proposed a parallel rectangular pipe jacking ground disturbance analysis model based on Mindlin’s solution and stochastic medium theory. Yen et al. [6], Xiao et al. [7], Li et al. [8], and Kong et al. [9] considered the pipe–soil side friction coefficient and established numerical models based on the displacement control method to predict the jacking force. Wen et al. [10] predicted the jacking force in frozen soil, compared a series of numerical simulations with theoretical values, and proposed a calculation model for jacking force in frozen soil. By introducing the effective friction coefficient, Ye et al. [11] established a calculation method of jacking force that takes into account lubrication and pipe–soil–slurry interaction. The coefficient reflects the contact state of pipe–soil–slurry and the influence of slurry resistance on reducing friction resistance. Additionally, research has shown that friction is closely related to the jacking force. Ma et al. [12] proposed a new rectangular pipe–soil contact model for friction prediction. Experiments performed by Namli et al. [13] showed that continuous injection of bentonite can significantly reduce the friction coefficient, while intermittent injection can reduce it to half of the value without injection. Zhou et al. [14] conducted a three-dimensional model test on the grouting effect of the pipe jacking tunnel in the silty soil layer. It was found that the viscosity and cohesion of the mud had a significant effect on reducing the friction resistance and ground settlement. The construction needs to pay more attention to the mud parameters. Zhong et al. [15] discovered through field tests that the friction between the pipe segment and the surrounding rock was influenced by the filler and lubricating slurry and derived a formula for the calculation of jacking friction. A comparison with measured data verified the applicability of this formula for rock strata. Ye et al. [16] proposed a new method for predicting slurry pipe jacking frictional resistance by fully considering the effects of lubrication, soil characteristics, and design parameters. In the study of deviations in jacking parameters, Wang et al. [17] analysed the jacking parameters and found that the greater the jacking speed, the greater the surface settlement, and that the stability of the jacking force has a significant impact on surface settlement. Pan et al. [18] analysed surface settlement during pipe jacking construction of a pipe curtain group and found that surface settlement is positively correlated with jacking speed. Xu et al. [19] studied the relationship between jacking parameters and ground settlement based on numerical simulations and concluded that the greater the jacking force, the greater the ground uplift value in front. In summary, most of the current research on pipe jacking focuses on the patterns and causes of ground deformation, surface settlement prediction, jacking force prediction, and friction calculation. However, relatively little research has been conducted on the response of strata to deviations in the jacking parameters.

This work, which is based on the Zhuyuan–Bailonggang sewage interconnection pipe project in Shanghai, investigated the deformation response of rock surrounding a tunnel when multiple parameter deviations occurred during pipe jacking construction. This was achieved through a combination of measured data and finite element numerical simulation. A sensitivity analysis of the jacking parameter deviations was conducted, and the Peck formula was used to predict surface settlement, The distribution interval of the correction coefficient $\alpha$ of the maximum surface settlement value and the correction coefficient $\beta$ of the width of the settlement trough are obtained. These results provide reference information for the improved construction of pipe jacking projects.

2. Project Overview

The Zhuyuan–Bailonggang sewage interconnection pipe project in Pudong New Area, Shanghai, starts at the Zhuyuan Sewage Treatment Plant and ends at the Bailonggang Sewage Treatment Plant, with a total length of 19.8 km. It is divided into two sections, and this paper focuses on the 0-50 ring segments of the first section from well 9 to well 7. The project employs a multi-cutterhead slurry balance pipe jacking machine for straight jacking. The cutterhead diameter of the jacking machine is 4.18 m. The jacking structure used precast concrete pipe segments with a concrete strength of C50. The inner diameter of the segments is 3.5 m, with a wall thickness of 0.32 m and a single segment length of 2.5 m. The launching and receiving shafts are protected by diaphragm walls. According to the survey report, the strata involved in pipe jacking include sandy silt and silty clay, which are prone to disturbances from jacking speed and friction, increasing the risk of face instability during excavation.

3. Analysis of Monitoring Results

3.1. Monitoring Point Layout

The layout of the surface settlement monitoring points at the construction site is shown in Figure 1. Along the jacking direction of the tunnel, 10 longitudinal settlement monitoring points, labelled B10 to B1, are arranged. A transverse monitoring section D₀ is set up at B8, which includes settlement monitoring points D1–D6. The launch line is located at a distance of 17.5 m from the transverse monitoring section.

