energies

Article Study on the Tri-axial Time-Dependent Deformation and Constitutive Model of Glauberite Salt Rock under the Coupled Effects of Compression and Dissolution

Mengtao Cao 1,2,* and Shunde Yin 3 1 College of Mining Engineering, Taiyuan University of Technology, Taiyuan 030000, China 2 Key Laboratory of In-Situ Modified Mining of Ministry of Education, Taiyuan University of Technology, Taiyuan 030000, China 3 Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, ON 50701, Canada; [email protected] * Correspondence: [email protected]; Tel.:+86-187-3413-2863



Received: 22 January 2020; Accepted: 28 March 2020; Published: 8 April 2020


Abstract: Solution mining for glauberite salt rock is a long-term process that takes several years to several decades. Therefore, deposit deformations and subsidence of ground surfaces are time-dependent deformation problems that should consider the effect of water dissolution. In order to investigate the time-dependent deformation characteristics of glauberite salt rock, tri-axial time-dependent deformation tests were conducted under the condition of 4 MPa confining pressure and 5 MPa axial pressure with infiltration pressures of 3, 2, 1, and 0 MPa, respectively, and the micro-CT scan system was used to scan the glauberite specimens before and after the experiment in order to study the fracture evolution inside the specimen, and a damage constitutive model was established to fit the time-dependent deformation curves based on the damage mechanics and effective stress principle. To simulate the solution mining process, the time-dependent deformation process of glauberite salt rock was divided into three stages: hydraulic connection stage, water-saturated stage, and drainage stage. The results demonstrate that the hydraulic connection time for glauberite salt rock decreases with increasing infiltration pressure. The time-dependent deformations of the specimens at the hydraulic connection and saturated-water stages are significantly affected by the effective stress and continual mineral dissolution. At the drainage stage, the softening degree of the solid skeleton mechanical properties, which is caused by the dissolution effect and infiltration pressure loading history, decides the deformation of glauberite salt rock. In addition, the degree of softening inside glauberite salt rock caused by dissolution becomes more severe with increasing infiltration pressure using the micro-CT scan technology. Lastly, the time-dependent damage constitutive model is able to describe the tri-axial time-dependent deformation behavior of glauberite salt rock, and the variations of time-dependent deformation parameters further indicate the damage evolution of the solid skeleton mechanical properties of glauberite caused by infiltration pressure and dissolution effect.

Keywords: glauberite salt rock; time-dependent deformation; rock mechanics; dissolution effect; damage constitutive model

1. Introduction Glauberite is an unusual salt rock that mainly consists of sodium sulfate and calcium sulfate, i.e., Na2Ca(SO4)2 (anhydrous sodium calcium sulfate), in an anhydrous state [1]. Sodium sulfate is highly soluble in water with the solubility of 192.32 g per liter of water at saturation (20 ◦C), while calcium sulfate has a low water solubility with 2.02 g per liter of water at saturation (20 ◦C) [2]. Sodium sulfate, an important raw material in the chemical industry, can be extracted from glauberite deposits using

