Numerical modeling of time-dependent deformation and induced stresses in concrete pipes constructed in Queenston shale using micro-tunneling technique

Hyder Mohmmed Slim Al-Mmori,M.Heshm El Nggr,Silvn Micic

aDepartment of Civil and Environmental Engineering,Western University,London,N6A 5B9,Canada

bGeotechnical Research Center,Western University,London,N6A 5B9,Canada

1.Introduction

The time-dependent deformation(TDD)behavior of shales in southern Ontario,Canada was extensively investigated during the past fourdecades(e.g.Lo et al.,1975,1978;Yuen,1979;Lo and Yuen,1981;Lo and Lee,1990;Lee and Lo,1993;Hefny et al.,1996;Lo and Hefny,1996;Hawlader et al.,2003).These studies provided good insights into the swelling phenomenon of these shales,and its measurements,causes and controlling mechanism.Lo and Yuen(1981)developed a closed-form visco-elastic solution to predict deformations and stresses in concrete lining of circular tunnels in swelling rocks.Hefnyet al.(1996)extended this solution to account for long-term swelling and critical stress,defined as the minimum stress required to stop rock swelling.Both solutions were utilized to analyze many tunnels in southern Ontario,as indicated by Lo and Hefny(1996).

The strength of shales in southern Ontario was also investigated(Lo and Hori,1979;Yuen,1979;Lo and Yuen,1981;Wai et al.,1981;Lo et al.,1987;Lee,1988).The strength was found to be anisotropic with respect to the rock bedding.Investigations on similar rock types demonstrated that increasing their moisture content reduced their strength significantly(Colback and Wiid,1965;Paterson,1978;Baud et al.,2000;Claesson and Bohloli,2002;Paterson and Wong,2005;Gorski et al.,2007;Liang et al.,2012;Dan et al.,2013;Wasantha and Ranjith,2014).

Al-Maamori(2016)investigated the impact of lubricant fluids(LFs)used in micro-tunneling applications on both of TDD and strength of the Queenston shale(QS).The TDD behavior of QS in LFs was found to be different from that in water,and its strength decreased with different percentages after being soaked in water and LFs.Given that the TDD behavior of QS and its strength degradation are different in LFs compared to those in water,it is necessary to investigate the effects of LFs used in micro-tunneling technique on the constructed pipe or tunneling.

Several numerical studies were conducted to investigate the TDD of swelling rock employing the finite element(FE)method.Hawlader et al.(2003)developed a plane-strain FE model to predict deformations and stresses induced in tunnel lining due to swelling rock.Kramer and Moore(2005)used the swelling model proposed by Lo and Hefny(1996)and developed a plane-strain FE model to calculate the TDD of the rock and stresses due to rock-tunnel lining interaction.Heidkamp and Katz(2002)proposed an implicit integration scheme using Grob(1972)’s swelling law to predict volume increase of gypsum and clay minerals and implemented it in a FE program.Wittke-Gattermann and Wittke(2004)developed a constitutive elasto-plastic law to describe swelling of non-leached gypsum,considering its anisotropic behavior,and implemented it in three-dimensional(3D)FE program.Sch?dlich et al.(2012)extended this model to a general rock swelling model using Grob’s swelling law.Sch?dlich et al.(2013)implemented the rock swelling model in PLAXIS computer program to back-analyze the in situ measurements of the Pf?ender railway tunnel in Austria.

All the previous studies investigated the TDD effects due to soaking of rocks in water.However,the TDD behavior of shale in the LFs was found to be significantly different.Compared to water,the polymer and bentonite solutions have caused different amounts of decrease in the TDD of the QS(Al-Maamori,2016;Al-Maamori et al.,2016).When these effects are considered,they may lead to a safer design of pipes and tunnels constructed in shales using the micro-tunneling technique.

The main objective of this research is to investigate the TDD and stresses induced in pipes or tunnels constructed using the microtunneling technique in QS.This objective is achieved through numerical modeling employing the FE analysis program PLAXIS 2D(PLAXIS,2016).The rock swelling model developed and implemented in PLAXIS 2D environment is based on Grob’s swelling law,which is mathematically identical to the Lo and Hefny(1996)’s swelling model.The results are envisioned to aid designers and contractors to determine whether micro-tunneling technique is a feasible construction technique for pipelines and tunnels in the QS of southern Ontario,Canada.

2.Geological background

Fig. 1 shows the geological map of southern Ontario.As can be observed from Fig. 1,most of southern Ontario region is located in the Appalachian sedimentary basin,which is bounded by the Precambrian basement highs in the west,Taconic mountain range in the east and south,and the Frontenac arch in the north(Perras,2009).The QS layer in this basin is an argillaceous sedimentary rock formed in the upper Ordovician age in south-west region of Ontario.It forms most of south and west shores of Lake Ontario and extends to the north towards the Georgian Bay.Together with other rocks in the Appalachian basin,the QS layer dips 6 m/km to the south(Yuen et al.,1992),and it becomes thinner and overlain by other rocks from the Silurian and Devonian ages,such as Grimsby and Eramosa shales,dolomite,and limestone of different formations.This layer forms the host ground for many important engineering projects in Ontario,such as the Niagara tunnel(Perras,2009).More recently,several pipelines and tunnels are constructed in the QS layer using the micro-tunneling technique.

