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Theoretical Analysis of Safety Thickness of theWater-Resistant Rock Mass of Karst Tunnel FaceTaking Into Account Seepage EffectJiaqi GUO
Henan polytechnic university https://orcid.org/0000-0002-0093-7923Wenlong Wu ( [email protected] )
Henan Polytechnic UniversityXiliang Liu
Henan Polytechnic UniversityXin Huang
Henan Polytechnic UniversityZhengguo Zhu
Shijiazhuang Tiedao University
Research Article
Keywords: Karst tunnel, Water-resistant rock mass, Safety thickness, Upper limit analysis, Seepage effect
Posted Date: April 19th, 2021
DOI: https://doi.org/10.21203/rs.3.rs-384740/v1
License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License
Version of Record: A version of this preprint was published at Geotechnical and Geological Engineeringon July 8th, 2021. See the published version at https://doi.org/10.1007/s10706-021-01916-7.
Theoretical analysis of safety thickness of the water-1
resistant rock mass of karst tunnel face taking into 2
account seepage effect 3
4
Jiaqi Guo1, 2, 3, Wenlong Wu1, 2*, Xiliang Liu1, 2, Xin Huang1, 2, 3, Zhengguo Zhu3 5
1 School of Civil Engineering, Henan Polytechnic University, Jiaozuo 454000, Henan, China 6
2 Key Laboratory of Henan Province for Underground Engineering and Disaster Prevention, Jiaozuo 454000, Henan, 7
China 8
3 State Key Laboratory of Mechanical Behavior and System Safety Traffic Engineering Structures, Shijiazhuang 9
Tiedao University, Shijiazhuang 050043, Hebei, China 10
*Correspondence e-mail: [email protected] 11
12
Abstract: This paper took into account the adverse influence of the karst water seepage effect on 13
the water-resistant rock mass. Based on the upper-bound theorem of limit analysis and the Hoek-14
Brown failure criterion, through a series of formula derivation, the expression of critical safety 15
thickness of water-resistant rock mass of karst tunnel face was finally obtained. The paper carried 16
out a feasibility analysis, an analysis of influencing factors and a comparative analysis with previous 17
related research achievements of this method. The results showed that: (1) With the decrease of 18
surrounding rock grade, the safety thickness of water-resistant rock mass gradually increased, and 19
the safety thickness of surrounding rock at all grades remained within a reasonable range. (2) The 20
safety thickness decreased as the compressive strength, the tensile strength and parameter A 21
increased, and it increased as the karst water pressure, the tunnel excavation height, and parameter 22
B increased. (3) The change trend of the safety thickness with the influencing factors was completely 23
consistent under the two conditions of considering and without the seepage effect, and the safety 24
thickness with considering the seepage force was greater than that without considering the seepage 25
force. Taking the Yunwushan tunnel of Yiwan railway as an example, the critical safety thickness of 26
the water-resistant rock mass was calculated and the calculated value was in good coincidence with 27
the safety thickness adopted in the actual project. The research results are of great significance to 28
prevent the occurrence of high pressure filling karst geological disasters such as water inrush in 29
tunnels. 30
Keywords: Karst tunnel; Water-resistant rock mass; Safety thickness; Upper limit analysis; Seepage 31
effect 32
33
1 Introduction 34
China has a vast territory, 70% of which are mountainous areas. The geographical and 35
geological conditions are very complex. It is the country with the widest distribution of karst in the 36
world. With the rapid development of highway, railway and other transportation infrastructure 37
construction in southwest China, karst tunnel water inrush disasters are increasingly frequent. Water 38
pressure filling cavity and other karst structural forms are often developed in front of karst tunnel 39
face. In tunnel engineering, when the excavation reaches a certain thickness behind the high-40
pressure water-rich karst cavity of the tunnel face, if the construction is not stopped or measures 41
such as pre-reinforcement are not taken, the combined effects of the hydraulic pressure and the 42
excavation disturbance likely will cause the water-resistant rock mass of the tunnel face to collapse, 43
resulting in a water-inrush disaster (Guo et al. 2017). Therefore, the safety thickness of the water-44
resistant rock mass of face is an important technical parameter to ensure the safe construction of 45
karst tunnel. 46
Scholars at home and abroad have calculated the safety thickness of water-resistant rock mass 47
with various methods and given the corresponding calculation formulas. Li et al. (2015) deduced 48
