An Zhang ,Mingjing Jiang ,Wenhao Du

a School of Civil Engineering,Suzhou University of Science and Technology,Suzhou,Jiangsu,215009,China

b Shandong Provincial Key Laboratory of Marine Environment and Geological Engineering,Ocean University of China,Qingdao,266000,China

c State Key Laboratory for Disaster Reduction in Civil Engineering,Tongji University Shanghai,200092,China

d Department of Geotechnical Engineering,College of Civil Engineering,Tongji University,Shanghai,200092,China

e Department of Civil Engineering,Tianjin University,Tianjin,300072,China

f Tongji Architectural Design(Group)Co.,Ltd.,No.1230 Siping Road,Shanghai,200092,China

ABSTRACT In this study,the Discrete Element Method(DEM)was employed to investigate numerically the effects of hydrate cementation and intermediate principal stress on the stress-dilatancy relation of graincementing type methane hydrate-bearing sediment(MHBS)by conducting a series of conventional and true triaxial tests.A novel 3D thermo-hydro-mechanical-chemical(THMC)contact model for MHBS was employed.The numerical results show that with increasing hydrate saturation and back pressure,or decreasing confining pressure,temperature and salinity,the stress-dilation relation of grain-cementing type MHBS evolves from dilation-dominant to bond-dominant.For the clean sand samples,the relationship between the normalized stress ratio η/Mcr and the dilatancy rate d is close under different intermediate principal stress coefficients.However,for the MHBS samples,this relationship is still affected by the intermediate principal stress coefficient b,due to the effect of hydrate cementation.

Keywords:Stress-dilatancy Cementation Methane hydrate bearing sediment Discrete element method Intermediate principal stress

1.Introduction

The stress-dilatancy relation is the key issue in soil mechanics and plays a significant role in the construction of constitutive models.Regarding the methane hydrate-bearing sediments(MHBS),their dilatancy behavior is more complicated because of their complex hydrate occurrence[1]and the external temperature-pressure-chemical environment.

In order to obtain the mechanical properties of MHBS,a series of laboratory experiments have been performed including conventional triaxial compression[1-10],plane strain[11]and direct shear tests[12].However,the research on the effects of methane hydrate(MH)on the stress-dilatancy relation is still limited.Moreover,there is still a controversy regarding microscopic forms of hydrate cementation in MHBS and its effect on the stressdilatany relation.Some researchers[2,7,13]claimed that the hydrate could be formed between particles and firmly bond them together(Fig.1).However,other researchers[14-16]found that there is a layer of water film between particles and hydrates observed with synchrotron X-ray computed tomographic microscopy,which prevents the hydrates from directly contacting the particle surfaces.Pinkert[17]found that the stress-dilatancy relation of water-saturated MHBS is less affected by the hydrate saturation,while the gas-saturated MHBS results,in contrast,showed higher stress-dilatancy relations for higher hydrate saturation.Pinkert[17]speculated that macro cohesiveness shown in the stress-dilatancy relation of gas-saturated MHBS may not arise from cementation interactions between soil particles and hydrates but come from suction responses in water films or a continuous hydrate bridge throughout the pores or from both.

Furthermore,the true triaxial shear test has not been conducted on MHBS yet due to the high requirements of the equipment(low temperature and high pressure),but the effect of the intermediate principal stress on the stress-dilatancy relation of MHBS is quite important.Fortunately,such laboratory experiments can be numerically complemented by the Discrete Element Method(DEM)[18],by which complex loading paths can be realized.In comparison to continuous medium methods,DEM is capable of simulating the actual macroscopic behavior of soils by using only a few model parameters based on the microscopic contact relationship of the soil,and each model parameter can have a physical meaning in terms of its physical characteristics.DEM is also capable of obtaining detailed micro-information at the grain scale level,and has been widely used in the recent research of MHBS[19-27].

The main purpose of this study is to investigate the effect of hydrate cementation and the intermediate principal stress on the stress-dilatancy relation of grain-cementing type MHBS.A series of conventional and true triaxial shearing tests were simulated using Discrete Element Method(DEM)with a novel thermo-hydromechanical-chemical(THMC)contact model for MHBS proposed by Jiang et al.[28].The results of this study may be useful in identifying the microstructure of MHBS,comparing laboratory tests,and developing constitutive models for MHBS.

2.THMC contact model for MHBS

Details of the contact law and validation of the THMC contact model have been introduced in Jiang et al.[28].Only a brief introduction to the model is repeated here.