3.Locations of investigated Queenston shale

The QS samples investigated in this research were collected from two boreholes located in Milton and Niagara Falls regions,as shown in Fig. 1.The rock quality designation(RQD)of the QS layer from Milton increased with depth from 50%to 99.2%,indicating fair to excellent rock condition.The bedrock in this region starts at a depth of 15.5 m below ground level(BGL).Based on the RQD value and the occurrence of joints,the QS layer of Milton can be divided into three sublayers:(i)upper sublayer(from 15.5 m to 22.5 m BGL)with an average RQD value of 68.9%,and 2-5 mm compacted clay joints occurring at an average of 4 joints/m,which corresponds to geological strength index(GSI)value of 42(Hoek et al.,1995);(ii)middle sublayer(from 22.5 m to 26.5 m BGL)with an average RQD value of 89.9%,and 2-5 mm compacted clay joints occurring at an average of 1 joint/m,which corresponds to GSI value of 54;and(iii)lower sublayer(from 26.5 m to 34.52 m BGL)with an average RQD value of 96.8%,and compacted clay joints that occur at 1 joint/(4 m),corresponding to GSI value of 59.The Niagara QS was collected from a bore hole drilled from the invert of the Niagara tunnel at its lowest part at a depth of 125-137 m BGL.The recovered samples were approximately 12 m long.The suggested GSI value for this layer is 59.Based on the GSI value,the strength envelope of each layer was developed utilizing RocLab software(Rocscience,2016).The modulus,cohesion and frictional angle of the rock mass of each layer were derived from the developed strength envelopes.The parameters were derived for intact rock(i.e.as collected from site)and after soaking the QS samples in water and LFs for 100 d to account for the strength degradation of the shale near the excavation.The strength envelopes were established based on the results of an experimental study that was conducted to evaluate the strength degradation of the QS after being exposed to water and LFs(Al-Maamori,2016).

4.Finite element PLAXIS 2D rock swelling model

The computer program PLAXIS 2D has two-dimensional(2D)user-defined constitutive model that can simulate rock swelling behavior(PLAXIS,2016).This constitutive model simulates the TDD behavior of rocks based on swelling clay minerals that exist in their micro-structure.The mathematical formulation of the TDD behavior of rocks in this constitutive model was developed essentially based on Grob(1972)’s swelling law,which is shown in Fig. 2a.In this model,the swelling mechanism of rocks is related to the osmotic swelling and the inner-crystalline swelling of clay minerals(Madsen and Müller-Vonmoss,1989).The osmotic swelling is caused by the differences in cation concentration in the clay and in the free pore water,and it occurs after the completion of the innercrystalline swelling.This swelling is caused by the increase in the repulsive forces between the negatively charged neighboring clay layers,which in turn increases the distance between these layers.The inner-crystalline swelling occurs first,and it can result in 100%of volume increase of clay particles in the case of montmorillonite.This swelling occurs due to the integration of water molecules into the clay mineral crystals when the existing cations hydrate in the presence of water.When the energy released in the process of cations hydration exceeds the anion-cation bond within the clay mineral,swelling occurs.The swelling pressure in this process depends on the nature of the existing cations,where Na+cations result in larger swelling pressure than Ca2+cations(PLAXIS,2014).The stress-dependency of the TDD of rock follows a semilogarithmic relation,as indicated in Fig. 2a.The TDD decreases with increasing applied stress in a logarithmic scale.The mathematical expression of the developed rock swelling model in PLAXIS(2014)is given by

Fig. 1.Geological map and sectional profile of southern Ontario showing rock layers forming this region and the locations of sampling boreholes.

To avoid excessive swelling strains at low or tensile stresses,the swelling curve is limited atσi=σc.The swelling strain at any time is calculated from Eq.(2),where the parameterηqidentifies how fast the swelling strain is achieved and it depends on the slope of the swelling curve in the free swell test.For further details,please see PLAXIS(2014).

Fig. 2b shows the mathematical model expressing the TDD of swelling rocks developed by Lo and Hefny(1996),which is identical to Grob’s law adopted in PLAXIS 2D.The TDD of swelling rocks in southern Ontario follows the free swelling up to a threshold stress,defined as the minimum stress that can cause suppression influence on the swelling deformation of the rock(Lo and Hefny,1996).Beyond this stress,the TDD follows a straight line with the stress applied in the same direction of swelling in a logarithmic scale until it reaches zero swelling at a specific applied stress called the critical stress.This stress is defined as the minimum stress that supresses the swelling of the tested rock completely.The critical stress can be measured by the null swell test(Lo and Lee,1990).

Fig. 2.Rock swelling models:(a)Grob(1972)’s swelling law;and(b)Lo and Hefny(1996)’s swelling model.σcis the minimum axial stress to limit excessive swelling(i.e.threshold stress in the Lo and Hefny(1996)’s model).

These two swelling expression models,however,differ in terms of the swelling mechanism.In southern Ontario,swelling occurs due to the following reasons:(i)the relief of initial in situ stress;(ii)the accessibility to water;and(iii)an outward gradient of the pore fluid salinity of the rock to the ambient fluid(Lee and Lo,1993).Meanwhile,the swelling modeled by the Grob’s law is caused by osmosis pressure and swelling of clay minerals existing in the rock as explained earlier.

5.verification of finite element PLAXIS 2D rock swelling model

The rock swelling model in PLAXIS 2D was verified using the results of the TDD tests(i.e.free swell,semi-confined swell and null swell tests)performed on Milton and Niagara QS samples.The setups for these tests are shown in Fig. 3.In these tests,water and LFs(i.e.bentonite and polymer solutions)used in micro-tunneling applications were utilized as the ambient fluids.The experimentally derived TDD parameters of the QS from Milton and Niagara regions are summarized in Table 1.The values presented in Table 1 are the average of measured parameters from the TDD tests(Al-Maamori et al.,2016).These parameters are used in the verification stage of the PLAXIS 2D and in the parametric study afterwards.

5.1.Finite element model of free swell test

The free swell test was simulated using an axi-symmetrical model that represented half cylinder of the test specimen.The initial dimensions of the FE model represented the actual dimensions of the test specimen as indicated in Fig. 3a(i.e.diameter of 63 mm and height of 63 mm).The FE model simulated the behavior of specimens submerged in water and LFs for 100 d.This is a standard test period usually used to measure the TDD of the tested rock in southern Ontario(Lo et al.,1975,1978).

