Laboratory of Mechanics of Lille,University of Lille,Villeneuve d'Ascq 59655,France

Experimental study and modeling of hydromechanical behavior of concrete fracture

He Yang,Shou-yi Xie,Jean Secq,Jian-fu Shao*

Laboratory of Mechanics of Lille,University of Lille,Villeneuve d'Ascq 59655,France

Abstract

In this study,the hydromechanicalbehaviorof a concrete fracture under coupled compressive and shear stresseswas investigated.A special experimentaldevicewasdesigned to createa planar fracture in a cylindricalsampleand to carry outdifferentkindsof hydromechanical testson the fracture.Four series of laboratory testswere performed on an ordinary concrete sample.Hydrostatic compression testswere fi rst conducted to characterize the normal compressibility of the fracture.In the second series,directshear testswere conducted on the fracture under different normal stresses.Themaximal shear stress of the fracturewas determ ined as a function of the normal stress.In the third series,fluid flow tests were carried out in view of characterizing the overallhydraulic conductivity of the fracture as a function of its opening and closure.Shear tests with a constant fluid pressure were finally performed to investigate the influence of fluid pressure on the deformation behavior of concrete fractures.Based on the experimental investigation,an elastoplastic model is proposed.Thismodel takes into account the nonlinear elastic behavior of a fracture under normal compression and the plastic deformation and failure due to shear stress.Themodelwas coupled w ith the classical Darcy's law to describe the fluid flow along the fracture by considering the variation of permeability w ith fracture aperture.Numerical results agreew ith experimental data from various laboratory tests.

?2017 Hohai University.Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license(http:// creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords:Concrete fracture；Direct shear；Hydromechanical coupling；Hydraulic conductivity；Elastoplasticmodel

1.Introduction

Concrete is w idely used in different engineering fields, including hydraulic constructions.In many cases,concrete materials are subjected to mechanical and hydraulic solicitations aswellas chemical degradation.Among variousaspects related to the performance of concrete structures,the damage induced by m icrocracks is an essential process of inelastic deformation and failure of concretematerials.In many cases,the failure of concrete structures is induced by the coalescence of microcracks,leading to localized macroscopic fractures. The description of the transition from diffused damage to localized fracture remains so far themost important challenge of durability analysisof concrete structures.A greatnumberof experimental studies have been performed onmechanicaland transport properties of various types of concrete in relation to damage evolution.It has been found thatmicrocrack-induced damage affects not only mechanical behaviors butalso transport and diffusion properties.Themain consequences include deterioration of elastic properties,induced anisotropy,unilateral effects,friction-damage coupling,irreversible deformation and hysteretic response,and significantmodification of permeability and thermal conductivity(Kermani,1991；Sugiyama et al.,1996；Wang et al.,1997；Abbas et al., 1999；Baroghel-Bouny et al.,1999；Picandet et al.,2001；Choinska et al.,2007；Hoseini et al.,2009；Yurtdas et al.,2011).These studies have clearly shown an inherent relationship between concrete permeability evolution and m icrocrack grow th.However,mostexistingworksare so far lim ited to concrete materials w ith diffused m icrocracks or w ithout fully connected localized cracks.There have been very few studies devoted to hydromechanical behaviors of individual concrete fractures,in particular when such fractures are subjected to both normaland shearstresses.Experimentaldataare obviously necessary formodeling progressive propagation of localized fractures,the key phenomenon of the failure process in concrete structures(Bourgeois et al.,2002；Chen et al., 2011).For this purpose,we intended to obtain more experimental data in this study.Original and quite comprehensive laboratory investigations are proposed in order to characterize bothmechanicaland hydraulic propertiesof concrete fractures in different loading conditions,and a fresh fracture in CEM I concrete was investigated.A specific homemade device was designed,allow ing the creation of a planar fracture in a standard cylindrical sample(Shao,2016).With this device, different kinds of laboratory tests were performed on the fracture:the hydrostatic compression test,direct shear test, and water flow test.The obtained results allow the characterization of elastic deformation,plastic deformation,failure properties,and permeability variation of the fracture.

