Physical model simulation of rock-support interaction for the tunnel in squeezing ground

Ketan Arora,Marte Gutierrez,Ahmadreza Hedayat

Civil and Environmental Engineering,Colorado School of Mines,Golden,CO,80401,USA

Keywords:Squeezing ground Tunnel support system Longitudinal displacement pro file Rock-support interaction

ABSTRACT The existence of squeezing ground conditions can lead to signi ficant challenges in designing an adequate support system for tunnels.Numerous empirical,observational and analytical methods have been suggested over the years to design support systems in squeezing ground conditions,but all of them have some limitations.In this study,a novel experimental setup having physical model for simulating the tunnel boring machine(TBM)excavation and support installation process in squeezing clay-rich rocks is developed.The observations are made to understand better the interaction between the support and the squeezing ground.The physical model included a large true-triaxial cell,a miniature TBM,laboratoryprepared synthetic test specimen with properties similar to natural mudstone,and an instrumented cylindrical aluminum support system.Experiments were conducted at realistic in situ stress levels to study the time-dependent three-dimensional tunnel support convergence.The tunnel was excavated using the miniature TBM in the cubical rock specimen loaded in the true-triaxial cell,after which the support was installed.The con fining stress was then increased in stages to values greater than the rock’s uncon fined compressive strength.A model for the time-dependent longitudinal displacement pro file(LDP)for the supported tunnel was proposed using the tunnel convergence measurements at different times and stress levels.The LDP formulation was then compared with the unsupported model to calculate the squeezing amount carried by the support.The increase in thrust in the support was backcalculated from an analytical solution with the assumption of linear elastic support.Based on the test results and case studies,a recommendation to optimize the support requirement for tunnels in squeezing ground is proposed.

1.Introduction

After excavation,tunnel wall convergence is a function of the in situ stresses and the behavior of the rock mass at the tunnel location.In squeezing ground conditions,the tunnel converges significantly and continuously with time during or even after the tunnel excavation.As the tunnel converges with time,the loads applied to the tunnel support system increase,resulting in severe stability issues in some cases.Therefore,estimation of the interactions between the rock mass and the tunnel support system(or rocksupport interaction)with time is essential to counter the squeezing.

In tunneling,rock-support interaction is used as a preliminary tool for assessing the tunnel support system’s behavior during both the design and construction process(Panet,1995;Carranza-Torres and Fairhurst,2000;Gschwandtner and Galler,2012).Several practices were reported in the literature to carry out rock-support interaction(Fenner,1938;Pacher,1964;Lombardi,1975;Brown et al.,1983;Corbetta et al.,1991;Duncan-Fama,1993;Panet,1993,1995;Peila and Oreste,1995;Carranza-Torres and Fairhurst,2000;Alejano et al.,2009;Vrakas and Anagnostou,2014;Cai et al.,2015;Cui et al.,2015;Lüet al.,2017;Pandit and Babu,2021;Zhang and Nordlund,2021).Most of the proposed solutions are based on simplifying the three-dimensional(3D)into a twodimensional(2D)plane-strain problem.A typical example of this approach is the convergence-con finement method(CCM),which has three basic components:(i)the ground reaction curve(GRC),(ii)the longitudinal displacement pro file(LDP)of the tunnel,and(iii)the support characteristic curve(SCC).The GRC de fines the relationship between the decreasing internal pressure in the tunnel and the increase in tunnel convergence.The LDP is the plot of the tunnel convergence that occurs along the tunnel axis as a function of the distance from the tunnel face.The SCC expresses the relationship between the increasing stresses on the tunnel support and the increasing support displacement.

Conventional CCMconsiders the tunnel deformation only based on the tunnel’s advancement and the change in the internal pressure.However,the tunnel also converges with time and even after the excavation is complete in squeezing ground conditions.This time-dependent convergence can generate additional stresses in the tunnel support system and can ultimately cause unpredictable support failures,construction hazards,cost overruns,and project delays(Paraskevopoulou and Benardos,2013).