3.2. Analysis of the Measured Settlement Results

The development trends of lateral and longitudinal surface settlement during pipe jacking construction are shown in Figure 2. Figure 2a shows that the shape of the transverse surface settlement trough conforms to the Peck settlement curve. Before the cutterhead passes the monitoring section, the soil in front of the face is compressed, causing the ground to heave slightly; as the cutterhead approaches the monitoring section, the heave increases, reaching a maximum value of 1.1 mm. After the tail of the machine passes the monitoring section, due to ground loss, the surface deformation begins to change from heave to settlement, which continues to increase throughout the construction process. The maximum settlement occurs at D1, with a settlement of 22 mm. Figure 2b shows that the settlement at all monitoring points increases with the jacking process. Owing to the presence of a 7.5 m-long ground reinforcement area, the soil near the launching shaft shows relatively small displacement changes within the range of ±5 mm. The maximum surface settlement is observed at B8, with a maximum settlement value of 26.49 mm. Moreover, the curves in the figure indicate a significant increase in settlement at B1 and B3. According to the survey report, weak interlayers were found in the soil at boreholes B1 and B3, which may lead to excessive jacking parameters when the pipe jacking machine reaches these positions.

3.3. Jacking Parameter Analysis

Figure 3 shows the fluctuations in the jacking force, jacking speed, and face pressure of the pipe jacking machine with increasing number of jacking rings. When the pipe jacking machine is jacking in the reinforcement area of the starting well, the values of the three parameters clearly increase. When the pipe jacking machine moves out of the reinforcement area, the pressure of the working face begins to fluctuate within the range of 100–140 kPa, and the jacking speed is stable at 55 mm/min, while the jacking force continues to grow and then fluctuates around 4700 kN. When the pipe jacking machine is jacked into rings 32–45, the jacking force and pressure of the tunnel face fluctuate violently, and there is a trend of first increasing and then decreasing, which will lead to the trend of the ground settlement first increasing and then decreasing. When the pipe jacking machine is jacked into rings 45–50, the jacking force and the pressure of the tunnel face increase, and the ground settlement also increases. The description of the above two stages is similar to the change trend of B3 and B1 in Figure 2b, confirming that the soil layer may lead to the jacking parameters being too large when the pipe jacking machine is jacking to the corresponding position, and further illustrating the necessity of research on the response of the jacking parameters to the stratum.

4. Numerical Simulation Study

4.1. Numerical Modelling

Using the ABAQUS finite element software (v.5.4), a three-dimensional numerical model that considered fluid–solid coupling was established for the Zhuyuan–Bailonggang sewage interconnection pipe jacking project in Shanghai, as shown in Figure 4. The pipe jacking process was simulated using the equivalent layer method [20] and the displacement control method [6]. The dimensions of the model’s computational domain were obtained from Lambrughi et al. [21] and Chapman et al. [22], with a longitudinal (Y-axis) length of 130 m, a lateral (X-axis) length of 50 m (12D), and a vertical (Z-axis) length of 30 m (7D). According to the verification of multiple simulation results, the analysis of formation deformation results at the boundary was not affected and the deformation was stable (see the simulation results of parameter deviation later). Therefore, the size of the model calculation domain was set reasonably to ensure that the boundary effect was effectively eliminated.

The soil constitutive model adopted the modified Cam-clay model. Based on the field measurements and investigations, the physical and mechanical parameters of the soil layer can be obtained as shown in Table 1. The soil, equivalent layer, segment, and pipe jacking machine were simulated by solid elements. The length of each step of excavation was the length of a ring segment (2.5 m), and the length of the selected transverse monitoring section was four segments from the starting position. The simulation process was as follows: (1) The soil at the head position was killed, the pipe jacking head was generated at the same time, surface contact between the pipe jacking head and the soil was established, and the friction coefficient was 0.3 [20]; (2) Simultaneously with soil excavation, the field-measured soil pressure in the soil chamber was applied to the working face as the supporting pressure; and (3) the displacement control method was used to realize the dynamic simulation of pipe jacking. The lateral displacement of the over excavated part of the soil was constrained at the first ring segment after the tail of the pipe jacking machine, and was cancelled when the grouting body of the next ring segment was generated. The pipe–soil friction coefficient was 0.2 according to the test statistical data [23], and the dynamic friction coefficient of the pipe–slurry contact interface was in the range of 0.01–0.1 [24,25]. Steps 2 and 3 were repeated until the end of excavation.