Based on the experimental study,an elastoplastic model was formulated to describe the mechanical behavior of the concrete fracture.The formulation of the model is based on previousworks performed on rock joints.For instance,Bandis (1980),Gentier(1986),Plesha(1987),Bart(2000),Barton et al.(1985),Yeo et al.(1998),and M isra(2002)proposed various constitutive models for rock joints under normal compressive and shear stresses.On the other hand,a number of studies have been performed to investigate hydraulic properties through a single fracture or joint(Tsang and Witherspoon,1981,1983；Moreno et al.,1990；Olsson and Brown,1993).The present model describes the nonlinear elastic behavior of the fracture under normal compressive stress and the plastic deformation due to shear stress.The effect of fluid pressure on the fracture deformation was also investigated.Finally,using the classic Darcy's law and the massbalance equation,water flow along the concrete fracture was studied.Comparisons of numerical and experimental results are presented and discussed.

2.M aterial and experim ental procedure

Thestudiedmaterialwasaso-called CEM Iconcrete,which was chosen by ANDRA(the French National Agency for Radioactive Waste Management)for potential use in underground structures fornuclearwaste disposal(Camps,2008). This concrete is composed of cement(CPA,CEM I 52.5, PM,ES,and CP2),sand(washed limestone w ith a size of 0-4 mm),fine gravel(washed limestone w ith a size of 5-12.5mm),adjuvant,andwater.Thewater-cement ratiowas 0.43,and the gravel-sand ratio was 1.1.The concrete was molded inside a coffering beam w ith a length of 56 cm and a section of 14 cm×14 cm.After sixmonths ofmaturation in limewater ata temperature of 20°C,cylindrical sampleswere cored inwater,and cutup and rectified to reach a desired size.

All laboratory tests were conducted using a self-designed autonomous triaxial system(Liu et al.,2015,2016).This testing system was composed of a conventional triaxial cell, used to apply a confining stress(Pc)by injection ofwater into the cell chamber,an upper pressure chamber to generate an axial stress through the piston,and a circuit for an interstitial fluid flow.This testing system was also equipped w ith a compensation chamber inside the upper part of the cell.The liquid used to generate the confining stress was connected to this chamber.Therefore,the application of confining stress would not generate any force on the piston of the cell.The axial stress applied on the piston by the upper pressure chamber would generate a pure deviatoric stress.The axial stressand confining stresswere both applied andmonitored by an independent pressure generator.The axial strain was measured by two LVDT sensors,while the radial strain was measured by a self-designed circum ferential strain ring.All monitoring datawere collected in an acquisition computer.

In order to create a planar fracture in a cylindrical sample and subsequently to carry out a direct shear test on the crated fracture,a new experimental device was designed.The principle of the device is illustrated in Fig.1.This device is composed of two cylindrical shearing discs w ith the same diameter as the sample.Each disc consists of two sem i-discs made of different materials(A and B)w ith different stiffness.The two sem i-discs are placed in opposite positions at the top and bottom surfaces of the sample.Due to a large stiffness difference between the two semi-discs,an axial displacementgenerated by the cellpiston createsa shearstress along the diameter plane of the sample,leading to a planar fracture,and then the shear stress(τ)is applied along the fracture.Note thata rotation force can begenerated before the creation of a fracture due to the difference of forcesapplied to two semi-discs.After a series of preliminary tests,it was found that the effect of such a rotation forcewas very small, and no sample rotation was observed.

Fig.1.Schematization of experimental device for creation of fractures and direct shear tests.

A narrow cylindrical samplewas chosen.The diameter and height of the sample were 37 and 45 mm,respectively.The samplewas placed between two composite discs and inside a plastic jacket.The creation of a fracture isalways done under a confining stress,for instance 5 MPa,in order to prevent apossible rotation of the sample.The axial stress was applied undera displacementcontrolled condition until the creation of a fracture along the diameter plane of the sample.Furthermore,due to the fact that the stiffnessof the softsem i-discwas much smaller than thatof the hard one,the force applied to the half cylindrical sample under the soft sem i-disc was very small.Therefore,the applied shear stress along the diameter plane of the sample was quasi-equal to the deviatoric stress applied to the concrete sample.The axial and radial strains were measured as functions of the generated shear stress during the creation of a fracture.An example of stress-strain curves during the creation of a fracture is presented in Fig.2, whereε1andε3are the axial and radial strains,respectively. The peak shear stresses correspond to the point of fracture creation in the sample.A fter the creation of a fracture using this device,four different categories of tests were performed according to the follow ing procedures.