The only approach reported in the literature that developed CCM for squeezing ground conditions incorporates a timedependent rock mass model for a tunnel(Paraskevopoulou and Diederichs,2018).The time-dependent behavior was explicitly studied for the rock mass(Paraskevopoulou and Diederichs,2018)or indirectly by monitoring the change in stresses on the tunnel support(Gschwandtner and Galler,2012).The former approach modi fies the LDP with time using a time-dependent constitutive model while assuming the rock mass’s visco-elastic behavior.On the other hand,the latter approach predicts the increase in stresses applied to the support system in the excavated tunnel based on the field monitoring data.Given that the change in stresses applied to the tunnel support system is monitored for a real project,this latter approach can be more accurate than the first approach.However,due to high levels of applied loads and deformations,the field sensors(just like the tunnel support system themselves)experience extreme environments and often fail before meaningful data can be measured once the squeezing has started.Without reliable field data,it is challenging to develop a CCMto calculate the tunnel support stresses.This is the primary reason why tunnel designers mostly rely on empirical equations derived from limited field observations for tunnels in squeezing ground conditions,as discussed in Arora et al.(2021c).Since in situ monitoring is often limited in squeezing environments,realistic physical model tests can provide valuable information to evaluate better the interaction between the support and the squeezing ground(Lin et al.,2015).

This paper aims to provide a comparison between the LDPs of supported and unsupported tunnels using carefully conducted laboratory-scale physical model tests in squeezing ground.The experiment involved tunnel boring machine(TBM)excavation and support installation in a synthetic mudstone specimen,with squeezing and time-dependent behavior,under increasing in situ stresses.The physical model test explored the critical aspects of tunnel excavation,such as the tunnel advancement,the 3D effects,the support installation,and the highly plastic and ductile timedependent behavior of the ground.The experimental observations and results are synthesized in terms of convergence and support stresses at different times and loading levels.Case studies of the tunnels’support systems in squeezing ground conditions are also presented as well as a proposed methodology for the selection of the tunnel support system.

2.Experimental setup

The setup used in this study physically simulated the TBM excavation and tunnel support installation procedure in a cubical rock specimen at field stress levels(up to 550 m overburden due to weight of mudstone)in a true-triaxial cell.More features of the experimental setup are available in Arora et al.(2019,2020a,b,2021a).The temporal changes in the liner deformation were monitored and analyzed throughout the experiment.

Fig.1.Experimental arrangement to study tunnels in squeezing ground conditions:(a)Schematic diagram,and(b)Actual setup at Colorado School of Mines.

The experimental setup is illustrated in Fig.1a,using a schematic diagram.A photograph of the experimental setup at the Colorado School of Mines,where this research was conducted,is provided in Fig.1b.The experimental setup consists of a truetriaxial cell,a miniature TBM,a cubical synthetic mudstone specimen as the mudstone-like material,the tunnel support system,and the monitoring unit.The important aspects of the test setup and the loading,excavation and support installation details are discussed in the following sections.

2.1.Cubical rock specimen

It is challenging to obtain natural,homogenous and undisturbed specimens of mudstone from the field.Therefore,this study incorporated a synthetic mudstone prepared in the laboratory by mixing cement,clay and water following Arora et al.(2020c).The test material exhibits the characteristics of squeezing clay-rich rock(i.e.ductility even at low con fining stresses,extensive plastic deformation,and pronounced time and loading rate dependency).

Arora et al.(2020c)reported that,based on laboratory tests,the mean uncon fined compressive strength(UCS)σcwas 4.47 MPa with a standard deviation of 0.15 MPa,the mean Young’s modulusEwas 0.65 GPa with a standard deviation of 0.02 GPa with,and the Poisson’s ratioνwas 0.13.For the con fining stresses varying from 1 MPa to 4 MPa and using the Mohr-Coulomb linear fit,the effective cohesioncand internal friction angleφwere 2.06 MPa and 10°,respectively.During the creep tests,the synthetic mudstone was observed to develop a considerable amount of creep strain with time(Arora et al.,2020c).The laboratory tests conducted by Arora et al.(2020c)are discussed in Appendix A.

