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Study on the construction mechanical response of the double-shield TBMs of a long diversion tunnel crossing a fault fracture zone

The Bheri Babai Diversion Multipurpose Project (BBDMP) in Nepal is located in the southern foothills of the Himalayas. The tunnel of this project passes through the Siwalik rock stratum. In this paper, the mechanical numerical model of the double-shield tunnel boring machine (TBM) crossing the fault fracture zone is established based on the morphology of the developed joint in the Bheri fault zone of the BBDMP. By analyzing the influence of joint group on double-shield TBM tunneling, the variation rules of the surrounding rock displacement, plastic zone distribution, and internal force of TBM crossing the fault fracture zone are obtained. The results of the research demonstrate the following: (1) when the single-shield tunneling mode is used in the fault fracture zone, the surrounding rock displacement is larger than that in intact rock mass, and the dissymmetry of joint geometry leads to the dissymmetry of the displacement distribution. (2) The plastic zone of the surrounding rock is mainly distributed along the geometric plane of two joints. The maximum value appears in the area from the left arch waist to the left arch shoulder and from the right arch waist to the right foot of the wall. The joints at all the other sites reach the yield state, except for the area around the vault and inverted arch, with a range of about 0.3 times the tunnel diameter. (3) The existence of joint surface leads to the asymmetrical distribution of the internal force in the pipe segment. The minimum safety coefficient appears at the left shoulder and right foot of the wall.


Introduction
Tunnel boring machines (TBMs) have been increasingly used in recent years owing to its high efficiency and safety. However, its adaptability to bad geological conditions, such as fault fracture zones, is poor [1,2]. The main challenges in underground excavation through fault rock are extremely short stand-up time, tunnel face instabilities, and high radial convergence [3][4][5][6][7]. TBM tunneling in the fault zone may cause numerous problems, which include high ground pressure, TBM jamming, tunnel face instability or collapse, and a high or sudden inflow of water [8]. TBM tunneling in zones with poor quality has been quite problematic, for example, in the case of the Tuzla tunnel in Turkey [9], the Kargi Kizilirmak headrace tunnel in Turkey [10], the Water Transmission Tunnel in Iran [11], and the Zagros water-transfer tunnel through the Zagros Mountains in Iran [12]. Therefore, in engineering, studying the mechanical response of TBM crossing the fault fracture zone is important for the improvement of the performance of TBM and efficiency of tunneling. Evidently, having proper knowledge of the determination of the rock mass geomechanical characteristics in the fault fracture zone is essential. Hao and Azzam [13] conducted a parametric study with the use of numerical method (UDEC) and identified three fault parameters (fault dips, fault shear strength, and fault locations relative to the underground structure) that are critical to the stability of the underground structure. Kahraman and Alber [14] investigated a fault breccia with clasts weaker than the matrix. They found that its strength was related to the proportion of the blocks and the size of the specimen. Zhao et al. [15] conducted a 3D simulation of TBM excavation in brittle rock associated with fault zones and found that the condition of the local rock mass and the complex interaction between the rock mass, TBM components, and tunnel support play a role in the characterization of instability phenomenon. However, the mechanical behavior of the rock mass in a fault fracture zone is difficult to characterize and is still poorly understood. Numerous scholars have investigated the interaction between the surrounding rock and tunnel structure in a fault fracture zone. Shrestha and Panthi [16] [19], and Gao et al. [20] investigated the influence of fault on the stability of the surrounding rock of a tunnel. When the TBM tunnel passes through the fault fracture zone, the stratum characteristics exhibit a more prominent effect on the mechanical response of the TBM tunnel construction. The strength and deformation characteristics of the broken rock mass are mainly controlled by joint cracks, and its mechanical properties are largely dependent on the mechanical characteristics, geometric characteristics, and distribution laws of joint cracks. Thus, selecting a proper constitutive model for numerical simulation is extremely important. The ubiquitous joint model can simultaneously consider the role of the two media types in joint rock mass and truly simulate the strength and deformation characteristics of the rock mass. Moreover, this model has been widely studied [21]. Wang and Huang [22] proposed a constitutive model for the deformation of a rock mass containing sets of ubiquitous joints, which can present the joint-induced anisotropy in the strength and deformation of rock mass. Ismael and Konietzky [23] proposed a ubiquitous joint constitutive model that links two failure criteria, the sliding and non-sliding failure modes, for inherent anisotropic rocks. Das et al. [24] evaluated the stability of underground workings for the exploitation of an inclined coal seam by the ubiquitous joint model. In this paper, a 3D numerical model of the double-shield TBM passing through the Babai fault fracture zone was established based on the Bheri Babai Diversion Multipurpose Project (BBDMP) in Nepal. The ubiquitous joint model of the Abaqus software was utilized to simulate the stratum in the fault fracture zone. Moreover, the influence of joints on the double-shield TBM tunneling in the fault fracture zone was analyzed. Finally, the variation laws of the surrounding rock displacement, plastic zone distribution, and segment internal force of the TBM crossing the fault fracture zone were obtained.

