A New Unified Solution for Deep Tunnels in Water-Rich Areas considering Pore Water Pressure

Pore water pressure has an important influence on the stresses and deformation of the surrounding rock of deep tunnels in water-rich areas. In this study, a mechanical model for deep tunnels subjected to a nonuniform stress field in water-rich areas is developed. Considering the pore water pressure, a new unified solution for the stresses, postpeak zone radii, and surface displacement is derived based on a strain-softening model and the Mogi-Coulomb criterion. Through a case study, the effects of pore water pressure, intermediate principal stress, and residual cohesion on the stress distribution, postpeak zone radii, and surface displacement are also discussed. Results show that the tangential stresses are always larger than the radial stress. The radial stress presents a gradually increasing trend, while the tangential stress presents a trend of first increasing and then decreasing, and the maximum tangential stress appears at the interface between the elastic and plastic zones. As the pore water pressure increases, the postpeak zone radii and surface displacement increase. Because of the neglect of the intermediate principal stress in the Mohr-Coulomb criterion, the postpeak zone radii, surface displacement, and maximum tangential stress solved by the Mohr-Coulomb criterion are all larger than those solved by the Mogi-Coulomb criterion. Tunnels surrounded by rock masses with a higher residual cohesion experience lower postpeak zone radii and surface displacement. Data presented in this study provide an important theoretical basis for supporting the tunnels in water-rich areas.


Introduction
Tunnels are one of the most basic facilities in water conservancy engineering, civil engineering, and mining engineering [1][2][3]. The stress distribution of the rock mass changes with the excavation of the tunnel, and the surrounding rock of the tunnel will be deformed if the redistributed stress exceeds the peak strength of the rock mass. Therefore, accurate calculation for the stress of the surrounding rock plays an important role in stability evaluation and support design of the tunnel [4][5][6].
In the past decades, a series of unified solutions based on different strength criteria have been proposed. Some common strength criteria, such as the Mohr-Coulomb criterion [7][8][9][10] and the Hoek-Brown criterion [11][12][13][14], were widely used to calculate the stress in the tunnel surrounding rock.
However, the intermediate principal stress is not considered in these criteria, which caused an inaccurate result. In fact, the surrounding rock of the tunnel is always in a true triaxial stress environment [15,16], and the intermediate principal stress has a nonnegligible influence on the strength of the rock mass. Therefore, it is of great significance that the intermediate principal stress be taken into consideration in the unified solution for deep tunnels.
Because of the underground faults, folds, and other special structures, the ratio of the vertical stress to horizontal stress is usually not equal to one. In this case, Galin [17] first analyzed the tunnel in a nonuniform stress field and deduced the radius of the plastic zone. However, Galin's solution only applies to frictionless rock mass. Detournay [18][19][20] extended Galin's result to other materials and obtained the boundary of the elastic and plastic zones. Tokar [21], Leitman and Villaggio [22], and Ochensberger et al. [23] also presented a series of analytical solutions for a circular wellbore in some certain cases based on Galin's solution.
It is known that the flowing water always exists in the underground rock mass, which has a certain impact on the stress distribution and deformation of the surrounding rock of the tunnel. Therefore, the influence of the pore water pressure should be considered in the elastic-plastic analysis for deep tunnels in water-rich areas [24][25][26]. In the present study, a mechanical model for deep tunnels subjected to a nonuniform stress field in water-rich areas is first established. Considering the pore water pressure, a new unified solution for the stresses, postpeak zone radii, and surface displacement is derived based on strain-softening model and Mogi-Coulomb criterion. Through a case study, the sensitivity of pore water pressure, intermediate principal stress, and residual cohesion on the stress distribution, postpeak zone radii, and surface displacement is analyzed.