The default“medium refined”FE mesh of PLAXIS 2D shown in Fig. 3b was used in the test model,which produced reasonable results.A total of 196 15-noded triangular elements were used in the FE mesh to model the free swell test.The axis of symmetry was fixed in the horizontal direction(x-axis)and the bottom of the FE model was fixed in the vertical direction(y-axis),while all other boundaries were free to move.In order to simulate TDD behavior of QS,the TDD parameters of the rock swelling model for QS,summarized in Table 1,were defined in the developed FE model utilizing the coupled f l ow and deformation analysis approach available in PLAXIS 2D.

In the actual laboratory frees well test,the swelling strains of the specimen were measured every day along the specimen’s vertical and horizontal axes,over the duration of the test(i.e.100 d),and were plotted vs.time in semi-logarithmic scale to derive the TDD parameters(i.e.vertical and horizontal swelling potentials),as described in Al-Maamori et al.(2016).In order to simulate the physical test conditions,the numerical analysis was performed in two phases:initial phase and swelling phase that lasted for 100 d.The deformations were monitored in the analysis at the top and mid-height of the free edge of the model,which simulated the measuring locations in the actual free swell test.

Three cases of free swell test of Milton QS were modeled:swelling in water,swelling in polymer solution,and swelling in bentonite solution.Fig. 4a and b compares the computed TDDs from the FE models and the measured TDDs from free swell tests performed on Milton QS soaked in polymer solution.As can be noted from Fig. 4,there is an excellent agreement between the calculated and measured TDDs.Similar agreement was observed between the calculated and measured TDDs for specimens soaked in water and bentonite solution.The excellent agreement between the calculated and measured TTD values verified the ability of the FE model to simulate the TDD behavior of QS under free swell conditions.

5.2.Finite element model of semi-confined swell test

The semi-confined swell test was modeled in a similar manner to the free swell test,except that the model was shorter(i.e.its height is 45 mm,similar to the height of laboratory test specimen),and a uniform pressure,with the same value of the applied pressure used in the actual semi-confined swell test,was applied to the top of the FE model.In the numerical model,the FE mesh comprised 270 15-noded triangular elements.

Fig. 3.Schematic illustration of rock swelling tests developed earlier at the University of Western Ontario and their finite element models.

In the actual semi-confined swell test,the swelling strain of the specimen in the same direction of the applied pressure is measured daily for 100 d(Lo et al.,1978).Six different FE models were used to simulate the actual semi-confined swell tests of Milton QS in water,polymer solution and bentonite solution in the vertical and horizontal directions with respect to rock bedding,using the average measured values of TDD derived from laboratory tests(Al-Maamori et al.,2016).The TDD parameters used in these FE models are listed in Table 1.The analysis was performed in two phases:initial phase and swelling phase that lasted for 100 d under the specific applied pressure to simulate the test duration.Fig. 4c and d compares the computed and measured swelling strains of vertical and horizontal Milton QS specimens in bentonite solution under applied pressures of 1 MPa and 0.7 MPa,respectively.It is observed from Fig. 4c and d that the computed strains are in good agreement with the measured values.It should be noted that the TDD parameters listed in Table 1 are the average of the measured values of the QS.Therefore,the calculated TDD curves lie within the upper and lower bounds of the measured TDD curves.It should also be noted that the measured TDD curves in the semi-confined swell test usually follow a stepwise deformation pattern due to mechanical reasons related to the measuring device or due to other reasons as explained by Hefny et al.(1996)and Al-Maamori et al.(2016).On the other hand,the computed strains are smooth.Similarobservations were made from the analysis of the QS samples soaked in water and polymer solution.The good agreement between the computed and measured swelling curves verifies the ability of the numerical model to simulate the stress-dependency of the TDD behavior of QS in the three fluids used.

Table 1Time-dependent deformation parameters of Milton and Niagara Queenston shales based on Lo and Hefny(1996)’s swelling model from Al-Maamori et al.(2016).

5.3.Finite element model of the null swell test

The FE model of the null swell test simulated the geometry of the actual null swell test specimen shown in Fig. 3e.Due to symmetry,only half of the sample was simulated utilizing an axisymmetrical model as indicated in Fig. 3f.The numerical model comprised 270 15-noded triangular elements.Fixed boundaries were applied at the top and bottom of the model to prevent vertical swelling(i.e.alongy-axis),while horizontal swelling(i.e.alongxaxis)was allowed by assigning rollers at both sides.These boundary conditions represented the actual laboratory setup of the null swell test.The time required for developing the maximum pressure in the null swell test of QS was found to be 45 d(Al-Maamori et al.,2016).Therefore,the FE analyses involved 45 phases,with one day time interval for each phase.Six numerical models were analyzed for Milton QS soaked in water,bentonite solution,and polymer solution considering vertical and horizontal specimens.The parameters listed in Table 1 were used in these analyses.The swelling pressure in each phase of the analyses was considered to be the maximum developed pressure in the FE model in they-direction,analogous to the measured swelling pressure in the laboratory test.

Fig. 4e and f compares the developed swelling pressure obtained from the FE models with those measured from the laboratory tests after 45 d of swelling for vertical and horizontal Milton QS specimens submerged in water.As can be noted from Fig. 4e and f,there is an excellent agreement between the computed and measured pressure values from the null swell test.Similar excellent agreement was achieved from the analysis of the null swell tests for QS specimens soaked in bentonite and polymer solutions,which confirms the ability of the numerical model to capture the behavior of induced swelling pressure of QS with time.

The excellent agreement between the numerical and experimental results of different swelling tests as discussed above verifies the ability of the developed FE model to simulate the observed behavior of QS in laboratory swelling tests.The results confirm that the model can accurately simulate different aspects of the swelling behavior of QS and can correctly predict its TDD behavior.The stress-dependency of the TDD of QS was also accurately simulated using the developed models of semi-confined swell and null swell tests.Thus,it is concluded that the FE model can be used to analyze different infrastructure projects constructed in QS of the southern Ontario.In addition,the FE model will be further verified against some documented infrastructure cases in the literature.These cases documented the observed behaviors of actual projects including excavations in swelling rocks and the analysis of circular tunnel constructed in QS.