Energies 2020, 13, 1797; doi:10.3390/en13071797 www.mdpi.com/journal/energies Energies 2020, 13, 1797 2 of 20 solution mining technologies [3]. Solution mining, commonly used in salt fields, involves drilling wells to subsurface ores and dissolving the ore to obtain the minerals by injecting fresh water and pumping the salt solution to the surface [4]. The solution mining method has been reported to be safer and more economical compared to conventional underground mining. Solution mining is a long process of ore body dissolution that takes several years to decades. Based on the principle of effective stress, subtracting the total stress to pore water pressure gives the effective stress. Consequently, during this process, ore bodies are subjected to mineral dissolution and dynamic effective stress changes, and time-dependent behaviors of ore bodies occur with the gradual evolution of the pore structure. The progress of dissolution of the ore bodies affects the production efficiency of solution mining, and the time-dependent dissolution process also triggers deformations and fractures of ore bodies and the subsequent surface subsidence [5,6]. For the long-term stability assessment of solution mining, it is essential to understand the time-dependent behaviors of glauberite salt rock. The time-dependent properties of salt rock are of fundamental importance in predicting long-term evolution of strategic storage caverns for oil and gas and as well as nuclear waste disposal in China due to their characteristics of low permeability and porosity [7,8]. As salt rock exhibits pronounced creep characteristics even under relatively low loads, many efforts have been devoted to investigating the mechanical creep behavior of salt rock through laboratory experiments and theories [9–15]. Creep test is the direct method to research the creep properties of salt rock, and it has been adopted by scholars to observe the creep deformation and rock characteristics. References [13,14] analyzed the influence of confining pressure on the creep properties based on larger amounts of uniaxial and tri-axial experimental data. The author of [16] compared and analyzed the creep deformation of glauberite, anhydrite, and argillaceous salt rock based on tri-axial creep test. Considering the factor of temperature effect, [17] found that temperature strongly influences the creep properties of salt. To better understand the creep mechanism and predict the long-term deformation and stability of salt caverns, lots of constitutive equations have been established and proposed. The authors of [18] built a new elastic/viscoplastic transient creep model according to the true tri-axial test results. Reference [19] proposed an improved Kelvin–Hooke model to precisely describe the creep behavior of thenardite, while [20] established the creep model to investigate the time-dependent behavior of salt based on fractional calculus. Particularly, in the process of solution mining from glauberite ore, original impermeable glauberite gradually evolves into a porous medium because dissolution of glauberite leads to removal of sodium sulfate in solution and precipitation of gypsum (CaSO4.2H2O) or anhydrite (CaSO4) depending on total solution composition and temperature. The authors of [3] studied the evolution characteristics of pores and residual porous skeleton of glauberite dissolved in fresh water using the micro-computed tomography. Reference [21] studied the microstructure development of glauberite under the influence of chemistry and temperature, while [1] investigated the chemical dissolution and seepage characteristics of glauberite salt rock. As the evolving porous medium is generated, it results in the variation of all the mechanical properties with interior structure alteration, such as strength, deformation moduli, and acoustic velocity. In addition, pressure solution also plays an important role in the deformation mechanism [22] of time-dependent deformation with the presence of saturated solution [2]. It involves the dissolution of grains at highly stressed boundaries, diffusion of material through the grain boundary fluid or fluid-filled channel network, and crystallization of material at interfaces under low normal stress. These three transport steps occur in series, and the slowest controls the rate of creep [23–26]. Due to the coupled effects of a continuously evolving solid skeleton and pore water pressure, the time-dependent characteristics of glauberite salt rock are very unusual. Consequently, the dissolution and erosion effects of water need to be considered when investigating the time-dependent characteristics of glauberite salt rock, which has theoretical and practical engineering significance. These previous studies mentioned above covered the mechanical characteristics of creep in salt rock that are influenced by various loading conditions (axial pressure, confining pressure, and loading rate) and temperatures. However, up to now, little research has been conducted on the Energies 2020, 13, x FOR PEER REVIEW 3 of 20