The laboratory test results showed that both the synthetic mudstone as tested and the naturally occurring and squeezingmudstone have similar shear strength,elastic modulus,ductility,strain-rate sensitivity and creep behavior.

The 300 mm×300 mm×300 mm cubical synthetic mudstone specimen was prepared using the same material,composition and procedure given in Arora et al.(2020c).The test specimen made of the mixture of clay,cement and water was removed from the mold after 24 h and further cured in distilled water for 28 d before the testing commenced.

2.2.The loading unit

The physical model test was carried out using a true-triaxial cell developed by Frash et al.(2014)at the Colorado School of Mines.The unique feature is that the drilling activity is carried out in the rock specimen as it is loaded in a true-triaxial stress state,making it suitable to simulate in situ stress conditions.The true-triaxial cell,originally designed to carry out the thermo-mechanical simulation of enhanced geothermal systems,was modi fied to study TBM-excavated tunnels in squeezing ground conditions by Arora et al.(2020a).The true-triaxial cell applies the con fining pressure in each direction using the combination of flexible membrane flat jack and steel platens.

In each one of the loading directions,one face of the rock specimen(active face)makes direct contact with the flat jack while the other face(de fined as the passive face)gives the reaction with a rigid plane.The active face has an active hydraulic jack,300 mmdiameter circular steel platens on each faces of the flat jack,and a 300 mm×300 mm rectangular steel platen sandwiched between the specimen and the circular steel platen.The rigid passive platen is stationary and made up of concrete.With this arrangement,the apparatus is ef ficient of applying three independently regulated principal stresses up to 13 MPa to a 300 mm cubical rock specimen.The top lid of the true-triaxial cell is made of steel and equipped with two circular ports of 63 mm and 75 mm in diameter.The smaller 63 mm-diameter port is a pass-through for the sensor cable and hydraulic lines while the larger 75 mm port at the center of the lid provides entry to the surface of the rock specimen for the tunnel excavation using the miniature TBM.Further details about this true-triaxial apparatus can be found in Frash et al.(2014).The revisions for the tunneling experiment presented in this study are given in Arora et al.(2019,2020a,2021a).

2.3.Tunnel excavation unit

Numerous researchers have investigated tunnel excavation process using physical model tests by developing a miniature TBM(Nomoto et al.,1999;Wang et al.,2019;Liu et al.,2020).A miniature soft ground open-face TBMwas mounted on the true-triaxial cell to excavate a tunnel.At the same time,the cubical rock specimen was loaded in the true-triaxial cell.As the tunnel excavation needed to occur while the rock was subjected to the in situ stress levels,the miniature TBM was designed to provide a suf ficient cutter head thrust and torque.The miniature TBM mounted on the top of the true-triaxial cell is given in the conceptual diagram in Fig.1.

The four key elements of the miniature TBM are the thrust unit,torque unit,cutting unit and supporting unit.The diameter of the cutter head disc is 48 mm,and the maximum 150 mm long tunnel can be excavated with this miniature TBM.A set of Teledyne ISCO syringe pumps and a hydraulic cylinder ful filled the required face pressure(thrust)for the excavation to the cutter head.The apparatus can apply a face pressure up to 10 MPa.The torque is supplied by a 0.5 hp,115 V,single-phase,constant speed,alternating current(AC)motor through a chain drive,as presented in Fig.1.The motor spins at a continuous speed of seven revolutions per minute and is capable of providing a maximum torque of 400 N m.

As shown in Fig.1,the cutting unit is comprised of a shaft with a cutter head connected at the end.The other end of the shaft is connected to a hydraulic jack by employing a thrust bearing.The miniature TBM was constructed for the excavation of soft rock(e.g.mudstone)and the drill bit had a cutter head with nine button-type tungsten carbide bits.The complete design and working mechanism of the miniature TBM can be found in Arora et al.(2021a).