5.4.Case histories for excavation in rocks

The reported cases of excavations in rocks with long-term in situ measurements are limited.Two well-documented cases are studied herein:(i)the Niagara wheel pit excavation(Lee and Lo,1976),and(ii)the Scotia Plaza foundation excavation(Trow and Lo,1989).

5.4.1.The Niagara wheel pit excavation

The TDD of the excavated rock for the Niagara wheel pit was recorded over 72 years.The Canadian Niagara Generation Station was constructed in 1902-1906 to generate clean energy utilizing the water head loss at the Niagara Falls on the Niagara River.As part of the construction requirements to host the penstocks and the turbines units,a deep and narrow slot was excavated in the rock between 1902 and 1903.The slot was 174 m long,5.49 m wide and 50.3 m deep.The overburden soil layer was 4.6 m of gravel underlain by four types of rocks:21.3 m of Lockport dolomite,7.2 m of Gasport and Decew limestone,17.1 m of Rochester shale,and the lower 45 m of Grimsby limestone and Power Glen shale(Morison,1957;Lee and Lo,1976).Inward movement of around 38 mm in the Rochester shale layer was noticed during the excavation period(Smith,1905;Lee and Lo,1976).This inward movement continued after the excavation was completed,and caused damage to some of the turbines and draft tubes(Lee and Lo,1976).Extenso meters were installed in 1905 at different elevations to measure the inward movement of the sides of the slot,especially in the Rochester shale layer where the maximum inward movement was reported.The in situ horizontal stress of the Niagara Falls was 6.9 MPa(Lee and Lo,1976).

This case was modeled using the verified rock swelling model in PLAXIS 2D.The TDD parameters used in this analysis for the rock layers are summarized in Table 2.Fig. 5a presents the numerical model of the slot and the rock block,which is 75 m in height and 125 m in width to account for the high horizontal in situ stress effect.Both sides of the model were assigned to horizontal fixity(i.e.fixed along thex-axis)and its bottom was assigned vertical and horizontal fixities while its top and the excavated slot were free to deform.The water table was located at the top of the model.A fine FE mesh was assigned for the numerical model with a total of 929 15-noded elements.The analysis was conducted into 18 phases covering time from 1902 to 1972.The excavation was modeled in the first phase,which lasted for 581 d(from March 1902 until October 1903).The duration of subsequent phases was selected to cover the following periods:1903 to 1905;1905 to 1907;1907 to 1910;five years intervals from 1910 to 1970;and finally,two years from 1970 to 1972 to match the total period of monitoring the extensometer readings(Lee and Lo,1976).

Fig. 4.Computed finite element results and laboratory test results of free swell tests,semi-confined swell test and null swell test of Milton Queenston shale(MQS)in vertical and horizontal directions under different ambient fluids.

The results of FE analysis are illustrated in Fig. 5,which displays the computed maximum inward movement.Fig. 5a indicates that the maximum inward deformation occurring at the Rochester shale layer.The inward deformation arrows of the FE model shown in Fig. 5b indicate the magnitude of the TDD of the Rochester shale compared to that of the other rock layers.This behavior is quite similar to that reported based on in situ measurements(Lee and Lo,1976).The computed inward TDD of the Rochester shale at the turbine deck level from 1905 to 1972 is compared in Fig. 6 with the in situ measured values from the extensometers.Fig. 6 demonstrates the excellent agreement between the computed and measured in situ deformations.Beyond 1952,a discrepancy can be noted between the computed and the measured TDD curves,which was due to another tunnel excavated in 1952 at 152 m away from the Niagara wheel pit(Lee and Lo,1976).The overall agreement between the FE calculations and actual measurements from 1905 till 1952 is very evident.This example confirms the capability of therock swelling model to predict the TDD behavior of different rock layers in case of an unsupported vertical excavation in the rock mass.

Table 2Parameters used in the analysis of the Niagara wheel pit(Lee and Lo,1976).

Fig. 5.Finite element model of the Niagara wheel pit excavation.

Fig. 6.Inward time-dependent deformation of the Rochester shale layer of the Niagara wheel pit excavation.The extensometer readings and locations were collected from Lee and Lo(1976).

5.4.2.The Scotia Plaza foundation excavation

This case history illustrates a supported vertical excavation made in the rock with high initial horizontal in situ stresses,and the associated inward deformations at different locations away from the excavation.The relief of high in situ stresses due to an excavation causes elastic deformation and TDD of the rock,which was the case for Scotia Plaza foundation excavation.Relative displacement that may occur on the rock surface due to the excavation can seriously impact the adjacent buildings.

The Scotia Plaza is a 69-story high-rise complex building in downtown Toronto,and it is surrounded by existing tall buildings(Trow and Lo,1989).The critical section of excavation was the east west section,40 m wide and 24 m deep,14 m of which passed through rock.The upper 2 m of the Georgian Bay shale at the site was fractured and it had different elastic modulus from the lower layer,as indicated in Table 3.Both upper and lower Georgian Bay shales were assigned the same TDD parameters(i.e.free swell potential,threshold and critical stress)but different elastic moduli.The water table was located at 1 m below the rock top surface.Monitoring devices were installed at the excavation face and at distances away from the excavation on the ground surface to measure the relative displacement of the ground during and after the excavation.The adjacent building consisted of a bearing wall located on the face of the excavation,having line load of 820 kN/m.At 7.3 m away from the bearing wall,there were five rows of columns distributed at constant spacing of 5.5 m,supporting 5650 kN of load on each column.The bottom of the excavation was subjected to 0.19 MPa distributed load of equipment during construction.As the excavation progressed,a 0.4 MPa shoring support of uniformly distributed anchors was gradually applied on the side of the excavation.

