Research ArticleAnalytical Solution of Seepage Field in Karst Tunnel

Chong Jiang Han-song Xie Jia-li He Wen-yan Wu and Zhi-chao Zhang

School of Resources and Safety Engineering Central South University Changsha 410083 Hunan China

Correspondence should be addressed to Chong Jiang jiang4107sohucom

Received 17 May 2018 Revised 15 August 2018 Accepted 27 August 2018 Published 20 September 2018

Academic Editor Zhongwei Chen

Copyright copy 2018 Chong Jiang et al )is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An analytical solution for the seepage field in water-filled karst tunnel is derived based on the inversion of complex function andgroundwater hydraulics theory )e solution considers the distance between the tunnel and the cavern the size of the cavern andthe properties of the lining structure such as the permeability coefficient as well as the radius of the grouting ring )is paper alsoperformed numerical simulations for two cases the application of gravity and the absence of gravity )e numerical solution wasobtained to verify the analytical solution and a good agreement was found )en the effect of parameters is discussed in detailincluding the distance between the tunnel and the cavern the radius of the cavern the grouting ring and the initial support )eresults show that when the radius of the cavern is constant the pressure head and seepage flow decrease as the distance betweenthe tunnel and the cavern increases When the distance is constant the pressure head and seepage flow increase with the increaseof the radius of the cavern In addition the pressure head and the seepage flow decrease with the increase of the thickness of thegrouting ring and decrease with the decrease of the permeability coefficient As the thickness of the initial support increases thepressure head gradually increases and the percolation decreases Furthermore due to the great influence of the grouting ring andinitial support on the pressure head and seepage flow the thickness and permeability coefficient of the grouting ring and initialsupport should be taken into account carefully during construction

1 Introduction

In recent years with the development of the Belt and Roadstrategy the construction of tunnels has developed rapidlyAnd it is possible to build a tunnel in the karst zone insouthwestern China)e karst water has a great influence onthe construction of the tunnel Its seepage causes karsttunnel water gushing and it is easy to cause tunnel de-formation and other disasters when the water volume is toolarge )erefore it is important to study the distribution ofseepage field in karst tunnels

)e analysis of seepage field in karst tunnels generallyuses analytical method and numerical simulation Based onwell mapping theory and superposition principle Zhang [1]deduced the potential function and distribution of waterhead in the karst tunnel but did not study the influence ofkarst water on tunnel lining structure Zhang [2] derived theanalytical solution of the seepage field of a semiinfiniteparallel double-hole tunnel through the seepage mechanicsand image method Yang et al [3] studied the effect of

seismic loading on the double-sided slope of high watercontent sandy clay Wu [4] deduced the analytic solution ofthe complex variable function for the distribution of highwater pressure seepage field according to the conformalmapping theory Park et al [5] compared two existinganalytical solutions for the steady-state groundwater inflowinto a drained circular tunnel in a semiinfinite aquifer Someresearchers used the theory of hydraulics and complexfunctions to solve the analytical solution of seepage field ofunderwater tunnel composed of surrounding rocks andlining structures [6ndash9] Based on the theory of the growth ofpore pressure caused by linear waves Xiong et al [10]developed an analytical solution of the seepage field ofa tunnel affected by the wave in the semiinfinite aquifer Maet al [11 12] carried out some experimental investigationson hydraulic characteristics and seepage characteristics Lei[13] acquired an analytical solution for steady flow intoa tunnel based on the image method Huangfu et al [14]studied analytical solutions for steady groundwater flowinginto a horizontal tunnel in the semiinfinite aquifer

HindawiAdvances in Civil EngineeringVolume 2018 Article ID 9215472 9 pageshttpsdoiorg10115520189215472

However most of the aforementioned literature havestudied high hydraulic pressure tunnels or high hydraulicparallel tunnels in a semiinfinite plane )ere is less ana-lytical formula for the distribution of seepage field in karsttunnels )erefore based on the inversion of complexfunction and groundwater hydraulics theory this paperdeduces the distribution of seepage field in karst tunnels andcalculates the amount of seepage )e formula related topressure head and tunnel seepage is deduced from the re-lationship between the grouting ring initial support andsecondary lining )e results are compared with the nu-merical solutions of the two cases of gravity and no gravity)e change of lining structure characteristics such asgrouting circle and initial support is discussed in detail It ishelpful for us to have a deeper understanding of un-derground water seepage in karst tunnels and to take ef-fective measures to reduce the harmful effects of karst wateron tunnels so as to provide references for such problems inthe future

2 Calculation Model

)e schematic diagram of the karst tunnel in an infiniteplane is shown in Figure 1 Here q is the water pressure inthe karst cavern a is the tunnel radius and d is the distancebetween the center of the tunnel and the karst cavernAdditionally the basic assumptions are as follows

(1) )e surrounding rock of the tunnel is homogeneouswith isotropic permeability [5]

(2) )e aquifer and karst water are incompressible )eflow is in a steady state and is governed by Darcyrsquoslaw [13]

(3) Due to the long distance from the tunnel to thesurface it is assumed that the plane where the karsttunnel is located is an infinite plane and the seepagefield is only affected by karst water

(4) When studying the lining structure the pore pres-sure is constant on the same ring and the water-filling hole is equal to the water pressure head [7]

3 Analytical Solution of Seepage Field

31 Analytical Method Considering the characteristics ofkarst tunnels the inversion of complex functions andgroundwater hydraulics is used to analyze the seepage fieldof karst tunnels )e solution steps are as follows

(a) According to the complex potential function of thepoint source and the inverse transformation relationthe mirror image points a2z0 of z0 in the tunnel areobtained and the complex potential function at anyplace in the seepage plane of the karst tunnel issolved

(b) )rough the resetting potential at any place in theseepage plane of the karst tunnel the potentialfunction flow function and complex velocity of theseepage field in the karst tunnel are solved

(c) )e seepage flow and pressure head are obtained bythe boundary conditions and the head pressure andthe seepage flow in each lining structure were solvedaccording to the relationship of the lining structures

32 Analysis of Complex Potential Function )e water-filledcavern existing outside the tunnel is treated as a constantpoint source and the inversion of the cavern and tunnel isshown in Figure 2 In the infinite plane where the tunnelradius is a there is a cavern with the water pressure q atz0 x0 + iy0 outside the tunnel )e cavern is considered asa point source After the inversion transformation a cavernwith the same intensity is generated at a2z0 which iscalculated as a point sink )e complex potential function ofthe point z x + iy in the infinite plane of the point sourcez0 x0 + iy0 is

f(z) q

2πln z

q

2πln ae

iθ1113872 1113873 (1)

where the potential function is ϕ q2π ln a and the flowfunction is φ q2πθ

Based on the complex potential function of the pointsource in the infinite plane and the inversion transformationrelationship the conjugate complex potential function of thepoint sink is obtained

fa2

z1113888 1113889 minus

q

2πln z minus

a2

z01113888 1113889 (2)

)e point source in the unbounded fluid has a resetpotential off(z) and the conjugate reset potential of thepoint converted according to the inversion isf(a2z) All thepoints are in the region of |z|gt a outside the tunnel In theflow field behind the tunnel |z| a the tunnel wall can bea resetting potential of a streamline According to the su-perposition principle of resetting potential in seepage me-chanics the complex potential function of the point seepingin the infinite plane of the tunnel is obtained

BC

A

ad

rw

Figure 1 Schematic diagram of karst tunnel in infinite plane Atunnel B surrounding rocks C cavern

2 Advances in Civil Engineering

F(z) f(z) + fa2

z1113888 1113889 (3)

33 Analysis of Seepage Field According to the complexpotential formula (3) of the point in the infinite plane of thetunnel formulas (1) and (2) are plugged into (3) as shown inthe following

F(z) q

2πln zminus z0( 1113857minus

q

2πln zminus

a2

z01113888 1113889 (4)

For simplicity without loss of generality the pointsource can be selected to the right of the horizontal directionof the tunnel that is the x-axis passes through the pointsource)en there is z0 z0 x0 equiv d as shown in Figure 3)erefore Formula (4) can be written as

F(z) q

2πln(zminus d)minus

q

2πln zminus

a2

d1113888 1113889

q

2πln

(xminusd)2 + y21113969

minus iq

2πarctg

y

xminusd

minusq

2πln

xminusa2

d1113888 1113889

2

+ y2

11139741113972

+ iq

2πarctg

y

xminus a2d( )

(5)

Complex velocity

W(z) dF

dz

(q2π)ln(zminusd)minus(q2π)ln zminus a2d( 1113857( 1113857

dz

q

2π1

zminusdminus

1zminus a2d( )

1113888 1113889

(6)

)e potential function of the point source in the infiniteplane of the tunnel

ϕ q

4πln

(xminusd)2 + y2

xminus a2d( )( )2

+ y2 (7)

)e flow function of the point source in the infinite planeof the tunnel

φ minusq

2πarctg

y

xminus d+

q

2πarctg

y

xminus a2d( ) (8)

Boundary conditionsCavern boundary

Φw minusq

2πln

rw

l (9)

Tunnel boundary

Φs minusq

2πln

d

a (10)

where l is the distance between the mirror point a2z0 andz0 rw is the radius of the cavern Φw is the cavern potentialfunction and Φs is the tunnel potential function

According to the boundary conditions and the steady-state percolation of the single-phase liquid [15] it can beobtained that

q 2π Φw minusΦs( 1113857

minusln arwld( 1113857

2π Φw minusΦs( 1113857

ln d2 minus a2arw( 1113857

Hw pw

ρg Hs

ps

ρg

Q 2πKρg Hw minusHs( 1113857

ln d2 minus a2arw( 1113857

(11)

where K is the permeability Pw and Ps are the headpressures of the cavern and tunnel respectively Q is the unitpermeate flow Hw is the head pressure of the cavern and Hs

is the head pressure of the tunnel boundary that is the headpressure of the surface of the grouting ring

34 Analysis of Seepage Field of Lining Structure )e seepagefield in the lining and grouting circle cannot be solved by thecomplex function but it can be solved by using thegroundwater seepage mechanics and the shaft theory [6]

x

y

CavernSecondary lining

Surrounding rocks

Initial supportGrouting ring

o

Figure 3 Schematic diagram of the relationship between eachlining structure

a

z 0

q

x

y

ndashq

a2 z 0

o

Figure 2 Schematic diagram of inversion of caverns and karsttunnels

Advances in Civil Engineering 3

)e relationship between the initial support the secondarylining and the grouting ring pressure head is shown inFigure 3 and the formulas are as follows

Hs minusH1 Q

2πk1ln

a

r1

H1 minusH2 Q

2πk2ln

r1

r2

H2 minusH3 Q

2πk3ln

r2r3

(12)

where Hs is the head pressure of the surface of the groutingring H1 H2and H3 are the head pressures of the joints ofthe grouting ring and the initial support the initial supportand the secondary lining joint and the secondary lininginner surface respectively r1 r2 and r3 are the radius of thegrouting ring initial support and internal of secondarylining respectively k1 k2 and k3 are the permeability co-efficients of the grouting circle initial support and sec-ondary lining respectively

Because of the same seepage the above equations can beintegrated to obtain the tunnel seepage when the groutingcircle the initial support and the two lining are completed

Q 2π Hw minusH3( 1113857

1k2( 1113857ln r1r2( 1113857 + 1k1( 1113857ln ar1( 1113857 + 1k3( 1113857ln r2r3( 1113857 +(1k)ln d2 minus a2arw( 1113857 (13)

Head pressure of initial support and secondary liningjoint

H2 H3 +Hw minusH3( 1113857

k3Aln

r2

r3 (14)

Head pressure of grouting circle and initial supportconnection

H1 H3 +Hw minusH3( 1113857

k3Aln

r2

r3+

Hw minusH3( 1113857

k2Aln

r1

r2 (15)

In the formula

A 1k2

lnr1

r21113888 1113889 +

1k1

lna

r11113888 1113889 +

1k3

lnr2

r31113888 1113889 +

1kln

d2 minus a2

arw

1113888 1113889

(16)

where k is the permeability coefficients of the surroundingrock

According to different stages of construction such astunnel holes and grouting the different water pressuredistribution can be obtained When tunnel grouting andinitial support are completed the tunnel seepage is

Q 2π Hw minusH2( 1113857

1k2( 1113857ln r1r2( 1113857 + 1k1( 1113857ln rr1( 1113857 +(1k)ln d2 minus a2arw( 1113857

(17)

Head pressure of grouting circle and initial supportconnection

H1 H2 +Hw minusH2( 1113857ln r1r2( 1113857

ln r1r2( 1113857 + k2k1( 1113857ln rr1( 1113857 + k2k( 1113857ln d2 minus a2arw( 1113857

(18)

4 Verification and Discussion

41 Numerical Verification Based on the complex functionand groundwater hydraulics theory the distribution ofseepage field suitable for homogeneous and isotropic karsttunnels and lining structures were deduced In order to

ensure the accuracy of the analytical solution of the karsttunnel the numerical solution simulated by the finite dif-ference software FLAC3D is compared with the theoreticalsolution obtained by the method used in this paper

)e calculation conditions are listed as follows the headpressure of the cavern is 54m the radius of the cavern is 4mthe radius of the tunnel is 725m the radius of the groutingcircle is 225m and the initial support radius is 025m thepermeability coefficient of surrounding rocks is15times10minus6 cmmiddotsminus1 the permeability coefficient of groutingring is 10minus7 cmmiddotsminus1 the initial support permeability co-efficient of initial support is 10minus8 cmmiddotsminus1

