Seismic Response Analysis of Secondary Lining Polymer Grouting Debonding Repair for Tunnel Construction Based on Parameter Inversion

,


Introduction
Over the past few decades, there have been many reports of varying degrees of tunnel damage caused by earthquakes, such as the Tokachi-Oki (Japan) earthquake in 1952, the Tonghai (China) earthquake in 1970, the Chi-Chi (Taiwan, China) earthquake in 1999, the Niigataken-Chetu (Japan) earthquake in 2004, the Wenchuan (China) earthquake in 2008, and the Kumamoto (Japan) earthquake in 2016 [1]. Scholars such as Ye et al. [2][3][4][5] found that voids behind the lining are common in tunnels and have become the main cause of tunnel disease. Xin et al. studied the seismic response and damage patterns of tunnels with/without voids behind the linings by shaking table tests [6]. Yasuda et al. studied the seismic response of cylindrical tunnels with void areas under 3D earthquakes, and the results showed that the large stress concentration on the lining caused by voids under the action of the earthquake resulted in the destruction of the tunnel [7]. Min et al. studied the efect of voids behind linings on the cracking performance of asymmetric double-arch tunnels, and the results show that due to the infuence of the void area, the cracking of the tunnel area opposite the void area is more serious [8]. Te research of the above scholars shows that the void area behind the tunnel lining seriously reduces the seismic performance of the tunnel, the lining in the void area generates tensile stresses under seismic action, and the surrounding rock around the void area deforms severely in plasticity, leading to rockfall impacting the lining and serious lining damage.
Te demolition and reconstruction method, backfll grouting method, and shotcrete concrete combined with the steel arch reinforcement method are often used to repair and fll void areas [9]. Jinlong et al. used secondary grouting to repair the void area of the tunnel. During the secondary grouting process, it was found that higher grouting pressure, a larger grout amount, and improper grouting position would lead to cracks in the tunnel [10]. In order to solve the problem of tunnel disease caused by cracks in the tunnel lining, Zhou et al. adopted the plate-short bolt assembly structure to reinforce the tunnel cracks [11]. Han et al. explored the reinforcement efect of the fber-reinforced plastic (FRP)-polymer cement mortar (PCM) method on the void area of the tunnel [12]. Liu et al. used crack grouting, shotcrete support, and reconstruction of the secondary lining to repair the damaged lining [13]. However, due to the inherent defects of concrete, which limit the efect of tunnel disease management, many scholars have applied polymer grouting technology to tunneling projects in view of the advantages of early strength, lightweight, and good durability of polymer materials [14,15]. At present, the research in this feld mainly focuses on the rapid repair of buried pipeline leakage and settlement, railway track settlement, and pavement voids and collapse. Wang et al. repaired the void under the pipeline with polymer, and the results showed that polymer grouting can efectively restore the strength and resilience of the pipeline to its normal state [16,17]. Li et al. developed an innovative trenchless concrete pipeline corrosion and void defect pretreatment technology called membrane bag pile polymer grouting pretreatment (MBP-PGP) technology, which can be used for pipeline corrosion and voiding repair [18]. Fang et al. showed through experiments that polymers can efectively repair the settlement of high-speed railways and provide sufcient long-term durability under dynamic train loads [19,20]. Te research of Li et al. shows that high polymer can efectively repair cracks and voids in pavement and efectively prevent pavement settlement [21,22]. Te polymer grouting material can sufciently and densely fll the damaged area, thereby controlling the unfavorable deformation and better restoring the structural integrity. Terefore, the combination of polymer grouting repair technology and tunnel nondestructive testing can quickly fnd and repair tunnel disaster problems.
Te abovementioned study did not investigate the efect of the seismic response of the tunnel after polymer grouting repair. Tis paper takes the Longmenshan tunnel project as an example. In order to simulate the Longmenshan tunnel more precisely, this paper deduces the inversion formula of the three-dimensional parameters of the physical mechanics of the surrounding rock of the tunnel based on the basic theory of system identifcation and improves the inversion method with the help of numerical simulation software. Te improved inversion results are used to optimize the mechanical parameters of the surrounding rock of the tunnel and then study the repair efect and seismic capacity of the Longmenshan tunnel using polymer grouting to repair the void area behind the lining under the action of an earthquake load so as to provide the tunnel void space. Polymer grouting repair provides a theoretical basis.