Energies 2020, 13, 1797 3 of 20 loading rate) and temperatures. However, up to now, little research has been conducted on the time- dependent deformation of salt rock that is coupled to the hydraulic phenomenon under tri-axial stresstime-dependent conditions deformation [26,27,28], especially of salt rock for that glauberite is coupled salt torock. the hydraulic phenomenon under tri-axial stressIn conditions this paper, [26 to– 28investigate], especially the for time-dependent glauberite salt rock.deformation characteristics of glauberite salt rock Inunder this the paper, combined to investigate compression the time-dependent and dissolutio deformationn effects, tri-axial characteristics time-dependent of glauberite deformation salt rock testsunder were the combinedconducted compression on glauberite and salt dissolution rock with ethreeffects, different tri-axial time-dependentinfiltration pressures. deformation These tests consistedwere conducted of three on stages glauberite in sequence: salt rock hydraulic with three connection different infiltrationstage, water-saturated pressures. These stage, tests and consisteddrainage stage.of three Based stages on in the sequence: concept hydraulicof damage connection mechanics stage,and effective water-saturated stress principle stage, and to the drainage generalized stage. KelvinBased onmodel, the concepta time-dependent of damage da mechanicsmage constitutive and effective model stress was propos principleed to to explain the generalized the results Kelvin from thesemodel, laboratory a time-dependent tests. These damage experimental constitutive and theo modelretical was studies proposed are toexpected explain to the provide results a from reference these forlaboratory analyzing tests. cavern These stability experimental and surface and subsidence theoretical studiesin solution are mining. expected to provide a reference for analyzing cavern stability and surface subsidence in solution mining. 2. Materials and Methods 2. Materials and Methods 2.1. Preparation of Specimen for Tri-axial Compression Test 2.1. Preparation of Specimen for Tri-axial Compression Test The tested glauberite salt rock samples were from a glauberite deposit approximately 200 m undergroundThe tested located glauberite in Pengshan, salt rock samplesSichuan wereProvince, from aChina. glauberite The depositirregular approximately original glauberite 200 m samplesunderground were locatedblasted inand Pengshan, transported Sichuan to the Province, laboratory China. of Taiyuan The irregular University original of glauberiteTechnology; samples then, fourwere standard blasted and column transported specimens to the with laboratory dimensions of Taiyuan of Φф University50 × 100 mm of Technology; (Figure 1) then,were fourformed standard from column specimens with dimensions of Φ50 100 mm (Figure1) were formed from these samples. these samples. The physical and mechanical ×properties of glauberite salt rock are shown in Table 1. The physical and mechanical properties of glauberite salt rock are shown in Table1. In the glauberite In the glauberite specimens, the contents of sodium sulfate (Na2SO4) and calcium sulfate (CaSO4) are 40specimens, wt% and the35 wt%, contents respectively. of sodium As sulfate shown (Na in 2TableSO4) and2, the calcium main components sulfate (CaSO of 4the) are tested 40 wt% glauberite and 35 sampleswt%, respectively. are glauberite As shown(anhydrous in Table sodium2, the calcium main components sulfate), quartz, of the chlorite, tested glauberitemica, montmorillonite, samples are andglauberite illite [21]. (anhydrous sodium calcium sulfate), quartz, chlorite, mica, montmorillonite, and illite [21].

(a) (b)

Figure 1 1.. The glauberite specimens before and after testing. ( (aa)) the the specimens before before testing; ( (b)) the specimens after testing.

TableTable 1. Basic physical properties and solubility of glauberite at room temperature (20 ◦°C).C).

SolubilitySolubility of Sodium of SolubilitySolubility of Calcium of Elastic Modulus/GPa UniaxialUniaxial Compressive Density/(g/cm3) Elastic Density/(g/cm3) Sulfate/(g/LSodium) [2] Sulfate/(Calciumg/L) [2] [29] Strength/MPaCompressive [29] Modulus/GPa [29] 2.78 192.32Sulfate /(g/L) [2] Sulfate 2.02 /(g/L) [2] 4.0 Strength 16.0/MPa [29] 2.78 192.32 2.02 4.0 16.0 Table 2. The composition of glauberite ore [21].