Six groups of 100mtimes 100m models of caverns on theright side of the tunnel were established and calculated withFLAC3D )e pore water pressure of the tunnel with gravityand without gravity was obtained respectively )e sixgroups of models are d 15 m d 20 m d 25 md 30 m d 35 m and d 40 m When d 20 m thekarst tunnel model consists of 4236 elements as shown inFigure 4 Figures 5 and 6 are contours of zone pore pressurewith gravity and without gravity respectively

)e pore water pressures extracted from the six groups ofnumerical simulation data in Figure 5 with gravity andFigure 6 without gravity are compared with the theoreticalsolutions )e results are shown in Figure 7 )e law ob-tained by numerical and theoretical solutions is that whenthe grouting ring and the initial support are completed andthe secondary lining is not yet completed H1 decreases withthe increase of d Moreover the curves have a good con-sistency When d 15 m the analytical solution for H1 is2578m the numerical solution with gravity is 2567m andthe numerical solution without gravity is 2577m)ereforethe analytical method in this paper is suitable for solving thedistribution of seepage field in karst tunnels

420eEffect ofDistance andCavernRadius Figure 8 showsthe relationship between H1 and d Figure 9 shows therelationship between Q and rw In the infinite plane whenHs is constant the main factors affecting H1 and Q in the

4 Advances in Civil Engineering

tunnel are rw and dWhen rw is constant H1 and Q decreasewith the increase of d When d is constant H1 and Q

gradually increase as rw increases

43 0e Effect of Grouting Ring When the diameter of thegrouting ring and the initial support radius is constant thetunnel radius and the thickness of the grouting ring increaseFigures 10ndash13 show that H1 and Q gradually decrease withthe increase of the thickness of the grouting ring when theratio between the distance and the permeability coefficient isconstant When the thickness of the grouting ring was025m and a reached 6m Q and H1 decreased to 4194and 4186 respectively When a is constant H1 and Q

gradually decrease as the permeability coefficient decreasesWhen a increases to 7m the rate of H1 and Q decreasegradually From Figures 12 and 13 when a is constant H1

becomes smaller as the seepage coefficient of the groutingring decreases When kr kk1 increases to 80 Q decreasesto 224 of Q when kr 5 and the Q reduction rategradually decreases In summary increasing the thickness ofthe grouting ring and reducing the permeability coefficientof the grouting ring can reduce Q and H1 However ex-cessively increasing the thickness of the grouting ring willnot only increase the cost but also reduce the impact on thepressure head and seepage flow )erefore a reasonablegrouting material and construction process should be se-lected to ensure the thickness of the grouting ring when thepermeability coefficient of the grouting ring is less than thespecified value

440e Effect of Initial Support When the inner diameter r1of the grouting ring is constant increasing r2 is to reduce the

52900E + 0552500E + 05

47500E + 05

42500E + 0545000E + 05

50000E + 05

40000E + 0537500E + 05

32500E + 0535000E + 05

30000E + 0527500E + 05

22500E + 0525000E + 05

20000E + 0517500E + 05

12500E + 0515000E + 05

11674E + 05

Figure 6 Contour of pore water pressure without gravity ford 20 a 725 and rw 4

15 20 25 30 35 40240

242

244

246

248

250

252

254

256

258

H1 (

m)

d (m)

Unweighted numerical solutionWeighted numerical solutionTheoretical analytical solution

Figure 7 Comparison of numerical solution and theoretical so-lution for a 725 and rw 4

100 m10

0 my

x

Figure 4 Finite element model with circular karst tunnel

97843E + 0595000E + 0590000E + 0585000E + 0580000E + 0575000E + 0570000E + 0565000E + 0560000E + 0555000E + 05

45000E + 05

35000E + 05

25000E + 05

15000E + 05

50000E + 05

40000E + 05

30000E + 05

20000E + 05

10000E + 0550000E + 0420664E + 04

Figure 5 Contour of pore water pressure with gravity for d 20a 725 and rw 4

Advances in Civil Engineering 5

thickness of the initial support Figures 14 and 15 show thatas the thickness of the initial support decreases H1 decreasesand Q increases gradually When r2 225 m Q and H1 are133 of Q and 207 of H1 when r2 475 m respectivelyIt shows that the initial support thickness has a great in-fluence on the pressure head During construction weshould pay attention to the safety range of the initial supportthickness

If the initial diameter r2 and a are constant increasing r1means increasing the thickness of the initial support andreducing the thickness of the grouting ring In Figures 16

and 17 with the increase of r1 H1 gradually increases and Q

gradually decreasesAs shown in Figures 18 and 19 the permeability

coefficient of grouting ring is constant and H1 graduallydecreases and Q gradually increases with the increase ofk2 )e change rate of Q and H1 is larger whenk2 lt 10minus8 cmmiddotsminus1 When k2 2 times 10minus8 cmmiddotsminus1 both Q andH1 tend to be stable When the other conditions areconstant the change of k2 on the magnitude of 10minus8 haslittle effect on Q and H1 but the influence on themagnitude of 10minus9 is greater

24

26

28

30

32

34

36

38

40

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

r1 = 225m r2 = 2m Hw = 54mk1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

3 4 5 6 7 8a (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 10 Effect of d on the relationship between pressure headand tunnel radius

3 4 5 6 7128

130

132

134

136

138

140

142

144

Q (m

3 ∙sndash1∙m

ndash1)

rw (m)

d = 15 md = 25 md = 35 m

d = 20 md = 30 md = 40 m

(times10ndash5)

a = 725 mk1 = 10ndash7 mmiddotsndash1

r1 = 225 m

k = 15 times 10ndash7 mmiddotsndash1

k2 = 10ndash8 mmiddotsndash1

Hw = 54 mr2 = 2 m

Figure 9 Relationship between seepage flow and cavern radius

10 15 20 25 30 35 40 45 50 55235

240

245

250

255

260

265

rw = 3 mrw = 4 mrw = 5 m

rw = 6 mrw = 7 m

H1 (

m)

d (m)

a = 725 mk1 = 10ndash7 mmiddotsndash1

r1 = 225 m

k = 15 times 10ndash7 mmiddotsndash1

k2 = 10ndash8 mmiddotsndash1

Hw = 54 mr2 = 2 m

Figure 8 Relationship between head pressure and distance

3 4 5 6 7 8

13

14

15

16

17

18

19

2

21

Q (m

3 ∙sndash1∙m

ndash1)

a (m)

rw = 4m k = 15times10ndash7m∙sndash1

r1 = 225m r2 = 2m Hw = 54m

d = 15md = 20md = 25m

d = 30md = 35md = 40m

k1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

(times10ndash5)

Figure 11 Relationship between seepage flow and tunnel radius

6 Advances in Civil Engineering

5 Conclusion

Based on the inversion of complex function and ground-water hydraulics theory this paper derives the analyticalsolution for the karst seepage field in the karst tunnel in aninfinite plane )e potential function the flow function andthe complex velocity at any place in the infinite plane aresolved According to the distribution of the seepage field andthe boundary conditions the seepage flow of the karst tunnelis obtained )e seepage field of the lining structure is de-duced from the relationship between the grouting ring theinitial support and the secondary lining )e new solution

was verified with numerical simulation via the softwareFLAC3D )e conclusions obtained from this study aresummarized as follows

(1) When Hs is constant for karst tunnels rw and d arethe main factors affecting the pressure head andseepage flow of the tunnel When rw is constant H1and Q decrease with the increase of d When d isconstant H1 and Q increase with the increase of rw

(2) When the initial support is completed and thesecondary lining has not been constructed yet H1

3 4 5 6 7 85

10

15

20

25

30

35

40

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

k2 = 10ndash8m∙sndash1 d = 15mr1 = 225m r2 = 2m Hw = 54m

a (m)

kr = 20kr = 100

kr = 10kr = 50

Figure 12 Effect of kr on the relationship between pressure headand tunnel radius

20

25

30

35

40

45

50

55

H1 (

m)

rw = 4m k = 15times10ndash7m∙sndash1

a = 725m r1 = 5m Hw = 54mk1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

20 25 30 35 40 45 50r2 (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 14 Relationship between pressure head and initial supportradius

20 25 30 35 40 45 500

5

10

15

20

25

30

35

rw = 4m k = 15 times 10ndash7m∙sndash1

a = 725m r2 = 2m Hw = 54mk1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

Q (m

3 ∙sndash1∙m

ndash1)

r2 (m)

(times10ndash6)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 15 Relationship between seepage flow and initial supportradius

0 20 40 60 80 1002

4

6

8

10

12

14

16

18

20

22

kr

(times10ndash6)

rw = 4m k = 15times10ndash7m∙sndash1

r1 = 225m r2 = 2m Hw = 54m

Q (m

3 ∙sndash1∙m

ndash1)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

k2 = 10ndash8m∙sndash1 d = 15m

Figure 13 Relationship between seepage flow and permeabilitycoefficient

Advances in Civil Engineering 7

and Q will gradually decrease with the increase of thethickness and the decrease of the permeability co-efficient of the grouting ring When the permeabilitycoefficient of grouting ring is less than the specifiedvalue reasonable grouting material and constructiontechnology should be selected to ensure the thicknessof grouting ring

(3) H1 gradually decreases with the decrease of thethickness and the increase of k2 However Q

gradually increases with the decrease of the thicknessand the increase of k2 During construction we

should pay attention to the safety range of the initialsupport thickness

Data Availability

Based on the ldquohistoryrdquo function of FLAC3D software thedata which recorded the pore water pressure during thestable period of the monitoring point were obtainedAccording to the simulation conditions in Chapter 4 readerscan complete the modeling process and obtain the contourof pore water pressure such as Figures 5 and 6 )e data ofthe pore water pressure in the stable period of the moni-toring point were compared with the analytical solution to

20

25

30

35

40

45

50

55

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

a = 725m r2 = 2m Hw = 54mk1 = 10ndash7m∙sndash1 k2=10ndash8m∙sndash1

2 3 4 5 6 7r1 (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 16 Relationship between pressure head and radius ofgrouting ring

Q (m

3 ∙sndash1∙m

ndash1)

rw = 4m k = 15 times 10ndash7m∙sndash1

a = 725m r2 = 2m Hw = 54m

(times10ndash6)

14

12

10

8

6

4

2

2 3 4 5 6 7r1 (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

k1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

Figure 17 Relationship between seepage flow and radius ofgrouting ring

00

1 2 3 4 5

10

20

30

40

50

k2 (msndash1)

kr = 20kr = 10

kr = 100kr = 50

(times10ndash8)

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

r1 = 225m d = 15ma = 725m r2 = 2m Hw = 54m

Figure 18 Relationship between pressure head and permeabilitycoefficient of grouting ring

0 1 2 3 4 5

5

10

15

20

25

30

Q (m

3 ∙sndash1∙m

ndash1)

k2 (msndash1)

rw = 4m k = 15 times 10ndash7m∙sndash1

r1 = 225m d = 15ma = 725m r2 = 2m Hw = 54m

kr = 20kr = 100

kr = 10kr = 50

(times10ndash6)

(times10ndash8)

Figure 19 Relationship between seepage flow and permeabilitycoefficient of grouting ring

8 Advances in Civil Engineering

verify the accuracy )e author only recorded data of thecorresponding monitoring point in corresponding area forresearch needs In addition all the sharing data were in-volved in this paper and there were no redundant data )eparameters of the lining structure in the numerical simu-lation conditions come from [6]

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (Grant nos 51478479 and 51678570)and Hunan Transport Technology Project (Grant no201524)

References

[1] M D Zhang Study on the Characteristics of Seepage Field andWater Pressure on Tunnel Lining in Karst Ground BeijingJiaotong University Beijing China 2008

[2] B Q Zhang ldquoAnalytical solution for seepage field of twin-parallel tunnel in semi-infinite planerdquo Journal of 0e ChinaRailway Society vol 39 no 1 pp 125ndash131 2017

[3] B Yang F Gao and D Jeng ldquoFailure mode and dynamicresponse of a double-sided slope with high water content ofsoilrdquo Journal of Mountain Science vol 15 no 4 pp 859ndash8702018

[4] J G Wu A Coupled Fluid-Mechanical Study on Seepage Fieldof Tunnels with High Hydraulic Pressure Beijing JiaotongUniversity Beijing China 2006

[5] K H Park A Owatsiriwong and J G Lee ldquoAnalytical so-lution for steady-state groundwater inflow into a drainedcircular tunnel in a semi-infinite aquifer A revisitrdquo Tunnellingand Underground Space Technology vol 23 no 2 pp 206ndash209 2008

[6] C W Du M S Wang Z S Tan et al ldquoAnalytic solution forseepage field of subsea tunnel and its applicationrdquo ChineseJournal of Rock Mechanics and Engineering no s2pp 3567ndash3573 2011

[7] H W Ying C W Zhu H W Shen and X N Gong ldquoSemi-analytical solution for groundwater ingress into lined tunnelrdquoTunnelling and Underground Space Technology vol 76pp 43ndash47 2018

[8] L Tong K Xie M Lu et al ldquoAnalytical study of seepage flowinto a lined tunnel in a semi-infinite aquiferrdquo Rock and SoilMechanics vol 32 no 1 pp 304ndash308 2011

[9] D Ma Z Zhou J Wu Q Li and H Bai ldquoGrain size dis-tribution effect on the hydraulic properties of disintegratedcoal mixturesrdquo Energies vol 10 no 5 p 612 2017