This excavation and the critical loads from the adjacent building on the east side were modeled using a 60 m deep and 140 m wide model from the center line of the excavation towards the east.The FE mesh comprised 1268 15-noded triangular elements.The overburden was simulated as distributed load of 0.19 MPa.The column loads were represented as equivalent line load to account for the other column loads in the perpendicular direction to the model.The analysis was performed in seven phases covering 180 d from the beginning of excavation.Each phase consisted of 2 m of excavation.The shoring pressure was applied gradually in these phases similar to the actual case,until the maximum shoring pressure of 0.4 MPa was achieved.The sides of the FE model were fixed in thex-direction,the bottom boundary was fixed inx-andy-directions,and the top and excavation boundaries were free to move.

The maximum surface displacement measured in the field at the face of the excavation was 25 mm(Trow and Lo,1989),while the computed maximum deformation at the same location was 24 mm.The deformation with time at the excavation face and at different intervals along the rock surface are presented in Fig. 7a and b,together with the deformations of other numerical analysis(Trow and Lo,1989).It can be noted from Fig. 7b that the computed deformations are in good agreement with the extensometers readings and they are also consistent with other numerical analyses.This agreement confirms the ability of the rock swelling model to predict the swelling behavior of rocks in supported excavation.

Table 3Parameters used in the analysis of the Scotia Plaza foundation excavation(Trow and Lo,1989).The shoring pressure is 0.4 MPa.

Fig. 7.Scotia Plaza excavation:(a)Finite element model;and(b)Measured and calculated deformations of excavation face.

5.5.Modeling circular tunnel in rock and comparison with closed form solution

This case is presented to verify the FE model against the closed form solution proposed by Lo and Hefny(1996)for a circular tunnel in swelling rock.The considered tunnel was constructed in QS with 6.8 m radius and with concrete lining of 0.55 m thick,similar to the tunnel in the Sir Adam Beck Niagara Generation Station No.3 Project.The TDD and strength parameters of the QS and concrete lining are given in Table 4.The depth of the analyzed tunnel was 200 m.The in situ horizontal stress measured from hydraulic fracturing test at the level of the tunnel was 21 MPa in average.The vertical in situ stress was 5.2 MPa,indicating horizontal to vertical stress ratio(Ko)of 4.The full details of the adopted tunnel are given in Lo and Hefny(1996).

Only a quarter of the tunnel is modeled in consistence with the studied example.The FE model simulated 200 m block of QS,which was discretized using 1068 15-noded triangular elements.The side boundaries were fixed horizontally,and the bottom boundary was fixed vertically,while the top of the model was free to move.The water table was set in the model at the top of the QS layer.The rock layer was assigned the parameters presented in Table 4.The concrete lining was modeled as a plate element with parameters of the concrete lining given in Table 4.

The waiting period prior to installing the lining was 30 d(Lo and Hefny,1996).Accordingly,the plate representing the concrete lining in the FE model was activated after phase 1,which lasted for 30 d.Phase 2 was assigned a period of 150 years of swelling,which was assumed to correspond to inf i nity in the closed-form solution.The axial thrust acting in the concrete lining after 150 years(i.e.axial force divided by the cross-sectional area of the concrete lining)was obtained from the FE analysis.Two cases of analysis were considered:(i)full slip is allowed at the interface between the concrete lining and rock,and(ii)no slip at the interface between the concrete lining and rock.

Table 4Parameters used in the analyses of the Lo and Hefny(1996)’s model.

Fig. 8.Axial thrust and bending moment in concrete lining computed based on Lo and Hefny(1996)’s closed-form solutions and calculated from FE model:(a,b)full slip at the interface;and(c,d)no slip at the interface.

Fig. 8 compares the axial thrust values calculated from the FE analyses with those obtained from the closed-form solution for the full slip and no-slip conditions.For the case of full slip at the interface(Fig. 8a and b),the computed bending moment is in reasonable agreement with the value calculated from the closed form solution.The bending moment is positive at the spring line,which causes tension in the inner fibers of the concrete lining.The bending moment decreases gradually and becomes zero close to the tunnel shoulder,and reaches its maximum negative value at the tunnel crown.This negative bending moment causes tension in the outer fibers of the concrete lining.The axial thrust is positive and it causes compression in the concrete lining along the tunnel.It can be noted that the thrust value obtained from the FE model is close to the thrust calculated from the closed-form solution.

Table 5Properties of concrete pipes or tunnel lining used in micro-tunneling applications(provided by the manufacturer).

For case of no slip at interface,the trends of both axial thrust and the bending moment computed from the FE analyses are similar to those of the full slip case but they have different magnitudes.Fig. 8c and d shows that the FE prediction of axial thrust at the crown is very close to the value calculated from the closed-form solution.However,the thrust at the springline obtained from FE analysis is higher than the prediction of the closed-form solution.Similar observations can be made for the bending moment,for which the FE prediction is higher than the value obtained from the closed form solution.These minor discrepancies between the FE computations and the closed-form solution calculations may be attributed to the following reasons:(i)the FE model has limited size of 200 m block of swelling rock while Lo and Hefny(1996)’s closed-form solution considers an infinite medium of swelling rock,which increases the swelling in the horizontal direction and induced stresses;and(ii)the time used in the FE analyses was 150 years;while in the closed-form solution,the swelling time is assumed to be infinite.

The acceptable predictions of the FE model for TDD behavior and induced stresses of tunnel segments in the discussed cases confirm the suitability of the FE model to be used for analyzing pipe or tunnel segments constructed in QS using the micro-tunneling technique.Thus,the verified numerical model will be used to conduct further analyses of reinforced concrete pipe or tunnel segments in QS.In the following section,only the TDD effects of the QS on the constructed pipe or tunnel will be considered.