Glauberite Quartz TableChlorite 2. The compositionMica of glauberiteMontmorillonite ore [21]. Illite Others 75% 4% 5% 4% 2% 6% 4% Glauberite Quartz Chlorite Mica Montmorillonite Illite Others

2.2. Microstructural75% Observation 4% of 5%Glauberite 4% 2% 6% 4% Microstructure analysis was carried out by means of optical microscopy performed on thin and 2.2. Microstructural Observation of Glauberite ultra-thin sections of epoxy-impregnated samples. Examination of the material showed that the glauberiteMicrostructure minerals exist analysis as granular was carried plate-shaped out by means texture of with optical the microscopy grain size mostly performed between on thin 0.5 mm and andultra-thin 2 mm, sections and they of epoxy-impregnatedare distributed non-uniformly samples. Examination in the sample of the(Figure material 2). showedIn addition, that the

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Energiesglauberite 2020, minerals13, x FOR PEER exist REVIEW as granular plate-shaped texture with the grain size mostly between4 of 0.5 20 mmEnergies and 2020 2, mm,13, x FOR and PEER they REVIEW are distributed non-uniformly in the sample (Figure2). In addition,4 theof 20 observation of crystal boundary structure structure and and crysta crystall size size distribution distribution was was done done by by reflected reflected light- light- observation of crystal boundary structure and crystal size distribution was done by reflected light- and scanning electron microscopy (SEM), and the glauberite sample’s surface was carefully ground and scanning electron microscopy (SEM), and the glauberite sample’s surface was carefully ground and polished before the examination. Figure 33 showsshows thethe shapeshape andand sizesize ofof glauberiteglauberite crystals,crystals, withwith and polished before the examination. Figure 3 shows the shape and size of glauberite crystals, with differentdifferent scale bars, obtained inin thethe sample.sample. different scale bars, obtained in the sample.

. . FigureFigure 2. TypicalTypical optical microstructure of glauberite rock salt samples. Figure 2. Typical optical microstructure of glauberite rock salt samples.

Figure 3. GlauberiteGlauberite crystal crystal images by scanning electron microscopy (SEM). Figure 3. Glauberite crystal images by scanning electron microscopy (SEM). 2.3. Testing Testing ApparatusApparatus 2.3. Testing Apparatus 2.3.1. Tri-axial Tri-axial Apparatus 2.3.1. Tri-axial Apparatus As shown in Figure 44,, thethe mainmain apparatusapparatus usedused forfor thisthis testtest isis thethe tri-axialtri-axial apparatusapparatus inin thethe KeyKey As shown in Figure 4, the main apparatus used for this test is the tri-axial apparatus in the Key Laboratory of In-Situ Modified Modified Mining Mining of of Ministry Ministry of of Education Education at at Taiyuan Taiyuan University University of of Technology, Technology, Laboratory of In-Situ Modified Mining of Ministry of Education at Taiyuan University of Technology, and the rock specimen with 50 mm diameter an andd 100 mm height was enclosed in a rubber sealing and the rock specimen with 50 mm diameter and 100 mm height was enclosed in a rubber sealing jacket between porous discs at top and bottom of the specimen. specimen. The The confining confining pressure pressure was was provided provided jacket between porous discs at top and bottom of the specimen. The confining pressure was provided in the cell pressurepressure toto causecause three-dimensionalthree-dimensional deformation deformation of of the the rock rock specimen, specimen, with with the the precision precision of in the cell pressure to cause three-dimensional deformation of the rock specimen, with the precision of0.01 0.01 MPa. MPa. The The axial axial load load was was applied applied through through the the loading loading ram, ram, keeping keeping the the loading loading constant constant with with the of 0.01 MPa. The axial load was applied through the loading ram, keeping the loading constant with the precision of 0.01 KN. The pore water pressure was applied on the bottom of the specimen through the precision of 0.01 KN. The pore water pressure was applied on the bottom of the specimen through the porous disc, and the water escaped at the top through the whole length of the specimen. the porous disc, and the water escaped at the top through the whole length of the specimen.

Energies 2020, 13, 1797 5 of 20 precision of 0.01 KN. The pore water pressure was applied on the bottom of the specimen through the Energies 2020, 13, x FOR PEER REVIEW 5 of 20 porous disc, and the water escaped at the top through the whole length of the specimen.