2.1.Hydrostatic compression test

A fter the creation of a fracture,two cylindrical shearing discs were replaced by two draining discs.The fractured sample was subjected to a hydrostatic stress up to a desired value by increasing the confining stress.In this way,the created fracturewas subjected to a uniform normal stress.The main objective of this testwas to establish a relationship between the normal deformation of the fracture(opening or closure)and the applied normal stress.For this purpose,a comparative hydrostatic compression testwas fi rst performed on a sound sample(w ithout fracture)under the same loading condition.By comparing the radial strains between the sound and fractured samples,it is possible to determ ine the opening or closure of the fracture as a function of the normal stress.

2.2.Direct shear test under confining stress

A confining stress up to a desired value,for instance 5,10, and 15 MPa,was fi rst loaded to the sample.The shear stress was then generated by increasing axial displacementw ith the help of the specific shearing device described above.The objective of this testwas to characterize the shear strain and normal strain of the fracture versus shear stress for different values of confining stress.

Fig.2.Radial and axial strains versus shear stress during fracture creation under confining stress of 5MPa.

2.3.Hydraulic flow test

The purpose of this testwas to determ ine the relationship between thewater flow rate and injection pressure through the fracture under different confining stresses.Under a constant confining stress,waterwas injected into the fracture from the bottom surfaceof the sampleata flow rateof0.1-0.8 cm3/m in, while a constantoutletpressure of 0.5MPawaskeptat the top surface of the sample.The injection pressure corresponding to each flow rate aswellas the normal strain of the fracturewas measured during injection.

2.4.Shear testwith constant fluid pressure

This testwas complementary to the direct shear test,and the objective of this testwas to identify the influence of the fluid pressure inside the fracture on itsmechanicalbehavior.In this test,a uniform fluid pressure was prescribed inside the fracture,for instance 5MPa,by injection ofwater through the injection hole of the cell,while the outlet hole was closed. Two confining stresses,10 and 15 MPa,were chosen so that the effective confining stresseswere equal to those used in the directshear test.The comparison between the results obtained from the two series of tests allows for the study of the fluid pressure effect.

3.Experimental results

In this section,the main results obtained from different series of testsare presented and discussed.These resultswere used for the formulation and verification of the constitutive model presented in the next section.

3.1.Hydrostatic compression test

In Fig.3,the typical radial strain versus confining stress curves are presented for both sound and fractured samples. One can see that the strain curve is nearly linear for the sound sample,while thatof the fractured sample is clearly nonlinear and concave during the loading phase.During the unloading phase,there is a significant irreversible strain in the fractured sample while the radial strain of the sound sample is quasi reversible.Obviously,the nonlinearity and irreversibility of radial strain in the fractured sample are related to inelastic evolution of the fracture surface under confining stress.The confining stress here is equal to the normal stress applied to the fracture surface.By calculating the difference in radial strain between the sound and fractured samples,we could obtain the normal strain or normal displacement of fracture, which w illbe shown later in Section 4.1.Itwas found that the fracture closure occursessentially under the confining stressof 0-5 MPa.A fter this stage,the slope of the radial strainconfining stress curve of the fractured sample is quasiidentical to that of the sound sample.Therefore,the closurepressure of the concrete fracture could be identified asaround 5MPa in the present case.

Fig.3.Typical radial strain versus confining stress curves for sound and fractured samples.

3.2.Direct shear test

Three shear testswere performed under confining stresses of 5,10,and 15 MPa.Experimental data are presented in Fig.4,together w ith the numerical results described later in Section 4.4,whereuis the shear displacement,andvis the normaldisplacement.Asobserved in rock joints(Bandisetal., 1983；Barton and Choubey,1977；Bart et al.,2004),the peak shear stress significantly increases w ith the confining stress, while the shearmodulus remains nearly constant.For all the confining stresses considered,a softening regimewasobtained after the peak stress.This kind of behavior isgenerally related to the progressive destruction of asperities on the fracture surface,leading to a significant decrease of the friction angle. For a relatively high confining stress,one can obtain a relatively smooth softening regime.However,itbecomes sharper under a lower confining stress.On the other hand,the radial strain measured on the fractured sample corresponds to the fracture normal deformation(opening or closure)induced by shear stress(Fig.4(a)).Obviously this strain playsan essential role in theevolution ofmechanicaland hydraulic propertiesof the fracture.One observes that the normal displacement remains relatively smallwhen the shear stress is low relative to the peak value.In the case under a high confining stress,a compressive normal displacement(denoted as a positive value)can even be obtained due to the progressivematching of fracture asperities.When the shear stress approaches its peak value,the normal displacementquickly increases,indicating a significant opening of the fracture(denoted as a negative value).The latter is physically dependent on the roughness of fracture.