The 3D contact model incorporating rolling and twisting resistances[29]developed from Jiang's 2D complete contact model incorporating rolling resistance[30]was employed in the uncemented contact part of THMC contact model.It assumes that two spheres microscopically interact at a contact over circular flat area with a radius of,wherer=2R1R2／(R1+R2)is the common radius of two spheres in contact.The angularity effect is reflected by the shape parameter β since higher angularity leads to larger contact area and greater rolling/twisting resistances.Previous studies have proven that the major behavior of granular soils can be successfully captured by this model[29,31-35].In this study,the unbonded contact model parametersEp= 0.7 GPa,ξ = 5,μ = 0.5,β = 0.25,and ζc= 4 are used insubsequent simulations.Here,Eprepresents the modulus of the particle material;ξ is the ratio of the normal to the tangential contact stiffness;μ is the inter-particle friction coefficient;ζcis the local crushing parameter describing the local asperity crushing effects.

The model incorporates complete bonding interactions of two particles in the normal,tangential,rolling and torsional directions.A 3D bond failure criterion obtained from 3D experiments[36]is incorporated in this model,which can describe the combining effect of normal force Fn,shear force Fs,rolling moment Mr,and torque Mton bond failure,in association with bond geometry.The bond failure criterion can be practically described as follows:

whereRs,b,Rr,bandRt,brepresent shear,bending and torsional resistances of bonded material,respectively.

Among the most important features of this model is the introduction of a conditional parameterL(T,P,w)that can describe comprehensively the effects of temperatureT,pressurePand salinitywon the mechanical properties of hydrate cementation.The conditional parameterLis defined as the minimum distance between a test condition point and the phase equilibrium line of MH(see Fig.2),and can be calculated as follows:

whereT0= 273K,P0= 2.23 MPa andw0= 3wt%.Eq.(2)indicates that the value ofLincreases with the increase of water pressure,but with the decrease of temperature and salinity.

The compressive strength σc,tensile strength σtand elastic modulusEmhof MH can be expressed as follows:

where σ1,maxis the peak major principal stress when MH is exposed to a confining pressure ofP.αs= 30 is a size-effect correction coefficient,which is calibrated by DEM simulations and Weibull theory in order to take into account size effects on bond breakage[28].In Eq.(4),the tensile failure is assumed to occur when the pore pressurePacts as the major principal stress and the minor principal stress reaches its minimum σ3,max.

3.Sample preparation and the simulation procedures

In this study,the conventional and constant-ptrue drained triaxial tests were simulated using the commercial code PFC3D[37]with the THMC contact model.Fig.3 illustrates the particle size distribution used for the DEM samples.Fig.4 illustrates the configurations of the DEM samples.Prismatic samples are used in conventional tests,while cubic samples are employed in true triaxial tests.Each sample has about 40000 particles.To prepare the MHBS samples,the host sand samples with the initial void ratio of 1.1 were first generated.Then,the samples are isotropically compressed with a confining pressure of 200 kPa until it arrives at the equilibrium state,after which MH was“synthesized”by applying the THMC contact mode.Table 1 summarizes the test program in this study.The true triaxial shear tests were conducted on the DEM samples by keeping the mean stress(500 kPa)constant.The value of intermediate principal stress ratio,specified byb=(σ2-σ3)／(σ1-σ3),is set to 0.0,0.2,0.4,0.6 0.8 and 1.0,respectively.For all cases,MH saturationSMHis defined as follows:

Table 1 Summary of the test program.

whereVVis the volume of void.VMHis the volume of MH,which is controlled by particle size distribution,fabric at the moment of bonding,and the value of the hydrate radius multiplier λ.For a given collection of particles in a DEM specimen,VMHis uniquely linked toSMHthrough λ.In this study,the numerical MHBS samples withSMH= 20%,30% and 40% were generated,and the corresponding hydrate radius multipliers λ are 0.774,0.846 and 0.9,respectively.