[10] H Xiong K Zhao G Chen et al ldquoAnalysis of seepage field ofunderwater tunnel in a semi-infinite aquifer under waveactionrdquo China Earthquake Engineering Journal vol 36 no 4pp 919ndash923 2014

[11] D Ma Q Li M Hall and YWu ldquoExperimental investigationof stress rate and grain size on gas seepage characteristics ofgranular coalrdquo Energies vol 10 no 4 p 527 2017

[12] D Ma X Cai Z Zhou and X Li ldquoExperimental investigationon hydraulic properties of granular sandstone and mudstone

mixturesrdquo Geofluids vol 2018 Article ID 9216578 13 pages2018

[13] S Lei ldquoAn analytical solution for steady flow into a TtonnelrdquoGround Water vol 37 no 1 pp 23ndash26 1999

[14] M Huangfu M S Wang Z S Tan and X Y Wang ldquoAn-alytical solutions for steady seepage into an underwater cir-cular tunnelrdquo Tunnelling and Underground Space Technologyvol 25 no 4 pp 391ndash396 2010

[15] Y Kong Higher Fluid Mechanics University of Science andTechnology of China Press Hefei China 2010

Advances in Civil Engineering 9

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

F(z) f(z) + fa2

z1113888 1113889 (3)

33 Analysis of Seepage Field According to the complexpotential formula (3) of the point in the infinite plane of thetunnel formulas (1) and (2) are plugged into (3) as shown inthe following

F(z) q

2πln zminus z0( 1113857minus

q

2πln zminus

a2

z01113888 1113889 (4)

For simplicity without loss of generality the pointsource can be selected to the right of the horizontal directionof the tunnel that is the x-axis passes through the pointsource)en there is z0 z0 x0 equiv d as shown in Figure 3)erefore Formula (4) can be written as

F(z) q

2πln(zminus d)minus

q

2πln zminus

a2

d1113888 1113889

q

2πln

(xminusd)2 + y21113969

minus iq

2πarctg

y

xminusd

minusq

2πln

xminusa2

d1113888 1113889

2

+ y2

11139741113972

+ iq

2πarctg

y

xminus a2d( )

(5)

Complex velocity

W(z) dF

dz

(q2π)ln(zminusd)minus(q2π)ln zminus a2d( 1113857( 1113857

dz

q

2π1

zminusdminus

1zminus a2d( )

1113888 1113889

(6)

)e potential function of the point source in the infiniteplane of the tunnel

ϕ q

4πln

(xminusd)2 + y2

xminus a2d( )( )2

+ y2 (7)

)e flow function of the point source in the infinite planeof the tunnel

φ minusq

2πarctg

y

xminus d+

q

2πarctg

y

xminus a2d( ) (8)

Boundary conditionsCavern boundary

Φw minusq

2πln

rw

l (9)

Tunnel boundary

Φs minusq

2πln

d

a (10)

where l is the distance between the mirror point a2z0 andz0 rw is the radius of the cavern Φw is the cavern potentialfunction and Φs is the tunnel potential function

According to the boundary conditions and the steady-state percolation of the single-phase liquid [15] it can beobtained that

q 2π Φw minusΦs( 1113857

minusln arwld( 1113857

2π Φw minusΦs( 1113857

ln d2 minus a2arw( 1113857

Hw pw

ρg Hs

ps

ρg

Q 2πKρg Hw minusHs( 1113857

ln d2 minus a2arw( 1113857

(11)

where K is the permeability Pw and Ps are the headpressures of the cavern and tunnel respectively Q is the unitpermeate flow Hw is the head pressure of the cavern and Hs

is the head pressure of the tunnel boundary that is the headpressure of the surface of the grouting ring

34 Analysis of Seepage Field of Lining Structure )e seepagefield in the lining and grouting circle cannot be solved by thecomplex function but it can be solved by using thegroundwater seepage mechanics and the shaft theory [6]

x

y

CavernSecondary lining

Surrounding rocks

Initial supportGrouting ring

o

Figure 3 Schematic diagram of the relationship between eachlining structure

a

z 0

q

x

y

ndashq

a2 z 0

o

Figure 2 Schematic diagram of inversion of caverns and karsttunnels

Advances in Civil Engineering 3

)e relationship between the initial support the secondarylining and the grouting ring pressure head is shown inFigure 3 and the formulas are as follows

Hs minusH1 Q

2πk1ln

a

r1

H1 minusH2 Q

2πk2ln

r1

r2

H2 minusH3 Q

2πk3ln

r2r3

(12)

where Hs is the head pressure of the surface of the groutingring H1 H2and H3 are the head pressures of the joints ofthe grouting ring and the initial support the initial supportand the secondary lining joint and the secondary lininginner surface respectively r1 r2 and r3 are the radius of thegrouting ring initial support and internal of secondarylining respectively k1 k2 and k3 are the permeability co-efficients of the grouting circle initial support and sec-ondary lining respectively

Because of the same seepage the above equations can beintegrated to obtain the tunnel seepage when the groutingcircle the initial support and the two lining are completed

Q 2π Hw minusH3( 1113857

1k2( 1113857ln r1r2( 1113857 + 1k1( 1113857ln ar1( 1113857 + 1k3( 1113857ln r2r3( 1113857 +(1k)ln d2 minus a2arw( 1113857 (13)

Head pressure of initial support and secondary liningjoint

H2 H3 +Hw minusH3( 1113857

k3Aln

r2

r3 (14)

Head pressure of grouting circle and initial supportconnection

H1 H3 +Hw minusH3( 1113857

k3Aln

r2

r3+

Hw minusH3( 1113857

k2Aln

r1

r2 (15)

In the formula

A 1k2

lnr1

r21113888 1113889 +

1k1

lna

r11113888 1113889 +

1k3

lnr2

r31113888 1113889 +

1kln

d2 minus a2

arw

1113888 1113889

(16)

where k is the permeability coefficients of the surroundingrock

According to different stages of construction such astunnel holes and grouting the different water pressuredistribution can be obtained When tunnel grouting andinitial support are completed the tunnel seepage is

Q 2π Hw minusH2( 1113857

1k2( 1113857ln r1r2( 1113857 + 1k1( 1113857ln rr1( 1113857 +(1k)ln d2 minus a2arw( 1113857

(17)

Head pressure of grouting circle and initial supportconnection

H1 H2 +Hw minusH2( 1113857ln r1r2( 1113857

ln r1r2( 1113857 + k2k1( 1113857ln rr1( 1113857 + k2k( 1113857ln d2 minus a2arw( 1113857

(18)

4 Verification and Discussion

41 Numerical Verification Based on the complex functionand groundwater hydraulics theory the distribution ofseepage field suitable for homogeneous and isotropic karsttunnels and lining structures were deduced In order to

ensure the accuracy of the analytical solution of the karsttunnel the numerical solution simulated by the finite dif-ference software FLAC3D is compared with the theoreticalsolution obtained by the method used in this paper

)e calculation conditions are listed as follows the headpressure of the cavern is 54m the radius of the cavern is 4mthe radius of the tunnel is 725m the radius of the groutingcircle is 225m and the initial support radius is 025m thepermeability coefficient of surrounding rocks is15times10minus6 cmmiddotsminus1 the permeability coefficient of groutingring is 10minus7 cmmiddotsminus1 the initial support permeability co-efficient of initial support is 10minus8 cmmiddotsminus1

Six groups of 100mtimes 100m models of caverns on theright side of the tunnel were established and calculated withFLAC3D )e pore water pressure of the tunnel with gravityand without gravity was obtained respectively )e sixgroups of models are d 15 m d 20 m d 25 md 30 m d 35 m and d 40 m When d 20 m thekarst tunnel model consists of 4236 elements as shown inFigure 4 Figures 5 and 6 are contours of zone pore pressurewith gravity and without gravity respectively

)e pore water pressures extracted from the six groups ofnumerical simulation data in Figure 5 with gravity andFigure 6 without gravity are compared with the theoreticalsolutions )e results are shown in Figure 7 )e law ob-tained by numerical and theoretical solutions is that whenthe grouting ring and the initial support are completed andthe secondary lining is not yet completed H1 decreases withthe increase of d Moreover the curves have a good con-sistency When d 15 m the analytical solution for H1 is2578m the numerical solution with gravity is 2567m andthe numerical solution without gravity is 2577m)ereforethe analytical method in this paper is suitable for solving thedistribution of seepage field in karst tunnels

420eEffect ofDistance andCavernRadius Figure 8 showsthe relationship between H1 and d Figure 9 shows therelationship between Q and rw In the infinite plane whenHs is constant the main factors affecting H1 and Q in the

4 Advances in Civil Engineering

tunnel are rw and dWhen rw is constant H1 and Q decreasewith the increase of d When d is constant H1 and Q

gradually increase as rw increases

43 0e Effect of Grouting Ring When the diameter of thegrouting ring and the initial support radius is constant thetunnel radius and the thickness of the grouting ring increaseFigures 10ndash13 show that H1 and Q gradually decrease withthe increase of the thickness of the grouting ring when theratio between the distance and the permeability coefficient isconstant When the thickness of the grouting ring was025m and a reached 6m Q and H1 decreased to 4194and 4186 respectively When a is constant H1 and Q

gradually decrease as the permeability coefficient decreasesWhen a increases to 7m the rate of H1 and Q decreasegradually From Figures 12 and 13 when a is constant H1

becomes smaller as the seepage coefficient of the groutingring decreases When kr kk1 increases to 80 Q decreasesto 224 of Q when kr 5 and the Q reduction rategradually decreases In summary increasing the thickness ofthe grouting ring and reducing the permeability coefficientof the grouting ring can reduce Q and H1 However ex-cessively increasing the thickness of the grouting ring willnot only increase the cost but also reduce the impact on thepressure head and seepage flow )erefore a reasonablegrouting material and construction process should be se-lected to ensure the thickness of the grouting ring when thepermeability coefficient of the grouting ring is less than thespecified value

440e Effect of Initial Support When the inner diameter r1of the grouting ring is constant increasing r2 is to reduce the

52900E + 0552500E + 05

47500E + 05

42500E + 0545000E + 05

50000E + 05

40000E + 0537500E + 05

32500E + 0535000E + 05

30000E + 0527500E + 05

22500E + 0525000E + 05

20000E + 0517500E + 05

12500E + 0515000E + 05

11674E + 05

Figure 6 Contour of pore water pressure without gravity ford 20 a 725 and rw 4

15 20 25 30 35 40240

242

244

246

248

250

252

254

256

258

H1 (

m)

d (m)

Unweighted numerical solutionWeighted numerical solutionTheoretical analytical solution

Figure 7 Comparison of numerical solution and theoretical so-lution for a 725 and rw 4

100 m10

0 my

x

Figure 4 Finite element model with circular karst tunnel

97843E + 0595000E + 0590000E + 0585000E + 0580000E + 0575000E + 0570000E + 0565000E + 0560000E + 0555000E + 05

45000E + 05

35000E + 05

25000E + 05

15000E + 05

50000E + 05

40000E + 05

30000E + 05

20000E + 05

10000E + 0550000E + 0420664E + 04

Figure 5 Contour of pore water pressure with gravity for d 20a 725 and rw 4

Advances in Civil Engineering 5

thickness of the initial support Figures 14 and 15 show thatas the thickness of the initial support decreases H1 decreasesand Q increases gradually When r2 225 m Q and H1 are133 of Q and 207 of H1 when r2 475 m respectivelyIt shows that the initial support thickness has a great in-fluence on the pressure head During construction weshould pay attention to the safety range of the initial supportthickness

If the initial diameter r2 and a are constant increasing r1means increasing the thickness of the initial support andreducing the thickness of the grouting ring In Figures 16

and 17 with the increase of r1 H1 gradually increases and Q

gradually decreasesAs shown in Figures 18 and 19 the permeability

coefficient of grouting ring is constant and H1 graduallydecreases and Q gradually increases with the increase ofk2 )e change rate of Q and H1 is larger whenk2 lt 10minus8 cmmiddotsminus1 When k2 2 times 10minus8 cmmiddotsminus1 both Q andH1 tend to be stable When the other conditions areconstant the change of k2 on the magnitude of 10minus8 haslittle effect on Q and H1 but the influence on themagnitude of 10minus9 is greater

24

26

28

30

32

34

36

38

40

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

r1 = 225m r2 = 2m Hw = 54mk1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

3 4 5 6 7 8a (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 10 Effect of d on the relationship between pressure headand tunnel radius

3 4 5 6 7128

130

132

134

136

138

140

142

144

Q (m

3 ∙sndash1∙m

ndash1)

rw (m)

d = 15 md = 25 md = 35 m

d = 20 md = 30 md = 40 m

(times10ndash5)

a = 725 mk1 = 10ndash7 mmiddotsndash1

r1 = 225 m

k = 15 times 10ndash7 mmiddotsndash1

k2 = 10ndash8 mmiddotsndash1

Hw = 54 mr2 = 2 m

Figure 9 Relationship between seepage flow and cavern radius

10 15 20 25 30 35 40 45 50 55235

240

245

250

255

260

265

rw = 3 mrw = 4 mrw = 5 m

rw = 6 mrw = 7 m

H1 (

m)

d (m)

a = 725 mk1 = 10ndash7 mmiddotsndash1

r1 = 225 m

k = 15 times 10ndash7 mmiddotsndash1

k2 = 10ndash8 mmiddotsndash1

Hw = 54 mr2 = 2 m

Figure 8 Relationship between head pressure and distance

3 4 5 6 7 8

13

14

15

16

17

18

19

2

21

Q (m

3 ∙sndash1∙m

ndash1)

a (m)

rw = 4m k = 15times10ndash7m∙sndash1

r1 = 225m r2 = 2m Hw = 54m

d = 15md = 20md = 25m

d = 30md = 35md = 40m

k1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

(times10ndash5)