the calculation formula of minimum safety thickness of cracked rock mass under blasting excavation 49
disturbance and water pressure, which was verified by engineering examples. Zhang et al. (2018) 50
proposed the calculation method of minimum safety thickness of stable rock mass of tunnel face by 51
combining theoretical calculation, numerical simulation and engineering practice. Based on two 52
band theory and critical water pressure, Guo et al. (2019) established the safety thickness of water-53
resistant strata with multi fractures ahead of karst tunnel. Meng et al. (2020) established models of 54
water inrush in fault fracture zone and obtained the calculation formula of minimum safety thickness. 55
These achievements are mostly based on specific engineering cases, and each research direction has 56
its own emphasis, different mechanical models, influencing factors, and judgment standards. In 57
recent years, the limit analysis principle has been widely used in solving the problem of the safety 58
thickness of water-resistant rock mass of tunnel face. By means of the upper bound theorem and 59
variation principle, Yang et al. (2016) deduced the calculation processes of rock plug thickness and 60
obtained the expression of minimum safe thickness. On the basis of a plane failure pattern of rock 61
plug, Yang et al. (2017) employed the limit analysis principle and derived the expressions of 62
detaching curve and rock plug thickness. None of the above studies considered the adverse influence 63
of karst water seepage effect on water-resistant rock mass. 64
On the basis of the upper-bound theorem of limit analysis and the Hoek-Brown failure criterion, 65
we derived and obtained the expression of the critical safety thickness of the water-resistant rock 66
mass of karst tunnel face considering the adverse influence of karst water seepage effect on the 67
water-resistant rock mass. The feasibility of the application of the method was analysed, the analysis 68
of influencing factors affecting the safety thickness of water-resistant rock mass was executed, and 69
the comparative analysis was carried out with the previous related research achievements. We took 70
the Yunwushan tunnel of Yiwan railway as an example for an engineering application to verify the 71
rationality and practicability of the method proposed in this paper. The research results are of great 72
significance to prevent the occurrence of high pressure filling karst geological disasters such as 73
water inrush in tunnels. 74
75
2 Basic principles 76
2.1 Upper limit theorem of the limit analysis considering the seepage effect 77
The upper limit theorem can be stated as follows (Chen 2013): If the assumed maneuvers 78
allowable velocity field satisfies the displacement boundary conditions, the load determined by 79
equating the internal energy dissipation rate to the external work rate must be greater than or equal 80
to the actual load in the limit state. When considering the adverse influence of the seepage effect on 81
the water-resistant rock mass of a tunnel face, the seepage force is regarded as the external force 82
acting on the rock and the soil skeleton. Therefore, the mathematical expression of the upper limit 83
theorem is 84
(1) 85
where D is the internal energy dissipation rate of the plastic strain rate vector; Ti and Fi are the 86
area force and volume force, respectively; vi is the velocity vector in the maneuvering velocity field; 87
S and Ω are the surface area and volume of the research object, respectively; and fs is the seepage 88
force. 89
2.2 Hoek-Brown nonlinear failure criterion 90
Since Hoek and Brown proposed the Hoek-Brown nonlinear failure criterion in 1980 (Hoek 91
and Brown 1980a, b), their criterion has been widely used by scholars in the industry, and the Hoek-92
Brown failure criterion has been demonstrated to be an appropriate yield function for evaluating the 93
s( )d d d dij i i i i i
SD Tv S Fv f v
≥
( )ij
strength and stability of rock masses in karst regions (Huang et al. 2017). The two classical 94
expressions of the Hoek-Brown failure criterion (H-B criterion) (Hoek and Brown 1997) are 95
expressed (1) using the large principal stress and the small principal stress, and (2) using the normal 96
stress and the shear stress. The normal stress and shear stress parameters of the unit body should be 97
considered when calculating the internal energy dissipation. Thus, it is convenient to use the second 98
form of the Hoek-Brown failure criterion. Its mathematical expression is as follows: 99
(2) 100
where τn and σn are the normal stress and shear stress, respectively; and the parameters A and B are 101
dimensionless material constants that can be obtained through triaxial compression tests. When B = 102