Fig.5 shows some concrete fractures after direct shear tests.There are some strongly sheared zonesw ith a significant destruction ofasperities.Such zonesaremore developed under a high confining stress due to a stronger coupling between the confining stress and shear stress.

3.3.Hydraulic flow tests

Fig.6 shows relationships between the injection pressure (pin)and water flow rate(Q)under three different confining stresses.The injection pressure increases quasi-linearly w ith the water flow rate for each confining stress.On the other hand,as expected,the water flow rate is lower when the confining stress is higher for the same value of injection pressure.This is due to the fact that the fracture opening issmaller for a higher confining stress.Using the classical Darcy's law,it is possible to estimate the overall transmissibility(called permeability hereafter for convenience)at each point of injection.As expected again,the overall permeability(K)of the fracture increases w ith the injection pressure,as shown in Fig.7.

Fig.4.Experimental data and numerical simulation of direct shear tests under different confining stresses.

Fig.5.Profi les of fracture surfacew ith asperity destruction zones.

Fig.6.Water flow rate versus injection pressure in fluid flow tests under different confining stresses.

With themeasured radial strain of a fracture during water injection,the fracture opening is evaluated.The variation of overall permeability is expressed as a function of fracture opening as shown in Fig.8.It is very interesting to observe that there is a nearly unique relationship between the permeability and fracture opening for all confining stresses.As a conclusion,the fracture overall permeability is essentially controlled by the fracture opening and closure.

3.4.Shear testwith constant fluid pressure

Fig.7.Relationships between overall permeability and injection pressure for different confining stresses.

Two shear tests were performed,in which the effective confining stress(the difference between the confining stress and fluid pressure,p)inside the fracture was equal to 5 and 10 MPa,respectively,which were the values of confining stress used in the direct shear test w ithout fluid pressure, described in Section 3.2.The objective of this test was to compare the results between the tests w ith and w ithout fluid pressure.Note that the effective confining stresses are the same for both series in the sense of Terzaghi's concept.The results foran effective confining stressof 5MPaare presented in Fig.9.One can see that the results from the two series of tests are very close in terms of the peak shear stressand radial strain.The shear strain before the peak stress is slightly smaller in the testw ithout fluid pressure.Sim ilar resultswere obtained for an effective confining stress of 10 MPa.These results seem to suggest that the effective confining stress defined w ith Terzaghi's concept can be used to study the fluid pressure effect in concrete fractures in this study.

4.Elastop lasticmodel

Based on experimental data presented above,an elastoplastic model is proposed for description of the mechanical behaviorof concrete fracturessubjected to normalstress,shear stress,and fluid pressure.For the sakeof clarity and in order to avoid any confusion,itisuseful to pointoutthatduring adirect shear test on a cylindrical sample as described above,the measured axialdisplacementof thesample isequal to theshear displacementof the fracture,while the normaldisplacementof the fracture(closureoropening)isobtained from themeasured radial displacement of the sample.At the same time,the confining stress applied to the sample is equal to the normal stress applied to the fracture,while the axial stress of the sample isequal to theshearstressof the fracture.Therefore,for the formulation of the elastoplasticmodel of the fracture presented below,we used normal stress,normal displacement, shear stress,and shear displacement for the fracture.

4.1.Fracture behavior under normal stress

Fig.8.Relationships between overall permeability and fracture opening for different confining stresses.

Fig.9.Comparison between shear testsw ith and w ithout fluid pressure under effective confining stress of 5MPa.

Fig.10.Numerical fi tting of nonlinear stress-strain curve in hydrostatic compression test.

As shown in Figs.3 and 10,under a confining stress,the fracture exhibits a nonlinear mechanical response,and an irreversible normal strain is observed after unloading.Therefore,the fracture exhibits an inelastic behavior under normal stress.However,putting the emphasis on the mechanical behavior of the fracture under coupled normal and shear stresses and for the sake of simplicity,a simple incremental elastic modelwas developed formodeling the fracture deformation under the normal stress by neglecting the irreversible strain during the unloading process.Furthermore,according to theeffectivenormalstressconceptmentioned above,when the fracture issaturated w ith a fluid,the variation of fluid pressure w ill also induce a normal deformation(opening or closure)in the fracture.Assum ing that the effectof fluid pressure on the fracture deformation can be described using the classical Terzaghi's effective stress concept,the follow ing incremental elasticmodel is given:

whereσnis the normal stress；is the normalized normal displacement,w ithwhereb0is the initial hydraulic aperture of the fracture,and a positive value ofvdenotes a closureof the fracture；and~Kn,asa function of~v,is the normal modulus of fracture(MPa),w ith=Knb0,whereKnis the normal stiffness(MPa/mm).Based on experimental data and according to Bandis et al.(1983),the following relation is given:

whereKniis the initial normal stiffness,andvmis the maximum closure of the fracture at its completemechanical closure state.

4.2.Elastoplastic behavior under shear test

Assum ing that dvis the incremental normal displacement and d u is the incremental shear displacement vector,each componentof displacement isdivided into an elastic part(with subscript e)and a plastic part(w ith subscript p):

The incremental stress-strain relations can bew ritten as

whereτis the shear stress vectoron the fracture,andKtis the shearmodulusof fracture(MPa/mm).The plastic strain should be determined by defining a plastic yield function and aplastic potential in case of a non-associated flow rule.According to experimental data,the plastic strain(irreversible opening or closure and sliding)is induced by the shear stress and influenced by the effective normal stress defined as=σn-p. The plastic strain rate is dependent not only on the friction angle but also on the inclination angle of asperities of the fracture surface.To this end,a simplified m icrostructural morphology of a rough fracture is considered and illustrated in Fig.11,whereθ0is the initial inclination angle of asperities.

During the shearing process,asperities are progressively destructed,leading to a reduction of the asperity inclination angleθ.When all asperities are fully damaged,θtends to be zero.On the other hand,the shear strength of the fracture is also affected by the surface roughness,represented by a friction angleφr.Furthermore,in some situations,fracture surfaces may be cemented,and this cementation provides the shear strength in unconfined conditions(=0),which is described by the cohesion parameterC0.Considering all these features and making the projection of the effective normal stress and shear stress along the inclined fracture surface,a generalized Mohr-Coulomb criterion is used as the plastic yield function of fracture,F,as follows:

Due to the presenceof such asperities,the tangentialplastic sliding can induce a normal plastic closure or opening.Note that the Mohr-Coulomb criterion adopted in this study is formally equivalent to the w idely used Patton criterion.In order to properly describe the normal plastic deformation induced by shear stress,a non-associated plastic flow rule is necessary.The plastic potential,G,is as follows:

The plastic deformation(dvpand d up)can be expressed by the follow ing plastic flow rule forF=0 and=0:

Fig.11.Illustration of fracture m icrostructure w ith initial asperity angle and damaged asperity angle.

whereλis the plasticmultiplier,which verifies the follow ing loading-unloading condition:

The plasticmultiplierλcan be determ ined from the plastic consistency condition as

The scalar valued functionHdenotes the plastic hardening modulus,which is expressed as a function of plastic strain energy:

wherewpis the plastic strain energy,andwp=∫τ$d up. During the plastic sliding process,asperities of fracture are progressively destroyed.In order to describe this kind of surface evolution,each asperity is idealized by a saw tooth form.The degradation process is represented by a decrease of its inclination angle w ith the plastic energy.According to Plesha(1987),the follow ing exponential form is adopted:

wherecis the parameter controlling the rate of asperity degradation.It is found that this parameter usually dependson the effective normal stress(Benjelloun,1991).Therefore,the following relation is adopted(Nguyen and Selvadurai,1998):

wherePatmis the atmospheric pressure；andaandbare two model's parameters,witha＞0 andb＞0.

4.3.Determination ofmodel's parameters

The proposed elastoplastic model for saturated concrete fracture contains three parameters for mechanical behavior under normal stress and six parameters for mechanical behavior under shear stress.The initial normal stiffnessKniand the maximal closurevmcan be identified from the stress-strain curve under a hydrostatic compression test,as shown in Fig.10.The initial hydraulic apertureb0can be estimated from thepermeabilitymeasured at the initialstateof fracture using the cubic law,presented in Section 4.4,together w ith the classicalDarcy's law.The shearmodulus is calculated byKt=τpeak/upeakw ithτpeakandupeakbeing the values of shear stress and shear displacement in the peak state.The friction angle of the fracture surfaceφr,the fracture cohesion coefficientC0,and the initial inclination angle of asperityθ0can be determ ined by draw ing the peak shear stress versus normalstress,asshown in Fig.12.The parametersaandbcan be defined by fi tting the shear stress-displacement curve at the post-peak regime.The typical values of parameters for the concrete fracture are listed in Table 1.

Fig.12.Peak shear stress versus normal stress in direct shear tests.

4.4.Verification ofmodel

The verification of the proposed model was carried out through the simulation of mechanical tests and fluid flow tests.Formechanical tests,for the sake of simplicity,itwas assumed that stresses and strains were uniform along the whole surface of the fracture.Furthermore,the shear strain occurred only in the axial direction.It consisted in a direct integration of constitutiveequations for theshearstrain-stress relations,respectively,in the normal and shear directions, w ithout requiring the solution of a boundary value problem. A ll calculations were conducted in strain-controlled conditions.For a normal compression test on the fracturew ithout occurrence of shear strain,given a normal strain,the corresponding normal stress could be easily calculated according to Eq.(1)w ith dp=0.For a shear test under a constant effective normal stress w ith both elastic and plastic strains produced,given a totalshear strain,the corresponding plastic shear stress and normal displacement could be calculated according to Eqs.(4)and(7),respectively.Due to the coupling between the normaland shear stressesor strains,an iterative procedure is needed.In this study,we employed the classical operator splitting and return mapping methods w idely used in plasticmodels(Simo and Hughes,1998).The numericalalgorithm wascomposed ofan elastic predictorand a plastic corrector.Elastic trial stresseswere fi rstdeterm ined by considering the elastic behaviorof the fracture.W ith thesetrial stresses,the plastic yield function given in Eq.(5)was checked.If the plastic yield condition was verified,we proceeded to the plastic corrector by calculating incremental plastic strains w ith Eqs.(7)and(9).Then,the trial stresses were corrected taking into account the plastic strains according to Eq.(4).The corrected stresseswere comparedw ith the prescribed ones,for example the constant normal stress. Their differences represent the residual stresses used as unbalanced stresses for the next iteration step.The iterationwas stopped when the residual stresses were smaller than a tolerance value.

Table 1Typical values ofmechanical parameters for concrete fracture.

In Fig.4,the stress-displacementcurves in shear testsunder three different confining stresses are presented.There is a strong agreementbetween theexperimentaldataand numerical results.In particular,the fracture softening due to asperity damage in the post-peak regime is correctly reproduced.The normal displacement induced by the shear stress remains very small until the peak stress is reached.In the post-peak regime, asperity damage generates a normal opening during the shear tests.The shear testsw ith a constant fluid pressure inside the fracture,as presented above,were also simulated w ith the proposed model using the effective normal stress concept(but are notshown here to avoid redundancy).There isalso a strong agreement between the numerical results and experimental data.This confi rms that the concept of effective normal stress can be used for description of fluid pressure effects in saturated concrete fractures.However,due to the fact that a simplified plastic softening law is used in the presentmodel,as given in Eq.(11),the plastic deformation occurs only when the peak stress is reached.One obtains a sharp change of stress-strain curve at the peak point.This feature can be improved in the future by introducing a smooth plastic hardening(before the peak point)and softening law(after the peak point).

For fluid flow tests,the fluid pressure distribution was not uniform along the fracture.Therefore,a boundary value problem needed to be solved.However,the objective of the present study was to establish a relationship between the pressure gradientand water flow rate in the hydraulic flow test. Some simplifications were retained.It was assumed that the fluid flow occurred only along the axial direction.Furthermore,one assumed that the permeability of the concretematrix wasmuch smaller than thatof the fracture so that the fluid flow occurred entirely through the fracture.Therefore,it consisted of solving a one-dimensional flow problem along a planar fracturew ith a varying opening as illustrated in Fig.13. Using the cubic law,the one-dimensional hydraulic diffusion equation along the fracture can be expressed as follows:

Fig.13.Division of fracture in finite difference calculation.

whereμis the dynam ic viscosity of fluid；ehis the hydraulic aperture；WandLare the w idth and length of the fracture, respectively；andΔpis the difference in fluid pressure between the injection and outlet points.Because the fluid pressure distribution was not uniform along the fracture,the deformation was not uniform and should be correlated w ith the pressure distribution.However,as there was no shear stress applied to the fracture,no plastic strainsoccurred.Only the normal strain (relative displacement)was coupled w ith the fluid pressure according to the elastic stress-strain relation given in Eq.(4) with dvp=0.Itwas then convenient to express the hydraulic aperture as a function of normal opening or closure,i.e.,eh=eh(v).The latter could be calculated using the elastic stress-strain relation mentioned above.Due to the nonlinear relation between the hydraulic aperture and fluid pressure,it wasnoteasy to establish an analyticalsolution to the diffusion equation.Therefore,a simple finite difference method was used to obtain a numerical solution.The fracturewas divided intoNintervalsalong its length(Fig.13),denoted as elementsEi(i=1,2,…,N).The diffusion equation for each element isw ritten as

whereQiandehiare thewater flow rate and hydraulic aperture ofEi,respectively；andpinandpoutare the fluid pressures at the injection and outlet points,respectively.Moreover,the empirical relation proposed by Barton etal.(1985)is used to define the relation betweenehiandvi:

whereviis the average normal displacementof the elementEi. The fracture roughness coefficient,JRC,is a morphological parameter related to the asperity distribution of fracture.The fluid pressure in the elementEiis calculated w ithpi=pin-which is used in the elastic stress-strain relation given in Eq.(4)to calculatevifor the elementEi.The fluidmassbalance leads to the continuity equation,as follows:

The water flow rate of fracture is finally determ ined as a function of injection fluid pressure by

Fig.14.Comparison between numerical resultsand experimentaldata forwater flow rate versus injection pressure.

Fig.14 presents the water flow rate as a function of injection pressure for three fluid flow tests w ith different confining stresses.One can see that the numerical results are very close to the experimental data.The flow rate is higher when the confining stress is lower due to the lower normal opening of the fracture.There isa strong coupling between the normal deformation of fracture and fluid flow process.

5.Conclusions

Originalmechanical and hydromechanical tests on a fresh concrete fracture were performed w ith a specific testing device.Itwas found that the concrete fracture exhibits nonlinear behavior under a hydrostatic stress.Themechanical behavior of fractures under shear stress is strongly affected by the normal stress.The maximal shear stress increases w ith the confining stress.The fracture showsa softening response after the peak shear stress due to the progressive damage of asperities.The normalopening of a fracture induced by the shear stress remains small before the peak shear stress is reached. There isa clear correlation between theoverallpermeability of a fracture and its normal opening/closure.Based on the experimental data,an elastoplastic model is proposed for modeling the mechanical behavior of a saturated fracture under both normal and shear stresses.The normal elastic stiffness depends on the normal closure/opening of the fracture.The yield function under the shear stress is described based on a Mohr-Coulomb criterion.The fracture softening is taken into account through the damage law of the asperity inclination angle.A non-associated plastic flow rule isadopted in order to correctly describe the normal closure/opening induced by the shear stress.There is an overall strong agreementbetween the numerical resultsand experimental data for both mechanical tests and hydraulic flow tests.The main features of hydromechanical responses of fractures are correctly reproduced.It is possible to use an effective normal stress defined by the classical Terzaghi's concept to describe the effectof fluid pressure on the deformation of the saturated fracture.The water flow rate through the fracture increases w ith the decrease of the effective normal stress.

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Received 2 January 2017；accepted 23 March 2017 Available online 13 June 2017

Thiswork was supported by the National Key Basic Research Program of China(Grant No.2006CB400502),the French National Agency for RadioactiveWaste Management(Grant No.51992),and the European Comm ission through the Collaborative Project Cebama(Grant No.662147).

*Corresponding author.

E-mail address:jian-fu.shao@polytech-lille.fr(Jian-fu Shao).

Peer review under responsibility of Hohai University.

http://dx.doi.org/10.1016/j.wse.2017.06.002

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