2.4.Tunnel support system

A thin cylinder made of 6061 aluminum alloy,with 44.5 mm in outer diameter(OD)and 1.65 mm in thicknessts,was used as the tunnel support system.The ultimate tensile strength,Young’s modulusEsand Poisson’s ratioνsof the 6061 aluminum were 276 MPa,68.9 GPa and 0.35,respectively(Zwilsky and Langer,2001).The annular gap between the support system and the tunnel boundary measured 1.5 mm and was filled with non-shrinking grout and cured for 3 d.The 3-d compressive strength of this grout was 1.12 MPa,which is the same asσc.It is reasonable to assume that the grout,similar to the real back fill grout in tunnels,played its intended role in transferring the ground loads to the support system considering the strength difference between the grout and the aluminum material.

Fig.2a schematically shows the cubical rock specimen with the tunnel loaded in the true-triaxial cell.Fig.2b shows the support system with strain gauges in the tangential direction.The strain gauges’position on the tunnel support from the longitudinal view is shown in Fig.2c,and the cross-sectional view is shown in Fig.2d.

3.Test procedures

After 28 d of curing in water,the cubical synthetic mudstone specimen was loaded in the true-triaxial cell.The rock specimen was first subjected to a low isotropic compressive stresspo=0.5σc.Then,a tunnel of 48 mm in diameter and 140 mm in length was excavated in the rock specimen using the miniature TBM with longitudinal axis along thex-axis of the coordinate system,as shown in Fig.2a.

After the grout cured for 3 d,the isotropic stresspowas applied in five stages over 605 h,as shown in Fig.3.The complete time of loading and monitoring was decided based on the beginning of the steady-state(i.e.the rock specimen was loaded until the strain in the tunnel support hit the steady-state).The five loading stages of this test are de fined as follows:

(1)Loading stage I:After excavating the tunnel and installing the tunnel support,powas increased from 0.5σctoσc.As shown in Fig.3,pointsAandBare the beginning and end of the loading stage I,respectively.The total loading stage time was 125 h andpowas equal toσc.

(2)Loading stage II:At the end of the loading stage I,powas increased rapidly fromσc(pointB)to 1.5σc(pointC),as shown in Fig.3.PointCindicates the beginning of loading stage II,and after 120 h,pointDmarks the end,as shown in Fig.3.

(3)Loading stage III:The isotropic stress on the cubical rock specimen was further increased by 0.5σc,makingpo=2σc.The total time of loading for this stage was 120 h.The beginning and the endpoint of this loading stage are pointsEandFin Fig.3.

Fig.2.(a)Schematic diagram of cubical rock specimen in true-triaxial stress state with tunnel,(b)Aluminum tunnel support with strain gauges in tangential direction,(c)Longitudinal view of the tunnel support system along with position of strain gauges,and(d)Cross-section of the tunnel support system.

Fig.3.(a)Conceptual diagram of the cubical rock specimen with tunnel loaded in true-triaxial stress,(b)Support placed inside the tunnel,and(c)Isotropic stress ratio applied on a cubical rock specimen during the five loading stages.

(4)Loading stage IV:In this loading stage,powas increased by 0.5σcand kept constant for another 120 h while monitoring tunnel support behavior.

(5)Loading stage V:In this final loading stage,powas further increased from 2.5σcto 3σc.The rock specimen was loaded for 120 h and finally unloaded to reachpo=0 MPa.

4.Experimental observations and results

4.1.Tunnel excavations in three phases

Fig.4 shows the 48-mm diameter tunnel response after it was excavated in three phases,lasting 19.5 h.As shown in Fig.4a,themean TBMface pressurepfapplied to the tunnel face via the cutter head throughout the excavation stage was about 1.12 MPa with a standard deviation of 0.05 MPa.The isotropic stresspoapplied on the cubical rock specimen was 0.5σc,roughly two times that ofpf.The miniature TBM shaft rotated at a constant speed of 13 revolutions per minute,and the torque output was continuously monitored,as shown in Fig.4b.

