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

Authors: Hao Fan; Lianguo Wang; Shuai Wang; Chongyang Jiang
Journal: Geofluids
 (2021)

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## Abstract

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.


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## Body

## 1. Introduction

Tunnels are one of the most basic facilities in water conservancy engineering, civil engineering, and mining engineering [1–3]. 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 [4–6].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 [7–10] and the Hoek-Brown criterion [11–14], 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 [18–20] 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 [24–26]. 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.


## 2. Definition of the Problem

### 2.1. Mechanical Model of a Circular Tunnel

A circular tunnel of radiusR0 was excavated in an infinite rock mass (Figure 1). The vertical and horizontal stresses are σ0 and λσ0, respectively, where λ is the lateral stress coefficient. A support pressure (Pi) 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 Re, Rp, and Rd, respectively.Figure 1
Mechanical model of tunnels subjected to a nonuniform stress field in water-rich area.We assume that there is a pore water pressure (P0) outside the elastic zone of the tunnel. Based on Darcy’s law, the continuous differential equation of seepage is
(1)d2Pwdr2+1rdPwdr=0,where Pw is the pore water pressure at any point of the tunnel surrounding rocks.Combined with the boundary condition ofPw=0 at r=R0 and Pw=P0 at r=Re, the pore water pressure can be derived by solving Equation (1):
(2)Pw=P0lnR0/rlnR0/Re.


### 2.2. Strain-Softening Model

As shown in Figure2, 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.Figure 2
Total stress-strain curve of rock mass.


## 2.1. Mechanical Model of a Circular Tunnel

A circular tunnel of radiusR0 was excavated in an infinite rock mass (Figure 1). The vertical and horizontal stresses are σ0 and λσ0, respectively, where λ is the lateral stress coefficient. A support pressure (Pi) 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 Re, Rp, and Rd, respectively.Figure 1
Mechanical model of tunnels subjected to a nonuniform stress field in water-rich area.We assume that there is a pore water pressure (P0) outside the elastic zone of the tunnel. Based on Darcy’s law, the continuous differential equation of seepage is
(1)d2Pwdr2+1rdPwdr=0,where Pw is the pore water pressure at any point of the tunnel surrounding rocks.Combined with the boundary condition ofPw=0 at r=R0 and Pw=P0 at r=Re, the pore water pressure can be derived by solving Equation (1):
(2)Pw=P0lnR0/rlnR0/Re.


## 2.2. Strain-Softening Model

As shown in Figure2, 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.Figure 2
Total stress-strain curve of rock mass.


## 3. Analytical Solution

### 3.1. Basic Equations

The Mogi-Coulomb criterion can be expressed as [27–29]
(3)σθi=Mσri+Ni,where σθi and σri are the tangential and radial stresses in the “i” region, respectively; M=3+2sinφ/3−2sinφ; φ is the internal friction angle; Ni=4cicosφ/3−2sinφ; and ci 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
(4)dσridr+σri−σθir+ηdPwdr=0,where η is the pore water pressure coefficient.The geometric equation can be written as(5)εri=duidr,εθi=uir,where εri and εθi are the radial and tangential strains in the “i” zone, respectively, and ui represents the displacement in the “i” zone.The constitutive equations can be denoted as(6)εr=1−μ2Eσr−μ1−μσθ,εθ=1−μ2Eσθ−μ1−μσr,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:(7)εri+βiεθi=0,where βi=1+sinψi/1−sinψi; ψi is the dilation angle in the “i” zone.


