1. SLOW CREEP RATES 1.1. Introduction
ARMA 14-7052 A very slow creep test on an Avery Island salt sample Bérest P., Béraud J.F. and Gharbi H. LMS, Ecole Polytechnique, Palaiseau, France Brouard B. Brouard Consulting, Paris, France DeVries K. RESPEC, Rapid City, South Dakota, USA Copyright 2014 ARMA, American Rock Mechanics Association This paper was prepared for presentation at the 48th US Rock Mechanics / Geomechanics Symposium held in Minneapolis, MN, USA, 1-4 June 2014. This paper was selected for presentation at the symposium by an ARMA Technical Program Committee based on a technical and critical review of the paper by a minimum of two technical reviewers. The material, as presented, does not necessarily reflect any position of ARMA, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of ARMA is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 200 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgement of where and by whom the paper was presented. ABSTRACT: A very slow creep test was performed on an Avery island salt sample. The testing device was set in a remote gallery of the Varangéville Mine to take advantage of the very stable temperature there. This multi-step test was 30-month long. The applied load was 0.1, 0.2 and 0.3 MPa, successively. Steady-state strain rates are of the order of 10-12 s-1, significantly faster than what can be extrapolated from creep tests performed under larger load, Rates are an increasing function of the applied load. . 1. SLOW CREEP RATES 1.1. Introduction An abundant literature has been dedicated to various aspects of the mechanical behavior of salt. Consider a cylindrical sample submitted to a triaxial compressive load, 0 2 3 1 — i.e., a deviatoric stress deviatoric stress is σ = 10 MPa, the steady-state strain rate typically is ss 1010 s 1. For instance, Avery Island salt has been studied extensively by RESPEC; the results of 55 creep tests are represented in a log log plot in Fig. 1. 2 1 3J 2 , where sij ij kk ij / 3 is the deviatoric stress tensor, and J 2 sij s ji 2 is its second invariant. It generally is accepted [1] that, when this deviatoric stress is kept constant, a steady-state strain rate, ss , is reached after several weeks or months. This rate is a non-linear function of the applied deviatoric stress and is highly sensitive to temperature. The volumetric strain rate is nil. The main features of such steady-state behavior are captured by the Norton-Hoff law: ss A exp Q RT n tr ss 0 (1) where A, n and Q/R are three constants, with n in the 3-6 range and the thermal constant, Q/R, in the range 3000 to 10,000 K domain. At ambient temperature, when the Fig. 1. Avery Island salt steady-state strain rate as a function of deviatoric stress and temperature. In fact, most tests are performed in the range 5 20 MPa . For instance, when n 4 is assumed, the Norton-Hoff law predicts that, when 1 MPa, the ARMA 14-7052 steady-state strain rate should be ss 1014 s 1 , a rate which, as explained below, is too slow to be measured. 1.2. Deviatoric Stresses at the Wall of a Cavern When the Norton-Hoff law is used for computing the behavior of a salt cavern, it is observed that the deviatoric stresses at the vicinity of the cavern are smaller than 5 MPa, at least when the cavern is not too deep. Consider, for instance, the cavern having the shape represented in Fig. 2. by 5–20 MPa rectangle is the domain inside which most laboratory tests are performed. In fact, the micromechanisms that govern creep in the domain σ = 0–5 MPa, which is of primary interest, are unknown. In other words, prediction of the mechanical behavior of salt in this domain is based on extrapolation of purely empirical data and cannot be supported by theoretical consideration. This cavern is 750-m deep. The behavior of the salt is elasto-viscoplastic, and the Norton-Hoff law is considered. During the leaching period, which lasts 600 days, cavern pressure is lowered progressively from geostatic pressure to halmostatic pressure (i.e., the cavern pressure when the access borehole is filled with saturated brine of density 1200 kg/m3). Pressure then is kept constant for 2400 additional days; stress distribution at that time is not far from steady-state distribution. It can be observed that deviatoric stresses, which were slightly larger than 5 MPa at the cavern wall when leaching was completed, are smaller than 5 MPa in almost all the rock mass, except in some overhanging parts