1, the H-B criterion becomes the M-C linear yield criterion. σci is the uniaxial compressive strength 103
of the intact rock, which can be obtained through experiments. σtm is the tensile strength of the rock 104
mass, and its value can be calculated by the following formula: 105
(3) 106
where mi is a parameter reflecting the degree of hardness and softness of rock; GSI is a geological 107
strength index, Di is the disturbance parameter of jointed rock mass. 108
109
3 Critical safety thickness of the water-resistant rock mass of karst 110
tunnel face 111
3.1 Failure mode of the water-resistant rock mass 112
In this paper, we established the failure mode of the water-resistant rock mass (Fig. 1) 113
according to the model of Fraldi and Guarracino (2009). 114
115
116
In Fig. 1, D is the height of the excavated tunnel section; w is the separation layer thickness 117
when the water-resistant rock mass is broken; S is the safety thickness of the water-resistant rock 118
mass, which is an unknown quantity to be determined according to the limit analysis theory; fs is 119
the seepage force on the rock mass; p is the karst cavity pressure, which is equal to the hydrostatic 120
pressure pw; and L is the effective range of the karst cavity pressure. In addition, it is assumed that 121
the karst cavity pressure is uniformly distributed perpendicular to the surface of the water-resistant 122
rock mass, and the stress surface is simplified into a plane, that is, the dotted line in Fig. 1(a) 123
n tmn ci
ci
B
A
2
citm i i
i i i
100 100 100exp exp 4exp
2 28 14 28 14 9 3
B
GSI GSI GSIm m
D D D
Fig. 1 Failure mode of the water-resistant rock mass
coincides with the x-axis. 124
As shown in Fig. 1, the water-resistant rock mass of the tunnel face exhibits an overall 125
squeezing failure mode at a speed v under various external forces, and the failure surface is 126
composed of two separation curves f(x) and f(-x), which are symmetric about the y-axis. To simplify 127
the calculation, we selected the water-resistant rock mass in the positive direction of the x-axis as 128
the research object. We established a local coordinate system s-n (Fig. 1(b)) on the micro-unit of 129
the separation layer with the tangent and the outer normal as the positive directions. Thus, the 130
algebraic relationship between the angle θ and f '(x) is as follows: 131
(4) 132
3.2 Calculation of the critical safety thickness of the water-resistant rock mass 133
3.2.1 Objective function 134
As shown in Fig. 1(b), the micro-unit moves from the dotted line (state 1) to the solid line 135
(state 2) with speed v, and the separation layer exhibits shear deformation along the tangential 136
direction and tensile deformation along the normal direction. Therefore, the energy dissipation rate 137
of the micro-unit can be expressed as follows: 138
n n( ) ( )ij
D l (5) 139
where l is the area of the corresponding separation layer, which has a value of w; and are the 140
positive and tangential strain rates, respectively. According to the definition of strain rate and the 141
trigonometric function, the mathematical expressions for and are as follows: 142
(6) 143
where vn and vs are the normal velocity and tangential velocity of the micro-unit respectively. In 144
addition, soil mechanics stipulates that when shear failure occurs in geotechnical materials, the 145
compressive stress is positive and the tensile stress is negative. When the tensile deformation of the 146
separation layer occurs along the normal direction, the value is considered to be negative. 147
On the basis of the aforementioned basic theorem, in this paper, we used the H-B failure 148
criterion and considered the influence of the correlation flow rule. This rule not only meets the basic 149
assumption of the limit analysis but also ensures that the yield surface coincides with the plastic 150
potential surface. Thus, the plastic potential function g(σn, τn) can be expressed by the yield function 151
1
2 2
1
2 2
sin ( )[1 ( ) ]
cos [1 ( ) ]
f f
f
x x
x
1
n
1
s
2 2
2 2
[1 ( ) ]
( )[1 ( ) ]
vx
l w
vx x
l
vf
vf
wf
f(σn, τn) when the material yields, and its mathematical expression is 152
(7) 153
According to the orthogonal flow law, the normal strain rate and the tangential strain rate 154
can also be expressed as 155
1
n tm
ci
n
n
B
gAB
g
(8) 156
where λ is the plastic ratio coefficient greater than 0. The expression for the normal stress σn can be 157
obtained by eliminating λ and w in simultaneous Eqs. (6) and (8). 158
(9) 159
By substituting Eqs. (3), (6), and (9) into Eq. (5), the internal energy dissipation rate of the 160
micro-unit on the separation curve f(x) can be obtained, as follows: 161
1 1
2 12 1n tm cin( ) ( ) [1 ( ) ] (1 ) ( ) B