During Phase I of the excavation,a 42-mm long tunnel was excavated in 5 h,followed by removing the cut material.It was observed that when the cut material was not removed ef ficiently,the tunnel became clogged,and the advance rate signi ficantly slowed down,as shown in Fig.4c.Phase II of the tunnel excavation began after the cut material of Phase I was removed.In this stage,a 38-mm long tunnel was excavated in 5.3 h,making the total length of the tunnel 80 mm.In the last stage of the excavation,the tunnel further advanced by 60 mm in 9.2 h,making the full length of the tunnel 140 mm.

4.2.Support deformations

Throughout the experiment,the tunnel support behavior was monitored by the tangential strain gauges on the support system,as shown in Fig.2c.The strain gauges were installed on the outer surface of the cylinder in contact with the grout.It was assumed that at the point of contact,the strains in the rock and support were the same,considering the continuity of the tangential strain at this interface.The tangential strain at the tunnel boundary is the ratio of tunnel convergenceuto its radiusR.Hence,the strain gauges on the support provide the LDP of the supported tunnel at any givenpoand time.

This section discusses the change in the LDP of the supported tunnel with time during the five loading stages shown in Fig.3.Displacementuand the longitudinal distance from the tunnel facexfwere normalized with tunnel radiusRto present the most common form results.Ratiou/R(in%)is effectively the tunnel wall radial strain,and parameterX=xf/Ris the normalized longitudinal distance from the tunnel face.

Fig.5 shows the selected but representative LDP readings(u/RversusXplot)of the tunnel over the time for all five loading stages.Due to squeezing,the LDP of the tunnel was observed to shift upwards with time.Of the five tunnel sections studied,the LDP’s maximum and minimum upward shifts with time were recorded atX=4.5 and 0.9,respectively,throughout the test.The corresponding increases inu/RatX=4.5 for loading stages I to V due to squeezing were 0.005%,0.125%,0.13%,0.166%and 0.452%,respectively.However,the increases inu/RatX=0.9 were relatively low and were recorded as 0.001%,0.038%,0.055%,0.074%and 0.091%during loading stages I to V,respectively.

5.A time-dependent LDP

The most widely used LDPs for unsupported tunnels were proposed by Panet(1995)and Vrakas and Anagnostou,2014.Due to the rock’s time-dependent behavior,the stresses on the tunnel support increase with time in squeezing ground conditions.The complications reported in the literature regarding the application of CCMin squeezing ground were discussed in this paper.However,the study presented here deals with the direct monitoring of the tunnel support response in squeezing ground conditions subjected to different in situ stress levels.Hence,a time-dependent LDP of the supported tunnel was chosen to simplify the study of tunnel support behavior in squeezing ground conditions.

Fig.5.(a)Longitudinal tunnel excavated in isotropic stress state,and LDP of the tunnel at the different times for the loading stages(b)I,(c)II,(d)III,(e)IV and(f)V.

Fig.6 depict the tangential strain changes as a function of the loading stage timetfor the five loading stages for five equally spaced longitudinal tunnel sections ranging fromX=0.9 to 4.5.It also can be seen that for any tunnel section and during any loading stage,the correlation between tunnel strainu/Rin%and timetin h could be expressed as an asymptotic model and given by the following equation:

whereas,bsanddsare the model parameters determined from the best fitting curve having the maximum coef ficient of determination with the experimental data.The best fit values of the model parametersas,bsanddsfor all the plots and the coef ficient of determination are shown in Table 1.