4.2. Verification of the Simulation Results

The pipe jacking machine passed through monitoring section D₀ on 6 October. Figure 5 shows the comparison between the numerical simulation results and the measured results of the surface transverse settlement trough, after the tail of the pipe jacking machine passed through monitoring section D₀ 3 and 4 d later. The RMS (root mean square) error of the monitoring point was calculated to be 3.04 mm. This is because the measured site on the side of the monitoring point D4 is a road, which causes the measured settlement value to be small under the long-term vehicle load. However, it can be seen that the trend and value of the settlement numerical calculation results are generally consistent with the measured results, which indicates that the model simulation results are reliable and can provide a guarantee for the subsequent jacking parametric analysis.

5. Analysis of the Influence of Jacking Parameters on Land Subsidence

In the process of pipe jacking construction, key construction parameters such as the jacking speed $(v)$, face pressure $(p_N)$, and friction coefficient $(\mu )$ should be dynamically adjusted according to the actual stratum conditions. Table 2 lists the average value and value range of each parameter in the normal jacking state of the pipe jacking machine in this project, which were applied in the verification model and parametric analysis model of the finite element calculation. The average value of the jacking speed and the pressure of the working face were obtained according to the average value of the parameters that appear many times in the construction. For the value of the slurry friction coefficient, according to the study of Ye et al. [11], when only the slurry contact is considered, the slurry friction coefficient is uniform under the package of the slurry sleeve. On this basis, according to Yu et al. [25], when the slurry–soil mixture changes from pure thixotropic slurry to pure clay, the friction coefficient of pipe slurry is between 0.01 and 0.09. Based on this interval, the numerical simulation inversion is carried out according to the measured total thrust, and finally the average friction coefficient of the pipe slurry is 0.03.

5.1. Analysis of the Formation Response After Jacking Speed Deviation

Owing to the complexity of Shanghai’s soft soil layer conditions, the jacking speed was set to fluctuate around 5 ring/d. Figure 6 shows the vertical displacement of the surface (B8) directly above the tunnel axis at different construction stages when the jacking speed was 2, 5, 7, and 10 ring/d. With increasing jacking speed, the extrusion effect of the cutter head on the soil in front of the tunnel face also increases, and the surface settlement value gradually decreases, showing that an appropriate increase in the jacking speed can reduce the surface settlement; the slope of each curve also decreases with increasing jacking speed. When the jacking speed exceeds 7 ring/d, the influence of the change in the jacking speed on the surface is very small, indicating that the increase in the jacking speed has only a limited impact on surface subsidence control.

When the tail passes through the 16th ring of the monitoring section, the maximum settlement values of the ground surface at each jacking speed are 8.54, 7.02, 6.40, and 6.13 mm, respectively, and the maximum fluctuation range of settlement is 2.41 mm. When the cutter head has not passed the monitoring section, the maximum fluctuation range of settlement is only 0.74 mm, which also shows that timeliness is important in controlling the jacking speed. When the tail passes through the section with strict surface deformation, the lower jacking speed aggravates the influence of surface settlement in this section. At this time, the lower limit of the jacking speed should be more strictly controlled.

Figure 7 shows the change in the surface transverse settlement trough when the tail passes through the rear 16th ring. Compared with the settlement’s extreme value under the average jacking speed, the settlement’s extreme value under the maximum positive deviation is reduced by 12.8%, and the settlement’s extreme value under the maximum negative deviation is increased by 21.6%; that is, the deviation of the jacking speed is negatively correlated with the fluctuation of the settlement’s extreme value. The deviation in the thrust velocity has little effect on the distribution of subsidence and uplift areas. Taking the average jacking speed as an example, the surface within the range of approximately 3.8 D from the central axis of the tunnel exhibits subsidence, and the areas on both sides show uplift. With increasing jacking speed, the settlement area decreases only slightly.