### 3.2. Elastic Zone

The stress state of the circular tunnel in a nonuniform stress field can be decomposed into two parts (see Figure3). In state I, the tunnel is subjected to a uniform pressure (0.51+λσ0), pore water pressure (Pw), and support pressure (Pi); the differential equation can be obtained by substituting Equations (2), (5), and (6) into Equation (3):
(8)d2ue1dr2+1rdue1dr−ue1r2=FvrE,where F=ηP0/lnR0/Re and v=1+μ1−2μ/1−μ.Figure 3
Mechanical model of surrounding rock in water-rich coal tunnel.Solving Equation (8), the displacement and strains in state I can be easily obtained as
(9)ue1=C1r+C2r+Fvrlnr2E,εre1=C1−C2r2+Fv2Elnr+1,εθe1=C1+C2r2+Fvlnr2E,where C1 and C2 are integral constants.The radial and tangential stresses can be derived by integrating Equations (6) and (9):
(10)σre1=EC11+μ1−2μ−EC21+μr2+ηFlnr21−μ+ηF2,σθe1=EC11+μ1−2μ+EC21+μr2+ηFlnr21−μ+μηF21−μ.Considering the boundary conditionσr=0.51+λσ0+P0 at r=R0 and σr=σre−p at r=Rp, the integral constants can be solved as follows:
(11)C1=ν1-μE121+λσ0+P0+Rp2Re2-Rp2v1-μE121+λσ0+P0-σre-p+ηFv2ERp2Re2-Rp2lnRpRe-ηFv1-μ2E-ηFv2ElnRe,C2=1+μERe2Rp2Re2-Rp2121+λσ0+P0-σre-p+ηF1+μ2E1-μRe2Rp2Re2-Rp2lnRpRe.The stresses in state I can be determined by substituting Equation (11) into Equation (10):
(12)σre1=121+λσ0+P0+ηF21−μlnrRe+Rp2Rp2−Re2Re2r2−1,121+λσ0+P0−σre−p+ηF21−μlnRpRe,σθe1=121+λσ0+P0+ηF2lnr/Re+2μ−11−μ−Rp2Rp2−Re2Re2r2+1,121+λσ0+P0−σre−p+ηF21−μlnRpRe.In state II, the surrounding rock is subjected to a horizontal tension (0.51−λσ0) and a vertical pressure (0.51−λσ0). At r=Rs, the boundary condition can be written as follows:
(13)σx=−0.51−λσ0,σy=0.51−λσ0,τrθ=0.Though coordinate transformation, Equation (13) can be re written as follows:
(14)σr=−0.51−λσ0cos2θτrθ=0.51−λσ0sin2θAtr=R0,σx=τrθ=0. Therefore, using semi-inverse method, the stresses in state II can be deduced as follows:
(15)σre2=−121−λσ01−4Rp2r2+3Rp4r4cos2θ,σθe2=121−λσ01+3Rp4R4cos2θ.Therefore, the stresses in the elastic zone considering pore water pressure can be obtained by superimposing Equations (12) and (15):
(16)σre=121+λσ0+P0+ηF21-μlnrRe+Rp2Rp2-Re2Re2r2-1·121+λσ0+P0-σre-p+ηF21-μlnRpRe-121-λσ01-4Rp2r2+3Rp4r4cos2θ,σθe=121+λσ0+P0+ηF2lnr/Re+2μ-11-μ-Rp2Rp2-Re2Re2r2+1·121+λσ0+P0-σre-p+ηF21-μlnRpRe+121-λσ01+3Rp4r4cos2θ.At the interface between the elastic and plastic zones, the radial and tangential stresses should satisfy Equation (3). Thus, σre−p can be derived by substituting Equation (16) into Equation (3):
(17)σre−p=−2Re21/21+λσ0+P0+ηF/21−μRp2−Re2lnRp/Re+2μ−1−Rp2+Re2lnRp/Re+21−λσ0cos2θRp2−Re2−NeRp2−Re2MRp2−Re2−Rp2−Re2.The radial and tangential strains in the elastic zone can be derived by substituting Equation (14) into Equation (6):
(18)εre=C1−C2r2+ηFv2Elnr+1−1+μ2E1−λσ01−41−μRp2r2+3Rp4r4cos2θ,εθe=C1+C2r2+ηFvlnr2E+1+μ2E1−λσ01−4μRp2r2+3Rp4r4cos2θ.Using Equation (5), the displacement in the elastic zone can be easily obtained as
(19)ue=C1r+C2r+ηFvrlnr2E+1+μ2E1−λσ0r−4μRp2r+3Rp4r3cos2θ.