Parameterdsgoverns the supported tunnel’s time-dependent response,and from the test observations,it can be considered constant asds=0.98.As shown in Table 1,the parameterasincreases with increases inXandpo/σc.Table 1 also indicates that parameterbsincreases with an increase inpo/σcbut shows a weak correlation withX.Hence,theascan be expressed as the function ofpo/σcandXas follows:

wherefis the function ofpo/σcandX.Fig.7 shows the dependency of parameterasonpo/σcandX.It can be observed in Fig.7 that for a givenpo/σc,asincreases almost linearly with increase inX.The parameterbswas found increase with increase inpo/σcand can be expressed as

wheregis the function ofpo/σc.Eq.(1)can be rewritten as

According to Eq.(1),parametera(fords＜1)can be de fined for any tunnel section as the tunnel strain at an in finite time as follows:

Also,according to Eq.(1),the difference between parametersasandbsis the tunnel strain at the beginning of any loading stage(t=0):

5.1.Effect of support system on the LDP

In addition to the in fluences ofpo/σcandX,the LDP was also in fluenced by the support systems by comparing the LDP of the supported and unsupported tunnels,as discussed in Arora(2020)and Arora et al.(2021b).It was determined that the support system in fluences the instantaneous and long-term convergences of the tunnel.

Fig.6.Tunnel strain u/R(in%)with loading stage time t for X varying from 0.9 to 4.5 for loading stages(a)I,(b)II,(c)III,(d)IV and(e)V.

The change inu/Rin percentage att=0 due to support installation,calculated using Eq.(6),is given by

Similarly,the percentage change inu/Rat an in finite time due to support installation,calculated using Eq.(5),is given by

Parametersa,banddfor an unsupported tunnel for different loading stages are provided in Table 1,as discussed in Arora(2020).The parameters in Table 1 were used to calculateΔ(u/R)instantand Δ(u/R)ultimate.The installed tunnel support reduced the instantaneous and ultimate tunnel convergences by 96%-98%for all crosssections and loading conditions.

Table 1LDP model parameters for supported and unsupported tunnels.

5.2.Load applied to the support system

Throughout the experiment,the tunnel support strains increased with time due to squeezing ground.Given its modulus and dimensions,the support system did not reach the yielding stage and remained linear elastic throughout the test.Hence,it can be stated that the stresses on the tunnel support also increasedlinearly with the increase in measured strains.These stresses can be calculated from the analytical solution of the tunnel support system given by Schwartz and Einstein(1980),which considers that the tunnel boundary transfers bending and thrust forces to the tunnel support.The convergence of the tunnel support is a combined effect of the bending and thrust given by the following equations for the full slip condition(no shear stress)between the rock and the support,and the tunnel excavated in an isotropic stress state:

Fig.7.Change in the model parameter a s with change in p o/σc and normalized distance from the tunnel face.

wherePis the ground stress acting on the support,Tis the thrust force on the support,andMis the bending moment on the support and is zero in isotropic stress conditions(Schwartz and Einstein,1980).The parametersEgandνgare the Young’s modulus and Poisson’s ratio of the rock,respectively.Parametera*ois the function of compressibility ratioC*and flexibility ratioF*of the rock and support given by the following equations:

whereEsandνsare the Young’s modulus and Poisson’s ratio of the support,respectively;andIsandAsare the moment of inertia per unit length of the support cross-section about its centerline and the area of the support per unit length experiencing the thrust and moment,respectively.

Combining Eqs.(9)and(11)to calculateTfor a givenuin the following equation is

Using Eq.(17),thrust forcesTat the start and end of each loading cycle are presented in Table 2.As can be seen,except during loading stage I,the thrust on the support increased signi ficantly during loading stages II through V.This additional increase inTwith time was the result of the added squeezing of the rock.

6.Case studies

Parametersas,bsanddsof the LDP determined from this model study depend on the rock elastic properties,the support elastic properties and its geometry.However,exploring the individual effects of the involved parameters requires numerous tests,which may not always be feasible.One feasible approach to account for the rock and support systemvariations is to utilize the compressibility ratioC*and flexibility ratioF*concept,given in Eqs.(13)and(14),respectively.