at the cavern wall. Consider, also, an idealized cylindrical cavern whose internal pressure abruptly decreases at t = 0 from geostatic pressure, P P , to halmostatic pressure, or P Ph . At t 0 and at t (steady state), the deviatoric stress is 3J 2 3J 2 t 0 ss 3 P Ph a r 3 P Ph a r 2/ n 2/ n 2 (2) 2n In addition, the steady-state deviatoric stress is largest at the cavern wall: it is smaller than the initial deviatoric stress by a factor of n, and smaller when the exponent of the Norton-Hoff law, n, is larger. In a 750-m deep cavern, the deviatoric stress at the cavern wall is 6.5 MPa at t = 0. When steady state is reached, it is 1.6 MPa. In other words, standard laboratory tests are not performed in the range of deviatoric stresses that are relevant when computing the behavior of a salt cavern. 1.3. Deformation Mechanism Langer [2] stated that reliable extrapolation of the creep equations at low deformation rates can be carried out only on the basis of deformation mechanisms. The micromechanisms that govern salt creep have been discussed by Langer [2], Munson and Dawson [3] and Blum & Fleischman [4]. A deformation-mechanism map (adapted from [3]) is presented in Fig. 3. The 0–120 °C Fig. 2. Deviatoric-stress contours after 600 days (top) and 2000 days (bottom). ARMA 14-7052 2.2. Temperature and Hygrometry When the creep rate is 1012 s 1 , a test lasting 12 days results in a cumulated strain of 106. The thermal expansion coefficient of salt is 4 105 /°C — and temperature variations T of a couple of degrees Celsius generate thermoelastic deformations, or T , which, in many cases, are larger than the signal to be measured (i.e., sample average deformation originated by creep proper). The same can be said of small hygrometric variations ([9], [10], [11]). In [10], for instance, it is suggested that, when hygrometry, Φ, in RH (%) is taken into account, the steady-state creep rate must be corrected as follows: ss A 1 w sinh q exp Q RT n Fig. 3. Mechanism map (after [3]). However, Spiers et al. [5] and Uraï & Spiers [6] observed that in the low-stress domain, pressure-solution creep, an important deformation mechanism of most rocks in the earth’s crust, is especially rapid in the case of rock salt. Theoretical findings strongly suggest that, for this mechanism, the relation between deviatoric stress and strain rate is linear, with important consequences for the computation of both geological processes and underground works operation. 2. PROBLEMS CREEP TESTS RAISED BY SLOW-RATE 2.1. Literature Small strain rates ( 10 14 to 1011 s 1 ) have not been investigated widely in the laboratory. Hunsche [7] describes the measurement of creep in rock salt at small strain rates using a special testing device. The test lasted approximately one week, during which a strain rate of 7 1012 s 1 was “the lowest reliably determined deformation rate”. Bérest et al. [8] performed a series of uniaxial compression tests during which the applied stress was zz 0.1 MPa. The steady-state creep rate was 1012 s 1 . The experimental system was basically the same as that used for the tests described in this paper. The limited available literature is inherent to the particular problems raised by long-term, slow-rate creep tests, as noted below. (3) where is the room hygrometry in % RH; q 0.1 and w 0.1 are two constants. A change in room hygrometry from Φ = 55% RH to Φ = 75% RH leads to a multiplication of the steady-state rate by a factor of 7. 2.3. Loading Slow creep rates are obtained when small mechanical loadings are applied. Most creep-test devices are designed to operate in the deviatoric stress range of 5 20 MPa, and stress control usually is poor when the applied stress is smaller than 5 MPa. 2.4. Deformation Measurement Creep rate is computed by comparing the strains 1 , and 2 measured at two different times, t1 and t2 , where ( 2 1 ) (t2 t1 ). When, for instance, t2 t1 105 s (one day) and 1012 s 1 , then 2 1 107. Therefore, a reasonable assessment of daily strain rate demands that strain be measured with an accuracy of 108. 