### 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(20)εr=εrp+εre−p,εθ=εθp+εθe−p.The displacement differential equation in the plastic zone can be obtained by integrating Equations (5), (7), and (20):
(21)dupdr+βpupr=εre−p+βpεθe−p.With the boundary condition ofup=ue−p at r=Rp, the displacement in the plastic zone can be deduced by solving Equation (21):
(22)up=ue−p−εre−pRpβp+1Rprβp+εre−p+βpεθe−prβp+1.Using Equation (5), the strains in the plastic zone can be obtained as
(23)εrp=βpεre−p−εθe−pβp+1Rprβp+1+εre−p+βpεθe−pβp+1,εθp=εθe−p−εre−pβp+1Rprβp+1+εre−p+βpεθe−pβp+1.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 Figure4), the cohesion at any point in the plastic zone can be expressed as
(24)cp=c0−αεθp−εθe−p=c0−αεθe−p−εre−pβp+1Rprβp+1+εre−p+βpεθe−pβp+1−εθe−p=c0−αεθe−p−εre−pβp+1Rprβp+1−1,where c0 is the initial cohesion and α is the softening coefficient of the cohesion.Figure 4
Softening model of the cohesion.The equilibrium differential equation in the plastic zone can be rewritten by substituting Equations (2), (3), and (24) into Equation (4).
(25)dσrdr+1−Mσrr−4cosφ3−2sinφ⋅c0−αεθe−p−εre−p/βp+1Rp/rβp+1−1r−ηFr=0Combining the boundary condition ofσr=σre−p at r=Rp, the radial stress in the plastic zone can be obtained by solving Equation (25) as
(26)σrp=σre−p−Ne+ηF1−MRpr1−M+4cosφ/3−2sinφαεθe−p−εre−pβp+1βp+M⋅Rpr1+βp−Rpr1−M+4cosφ/3−2sinφαεθe−p−εre−pβp+11−M⋅1−Rpr1−M+Ne+ηF1−M.


### 3.4. Damage Zone

In the damage zone, the total strains of the surrounding rock are also composed of two parts as:(27)εr=εrd+εrp−dεθ=εθd+εθp−dwhere εrp−d and εθp−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):
(28)duddr+βdudr=εrp−d+βdεθp−d.Considering the boundary condition ofud=up−d at r=Rd, the displacement in the plastic zone can be deduced by solving Equation (28):
(29)ud=up−d−εrp−dRdβd+1Rdrβd+εrp−d+βdεθp−drβd+1.Using Equation (5), the strains in the damage zone can be achieved as follows:
(30)εrd=βdεrp−d−εθp−dβp+1Rdrβp+1+εrp−d+βdεθp−dβd+1,εθd=εθp−d−εrp−dβd+1Rdrβd+1+εrp−d+βdεθp−dβd+1.The equilibrium differential equation in the damage zone can be rewritten by substituting Equations (2) and (3) into Equation (4) as
(31)dσrdr+1−Mσr−Ndr−ηFr=0.Combining the boundary condition ofσr=Pi at r=R0, the radial stress in the plastic zone can be obtained by solving Equation (31) as
(32)σrd=pi−Nd+ηF1−MR0r1−M+Nd+ηF1−M,σθd=Mpi−Nd+ηF1−MR0r1−M+MNd+ηF1−M.


### 3.5. Radius of Postpeak Failure Zone

Because of the continuity of radial stress in the surrounding rock of the tunnel, the relationship betweenRp and Rd can be established by combining with (26) and (32).
(33)σre−p−Ne+ηF1−MRpRd1−M+Ne+ηF1−M+4cosφ/3−2sinφαεθe−p−εre−pβp+1βp+M⋅RpRd1−βp−RpRd1−M+4cosφ/3−2sinφαεθe−p−εre−pβp+11−M1−RpRd1−M=pi−Nd+ηF1−MR0Rd1−M+Nd+ηF1−M.According to Equation (24), the cohesion at r=Rd can be expressed as follows:
(34)cd=c0−αεθe−p−εre−pβp+1RpRdβp+1−1.Subsequently, the radiiRp and Rd can be derived by integrating (33) and (34).