The support system used in this study reduced the timedependent convergence of the tunnel by 96%compared to the unsupported tunnel.TheC*andF*for the conducted test determined from Eqs.(13)and(14)are 0.11 and 0.24,respectively.For the unsupported tunnels,C*andF*both approached in finity.Additional information on the convergence of the supported tunnels with differentC*andF*values would be required to optimize the support requirements.In this paper,the knowledge is expanded by carrying out the case studies of the tunnel support responseunder squeezing conditions.The case studies are summarized in Table 3 and presented as follows.

Table 2Thrust in the tunnel support at the beginning and end of different loading stages.

6.1.Laodongshan railway tunnel in China

This tunnel,a part of the Guangzhou-Kunming railway in China,experienced squeezing failure in its early construction stage due to excavation in weak mudstone at high in situ stresses(Cao et al.,2018).Four primary tunnel support systems were tested at a critical cross-section.The in fluence of the long-term convergence at the crown was studied to optimize the support requirements.The four support systems in the order of decreasingC*(see Table 3)were I20a steel at 0.8-m spacing,I22b steel at 0.6-m spacing,H175 steel at 0.6-m spacing and H200 steel at 0.6-m spacing.It was concluded from the field monitoring of Cao et al.(2018)that the tunnel cross-section with a support system having higherC*experienced less convergence due to squeezing.Table 3 presents the percent change in tunnel convergenceΔ(u/R)ultimatefor the four tested tunnel support systems calculated from the observations of Cao et al.(2018).

6.2.John street pumping station tunnel

The John street pumping station tunnel,built in 1925,supplies the water to downtown Toronto,Canada.A new tunnel was constructed to help with a water supply and the existing tunnels(Czurda et al.,1973).It was constructed in weak squeezing mudstone with the horizontal stresses between 13 and 30 times the vertical stresses(Lo et al.,1987).A 39.5-cm thick steel tube and the concrete composite liner was installed to prevent the considerable convergence due to squeezing.Table 3 presents theC*andΔ(u/R)ultimatecalculated for the John street pumping station tunnel from Lo et al.(1987).

6.3.‘Wished-in-place’tunnel with polyvinyl chloride(PVC)liner

Arora et al.(2020b)carried out a physical model test on a synthetic mudstone specimen using the true-triaxial setup discussed earlier in this paper.However,unlike the present study,the cubical sample of the synthetic mudstone had a‘wished-in-place’tunnel with PVC liner as a support system.Based on the experimental observations in Arora et al.(2020b),it was observed that the PVC liner reduced the convergence of the tunnel by 87%.The PVC liner properties,C*andΔ(u/R)ultimateare shown in Table 3.

Table 3Case studies of the tunnel support in squeezing ground conditions.

7.Support selection criterion

Fig.8 shows the plot ofΔ(u/R)ultimateversusC*for the discussed case studies and the experimental observations in this paper.Based on the observations,the relationship betweenΔ(u/R)ultimateversusC*can be expressed by

For a very stiff support(C*=0),from Eq.(18),Δ(u/R)ultimateis 100.Also,for unsupported tunnels,C*is in finity and the correspondingΔ(u/R)ultimateis 0.

The methodology for the selection of the ef ficient and economic tunnel support system in squeezing ground conditions using Eq.(18)is as follows:

(1)Step 1:Determine the ultimate convergence of the unsupported tunnel(u/R)unsupportedfrom the field monitoring.However,in most of the squeezing tunnels,it is almost impossible to obtain reliable data from the field monitoring.Therefore,in such cases,(u/R)unsupportedcan be obtained from the in situ stresspoand compressive strength of the rock mass at the tunnel location using the equation proposed by Hoek and Marinos(2000)as

(2)Step 2: Determine Δ(u/R)ultimateusing the following equation:

where(u/R)desiredis the maximum allowable convergence selected by the tunnel designer.

(3)Step 3:From Eq.(18),determineC*for theΔ(u/R)ultimateobtained in Step 2.