Project Overview
Longmenshan Mountain tunnel is located at the junction of Tangchi town of Liuan city and Daguan town of Tongcheng city, Anhui province, China. It is a detached tunnel, buried at a depth of 50 m−300 m, with starting pile number K81 + 834 (ZK81 + 830) and ending pile number K84 + 47 (ZK84 + 450), a total length of 2644 (2620 m). Te surrounding rock grade of the tunnel is mainly III, IV, and V grades surrounding rock, where III and IV grades surrounding rock is mainly medium weathered granite, and V grade surrounding rock is mainly full weathered granite.

Tunnel Construction Monitoring and Measurement.
As is shown in Figure 1, the measurement items of the Longmenshan tunnel include surface subsidence observation, vault subsidence observation, and peripheral convergence measurement. As shown in the arch settlement diagram (Figure 2) of the K83 + 020 void section of the Longmenshan tunnel, the surrounding rock basically reached a stable state after 40 days, the accumulated settlement value was kept below 10 mm, and the development trend of the arch settlement curve was approximately a logarithmic function curve, which was fnally kept at 9.4 mm. As is shown in Figure 3. Te settlement rate of the vault also tended to decrease slowly on the whole, and after 40 days, the settlement rate remained relatively stable, between 0.1 mm/d and 0 mm/d. Te maximum settlement rate was 0.6 mm/d on day 5, which occurred during the excavation of the lower step. Due to the relatively short duration of the maximum change in the settlement rate, it would not have much impact on the overall stability and safety of the tunnel excavation process. Overall, the vault deformation of this section is in a reasonable range at all stages; no abnormality is seen, the surrounding rock is basically stable, and the excavation method is reasonable.
As is shown in Figure 4, analysis of the convergence deformation data around the K83 + 020 void section of the Longmenshan tunnel right line shows that the accumulated peripheral convergence value of this section is smaller than the arch settlement value, and the regularity of its deformation process is more scattered than that of the arch deformation, and the result still tends to convergence. As is shown in Figure 5, analysis of the convergence deformation data around the K83 + 020 void section of the Longmenshan tunnel right line shows that during the whole process of convergence deformation, the convergence rate is higher in the frst period, the peak reaches 0.4 mm/d, the duration is still short, and then drops sharply, the convergence rate is stabilized between 0.3 mm/d and 0.1 mm/d, the phenomenon of negative growth of convergence deformation appears in the 25th day, and there is no obvious sign of damage to the cave body in this process. After the 40th day, the convergence value of the cave body was kept at about 7 mm, and it could be afrmed as basic convergence when it reached 7.3 mm. Te convergence rate was stabilized within 0.1 mm/ d at the end, and the cave was basically stable without any abnormalities.