ijD l v x B xf A fB
(10) 162
According to the lateral area integration formula, Eq. (10) was integrated along the total 163
separation line, and the internal energy dissipation rate of the water-resistant rock mass was obtained: 164
1/2
11
tm ci/2
( )d (1 ) ( ) dD
BD ij
LW D B AB xf x v
(11) 165
Then, we calculated the work power of the external force. Under this failure mode, the external 166
load can be divided into the karst cavity pressure, the seepage force, and the rock mass gravity. The 167
karst cavity pressure is the surface force, and the work power can be expressed as 168
w w
1 1d
2 2p i i
SW T v S pvL p vL (12) 169
The gravity of the rock mass is the volume force, and it is always perpendicular to the 170
movement direction of the rock mass. Therefore, throughout the failure process, the work done by 171
gravity is equal to 0, that is 172
(13) 173
The seepage of water occurs in the pores of the water-resistant rock mass. Under the action of 174
the water head difference, the groundwater flows through the pores of the water-resistant rock mass. 175
Generally, the horizontal component of the tunnel face seepage force is far greater than the vertical 176
component, so we considered only the influence of the horizontal component of the seepage force 177
nn tm
n n n n ci
ci
( )= ( )
B
Ag f
, ,
1
n tm1
ci ( ) BA xfB
d 0i i
W Fv
on the stability of the tunnel face (Lee and Nam 2001). The seepage force of the rock and soil mass 178
per unit volume along the streamline direction is 179
s w w
hf i
S
(14) 180
where γw is the volumetric weight of the groundwater; i is the hydraulic gradient; ∆h is the water 181
head difference; and S is the seepage length, that is, the safety thickness of the water-resistant rock 182
mass. For a rock and soil mass with volume V, the seepage force along the streamline direction is 183
(15) 184
Therefore, the power of the work done by the internal seepage force of water-resistant rock 185
mass is 186
s
/2
s w/2
d ( ) dH
f iL
W f v i f x v x
(16) 187
By substituting Eqs. (11), (12), (13), and (16) into Eq. (1), the objective function, including 188
the separation curve f(x), is obtained 189
1/21
1tm ci w
/2
/2 /2
w tm w/2 /2
1( ) (1 ) ( )( ) d
2
1 1( ) d ( ) d )) (
2(
2
DB
L
D D
L L
f x x B AB x v x p vLf x f
fi f x v x f x x v x v D L p vx L
, ,
, ,
(17) 190
Where 191
(18) 192
3.2.2 Separation curve equation 193
As evident from Eqs. (16) and (17), the extreme value of the objective function ξ is completely 194
determined by the function φ. Therefore, solving for the extreme value of ξ also solves for the 195
extreme value ofφ. According to the variational principle, it can be transformed into a Euler 196
equation, so that the extreme value problem can be transformed to solve the definite solution of the 197
Euler equation under the boundary conditions. The Euler equation of φ is 198
(19) 199
According to Eq. (18), the following equation can be obtained by combining Eq. (18) with Eq. 200
(19): 201
(20) 202
s w s=F iV f V
1
11
ci w( ) (1 ) ( ) ( )( ) Bf x x B AB x if xf x f , ,
d0
( ) d ( )f xf x x
w=( )
if x
(21) 203
(22) 204
Eqs. (20) and (22) are substituted into Eq. (19), and after sorting and simplifying, the following 205
equation is obtained: 206
(23) 207
By solving this second-order differential equation, the expression for the separation curve f(x) 208
can be obtained: 209
(24) 210
where C1 and C2 are undetermined constants. Then the mathematical expression of f '(x) can be 211
obtained : 212
(25) 213
It can be seen from Fig. 1 that the separation curve f(x) is symmetric about the y-axis and 214
continuously differentiable in its domain, which is an even function. This indicates that when x=0, 215
f '(x)=0, namely: 216
(26) 217
It can be seen that the value of C1 is 0. Therefore, the expression of f(x) can be simplified to: 218
(27) 219
Taking the separation curve as a generatrix rotating around the y-axis, the expression of 220
discontinuity caused by the failure of water-resistant rock mass can be obtained: 221
(28) 222
1
1ci 1
( )
(= (
))
BBB
AB
fx
x Bf
12 11
ci 1( )d
( )( )
= ( )d 1
BBBf
Af x
f
B
xx
x B
2 1
w11
1ci
( 1)( )( )
( )
B
B
B
i Bx
AB
f f x
1
1ci w
1
w 1ci
1
1 2
( )( )
( )
BB
B
f xAB iB
iAB
x C C
w
1
1ci
1
1
( )
( )B
B
B
iB
AB
f x x C
1
1(0) ( ) 0B
Bf C
ci
11
21 1
)(( )
B
B
wB
B
B B
i
A
f x x C
11
2 2 221 1
ci
((
), ) ( )
B
BB
B
B B
wf x z x z Ci
A
3.2.3 Expression of critical safety thickness of water-resistant rock mass 223
According to the geometric relationship of the failure mode in Fig. 1(a) and the expression of 224
the separation curve f(x), the following equations can be obtained: 225
1 1
21
c
1
c
1
21
i
1
1
i
(0
2 2
(
2
)
2
)B
B B
B
B B
B
B B
B
B B
w
w
L Lf C
D Df C
i
A
Si
A
(29) 226