4.Results and discussion

4.1.Conventional triaxial test

Fig.5 presents the evolutions of stress ratio,dilation rate,bond breakage ratio and rate of bond breakage ratio with axial strain(SMH= 30%,L= 0.0249)under different confining pressures in conventional drained triaxial tests.The dilatancy rate is defined asd= -dεv／dεq,wheredεvanddεqare the increments of the total volumetric and deviatoric strains,respectively.The positive value of dilation rate denotes compression,while the negative value denotes dilation.The bond breakage ratio is defined as the ratio of broken bond number over the initial bond number.The rate of bond breakage ratio is defined as the tangent slope of the‘‘bond breakage ratio-axial strain”curve.Points A,B and C in Fig.5 locate the initial bond breakage,the peak strength and the maximum dilatancy rate,respectively.As shown in Fig.5,the rate of bond breakage ratio increases first and then decreases with the axial strain.Under the confining pressure of 1 MPa,the peak stress ratio and the maximum rate of bond breakage ratio occur almost simultaneously and prior to the maximum dilatancy rate.This indicates that the peak strength of MHBS is mainly controlled by the cemented contact.Under the confining pressures of 3 MPa and 5 MPa,the maximum rate of bond breakage ratio occurs prior to the peak stress ratio.Before reaching peak strength,a large number of bonds have been broken,indicating that the peak strength is mainly controlled by the sliding friction and interlocking effect between soil particles.Therefore,grain-cementing MHBS undergo a change in mechanical behavior from being bond-dominant,where the bonds between parts play a dominant role,to being dilationdominant,where the sliding friction and interlocking effect play a dominant role.

Figs.6 and 7 show the effects of MH saturation and conditional parameterLon the stress-dilatancy relations of grain-cementing type MHBS under different confining pressures in conventional drained triaxial tests,respectively.It shows that peak strength and maximum dilatancy rate increase with hydrate saturation and conditional parameterL.This is because the number of bonded contacts increases with the hydrate saturation.In addition,the bond strength increases with hydrate saturation and condition parameterL.As confining pressure increases,the peak strength coincides with maximum dilation rate.The stress-dilatancy relation is influenced by both the initial void ratio and the degree of hydrate cementation.In general,the greater the void ratio or the lower the bonding strength,the greater the shear shrinkage or the smaller the shear dilatancy of the samples.As a result of hydrate cementation,the MHBS has a greater initial void ratio after isotropic compression compared to the clean sand.The larger contraction of MHBS compared to clean sand is mainly due to its large void ratio.As the hydrate saturation or the condition parameterLincreases,the bond strength increases,which results in a smaller shear contraction or greater shear dilatancy for the MHBS sample.In this case,the MHBS sample withL= 0.0099 exhibits a greater shear contraction than the clean sand sample or other MHBS samples with largerLvalues,as shown in Fig.7(b).

Fig.8 presents the stress-dilatancy relations of artificial MHBS under the air saturation condition(methane provides the back pressure)in the conventional triaxial tests conducted by Hyodro et al.[8].Comparing Figs.6 and 7 to Fig.8,it is found that the stressdilatancy relation derived from our DEM simulations is generally consistent with the relation obtained from laboratory tests conducted on gas-saturated MHBS.However,Pinkert[17]found that the stress-dilatancy relation of artificial MHBS under water saturation condition(where the free gas in the sample was replaced with water prior to shearing)is less affected by the MH saturation(as shown in Fig.9)and is governed mainly by the relative slip and rotation of sand particles and hydrates.Fig.10 presents the effect of cement content on the stress-dilatancy relation of cemented sand under a high confining pressure of 4 MPa[38].The stress-dilatancy curve of the cement-coated sands appeared to shift upward as cement content increased,consistent with the DEM simulation presented here.

According to Ref.[17],the reasons for the different stressdilatancy relations of MHBS under gas saturation and water saturation testing conditions may primarily relate to the following two factors:(1)a thin water film may exist between the soil particles and the hydrate in the pores,and the hydrate and the soil particles are not in direct contact.Under the condition of water saturation,the water film is connected to the water within the pores,while under the condition of gas-water saturation,tension forces can be generated between the particles and the hydrate due to the effect of the water film.Similar to the unsaturated soils,such tension forces may affect the overall mechanical stress-dilatancy relation of gassaturated MHBS.(2)Under a gas saturation condition,hydrates may not directly bond with soil particles,but if the hydrate saturation is high in porosity,a continuous hydrate bridge can be formed throughout the pore space.However,methane hydrate will dissolve and destroy the hydrate cement structure in the water-saturated MHBS during the process of replacing methane gas in the pore space with water before shearing.

Yoneda et al.(2015)conducted conventional drained triaxial tests under the water saturation condition on the natural MHBS sample(Msh= 63%)obtained from the Nankai Trough in the southeast Sea of Japan.The experimental tests showed that the MHBS sample presented a brittle failure mode,accompanied by softening and shear bands.Dilatancy occurred after the sample strength reached its peak value,reflecting the influence of hydrate cementation.Although there is still controversy about whether the hydrate and particles are directly in contact due to the existence of water film,it can be concluded from the results of triaxial tests on the natural samples and laboratory synthetic samples of MHBS that the hydrate cementation effect does exist.Further research is needed on the presence of water film between particles and hydrates and its effect on the mechanical properties of MHBS.