Figure 11 Relationship between seepage flow and tunnel radius

6 Advances in Civil Engineering

5 Conclusion

Based on the inversion of complex function and ground-water hydraulics theory this paper derives the analyticalsolution for the karst seepage field in the karst tunnel in aninfinite plane )e potential function the flow function andthe complex velocity at any place in the infinite plane aresolved According to the distribution of the seepage field andthe boundary conditions the seepage flow of the karst tunnelis obtained )e seepage field of the lining structure is de-duced from the relationship between the grouting ring theinitial support and the secondary lining )e new solution

was verified with numerical simulation via the softwareFLAC3D )e conclusions obtained from this study aresummarized as follows

(1) When Hs is constant for karst tunnels rw and d arethe main factors affecting the pressure head andseepage flow of the tunnel When rw is constant H1and Q decrease with the increase of d When d isconstant H1 and Q increase with the increase of rw

(2) When the initial support is completed and thesecondary lining has not been constructed yet H1

3 4 5 6 7 85

10

15

20

25

30

35

40

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

k2 = 10ndash8m∙sndash1 d = 15mr1 = 225m r2 = 2m Hw = 54m

a (m)

kr = 20kr = 100

kr = 10kr = 50

Figure 12 Effect of kr on the relationship between pressure headand tunnel radius

20

25

30

35

40

45

50

55

H1 (

m)

rw = 4m k = 15times10ndash7m∙sndash1

a = 725m r1 = 5m Hw = 54mk1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

20 25 30 35 40 45 50r2 (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 14 Relationship between pressure head and initial supportradius

20 25 30 35 40 45 500

5

10

15

20

25

30

35

rw = 4m k = 15 times 10ndash7m∙sndash1

a = 725m r2 = 2m Hw = 54mk1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

Q (m

3 ∙sndash1∙m

ndash1)

r2 (m)

(times10ndash6)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 15 Relationship between seepage flow and initial supportradius

0 20 40 60 80 1002

4

6

8

10

12

14

16

18

20

22

kr

(times10ndash6)

rw = 4m k = 15times10ndash7m∙sndash1

r1 = 225m r2 = 2m Hw = 54m

Q (m

3 ∙sndash1∙m

ndash1)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

k2 = 10ndash8m∙sndash1 d = 15m

Figure 13 Relationship between seepage flow and permeabilitycoefficient

Advances in Civil Engineering 7

and Q will gradually decrease with the increase of thethickness and the decrease of the permeability co-efficient of the grouting ring When the permeabilitycoefficient of grouting ring is less than the specifiedvalue reasonable grouting material and constructiontechnology should be selected to ensure the thicknessof grouting ring

(3) H1 gradually decreases with the decrease of thethickness and the increase of k2 However Q

gradually increases with the decrease of the thicknessand the increase of k2 During construction we

should pay attention to the safety range of the initialsupport thickness

Data Availability

Based on the ldquohistoryrdquo function of FLAC3D software thedata which recorded the pore water pressure during thestable period of the monitoring point were obtainedAccording to the simulation conditions in Chapter 4 readerscan complete the modeling process and obtain the contourof pore water pressure such as Figures 5 and 6 )e data ofthe pore water pressure in the stable period of the moni-toring point were compared with the analytical solution to

20

25

30

35

40

45

50

55

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

a = 725m r2 = 2m Hw = 54mk1 = 10ndash7m∙sndash1 k2=10ndash8m∙sndash1

2 3 4 5 6 7r1 (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 16 Relationship between pressure head and radius ofgrouting ring

Q (m

3 ∙sndash1∙m

ndash1)

rw = 4m k = 15 times 10ndash7m∙sndash1

a = 725m r2 = 2m Hw = 54m

(times10ndash6)

14

12

10

8

6

4

2

2 3 4 5 6 7r1 (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

k1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

Figure 17 Relationship between seepage flow and radius ofgrouting ring

00

1 2 3 4 5

10

20

30

40

50

k2 (msndash1)

kr = 20kr = 10

kr = 100kr = 50

(times10ndash8)

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

r1 = 225m d = 15ma = 725m r2 = 2m Hw = 54m

Figure 18 Relationship between pressure head and permeabilitycoefficient of grouting ring

0 1 2 3 4 5

5

10

15

20

25

30

Q (m

3 ∙sndash1∙m

ndash1)

k2 (msndash1)

rw = 4m k = 15 times 10ndash7m∙sndash1

r1 = 225m d = 15ma = 725m r2 = 2m Hw = 54m

kr = 20kr = 100

kr = 10kr = 50

(times10ndash6)

(times10ndash8)

Figure 19 Relationship between seepage flow and permeabilitycoefficient of grouting ring

8 Advances in Civil Engineering

verify the accuracy )e author only recorded data of thecorresponding monitoring point in corresponding area forresearch needs In addition all the sharing data were in-volved in this paper and there were no redundant data )eparameters of the lining structure in the numerical simu-lation conditions come from [6]

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (Grant nos 51478479 and 51678570)and Hunan Transport Technology Project (Grant no201524)

References

[1] M D Zhang Study on the Characteristics of Seepage Field andWater Pressure on Tunnel Lining in Karst Ground BeijingJiaotong University Beijing China 2008

[2] B Q Zhang ldquoAnalytical solution for seepage field of twin-parallel tunnel in semi-infinite planerdquo Journal of 0e ChinaRailway Society vol 39 no 1 pp 125ndash131 2017

[3] B Yang F Gao and D Jeng ldquoFailure mode and dynamicresponse of a double-sided slope with high water content ofsoilrdquo Journal of Mountain Science vol 15 no 4 pp 859ndash8702018

[4] J G Wu A Coupled Fluid-Mechanical Study on Seepage Fieldof Tunnels with High Hydraulic Pressure Beijing JiaotongUniversity Beijing China 2006

[5] K H Park A Owatsiriwong and J G Lee ldquoAnalytical so-lution for steady-state groundwater inflow into a drainedcircular tunnel in a semi-infinite aquifer A revisitrdquo Tunnellingand Underground Space Technology vol 23 no 2 pp 206ndash209 2008

[6] C W Du M S Wang Z S Tan et al ldquoAnalytic solution forseepage field of subsea tunnel and its applicationrdquo ChineseJournal of Rock Mechanics and Engineering no s2pp 3567ndash3573 2011

[7] H W Ying C W Zhu H W Shen and X N Gong ldquoSemi-analytical solution for groundwater ingress into lined tunnelrdquoTunnelling and Underground Space Technology vol 76pp 43ndash47 2018

[8] L Tong K Xie M Lu et al ldquoAnalytical study of seepage flowinto a lined tunnel in a semi-infinite aquiferrdquo Rock and SoilMechanics vol 32 no 1 pp 304ndash308 2011

[9] D Ma Z Zhou J Wu Q Li and H Bai ldquoGrain size dis-tribution effect on the hydraulic properties of disintegratedcoal mixturesrdquo Energies vol 10 no 5 p 612 2017

[10] H Xiong K Zhao G Chen et al ldquoAnalysis of seepage field ofunderwater tunnel in a semi-infinite aquifer under waveactionrdquo China Earthquake Engineering Journal vol 36 no 4pp 919ndash923 2014

[11] D Ma Q Li M Hall and YWu ldquoExperimental investigationof stress rate and grain size on gas seepage characteristics ofgranular coalrdquo Energies vol 10 no 4 p 527 2017

[12] D Ma X Cai Z Zhou and X Li ldquoExperimental investigationon hydraulic properties of granular sandstone and mudstone

mixturesrdquo Geofluids vol 2018 Article ID 9216578 13 pages2018

[13] S Lei ldquoAn analytical solution for steady flow into a TtonnelrdquoGround Water vol 37 no 1 pp 23ndash26 1999

[14] M Huangfu M S Wang Z S Tan and X Y Wang ldquoAn-alytical solutions for steady seepage into an underwater cir-cular tunnelrdquo Tunnelling and Underground Space Technologyvol 25 no 4 pp 391ndash396 2010

[15] Y Kong Higher Fluid Mechanics University of Science andTechnology of China Press Hefei China 2010

Advances in Civil Engineering 9

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

)e relationship between the initial support the secondarylining and the grouting ring pressure head is shown inFigure 3 and the formulas are as follows

Hs minusH1 Q

2πk1ln

a

r1

H1 minusH2 Q

2πk2ln

r1

r2

H2 minusH3 Q

2πk3ln

r2r3

(12)

where Hs is the head pressure of the surface of the groutingring H1 H2and H3 are the head pressures of the joints ofthe grouting ring and the initial support the initial supportand the secondary lining joint and the secondary lininginner surface respectively r1 r2 and r3 are the radius of thegrouting ring initial support and internal of secondarylining respectively k1 k2 and k3 are the permeability co-efficients of the grouting circle initial support and sec-ondary lining respectively

Because of the same seepage the above equations can beintegrated to obtain the tunnel seepage when the groutingcircle the initial support and the two lining are completed

Q 2π Hw minusH3( 1113857

1k2( 1113857ln r1r2( 1113857 + 1k1( 1113857ln ar1( 1113857 + 1k3( 1113857ln r2r3( 1113857 +(1k)ln d2 minus a2arw( 1113857 (13)

Head pressure of initial support and secondary liningjoint

H2 H3 +Hw minusH3( 1113857

k3Aln

r2

r3 (14)

Head pressure of grouting circle and initial supportconnection

H1 H3 +Hw minusH3( 1113857

k3Aln

r2

r3+

Hw minusH3( 1113857

k2Aln

r1

r2 (15)

In the formula

A 1k2

lnr1

r21113888 1113889 +

1k1

lna

r11113888 1113889 +

1k3

lnr2

r31113888 1113889 +

1kln

d2 minus a2

arw

1113888 1113889

(16)

where k is the permeability coefficients of the surroundingrock

According to different stages of construction such astunnel holes and grouting the different water pressuredistribution can be obtained When tunnel grouting andinitial support are completed the tunnel seepage is

Q 2π Hw minusH2( 1113857

1k2( 1113857ln r1r2( 1113857 + 1k1( 1113857ln rr1( 1113857 +(1k)ln d2 minus a2arw( 1113857

(17)

Head pressure of grouting circle and initial supportconnection

H1 H2 +Hw minusH2( 1113857ln r1r2( 1113857

ln r1r2( 1113857 + k2k1( 1113857ln rr1( 1113857 + k2k( 1113857ln d2 minus a2arw( 1113857

(18)

4 Verification and Discussion

41 Numerical Verification Based on the complex functionand groundwater hydraulics theory the distribution ofseepage field suitable for homogeneous and isotropic karsttunnels and lining structures were deduced In order to

ensure the accuracy of the analytical solution of the karsttunnel the numerical solution simulated by the finite dif-ference software FLAC3D is compared with the theoreticalsolution obtained by the method used in this paper

)e calculation conditions are listed as follows the headpressure of the cavern is 54m the radius of the cavern is 4mthe radius of the tunnel is 725m the radius of the groutingcircle is 225m and the initial support radius is 025m thepermeability coefficient of surrounding rocks is15times10minus6 cmmiddotsminus1 the permeability coefficient of groutingring is 10minus7 cmmiddotsminus1 the initial support permeability co-efficient of initial support is 10minus8 cmmiddotsminus1

Six groups of 100mtimes 100m models of caverns on theright side of the tunnel were established and calculated withFLAC3D )e pore water pressure of the tunnel with gravityand without gravity was obtained respectively )e sixgroups of models are d 15 m d 20 m d 25 md 30 m d 35 m and d 40 m When d 20 m thekarst tunnel model consists of 4236 elements as shown inFigure 4 Figures 5 and 6 are contours of zone pore pressurewith gravity and without gravity respectively

)e pore water pressures extracted from the six groups ofnumerical simulation data in Figure 5 with gravity andFigure 6 without gravity are compared with the theoreticalsolutions )e results are shown in Figure 7 )e law ob-tained by numerical and theoretical solutions is that whenthe grouting ring and the initial support are completed andthe secondary lining is not yet completed H1 decreases withthe increase of d Moreover the curves have a good con-sistency When d 15 m the analytical solution for H1 is2578m the numerical solution with gravity is 2567m andthe numerical solution without gravity is 2577m)ereforethe analytical method in this paper is suitable for solving thedistribution of seepage field in karst tunnels

420eEffect ofDistance andCavernRadius Figure 8 showsthe relationship between H1 and d Figure 9 shows therelationship between Q and rw In the infinite plane whenHs is constant the main factors affecting H1 and Q in the

4 Advances in Civil Engineering

tunnel are rw and dWhen rw is constant H1 and Q decreasewith the increase of d When d is constant H1 and Q

gradually increase as rw increases

43 0e Effect of Grouting Ring When the diameter of thegrouting ring and the initial support radius is constant thetunnel radius and the thickness of the grouting ring increaseFigures 10ndash13 show that H1 and Q gradually decrease withthe increase of the thickness of the grouting ring when theratio between the distance and the permeability coefficient isconstant When the thickness of the grouting ring was025m and a reached 6m Q and H1 decreased to 4194and 4186 respectively When a is constant H1 and Q

gradually decrease as the permeability coefficient decreasesWhen a increases to 7m the rate of H1 and Q decreasegradually From Figures 12 and 13 when a is constant H1

becomes smaller as the seepage coefficient of the groutingring decreases When kr kk1 increases to 80 Q decreasesto 224 of Q when kr 5 and the Q reduction rategradually decreases In summary increasing the thickness ofthe grouting ring and reducing the permeability coefficientof the grouting ring can reduce Q and H1 However ex-cessively increasing the thickness of the grouting ring willnot only increase the cost but also reduce the impact on thepressure head and seepage flow )erefore a reasonablegrouting material and construction process should be se-lected to ensure the thickness of the grouting ring when thepermeability coefficient of the grouting ring is less than thespecified value