Project Overview
The BBDMP is the first interbasin water-transfer project in Nepal. The BBDMP primarily comprises of a tunnel with a total length of 12.2 km that transfers the water from the Bheri River, which is rich in water resources, to the Babai River-a local river with intermittent drought and flood. The purpose of this project is to irrigate 51,000 ha of farmlands in Bardiya and Banke in central and eastern Nepal.

Overview of Ubiquitous Joint Model
The Abaqus software provides a model that is suitable for materials rich in parallel joint planes. The model may be used for the simulation of regular fracture joint planes appearing in groups or rock mass with a large number of faults. The spacing between joints is sufficiently smaller than the structural size of the model. Therefore, the whole model can be combined with a continuous sliding system. The main characteristics of this model include the following: 1) elastic-plastic deformation is considered; 2) three relationships exist between the joint layers (frictional sliding, closed, and separated), and once the joint layers are separated, the materials immediately become orthogonal anisotropic bodies; 3) the damage caused by volumetric deformation based on the Drucker-Prager or Mohr-Coulomb model may be considered; and 4) the model provides a reasonable stress cycle, including joint opening/closing and shearing force cycle.

Mechanical Behavior of Joint Materials
The direction of a joint plane is mainly determined based on its normal direction by setting the normal vector of the joint plane a to a l nm n   =    , where l, m, and n denote the cosine in three directions of the normal vector. Two unit vectors on the joint plane together with a n form mutually orthogonal vectors as the local coordinate system describing the orientation of the joint plane, as presented in Figure 2.  Figure 2. Joint element and joint direction. The compressive stress perpendicular to the joint plane and the two shear stresses in the joint plane are calculated using the following equations: The shear stress is defined as: The strain perpendicular to the joint plane is defined as local strain: The strain at an angle  to the joint plane is defined as engineering shear strain: a a a a a n t t n Assuming that the total strain can be decomposed into the sum of elastic strain and plastic strain: The plastic strain d pl  consists of two parts: dd pl pl i i  =  (7) ib = denotes volume deformation, and ia = denotes joint deformation. When joints are closed, the materials become isotropic linear elastic bodies. Once joints are identified to be tensioned in the normal direction, i.e., 0 a p  , they are opened, and the materials become orthogonal anisotropic bodies. As long as Equation (8) In Equation (9) where   denotes the friction angle, and a d denotes the cohesion.
When 0 a f  , joints will not slide. When =0 a f , joints will side. The plastic strain on the joint plane consists of two parts, which are calculated using Equations (11) and (12): Equation (13) (13) In Equation (13) In Equation (14), b  denotes the friction angle of volumetric strain, and b d denotes the cohesion of volumetric strain. Once the volumetric strain meets the above criteria, volumetric plastic strain will occur: In Equation (15), plastic potential is calculated as follows:

Modeling and Parameter Selection
The conditions of the surrounding rock are extremely poor in the fault fracture zone. Thus, the singleshield tunneling mode shall be adopted for tunneling in this stratum. The geometric parameters of the front shield, rear shield, gripper shield, and segment are presented in Table 1. The surrounding rock includes three joint groups, i.e., N0°10°W, SW ∠ 70°80° and EW, S(N) ∠ 80°90°, mainly developed in the construction area of the BBDMP. The tunnel is located at N26.84°E lengthways.  Figure 5. Layout of characteristic points.
The surrounding rock of the Bheri fault zone is classified as Class V. Table 2 presents the mechanical parameters of the surrounding rock and joints. With regard to the tunneling parameters, the thrust is set to 3000 kN and the torque to 389 kN· m. The single-shield tunneling mode is adopted. The weight of the TBM component is presented in Table 3. The cutter head and shield are based on their actual weights. The backup structure is complex and has little influence on the mechanical response during tunneling. Thus, its mass is reflected by equivalent segment increase. The ideal elastoplastic constitutive model is adopted for soil, and the Mohr-Coulomb criterion is applied as the yield criterion. The pea gravels and segments are considered as elastic bodies. Due to the time effect of filling, the elastic modulus of the pea gravel is considered as two stages of soft and hard. Assuming that TBM will not shift or rotate during tunneling, the horizontal degree of freedom and three rotational degrees of freedom of TBM are constrained. The numerical model is presented in Figure 6, and the schematic of measures for supporting and hydraulic filling of pea gravels is illustrated in Figure 7.