Substitute hydraulic gradients i=Δh/S and Δh=pw/γw into Eq. (29) to simplify, The expression 227
of critical safety thickness of water-resistant rock mass based on the Hoek-Brown failure criterion 228
can be obtained: 229
1 11
w
1
ci 2=
2
B
BB B
B
pS
D L
A
(30) 230
where the parameter L is an unknown number, so the critical safe thickness of water-resistant rock 231
mass can be calculated according to the mechanical boundary conditions. 232
According to the mechanical boundary conditions, the micro-unit located at the interface 233
between the separation curve and the water-filled karst cave is selected for mechanical analysis, and 234
the mechanical equilibrium equation is established in the vertical direction. After derivation and 235
simplification, the mathematical expression of the shear stress τyx is: 236
(31) 237
Considering the assumption that the karst cavity pressure is uniform and vertically distributed, 238
there is no shear stress on the surface of the karst cavity, so τyx is 0 at (x=L/2, y=0). Substituting Eqs. 239
(3), (4), (9) into Eq. (31) to simplify: 240
(32) 241
Substituting the expression of f '(x) and x=L/2 into Eq. (32), the result is as follows: 242
(33) 243
When the basic parameters are determined, the critical safety thickness of the water-resistant 244
sin 2cos 2
2
nyx
1 1
1 1 1ci tm c
2 1
1 1 1i( ) ( ) ( ) ( )( )
B B B
B BB B BBAB Af x f x f x f xB
2
2 2 1 2 2
w w w tm
2 1 22 2
ci ci 1ci
1 11
2 2 2
B B B
B B B
BB B
B B B B B B
p L p L p L
S S SA B A B A B
Table 1 Critical safety thickness of the water-resistant rock mass with different surrounding rock grades
rock mass conforming to the Hoek-Brown failure criterion can be obtained using simultaneous Eqs. 245
(30) and (33). 246
247
4 Results analysis 248
4.1 Feasibility analysis 249
In order to verify the feasibility of the above method, we calculated the critical safety thickness 250
of the water-resistant rock mass under I–V grade surrounding rock. The parameter c and φ were 251
selected according to Standard for engineering classification of rock mass (GB/T 50218-2014). The 252
tunnel section height was 10 m, the karst water pressure was 1.0 MPa, the parameters A and B were 253
0.8 and 0.5 respectively, and tensile strength was 1/100 of compressive strength for calculation. The 254
calculation results are shown in Table 1. 255
256
257
It could be seen from the calculation results in Table 1 that with the decrease of surrounding 258
rock grades, the critical safety thickness of the water-resistant rock mass gradually increased, and 259
the safety thickness of surrounding rock at all grades were all within a reasonable range (Huang et 260
al. 2019), which verified the feasibility of the method in this paper to a certain extent. The specific 261
engineering case will be given in the second half of the paper. 262
4.2 Analysis of influencing factors 263
On the basis of the previous theoretical equation, the main factors affecting the safety thickness 264
of the water-resistant rock mass of tunnel face are the compressive strength σci and tensile strength 265
σtm of the water-resistant rock mass, the karst water pressure pw, the tunnel section height D, and 266
parameters A and B. The factors have different degrees of influence on the failure range of the water-267
resistant rock mass (Yang and Long 2015) and we analyzed the influences of the changes in these 268
factors on the safety thickness of the water-resistant rock mass. 269
In order to study the influence of single factor change on the safety thickness of water-resistant 270
rock mass, the safety thickness of water-resistant rock mass is obtained according to Eqs (30) and 271
(33) when other factors are determined. In order to more clearly determine how the safety thickness 272
of the water-resistant rock mass changes as the change of each factor, we combined the typical data 273
in calculation results with the separation curve and the separation surface equation, and drew the 274
failure shape and damage range of the water-resistant rock mass as diagrams (Fig. 2). 275
Fig. 3 Variation law of safety thickness under the influence of different factors
276
277
As shown in Fig. 2, the safety thickness decreased as the compressive strength, the tensile 278
strength of the water-resistant rock mass, and parameter A increased, and it increased as the karst 279
water pressure, the tunnel excavation height, and parameter B increased. Comparing the influence 280
law of each factor change on the safety thickness of the water-resistant rock mass, it was found that 281
the change of parameter A and B had the most significant influence on the safety thickness. 282
Therefore, in practical engineering application, according to the I–V grade of surrounding rock and 283