Tunnel Excavation Process
Simulation. Numerical simulation object for Longmenshan tunnel right line K83 + 010∼K83 + 90 section, excavation length 80 m, tunnel burial depth of about 100∼110 m, the terrain is relatively gentle, the section is IV grade surrounding rock, rock type for the medium weathering granite, the tunnel using the upand-down step method of boring. Te tunnel is excavated by the up-and-down step method. Te inner contour of the main tunnel of the tunnel is a three-centered circle with diameters of 10.5 m, 12.8 m, and 10.5 m, respectively. Te total height of the main tunnel is 7.45 m, and the total width is 11.93 m. For the creation of soil, according to Saint-Venant's principle, the stress redistribution caused by excavation is only within the range of 3−5 times the excavation width from the cavern, and will not cause too much impact on other areas. Terefore, the left and right boundaries of the model are three times the diameter of the tunnel, and the width is 100 m. Te lower boundary is 3.5 times the diameter of the hole. According to the real terrain, the height is selected to be 150-160 m. Te height of the tunnel from the bottom of the soil is 50 m, and the buried depth is 100-110 m. Combined with the tunnel ground investigation report, the soil model adopts the Mohr-Coulomb model in the elastoplastic principal model. Te physical parameters of the soil are its density of 1.8 g/cm 3 , modulus of elasticity of 2.6 GPa, Poisson's ratio of 0.32, friction angle of 34°, and cohesion of 0.5 MPa. Te overrun support structure of the tunnel is a Φ25 mm overrun hollow grouting anchor with a density of 7.8 g/cm 3 , a modulus of elasticity of 200 GPa, and Poisson's ratio of 0.2. Te lining is made of C25 plain waterproof concrete; the density is 2.4 g/cm 3 , the elastic modulus is 20 GPa, and Poisson's ratio is 0.2. Te inverted arch and the secondary lining are made of C30 concrete with a density of 2.5 g/cm 3    downward gravity load acts on the whole model. Te overall meshing of the tunnel model is sparse in the overall soil and dense in the boundary of the tunnel. Te neutral axis algorithm is used. Te element type of the anchor is the truss and the total number of mesh elements is 126,600. Te tunnel model of the lattice is shown in Figure 6.

Analysis of Results.
It can be seen from Figure 7 that the top of the arch is the maximum vertical displacement of the primary support structure, which reaches 6.267 mm. Te maximum lateral displacement of 2.456 mm occurred in the arch waist area, while the measured settlement value of the arch top and the convergence value around the target section were 9.4 mm and 7.3 mm, respectively, with a large diference of 3.133 mm in the accumulated vault settlement value and 2.31 mm in the accumulated convergence value. To better verify the accuracy of the numerical model, the vertical displacement at monitoring point 1 of the liner and the summation of the transverse displacement at monitoring points 2 and 3 were extracted (the locations of the points taken are shown in Figure 8), plotted as curves, and analyzed to compare the deformation trend of the measured vault subsidence value and the peripheral convergence value of the section where the liner is located. As shown in Figure 9, a comparison of the analysis of the settlement values of the vault (the vertical displacement value at monitoring point 1) of void section K83 + 020, the peripheral convergence values (the sum of the lateral displacement values at monitoring points 2 and 3), and the calculated values of the model simulation reveals that the analysis results obtained also difer from the real situation due to the diference between the selected soil mechanical parameters and the real surrounding rock mechanical parameters. However, the theoretical analysis of the change in the initial lining displacement during the numerical simulation of tunnel excavation is consistent with the deformation law of the real tunnel excavation process, indicating that the numerical simulation of the Longmenshan Mountain tunnel excavation process is consistent with the real excavation process and the diference in stress and strain is caused by the unreasonable setting of the mechanical parameters of the tunnel surrounding rock during the simulation process, which can be analyzed by the inversion of the surrounding rock mechanical parameters to obtain Te inversion analysis of the mechanical parameters of the surrounding rock can be used to obtain the mechanical parameters of the surrounding rock that are close to the real situation.