Contact Settings
(1) Contact between the shield and surrounding rock Setting of the contact relation between the shield and surrounding rock is extremely important for this model. At the start of the calculation, the shield and surrounding rock are in a non-contact state and have uneven gaps, which greatly influences the convergence of the numerical solution. During tunneling, the shield and surrounding rock are in a sliding friction state, and the tangential behavior of the contact surface is considered as dynamic friction. In this model, the exponential decay method is employed to describe the relationship between static friction coefficient and dynamic friction coefficient and expressed by Equation (16)  (4) Contact between the pea gravel and segment The inner side of the pea gravel is in contact with the outer side of the segment. The pea gravels are filled as force transmission media for the transmission of the load released by the excavated surrounding rock to the segment, as well as for load sharing. In the numerical model, the Tie Contact is also set between the inner side of the pea gravel and the segment for deformation together. Figures 8 and 9 present the nephograms for the vertical displacement and horizontal displacement of the surrounding rock when the fault fracture zone is penetrated.  . Nephogram for horizontal displacement following penetration. As can be seen from the figures, the displacement distribution in the vertical direction is symmetrical on the whole. After penetration by TBM, the vault settlement is 9.429 cm, and the bottom uplift is 7.288 cm. The horizontal convergence displacements at all nodes exhibit convergence toward the inside of the tunnel. The maximum value appears around the left arch shoulder and the right arch foot, which have horizontal convergence displacements of 5.518 and 5.347 cm, respectively. Table 4 presents both the vault settlement and inverted arch uplift values after tunneling for the total length of 60 m. Compared with the result of tunneling in intact rock mass, the trend of vertical displacement distribution is unchanged, and the specific values increase to a certain extent for tunneling in the fault fracture zone.

Plastic Zone Distribution and Joint Yield of the Surrounding Rock
Since the mechanical properties of the joint plane also follow the D-P criterion, the rock mass will generally be subject to shear sliding along the structural plane during tunneling, which results in a certain yield on certain joint positions. From Figure 10, it can be seen that the contour line of the node yield point is about 0.3 times the diameter of the tunnel. In addition to the joints, the rock mass will also be yielded to a certain extent. The nephogram for the plastic zone distribution of the surrounding  Figure 11. The plastic zone is mainly distributed along the geometric plane of two joints, as well as in the area from the left arch waist to the left arch shoulder and from the right arch waist to the right foot of the wall. The maximum plastic strain is 0.1667 and is located between the left arch waist and the vault, which is close to the arch shoulder. Table 5 presents the maximum plastic strain of the surrounding rock after tunneling for a total length of 60 m. The plastic zone due to tunneling in the fault fracture zone is obviously larger than that due to tunneling in intact rock mass.

Internal Force Analysis of Segment
The internal force and the safety coefficient of the segments during TBM tunneling to 20 m, 30 m, and total length in fault fracture zone are presented in Table 6.  Table 6 demonstrates that: (1) The dissymmetry of the mechanical properties of joint rock mass leads to the dissymmetry of the internal force distribution of the segment.
(2) Due to the inherent properties of joint rock mass, "potential" shear sliding surface appears on the structural plane following excavation, and the internal force distribution law of the segment changes. With regard to the axial force and bending moment, the maximum values are roughly located at the left arch shoulder and the right foot of the wall. The axial force and bending moment also exhibit an increasing trend during the whole process of excavation.
(3) The minimum safety coefficient is located at the left arch shoulder and the right foot of the wall. After the tunnel penetration, the safety coefficient of the left arch waist is decreased, with the minimum value up to 2.02. Compared with the calculation results of tunneling in intact rock mass, the internal force of the segment in joint rock mass is much larger, but the safety coefficient is much smaller. The maximum internal force and the minimum safety coefficient and their positions are compared and analyzed, as presented in Table 7.  Table 7 demonstrates that the internal force of the segment during tunneling in the fault fracture zone significantly increases while the safety coefficient decreases. For the specific maximum value alone, the maximum axial force during tunneling in the fault fracture zone increases by 78.8% compared with that in the sandstone stratum and 69.6% compared with that in the mudstone stratum. The maximum bending moment increases by 60.8% and 32.4%, respectively, but the safety coefficient decreases by 20.3% and 14.4%, respectively.

Conclusion
This paper takes the well-developed joints in the BBDMP as the study objects to create a dynamic TBM construction model. This model is expected to be used for studying the construction mechanical response of TBM tunneling in the fault fracture zone of the Siwalik rock stratum of the Himalayas. The main conclusions are as follows: (1) When the single-shield tunneling mode is used in the fault fracture zone, the displacement of the surrounding rock is larger than that in intact rock mass. The displacement distribution is subject to spatial asymmetry mainly because of the asymmetry of the geometric parameters of joints.
(2) In the fault fracture zone, the plastic zone of the surrounding rock is mainly distributed along the geometric plane of two joints. The maximum value appears in the area from the left arch waist to the left arch shoulder and from the right arch waist to the right foot of the wall. The maximum plastic strain reaches 0.1667. Some joints also reach the yield state, with a range of about 0.3 times the diameter of the tunnel. The joints at all the other sites reach the yield state, except for the area around the vault and inverted arch.
(3) The existence of joints in the fault fracture zone leads to the asymmetrical distribution of the internal force in the segment. The minimum safety coefficient appears at the left arch shoulder and the right foot of the wall.