2.5. Requirements When Performing Slow Creep Tests In other words, accurate long-term creep tests are possible only when the following applies. The temperature and hygrometry experience very small changes The applied load can be control with a high accuracy. The sample length change can be measured with a high resolution. How these problems were tackled is described in the following section. ARMA 14-7052 3. TESTING DEVICE, MECHANICAL LOAD 3.1. Samples and Loading Uniaxial creep tests are being performed on cylindrical salt samples with D 70 mm and H 140 mm. The sample is set between two duralumin plates (Fig. 4). Dead weights are set on the lower part of a rigid frame below the sample. The frame weight is transmitted to the upper duralumin plate through a small metallic ball. (This ball raised some concern, as stresses at the ball plates are high, and it was feared that punching of the plates by the ball would lead to wear. To lessen this fear, grease was set between the ball and the plates.) The applied stress is calculated by dividing the overall weight of the steel frame by the initial cross-section of the salt cylinder. (Strains are small, and no correction of the sample cross-sectional area was deemed necessary.) The range of stresses that can be applied to a sample is zz 0.05 to -1 MPa. room. The corresponding offsets are erased in the strainvs-time curves. Strictly speaking, only three sensors are needed to allow both the relative rotation and the vertical displacement of the upper plate to be measured. However, four sensors are used to provide some redundancy. 3.3. Rotation of the Upper Plate An example of plate rotation measured by the apparatus is provided in Fig. 5. The displacements, u1 , u2 , u3 and u4 , of the four vertical sensors were measured during an 8-week-long period at the beginning of the test performed on an Avery Island salt sample. It can be observed that u2 u4 u1 u3 2 , proving that the four measured displacements are consistent and strongly suggesting that the upper plate is rotating along the 2–4 horizontal axis. This consistency also suggests that sensor drift is small. Fig. 4. Testing device and salt sample below the upper plate. 3.2. Sensors During a test, four (C1 to C4) high-resolution displacement sensors (Solartron linear encoders) are positioned in two vertical planes at 90° angles (Fig. 4). Sensor accuracy is 0.5 µm, and its resolution is 0.0125 µm (1/80 µm). The encoders operate on the principle of interference between two diffraction gratings. Gratings are deposited on a quartz substrate. The gratings are composed of black and white rectangles with length 10 µm. A first grating is illuminated by a light-emitting diode. A second grating is used to scan the modulated light intensity generated by the first grating when this grating moves as a consequence of sample deformation. The system computes the number of rectangles that have crossed through a fixed line. One drawback of this system is that, in case of an electric cut, the counting is reset to zero. There were a couple of electric cuts when members of the staff operated in the Fig. 5. Vertical displacements measured by the four sensors. 3.4. Sensor Drift In order to assess sensor drift, a duralumin cylindrical sample (instead of a salt sample) was set in the testing device from October 20, 2010 to November 30, 2010; a 0.15-MPa load was applied. The strain rate of the duralumin sample at the end of November 30 is less than 3 1013 s 1 , but it was faster at the beginning of this ARMA 14-7052 test. The test was not long enough, as crushing of small irregularities of the duralumin-platen interfaces at the beginning of the test is suspected. A longer test is planned. 3.5. Temperature and Hygrometry Fluctuations As explained above, temperature changes during a longterm creep test must be minimized, as they are the main source of strain fluctuations. In a laboratory room, daily temperature fluctuations are hardly smaller than 1 °C, generating thermoelastic strains that are much larger than the signal to be measured during a slow creep test. For this reason, the tests are performed in a deep underground room, where temperature is much more constant than in any surface facility. With the kind support of the Compagnie des Salins du Midi et Salines de l’Est, tests were performed at the dead end of a 700-m-long, 160-m-deep gallery of the Varangéville salt mine in eastern France (Fig. 6). This gallery is remote from the area of present salt extraction. misinterpretation of the test results. More erratic daily temperature changes also can be observed. It was believed, first, that they were correlated to atmospheric pressure changes; however, in fact, heat resulting from air compression or extension is dissipated rapidly through the gallery walls. Fig. 7. Temperature in the gallery from day 260 