3D Mechanical Parameter Inversion Analysis of Surrounding Rock.
Some scholars [23][24][25][26][27][28][29] have already applied the system identifcation sensitivity analysis method in the inverse calculation of layered pavement structure, foundation excavation deformation analysis, and asphalt pavement aging inversion analysis, but at present, the system identifcation sensitivity analysis method is mainly applied to the inversion of two-dimensional parameters of tunnel envelope, and less tunnel monitoring data are used in the inversion analysis. In this paper, based on the forward model established for the Longmenshen Mountain tunnel as the Mohr-Coulomb 3D numerical model, a system identifcation method is introduced in the feld of 3D parameter inversion analysis of the tunnel envelope. As shown in Figure 10, the three parameters of static elastic modulus, Poisson's ratio, and internal friction angle of the surrounding rock are calculated by inversion based on the measured peripheral convergence value S 1 ′ , vault settlement value S 2 ′ , and vault settlement value S 3 ′ of the measured section of a tunnel, which can realize the high-precision and efcient automation of the model parameter adjustment process, and the basic process is as follows:  Advances in Civil Engineering (1) Setting the initial surrounding rock mechanical parameters Te initial internal friction angle φ 0 , the modulus of elasticity E 0 , and the initial Poisson's ratio μ 0 are assumed. φ 0 , E 0 , and μ 0 are input into the fnite element forward model to calculate the displacement values of the section corresponding to the measured values (2) Comparing the calculated value s { } with the measured value s ′ If the absolute value of the diference between the calculated value and the measured value is small, that is, the calculated result meets the requirements, take max ∆s { } T ∆s { } ≤ e (e is the calculation accuracy), immediately terminate the inverse analysis calculation, at this time the model elastic modulus E, Poisson's ratio μ and internal friction angle φ is the actual surrounding rock material static elastic modulus E s , Poisson's ratio μ and internal friction angle φ.

Advances in Civil Engineering
(4) Calculating the parameter adjustment vector ∆x between the measured values and the calculated values under the initial modulus of elasticity E 0 , the initial Poisson's ratio μ 0 , and the initial angle of internal friction φ 0 . Ten, construct the sensitivity equation where the sensitivity matrix D � zs 1 /zE zs 1 /zμ zs 1 /zφ zs 2 /zE zs 2 /zμ zs 2 /zφ zs 3 /zE zs 3 /zμ zs 3 /zφ Te diference {∆s} between the measured values and the calculated values under the initial elastic modulus E 0 , the initial Poisson's ratio μ 0 , the initial internal friction angle φ 0 , and the sensitivity matrix D is input to the prepared numerical analysis program, which will calculate the parameter adjustment vector ∆x { }.

Advances in Civil Engineering
Finding the calculated values s under the modulus of elasticity E 1 , Poisson's ratio μ 1 , and the angle of internal friction φ 1 , with the upper mark "(1)" representing the frst parameter adjustment, and then go back to step 2 until the requirements are met.

Inversion of Mechanical Parameters of Longmenshan
Tunnel Enclosure. After several parameter adjustments, assuming that the initial mechanical properties of the surrounding rock parameters E 0 � 2.0 GPa, μ 0 � 0.32, φ 0 � 32°, the inversion calculation is carried out according to the 3D mechanical parameters inversion method proposed in this paper, and four iterations are carried out to obtain E � 1.82 GPa, μ � 0.32, φ � 35°, at which time the tunnel fnite element model calculates the peripheral convergence value s 1 as 7.2554 mm, s 2 is 9.5317 mm, and s 3 is 8.8206 mm. According to the measured cumulative peripheral convergence value s 1 ′ is 7.3224 mm, the measured cumulative vault settlement value s 2 ′ is 9.4669 mm, and the measured cumulative vault settlement value s 3 ′ is 8.7128 mm, it can be found that the simulated value is very close to the measured value as well. According to the model results, the measured results of the tunnel can be obtained, as shown in Table 1. At this point ∆s T ∆s { } � 0.0466 mm, satisfying the condition max max ∆s { } T ∆s { } ≤ 1.0E − 5m, the real surrounding rock mechanical parameters of the grade IV section of the Longmenshan tunnel numerically simulated are static elastic modulus E s � 1.82 GPa, Poisson's ratio μ � 0.32, and internal friction angle φ � 35°. Based on the system identifcation sensitivity analysis method for inversion to obtain the fnal surrounding rock parameters to establish the forward model, extract the simulated value of displacement change of its tunnel right line K83 + 020 section and the actual measurement to draw the ftting curve as shown in Figure 11.
Comparing the simulated and measured displacement variation curves of the target sections, it can be seen that the mechanical parameters obtained by applying the system identifcation sensitivity analysis method to the mechanical parameters of the tunnel envelope after displacement measurement inversion analysis and then substituting into the fnite element model for the forward evolution have small error between the simulation results and the measured values. In each stage of tunnel construction, the error values of vault deformation and peripheral convergence of each target section in the numerical model and the actual monitoring and measurement results are small, within 0.3 mm. Trough the inversion analysis of the Longmenshan tunnel, the actual surrounding rock parameters were reasonably determined for the section from K83 + 020 to K83 + 90, with a static elastic modulus E s � 1.82 GPa, Poisson's ratio μ � 0.32, and an internal friction angle φ � 35°. Te results all meet the requirements of highway tunnel design specifcations and prove the feasibility of the system identifcation sensitivity analysis method, which provides important application value for the real-time detection and evaluation of the tunnel under construction in Longmenshan Mountain and provides a basis for subsequent related scientifc research.

