J. Cent. South Univ. (2019) 26: 1820-1829

DOI: https://doi.org/10.1007/s11771-019-4136-1

Catastrophe analysis of deep tunnel above water-filled caves

ZHANG Rui(张睿)

Department of Civil and Structural Engineering, University of Sheffield, UK

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: To explore the influence of karst cavity pressure on the failure mechanisms of rock layers above water-filled caves, novel blow-out and collapse mechanisms are put forward in this study. The proposed method uses the nonlinear optimization to obtain the failure profiles of surrounding layered rock with water-filled cave at the bottom of the tunnel. By referring to the functional catastrophe theory, stability analysis with different properties in different rock layers is implemented with considering the incorporation of seepage forces since the groundwater cannot be ignored in the catastrophe analysis of deep tunnel bottom. Also the parametric analysis is implemented to discuss the influences of different rock strength factors on the failure profiles. In order to offer a good guide of design for the excavation of deep tunnels above the water-filled caves, the proposed method is applied to design of the minimum effective height for rock layer. The results obtained by this work agree well with the existing published ones.

Key words: concealed karst cavity; collapse mechanism; blow-out failure mechanism; functional catastrophe; layered rock mass

Cite this article as: ZHANG Rui. Catastrophe analysis of deep tunnel above water-filled caves [J]. Journal of Central South University, 2019, 26(7): 1820-1829. DOI: https://doi.org/10.1007/s11771-019-4136-1.

1 Introduction

Karst disaster is a major threat to the stability of mountain tunnels. Geological disasters such as water inrush from karst tunnels pose great threats to tunnel construction and operation safety, and have attracted widespread attention. How to reduce the karst damage of tunnels by using appropriate karst treatment measures for different types of karst tunnels under different geological conditions is a major problem to be solved. Most of the mountainous areas in western part of China are carbonate rock formations, and the climate is humid, resulting in the formation of karst depressions on the surface. The karst structures such as caves and underground rivers in the mountains are developed. The existence of a large number of karst caves can easily lead to large-scale geological disasters in the mountain areas under tunneling disturbances, posing a serious threat to construction safety. This failure leads to economical loss. Some scholars have focused on the stability of geotechnical engineering [1-3]. Especially, during the excavation of the deep tunnels, the collapse would happen because of the cavity fillers pressure when the rock layer is thin [4].

It is very difficult to obtain the accurate mathematical solutions for describing the geomechanical changes in nonlinear plastic media, especially, the geomechanical properties of rock mass are very complex during the collapse process of the surrounding rock around the water-filled cavity roof. The functional catastrophe theory is a useful tool to predict the most possible failure profiles of rock layers because of the employed total potential in the form of a function with variable. The catastrophe theory has been widely used to tackle the practical engineering in the construction of deep tunnels. Seepage forces cannot be ignored for the failure analysis of the rock layer between bottom of the tunnel and karst cavity roof. In this study, for investigating the effects of groundwater in karst area, the pore-pressure coefficient is incorporated into the upper bound formulation to derive the failure surface in collapse mechanism and blow-out mechanism, as shown in Figure 1. The surrounding layered rock mass around karst cave is also considered because of the field observations. By introducing a novel attribute synthetic evaluation system, YANG et al [5] assessed the stability of karst tunnels under water inrush. AUGARDE et al [6] combined the numerical approach and limit analysis theorem to obtain the limit loads of sinkholes. ERIC et al [7] developed a dimensionless stability chart and the geometric study for potential voids in the residual soil. According to a large amount of tests, the stress-strain behavior tends to be nonlinear. Therefore, the generalized NL failure form is adopted in this study. Based on the published works which introduce the functional catastrophe theory, this work used upper bound theory to derive the failure profile of the rock layer with the kinematic approach, which has similarity to the collapse mechanism of shallow tunnel [8].

Figure 1 Failure profiles of layered rock mass:

On the basis of the combination of functional catastrophe theory and limit analysis theory, new collapse and blow-out mechanisms are put forward in this study. By the nonlinear optimization, failure surface of rock layer between bottom of deep tunnel and water-filled cavity roof are derived. The kinematic analysis of failure pattern is implemented with considering the incorporation of seepage forces since the effects of groundwater cannot be ignored in the catastrophe analysis of deep tunnel. The parametric study is implemented to discuss the influences of different rock parameters on the failure profiles. Lastly, the proposed method is applied to the design of the minimum safety thickness for rock layer.

2 General method of analysis

2.1 NL failure form

Based on the previous work [9-15], it is necessary to construct velocity field for upper bound analysis. According to a large amount of field tests, the nonlinearity of geotechnical media is obvious by observations. In this study, in order to describe the arch collapse profiles of the rock layers, the nonlinear NL yield rule is adopted [16, 17], which gives

                         (1)

where τ_n and σ_n are shear and normal stress, respectively; A is shear strength coefficient; P_a is atmospheric pressure; T represents a non- dimensional tensile strength;is the tensile strength; n stands for the nonlinear coefficient.