## 3.1. Basic Equations

The Mogi-Coulomb criterion can be expressed as [27–29]
(3)σθi=Mσri+Ni,where σθi and σri are the tangential and radial stresses in the “i” region, respectively; M=3+2sinφ/3−2sinφ; φ is the internal friction angle; Ni=4cicosφ/3−2sinφ; and ci 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
(4)dσridr+σri−σθir+ηdPwdr=0,where η is the pore water pressure coefficient.The geometric equation can be written as(5)εri=duidr,εθi=uir,where εri and εθi are the radial and tangential strains in the “i” zone, respectively, and ui represents the displacement in the “i” zone.The constitutive equations can be denoted as(6)εr=1−μ2Eσr−μ1−μσθ,εθ=1−μ2Eσθ−μ1−μσr,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:(7)εri+βiεθi=0,where βi=1+sinψi/1−sinψi; ψi is the dilation angle in the “i” zone.


## 3.2. Elastic Zone

The stress state of the circular tunnel in a nonuniform stress field can be decomposed into two parts (see Figure3). In state I, the tunnel is subjected to a uniform pressure (0.51+λσ0), pore water pressure (Pw), and support pressure (Pi); the differential equation can be obtained by substituting Equations (2), (5), and (6) into Equation (3):
(8)d2ue1dr2+1rdue1dr−ue1r2=FvrE,where F=ηP0/lnR0/Re and v=1+μ1−2μ/1−μ.Figure 3
Mechanical model of surrounding rock in water-rich coal tunnel.Solving Equation (8), the displacement and strains in state I can be easily obtained as
(9)ue1=C1r+C2r+Fvrlnr2E,εre1=C1−C2r2+Fv2Elnr+1,εθe1=C1+C2r2+Fvlnr2E,where C1 and C2 are integral constants.The radial and tangential stresses can be derived by integrating Equations (6) and (9):
(10)σre1=EC11+μ1−2μ−EC21+μr2+ηFlnr21−μ+ηF2,σθe1=EC11+μ1−2μ+EC21+μr2+ηFlnr21−μ+μηF21−μ.Considering the boundary conditionσr=0.51+λσ0+P0 at r=R0 and σr=σre−p at r=Rp, the integral constants can be solved as follows:
(11)C1=ν1-μE121+λσ0+P0+Rp2Re2-Rp2v1-μE121+λσ0+P0-σre-p+ηFv2ERp2Re2-Rp2lnRpRe-ηFv1-μ2E-ηFv2ElnRe,C2=1+μERe2Rp2Re2-Rp2121+λσ0+P0-σre-p+ηF1+μ2E1-μRe2Rp2Re2-Rp2lnRpRe.The stresses in state I can be determined by substituting Equation (11) into Equation (10):
(12)σre1=121+λσ0+P0+ηF21−μlnrRe+Rp2Rp2−Re2Re2r2−1,121+λσ0+P0−σre−p+ηF21−μlnRpRe,σθe1=121+λσ0+P0+ηF2lnr/Re+2μ−11−μ−Rp2Rp2−Re2Re2r2+1,121+λσ0+P0−σre−p+ηF21−μlnRpRe.In state II, the surrounding rock is subjected to a horizontal tension (0.51−λσ0) and a vertical pressure (0.51−λσ0). At r=Rs, the boundary condition can be written as follows:
(13)σx=−0.51−λσ0,σy=0.51−λσ0,τrθ=0.Though coordinate transformation, Equation (13) can be re written as follows:
(14)σr=−0.51−λσ0cos2θτrθ=0.51−λσ0sin2θAtr=R0,σx=τrθ=0. Therefore, using semi-inverse method, the stresses in state II can be deduced as follows:
(15)σre2=−121−λσ01−4Rp2r2+3Rp4r4cos2θ,σθe2=121−λσ01+3Rp4R4cos2θ.Therefore, the stresses in the elastic zone considering pore water pressure can be obtained by superimposing Equations (12) and (15):
(16)σre=121+λσ0+P0+ηF21-μlnrRe+Rp2Rp2-Re2Re2r2-1·121+λσ0+P0-σre-p+ηF21-μlnRpRe-121-λσ01-4Rp2r2+3Rp4r4cos2θ,σθe=121+λσ0+P0+ηF2lnr/Re+2μ-11-μ-Rp2Rp2-Re2Re2r2+1·121+λσ0+P0-σre-p+ηF21-μlnRpRe+121-λσ01+3Rp4r4cos2θ.At the interface between the elastic and plastic zones, the radial and tangential stresses should satisfy Equation (3). Thus, σre−p can be derived by substituting Equation (16) into Equation (3):
(17)σre−p=−2Re21/21+λσ0+P0+ηF/21−μRp2−Re2lnRp/Re+2μ−1−Rp2+Re2lnRp/Re+21−λσ0cos2θRp2−Re2−NeRp2−Re2MRp2−Re2−Rp2−Re2.The radial and tangential strains in the elastic zone can be derived by substituting Equation (14) into Equation (6):
(18)εre=C1−C2r2+ηFv2Elnr+1−1+μ2E1−λσ01−41−μRp2r2+3Rp4r4cos2θ,εθe=C1+C2r2+ηFvlnr2E+1+μ2E1−λσ01−4μRp2r2+3Rp4r4cos2θ.Using Equation (5), the displacement in the elastic zone can be easily obtained as
(19)ue=C1r+C2r+ηFvrlnr2E+1+μ2E1−λσ0r−4μRp2r+3Rp4r3cos2θ.