to day 380 (after October 19, 2010). Hygrometry also was measured (Fig. 8). It is close to Φ = 75%RH — a very high figure, as it is known that salt cannot withstand a hygrometry higher than 76% RH. Further measurements proved that a small offset might be suspected. Such a large hygrometry was a concern, as it is known that hygrometry may have a dramatic influence on salt creep rate (see Section 2.2). Fig. 6. The testing devices at the dead-end of the gallery. Gallery temperature must be measured precisely enough to allow correction of the raw strain data for thermoelastic strains. Temperature is measured by platinum sensors whose resolution is one-thousandth of a degree Celsius; however, their accuracy is not better than 0.5 °C. As an example, temperature evolutions during the July 2011 to November 2011 period are illustrated in Fig.7. Two temperature gauges are used. An offset clearly is visible; however, more important, temperature fluctuations are parallel, providing some confidence in the gauges resolution. Large temperature changes are visible when members of the staff are working in the gallery on day 266 (July 11, 2011). A slow temperature increase by 0.1 °C/yr can be observed by autumn. The resulting sample expansion rate, estimated to be T 1.2 1013 s 1 , should not lead to significant Fig. 8. Temperature and hygrometry of the gallery air from day 1 to day 260 (after October 19, 2010). 4. TEST RESULTS 4.1. A Multi-Step Creep Test A multi-step creep test on an Avery Island salt sample was initiated on July 11, 2011 (A on Fig. 9). The applied axial load was zz 0.1 MPa. This load was increased to zz 0.2 MPa on March 14, 2012 (B on Fig. 9) and Fig. 9. Strain, hygrometry and room temperature as a function of time during the test performed on the AI salt sample. to zz 0.3 MPa on November 7, 2013 (D on Fig. 9). The two first phases lasted 8 months; the third phase is longer. Both temperature and hygrometry fluctuations, together with average strain evolutions, are represented on Fig. 9. The hygrometry gauge was out of order at the beginning of the test; it was reinstalled on April 23-25, 2012 (C on Fig. 9). Later, hygrometry dropped from Φ = 75% RH to Φ = 72% RH. During this 30-month period, temperature fluctuations are smaller than ± 0.04°C (Fig. 9), except during short periods when members of the staff work in the gallery [April 23-25, 2012 (C on Fig. 9), November 7, 2013 (D on Fig. 9) and June 5, 2012 (E on Fig. 9)]. A jump can be observed on the strain-vs-time curve (Fig. 9, point F); its origin is unknown. 4.2. Temperature Fluctuations Strain fluctuations are well correlated with temperature fluctuations, as expected. On Fig. 10, the average temperature rate and average strain rate over a 20-day period, beginning on May 21, 2013 were subtracted from the actual measured values. Both curves were smoothened. A time-lag can be observed, and the empirical correlation coefficient is smaller than the coefficient of the thermal expansion of salt. This can be explained as follows. The sample diameter is D 70 mm. The thermal diffusivity of salt is k 3 106 m 2 /s , and the characteristic time for thermal conduction is tc D 2 k (or half an hour). It takes several hours for the sample to reach thermal equilibrium with the gallery air, and a time lag of approximately 4 to 5 hours can be observed. As the period of thermal fluctuations is one day, approximately, full thermal equilibrium with room temperature never is reached, and the apparent coefficient of thermal expansion ( T ) is smaller than its actual value. 4.3. Transient Strains Displacement offsets during electrical cuts (on April 2325, 2012 (point C) and June 5, 2012 (point E) were erased on Fig. 9, making the strain-vs-time curve smooth, as explained in Section 3.2. In addition, immediately after any load change (A, B and D on Fig. 9), large transient strains develop. The Norton-Hoff model fails to account for the rheological transient behavior: only steady-state behavior is described. Munson and Dawson [12] proposed an extension of the Norton-Hoff law that accounts for transient rheological behavior. When a constant deviatoric stress, σ, is applied to a sample, the viscoplastic strain rate is the sum of the steady-state rate plus a transient rate: vp ss (4) where vanishes to zero when steady-state is reached — i.e., when *t () h(T ) m ; m 3 is typical. This model predicts that, when applied stress is multiplied by 2, the cumulated transient strain, *t , is multiplied by 8. In other words, the cumulated transient strain should be much larger at the end of the second phase (when 0.2 MPa) than at the end of the first phase (when 0.1 MPa). In fact, it is not. It is believed that, at the beginning of the test, the upper and lower faces of ARMA 14-7052 the sample were not perfectly flat, and small irregularities were crushed when the first load is applied. 