6.2D finite element modeling of time-dependent deformation behavior and induced stresses in tunnels constructed using micro-tunneling technique in Queenston shales

The TDD behaviors and induced stresses in pipes or tunnels constructed using micro-tunneling in QS are analyzed in this section.Pipes considered in this parametric study are made of reinforced concrete.The mechanical properties of the reinforced concrete pipes,obtained from actual test results as provided by the manufacturer,are summarized in Table 5.Eight pipe inner diameters are considered:d=0.6 m,0.9 m,1.2 m,1.5 m,1.8 m,2 m,2.5 m and 2.7 m.The tensile strength of the pipes’concrete is assumed to be 10%of its compressive strength(Neville,1996).The pipe wall or tunnel lining was defined in the FE model as a plate having strength properties listed in Table 5,and a Poisson’s ratio of 0.2(Neville,1996).The cellular cement grout to be used to replace the LFs at the end of the construction period has a compressive strength of 0.5 MPa as indicated by the manufacturer and it was defined in the FE model as linear elastic material with elastic modulus of 22.7 MPa and Poisson’s ratio of 0.2(Nehdi et al.,2002).The unit weight of the overburden soil is assumed to be 20 kN/m3.

The TDD and strength properties of Milton and Niagara QSs used in these analyses are summarized in Table 6.Three values of the stress ratio(i.e.in situ horizontal stress/vertical overburden stress)wereused:Ko=5,10 and 20(Al-Maamori et al.,2014).The length of construction period of a pipeline or a tunnel using the microtunneling technique depends on the site condition and the nature of hosting rock.This period is expected to be several days up to several weeks(Flint and Foreman,1992;Fritz,2007;Bezuijen,2009).Additional waiting time is suggested by Lo et al.(1987)to reduce the TDD effects on the constructed pipeline or tunnel.In the following discussion,“construction period”refers to the actual time required to install the pipeline in addition to the waiting time prior to replace the LF in the gap between the pipe and the excavated rock with the permanent cellular cement grout.Accordingly,the construction period(t)considered in this study wast=15 d,30 d,60 d,100 d,150 d and 200 d.

The following sign convention will be followed:negative axial force acting along the pipe diameter causes tangential compressive stress while positive axial force causes tangential tensile stress;and positive bending moment causes tension at the inner fibers while negative bending moment causes compression at the inner concrete fibers of the pipe.As shown in Fig. 9,due to symmetry,only half of the pipe or tunnel was modeled considering 35 m?35 m block of the QS.The FE mesh was refined at the pipe or tunnel and 15-noded triangular elements were used to simulate the rock layer and pipe(or tunnel lining).

Al-Maamori(2016)demonstrated that the QS would experience some degradation of its strength after being continuously exposed to water and LFs.The degradation was attributed to the TDD occurring as the fluids penetrated into the QS.Al-Maamori et al.(2017)developed a correlation to predict the long-term penetration depth of water and LFs into the QS.This correlation was derived based on long-term penetration tests performed on the QS accounting for its TDD effects,and is adopted in the FE analysis to specify the depth over which LFs(i.e.polymer and bentonite solutions used in micro-tunneling technique)can reach during the construction period.This correlation is given by

wheredis the depth of percolation of the applied fluid into the QS in m;Cf lis the fluid constant in m/d0.588,and it equals 0.012 m/d0.588for bentonite solution and 0.0102 m/d0.588for polymer solution(Al-Maamori et al.,2017);tis the time of fluid application(i.e.the construction period)in days.

Fig. 9.Typical finite element mesh used to model time-dependent deformation effects of the Queenston shale on the pipe or tunnel constructed using micro-tunneling technique.

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In the FE models,the QS was assigned strength and TDD parameters over this depth,which was different from the rest of the rock(denoted as intact rock)in order to account for strength degradation associated with exposure of QS to drilling fluids.The strength and TDD parameters are summarized in Table 6.

The FE analyses also accounted for the strength and TDD properties anisotropy of the QS by assigning different parameters in the horizontal and vertical directions with respect to the rock bedding.The FE analyses were performed for both Milton and Niagara QSs using the relevant parameters for either polymer or bentonite solution used during the construction period.The parametric study covers the variation in:(i)in situ stress ratio(Ko),(ii)depth of construction below the top of QS layer,(iii)diameter of the pipe or tunnel lining,and(iv)construction period.In this study,30 years were considered as a lifetime for the constructed pipeline or tunnel.The analyses were performed in 16 phases up to 30 years where the first phase represents the construction period followed by the rest of swelling phases with gradually increased duration.

In order to verify the sensitivity of the FE model to the size of the FE mesh,the analysis was performed for 1.8 m diameter pipe constructed at 1.7 m below the top of the QS layer in 30 d,using bentonite solution as LF during the construction period.Based on the default size of FE mesh in PLAXIS 2D,four sizes of the FE mesh were used:very coarse mesh with 218 elements,very fine mesh with 2297 elements,medium mesh with 860 elements,and medium mesh refined at the pipe with 1750 elements.The analyses were performed for 30 years and the pipe inward movement at its springline was calculated.Fig. 10 shows this movement for the four cases analyzed.It can be observed that changing the mesh size from very fine to very coarse can cause remarkable difference in the inward movement of the pipe.Increasing the FE mesh size produces higher inward movement at the springline.However,the medium FE mesh refined at the pipe or tunnel produced almost identical results to those obtained from the model with very fine mesh.Therefore,it was decided to use FE model with the default medium FE mesh refined near the pipe to achieve optimum balance between accuracy and computing efficiency in the following analyses.

Fig. 10.Inward time-dependent deformation at the springline of 1.8 m diameter pipe after 30 years of construction.

6.1.influence of stress ratio(Ko)

Fig. 11 shows the maximum tensile stress developed at the springline of the pipe or tunnel segment constructed in Milton QS using bentonite solution as LF in micro-tunneling construction technique.The results are presented for three pipe sizes(d=0.6 m,1.2 m and 2.7 m)considering shallow,moderate and deep pipes along with the stress ratioKo=5,10 and 20.From Fig. 11,the following observations can be made:

(1)The developed tensile stress in the pipe at the springline decreases as the stress ratio(Ko)increases for the same depth.The stressstate at the springline changes fromtension at low value ofKoto compression at high value ofKo.