the good to bad lithological conditions, it was suggested that the value of A should be 0.9–0.4 and 284
the value of B should be 0.5–0.8, so as to ensure that the thickness of the water-resistant rock mass 285
could have a certain safety reserve. 286
4.3 Comparative analysis with previous research achievements 287
In order to verify the effectiveness and consistency of the method in this paper, it is necessary 288
to make a comparative analysis with previous related research achievements. For example, Yang et 289
al. (2017) proposed a calculation method for the thickness of the karst tunnel rock wall based on the 290
Hoek-Brown criterion without considering the seepage effect. In this paper, “whether or not to 291
consider the influence of seepage effect on the safety thickness of the water-resistant rock mass in 292
water-rich area” was taken as the breakthrough point to compare with the research achievements 293
(Yang et al. 2017) to verify the correctness of this method. In order to ensure the reliability of the 294
analysis results, the selection of the calculation parameters and the value of the data should be 295
consistent with the comparison literature as much as possible to meet the requirements of the 296
homologous feature parameters. The influence law of each factor change on the critical safety 297
thickness of the water-resistant rock mass with or without seepage effect is plotted as a graph for 298
comparison, as shown in Fig. 3. 299
300
301
Different safety thickness calculation formulas have different value categories for the 302
parameters A and B, so the values of A and B in figures (e) and (f) are not exactly same. It could be 303
clearly seen from Fig. 3 that the safety thickness of the water-resistant rock mass had the same trend 304
with the change of factors in the case of considering and not considering the seepage effect, that is, 305
the safety thickness decreased with the increase of compressive strength, tensile strength and 306
Fig. 2 Influence of each factor on the failure shape and damage range of the water-resistant rock mass
parameter A, and increased with the increase of karst water pressure, excavation height and 307
parameter B. It could also be concluded from Fig. 3 that in the case of considering the seepage effect, 308
the safety thickness of the water-resistant rock mass was greater than that without considering the 309
seepage effcet. Combining with the specific data, under the influence of the excavation section 310
height, the maximum difference of the critical safety thickness was 4.16 m, and under the influence 311
of parameter A, the minimum difference of the critical safety thickness was 0.98 m. That is, in the 312
process of tunnel excavation, when the seepage effect of groundwater is considered, the reserved 313
safety thickness of the water-resistant rock mass is at least 1 m or even 4 m more than that without 314
considering seepage effect. It also showed that the seepage effect had a significant influence on the 315
stability of surrounding rock of underground tunnel. Therefore, the seepage effect of groundwater 316
must be fully considered in tunnel construction in areas rich in groundwater. 317
5 Engineering application 318
The entrance of Yunwushan tunnel of Yiwan railway line is located in Ginguba Town of Enshi 319
city, and the exit is located in Zhoujiawan of Tunbao Township of Enshi City. The total length of 320
tunnel line I is 6640 m with the mileage of DK242+084~DK248+724 and the total length of tunnel 321
line II is 6682 m with the mileage of IIDK242+084~IIDK248+766. The line II tunnel is located on 322
the left side of the line I tunnel with a distance of 30 m between the two lines. On October 12, 2008, 323
IIDK242 + 526 karst cavity was encountered in front of the working face of line II at the entrance 324
of Yunwushan Tunnel, and the karst cavity was filled with mud and sand. During the detection 325
period, the protruding mud sand was about 250 m3, and the water inflow was about 90 m3/h. The 326
location of the karst cavity was shown in Fig. 4 (Guo 2011). According to Yunwushan Tunnel 327
Engineering Geology Report (Qiao 2009), Yunwushan Tunnel passes through strata dominated by 328
limestone and argillaceous dolomite, the basic grade of surrounding rock is grade III, and the 329
average value of saturated uniaxial compressive strength is 67.34 MPa. There are fault fracture 330
zones and influence zones in the range of DK245+504.8~DK245+633.8 with rock fragmentation 331
lithification and breccification, karst water development, and cavity water pressure of 0.8 MPa. The 332
depth of the tunnel is more than 500 m, the height of the section is 9.8 m, and the span is 7 m. 333
334
335
The karst cave of the face was divided into four areas: the thickness of water-resistant rock 336