440e Effect of Initial Support When the inner diameter r1of the grouting ring is constant increasing r2 is to reduce the

52900E + 0552500E + 05

47500E + 05

42500E + 0545000E + 05

50000E + 05

40000E + 0537500E + 05

32500E + 0535000E + 05

30000E + 0527500E + 05

22500E + 0525000E + 05

20000E + 0517500E + 05

12500E + 0515000E + 05

11674E + 05

Figure 6 Contour of pore water pressure without gravity ford 20 a 725 and rw 4

15 20 25 30 35 40240

242

244

246

248

250

252

254

256

258

H1 (

m)

d (m)

Unweighted numerical solutionWeighted numerical solutionTheoretical analytical solution

Figure 7 Comparison of numerical solution and theoretical so-lution for a 725 and rw 4

100 m10

0 my

x

Figure 4 Finite element model with circular karst tunnel

97843E + 0595000E + 0590000E + 0585000E + 0580000E + 0575000E + 0570000E + 0565000E + 0560000E + 0555000E + 05

45000E + 05

35000E + 05

25000E + 05

15000E + 05

50000E + 05

40000E + 05

30000E + 05

20000E + 05

10000E + 0550000E + 0420664E + 04

Figure 5 Contour of pore water pressure with gravity for d 20a 725 and rw 4

Advances in Civil Engineering 5

thickness of the initial support Figures 14 and 15 show thatas the thickness of the initial support decreases H1 decreasesand Q increases gradually When r2 225 m Q and H1 are133 of Q and 207 of H1 when r2 475 m respectivelyIt shows that the initial support thickness has a great in-fluence on the pressure head During construction weshould pay attention to the safety range of the initial supportthickness

If the initial diameter r2 and a are constant increasing r1means increasing the thickness of the initial support andreducing the thickness of the grouting ring In Figures 16

and 17 with the increase of r1 H1 gradually increases and Q

gradually decreasesAs shown in Figures 18 and 19 the permeability

coefficient of grouting ring is constant and H1 graduallydecreases and Q gradually increases with the increase ofk2 )e change rate of Q and H1 is larger whenk2 lt 10minus8 cmmiddotsminus1 When k2 2 times 10minus8 cmmiddotsminus1 both Q andH1 tend to be stable When the other conditions areconstant the change of k2 on the magnitude of 10minus8 haslittle effect on Q and H1 but the influence on themagnitude of 10minus9 is greater

24

26

28

30

32

34

36

38

40

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

r1 = 225m r2 = 2m Hw = 54mk1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

3 4 5 6 7 8a (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 10 Effect of d on the relationship between pressure headand tunnel radius

3 4 5 6 7128

130

132

134

136

138

140

142

144

Q (m

3 ∙sndash1∙m

ndash1)

rw (m)

d = 15 md = 25 md = 35 m

d = 20 md = 30 md = 40 m

(times10ndash5)

a = 725 mk1 = 10ndash7 mmiddotsndash1

r1 = 225 m

k = 15 times 10ndash7 mmiddotsndash1

k2 = 10ndash8 mmiddotsndash1

Hw = 54 mr2 = 2 m

Figure 9 Relationship between seepage flow and cavern radius

10 15 20 25 30 35 40 45 50 55235

240

245

250

255

260

265

rw = 3 mrw = 4 mrw = 5 m

rw = 6 mrw = 7 m

H1 (

m)

d (m)

a = 725 mk1 = 10ndash7 mmiddotsndash1

r1 = 225 m

k = 15 times 10ndash7 mmiddotsndash1

k2 = 10ndash8 mmiddotsndash1

Hw = 54 mr2 = 2 m

Figure 8 Relationship between head pressure and distance

3 4 5 6 7 8

13

14

15

16

17

18

19

2

21

Q (m

3 ∙sndash1∙m

ndash1)

a (m)

rw = 4m k = 15times10ndash7m∙sndash1

r1 = 225m r2 = 2m Hw = 54m

d = 15md = 20md = 25m

d = 30md = 35md = 40m

k1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

(times10ndash5)

Figure 11 Relationship between seepage flow and tunnel radius

6 Advances in Civil Engineering

5 Conclusion

Based on the inversion of complex function and ground-water hydraulics theory this paper derives the analyticalsolution for the karst seepage field in the karst tunnel in aninfinite plane )e potential function the flow function andthe complex velocity at any place in the infinite plane aresolved According to the distribution of the seepage field andthe boundary conditions the seepage flow of the karst tunnelis obtained )e seepage field of the lining structure is de-duced from the relationship between the grouting ring theinitial support and the secondary lining )e new solution

was verified with numerical simulation via the softwareFLAC3D )e conclusions obtained from this study aresummarized as follows

(1) When Hs is constant for karst tunnels rw and d arethe main factors affecting the pressure head andseepage flow of the tunnel When rw is constant H1and Q decrease with the increase of d When d isconstant H1 and Q increase with the increase of rw

(2) When the initial support is completed and thesecondary lining has not been constructed yet H1

3 4 5 6 7 85

10

15

20

25

30

35

40

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

k2 = 10ndash8m∙sndash1 d = 15mr1 = 225m r2 = 2m Hw = 54m

a (m)

kr = 20kr = 100

kr = 10kr = 50

Figure 12 Effect of kr on the relationship between pressure headand tunnel radius

20

25

30

35

40

45

50

55

H1 (

m)

rw = 4m k = 15times10ndash7m∙sndash1

a = 725m r1 = 5m Hw = 54mk1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

20 25 30 35 40 45 50r2 (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 14 Relationship between pressure head and initial supportradius

20 25 30 35 40 45 500

5

10

15

20

25

30

35

rw = 4m k = 15 times 10ndash7m∙sndash1

a = 725m r2 = 2m Hw = 54mk1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

Q (m

3 ∙sndash1∙m

ndash1)

r2 (m)

(times10ndash6)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 15 Relationship between seepage flow and initial supportradius

0 20 40 60 80 1002

4

6

8

10

12

14

16

18

20

22

kr

(times10ndash6)

rw = 4m k = 15times10ndash7m∙sndash1

r1 = 225m r2 = 2m Hw = 54m

Q (m

3 ∙sndash1∙m

ndash1)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

k2 = 10ndash8m∙sndash1 d = 15m

Figure 13 Relationship between seepage flow and permeabilitycoefficient

Advances in Civil Engineering 7

and Q will gradually decrease with the increase of thethickness and the decrease of the permeability co-efficient of the grouting ring When the permeabilitycoefficient of grouting ring is less than the specifiedvalue reasonable grouting material and constructiontechnology should be selected to ensure the thicknessof grouting ring

(3) H1 gradually decreases with the decrease of thethickness and the increase of k2 However Q

gradually increases with the decrease of the thicknessand the increase of k2 During construction we

should pay attention to the safety range of the initialsupport thickness

Data Availability

Based on the ldquohistoryrdquo function of FLAC3D software thedata which recorded the pore water pressure during thestable period of the monitoring point were obtainedAccording to the simulation conditions in Chapter 4 readerscan complete the modeling process and obtain the contourof pore water pressure such as Figures 5 and 6 )e data ofthe pore water pressure in the stable period of the moni-toring point were compared with the analytical solution to

20

25

30

35

40

45

50

55

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

a = 725m r2 = 2m Hw = 54mk1 = 10ndash7m∙sndash1 k2=10ndash8m∙sndash1

2 3 4 5 6 7r1 (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 16 Relationship between pressure head and radius ofgrouting ring

Q (m

3 ∙sndash1∙m

ndash1)

rw = 4m k = 15 times 10ndash7m∙sndash1

a = 725m r2 = 2m Hw = 54m

(times10ndash6)

14

12

10

8

6

4

2

2 3 4 5 6 7r1 (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

k1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

Figure 17 Relationship between seepage flow and radius ofgrouting ring

00

1 2 3 4 5

10

20

30

40

50

k2 (msndash1)

kr = 20kr = 10

kr = 100kr = 50

(times10ndash8)

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

r1 = 225m d = 15ma = 725m r2 = 2m Hw = 54m

Figure 18 Relationship between pressure head and permeabilitycoefficient of grouting ring

0 1 2 3 4 5

5

10

15

20

25

30

Q (m

3 ∙sndash1∙m

ndash1)

k2 (msndash1)

rw = 4m k = 15 times 10ndash7m∙sndash1

r1 = 225m d = 15ma = 725m r2 = 2m Hw = 54m

kr = 20kr = 100

kr = 10kr = 50

(times10ndash6)

(times10ndash8)

Figure 19 Relationship between seepage flow and permeabilitycoefficient of grouting ring

8 Advances in Civil Engineering

verify the accuracy )e author only recorded data of thecorresponding monitoring point in corresponding area forresearch needs In addition all the sharing data were in-volved in this paper and there were no redundant data )eparameters of the lining structure in the numerical simu-lation conditions come from [6]

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (Grant nos 51478479 and 51678570)and Hunan Transport Technology Project (Grant no201524)

References

[1] M D Zhang Study on the Characteristics of Seepage Field andWater Pressure on Tunnel Lining in Karst Ground BeijingJiaotong University Beijing China 2008

[2] B Q Zhang ldquoAnalytical solution for seepage field of twin-parallel tunnel in semi-infinite planerdquo Journal of 0e ChinaRailway Society vol 39 no 1 pp 125ndash131 2017

[3] B Yang F Gao and D Jeng ldquoFailure mode and dynamicresponse of a double-sided slope with high water content ofsoilrdquo Journal of Mountain Science vol 15 no 4 pp 859ndash8702018

[4] J G Wu A Coupled Fluid-Mechanical Study on Seepage Fieldof Tunnels with High Hydraulic Pressure Beijing JiaotongUniversity Beijing China 2006

[5] K H Park A Owatsiriwong and J G Lee ldquoAnalytical so-lution for steady-state groundwater inflow into a drainedcircular tunnel in a semi-infinite aquifer A revisitrdquo Tunnellingand Underground Space Technology vol 23 no 2 pp 206ndash209 2008

[6] C W Du M S Wang Z S Tan et al ldquoAnalytic solution forseepage field of subsea tunnel and its applicationrdquo ChineseJournal of Rock Mechanics and Engineering no s2pp 3567ndash3573 2011

[7] H W Ying C W Zhu H W Shen and X N Gong ldquoSemi-analytical solution for groundwater ingress into lined tunnelrdquoTunnelling and Underground Space Technology vol 76pp 43ndash47 2018

[8] L Tong K Xie M Lu et al ldquoAnalytical study of seepage flowinto a lined tunnel in a semi-infinite aquiferrdquo Rock and SoilMechanics vol 32 no 1 pp 304ndash308 2011

[9] D Ma Z Zhou J Wu Q Li and H Bai ldquoGrain size dis-tribution effect on the hydraulic properties of disintegratedcoal mixturesrdquo Energies vol 10 no 5 p 612 2017

[10] H Xiong K Zhao G Chen et al ldquoAnalysis of seepage field ofunderwater tunnel in a semi-infinite aquifer under waveactionrdquo China Earthquake Engineering Journal vol 36 no 4pp 919ndash923 2014

[11] D Ma Q Li M Hall and YWu ldquoExperimental investigationof stress rate and grain size on gas seepage characteristics ofgranular coalrdquo Energies vol 10 no 4 p 527 2017

[12] D Ma X Cai Z Zhou and X Li ldquoExperimental investigationon hydraulic properties of granular sandstone and mudstone

mixturesrdquo Geofluids vol 2018 Article ID 9216578 13 pages2018

[13] S Lei ldquoAn analytical solution for steady flow into a TtonnelrdquoGround Water vol 37 no 1 pp 23ndash26 1999

[14] M Huangfu M S Wang Z S Tan and X Y Wang ldquoAn-alytical solutions for steady seepage into an underwater cir-cular tunnelrdquo Tunnelling and Underground Space Technologyvol 25 no 4 pp 391ndash396 2010

[15] Y Kong Higher Fluid Mechanics University of Science andTechnology of China Press Hefei China 2010

Advances in Civil Engineering 9

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tunnel are rw and dWhen rw is constant H1 and Q decreasewith the increase of d When d is constant H1 and Q

gradually increase as rw increases

43 0e Effect of Grouting Ring When the diameter of thegrouting ring and the initial support radius is constant thetunnel radius and the thickness of the grouting ring increaseFigures 10ndash13 show that H1 and Q gradually decrease withthe increase of the thickness of the grouting ring when theratio between the distance and the permeability coefficient isconstant When the thickness of the grouting ring was025m and a reached 6m Q and H1 decreased to 4194and 4186 respectively When a is constant H1 and Q

gradually decrease as the permeability coefficient decreasesWhen a increases to 7m the rate of H1 and Q decreasegradually From Figures 12 and 13 when a is constant H1

becomes smaller as the seepage coefficient of the groutingring decreases When kr kk1 increases to 80 Q decreasesto 224 of Q when kr 5 and the Q reduction rategradually decreases In summary increasing the thickness ofthe grouting ring and reducing the permeability coefficientof the grouting ring can reduce Q and H1 However ex-cessively increasing the thickness of the grouting ring willnot only increase the cost but also reduce the impact on thepressure head and seepage flow )erefore a reasonablegrouting material and construction process should be se-lected to ensure the thickness of the grouting ring when thepermeability coefficient of the grouting ring is less than thespecified value