(2)At shallow depth,the developed stress at the springline of the most used pipes or tunnels is tensile regardless of pipe size andKovalue.

(3)Increasing the pipe depth decreases the tensile stress at the springline and increases the compressive stress at the crown.The stress state at the springline changes from tension to compression with increasing depth.

(4)Increasing the diameter of the pipe or tunnel causes higher tensile stresses developed at the springline.

(5)High tensile stress at the springline of pipes or tunnels occurs when short construction period(i.e.30 d)is used.

Based on these observations,it can be concluded that the critical state of the concrete pipe or tunnel lining can occur when they are constructed at shallow depths over a short construction period(i.e.30 d).At this state,the tensile stresses at the springline are the highest.Accordingly,the following analyses are performed on pipes constructed in Milton and Niagara QSs using 30-d construction period and stress ratioKo=5 as these two values are expected to produce the highest tensile stress at the springline of the pipe.

Fig. 11.Variation in the tangential stress at the springline of pipes constructed in Milton Queenston shale with increasing stress ratio(Ko)and construction period:(a)Pipe at 1.7 m depth;(b)Pipe at 10 m depth;and(c)Pipe at 20 m depth below the top of the rock layer.

Fig. 12.The maximum tangential stresses at springline and crown of a concrete pipe constructed in Milton Queenston shale with in situ stress ratio(Ko)of 5:(a,c)using polymer solution,and(b,d)using bentonite solution.

6.2.influence of pipe diameter

The results of FE analyses performed on pipes of different diameters constructed in Milton and Niagara QSs are presented in Figs.12 and 13,respectively.The analyses were performed using bentonite and polymer solutions as LFs during the construction period.The results are presented for shallow,moderate and deep pipes constructed in QSs.

Fig. 12 shows that for pipes constructed in Milton QS,the tensile stress acting at the springline increases substantially as the pipe diameter increases.The tensile stress reaches its peak value at 1.8 m pipe diameter and then slightly decreases with increasing diameter.The compressive stress acting at the crown does not change at small diameter.However,it remarkably increases for pipe diameter greater than 1.2 m and remains almost constant for larger diameters.The compressive stress at the crown is generally well below the concrete compressive strength.This behavior may be attributed to the zone in which the initial in situ stresses are relieved upon excavation.With increasing pipe or tunnel diameter,this zone becomes larger and triggers larger swelling area.On the other hand,the effects of ground confinement are greater in pipe or tunnel with smaller diameter.In other words,at small diameters,the swelling zone is relatively small and the ground imposes higher confinement effects on the whole section of the pipe or tunnel,while the opposite is true at large diameters.This may cause an increase in the swelling stresses at the springline and the crown of the pipe,as noticed in Figs.12 and 13.It seems that these effects decrease when the diameter is greater than 1.8 m and 2.1 m,respectively.The concretetensile strength of pipes with diameter of 0.6-1.8 m is 6 MPa,while it is 4.8 MPa for pipes with diameter larger than 1.8 m.The calculated tensile stress acting on the springline of all pipes considered in the analysis does not exceed the tensile strength of the concrete when polymer solution is used.In other words,the pipes do not experience tensile cracks during a lifetime of 30 years.However,the developed tensile stress at the springline of pipes with the diameter greater than 1.8 m may exceed the tensile strength of the concrete when they are constructed at shallow depth(i.e.1.7 m)and using bentonite solution as LF during the construction period.This means that tensile cracks may appear in these pipes only when they are constructed under these conditions.Deeper pipes did not experience stresses higher than their tensile strength,which indicates undamaged pipes throughout the lifetime of 30 years considered in this study.

Fig. 13.The maximum tangential stresses at springline and crown of a concrete pipe constructed in Niagara Queenston shale with in situ stress ratio(Ko)of 5:(a,c)using polymer solution;and(b,d)using bentonite solution.

Due to the higher swelling potentials of Niagara QS,the tensile strength of the concrete may be reached for pipes constructed at shallow and moderate depths below the top of the QS layer.Fig. 13 presents the behaviors of pipes constructed in Niagara QS.The TDD behavior in this case is similar to that observed for Milton QS;however,the shape of curves is slightly different.For pipes with the diameter greater than 1.8 m constructed at shallow depth using bentonite or polymer solution,the stresses at the springline have reached the concrete tensile strength.The stress at the springline in pipes constructed at moderate depths has also reached the tensile strength of concrete when bentonite solution is used.This indicates that the construction period of 30 d is not suitable for these pipes and has to be increased to a longer period or the pipes have to have higher compressive and tensile strengths(i.e.compressive and tensile strengths higher than 60 MPa and 4.8 MPa,respectively).In general,the predicted compressive stress at the crown is well below the compressive strength of the concrete and its value ranges between 4.4 MPa and 10.3 MPa.

6.3.influence of pipe depth

The influence of the pipe depth on the developed tensile stresses at the springline and the developed compressive stresses at the crown of the pipe is also illustrated in Figs.12 and 13.In general,increasing the pipe depth causes significant decrease in the developed tensile stress at the springline of the pipe and causes an increase in the compressive stress developed at the crown for both Milton and Niagara QSs using bentonite or polymer solution.The decrease in the tensile stress at the springline with increasing diameter is more evident for pipes of small diameter constructed in Niagara QS and it becomes smaller for large pipe diameters.In Milton QS,the decrease in the tensile stress developed at the springline of the pipe is almost constant for all of pipe diameters.The decrease in the tensile stress at the pipe spring line may be attributed to the confinement of the constructed pipe,where deeper pipes sustain more confinement in all directions compared to shallow pipes.Deep pipes overlain by thicker shale produce more swelling pressure on the pipe in the vertical direction compared to shallow pipes.This may cause more uniform stress distribution around the pipe in all directions which may result in less tensile stress at the springline.The stress state at the springline changes from tension at shallow depth to compression at high depths in both Milton and Niagara QSs.The developed compressive stress at the crown of the pipe increases with increasing pipe depth for both Milton and Niagara QSs using both LFs.This increase can also be attributed to the confinement effect of deep pipes compared to shallow pipes of the same diameter.