According to the concept of the stress change ratio R proposed by Lee et al. [26], the R value at the arch foot is greater than 0, and the R value below the soil arch is less than 0. Figure 8 shows the vertical stress soil arching in the surrounding rock under a low jacking speed and under a high jacking speed. The soil in front of the tunnel face is in a shear disturbance zone [27]. The soil is squeezed to produce shear deformation, and excess pore water pressure is generated inside. Because the vertical total stress is unchanged, the effective stress is reduced, which is manifested as a blue area. The soil in the rear is a consolidation zone. The soil around the tunnel is disturbed by construction, resulting in excess pore water pressure. Then, it slowly dissipates, the effective stress increases, and the soil consolidates and settles, resulting in a yellow area.

The vertical stress soil arch is formed in the upper and lower surrounding rock of the tunnel after the pipe jacking machine is pulled out. The two arch feet in the upper surrounding rock are located directly above the third ring segment, and the pipe jacking machine after the pipe jacking machine is pulled out, which is shown as a red area. The formation of this kind of soil arch will cause the surface subsidence from the loss of the tail stratum to tend to be stable. This can also explain why the difference between different settlement curves decreases gradually after the tail passes through the 3rd ring in Figure 6.

Comparing Figure 8a,b, it can also be concluded that as the jacking speed changes from low to high, the stress change ratio near the surface monitoring section (the surface of the 8th ring to the 17th ring after the tail) changes from R > 0 to R < 0. The vertical stress in this range decreases, and the maximum surface settlement value also decreases accordingly. Therefore, the fast jacking mode is more conducive to controlling the development of the maximum surface settlement value, which is also consistent with the law described in Figure 6 and Figure 7.

5.2. Analysis of the Formation Response After Pressure Deviation of the Tunnel Face

In this project, the slurry balance pipe jacking machine head is equipped with a PLC system, which controls the pressure of the tunnel face more finely (average 105 kPa), and the deviation range (100–140 kPa) is relatively small. In Figure 9, when the pressure of the tunnel face increases, the surface settlement at B8 shows a slight decreasing trend. This trend is more pronounced when the tail passes through the monitoring section, and the maximum decrease in settlement is 0.27 mm. When the tail passes, the settlement change caused by the pressure deviation of the tunnel face is no longer obvious. The soil deformation caused by the pressure deviation of the tunnel face during the jacking process of the pipe jacking tunnel at a burial depth of 9.9 m is concentrated around the tunnel, which has little effect on the surface deformation. However, when the pipe jacking machine has just moved out of the strict deformation control area, the pressure of the tunnel face can be appropriately increased to achieve fine settlement control.

Figure 10 shows the distribution of vertical stress soil arching when the tail passes through the 16th ring after D₀ and the face pressures are $p_N$ = 100 kPa and 140 kPa. The range of the arch foot located at the upper and lower positions of the pipe jacking head decreases slightly with increasing $p_N$, while the range of the arch foot located at the third ring segment of the tail is almost unchanged, which is consistent with Saint-Venant’s principle. The results also reveal the most important origin of the difference in the surface settlement when the tail passes through the monitoring section in Figure 9. A comparison of Figure 10a,b shows that whether the pressure of the tunnel face is low or high, the stress change near the monitoring section after the tail is essentially the same as the positive and negative dividing lines of R, and the effective stress shows no evident change, indicating that the change in the settlement at the monitoring section is very small.

5.3. Analysis of the Formation Response After Friction Resistance Deviation

The magnitude of the pipe–soil friction gradually increases with an increasing jacking length of the tunnel. To reduce the limitation of the jacking length, a bentonite slurry was used to reduce the resistance in this project. Figure 11 depicts the variation trend of the vertical displacement of surface B8 at different construction stages under different pipe slurry friction coefficients (0.01~0.09). Before the tail passes through D₀, the jacking length of the tunnel is less than three rings, and the distance is short. The changes in the friction coefficient have no evident influence on the maximum settlement of the ground surface. After the tail passes through the D₀ 6 ring, as the number of jacking segments increases, the increase in the friction coefficient causes a greater increase in the jacking friction, and the maximum increase in the surface settlement is 7.5%. When the tail passes through the D₀ 16 ring, the maximum increase in surface settlement is 22.5%. At this time, the larger friction coefficient further aggravates the development of surface settlement. Therefore, even a small friction coefficient deviation can cause great surface subsidence fluctuations, particularly when the jacking distance increases to a certain extent.