2.2 Case study for circular cavity

In the practical engineering, the shape profiles of numerous concealed karst cave always tend to be circular. Therefore, this study adopts the circular cavity since it is similar to many possible cavity profiles. This work adopts two-layer rock mass to model layered rock mass. With reference to Figure 2, the curve of failure block is made up of two functions f₁(x) and f₂(x). H₁ is the distance from the layered position to the top of cavity and H₂ stands for distance from tunnel base to the layered position. In Figure 3, g₁(x) and g₂(x) represent the failure curves in different layers. H₃ and H₄ stand for the height from tunnel base to the layered position and the layered position to the top of cavity, respectively.

The mathematical form for the profile of the karst cavity can be,

             (2a)

             (2b)

Although the shape profiles of concealed karst cave beneath deep tunnel have many different possibilities, such as rectangular shape and irregular shape. For convenience, only the case for circular cavity is adopted in this work. The conclusions for this case have many similarities to other possible shapes of concealed karst cavity.

Figure 2 Blow-out failure profiles of rock layers

Figure 3 Collapse failure profiles of rock layers

3 Blow-out mechanism and collapse mechanism with seepage forces

3.1 Blow-out mechanism

Based on the associative flow rule, the plastic potential, ξ, could be expressed with referring to NL yield rule,

                      (3)

Based on the previous work [18-21], the normal and shear stresses at any point on the failure slip-line can be expressed,

                  (4)

                       (5)

in which

Considering the application of Greenberg minimum principle, the potential profile of failure blocks can be found, the internal dissipation density is,

                      (6)

whereis normal plastic strain rate;is shear plastic strain rate; w is the thickness of the plastic detaching zone; and is the velocity of the failure pattern. The energy dissipation can be obtained:

    (7)

where L₁ characterizes the half-width of failure blocks at layered position; L₂ and L_t are the half-widths of failure pattern on the tunnel base and the cavity roof, respectively. The subscripts of 1 and 2 of soil’s parameters A, n, T and ρ represent the lower and upper rock layers, respectively. The body force work rate is

 (8)

The work rate of cavity pressure q is

                         (9)

With considering the effects of seepage forces, the pore pressure coefficient is expressed as,

                               (10)

where z stands for the distance between bottom of deep tunnel and top of the cavity. Therefore, work rate caused by seepage force is,

                      (11)

Objective function should be established with equilibrium of the energy dissipation and power of external forces:

      (12)

where

                     (13)

                   (14)

The extremum of objective function Λ can be obtained when the extreme values of two functions ψ₁ and ψ₂ are obtained simultaneously. The Euler- Lagrange equations could be transferred to:

             (15)

             (16)

The expressions of the detaching curve f₁(x) and f₂(x) can be derived by integral calculation.

                  (17)

                  (18)

where a₁, a₂, a₃ and a₄ are constants determined by boundary conditions.

                           (19)

Based on the stress boundary condition, the following equation is obtained.

                         (20)

Then,and a₄=0 from above. Thus, the expression of failure block can be written as follows:

                     (21)

Based on the geometric conditions, the equations,should be satisfied. Another geometrical relationship should be satisfied. Thus, the expressions of a₁ and a₂ can be obtained,and in which

              (22)

The function of failure block in lower layer results.

         (23)

According to the requirement of upper bound theory, an equation can be obtained by equating the internal work rate to the external work rate, let the function equal to zero,

              (24)

Based on the geometrical conditions, it results in,

                     (25)

                        (26)

Submitting Eqs. (25) and (26) into Eq.(24), L₁, L₂ and L_t can be solved by using numerical process. Based on L₁, L₂ and L_t, the curves f₁(x) and f₂(x) can be derived.

3.2 Analogous derivation for collapse mechanism

For the derivation of expression of target function in active mode, by considering supporting force, Eq. (12) takes the form:

       (27)

in which

;

 (28)

L₄ characterizes the collapsing width of upper and lower blocks; L₃ and L′_t are the half-widths on the tunnel base and the failure distance of cave roof, respectively. The subscripts of 3 and 4 of soil’s parameters A, n, T and ρ represent the upper and lower soil layers.

As shown in Figure 3, the failure slip-line is required to satisfy the geometric and stress boundary conditions:

,

,

,

,

                     (29)

With mathematical calculations, the analytical expression of failure surface for collapse mechanism is:

,

        (30)

in which

,

,

                  (31)

Incorporatinginto equation , the values of L₃, L₄ and L_t can be obtained by numerical algorithm using MATLAB. Based on L₃, L₄ and L_t, the final forms of failure slip-lines g₁(x) and g₂(x) could be found.

4 Functional catastrophe analysis

Because of the fact that the functional catastrophe theory can describe complex nonlinear dynamic system with a few of control variables, many scholars adopt functional catastrophe theory to conduct the progressive collapse analysis for tunneling construction. Based on the precious work [22], this study put forward the potential function,

             (32)

By expanding the increment of function to a two-variable Taylor series in small perturbations of y. It results in:

      (33)

The specific boundary conditions are required based on the non-Morse critical point,

                     (34)

where Df and det(Hf) stand for the gradient and determinant of the Hesse matrix of potential function f(x₁, x₂), respectively.