mass in zone I was less than 2.5 m, the thickness of water-resistant rock mass in zone II was 2.5–337
4.5 m, the thickness of water-resistant rock mass in zone III was 4.5–9 m, and the thickness of water-338
Fig. 4. Engineering geological profile of the Yunwushan tunnel and the karst cave 526 location
resistant rock mass in zone IV was greater than 9 m, as shown in Fig. 5 (Guo 2011). Therefore, in 339
order to ensure the construction safety and smooth, it is necessary to open the karst cavity zone I 340
and II. 341
342
343
The parameters of Hoek-Brown failure criterion (Hoek and Brown 1997): mi= 9, GSI = 60, Di= 344
0.8, A= 0.5, B=0.8. By substituting the above parameters into the method discussed in this paper, 345
the critical safety thickness of water-resistant rock mass was calculated to be 3.71 m, which was 346
within the range of the adopted safety thickness (2.5–4.5 m) in the actual project of karst cavity 526 347
and was equivalent to the average value (3.5 m), so the result was relatively safe. Therefore, this 348
method is reasonable and feasible to calculate the safety thickness of water-resistant rock mass, 349
which can be used to guide the construction of tunnels in karst areas and provide a decision-making 350
reference for preventing water and mud inrush in karst tunnels. 351
352
6 Conclusion 353
(1) The current methods for calculating the safety thickness of the water-resistant rock mass of 354
karst tunnel face in the water-rich area do not consider the karst water seepage effect. This paper 355
took into account the adverse influence of the karst water seepage effect on the water-resistant 356
rock mass. Based on the upper-bound theorem of limit analysis and the Hoek-Brown failure 357
criterion, through a series of formula derivation, the undetermined constant was solved by using 358
the property of even function, and the expression of critical safety thickness of water-resistant 359
rock mass of face was finally obtained. 360
(2) We conducted a feasibility analysis and an analysis of influencing factors. The results showed 361
that with the decrease of surrounding rock grade, the safety thickness of water-resistant rock 362
mass gradually increased, and the safety thickness of surrounding rock at all grades remained 363
within a reasonable range; the safety thickness of the water-resistant rock mass of face 364
decreased as the compressive strength, tensile strength, and parameter A increased and it 365
increased as the karst water pressure, tunnel excavation height, and parameter B increased. 366
(3) We carried out a comparative analysis with the previous relevant research achievements, which 367
showed the change trend of the safety thickness with the parameters was completely consistent 368
under the two conditions of considering the seepage effect and without considering the seepage 369
effect, and the safety thickness with considering the seepage effect was greater than that without 370
Fig. 5 Boundary locking value of karst cavity 526 in Yunwushan Tunnel
considering the seepage effect. Taking the Yunwushan tunnel of Yiwan railway as an example, 371
the critical safety thickness of the water-resistant rock mass was calculate and the calculated 372
value was in good coincidence with the safety thickness adopted in the actual project. 373
374
Acknowledgments 375
This study was financially supported by the National Natural Science Foundation of China 376
(Grant No.: 51778215, U1810203), the National Key Basic Research and Development Plan (973 377
Plan) Project (Grant No.: 2013CB036003), the China Postdoctoral Science Foundation Fund (Grant 378
No.: 2018M631114), and the Doctoral Fund of Henan Polytechnic University (Grant No.: B2020-379
41). 380
381
Disclosure statement 382
The authors declare that they have no conflicts of interest. 383
384
References 385
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31(10):141–149, 219 427
428
429
430
431
432
433
434
x
y 0 (pw) LD
f(-x)S
f(x)
dxdyv
fsw 1
2
vvs
vnƟ
s
n
f(x)
p
Water
pressure
filling cavity
Tu
nn
el f
ace
Tunneling direction
(a) (b)
Separation
layer
Ɵ
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
Fig. 1 Failure mode of the water-resistant rock mass
(a) Failure mode (b) Deformation mode of the separation layer
012345678910-6
-4
-2
0
2
4
6
σtm=σci/100
pw=1.0 MPa
D=10 m
A=0.4
B=0.7
x (m
)
y (m)
σci=30 MPa
σci=50 MPa
σci=70 MPa
012345678910
-6
-4
-2
0
2
4
6
σci=20 MPa
pw=1.0 MPa
D=10 m
A=0.4
B=0.7
x (m
)
y (m)
σtm=0.286 MPa
σtm=0.154 MPa
σtm=0.067 MPa
472
473
0123456789101112-6
-4
-2
0
2
4
6
σci=30 MPa
σtm=0.3 MPa
H=10 m
A=0.4
B=0.7
x (
m)
y (m)
pw=1.5 MPa
pw=2.5 MPa
pw=3.5 MPa
012345678910
-8
-6
-4
-2
0
2
4
6
8
σci=30 MPa
σtm=0.3 MPa
pw=1.0 MPa