## 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(20)εr=εrp+εre−p,εθ=εθp+εθe−p.The displacement differential equation in the plastic zone can be obtained by integrating Equations (5), (7), and (20):
(21)dupdr+βpupr=εre−p+βpεθe−p.With the boundary condition ofup=ue−p at r=Rp, the displacement in the plastic zone can be deduced by solving Equation (21):
(22)up=ue−p−εre−pRpβp+1Rprβp+εre−p+βpεθe−prβp+1.Using Equation (5), the strains in the plastic zone can be obtained as
(23)εrp=βpεre−p−εθe−pβp+1Rprβp+1+εre−p+βpεθe−pβp+1,εθp=εθe−p−εre−pβp+1Rprβp+1+εre−p+βpεθe−pβp+1.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 Figure4), the cohesion at any point in the plastic zone can be expressed as
(24)cp=c0−αεθp−εθe−p=c0−αεθe−p−εre−pβp+1Rprβp+1+εre−p+βpεθe−pβp+1−εθe−p=c0−αεθe−p−εre−pβp+1Rprβp+1−1,where c0 is the initial cohesion and α is the softening coefficient of the cohesion.Figure 4
Softening model of the cohesion.The equilibrium differential equation in the plastic zone can be rewritten by substituting Equations (2), (3), and (24) into Equation (4).
(25)dσrdr+1−Mσrr−4cosφ3−2sinφ⋅c0−αεθe−p−εre−p/βp+1Rp/rβp+1−1r−ηFr=0Combining the boundary condition ofσr=σre−p at r=Rp, the radial stress in the plastic zone can be obtained by solving Equation (25) as
(26)σrp=σre−p−Ne+ηF1−MRpr1−M+4cosφ/3−2sinφαεθe−p−εre−pβp+1βp+M⋅Rpr1+βp−Rpr1−M+4cosφ/3−2sinφαεθe−p−εre−pβp+11−M⋅1−Rpr1−M+Ne+ηF1−M.