440e Effect of Initial Support When the inner diameter r1of the grouting ring is constant increasing r2 is to reduce the

52900E + 0552500E + 05

47500E + 05

42500E + 0545000E + 05

50000E + 05

40000E + 0537500E + 05

32500E + 0535000E + 05

30000E + 0527500E + 05

22500E + 0525000E + 05

20000E + 0517500E + 05

12500E + 0515000E + 05

11674E + 05

Figure 6 Contour of pore water pressure without gravity ford 20 a 725 and rw 4

15 20 25 30 35 40240

242

244

246

248

250

252

254

256

258

H1 (

m)

d (m)

Unweighted numerical solutionWeighted numerical solutionTheoretical analytical solution

Figure 7 Comparison of numerical solution and theoretical so-lution for a 725 and rw 4

100 m10

0 my

x

Figure 4 Finite element model with circular karst tunnel

97843E + 0595000E + 0590000E + 0585000E + 0580000E + 0575000E + 0570000E + 0565000E + 0560000E + 0555000E + 05

45000E + 05

35000E + 05

25000E + 05

15000E + 05

50000E + 05

40000E + 05

30000E + 05

20000E + 05

10000E + 0550000E + 0420664E + 04

Figure 5 Contour of pore water pressure with gravity for d 20a 725 and rw 4

Advances in Civil Engineering 5

thickness of the initial support Figures 14 and 15 show thatas the thickness of the initial support decreases H1 decreasesand Q increases gradually When r2 225 m Q and H1 are133 of Q and 207 of H1 when r2 475 m respectivelyIt shows that the initial support thickness has a great in-fluence on the pressure head During construction weshould pay attention to the safety range of the initial supportthickness

If the initial diameter r2 and a are constant increasing r1means increasing the thickness of the initial support andreducing the thickness of the grouting ring In Figures 16

and 17 with the increase of r1 H1 gradually increases and Q

gradually decreasesAs shown in Figures 18 and 19 the permeability

coefficient of grouting ring is constant and H1 graduallydecreases and Q gradually increases with the increase ofk2 )e change rate of Q and H1 is larger whenk2 lt 10minus8 cmmiddotsminus1 When k2 2 times 10minus8 cmmiddotsminus1 both Q andH1 tend to be stable When the other conditions areconstant the change of k2 on the magnitude of 10minus8 haslittle effect on Q and H1 but the influence on themagnitude of 10minus9 is greater

24

26

28

30

32

34

36

38

40

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

r1 = 225m r2 = 2m Hw = 54mk1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

3 4 5 6 7 8a (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 10 Effect of d on the relationship between pressure headand tunnel radius

3 4 5 6 7128

130

132

134

136

138

140

142

144

Q (m

3 ∙sndash1∙m

ndash1)

rw (m)

d = 15 md = 25 md = 35 m

d = 20 md = 30 md = 40 m

(times10ndash5)

a = 725 mk1 = 10ndash7 mmiddotsndash1

r1 = 225 m

k = 15 times 10ndash7 mmiddotsndash1

k2 = 10ndash8 mmiddotsndash1

Hw = 54 mr2 = 2 m

Figure 9 Relationship between seepage flow and cavern radius

10 15 20 25 30 35 40 45 50 55235

240

245

250

255

260

265

rw = 3 mrw = 4 mrw = 5 m

rw = 6 mrw = 7 m

H1 (

m)

d (m)

a = 725 mk1 = 10ndash7 mmiddotsndash1

r1 = 225 m

k = 15 times 10ndash7 mmiddotsndash1

k2 = 10ndash8 mmiddotsndash1

Hw = 54 mr2 = 2 m

Figure 8 Relationship between head pressure and distance

3 4 5 6 7 8

13

14

15

16

17

18

19

2

21

Q (m

3 ∙sndash1∙m

ndash1)

a (m)

rw = 4m k = 15times10ndash7m∙sndash1

r1 = 225m r2 = 2m Hw = 54m

d = 15md = 20md = 25m

d = 30md = 35md = 40m

k1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

(times10ndash5)

Figure 11 Relationship between seepage flow and tunnel radius

6 Advances in Civil Engineering

5 Conclusion

Based on the inversion of complex function and ground-water hydraulics theory this paper derives the analyticalsolution for the karst seepage field in the karst tunnel in aninfinite plane )e potential function the flow function andthe complex velocity at any place in the infinite plane aresolved According to the distribution of the seepage field andthe boundary conditions the seepage flow of the karst tunnelis obtained )e seepage field of the lining structure is de-duced from the relationship between the grouting ring theinitial support and the secondary lining )e new solution

was verified with numerical simulation via the softwareFLAC3D )e conclusions obtained from this study aresummarized as follows

(1) When Hs is constant for karst tunnels rw and d arethe main factors affecting the pressure head andseepage flow of the tunnel When rw is constant H1and Q decrease with the increase of d When d isconstant H1 and Q increase with the increase of rw

(2) When the initial support is completed and thesecondary lining has not been constructed yet H1

3 4 5 6 7 85

10

15

20

25

30

35

40

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

k2 = 10ndash8m∙sndash1 d = 15mr1 = 225m r2 = 2m Hw = 54m

a (m)

kr = 20kr = 100

kr = 10kr = 50

Figure 12 Effect of kr on the relationship between pressure headand tunnel radius

20

25

30

35

40

45

50

55

H1 (

m)

rw = 4m k = 15times10ndash7m∙sndash1

a = 725m r1 = 5m Hw = 54mk1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

20 25 30 35 40 45 50r2 (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 14 Relationship between pressure head and initial supportradius

20 25 30 35 40 45 500

5

10

15

20

25

30

35

rw = 4m k = 15 times 10ndash7m∙sndash1

a = 725m r2 = 2m Hw = 54mk1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

Q (m

3 ∙sndash1∙m

ndash1)

r2 (m)

(times10ndash6)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 15 Relationship between seepage flow and initial supportradius

0 20 40 60 80 1002

4

6

8

10

12

14

16

18

20

22

kr

(times10ndash6)

rw = 4m k = 15times10ndash7m∙sndash1

r1 = 225m r2 = 2m Hw = 54m

Q (m

3 ∙sndash1∙m

ndash1)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

k2 = 10ndash8m∙sndash1 d = 15m

Figure 13 Relationship between seepage flow and permeabilitycoefficient

Advances in Civil Engineering 7

and Q will gradually decrease with the increase of thethickness and the decrease of the permeability co-efficient of the grouting ring When the permeabilitycoefficient of grouting ring is less than the specifiedvalue reasonable grouting material and constructiontechnology should be selected to ensure the thicknessof grouting ring

(3) H1 gradually decreases with the decrease of thethickness and the increase of k2 However Q

gradually increases with the decrease of the thicknessand the increase of k2 During construction we

should pay attention to the safety range of the initialsupport thickness

Data Availability

Based on the ldquohistoryrdquo function of FLAC3D software thedata which recorded the pore water pressure during thestable period of the monitoring point were obtainedAccording to the simulation conditions in Chapter 4 readerscan complete the modeling process and obtain the contourof pore water pressure such as Figures 5 and 6 )e data ofthe pore water pressure in the stable period of the moni-toring point were compared with the analytical solution to

20

25

30

35

40

45

50

55

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

a = 725m r2 = 2m Hw = 54mk1 = 10ndash7m∙sndash1 k2=10ndash8m∙sndash1

2 3 4 5 6 7r1 (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 16 Relationship between pressure head and radius ofgrouting ring

Q (m

3 ∙sndash1∙m

ndash1)

rw = 4m k = 15 times 10ndash7m∙sndash1

a = 725m r2 = 2m Hw = 54m

(times10ndash6)

14

12

10

8

6

4

2

2 3 4 5 6 7r1 (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

k1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

Figure 17 Relationship between seepage flow and radius ofgrouting ring

00

1 2 3 4 5

10

20

30

40

50

k2 (msndash1)

kr = 20kr = 10

kr = 100kr = 50

(times10ndash8)

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

r1 = 225m d = 15ma = 725m r2 = 2m Hw = 54m

Figure 18 Relationship between pressure head and permeabilitycoefficient of grouting ring

0 1 2 3 4 5

5

10

15

20

25

30

Q (m

3 ∙sndash1∙m

ndash1)

k2 (msndash1)

rw = 4m k = 15 times 10ndash7m∙sndash1

r1 = 225m d = 15ma = 725m r2 = 2m Hw = 54m

kr = 20kr = 100

kr = 10kr = 50

(times10ndash6)

(times10ndash8)

Figure 19 Relationship between seepage flow and permeabilitycoefficient of grouting ring

8 Advances in Civil Engineering

verify the accuracy )e author only recorded data of thecorresponding monitoring point in corresponding area forresearch needs In addition all the sharing data were in-volved in this paper and there were no redundant data )eparameters of the lining structure in the numerical simu-lation conditions come from [6]

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (Grant nos 51478479 and 51678570)and Hunan Transport Technology Project (Grant no201524)

References

[1] M D Zhang Study on the Characteristics of Seepage Field andWater Pressure on Tunnel Lining in Karst Ground BeijingJiaotong University Beijing China 2008

[2] B Q Zhang ldquoAnalytical solution for seepage field of twin-parallel tunnel in semi-infinite planerdquo Journal of 0e ChinaRailway Society vol 39 no 1 pp 125ndash131 2017

[3] B Yang F Gao and D Jeng ldquoFailure mode and dynamicresponse of a double-sided slope with high water content ofsoilrdquo Journal of Mountain Science vol 15 no 4 pp 859ndash8702018

[4] J G Wu A Coupled Fluid-Mechanical Study on Seepage Fieldof Tunnels with High Hydraulic Pressure Beijing JiaotongUniversity Beijing China 2006

[5] K H Park A Owatsiriwong and J G Lee ldquoAnalytical so-lution for steady-state groundwater inflow into a drainedcircular tunnel in a semi-infinite aquifer A revisitrdquo Tunnellingand Underground Space Technology vol 23 no 2 pp 206ndash209 2008

[6] C W Du M S Wang Z S Tan et al ldquoAnalytic solution forseepage field of subsea tunnel and its applicationrdquo ChineseJournal of Rock Mechanics and Engineering no s2pp 3567ndash3573 2011

[7] H W Ying C W Zhu H W Shen and X N Gong ldquoSemi-analytical solution for groundwater ingress into lined tunnelrdquoTunnelling and Underground Space Technology vol 76pp 43ndash47 2018

[8] L Tong K Xie M Lu et al ldquoAnalytical study of seepage flowinto a lined tunnel in a semi-infinite aquiferrdquo Rock and SoilMechanics vol 32 no 1 pp 304ndash308 2011

[9] D Ma Z Zhou J Wu Q Li and H Bai ldquoGrain size dis-tribution effect on the hydraulic properties of disintegratedcoal mixturesrdquo Energies vol 10 no 5 p 612 2017

[10] H Xiong K Zhao G Chen et al ldquoAnalysis of seepage field ofunderwater tunnel in a semi-infinite aquifer under waveactionrdquo China Earthquake Engineering Journal vol 36 no 4pp 919ndash923 2014

[11] D Ma Q Li M Hall and YWu ldquoExperimental investigationof stress rate and grain size on gas seepage characteristics ofgranular coalrdquo Energies vol 10 no 4 p 527 2017

[12] D Ma X Cai Z Zhou and X Li ldquoExperimental investigationon hydraulic properties of granular sandstone and mudstone

mixturesrdquo Geofluids vol 2018 Article ID 9216578 13 pages2018

[13] S Lei ldquoAn analytical solution for steady flow into a TtonnelrdquoGround Water vol 37 no 1 pp 23ndash26 1999

[14] M Huangfu M S Wang Z S Tan and X Y Wang ldquoAn-alytical solutions for steady seepage into an underwater cir-cular tunnelrdquo Tunnelling and Underground Space Technologyvol 25 no 4 pp 391ndash396 2010

[15] Y Kong Higher Fluid Mechanics University of Science andTechnology of China Press Hefei China 2010

Advances in Civil Engineering 9

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

thickness of the initial support Figures 14 and 15 show thatas the thickness of the initial support decreases H1 decreasesand Q increases gradually When r2 225 m Q and H1 are133 of Q and 207 of H1 when r2 475 m respectivelyIt shows that the initial support thickness has a great in-fluence on the pressure head During construction weshould pay attention to the safety range of the initial supportthickness

If the initial diameter r2 and a are constant increasing r1means increasing the thickness of the initial support andreducing the thickness of the grouting ring In Figures 16

and 17 with the increase of r1 H1 gradually increases and Q

gradually decreasesAs shown in Figures 18 and 19 the permeability

coefficient of grouting ring is constant and H1 graduallydecreases and Q gradually increases with the increase ofk2 )e change rate of Q and H1 is larger whenk2 lt 10minus8 cmmiddotsminus1 When k2 2 times 10minus8 cmmiddotsminus1 both Q andH1 tend to be stable When the other conditions areconstant the change of k2 on the magnitude of 10minus8 haslittle effect on Q and H1 but the influence on themagnitude of 10minus9 is greater