Mechanical Model of a Circular Tunnel.
A circular tunnel of radius R 0 was excavated in an infinite rock mass ( Figure 1). The vertical and horizontal stresses are σ 0 and λσ 0 , respectively, where λ is the lateral stress coefficient. A support pressure (P i ) is uniformly distributed along the excavation surface. The surrounding rock of the tunnel is subdivided into an elastic zone ("e"), plastic zone ("p"), and damage zone ("d"). The radii of the elastic, plastic, and damage zones are denoted by R e , R p , and R d , respectively.
We assume that there is a pore water pressure (P 0 ) outside the elastic zone of the tunnel. Based on Darcy's law, the continuous differential equation of seepage is where P w is the pore water pressure at any point of the tunnel surrounding rocks. Combined with the boundary condition of P w = 0 at r = R 0 and P w = P 0 at r = R e , the pore water pressure can be derived by solving Equation (1): 2.2. Strain-Softening Model. As shown in Figure 2, the experimental stress-strain curve of the rock mass can be simplified into a three-section line. The three straight lines correspond to the elastic zone, plastic zone, and damage zone, respectively.

Analytical Solution
3.1. Basic Equations. The Mogi-Coulomb criterion can be expressed as [27][28][29] where σ θi and σ ri are the tangential and radial stresses in the "i" region, respectively; M = ð ffiffi ffi 3 p + 2 sin φÞ/ð ffiffi ffi 3 p − 2 sin φÞ; φ is the internal friction angle; N i = 4c i cos φ/ð ffiffi ffi 3 p − 2 sin φÞ; and c i represents the cohesion in different zones. The symbol "i" can be replaced by "e," "p," and "d." Taking the pore water pressure into consideration, the equilibrium differential equation in the "i" zone can be given as where η is the pore water pressure coefficient.

Geofluids
The geometric equation can be written as where ε ri and ε θi are the radial and tangential strains in the "i" zone, respectively, and u i represents the displacement in the "i" zone. The constitutive equations can be denoted as where μ and E are Poisson's ratio and Young's modulus of the rock mass, respectively.
In addition, the volume of the rock mass is always changing in the postpeak failure zone; the plastic-strain relationships can be developed based on the nonassociated flow rule as follows: 3.2. Elastic Zone. The stress state of the circular tunnel in a nonuniform stress field can be decomposed into two parts (see Figure 3). In state I, the tunnel is subjected to a uniform pressure (0:5ð1 + λÞσ 0 ), pore water pressure (P w ), and support pressure (P i ); the differential equation can be obtained by substituting Equations (2), (5), and (6) into Equation (3):

Geofluids
Solving Equation (8), the displacement and strains in state I can be easily obtained as where C 1 and C 2 are integral constants. The radial and tangential stresses can be derived by integrating Equations (6) and (9): Considering the boundary condition σ r = 0:5ð1 + λÞσ 0 + P 0 at r = R 0 and σ r = σ e−p r at r = R p , the integral constants can be solved as follows: The stresses in state I can be determined by substituting Equation (11) into Equation (10): In state II, the surrounding rock is subjected to a horizontal tension (0:5ð1 − λÞσ 0 ) and a vertical pressure (0:5ð1 − λÞσ 0 ). At r = R s , the boundary condition can be written as follows: Though coordinate transformation, Equation (13) can be re written as follows: At r = R 0 , σ x = τ rθ = 0. Therefore, using semi-inverse method, the stresses in state II can be deduced as follows: Therefore, the stresses in the elastic zone considering pore water pressure can be obtained by superimposing Equations (12) and (15): At the interface between the elastic and plastic zones, the radial and tangential stresses should satisfy Equation The radial and tangential strains in the elastic zone can be derived by substituting Equation (14) into Equation (6): Using Equation (5), the displacement in the elastic zone can be easily obtained as 3.3. Plastic Zone. In the plastic zone, the total strain of the surrounding rock consists of two parts: the plastic and the elastic part. Thus, the following equation can be easily given as The displacement differential equation in the plastic zone can be obtained by integrating Equations (5), (7), and (20): With the boundary condition of u p = u e−p at r = R p , the displacement in the plastic zone can be deduced by solving Equation (21): Using Equation (5), the strains in the plastic zone can be obtained as Previous studies indicated that the internal friction angle of the rock does not change significantly in the postpeak phase, and the rock strength is only related to cohesion. Assuming that the cohesion in the plastic zone decreases linearly (see Figure 4), the cohesion at any point in the plastic zone can be expressed as where c 0 is the initial cohesion and α is the softening coefficient of the cohesion.
Combining the boundary condition of σ r = σ r e−p at r = R p , the radial stress in the plastic zone can be obtained by solving Equation (25) as 3.4. Damage Zone. In the damage zone, the total strains of the surrounding rock are also composed of two parts as: where ε r p−d and ε θ p−d are the radial and tangential strains at the interface between the plastic and damage zones, respectively.
The displacement differential equation in the plastic zone can be obtained by integrating Equations (5), (7), and (27): Considering the boundary condition of u d = u p−d at r = R d , the displacement in the plastic zone can be deduced by solving Equation (28): Using Equation (5), the strains in the damage zone can be achieved as follows: The equilibrium differential equation in the damage zone can be rewritten by substituting Equations (2) and (3) into Equation (4) as Combining the boundary condition of σ r = P i at r = R 0 , the radial stress in the plastic zone can be obtained by solving Equation (31) as   Geofluids relationship between R p and R d can be established by combining with (26) and (32).
According to Equation (24), the cohesion at r = R d can be expressed as follows: Subsequently, the radii R p and R d can be derived by integrating (33) and (34).