After passing through the D₀ 16 ring, the change in the surface settlement trough is shown in Figure 12. When the friction coefficient of the pipe slurry increases from 0.01 to 0.09, the maximum ground settlement increases from 6.68 to 8.18 mm. Compared with the maximum settlement value under the average friction coefficient of 0.03, the maximum settlement increase under the maximum positive deviation is 16.5%, and the maximum settlement decrease under the maximum negative deviation is 4.9%. The distributions of the settlement and uplift areas are not strongly affected by the deviation of the friction coefficient. With an increasing friction coefficient, the settlement area increases only slightly.

Figure 13 shows the distributions of vertical rock stress and soil arching when the tail passes 16 rings beyond D₀, with pipe slurry friction coefficients of $\mu$ = 0.01 and 0.09. Near monitoring section D₀, the area with R < 0 decreases significantly, while the area with R > 0 increases significantly, indicating an increase in vertical stress, which also leads to increased settlement in that area. When the jacking distance extends to a certain point, the maximum surface settlement value becomes positively correlated with the pipe slurry friction coefficient. However, the distribution range and position of the arch foot, as well as the stress variation in the soil ahead of the face, are not significantly affected by the pipe slurry friction coefficient.

Deviations of the above three jacking parameters from their average values lead to the fluctuations in the maximum surface settlement to varying degrees. Figure 14 shows the normalized vertical displacement of the ground surface caused by the deviation of the jacking parameters, where $\delta$ and ${\delta }_{st}$ represent the vertical displacement of the soil under the average value of the parameters and the deviation value after the tail passes through the monitoring section D₀ 16 ring, respectively. The absolute values of the slopes of the three curves in the figure indicate that among the jacking parameters, the jacking speed has the greatest influence on the surface settlement, followed by the friction coefficient of the pipe slurry; the pressure of the tunnel face has the least effect. The surface settlement is negatively correlated with the jacking speed and pressure of the tunnel face and positively correlated with the friction coefficient of the pipe slurry.

6. Prediction of the Ground Settlement of Pipe Jacking

6.1. Peck Theoretical Formula Regression Analysis

To better predict the surface response when deviations occur in the jacking parameters of pipe jacking, and to combine the prediction with numerical simulation results to provide guidance for the subsequent construction of the project, the classic Peck prediction formula was selected for predicting surface settlement during pipe jacking. Peck [28] analysed a large amount of measured data and reported that, under undrained conditions, the lateral settlement trough caused by tunnel construction approximates a normal distribution. The primary reason for surface settlement is the loss of ground, described by the following formula:

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

(2)$S_{\max }={\displaystyle \frac{V_{loss}}{\sqrt{2\pi i}}}={\displaystyle \frac{\sqrt{\pi }{R}^2{v}_{l,s}}{\sqrt{2}i}}$

(3)$I={\displaystyle \frac{Z}{\sqrt{2\pi }\tan ({45}^{\circ }-\frac{\phi }{2})}}$

where $S(x)$ is the ground settlement amount at the monitoring point located at a distance from the tunnel axis, m; $S_{\max }$ is the maximum surface settlement amount directly above the tunnel axis, m; $V_{loss}$ is the volume of soil loss; $i$ is the trough width coefficient; $R$ is the excavation radius, m; $v_{l,s}$ is the formation loss rate; $Z$ is the distance from the tunnel axis to the surface, m; and $\phi$ is the internal friction angle of the soil.

Linear regression analysis was performed on the Peck formula, taking the logarithm of both sides to transform it into a linear equation.

(4)$\ln S(x)=\ln S_{\max }+{\displaystyle \frac{1}{i^2}}({\displaystyle \frac{-x^2}{2}})$

Let $\ln S(x)$ be the constant term and let ${\displaystyle \frac{1}{i^2}}$ be the linear coefficient, yielding

(5)$S_{\max }=\exp (a),i={\displaystyle \frac{1}{\sqrt{b}}}$

where:

(6)$b={\displaystyle \frac{{\displaystyle \sum [(\frac{-x^2}{2})\ln S(x)]-\frac{1}{n}{\displaystyle \sum (\frac{-x^2}{2}){\displaystyle \sum \ln S(x)}}}}{{\displaystyle \sum {(\frac{-x^2}{2})}^2-\frac{1}{n}({\displaystyle \sum \frac{-x^2}{2}{)}^2}}}}$

(7)$a={\displaystyle \frac{1}{n}}{\displaystyle \sum \ln S(x)-[}{\displaystyle \frac{1}{n}}{\displaystyle \sum (\frac{-x^2}{2}})]b$

Regression analysis was performed on the monitoring data at monitoring section D₀. The arrangement of the surface monitoring points is shown in Figure 1, and the calculations can be performed using Equations (6) and (7).