Seismic Response Analysis of Tunnel Void
Polymer Grouting Repair

Tunnel Void Inspection and Repair.
In the right line of the Longmenshan tunnel section from K83 + 140 to K83 + 350, a geological radar method was used to inspect the quality of lining construction, and it was found that there was a void area with a maximum depth of 20 cm and a longitudinal length of about 2∼3 m between the lining and the rock body at the top of the tunnel, as shown in Figures 12 and 13. According to the data provided by radar, grouting repair of the void area, compared with concrete grouting material, has excellent antiseepage performance, compressive performance, tensile performance, and corrosion resistance. After the polymer repair material reaction, it can not only rapidly occur volume expansion (volume expansion 10∼20 times), but also can automatically compact and reinforce the void area, and recompact and reinforce the structural disease area. More importantly, it is light in weight, high in construction efciency, low in time cost, and high in economy, and the polymer material can reach 90% sufcient strength    [16][17][18][19][20]. Te grouting material used is a nonwater-reactive polymer. By comparing the radar detection spectra before and after the polymer grouting in Figure 14, it is found that the grouting efect in the void area behind the lining is good and that the integrity and compactness of the lining structure are signifcantly improved. To better verify the efect of polymer material for tunnel debonding repair, this paper carries out a comparative analysis of the seismic response before and after the tunnel polymer grouting repair under seismic action.   Figure 15. Te seismic fortifcation standard of the Longmenshan tunnel is 0.1 g peak ground motion acceleration and 7-degree earthquake basic intensity. Te three natural seismic waves selected for conducting seismic analysis are the Northridge wave (Class I site), the Taft wave (Class II site), and the EL-Centro wave (Class III site), which correspond to a peak acceleration of 0.1 g, a time interval of 0.02 s, and an efective duration of 19.2 s.