Case Study
The stress distribution and deformation of the tunnel surrounding rock are of great importance for the stability evaluation and support design of the tunnel. In order to study the influence of pore water pressure, intermediate principal stress, and residual cohesion on the stresses and displacement of the tunnel, the mechanical and geometrical parameters of the rock mass are shown in Table 1.  Figure 5 shows the postpeak zone radii and surface displacement around the tunnel under different pore water pressure. The radii of plastic and damage zones and surface displacement all increase with the increase of pore water pressure. For example, as P 0 increases from 3 MPa to 6 MPa, the R p , R d , and u 0 values at the tunnel side increase by 0.85 m, 0.83 m, and 36.44 mm, with an increment of 20.48%, 22.61%, and 67.31%, respectively, and the R p , R d , and u 0 values at the tunnel crown increase by 0.77 m and 0.75 m, and 72.06 mm, with an increment of 13.62%, 14.24%, and 43.43%, respectively. Therefore, the pore water pressure exerts a crucial influence on the radii of plastic and damage zones and surface displacement.

Stress Distribution in Tunnel Surrounding
Rock. Taking the tunnel crown as an example, the stress distribution based on different pore water pressure is shown in Figure 6. It can be seen that the tangential stresses are always larger than the radial stress. The radial stress presents a gradually increasing trend, while the tangential stress presents a trend of first increasing and then decreasing, and the maximum tangential stress appears at the interface between the elastic and plastic zones. As the pore water pressure increases, the maximum tangential stress increases and is farther away from the center of the tunnel.

Stress Distribution in Tunnel Surrounding
Rock. The stress distribution at the tunnel crown based on two different criteria is shown in Figure 8. It can be seen that the intermediate principal stress has a significant effect on the stress distribution in the three zones. When the intermediate principal stress is ignored, the stress concentration and maximum tangential stress are larger, and the boundary between the plastic and elastic zones is closer to the tunnel center.  Figure 9 shows the postpeak zone radii and surface displacement around the tunnel under different types of residual  Figure 10. It can be seen that the radial stress is always in the increasing trend and the tangential stress is always larger than the radial stress, which are similar to those laws in Figures 6 and 9. As the residual cohesion increases, 8 Geofluids the maximum tangential stress increases slightly, but the boundary between the plastic and elastic zones moves farther away from the center of the tunnel.

Conclusions
Considering the pore water pressure, the stress distribution and postpeak zone radii in the surrounding rock of a deep tunnel in water-rich areas are deduced based on a strainsoftening model and the Mogi-Coulomb criterion. The influence of pore water pressure, intermediate principal stress, and residual cohesion on the stress distribution and postpeak zone radii is also discussed. The conclusions can be summarized as follows: (1) As for the stress distribution in the surrounding rock of a tunnel, the tangential stresses are always larger than the radial stress. The radial stress presents a gradually increasing trend, while the tangential stress