## 3.4. Damage Zone

In the damage zone, the total strains of the surrounding rock are also composed of two parts as:(27)εr=εrd+εrp−dεθ=εθd+εθp−dwhere εrp−d and εθp−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):
(28)duddr+βdudr=εrp−d+βdεθp−d.Considering the boundary condition ofud=up−d at r=Rd, the displacement in the plastic zone can be deduced by solving Equation (28):
(29)ud=up−d−εrp−dRdβd+1Rdrβd+εrp−d+βdεθp−drβd+1.Using Equation (5), the strains in the damage zone can be achieved as follows:
(30)εrd=βdεrp−d−εθp−dβp+1Rdrβp+1+εrp−d+βdεθp−dβd+1,εθd=εθp−d−εrp−dβd+1Rdrβd+1+εrp−d+βdεθp−dβd+1.The equilibrium differential equation in the damage zone can be rewritten by substituting Equations (2) and (3) into Equation (4) as
(31)dσrdr+1−Mσr−Ndr−ηFr=0.Combining the boundary condition ofσr=Pi at r=R0, the radial stress in the plastic zone can be obtained by solving Equation (31) as
(32)σrd=pi−Nd+ηF1−MR0r1−M+Nd+ηF1−M,σθd=Mpi−Nd+ηF1−MR0r1−M+MNd+ηF1−M.


## 3.5. Radius of Postpeak Failure Zone

Because of the continuity of radial stress in the surrounding rock of the tunnel, the relationship betweenRp and Rd can be established by combining with (26) and (32).
(33)σre−p−Ne+ηF1−MRpRd1−M+Ne+ηF1−M+4cosφ/3−2sinφαεθe−p−εre−pβp+1βp+M⋅RpRd1−βp−RpRd1−M+4cosφ/3−2sinφαεθe−p−εre−pβp+11−M1−RpRd1−M=pi−Nd+ηF1−MR0Rd1−M+Nd+ηF1−M.According to Equation (24), the cohesion at r=Rd can be expressed as follows:
(34)cd=c0−αεθe−p−εre−pβp+1RpRdβp+1−1.Subsequently, the radiiRp and Rd can be derived by integrating (33) and (34).


## 4. 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 Table1.Table 1
Geometrical and mechanical parameters.
ParametersValuesRadius of the tunnel,R0 (m)3Initial stress,σ0 (MPa)16.2Support pressure,Pi (MPa)0Lateral stress coefficient1.5Pore water pressure,P0 (MPa)5Pore water pressure coefficient,η1Re/R012Poisson’s ratio,μ0.25Young’s modulus,E (MPa)2400Initial cohesion,c0 (MPa)3.5Residual cohesion,cd (MPa)1.5Dilation angle,ψi (°)10


### 4.1. Effect of Pore Water Pressure

#### 4.1.1. Postpeak Zone Radii and Surface Displacement

Figure5 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 P0 increases from 3 MPa to 6 MPa, the Rp, Rd, and u0 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 Rp, Rd, and u0 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.Figure 5
Postpeak zone radii and surface displacement around the tunnel under different pore water pressures.
(a)
Rp(b)
Rd(c)
u0


#### 4.1.2. 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 Figure6. 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.Figure 6
Comparison of stress distribution at the tunnel crown based on different pore water pressures.


### 4.2. Effect of Intermediate Principal Stress

#### 4.2.1. Postpeak Zone Radii and Surface Displacement

In order to research the influence of intermediate principal stress on the tunnel deformation, the current analytical results are compared with the data obtained based on the Mohr-Coulomb criterion. As shown in Figure7, because of the neglect of the intermediate principal stress in the Mohr-Coulomb criterion, the Rp, Rd, and u0 values solved by the Mohr-Coulomb criterion are all larger than those solved by the Mogi-Coulomb criterion. For example, the Rp, Rd, and u0 values at the tunnel side from the Mogi-Coulomb criterion are 4.73 m, 4.22 m, and 76.71 mm, respectively; however, the results from the Mohr-Coulomb criterion are 6.83 m, 5.95 m, and 130.69 mm, with an increment of 44.40%, 40.99%, and 70.37%, respectively.Figure 7
Postpeak zone radii and surface displacement around the tunnel under different strength criteria.
(a)
Rp(b)
Rd(c)
u0


#### 4.2.2. Stress Distribution in Tunnel Surrounding Rock

The stress distribution at the tunnel crown based on two different criteria is shown in Figure8. 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 8
Comparison of stress distribution at the tunnel crown based on different strength criteria.