$a=2.604,b=0.0078$

The linear function obtained after regression is as follows:

(8)$\ln S(x)=2.604+0.0078\times ({\displaystyle \frac{-x^2}{2}})$

Based on the results from Equation (8) and in conjunction with Equation (5), the value of the regression results for $S_{\max }$ and $i$ can be calculated as follows:

$S_{\max }=13.52mm,i=11.35$

The fitted curve is as follows:

(9)$S(x)=13.52\times \exp ({\displaystyle \frac{-x^2}{2\times {11.35}^2}})$

According to the experience of O ’Reilly and News [29] in London, the relationship between the width coefficient of the settlement trough and the depth of the tunnel in the Peck formula is as follows:

(10)$i=K{z}_0$

According to the study of the Shanghai area [30], when the settlement trough width parameter $K$ = 0.5, the formation loss rate $v_{l,s}$ = 1.25% can be substituted into Equations (2) and (10) to obtain:

(11)$S_{\max }={\displaystyle \frac{Vloss}{\sqrt{2\pi }i}}={\displaystyle \frac{\sqrt{\pi }{R}^2{v}_{l,s}}{\sqrt{2\pi }i}}=11.40mm$

(12)$i=K{z}_0=6m$

The Peck empirical formula can be obtained by substituting it into Equation (1):

(13)$S(x)=11.4\times \exp ({\displaystyle \frac{-x^2}{2\times 6^2}})$

The measured data of surface subsidence at monitoring section D₀ are compared with the Peck fitting curve and the Peck empirical formula prediction curve in Figure 15.

Figure 15 shows that the Peck regression linear function curve fits the measured data curve well, while a large difference is still observed between the Peck empirical formula curve and the measured data curve. The main reason is that the Peck empirical formula is used to predict the settlement during shield construction. The pipe–soil friction must be considered during pipe jacking construction, and the jacking force is relatively large. Therefore, the settlement value of the Peck empirical formula prediction curve is smaller than that of the fitting formula curve, and the Peck empirical formula must be corrected.

6.2. Peck Empirical Formula Correction

On the basis of the Peck fitting curve of the regression analysis of the measured data, the correction coefficient was introduced to correct the maximum settlement value of the Peck empirical formula and the width coefficient of the settlement trough. The correction expression is as follows:

(14)$S(x)=\alpha S_{\max }\exp [{\displaystyle \frac{-x^2}{2{(\beta i)}^2}}]$

Taking the logarithm of both sides, we obtain:

(15)$\ln S(x)=\ln (\alpha S_{\max })+{\displaystyle \frac{1}{{(\beta i)}^2}}({\displaystyle \frac{-x^2}{2}})$

According to the results, two coefficients can be obtained:

(16)$\alpha ={\displaystyle \frac{\exp (a)}{S_{\max }}}=1.19$

(17)$\beta ={\displaystyle \frac{1}{i\sqrt{b}}}=1.89$

The correction coefficient is substituted into Equation (14), and the Peck correction formula is obtained as follows:

(18)$S(x)=13.52\times \exp ({\displaystyle \frac{-x^2}{2\times {11.35}^2}})$

The corrected Peck empirical curve, Peck fitting curve, and measured data curve are shown in Figure 16. The corrected Peck empirical formula curve is consistent with the fitting curve and the measured data curve.

6.3. Statistical Analysis of the Peck Correction Coefficient

Based on Equations (16) and (17), the sections of three sections of well 7–4, well 11–9, and well 13–14 of the Bailonggang sewage connection pipe project in Zhuyuan, Shanghai were selected. A total of 30 sets of tunnel construction monitoring data were collected, and many correction coefficients α and β were statistically analysed, as shown in Figure 17a,b.