The formulations of function J[y] result in,

   (35)

The catastrophic function J[y] can be expressed below,

  (36)

Combining Eqs. (23), (25) and (32) into Eq. (36), the constraints for the variables should be given,

 (37)

where i=1, 2, 3, 4. The value of n_i in Eq. (37) must be 0.5 when the x could be set any values, which agrees well with previous published work [22, 23].

5 Sensitivity analysis and discussions

5.1 Effects of parameters

During the construction of deep tunnel above the concealed karst cavity, the effects of geomechanical properties should be considered, the parametric study is conducted for the rock properties corresponding to A=0.5-0.9, r_u=0, and ρ=22-25 kN/m³, as shown in Figures 4 and 5.

From the figure, the changing rule could be summarized as: the scope of the failure pattern increases with the decrease of A both in blow-out mechanism and collapse mechanism. And the scope of the failure pattern increases with increase of ρ both in blow-out mechanism and collapse mechanism. From the perspective of engineering, the range of the failure block of the rock layer could be reduced by increasing the values of A and decreasing the value of ρ.

5.2 Effects of seepage forces

During the construction of deep tunnel above the concealed karst cavity, the effects of seepage forces should be considered, the parametric study is conducted for the rock properties corresponding to A=0.9, r_u=0.0-0.3, P_a=100 kPa and ρ=25 kN/m³,as shown in Figure 6. For convenience, the parameter r_u is introduced to describe the effects of seepage force on failure scope of layered rock mass in collapse and blow-out failure mode.

Figure 4 Predictions for blow-out failure scope of rock layer:

Figure 5 Predictions for collapse failure scope of rock layer:

From the figure, the changing rule could be summarized as: the failure scope decreases with decrease of pore-pressure coefficient r_u both in blow-out mechanism and collapse mechanism, which agrees well with the observations in the practical engineering. From the perspective of engineering, the range of the failure block of the rock layer could be reduced by decreasing the values of r_u.

Figure 6 Effects of pore-pressure coefficient on failure scope of rock layer:

6 Minimum safety thickness

During the tunneling construction process, the design of effective height between tunnel base and cavity roof for the sake of protections for the rock layers around tunnel is significant for the drilling and blasting method used in deep tunnel excavation. According to previous work which defined the effective height between tunnel base and cavity roof to avoid the downward movements of the collapsing rock layer around the tunnel, the work proposed a different approach to predict the effective height between tunnel base and cavity roof.

From the perspective of practical engineering, the upper width of the failure block decreases with the increase of the height between the bottom of tunnel and the cavity roof in active collapse failure mechanism. The effective height H_cr is reached, L₁ will be zero. According to Table 1, the comparison in the effective height between tunnel base and cavity roof is made between this work and previous published work, which shows a good consistency.

Table 1 Comparison in minimum safety thickness

7 Conclusions

In this study, both blow-out and collapse shapes of rock mass above the karst cavity are put forward when the surrounding rock cannot bear the pressure caused by the water. Within the framework of upper bound theory, a new convenient way to include seepage effects is also presented and implemented, which can be used to explain the changing laws for scopes of failure profiles with varying pore pressure coefficients. The main research findings include:

1) The rock strength factors A and γ have influences on the failure profile. The scope of the failure pattern decreases with the increase of A both in blow-out mechanism and collapse mechanism. And the scope of the failure pattern increases with increase of ρ both in blow-out mechanism and collapse mechanism.

2) Seepage forces could not be ignored during the failure analysis of the deep tunnel constructed in the karst areas. The failure scope of rock layer decreases with decrease of pore-pressure coefficient r_u both in blow-out mechanism and in collapse mechanism, which agrees well with the observations in the practical engineering.

3) The procedure for predicting the effective height between tunnel base and cavity roof to avoid the failure of rock layers above the karst cavity is proposed, and the results obtained show good agreements with those of previous work well.

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(Edited by HE Yun-bin)

中文导读

深埋隧道隐伏岩溶洞穴的灾变分析

摘要：为了探讨岩溶洞穴压力对充水洞穴中隧道基岩围岩稳定性的影响，本研究提出了新的岩层坍塌和爆破机制。该方法利用非线性优化方法得到隧道底部与充水洞顶之间层状围岩的破坏面，并对隧道底部充水洞穴围岩破坏过程的预测进行了数值优化。因为地下水在岩层破坏中起着关键作用，基于功能突变理论，本文提出的方法结合渗流力作用对层状围岩破坏模式进行了运动学分析。本文同时进行参数研究以分析不同岩石参数对破坏形状的影响。该方法为充水洞穴上方的深埋隧道开挖提供良好的设计指导，可应用于岩层的最小安全厚度设计。这项工作获得的结果与现有的已发表的结果很好地吻合。

关键词：隐伏岩溶洞穴；坍塌破坏机制；爆破破坏机制；功能突变理论；层状岩石

Received date: 2019-04-04; Accepted date: 2019-05-14

Corresponding author: ZHANG Rui, PhD; Tel: +86-18538707962; E-mail: rzhang32@sheffield.ac.uk; ORCID: 0000-0002-1196-648X