### 4.3. Effect of Residual Cohesion

#### 4.3.1. Postpeak Zone Radii and Surface Displacement

Figure9 shows the postpeak zone radii and surface displacement around the tunnel under different types of residual cohesion. The radii of plastic and damage zones and surface displacement all decrease with the increase of residual cohesion. For example, as cd increases from 1.5 MPa to 2.5 MPa, the Rp, Rd, and u0 values at the tunnel side decrease by 1.34 m, 0.86 m, and 37.39 mm, with a reduction of 24.45%, 18.07%, and 37.59%, respectively, and the Rp, Rd, and u0 values at the tunnel crown increase by 2.41 m and 1.94 m, and 164.56 mm, with a reduction of 32.01%, 28.24%, and 52.63%, respectively. Therefore, some measures, such as grouting, can be used to increase the residual cohesion of the rock mass and reduce the deformation of the tunnel.Figure 9
Postpeak zone radii and surface displacement around the tunnel under different types of residual cohesion.
(a)
Rp(b)
Rd(c)
u0


#### 4.3.2. Stress Distribution in Tunnel Surrounding Rock

The stress distribution based on different types of residual cohesion is shown in Figure10. 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, the maximum tangential stress increases slightly, but the boundary between the plastic and elastic zones moves farther away from the center of the tunnel.Figure 10
Comparison of stress distribution at the tunnel crown based on different types of residual cohesion.


## 4.1. Effect of Pore Water Pressure

### 4.1.1. Postpeak Zone Radii and Surface Displacement

Figure5 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 P0 increases from 3 MPa to 6 MPa, the Rp, Rd, and u0 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 Rp, Rd, and u0 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.Figure 5
Postpeak zone radii and surface displacement around the tunnel under different pore water pressures.
(a)
Rp(b)
Rd(c)
u0


### 4.1.2. 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 Figure6. 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.Figure 6
Comparison of stress distribution at the tunnel crown based on different pore water pressures.


## 4.1.1. Postpeak Zone Radii and Surface Displacement

Figure5 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 P0 increases from 3 MPa to 6 MPa, the Rp, Rd, and u0 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 Rp, Rd, and u0 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.Figure 5
Postpeak zone radii and surface displacement around the tunnel under different pore water pressures.
(a)
Rp(b)
Rd(c)
u0


## 4.1.2. 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 Figure6. 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.Figure 6
Comparison of stress distribution at the tunnel crown based on different pore water pressures.


## 4.2. Effect of Intermediate Principal Stress

### 4.2.1. Postpeak Zone Radii and Surface Displacement

In order to research the influence of intermediate principal stress on the tunnel deformation, the current analytical results are compared with the data obtained based on the Mohr-Coulomb criterion. As shown in Figure7, because of the neglect of the intermediate principal stress in the Mohr-Coulomb criterion, the Rp, Rd, and u0 values solved by the Mohr-Coulomb criterion are all larger than those solved by the Mogi-Coulomb criterion. For example, the Rp, Rd, and u0 values at the tunnel side from the Mogi-Coulomb criterion are 4.73 m, 4.22 m, and 76.71 mm, respectively; however, the results from the Mohr-Coulomb criterion are 6.83 m, 5.95 m, and 130.69 mm, with an increment of 44.40%, 40.99%, and 70.37%, respectively.Figure 7
Postpeak zone radii and surface displacement around the tunnel under different strength criteria.
(a)
Rp(b)
Rd(c)
u0


### 4.2.2. Stress Distribution in Tunnel Surrounding Rock

The stress distribution at the tunnel crown based on two different criteria is shown in Figure8. 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 8
Comparison of stress distribution at the tunnel crown based on different strength criteria.