(4)Step 4:Using Eq.(13),select the most suitable combination of Young’s modulus of supportEsand cross-sectional areaAsthat will not giveC*more than obtained in Step 3.

8.Discussions

8.1.Observation time for squeezing

The observation time in each loading stage presented in this paper was 7 d while squeezing may continue for weeks or even months after the excavation.The criterion proposed is based on 7 d of monitoring and the observations are extrapolated to in finite time.However,in future,the developed technique will need to be calibrated with the long-term field observations of the tunnels excavated in squeezing ground conditions.

8.2.Application of the proposed criterion

The LDP for the supported tunnel presented is limited to tunnel excavated in the synthetic mudstone specimen and squeezing ground conditions.The scale effect considerations and the similitude theory explain that the tunnel deformations can be normalized with the tunnel radius and will be independent of the size of the tunnel(refer to Appendices A and B).The criterion proposed is only valid in case no damage occurs in the rock mass due to excavation.Fig.9 shows the post-test picture of the cubical rock specimen.It can be clearly seen that the rock around the excavated tunnel boundary was undamaged even after the five loading stages.

The LDP was found to be in good agreement with the experimental observations in conjunction with excellent coef ficients of determination.Future work will calibrate the proposed LDP against different rock properties and the long-term behavior of the squeezing tunnels particularly using field case histories.This will provide the effect of the change in excavation geometry,tunnel support system and rock properties on the LDP model parameters.

8.3.Stress path difference

In the tunnel construction process,the convergence at the boundary is due to an increase in the deviatoric component of stress,i.e.the difference between an increasing tangential stress component and a decreasing radial stress component due to the advancement of the tunnel.If the tunnel is constructed in squeezing ground conditions,the tunnel can converge gradually even after the excavation stage,and without any further stress release due to unloading at the tunnel boundary.

Fig.8.Plot ofΔ(u/R)versus support compressibility ratio for the case studies presented in Table 3 with the best fit curve.

Fig.9.Post-test image of a cubical rock specimen with the excavated tunnel and its support system.

The testing procedure described in this research allows for the application of the isotropic stress on cubical rock specimen having cylindrical cavity(tunnel)while monitoring the deformations with time.Although this testing procedure does not replicate the unloading at the tunnel boundary,it effectively simulates the squeezing conditions observed in tunnels.The primary focus of this paper was to monitor the convergence of tunnel with time and develop a criterion that could be applicable to the field,despite the differences in the stress paths.

9.Conclusions

The study presented in this paper developed a novel physical model test to observe supported tunnel closure over time in squeezing ground conditions.The physical model simulated the TBM excavation in a tunnel as it advanced,and time-dependent strains developed due to rock-support interaction after the excavation were monitored.Also,analysis of the data from the strain gauges on the liner provided a quantitative assessment of the LDP of the supported tunnel at different times and stress levels.

Based on the experimental observations,a model was proposed to express the LDP of the tunnel as a function of time.The LDP parameters were found to be a function of the ratio of the isotropic stress applied to the UCS of the rock specimen and the normalized distance from the tunnel face.The LDP parameters of supported and unsupported tunnels were compared to account for the in fluence of the support system.The back-calculation of thrust forces on the support system provided an estimate of the additional thrust forces in the support due to squeezing.A relationship between convergence,reduced by the support,and support compressibility,was established based on the literature and the present study’s observations.A methodology was proposed for the ef ficient design of the tunnel support system in squeezing ground conditions.

Declarationofcompetinginterest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to in fluence the work reported in this paper.

Acknowledgments

The authors gratefully acknowledge the financial support of the University Transportation Center for Underground Transportation Infrastructure(UTC-UTI)at the Colorado School of Mines under Grant No.69A3551747118 from the US Department of Transportation(DOT).The opinions expressed in this paper are those of the authors and not of the DOT.

AppendixA.Supplementarydata

Supplementary data to this article can be found online at https://doi.org/10.1016/j.jrmge.2021.08.016.