4.4. Steady-State Strain Rates The three phases of the test are shown on Fig. 11. For easy comparison, the origin of time is when each of the phases begins. After seven months (210 days), it can be assumed that, in principle, steady state is reached. Fig.11. Strains during the three phases of the test. 5. CONCLUSIONS The findings at this stage can be summarized as follows. 1 Creep tests were performed on an Avery Island salt sample. The testing device was set in a 160m-deep room of the Varangéville salt mine, where temperature and hygrometry are quite stable. Three phases were managed. The applied axial load was 0.1 MPa, 0.2 MPa and 0.3 MPa, respectively. Each phase was at least eight months long. 2 During each phase, after the load is changed, a several-month-long transient creep phase is exhibited. The cumulated transient creep is larger during the first phase — in fact, it is likely that small irregularities of the upper and lower faces of the samples were crushed during the first phase. 3 Hygrometry is quite high (close to 75% RH). It is believed, however, that the hygrometry influence is small: the applied loads are quite small, and no microfracturation is created, impeding any effective penetration of water vapor inside the sample. 4 After an 8-month-long period, the steady-state strain rate is of the order of ss 12 1 ss 12 1 1.110 s to 1.7 10 s . These strain rates are much greater than what can be extrapolated from standard creep tests performed under higher deviatoric stress. Strain rates are an increasing function of the applied stress, but no clear constitutive relation (a value of the exponent n of the power law) can be inferred from this test. Fig. 10. Temperature and strain fluctuations during a 20-day long period. The most striking result is that steady-state strain rates are faster than the strains that can be extrapolated from tests performed at larger stresses by a factor of 1001000. The strain rates are 1.1 1012 s 1 (when 0.1 MPa ), 1.6 1012 s 1 (when 0.2 MPa ) and 1.7 1012 s 1 (when 0.3 MPa ). They are of the same order of magnitude as the rate ( ss 1.4 1012 s 1 when 0.108 MPa ) observed during similar tests performed 10 years before ([8]) on an Etrez salt sample (rather than Avery Island salt) in the same gallery, when hygrometry was 55% RH instead of 75% RH. In addition to possible differences between the respective behaviors of the Avery Island and Etrez salts, this small difference in steady-state creep rates, which is not consistent with the drastic change in air humidity, suggests that strain rate might be less sensitive to air humidity when the applied deviatoric stress is small and when no dilation of the sample is expected, making the sample inaccessible to water vapor, [11], [13]. The steady-state strain rates are an increasing function of the applied stress, but no clear value of the exponent of the power law can be inferred from the three tests. The third phase is longer (17 months instead of 9 months). It can be observed that strain rates become slower during the five last months of this phase. This change in strain rate is puzzling. Temperature did not change during this period (Fig. 9). Although a drop in hygrometry can be observed, it was noted above that the hygrometry influence seems to be small. ARMA 14-7052 ACKNOWLEDGEMENTS rheology of rocksalt during long-term deformation. In Proc. 6th Conf. Mech. Beh. of Salt, 149–158. London: Taylor & Francis Group. Special Thanks to Kathy Sikora. 7. Bérest P. 2013. The mechanical behavior of salt and salt caverns. In Proc. Eurock 2013, 17-30. Rotterdam: Balkema. Hunsche, U. 1988. Measurement of creep in rock salt at small strain rates. In Proc. 2nd Conf. Mech. Beh. of Salt, eds. H. Reginald Hardy, Jr. and Michael Langer, 187196. Clausthal-Zellerfeld: Trans Tech Pub. 8. Langer M. 1984. The rheological behaviour of rock salt. In Proc. 1st Conf. Mech. Beh. of Salt, 201–240. Clausthal-Zellerfeld, Germany: Trans Tech Pub. Bérest, P., P.A. 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