## 4.2.1. Postpeak Zone Radii and Surface Displacement

In order to research the influence of intermediate principal stress on the tunnel deformation, the current analytical results are compared with the data obtained based on the Mohr-Coulomb criterion. As shown in Figure7, because of the neglect of the intermediate principal stress in the Mohr-Coulomb criterion, the Rp, Rd, and u0 values solved by the Mohr-Coulomb criterion are all larger than those solved by the Mogi-Coulomb criterion. For example, the Rp, Rd, and u0 values at the tunnel side from the Mogi-Coulomb criterion are 4.73 m, 4.22 m, and 76.71 mm, respectively; however, the results from the Mohr-Coulomb criterion are 6.83 m, 5.95 m, and 130.69 mm, with an increment of 44.40%, 40.99%, and 70.37%, respectively.Figure 7
Postpeak zone radii and surface displacement around the tunnel under different strength criteria.
(a)
Rp(b)
Rd(c)
u0


## 4.2.2. Stress Distribution in Tunnel Surrounding Rock

The stress distribution at the tunnel crown based on two different criteria is shown in Figure8. 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 8
Comparison of stress distribution at the tunnel crown based on different strength criteria.


## 4.3. Effect of Residual Cohesion

### 4.3.1. Postpeak Zone Radii and Surface Displacement

Figure9 shows the postpeak zone radii and surface displacement around the tunnel under different types of residual cohesion. The radii of plastic and damage zones and surface displacement all decrease with the increase of residual cohesion. For example, as cd increases from 1.5 MPa to 2.5 MPa, the Rp, Rd, and u0 values at the tunnel side decrease by 1.34 m, 0.86 m, and 37.39 mm, with a reduction of 24.45%, 18.07%, and 37.59%, respectively, and the Rp, Rd, and u0 values at the tunnel crown increase by 2.41 m and 1.94 m, and 164.56 mm, with a reduction of 32.01%, 28.24%, and 52.63%, respectively. Therefore, some measures, such as grouting, can be used to increase the residual cohesion of the rock mass and reduce the deformation of the tunnel.Figure 9
Postpeak zone radii and surface displacement around the tunnel under different types of residual cohesion.
(a)
Rp(b)
Rd(c)
u0


### 4.3.2. Stress Distribution in Tunnel Surrounding Rock

The stress distribution based on different types of residual cohesion is shown in Figure10. 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, the maximum tangential stress increases slightly, but the boundary between the plastic and elastic zones moves farther away from the center of the tunnel.Figure 10
Comparison of stress distribution at the tunnel crown based on different types of residual cohesion.


## 4.3.1. Postpeak Zone Radii and Surface Displacement

Figure9 shows the postpeak zone radii and surface displacement around the tunnel under different types of residual cohesion. The radii of plastic and damage zones and surface displacement all decrease with the increase of residual cohesion. For example, as cd increases from 1.5 MPa to 2.5 MPa, the Rp, Rd, and u0 values at the tunnel side decrease by 1.34 m, 0.86 m, and 37.39 mm, with a reduction of 24.45%, 18.07%, and 37.59%, respectively, and the Rp, Rd, and u0 values at the tunnel crown increase by 2.41 m and 1.94 m, and 164.56 mm, with a reduction of 32.01%, 28.24%, and 52.63%, respectively. Therefore, some measures, such as grouting, can be used to increase the residual cohesion of the rock mass and reduce the deformation of the tunnel.Figure 9
Postpeak zone radii and surface displacement around the tunnel under different types of residual cohesion.
(a)
Rp(b)
Rd(c)
u0


## 4.3.2. Stress Distribution in Tunnel Surrounding Rock

The stress distribution based on different types of residual cohesion is shown in Figure10. 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, the maximum tangential stress increases slightly, but the boundary between the plastic and elastic zones moves farther away from the center of the tunnel.Figure 10
Comparison of stress distribution at the tunnel crown based on different types of residual cohesion.


## 5. 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 strain-softening 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 presents a trend of first increasing and then decreasing, and the maximum tangential stress appears at the interface between the elastic and plastic zones(2)
The postpeak zone radii and surface displacement increase with the increasing pore water pressure and decreasing residual cohesion. The greater the pore water pressure, the farther the maximum tangential stress is from the center of the tunnel. Residual strength has little effect on the maximum tangential stress(3)
Because of the neglect of 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. Therefore, opportune consideration of the intermediate principal stress can lead to a more reasonable tunnel support design



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*Source: 6696757-2021-02-20.xml*