24

26

28

30

32

34

36

38

40

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

r1 = 225m r2 = 2m Hw = 54mk1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

3 4 5 6 7 8a (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 10 Effect of d on the relationship between pressure headand tunnel radius

3 4 5 6 7128

130

132

134

136

138

140

142

144

Q (m

3 ∙sndash1∙m

ndash1)

rw (m)

d = 15 md = 25 md = 35 m

d = 20 md = 30 md = 40 m

(times10ndash5)

a = 725 mk1 = 10ndash7 mmiddotsndash1

r1 = 225 m

k = 15 times 10ndash7 mmiddotsndash1

k2 = 10ndash8 mmiddotsndash1

Hw = 54 mr2 = 2 m

Figure 9 Relationship between seepage flow and cavern radius

10 15 20 25 30 35 40 45 50 55235

240

245

250

255

260

265

rw = 3 mrw = 4 mrw = 5 m

rw = 6 mrw = 7 m

H1 (

m)

d (m)

a = 725 mk1 = 10ndash7 mmiddotsndash1

r1 = 225 m

k = 15 times 10ndash7 mmiddotsndash1

k2 = 10ndash8 mmiddotsndash1

Hw = 54 mr2 = 2 m

Figure 8 Relationship between head pressure and distance

3 4 5 6 7 8

13

14

15

16

17

18

19

2

21

Q (m

3 ∙sndash1∙m

ndash1)

a (m)

rw = 4m k = 15times10ndash7m∙sndash1

r1 = 225m r2 = 2m Hw = 54m

d = 15md = 20md = 25m

d = 30md = 35md = 40m

k1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

(times10ndash5)

Figure 11 Relationship between seepage flow and tunnel radius

6 Advances in Civil Engineering

5 Conclusion

Based on the inversion of complex function and ground-water hydraulics theory this paper derives the analyticalsolution for the karst seepage field in the karst tunnel in aninfinite plane )e potential function the flow function andthe complex velocity at any place in the infinite plane aresolved According to the distribution of the seepage field andthe boundary conditions the seepage flow of the karst tunnelis obtained )e seepage field of the lining structure is de-duced from the relationship between the grouting ring theinitial support and the secondary lining )e new solution

was verified with numerical simulation via the softwareFLAC3D )e conclusions obtained from this study aresummarized as follows

(1) When Hs is constant for karst tunnels rw and d arethe main factors affecting the pressure head andseepage flow of the tunnel When rw is constant H1and Q decrease with the increase of d When d isconstant H1 and Q increase with the increase of rw

(2) When the initial support is completed and thesecondary lining has not been constructed yet H1

3 4 5 6 7 85

10

15

20

25

30

35

40

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

k2 = 10ndash8m∙sndash1 d = 15mr1 = 225m r2 = 2m Hw = 54m

a (m)

kr = 20kr = 100

kr = 10kr = 50

Figure 12 Effect of kr on the relationship between pressure headand tunnel radius

20

25

30

35

40

45

50

55

H1 (

m)

rw = 4m k = 15times10ndash7m∙sndash1

a = 725m r1 = 5m Hw = 54mk1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

20 25 30 35 40 45 50r2 (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 14 Relationship between pressure head and initial supportradius

20 25 30 35 40 45 500

5

10

15

20

25

30

35

rw = 4m k = 15 times 10ndash7m∙sndash1

a = 725m r2 = 2m Hw = 54mk1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

Q (m

3 ∙sndash1∙m

ndash1)

r2 (m)

(times10ndash6)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 15 Relationship between seepage flow and initial supportradius

0 20 40 60 80 1002

4

6

8

10

12

14

16

18

20

22

kr

(times10ndash6)

rw = 4m k = 15times10ndash7m∙sndash1

r1 = 225m r2 = 2m Hw = 54m

Q (m

3 ∙sndash1∙m

ndash1)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

k2 = 10ndash8m∙sndash1 d = 15m

Figure 13 Relationship between seepage flow and permeabilitycoefficient

Advances in Civil Engineering 7

and Q will gradually decrease with the increase of thethickness and the decrease of the permeability co-efficient of the grouting ring When the permeabilitycoefficient of grouting ring is less than the specifiedvalue reasonable grouting material and constructiontechnology should be selected to ensure the thicknessof grouting ring

(3) H1 gradually decreases with the decrease of thethickness and the increase of k2 However Q

gradually increases with the decrease of the thicknessand the increase of k2 During construction we

should pay attention to the safety range of the initialsupport thickness

Data Availability

Based on the ldquohistoryrdquo function of FLAC3D software thedata which recorded the pore water pressure during thestable period of the monitoring point were obtainedAccording to the simulation conditions in Chapter 4 readerscan complete the modeling process and obtain the contourof pore water pressure such as Figures 5 and 6 )e data ofthe pore water pressure in the stable period of the moni-toring point were compared with the analytical solution to

20

25

30

35

40

45

50

55

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

a = 725m r2 = 2m Hw = 54mk1 = 10ndash7m∙sndash1 k2=10ndash8m∙sndash1

2 3 4 5 6 7r1 (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 16 Relationship between pressure head and radius ofgrouting ring

Q (m

3 ∙sndash1∙m

ndash1)

rw = 4m k = 15 times 10ndash7m∙sndash1

a = 725m r2 = 2m Hw = 54m

(times10ndash6)

14

12

10

8

6

4

2

2 3 4 5 6 7r1 (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

k1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

Figure 17 Relationship between seepage flow and radius ofgrouting ring

00

1 2 3 4 5

10

20

30

40

50

k2 (msndash1)

kr = 20kr = 10

kr = 100kr = 50

(times10ndash8)

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

r1 = 225m d = 15ma = 725m r2 = 2m Hw = 54m

Figure 18 Relationship between pressure head and permeabilitycoefficient of grouting ring

0 1 2 3 4 5

5

10

15

20

25

30

Q (m

3 ∙sndash1∙m

ndash1)

k2 (msndash1)

rw = 4m k = 15 times 10ndash7m∙sndash1

r1 = 225m d = 15ma = 725m r2 = 2m Hw = 54m

kr = 20kr = 100

kr = 10kr = 50

(times10ndash6)

(times10ndash8)

Figure 19 Relationship between seepage flow and permeabilitycoefficient of grouting ring

8 Advances in Civil Engineering

verify the accuracy )e author only recorded data of thecorresponding monitoring point in corresponding area forresearch needs In addition all the sharing data were in-volved in this paper and there were no redundant data )eparameters of the lining structure in the numerical simu-lation conditions come from [6]

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (Grant nos 51478479 and 51678570)and Hunan Transport Technology Project (Grant no201524)

References

[1] M D Zhang Study on the Characteristics of Seepage Field andWater Pressure on Tunnel Lining in Karst Ground BeijingJiaotong University Beijing China 2008

[2] B Q Zhang ldquoAnalytical solution for seepage field of twin-parallel tunnel in semi-infinite planerdquo Journal of 0e ChinaRailway Society vol 39 no 1 pp 125ndash131 2017

[3] B Yang F Gao and D Jeng ldquoFailure mode and dynamicresponse of a double-sided slope with high water content ofsoilrdquo Journal of Mountain Science vol 15 no 4 pp 859ndash8702018

[4] J G Wu A Coupled Fluid-Mechanical Study on Seepage Fieldof Tunnels with High Hydraulic Pressure Beijing JiaotongUniversity Beijing China 2006

[5] K H Park A Owatsiriwong and J G Lee ldquoAnalytical so-lution for steady-state groundwater inflow into a drainedcircular tunnel in a semi-infinite aquifer A revisitrdquo Tunnellingand Underground Space Technology vol 23 no 2 pp 206ndash209 2008

[6] C W Du M S Wang Z S Tan et al ldquoAnalytic solution forseepage field of subsea tunnel and its applicationrdquo ChineseJournal of Rock Mechanics and Engineering no s2pp 3567ndash3573 2011

[7] H W Ying C W Zhu H W Shen and X N Gong ldquoSemi-analytical solution for groundwater ingress into lined tunnelrdquoTunnelling and Underground Space Technology vol 76pp 43ndash47 2018

[8] L Tong K Xie M Lu et al ldquoAnalytical study of seepage flowinto a lined tunnel in a semi-infinite aquiferrdquo Rock and SoilMechanics vol 32 no 1 pp 304ndash308 2011

[9] D Ma Z Zhou J Wu Q Li and H Bai ldquoGrain size dis-tribution effect on the hydraulic properties of disintegratedcoal mixturesrdquo Energies vol 10 no 5 p 612 2017

[10] H Xiong K Zhao G Chen et al ldquoAnalysis of seepage field ofunderwater tunnel in a semi-infinite aquifer under waveactionrdquo China Earthquake Engineering Journal vol 36 no 4pp 919ndash923 2014

[11] D Ma Q Li M Hall and YWu ldquoExperimental investigationof stress rate and grain size on gas seepage characteristics ofgranular coalrdquo Energies vol 10 no 4 p 527 2017

[12] D Ma X Cai Z Zhou and X Li ldquoExperimental investigationon hydraulic properties of granular sandstone and mudstone

mixturesrdquo Geofluids vol 2018 Article ID 9216578 13 pages2018

[13] S Lei ldquoAn analytical solution for steady flow into a TtonnelrdquoGround Water vol 37 no 1 pp 23ndash26 1999

[14] M Huangfu M S Wang Z S Tan and X Y Wang ldquoAn-alytical solutions for steady seepage into an underwater cir-cular tunnelrdquo Tunnelling and Underground Space Technologyvol 25 no 4 pp 391ndash396 2010

[15] Y Kong Higher Fluid Mechanics University of Science andTechnology of China Press Hefei China 2010

Advances in Civil Engineering 9

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

5 Conclusion

Based on the inversion of complex function and ground-water hydraulics theory this paper derives the analyticalsolution for the karst seepage field in the karst tunnel in aninfinite plane )e potential function the flow function andthe complex velocity at any place in the infinite plane aresolved According to the distribution of the seepage field andthe boundary conditions the seepage flow of the karst tunnelis obtained )e seepage field of the lining structure is de-duced from the relationship between the grouting ring theinitial support and the secondary lining )e new solution

was verified with numerical simulation via the softwareFLAC3D )e conclusions obtained from this study aresummarized as follows

(1) When Hs is constant for karst tunnels rw and d arethe main factors affecting the pressure head andseepage flow of the tunnel When rw is constant H1and Q decrease with the increase of d When d isconstant H1 and Q increase with the increase of rw

(2) When the initial support is completed and thesecondary lining has not been constructed yet H1

3 4 5 6 7 85

10

15

20

25

30

35

40

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

k2 = 10ndash8m∙sndash1 d = 15mr1 = 225m r2 = 2m Hw = 54m

a (m)

kr = 20kr = 100

kr = 10kr = 50

Figure 12 Effect of kr on the relationship between pressure headand tunnel radius

20

25

30

35

40

45

50

55

H1 (

m)

rw = 4m k = 15times10ndash7m∙sndash1

a = 725m r1 = 5m Hw = 54mk1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

20 25 30 35 40 45 50r2 (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 14 Relationship between pressure head and initial supportradius

20 25 30 35 40 45 500

5

10

15

20

25

30

35

rw = 4m k = 15 times 10ndash7m∙sndash1

a = 725m r2 = 2m Hw = 54mk1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

Q (m

3 ∙sndash1∙m

ndash1)

r2 (m)

(times10ndash6)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 15 Relationship between seepage flow and initial supportradius

0 20 40 60 80 1002

4

6

8

10

12

14

16

18

20

22

kr

(times10ndash6)

rw = 4m k = 15times10ndash7m∙sndash1

r1 = 225m r2 = 2m Hw = 54m

Q (m

3 ∙sndash1∙m

ndash1)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

k2 = 10ndash8m∙sndash1 d = 15m

Figure 13 Relationship between seepage flow and permeabilitycoefficient

Advances in Civil Engineering 7

and Q will gradually decrease with the increase of thethickness and the decrease of the permeability co-efficient of the grouting ring When the permeabilitycoefficient of grouting ring is less than the specifiedvalue reasonable grouting material and constructiontechnology should be selected to ensure the thicknessof grouting ring

(3) H1 gradually decreases with the decrease of thethickness and the increase of k2 However Q

gradually increases with the decrease of the thicknessand the increase of k2 During construction we

should pay attention to the safety range of the initialsupport thickness

Data Availability

Based on the ldquohistoryrdquo function of FLAC3D software thedata which recorded the pore water pressure during thestable period of the monitoring point were obtainedAccording to the simulation conditions in Chapter 4 readerscan complete the modeling process and obtain the contourof pore water pressure such as Figures 5 and 6 )e data ofthe pore water pressure in the stable period of the moni-toring point were compared with the analytical solution to

20

25

30

35

40

45

50

55

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

a = 725m r2 = 2m Hw = 54mk1 = 10ndash7m∙sndash1 k2=10ndash8m∙sndash1

2 3 4 5 6 7r1 (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 16 Relationship between pressure head and radius ofgrouting ring

Q (m

3 ∙sndash1∙m

ndash1)

rw = 4m k = 15 times 10ndash7m∙sndash1

a = 725m r2 = 2m Hw = 54m

(times10ndash6)

14

12

10

8

6

4

2

2 3 4 5 6 7r1 (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

k1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

Figure 17 Relationship between seepage flow and radius ofgrouting ring

00

1 2 3 4 5

10

20

30

40

50

k2 (msndash1)

kr = 20kr = 10

kr = 100kr = 50

(times10ndash8)