Figure 17a,b shows that when the correction coefficient $\alpha$ of the maximum surface settlement value is between 0.6 and 3.0, the distribution ratio is 86.67%, and when the correction coefficient $\beta$ of the settlement trough width is between 1.6 and 4.0, the distribution ratio is 90.00%. This shows that when the values of the correction coefficients α and β are distributed in these two intervals, it is reasonable to use the Peck formula to predict surface subsidence.

To further prove the feasibility of the modified parameter interval, two typical cases were selected for testing. The lower limits of $\alpha$ and $\beta$ are 0.6 and 1.6, and the upper limits are 3.0 and 4.0. The curves obtained using other values are between the upper and lower limits. Six sets of measured data were randomly selected from the four intervals of wells 7–4, 9–7, 11–9, and 13–14 to compare with the upper and lower limit curves predicted by the Peck formula. As shown in Figure 18, 88.90% of the measured data fall between the upper and lower limit correction prediction curves, indicating that the $\alpha$ and $\beta$ correction parameter intervals are feasible for this project and that the obtained correction curve can effectively predict the surface subsidence caused by the construction of the pipe jacking section of well 9–7. The value is also applicable to the prediction of surface subsidence in the other three intervals of the project.

7. Conclusions

On the basis of the Bailonggang sewage connecting pipe project in Zhuyuan, Shanghai, the effects of the jacking speed, tunnel face pressure, and pipe slurry friction coefficient deviation on stratum deformation were studied via field measurements and numerical simulation analysis. The conclusions are as follows:

• (1). Under the maximum positive deviation and negative deviation of the jacking speed, after the tail passes through the monitoring section D₀ 16 ring, the settlement’s extreme value at B8 increases by 21.6% and decreases by 12.8%, respectively. Increasing the jacking speed decreases the arch foot at the tail of the pipe jacking machine, increases the area with R < 0 at monitoring section D₀, and limits the influence of increasing the jacking speed on surface settlement control. When the pressure deviation of the tunnel face is small, only slight changes are observed when the tail passes through monitoring section D₀; the changes are less than 1%, the stress change ratio is essentially the same as the positive and negative boundaries of R, and the effective stress does not change significantly. These two jacking parameters are negatively correlated with surface subsidence.

• (2). Under the maximum positive deviation and negative deviation of the friction coefficient, after the tail passes through the monitoring section D₀ 16 ring, the settlement’s extreme value at B8 decreases by 4.9% and increases by 16.5%, respectively. The area with R < 0 at monitoring section D₀ decreases with an increasing friction coefficient, resulting in an increase in the surface settlement value. After the jacking distance increases to a certain extent, the maximum surface settlement value is positively correlated with the friction coefficient of the slurry.

• (3). Among the jacking parameters, the jacking speed has the strongest effect on surface subsidence, the pipe slurry friction coefficient has the second-strongest effect, and the tunnel face pressure has the weakest effect. When the tail of the machine passes through a section with strict surface deformation, the pressure of the tunnel face can be appropriately increased; after passing, the lower limit of the pushing speed should be strictly controlled.

• (4). The confidence intervals of α and β are 0.6~3.0 and 1.6~4.0, respectively. When the values of $\alpha$ and $\beta$ are in this range, the Peck-modified prediction curve is more suitable for the actual situation, and the prediction results are more reliable. The curve is also applicable to the prediction of surface subsidence in the other three intervals relevant to the project.

Author Contributions

Conceptualization, T.Z. and P.L.; methodology, T.Z.; software, G.C.; validation, G.C.; formal analysis, G.C.; investigation, T.Z. and D.N.; resources, T.Z. and D.N.; data curation, G.C.; writing—original draft preparation, G.C.; writing—review and editing, T.Z. and P.L.; visualization, G.C.; supervision, P.L.; project administration, P.L. and D.N.; funding acquisition, P.L. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Acknowledgments

The authors wish to express their gratitude to the anonymous reviewers whose constructive comments helped to improve the overall quality of this paper.

Conflicts of Interest

Author Dongqing Nie was employed by the company Shanghai Municipal Engineering Design General Institute (Group) Co., Ltd. The remaining authors declare that the re-search was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Footnotes

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Figures and Tables

Figure 1. Plane layout of monitoring points.

Figure 2. Surface subsidence monitoring curves: (a) lateral settlement; (b) longitudinal settlement.

Figure 3. Jacking parameter fluctuation diagram.

Figure 4. Finite element model and size.

Figure 5. Calculated curve and measured results of ground settlement at D0.

Figure 6. Influence of the jacking speed deviation on the ground settlement.

Figure 7. Change in the transverse settlement tank under different jacking speeds.

Figure 8. Comparison diagram of the vertical stress on the soil arch under two kinds of jacking speeds: (a) [Forumla omitted. See PDF.] = 2 ring/d; (b) [Forumla omitted. See PDF.] = 10 ring/d.

Figure 9. Influence of tunnel face pressure deviation on ground settlement.

Figure 10. Comparison diagram of the vertical stress on the soil arch under two kinds of tunnel face pressures: (a) [Forumla omitted. See PDF.]; (b) [Forumla omitted. See PDF.].

Figure 11. Influence of the friction coefficient deviation on ground settlement.

Figure 12. Change diagram of the transverse settlement tank under different coefficients of friction.

Figure 13. Comparison diagram of the vertical stress on the soil arch under two coefficients of friction: (a) [Forumla omitted. See PDF.]; (b) [Forumla omitted. See PDF.].

Figure 14. Normalized vertical surface displacement caused by parameter deviation.

Figure 15. Peck fitting, prediction, and comparison of the measured data curves.

Figure 16. Peck prediction correction, fitting, and comparison of measured data curves.

Figure 17. Distribution intervals of the α and β data: (a) [Forumla omitted. See PDF.] data distribution diagram; (b) [Forumla omitted. See PDF.] data distribution diagram.

Figure 18. Comparison of the corrected Peck upper and lower limit curves with the measured data.

Table 1

Physical and mechanical parameters of the soil layer.

Soil	$\mathbf{\Delta }\mathit{Z}\mathbf{(}\mathbf{m}\mathbf{)}$	${\mathit{\gamma }}^{\mathbf{\ast }}\mathbf{/}\mathbf{(}\mathbf{kN}\mathbf{\cdot }{\mathbf{m}}^{\mathbf{-}\mathbf{3}}\mathbf{)}$	$\mathit{k}\mathbf{(}\mathbf{m}\mathbf{\cdot }{\mathbf{day}}^{-1})$	${\mathit{e}}_{\mathbf{0}}$	$\mathit{\nu }$	$\mathit{\phi }\mathbf{\prime }\mathbf{/}\mathbf{(}\mathbf{°}\mathbf{)}$	$\mathit{M}$	$\mathit{\lambda }$	$\mathit{\kappa }$
Miscellaneous fill	1.5	19.2	5.51 × 10–2	0.843	0.35	32.3	0.574	0.0724	0.0091
Silty clay	1.5	19.3	4.10 × 10–4	0.855	0.32	29.5	1.178	0.1177	0.0100
Clayey silt	5.0	19.0	1.52 × 10−2	0.907	0.31	30.3	1.213	0.1005	0.0090
Silt	4.0	19.3	1.09 × 10−1	0.830	0.28	30.8	1.234	0.0395	0.0049
Muddy-silty clay	7.0	17.8	2.11 × 10−4	1.222	0.34	25.8	1.018	0.1356	0.0130
Clay	9.0	17.9	1.48 × 10−4	1.197	0.35	23.1	0.903	0.1081	0.0090
Silt	2.0	19.1	1.06 × 10−1	0.866	0.28	29.8	1.191	0.0290	0.0030

Note: $\Delta Z$ is the thickness of the layer; ${\gamma }^{*}$ the effective weight of the soil; $k$ is the permeability coefficient of the soil; $e_0$ is the void ratio of the soil; $v$ is the Poisson ratio of the soil; $\phi \prime$ is the angle of internal friction of the soil; $M$ is the stress ratio of the soil; $\lambda$ is the compression index; $\kappa$ is the rebound index of the soil.

Table 2

Pipe jacking parameters.

Parameter	Jacking Speed	Tunnel Face Pressure	Coefficient of Friction
Average value	5	105	0.03
Value range	[2,10]	[100,140]	[0.01,0.09]
Deviation area	[−60%,100%]	[−4.8%,33.3%]	[−66.7%,200%]

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