Figure 4. Tri-axial apparatus (schematic). 1—Loading piston, 2—Pore pressure outlet, 3—Valve, 4—Chamber, 5—Porous disc, 6—Confining pressure, 7—Rubber sealing jacket, 8—Rock specimen, Figure9—Air 4 release. Tri-axial outlet, apparatus 10—Cell (schematic). pressure inlet, 1— 11—PoreLoading piston, pressure 2— inlet,Pore 12—Dial pressure gauge. outlet, 3—Valve, 4— Chamber, 5—Porous disc, 6—Confining pressure, 7—Rubber sealing jacket, 8—Rock specimen, 9— 2.3.2. X-ray Micro-Computed Tomography (µCT) Air release outlet, 10—Cell pressure inlet, 11—Pore pressure inlet, 12—Dial gauge. The micro-CT scan system at Taiyuan University of Technology [30,31] was used to scan all the 2.3.2.glauberite X-ray specimens Micro-Computed before and Tomography after the experiment, (μCT) and the scans were conducted with 400 frames and aThe superimposed micro-CT scan frame system rate at of Taiyuan 2 fps, and University there were of 1500Technology scanned [30,31] layers. was In this used experiment, to scan all the parametersglauberite specimens used in the before CT procedureand after the were experime the scanningnt, and the voltage scans and were electric conducted current, with which 400 frames were and70 Kv a superimposed and 90 µA, respectively. frame rate The of 2 magnification fps, and there was were 7.2 1500 times, scanned and the layers. middle In layer this experiment, of the specimen the wasparameters selected used to analyze in the CT the procedure evolution were of fractures the sca insidenning voltage the specimens and electric under current, the coupled which e wereffects 70 of Kvcompression and 90 μA, and respectively. dissolution. The magnification was 7.2 times, and the middle layer of the specimen was selected to analyze the evolution of fractures inside the specimens under the coupled effects of 2.4. Experimental Methodology compression and dissolution. To investigate the long-term behavior of glauberite salt rock under the coupled effect of stress and 2.4.dissolution, Experimental three Methodology infiltration pressures were tested with the same axial and confining pressures as shown in Table3. The axial pressure and circular confining pressure loaded on the column specimens To investigate the long-term behavior of glauberite salt rock under the coupled effect of stress were 5.0 and 4.0 MPa, respectively. The infiltration pressures loaded on the bottom of the specimens and dissolution, three infiltration pressures were tested with the same axial and confining pressures with fresh water were 3.0, 2.0, 1.0 and 0 MPa, and the specimen with 0 MPa was a dry control as shown in Table 3. The axial pressure and circular confining pressure loaded on the column experiment, which can be regarded as an example of time-dependent deformation characteristics specimens were 5.0 and 4.0 MPa, respectively. The infiltration pressures loaded on the bottom of the without infiltration pressure. In addition, the micro-CT scan system was used to scan the glauberite specimens with fresh water were 3.0, 2.0, 1.0 and 0 MPa, and the specimen with 0 MPa was a dry specimens before and after the experiment in order to study the fracture evolution inside the specimen. control experiment, which can be regarded as an example of time-dependent deformation characteristics without infiltration pressure. In addition, the micro-CT scan system was used to scan the glauberite specimens before and after the experiment in order to study the fracture evolution inside the specimen.

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Table 3. Experimental conditions of samples tested.