4.2.True triaxial test

Fig.11 shows the stress-dilatancy relations of the clean sand DEM samples under true triaxial stress conditions.As shown in Fig.11(a),the relationship between the stress ratio η and dilatancy ratedis nonlinear.The dilatancy rate of the DEM sample reaches its maximum when the DEM sample reaches its peak strength,and then gradually decreases with the decline of the stress ratio and eventually fluctuates near the critical state.The relationship between stress ratio η and dilatancy ratedis clearly affected by the intermediate principal stress coefficientb,and the peak stress ratio and dilatancy stress ratio(the stress ratio atd=0)decrease with an increase inb.Note that the critical stress ratio of the sample varies with intermediate principal stress coefficients,and the higher the value ofb,the lower the critical stress ratio ofMcr.Fig.11(b)further shows the relationships between the normalized stress ratio η/Mcrand dilatancy ratedof the clean sand DEM sample.It can be seen that the relationship between the normalized stress ratio η/Mcrand the dilatancy ratedis close under different intermediate principal stress coefficients,and the normalized stress ratio η/Mcrat the shear dilatancy point is approximately the same.

A MHBS sample withSMH= 30% andL= 0.0249 is used to investigate the effect of intermediate principal stress coefficientbon the stress-dilatancy relation of MHBS numerically.Fig.12 presents the stress-dilatancy relations of MHBS samples in drained true triaxial tests.When the stress ratio is relatively small(i.e.,η ＜0.8),the MHBS samples mainly exhibit shear contraction characteristics and the relationship between stress ratio η and dilatancydis less affected by the intermediate principal stress coefficient,indicating that the hydrate cementation plays a dominant role in stress-dilatancy relation of MHBS samples.When the stress ratio η ＞0.8,the relationship between stress ratio η and dilatancydis different for different intermediate principal stress coefficients.This means that at high stress ratios,the effects of hydrate cementation decrease,due to bonds breaking.As with the clean sand samples,peak stress and critical stress ratioMcrof MHBS samples decrease as the intermediate principal stress coefficient increases.Fig.12(b)further shows the relationship between the normalized stress ratio η/Mcrand dilatancy ratedof MHBS samples.As shown in Fig.12(b),the relationship between η/Mcranddis still significantly affected by the intermediate principal stress coefficientb,which is different from that observed in the simulation of clean sand.Note that in the conventional triaxial tests,the MHBS samples show a larger contraction at the initial shearing stage compared to the true triaxial tests.It is due to the fact that the mean stress increases in conventional triaxial tests,while it remains constant in true triaxial tests.As a result,the volumetric contraction is larger due to increased mean stress in conventional triaxial tests.

5.Conclusions

In this study,a series of conventional and true triaxial tests were simulated using DEM to investigate the effects of hydrate cementation and intermediate principal stress coefficient on the stress-dilatancy relation of grain-cementing type MHBS.The following conclusions are reached.

(1)Due to the presence of the hydrate cementation,the peak stress can occur prior to the maximum dilatancy rate.As the confining pressure increases,the mechanical behavior of grain-cementing MHBS changes from bond-dominant to dilation-dominant.The peak strength and maximum dilatancy rate increase as MH saturation and condition parameterLincrease.The stress-dilatancy relation obtained by DEM simulation in this paper generally agrees with results of the laboratory tests conducted on artificial MHBS under the air saturation and natural MHBS under the water saturation condition,which differ from those conducted on artificial MHBS under the water saturation condition.

(2)In the DEM simulation of true drained triaxial tests,the peak and critical stress ratios of clean sand and MHBS samples decrease as the intermediate principal stress coefficientbincreases.The relationship between the stress ratio η and the dilatancy ratedof clean sand and MHBS samples is clearly affected by the medium principal stress coefficientb.For the clean sand samples,the relationship between the normalized stress ratio η/Mcrand the dilatancy ratedis close under different intermediate principal stress coefficients,but for the MHBS samples,this relationship is still affected by the intermediate principal stress coefficientb,due to the presence of the hydrate cementation.

Declaration of competing interest

We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

Acknowledgements

This research was financially supported by the National Natural Science Foundation of China(Grant No.51639008 and No.51890911),and State Key Lab.of Disaster Reduction in Civil Engineering(Grant No.SLDRCE14-A-04),which is greatly appreciated.