Displacement Analysis of Seismic Load Action.
Since the debris area is located at the top of the tunnel, the vertical displacement and vertical acceleration at the top node of the secondary lining can visually refect the improvement of the debris area by polymer grouting. It can be seen from Figures 16-19  .44 mm, respectively, and the maximum value of vertical acceleration is 0.91 m/s 2 , 1.04 m/s 2 , and 0.97 m/s 2 , respectively, compared with the normal condition, the increase of vertical displacement and vertical acceleration at the top of the secondary lining when the top is void is 29.6% and 14.3%. Compared with the normal condition, the increase of vertical displacement and vertical acceleration at the top of the secondary lining was 29.6% and 14.3%, and after the grouting repair, the increase of vertical displacement and vertical acceleration was 15.8% and 6.6% compared with the normal condition, which was 10.6% and 6.7% less than the increase when the top was void. From the above analysis, it can be seen that the void has a greater impact on the tunnel, which will signifcantly increase the seismic response of the tunnel. After repairing the tunnel with polymer grouting, the maximum tunnel displacement will be signifcantly reduced, coming close to the displacement of the normal tunnel. It refects the repair efect of polymer grouting on tunnel voids. It can be seen from Figure 19 that, under the action of seismic waves with the same peak value in diferent site conditions, there are obvious diferences in the vertical   Figure 14: Geological radar detection spectra before and after grouting: (a) geological radar detection spectrum before polymer grouting; (b) geopolymer grouting after georadar detection spectrum. small and close to the normal condition, which achieves the expected repair efect and makes the overall force deformation of the support structure safer and more stable.
It can be seen from Figure 23 that, under the action of seismic waves with the same peak value in diferent site conditions, there are obvious diferences between the maximum absolute value of the maximum principal stress and the maximum absolute value of the minimum principal stress at the top of the secondary lining. Te maximum absolute value of the maximum principal stress and the maximum absolute value of the minimum principal stress under Northridge wave (type I site) excitation are the largest,  and the maximum absolute value of the vertical maximum principal stress under the excitation of EL-Centro wave (type III site), Te maximum value of the absolute value of the minimum principal stress is the smallest; that is, as the site type changes from hard, medium hard, to medium soft, the stress efect on the top of the secondary lining gradually weakens, indicating that the dynamic response of the tunnel model has obvious sensitivity to the seismic wave spectrum.

Conclusion
In this paper, the construction process of the Longmenshan Mountain tunnel under construction was numerically simulated using the Longmenshan Mountain tunnel as the engineering background, and the inversion of the physical parameters of tunnel envelope mechanics was carried out for the grade IV envelope section using the system identifcation sensitivity analysis method. Finally, the analysis of the efect of high polymer grouting to repair the tunnel debonding was carried out under the seismic mechanical response, and the following conclusions were obtained: (1) Numerical simulation of the Longmenshan Mountain tunnel project using fnite element software, found that the deformation law and stress distribution law of its tunnel envelope and support structure are in line with the site construction, but the amount of deformation such as vault settlement and peripheral convergence is diferent from the actual monitoring and measurement data, and this diference is caused by the discrepancy between the surrounding rock mechanical parameters selected by the numerical model and the actual diference is mainly caused by the discrepancy between the surrounding rock mechanical parameters selected by the numerical model and the actual ones. (2) Based on the system identifcation sensitivity analysis method, the two-dimensional parameter inversion method is improved, the formula for three-dimensional physical and mechanical parameter inversion analysis of the surrounding rock is derived, and the three-dimensional mechanical parameter inversion analysis of the surrounding rock in the right line of the Longmenshan tunnel section from K83 + 010 to K83 + 90 is carried out. Te fnal mechanical parameters of the surrounding rock, in line with reality, are the static elastic modulus E s � 1.82 GPa, Poisson's ratio μ � 0.32, and the internal friction angle φ � 35°, proving the feasibility of the system identifcation sensitivity analysis method. It has an important reference value for the real-time detection of the Longmenshan tunnel project under construction and provides a guarantee for accurately establishing the fnite element model of the overall structure of the tunnel and surrounding rock and carrying out dynamic response analysis. (3) Te seismic mechanical response analysis verifed the signifcant efect of polymer grouting to repair the tunnel debonding. Te displacement and stress at the top of the tunnel's secondary liner debonding area were signifcantly reduced after the slurry repair, which was close to the normal condition. However, the stress and displacement of the top of the secondary lining under the action of seismic waves from the same peak at diferent site conditions are signifcantly diferent, and the efect on the displacement and stress response of the top of the secondary lining gradually decreases as the site type changes from hard, medium hard, to medium soft, indicating that the dynamic response of this tunnel model has obvious sensitivity to the seismic wave spectrum.

Data Availability
Te data presented in this study are available in the main text of the article.

Conflicts of Interest
Te authors declare that they have no conficts of interest regarding the publication of this paper.