6.4.influence of construction period

Fig. 14.Variations of tangential tensile stress developed at the springline of 1.8 m diameter concrete pipe constructed at 1.7 m depth below the top of the Queenston shale layer with different construction periods.

The variation in the tangential tensile stress developed at the springline of 1.8 m diameter concrete pipe constructed at 1.7 m depth below the top of the QS layer with different construction periods is illustrated in Fig. 14.Six construction periods were selected:t=15 d,30 d,60 d,100 d,150 d and 200 d.It can be noted from Fig. 14 that increasing the construction period significantly decreases the developed tangential tensile stress at the springline of the pipe.This impact is more distinct when polymer solution is used in micro-tunneling application compared to bentonite solution in Milton and Niagara QSs.The impact of increasing the construction period when bentonite solution is used seems to be limited to duration of100 d for Milton QS and to 60 d for Niagara QS;while this impact continues for longer construction periods for both shales when polymer solution is used.Longer construction period produces smaller tangential tensile stresses at the springline of the pipe for both shales.Increasing the construction period depends on the site-specific condition and the construction schedule.Accordingly,the right construction period can be specif i ed based on the shale type,pipe depth and pipe diameter.

Fig. 15.Bending moment at springline and crown of a concrete pipe after 30 years of construction in Milton Queenston shale with in situ stress ratio of 5:(a,c)using polymer solution;and(b,d)using bentonite solution.

6.5.Variation of bending moment at springline and crown

Figs.15 and 16 show the variations of the bending moment at the springline and the crown of pipes with different diameters constructed at different depths below the top surface of Milton and Niagara QS layers,respectively,using polymer and bentonite solutions,forKo=5 andt=30 d.Figs.15 and 16 show similar trends for TDD behavior in both shales.The positive bending moment developed at the springline and the negative bending moment developed at the crown of the pipe increase as the pipe diameter increases.The bentonite solution causes similar effects on the developed bending moments at the springline and the crown of the pipe constructed in Milton and Niagara QSs.For the same pipe diameter,increasing pipe depth reduces the positive bending moment at the springline and the negative bending moment at the crown.However,the decrease in bending moments is negligible for small diameter pipe and is significant for large diameter pipes,especially for Niagara QS.

6.6.Pipe inward convergence at the springline

Fig. 17 shows the inward convergence at the springline of a concrete pipe of 1.8 m diameter constructed at 1.7 m below the top surface of Milton and Niagara QS layers,respectively.The stress ratioKois 5 and the construction period(t)is 30 d.Fig. 17 shows that the majority of the inward convergence of the pipe wall at its springline occurs within the first 1000 d of the structure life.During this period,the inward convergence is remarkably built in both shales when polymer and bentionite solutions are used.In both Milton and Niagara QSs,using polymer solution as LF during the construction period results in slightly smaller convergence at the springline compared to bentonite solution.This difference between both fluids is more obvious in Niagara QS.The vertical dashed lines represent the time after construction where most of the inward convergence at the springline of the pipe has occurred.For pipes constructed in Milton QS,more than 98%of the inward convergence occurs during the first 500 d of the structure life.This period is slightly higher for pipes constructed in Niagara QS due to its higher swelling potentials.

Fig. 16.Bending moment at springline and crown of a concrete pipe after 30 years of construction in Niagara Queenston shale with in situ stress ratio of 5:(a,c)using polymer solution;and(b,d)using bentonite solution.

Fig. 17.Predicted inward convergence of 1.8 m diameter concrete pipe at the springline.

7.Summary and conclusions

The rock swelling model incorporated in PLAXIS 2D environment was utilized to perform FE analyses for simulating TDD effects on concrete pipes or tunnels constructed in the QS of southern Ontario using micro-tunneling technique.The rock swelling model was first verified against the TDD tests performed to measure and quantify the swelling parameters of shales in southern Ontario.Further validation of the model was carried out through modeling some actual projects,e.g.the Niagara wheel pit excavation and the Scotia Plaza excavation,which were constructed in similar swelling rocks.The model was also validated against Lo and Hefny(1996)closed-form solutions of circular tunnel constructed in swelling rock.The validation process showed the capability of the model to simulate the TDD behavior of shales in southern Ontario and to calculate the induced stresses in the constructed structure.

The verified model was then used to conduct a parametric study of time-dependent behaviors of pipes constructed using microtunneling technique in QS considering(i)the in situ stress ratio(Ko),(ii)the pipe diameter,(iii)the pipe depth,and(iv)the construction period.Based on the results of these analyses,the following conclusions can be drawn:

(1)The induced tensile stresses at the springline can be decreased when the waiting period before adding the permanent grout is reasonably extended.

(2)With low in situ stress ratio(Ko),the stresses at the springline are expected to be tensile and compressive at the crown and invert,respectively.With increasing stress ratio(Ko),the stress state at the springline gradually changes from tension to compression.

(3)For the concrete strength considered in this study,pipes of diameter greater than 1.5 m constructed at shallow depths in the Niagara QS may experience tensile cracks at their extreme fibers at the springline.

A general conclusion can be drawn from this study that the micro-tunneling as a construction technique can be considered workable and feasible to install pipelines and tunnels in the QS of southern Ontario.However,the acceptable performance of the constructed pipes and tunnels is mainly governed by the use of suitable concrete strength,appropriate LF,adopting reasonable waiting period and the location of the pipe or tunnel with respect to the existing structures.

Conflicts of interest

The authors wish to confirm that there are no known Conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Acknowledgment

This paper forms the last part of a comprehensive research performed in the Geotechnical Research Center at the University of Western Ontario to investigate the workability and feasibility of micro-tunneling technique in the QS of southern Ontario.The authors would like to acknowledge Ward and Burke Microtunneling Ltd.for its excellent technical and financial support.

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