A=0.4
B=0.7
x (
m)
y (m)
D=11 m
D=13 m
D=15 m
474
475
01234567-6
-4
-2
0
2
4
6
σci=30 MPa
σtm=0.3 MPa
pw=1.0 MPa
D=10 m
B=0.8
x (
m)
y (m)
A=0.5
A=0.7
A=0.9
01234567
-6
-4
-2
0
2
4
6
σci=30 MPa
σtm=0.3 MPa
pw=1.0 MPa
D=10 m
A=0.5
x (
m)
y (m)
B=0.6
B=0.7
B=0.8
476
477
478
479
480
481
482
483
484
485
486
487
488
(a) Influence of the compressive strength (b) Influence of the tensile strength
(c) Influence of the karst water pressure (d) Influence of the excavation section heights
(e) Influence of the value of parameter A (f) Influence of the value of parameter B
Fig. 2 Influence of each factor on the failure shape and damage range of the water-resistant rock mass
(a) Variation of safety thickness under the
influence of compressive strength
(b) Variation of safety thickness under the
influence of tensile strength
(c) Variation of safety thickness under the
influence of karst water pressure
(d) Variation of safety thickness under the
influence of excavation section height
(e) Variation of safety thickness under the
influence of parameter A
(f) Variation of safety thickness under the
influence of parameter B
Fig. 3 Variation law of safety thickness under the influence of different factors
30 40 50 60 70
1.0
1.5
2.0
2.5
3.0
3.5
4.0
σtm=σci/100
pw=1.0 MPa
D=10 m
A=0.4
B=0.7
Saf
ety
th
ick
nes
s S
(m
)
Compressive strength σci (MPa)
Considering seepage effect
Without considering seepage effect
0.0 0.1 0.2 0.3 0.42.4
2.8
3.2
3.6
4.0
4.4
4.8
5.2
Saf
ety
th
ick
nes
s S
(m
)
Considering seepage effect
Without considering seepage effect
σci=20 MPa
pw=1.0 MPa
D=10 m
A=0.4
B=0.7
Tensile strength σtm (MPa) 489
490
491
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.52.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
Saf
ety t
hic
kn
ess
S (
m)
Considering seepage effect
Without considering seepage effect
σci=30 MPa
σtm=0.3 MPa
D=10 m
A=0.4
B=0.7
Karst water pressure pw (MPa) 10 11 12 13 14 15 16
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
Safe
ty t
hic
kn
ess
S (
m)
Considering seepage effect
Without considering seepage effect
σci=30 MPa
σtm=0.3 MPa
pw=1.0 MPa
A=0.4
B=0.7
Excavation section height D (m) 492
493
0.4 0.5 0.6 0.7 0.8 0.9 1.00.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Saf
ety
th
ick
nes
s S
(m
)
Considering seepage effect
Without considering seepage effect
σci=30 MPa
σtm=0.3 MPa
pw=1.0 MPa
D=10 m
B=0.8
Parameter A 0.5 0.6 0.7 0.8 0.9
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Parameter B
Saf
ety t
hic
kn
ess
S (
m)
Considering seepage effect
Without considering seepage effect
σci=30 MPa
σtm=0.3 MPa
pw=1.0 MPa
D=10 m
A=0.5
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
∈1t
Total length: 6640 m
DK245+404
Ele
vat
ion (
m) 1600
1500
1400
1300
1200
1100
1000
900
800 DK
242+
084
F
DK
248+
724
∈2gn Middle Cambrian Guangzhuling Formation
∈2g+m Middle Cambrian Gaotai+Maoping Formation
∈1s Lower Cambrian Shilongdong Formation
∈1t Lower Cambrian Tianheban Formation
∈3hz Upper Cambrian Ratuo Group
01n Lower Ordovician Nanjinguan Formation 01f-g Lower Ordovician Fenxiang Group-Guniutan Formation 02m-3w Upper Ordovician Miaopo Formation-Wufeng Formation
karst cavity 526
Fig. 4. Engineering geological profile of the Yunwushan tunnel and the karst cave 526 location
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
Fig. 5 Boundary locking value of karst cavity 526 in Yunwushan Tunnel
▲The thickness of water-resistant rock
mass in zone I is less than 2.5 m.
▲The thickness of water-resistant rock
mass in zone II is 2.5–4.5 m.
▲The thickness of water-resistant rock
mass in zone III is 4.5–9 m.
▲The thickness of water-resistant rock
mass in zone IV is more than 9 m.
Zone I
(2.5 m)
(2.5 m) (3 m) (2.5 m)
(2.3 m)
Zone Ⅲ
Zone Ⅳ
(3 m)
(8.5 m)
(＞11 m) (4 m) (＞16 m)
(＞11 m) (＞9 m)
(6.5 m) (6.5 m) (＞9 m)
(X m)—the thickness of water-resistant rock mass
Zone II
Zone Ⅳ
Table 1 Critical safety thickness of the water-resistant rock mass with different surrounding rock grades 568
569
570
571
572
573
574
575
Surrounding rock
grades
Cohesion c
(MPa)
Internal friction
angle φ (°)
Compressive strength
σci (MPa)
Critical safety
thickness S (m)
Ⅰ ＞2.1 ＞60 ＞15.67 ＜1.03
Ⅱ 2.1–1.5 60–50 15.67–8.24 1.03–1.68
Ⅲ 1.5–0.7 50–39 8.24–2.93 1.68–3.13
Ⅳ 0.7–0.2 39–27 2.93–0.65 3.13–7.52
Ⅴ ＜0.2 ＜27 ＜0.65 ＞7.52
Figures
Figure 1
Failure mode of the water-resistant rock mass (a) Failure mode (b) Deformation mode of the separationlayer
Figure 2
In�uence of each factor on the failure shape and damage range of the water-resistant rock mass
Figure 3
Variation law of safety thickness under the in�uence of different factors
Figure 4
Engineering geological pro�le of the Yunwushan tunnel and the karst cave 526 location
Figure 5
Boundary locking value of karst cavity 526 in Yunwushan Tunnel