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

r1 = 225m d = 15ma = 725m r2 = 2m Hw = 54m

Figure 18 Relationship between pressure head and permeabilitycoefficient of grouting ring

0 1 2 3 4 5

5

10

15

20

25

30

Q (m

3 ∙sndash1∙m

ndash1)

k2 (msndash1)

rw = 4m k = 15 times 10ndash7m∙sndash1

r1 = 225m d = 15ma = 725m r2 = 2m Hw = 54m

kr = 20kr = 100

kr = 10kr = 50

(times10ndash6)

(times10ndash8)

Figure 19 Relationship between seepage flow and permeabilitycoefficient of grouting ring

8 Advances in Civil Engineering

verify the accuracy )e author only recorded data of thecorresponding monitoring point in corresponding area forresearch needs In addition all the sharing data were in-volved in this paper and there were no redundant data )eparameters of the lining structure in the numerical simu-lation conditions come from [6]

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (Grant nos 51478479 and 51678570)and Hunan Transport Technology Project (Grant no201524)

References

[1] M D Zhang Study on the Characteristics of Seepage Field andWater Pressure on Tunnel Lining in Karst Ground BeijingJiaotong University Beijing China 2008

[2] B Q Zhang ldquoAnalytical solution for seepage field of twin-parallel tunnel in semi-infinite planerdquo Journal of 0e ChinaRailway Society vol 39 no 1 pp 125ndash131 2017

[3] B Yang F Gao and D Jeng ldquoFailure mode and dynamicresponse of a double-sided slope with high water content ofsoilrdquo Journal of Mountain Science vol 15 no 4 pp 859ndash8702018

[4] J G Wu A Coupled Fluid-Mechanical Study on Seepage Fieldof Tunnels with High Hydraulic Pressure Beijing JiaotongUniversity Beijing China 2006

[5] K H Park A Owatsiriwong and J G Lee ldquoAnalytical so-lution for steady-state groundwater inflow into a drainedcircular tunnel in a semi-infinite aquifer A revisitrdquo Tunnellingand Underground Space Technology vol 23 no 2 pp 206ndash209 2008

[6] C W Du M S Wang Z S Tan et al ldquoAnalytic solution forseepage field of subsea tunnel and its applicationrdquo ChineseJournal of Rock Mechanics and Engineering no s2pp 3567ndash3573 2011

[7] H W Ying C W Zhu H W Shen and X N Gong ldquoSemi-analytical solution for groundwater ingress into lined tunnelrdquoTunnelling and Underground Space Technology vol 76pp 43ndash47 2018

[8] L Tong K Xie M Lu et al ldquoAnalytical study of seepage flowinto a lined tunnel in a semi-infinite aquiferrdquo Rock and SoilMechanics vol 32 no 1 pp 304ndash308 2011

[9] D Ma Z Zhou J Wu Q Li and H Bai ldquoGrain size dis-tribution effect on the hydraulic properties of disintegratedcoal mixturesrdquo Energies vol 10 no 5 p 612 2017

[10] H Xiong K Zhao G Chen et al ldquoAnalysis of seepage field ofunderwater tunnel in a semi-infinite aquifer under waveactionrdquo China Earthquake Engineering Journal vol 36 no 4pp 919ndash923 2014

[11] D Ma Q Li M Hall and YWu ldquoExperimental investigationof stress rate and grain size on gas seepage characteristics ofgranular coalrdquo Energies vol 10 no 4 p 527 2017

[12] D Ma X Cai Z Zhou and X Li ldquoExperimental investigationon hydraulic properties of granular sandstone and mudstone

mixturesrdquo Geofluids vol 2018 Article ID 9216578 13 pages2018

[13] S Lei ldquoAn analytical solution for steady flow into a TtonnelrdquoGround Water vol 37 no 1 pp 23ndash26 1999

[14] M Huangfu M S Wang Z S Tan and X Y Wang ldquoAn-alytical solutions for steady seepage into an underwater cir-cular tunnelrdquo Tunnelling and Underground Space Technologyvol 25 no 4 pp 391ndash396 2010

[15] Y Kong Higher Fluid Mechanics University of Science andTechnology of China Press Hefei China 2010

Advances in Civil Engineering 9

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

and Q will gradually decrease with the increase of thethickness and the decrease of the permeability co-efficient of the grouting ring When the permeabilitycoefficient of grouting ring is less than the specifiedvalue reasonable grouting material and constructiontechnology should be selected to ensure the thicknessof grouting ring

(3) H1 gradually decreases with the decrease of thethickness and the increase of k2 However Q

gradually increases with the decrease of the thicknessand the increase of k2 During construction we

should pay attention to the safety range of the initialsupport thickness

Data Availability

Based on the ldquohistoryrdquo function of FLAC3D software thedata which recorded the pore water pressure during thestable period of the monitoring point were obtainedAccording to the simulation conditions in Chapter 4 readerscan complete the modeling process and obtain the contourof pore water pressure such as Figures 5 and 6 )e data ofthe pore water pressure in the stable period of the moni-toring point were compared with the analytical solution to

20

25

30

35

40

45

50

55

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

a = 725m r2 = 2m Hw = 54mk1 = 10ndash7m∙sndash1 k2=10ndash8m∙sndash1

2 3 4 5 6 7r1 (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

Figure 16 Relationship between pressure head and radius ofgrouting ring

Q (m

3 ∙sndash1∙m

ndash1)

rw = 4m k = 15 times 10ndash7m∙sndash1

a = 725m r2 = 2m Hw = 54m

(times10ndash6)

14

12

10

8

6

4

2

2 3 4 5 6 7r1 (m)

d = 15md = 20md = 25m

d = 30md = 35md = 40m

k1 = 10ndash7m∙sndash1 k2 = 10ndash8m∙sndash1

Figure 17 Relationship between seepage flow and radius ofgrouting ring

00

1 2 3 4 5

10

20

30

40

50

k2 (msndash1)

kr = 20kr = 10

kr = 100kr = 50

(times10ndash8)

H1 (

m)

rw = 4m k = 15 times 10ndash7m∙sndash1

r1 = 225m d = 15ma = 725m r2 = 2m Hw = 54m

Figure 18 Relationship between pressure head and permeabilitycoefficient of grouting ring

0 1 2 3 4 5

5

10

15

20

25

30

Q (m

3 ∙sndash1∙m

ndash1)

k2 (msndash1)

rw = 4m k = 15 times 10ndash7m∙sndash1

r1 = 225m d = 15ma = 725m r2 = 2m Hw = 54m

kr = 20kr = 100

kr = 10kr = 50

(times10ndash6)

(times10ndash8)

Figure 19 Relationship between seepage flow and permeabilitycoefficient of grouting ring

8 Advances in Civil Engineering

verify the accuracy )e author only recorded data of thecorresponding monitoring point in corresponding area forresearch needs In addition all the sharing data were in-volved in this paper and there were no redundant data )eparameters of the lining structure in the numerical simu-lation conditions come from [6]

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (Grant nos 51478479 and 51678570)and Hunan Transport Technology Project (Grant no201524)

References

[1] M D Zhang Study on the Characteristics of Seepage Field andWater Pressure on Tunnel Lining in Karst Ground BeijingJiaotong University Beijing China 2008

[2] B Q Zhang ldquoAnalytical solution for seepage field of twin-parallel tunnel in semi-infinite planerdquo Journal of 0e ChinaRailway Society vol 39 no 1 pp 125ndash131 2017

[3] B Yang F Gao and D Jeng ldquoFailure mode and dynamicresponse of a double-sided slope with high water content ofsoilrdquo Journal of Mountain Science vol 15 no 4 pp 859ndash8702018

[4] J G Wu A Coupled Fluid-Mechanical Study on Seepage Fieldof Tunnels with High Hydraulic Pressure Beijing JiaotongUniversity Beijing China 2006

[5] K H Park A Owatsiriwong and J G Lee ldquoAnalytical so-lution for steady-state groundwater inflow into a drainedcircular tunnel in a semi-infinite aquifer A revisitrdquo Tunnellingand Underground Space Technology vol 23 no 2 pp 206ndash209 2008

[6] C W Du M S Wang Z S Tan et al ldquoAnalytic solution forseepage field of subsea tunnel and its applicationrdquo ChineseJournal of Rock Mechanics and Engineering no s2pp 3567ndash3573 2011

[7] H W Ying C W Zhu H W Shen and X N Gong ldquoSemi-analytical solution for groundwater ingress into lined tunnelrdquoTunnelling and Underground Space Technology vol 76pp 43ndash47 2018

[8] L Tong K Xie M Lu et al ldquoAnalytical study of seepage flowinto a lined tunnel in a semi-infinite aquiferrdquo Rock and SoilMechanics vol 32 no 1 pp 304ndash308 2011

[9] D Ma Z Zhou J Wu Q Li and H Bai ldquoGrain size dis-tribution effect on the hydraulic properties of disintegratedcoal mixturesrdquo Energies vol 10 no 5 p 612 2017

[10] H Xiong K Zhao G Chen et al ldquoAnalysis of seepage field ofunderwater tunnel in a semi-infinite aquifer under waveactionrdquo China Earthquake Engineering Journal vol 36 no 4pp 919ndash923 2014

[11] D Ma Q Li M Hall and YWu ldquoExperimental investigationof stress rate and grain size on gas seepage characteristics ofgranular coalrdquo Energies vol 10 no 4 p 527 2017

[12] D Ma X Cai Z Zhou and X Li ldquoExperimental investigationon hydraulic properties of granular sandstone and mudstone

mixturesrdquo Geofluids vol 2018 Article ID 9216578 13 pages2018

[13] S Lei ldquoAn analytical solution for steady flow into a TtonnelrdquoGround Water vol 37 no 1 pp 23ndash26 1999

[14] M Huangfu M S Wang Z S Tan and X Y Wang ldquoAn-alytical solutions for steady seepage into an underwater cir-cular tunnelrdquo Tunnelling and Underground Space Technologyvol 25 no 4 pp 391ndash396 2010

[15] Y Kong Higher Fluid Mechanics University of Science andTechnology of China Press Hefei China 2010

Advances in Civil Engineering 9

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

verify the accuracy )e author only recorded data of thecorresponding monitoring point in corresponding area forresearch needs In addition all the sharing data were in-volved in this paper and there were no redundant data )eparameters of the lining structure in the numerical simu-lation conditions come from [6]

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (Grant nos 51478479 and 51678570)and Hunan Transport Technology Project (Grant no201524)

References

[1] M D Zhang Study on the Characteristics of Seepage Field andWater Pressure on Tunnel Lining in Karst Ground BeijingJiaotong University Beijing China 2008

[2] B Q Zhang ldquoAnalytical solution for seepage field of twin-parallel tunnel in semi-infinite planerdquo Journal of 0e ChinaRailway Society vol 39 no 1 pp 125ndash131 2017

[3] B Yang F Gao and D Jeng ldquoFailure mode and dynamicresponse of a double-sided slope with high water content ofsoilrdquo Journal of Mountain Science vol 15 no 4 pp 859ndash8702018

[4] J G Wu A Coupled Fluid-Mechanical Study on Seepage Fieldof Tunnels with High Hydraulic Pressure Beijing JiaotongUniversity Beijing China 2006

[5] K H Park A Owatsiriwong and J G Lee ldquoAnalytical so-lution for steady-state groundwater inflow into a drainedcircular tunnel in a semi-infinite aquifer A revisitrdquo Tunnellingand Underground Space Technology vol 23 no 2 pp 206ndash209 2008

[6] C W Du M S Wang Z S Tan et al ldquoAnalytic solution forseepage field of subsea tunnel and its applicationrdquo ChineseJournal of Rock Mechanics and Engineering no s2pp 3567ndash3573 2011

[7] H W Ying C W Zhu H W Shen and X N Gong ldquoSemi-analytical solution for groundwater ingress into lined tunnelrdquoTunnelling and Underground Space Technology vol 76pp 43ndash47 2018

[8] L Tong K Xie M Lu et al ldquoAnalytical study of seepage flowinto a lined tunnel in a semi-infinite aquiferrdquo Rock and SoilMechanics vol 32 no 1 pp 304ndash308 2011

[9] D Ma Z Zhou J Wu Q Li and H Bai ldquoGrain size dis-tribution effect on the hydraulic properties of disintegratedcoal mixturesrdquo Energies vol 10 no 5 p 612 2017

[10] H Xiong K Zhao G Chen et al ldquoAnalysis of seepage field ofunderwater tunnel in a semi-infinite aquifer under waveactionrdquo China Earthquake Engineering Journal vol 36 no 4pp 919ndash923 2014

[11] D Ma Q Li M Hall and YWu ldquoExperimental investigationof stress rate and grain size on gas seepage characteristics ofgranular coalrdquo Energies vol 10 no 4 p 527 2017

[12] D Ma X Cai Z Zhou and X Li ldquoExperimental investigationon hydraulic properties of granular sandstone and mudstone

mixturesrdquo Geofluids vol 2018 Article ID 9216578 13 pages2018

[13] S Lei ldquoAn analytical solution for steady flow into a TtonnelrdquoGround Water vol 37 no 1 pp 23ndash26 1999

[14] M Huangfu M S Wang Z S Tan and X Y Wang ldquoAn-alytical solutions for steady seepage into an underwater cir-cular tunnelrdquo Tunnelling and Underground Space Technologyvol 25 no 4 pp 391ndash396 2010

[15] Y Kong Higher Fluid Mechanics University of Science andTechnology of China Press Hefei China 2010

Advances in Civil Engineering 9

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom