Effect of insoluble materials on the volumetric behavior of rock salt

Mejda Azabou, Ahmed Rouabhi, Laura Blanco-Martìn

MINES ParisTech, PSL Research University, Centre de Géosciences, 35 rue St Honoré, Fontainebleau, 77300, France

Keywords: Rock salt Dilatancy Material heterogeneity Natural variability Triaxial tests Virtual laboratory

ABSTRACT This paper focuses on the presence of nodules of insoluble materials within salt specimens, and their effect on the volumetric strain measurements and the dilatancy phenomenon. We analyzed experimental results of over 120 conventional triaxial compression tests,and found that in 20%of the cases,the volumetric strain measurements were atypical. We also noted that the natural variability of the specimens can lead to a non-negligible data scattering in the volumetric strain measurements when different specimens are subjected to the same test.This is expected given the small magnitude of those strains,but it occasionally implies that the corresponding specimens are not representative of the volumetric behavior of the studied rock. In order to understand these results, we numerically investigated salt specimens modeled as halite matrices with inclusions of impurities. Simulations of triaxial compression tests on these structures proved that such heterogeneities can induce dilatancy, and their presence can lead to the appearance of tensile zones which is physically translated into a micro-cracking activity. The modeling approach is validated as the patterns displayed in the numerical results are identical to that in the laboratory. It was then employed to explain the observed irregularities in experimental results. We studied the natural variability effect as well and proposed a methodology to overcome the issue of specimen representativity from both deviatoric and volumetric perspectives.

1. Introduction

Solution mined cavities in salt formations have been used for hydrocarbon storage for about 70 years (Bays, 1963). What made this technique conceivable in the first place is the excellent sealing capacity of rock salt due to its naturally low porosity and permeability (Popp and Kern,1998; Wang et al., 2019). However, underground cavity opening in undisturbed rocks results in the creation of disturbed zones around the underground facility(DeVries et al.,2002, 2005;Habibi, 2019), where the stress state is far from being isotropic and can present high levels of deviatoric stress (DeVries et al., 2002; Tsang et al., 2005).

Laboratory tests on salt specimens have proven that, under compressive loading and above some level of deviatoric stress,rock salt can undergo an irreversible volume increase resulting from a micro-cracking activity (DeVries et al., 2005). This phenomenon is known as dilatancy and is associated with material damage since the micro-cracking activity weakens the material and allows the development of flow paths (Stormont,1997; Schulze et al., 2001;DeVries et al., 2002). Fig. 1 shows the dilatancy onset during a triaxial test conducted under a confining pressure of 4 MPa and a constant axial strain rate of 5 ×10-5s-1.

Due to stress state distribution around the cavity, dilatancy is likely to occur, compromising the integrity of the facility (DeVries et al., 2002; Labaune, 2018; Wang et al., 2018), and hence the importance of the experimental measurements of volumetric strain. However, rock salt dilatancy usually occurs at small volumetric strains (typically below 0.4% (Roberts et al., 2015; Labaune,2018; Rouabhi et al., 2019)), which is so small that rock salt viscoplastic deformation was considered isochoric for years(Tijani et al.,1983;Munson and Dawson,1984;Heusermann et al.,2003).Thus,obtaining the accurate data is a delicate task as factors such as the natural variability, specimen heterogeneity, testing procedure or the used measurement techniques can significantly impact the results.Studies on some of those factors influencing the volumetric strain measurements have been carried out, for example, Medina-Cetina and Rechenmacher (2010) and DeVries and Mellegard(2010) investigated the effect of specimen preconditioning; Hou(2003), DeVries et al. (2005), and Rouabhi et al. (2019) tackled the effect of aspects of the loading conditions; and Rouabhi et al.(2019) studied the measurement techniques effect.

Fig.1. Experimental data of a triaxial compression test on a rock salt specimen (test conducted at Mines ParisTech). Compressive strains are positive.

Fig. 2. Results of a conventional triaxial compression test （P = 12 MPa， ˙εax = 5×10-5 s-1） representative of 80% of the tests performed.

Natural variability is inevitable when we are studying a geological material. As will be shown in this paper, the natural variability of the specimens has a slight effect on the deviatoric behavior, and the exhibited dispersion in axial strain measurements remains within the ranges of measurement errors.However,regarding the volumetric ones, due to their small magnitude as already mentioned,the measured volumetric behavior can show a non-negligible dispersion. This means that the used specimens could be non-representative of the corresponding rock salt on the volumetric level albeit they are so from a deviatoric point of view.

The work of Rouabhi et al.(2019)showed that if the dysfunction of the measurement techniques was excluded, volumetric strain data given by different measurement techniques can exhibit differences that cannot be attributed to the testing conditions alone.This raises concerns of material heterogeneity as it can lead to spatial heterogeneities in the stress and strain fields.

There are numerous aspectsofmaterialheterogeneity.Theycan be microscopic, i.e. related to the shape, size and orientation of the grains, fluid inclusions in grain boundaries, or the type of halite crystals(see Chemin,1990;Van-Hasselt,1991;Speranza et al.,2016,and references therein),or they can be macroscopic and visible on the specimen like the presence of nodules of anhydrite or clay, or even fracture planes.More details about the microscopic and macroscopic aspects of natural heterogeneity in rock salt can be found in the work of Speranza et al. (2016). The impurities presented in rock salt are usually clay, anhydrite and marl (Gillhaus et al., 2006). Their mass fractionisgenerallydeterminedintheprocessofsitecharacterization.This property varies from one rock salt to another over a large range,e.g. in the Tersanne cavern field in southeastern France, the mass fraction of impurities in the salt formation is less than 10%;while in other formations,this fraction can be greater than 50%(e.g.the Vauvert cavern field in south France).

To the best of the authors' knowledge, no studies have been carried out to investigate neither the effect of natural variability nor the impact of a specimen's spatial heterogeneity on the volumetric strain measurements, especially since they are used to determine design criteria for salt caverns.The main purpose of this paper is to understand the effect of material heterogeneity on the experimental measurements of volumetric strain and to propose a methodology to overcome the issue of specimen representativity from a volumetric perspective. We focused on the presence of macroscopic nodules of insoluble materials within a supposed pure halite matrix. Experimental investigation of such factors is a delicate task as it requires precise data and cannot tolerate measurement errors and perturbations that are difficult to be quantified.For this,we numerically analyzed our experimental investigations.The use of a virtual laboratory will allow a better understanding of what is observed in the laboratory and will offer a sound ground to investigate the dilatancy phenomenon with a structural approach without the limitations of experimental work.

We mainly found that the dilatancy phenomenon could be a consequence of material heterogeneity and that the presence of insoluble materials can explain some of the atypical observations in the lab. We also showed how sensitive the volumetric strain measurements are to the natural variability of the specimens and proposed a methodology to overcome the issue of non representativity. This paper is structured as follows. Section 2 is an experimental investigation that is focused on the volumetric strain measurements. Section 3 briefly presents the constitutive model used in the numerical study of section 4 during which the effect of heterogeneities on the volumetric behavior is investigated. This study is purely mechanical, neither the thermal nor the hydraulic aspects were taken into account.

Fig.3. Results of a triaxial compression test（P = 12 MPa， ˙εax = 5×10-5 s-1）and the corresponding tested salt specimen,an example of the irregularities observed in 20%of the performed tests. The local dilatancy onset is significantly lower than the global one.

Fig.4. Results of a triaxial compression test（P = 12 MPa， ˙εax = 5×10-5 s-1）and the corresponding tested salt specimen,an example of the irregularities observed in 20%of the performed tests. The local dilatancy onset is higher than the global one.

Fig.5. Results of a triaxial compression test（P = 12 MPa， ˙εax = 5×10-5 s-1）and the corresponding tested salt specimen,an example of the irregularities observed in 20%of the performed tests. The local and global volumetric strain measurements have different patterns.

2. Experimental data

We analyzed 11 experimental campaigns conducted at the Geosciences Department of Mines ParisTech on salt specimens from different locations in France and USA in 2004-2012. Each testing program is comprised of conventional uniaxial and triaxial compression tests and creep tests. As volumetric strain measurements were only provided in triaxial compression tests,we focused on those experiments(the total number of studied tests was about 123).The tested specimens are cylinders with height of 13 cm and a slenderness ratio of 2. A triaxial compression test consists in maintaining a constant confining pressure on the lateral surface of the specimen by injecting/withdrawing a confining fluid into a triaxial cell,while imposing a constant axial strain rate via loading platens in between which the specimen is placed (Wawersik and Hannum,1980; Mellegard et al., 2005; Ulusay and Hudson, 2007).

We distinguished two kinds of measurement techniques:“local”and “global”. The local measurements are given by strain gauges(axial and circumferential) placed at mid-height of the specimen.The global measurements use an LVDT (linear variable displacement transducer) attached to the loading platens (Rouabhi et al.,2019). For the axial strain, the local measurement is given by the axial strain gauge and the global one by the LVDT. The measurements of the axial and circumferential gauges are combined to record the local volumetric strain whereas the axial displacement between the loading platens(measured by the LVDT)is associated with the amount of fluid injected or withdrawn from the cell to register the global volumetric strain.

Fig. 6. Results of a triaxial compression test （P = 8 MPa， ˙εax = 5×10-5 s-1） and the corresponding tested salt specimen, an example of the irregularities observed in 20% of the performed tests. The local and global volumetric strain measurements have different patterns.

Fig. 7. Results of two identical triaxial compression tests （P = 8 MPa， ˙εax = 5×10-5 s-1） and the corresponding tested salt specimens with difference in depth of 2.25 m.

Fig. 8. Results of two identical triaxial compression tests （P = 12 MPa， ˙εax = 5×10-5 s-1） and the corresponding tested salt specimens with difference in depth of 2.3 m.

Fig. 9. Results of two identical triaxial compression tests （P = 10 MPa， ˙εax = 2×10-5 s-1） and the corresponding tested salt specimens with difference in depth of 21.4 m.

In the following,let Q be the applied axial stress,P the confining pressure, |Q - P| the deviatoric stress and ζ the volumetric strain defined as

where the subscripts g, l and i stand for global, local and initial,respectively;V is the volume of the specimen;and εθ and εzare the tangential and axial strains, respectively. Details on the given definitions can be found in Rouabhi et al. (2019).

2.1. Results and observations

The analysis of the experimental data and examination of the corresponding specimens showed that in about 80%of the tests,the difference between the local and global measurements of axial strain was small and stayed within the range of the uncertainty of the measurement techniques. Regarding the volumetric strain,there is always a difference between the global and local measurements as the strain gauges measure dilatancy onset before it is seen over the whole specimen,i.e.before the global measurement indicates it.In these classic cases,the dispersion between the local and global measurements is in the range of what is observed in the tests presented in Rouabhi et al.(2019).Fig.2 shows the results of a typical test conducted under a constant strain rate of 5 ×10-5s-1and a confining pressure of 12 MPa.In this figure, the evolution of the deviatoric stress is presented as a function of global and local measurements of the axial and volumetric strains. As can be seen,the dilatancy onset given by the global technique is about 2 times higher than the one measured by the local gauges, a ratio that is generally observed in a large majority of the tests.Nevertheless,the axial strain measurements present an acceptable dispersion.

Regarding the remaining 20% of the tests, the local volumetric strain measurements are atypical and their comparison with the global ones is out of the pattern described for 80% of the tests. In these tests, we have cases where gauges indicate a higher deviatoric stress for dilatancy onset than what the global technique does.We also see cases where the gauges measure only contractancy or dilatancy during the entire test period whereas the global technique shows that the specimen undergoes contractancy and then dilatancy. Figs. 3-6 give examples of these atypical observations and show the corresponding specimens as well. In Fig. 3, the deviatoric stress corresponding to the global dilatancy onset is 8 times higher than the one measured by the local technique. Fig. 4 indicates that when the deviatoric stress ranges between 35 MPa and 50 MPa, the specimen undergoes a global increase in volume and a local contractancy at the same time. Figs. 5 and 6 show that the local and global volumetric behaviors are significantly different during each of these tests.

We also studied the cases where the same test was conducted on two specimens in order to see to which extent the natural variability could affect the results and whether the resulting dispersion is acceptable. We observed that the natural variability did not have a significant effect on the axial strain, local or global,but it could significantly affect the volumetric behavior measurements. In Figs. 7-9, three of these comparisons along with the concerned specimens are shown.

2.2. Discussion

The analysis of these experimental data shows that it is possible for a specimen to display a local response that is significantly different from the global one. The malfunction of the measurement techniques is excluded:we analyzed the results and confirmed that the gauges had been fully functional, and we also assumed that there were no oil leaks to alter the global measurement. Rouabhi et al. (2019) proved that the friction with the loading platens can in fact cause the local measurements to deviate from the global ones but not to the observed extent. This can be interpreted by the presence of material heterogeneity close to the placement of the gauges which capture very local phenomena due to their size.

The specimens that are considered representative of a given rock salt can show a very similar deviatoric behavior when subjected to the same loading conditions, but a different volumetric behavior: the dispersion is always more pronounced in the local measurements. This implies that, in terms of the volumetric response, our specimens could be smaller than the required representative element volume (REV) for the studied rock salt.

Both of these observations can be traced back to the material spatial heterogeneity and among the many aspects of it (Chemin,1990; Van-Hasselt, 1991; Thiemeyer et al., 2015, 2016; Speranza et al., 2016; Mansouri et al., 2019), we chose to focus on the presence of nodules of impurities, which was similar to what can be seen on the specimen displayed in Fig. 4. The effect of such heterogeneity on the volumetric behavior and the representativity of the used specimens was investigated. However, rigorous experimental investigation of this aspect is problematic due to the following reasons:

(1) We do not have control over the material heterogeneity. As mentioned previously,various aspects of heterogeneity exist within a salt specimen, which means that we cannot attribute a certain behavior to one particular aspect.

(2) The measured volumetric strains are extremely small(Roberts et al., 2015; Labaune, 2018; Rouabhi et al., 2019),which means that the errors and uncertainties inherent to the laboratory work can be non-negligible and can affect the measurements and therefore alter our interpretations and analysis.

(3) In order to further investigate the representativity of the specimens, we need to perform typical triaxial compression tests on larger specimens. This is often complicated due to the limitations of the experimental facilities: typically a laboratory is equipped with cells that can host specimens with specific dimensions (Ulusay and Hudson, 2007), and it is unrealistic to have a new cell each time there is a need to characterize.

Consequently, we decided to use a virtual laboratory built on numerical modeling and simulation tools. This will allow investigating the effect of this particular aspect of heterogeneity on rock salt dilatancy, in an attempt to understand the mentioned observations.

3. Constitutive model

Fig. 10. Cutaway view of the modeled structure of a salt specimen with insoluble nodules.

In Eq. (11), the sign of 〈p/N〉n-γcvpindicates whether the behavior is contracting or dilating. Regarding the tensile part, we consider an evolution law of Perzyna-type:

In Eqs.(5)-(13),z,n,m,a,b,B,c,K,k,Λ,Rt,d,s,M,N and S are the material parameters.When this model is used to simulate a triaxial compression test,dilatancy is:

(1) taken into account on a constitutive level when we assume that the stress field within the specimen is homogeneous and all stresses are compressive(Eqs. (10) and (11));

(2) a result of a heterogeneous distribution of the stress field that leads to activating the tension mechanism(Eqs.(7)and(13)).

Table 1 Parameters used to simulate the behavior of the salt matrix.

4. Numerical investigations

We first present and validate our numerical approach which is then employed to investigate the volumetric strains measured by different measurement techniques on a salt specimen that contains nodules of impurities. The natural variability and its effect on the volumetric strain measurements and the specimen representativity are discussed in Section 4.3.

4.1. Structural approach for dilatancy

The micro-structure of a salt specimen is in reality very complex to be reproduced numerically(different grain sizes,fluid inclusions,grain boundaries, different types and shapes of heterogeneities,etc.). In this work, we assumed that salt is a two-phase medium composed of a pure halite matrix and nodules of insoluble materials.This simplified assumption is acceptable,since the aim of this research is to qualitatively understand the impact of the presence of heterogeneities on the macroscopic behavior. Thus, the numerical validation of the approach does not concern its ability to reproduce the complex geometry of real salt specimens,but rather the measurements of physical quantities.

A salt specimen is modeled as a cylindrical matrix of pure halite,with inclusions representing the insoluble nodules as shown in Fig.10.The insoluble nodules are modeled as spheres for the sake of simplicity and the specimens are generated as follows. First, a cube’s side length,the volume fraction of insoluble nodules within the cube,and the minimum and maximum radii(of the inclusions)are designated.Second,an iterative process is used to generate,via a uniform random number generator, the position of a sphere center (within the cube) and a radius (between the specified bounds).The specified volume fraction is checked at each iteration to ensure that it will not be exceeded.Finally,once the distribution of the spheres is generated within the cube,the position and size of the specimen to model are then chosen and the virtual sampling is performed.

The structure shown in Fig. 10 will be used as a reference specimen for our numerical investigations. It contains a volume fraction of inclusions of φ = 10.3%, and the density of the halite matrix is assumed to be 2160 kg/m3,which leads to a mass fraction of w=8.3%.Such mass fraction is within the usual ranges found in salt rocks (Gillhaus et al., 2006). The insoluble nodules are considered to have properties similar to clay: a density of ρ=1700 kg/m3,a Young’s modulus of E=5000 MPa and a Poisson’s ratio of ν = 0.32.

We assume the continuity of the displacement field within the structure, but a joint model for the halite-inclusions interface was not assumed.This simplified assumption was made because,on one hand, the joint parameters are difficult to be determined experimentally,and on the other hand,introducing additional parameters will further complicate the study, especially with the meshing requirement of the modeled structures. The assumption is valid since, as will be shown in this paper, the macroscopic aspects of rock salt behavior are reproduced.

The behavior of the insoluble nodules is assumed to be linearly elastic while the salt matrix is governed by the constitutive law presented in Section 3.The parameter set used for the salt matrix is presented in Table 1.This parameter set was adopted from Labaune(2018) where the distortion and dilatancy parameters were obtained by fitting triaxial compression and creep tests, and fitting a Brazilian test provided the ones relative to the tension mechanism.In this work, some of these parameters are changed. The tensile strength is set to Rt= 0 (Eq. (13)) in compliance with setting the cohesion c=0(Eq.(12)).Also,setting the parameter B=0(Eq.(12))means that we omit the mean pressure effect and setting z=0(Eq.(11)) implies that the viscoplastic volumetric strain due to compressive stresses is not considered. The tensile part is deactivated by setting the parameter Λ = 0 (Eq. (13)).

Fig.11. Cutaway (left) and transparent (right) views of the distribution of σ1 (in MPa)within the salt matrix at εax ≈2% (simulation of a typical triaxial compression test under a confinement of 5 MPa and an axial strain rate of 10-6 s-1, on the specimen shown in Fig.10).

We used VIPLEF3D, a finite element code developed in the Geosciences Department of Mines ParisTech (Tijani, 2008) to simulate a typical triaxial test on the structure shown in Fig.10.The simulated test was conducted under a constant strain rate of 10-6s-1and a confining pressure of 5 MPa.

The results showed that within the salt matrix and around the spherical inclusions, zones where the major principal stress σ1is positive appeared and continued to develop all along the simulation.Fig.11 shows the distribution of σ1within the salt matrix at an axial strain εax≈2%:the difference in rigidity of the inclusions and the halite matrix led to the appearance of tensile zones and to stress concentrations.

We re-conducted the same simulation except for activating the tensile part this time (the parameter Λ from Eq. (13) was set to 5×104).Activating this component of the viscoplastic strain tensor means that in the halite matrix,irreversible strains are of two kinds,i.e.those governed respectively by the deviatoric creep and tensile mechanisms, which obviously vanish under compressive loadings when the stress homogeneity usually assumed in analysis of laboratory tests is used. The obtained results of the two performed simulations in terms of global measurements are given in Fig.12.

The overall behavior of the structure has the same characteristics as those observed in the laboratory for salt specimens for this type of tests: viscoplastic hardening, contractancy and dilatancy.This is an important result because it shows that dilatancy can be a consequence of the material spatial heterogeneity. In fact in this study,the presence of heterogeneities leads to the development of stress concentrations and induced tensile stress zones which physically reflect the occurrence of a micro-cracking activity and therefore dilatancy. We used the reference specimen in Fig.10 to simulate a series of laboratory tests, as listed in Table 2.

The results of these simulations are shown in Figs.13 and 14 and are considered very satisfactory as both effects of loading rate and mean pressure are correctly predicted. In fact, the dilatancy limit increases with the mean pressure (equivalent to the confining pressure effect)and is significantly affected by the loading rate:the slower the test,the lower the dilatancy limit(Labaune et al.,2018).Also the creep behavior (Fig. 15) is very similar to that typically observed in the laboratory (Günther et al., 2015; Labaune et al.,2018).

These results have been obtained as well for other structures with different inclusion shapes, distributions, sizes, volume fractions and elastic constants.This approach is assumed to be valid to describe rock salt behavior in the laboratory.

Fig.12. Simulation results of a typical triaxial compression test under a confinement of 5 MPa and an axial strain rate of 10-6 s-1, on the specimen shown in Fig.10.

Table 2 Laboratory tests simulated on the specimen shown in Fig.10.

Fig.13. Simulation results of typical triaxial tests (P = 5 MPa) on the specimen shown in Fig.10: Effect of loading rate.

Fig.14. Simulation results of typical triaxial tests (˙εax = 10-6 s-1) on the specimen shown in Fig.10: Effect of confining pressure (P).

Fig.15. Simulation results of a typical triaxial multistage creep test(P=5 MPa)on the specimen shown in Fig.10.

4.2. Volumetric strain measurements on a heterogeneous specimen

We adopted the approach detailed in Section 4.1 and the same parameter set given in Table 1 for the numerical study hereafter in order to understand the experimental observations and qualitatively assess the effect of heterogeneities on the volumetric behavior of rock salt. We simulated on the reference specimen in Fig. 10, a conventional triaxial test under a confining pressure of 5 MPa and a constant axial strain rate of 10-6s-1(see Fig.13).We modeled the placement of two sets of strain gauges on the specimen in an attempt to measure local strain with the same procedure used in the laboratory. The position of the two sets of gauges was randomly chosen,and the only conditions were that the sets had to be placed at mid-height and on the salt matrix’s lateral surface.We investigated the effect of the friction with the platens as well so the simulation was conducted twice with two extreme conditions of specimen-platen contact:smooth contact with no friction between the specimen and the loading platens, and rough contact where there is a complete radial restraint on the specimen ends.The latter boundary condition causes the initially right cylinder specimen to deform into a barrel(triaxial compression)or an hourglass(triaxial extension) shape (Labaune, 2018; Rouabhi et al., 2019). Fig. 16 shows the obtained results. The results show that the friction with the platens significantly affects the global volumetric strain measurements but it has a slight impact on the axial and the local volumetric strain measurements. The sets of gauges give very similar results in terms of both axial and volumetric strains.In fact,this specimen does not have locations of inclusions concentration,instead the spheres are almost evenly distributed (see the transparent view in Fig.10). This leads to a sort of uniformity over the lateral surface:all gauges placed at mid-height of the specimen are likely to give very similar measurements, and no gauge could be significantly more exposed to inclusions than others.

Specimens with inclusions not evenly distributed are generally used,as shown in Fig.4.In this specimen,it can be noted that there are areas on the lateral surface covered with insoluble nodules and others that are almost clear. To study such cases, the same simulation is conducted on the specimen as presented in Fig.17.In this case,we placed the first set of gauges near the inclusion we see on the lateral surface of the specimen and the second set on the opposite side (relatively furthest from inclusions). The results are shown in Fig.17.

Similarly to that in the laboratory,the axial strain measurement is slightly affected, and the dispersion between global and local measurements is acceptable.Regarding the volumetric strains,the difference is out of the usual ranges and patterns (see Fig. 2). At a deviatoric stress level of ～4 MPa, the first set (closest to the inclusions)measures significantly important radial strains due to the local appearance of tensile stresses(see Fig.11),in association with a local dilatancy onset.On the other hand,the second set(furthest from the inclusions) only sees contractancy until the end of the simulation:the radial strains in that location of the specimen were not important because tensile stresses did not appear there.When the deviatoric stress reaches ～27 MPa, the whole specimen’s volume begins to increase. The information provided by the sets of gauges holds true but has unexpected patterns and dilatancy onset values because it captures very local phenomena.

In this investigation, a possible interpretation is given to cases like the ones presented in Figs.3 and 6.We also gave an explanation to one of the main conclusions of Rouabhi et al.(2019)which stated that the friction with the loading platens cannot explain the difference observed between local and global measurements of the volumetric strain.

Fig. 16. Simulation results of a typical triaxial compression test (P = 5 MPa and ˙εax = 10-6 s-1) on the specimen shown in Fig. 10. Curves in dashed and continuous lines correspond respectively to simulations with and without accounting for the friction with the loading platens.

Fig.17. Simulation results of a typical triaxial compression test (P = 5 MPa and ˙εax = 10-6 s-1) and the corresponding specimen.

4.3. Natural variability and specimen representativity

In this section,the representativity of specimens is investigated from a volumetric point of view. For this, a random distribution of spheres was generated, with radii ranging from 8 mm to 20 mm,which is within ～7%of the volume of a 20 cm side length cube.As shown in Fig.18, this cube is virtually sampled and four salt specimens are generated with dimensions of 13 cm in height and 6.5 cm in diameter. The inclusions and the halite matrix have the properties as mentioned in Section 4.1. The mass fraction of insoluble nodules in each specimen is indicated in Fig.18 as well.

For each of the four specimens, a conventional triaxial compression test was simulated, where the axial strain rate was fixed at 10-6s-1and the confining pressure at 5 MPa. The simulations continued until dilatancy onsets were globally observed.The results in terms of global measurements are shown in Fig.19.As shown in Fig.19, the axial strain measurements present slight deviations, while the volumetric ones are significantly different.Because of the size of the insoluble nodules and their distribution within the initial block, none of the sampled specimens was representative of the block from a volumetric perspective. Testing larger specimens might be an immediate and ideal solution for this issue. However, in practice, the tests may be constrained by the geometry of the available triaxial cells, as explained in Section 2.Thus, we may even face cases that the maximum testable size(diameter equal to the coring diameter and a slenderness ratio of 2)is not representative of the rock in question.Besides,testing larger specimens is likely to compromise the precision of the obtained strain measurements; in fact, the larger the specimen, the greater the required volume of confining oil, and the less accurate the obtained volumetric strain measurements.

Fig.18. A modeled salt block with virtually sampled specimens. The mass fraction of insoluble nodules is indicated for each specimen.

Fig.19. Results of a conventional triaxial compression test (P = 5 MPa and ˙εax = 10-6 s-1) on the specimens shown in Fig.18.

Fig. 20. A modeled salt block with virtually sampled specimens in the second case. The mass fraction of insoluble nodules is indicated for each specimen.

Fig. 21. Results of a conventional triaxial compression test (P = 5 MPa and ˙εax = 10-6 s-1) on the specimens shown in Fig. 20.

Fig. 22. Fitting of a conventional triaxial compression test (P = 5 MPa and ˙εax = 10-6 s-1) conducted on specimen C1 shown in Fig. 20.

Table 3 Parameters used to fit the triaxial test results of specimen C1 shown in Fig. 20.

For a second case scenario,a random distribution of spheres was generated,with radii ranging from 5 mm to 10 mm which is within～5%of the volume of a 20 cm side length cube.As shown in Fig.20,this cube is virtually sampled and 3 salt specimens are generated with size of 13 cm in height and 6.5 cm in diameter.Using the same properties of the inclusions and the halite matrix as the previous case, the mass fraction of insoluble nodules in each specimen is indicated in Fig.20.The same test is simulated as that performed on the specimens (see Fig.19) and the corresponding results are displayed in Fig. 21.

In this case, the inclusion size is small compared to the dimensions of the specimen.In fact,for all three of the cases,we have Rmax/Rspec≈0.29, where Rmaxis the radius of the largest sphere included within the halite matrix and Rspecis the radius of the specimen. Besides, there are more inclusions in these specimens(14 for C1, 12 for C2, and 13 for C3) than the ones from the first scenario (Fig. 18) and they are evenly distributed within the volumes.This leads to almost no dispersion either in the axial or in the volumetric strain measurements: we are addressing an equivalent homogeneous material. Consequently, in similar cases, we can safely adjust a phenomenological model and use the resulting parameter set for the concerned underground facility.

To illustrate this, we use the same constitutive model introduced in Section 3 to fit the curve of specimen C1 in Fig. 21. We assumed that the stress and strain fields within the specimen were homogeneous and all the stresses were compressive;therefore,the tensile mechanism cannot be activated.We set z ≠0 in order to reproduce the volumetric strain. In other words,the results shown in Fig. 21 for the specimen C1 are considered as experimental data to fit with the proposed model under the assumptions stated previously. Fig. 22 shows the fitting of these data with the parameter set given in Table 3. As can be seen, the phenomenological model, where dilatancy is taken into account on a constitutive level, matches very well the results.

5. Conclusions

In this paper, the volumetric strain measurements were experimentally investigated through the analysis of over 120 typical triaxial compression tests on salt specimens from different locations. The difference between local and global measurements exhibited in this analysis raised questions concerning the effect of material spatial heterogeneity on the volumetric behavior and consequently on the representativity of the used specimens. We focused on a specific aspect of material heterogeneity:the presence of nodules of insoluble materials within a pure halite matrix, in order to obtain a qualitative understanding of the issue.

A numerical study was carried out during which salt specimens were considered as cylindrical matrices of pure halite with spherical inclusions. The results show that dilatancy can be a consequence of the specimen’s material heterogeneity. The significant differences (in patterns and the measured dilatancy onsets) between local and global volumetric strain measurements were satisfactorily explained by the material spatial heterogeneity.

Finally, we found that the presence of nodules of insoluble materials within a salt rock could lead to representativity issues.For this, we proposed the following general methodology:

(1) Characterize one of the used specimens (known to be not representative): volume fraction of insoluble materials, nature of the inclusions (thus obtaining an estimation of their elastic constants),and concentration zones of heterogeneity.

(2) Model this specimen as a two-phase medium composed of a pure halite matrix and insoluble nodules.

(3) Perform a history matching of the experimental data with a given constitutive model(in which dilatancy results from the activation of a tensile mechanism induced by the presence of heterogeneities) on the modeled structure.

(4) Once a parameter set is obtained, model a representative specimen containing more inclusions of a negligible size compared to the dimensions of the specimen.

(5) Simulate the test conducted in the laboratory on the representative specimen using the same constitutive model from step 3 with the parameter set resulting from the history matching.

(6) The obtained results are supposed to be representative of the rock in question.They can be fitted with a given phenomenological constitutive model,with assumption of homogeneity of the stress field.Dilatancy in the chosen model should be taken intoaccountonaconstitutive level.Theresultingparameter set can then be used for the corresponding rock salt.

This methodology combines experimental work with numerical modeling in order to overcome specimen’s representativity issues and the limitations of laboratory testing procedures.

Declaration of competing 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.

Acknowledgments

The authors would like to thank their colleagues Michel Tijani and Faouzi Hadj-Hassen from Mines ParisTech for their valuable contributions.