Axial Confining Infiltration Number Diameter/mm Height/mm Pressure/MPa Pressure/MPa Pressure/MPa 0 49.80 99.90 5 4 0 1 49.76 99.94 5 4 1 2 49.78 99.96 5 4 2 3 49.78 99.92 5 4 3

Under the designed compression and infiltration pressure, the time required for pore and fissure coalescence in the specimen was recorded along with the axial deformation. During the time-dependent deformation experiments, the axial deformations of the specimens were recorded every half hour using a dial gauge with a resolution of 1 % mm, and the data were automatically collected and recorded by a computer. The procedures of the experiment were to (1) apply the designed axial stress and confining pressure on the specimen; (2) apply the infiltration pressure on the bottom end of specimen and record the axial deformation as zero simultaneously; (3) keep the pore pressure outlet closed when there was water flow from the outlet. This process is the hydraulic connection stage; (4) keep the state constant until 500 h. This process is the water-saturated stage; (5) unload the water pressure until 720 h. This process is the drainage stage. The reason for dividing this experiment into three stages is to model the in-situ solution mining process of glauberite ore. The hydraulic connection stage is the process where water is injected from an injection well into the glauberite deposit, but there is no hydraulic connected channel between the injection well and the production well; this is called the water-saturated stage after the hydraulic connection channel is created. The difference between the two previous stages is that the water-saturated stage has hydraulic connected channels in the glauberite ore, which indicates that the glauberite is in a water-saturated state. The drainage stage is to investigate the time-dependent deformation of glauberite after solution mining.

3. Results

3.1. Time-Dependent Deformation During Loading Figure5 shows the strain evolution of specimens 0, 1, 2, and 3 over time during the entire experiment. Within the figures, the letters represent different stages in the time-dependent experiments: “a” denotes hydraulic connection stage; “b” and “c” represent the water-saturated stage and drainage stage, respectively. The numbers of “1”, “2”, and “3” in “a-1”, “b-1”, “b-2”, etc., are specimens 1, 2, and 3, respectively. In these tests, specimen 1 was not hydraulically connected; therefore, it did not experience the water-saturated stage, unlike specimens 2 and 3. Figure5 shows that the deformations of specimens 0, 1, 2, and 3 generally increased with time during different stages. However, the degree of weakening of glauberite salt rock under the dissolution and erosion effects and the effective stress varied with different infiltration pressures, indicating obvious difference in the deformation mechanisms and characteristics of specimens 1, 2, and 3 at different stages. Comparing the time-dependent deformations of specimens 1, 2, and 3 with that of specimen 0, it is obvious that infiltration pressure causes degradation of mechanical properties of the glauberite specimen. At the hydraulic connection stage and water-saturated stage, the time-dependent deformations of the specimens were dominated by the effective stress and the degree of weakening of the solid skeleton caused by the dissolution effect. In addition, at the drainage stage, the time-dependent deformations of specimens 1, 2, and 3 were greater compared to previous stage for the same amount of time. Energies 2020, 13, 1797 7 of 20 Energies 2020, 13, x FOR PEER REVIEW 7 of 20

Figure 5. StrainStrain curves of specimens 0, 1, 2, and 3 during the entire loading process. 3.2. Time-Dependent Deformation During the Hydraulic Connection Stage 3.2. Time-Dependent Deformation During the Hydraulic Connection Stage Figure6 shows the time-dependent deformation strain of specimens 2 and 3 over time for Figure 6 shows the time-dependent deformation strain of specimens 2 and 3 over time for infiltration pressures of 2.0 and 3.0 MPa, respectively. It indicates that the two specimens have different infiltration pressures of 2.0 and 3.0 MPa, respectively. It indicates that the two specimens have time-dependent properties with different infiltration pressures. For specimen 3 with a dissolution and different time-dependent properties with different infiltration pressures. For specimen 3 with a infiltration pressure of 3 MPa, a sharp increase in strain occurred during the time-dependent test even dissolution and infiltration pressure of 3 MPa, a sharp increase in strain occurred during the time- though the strain was small. The axial strain of the specimen increased from 0.12% to 0.135% until the dependent test even though the strain was small. The axial strain of the specimen increased from 0.12% cylinder specimen was hydraulically connected from the bottom to the top face. This behavior showed to 0.135% until the cylinder specimen was hydraulically connected from the bottom to the top face. the gradual softening process of glauberite during the dissolution of the dissolvable components. This behavior showed the gradual softening process of glauberite during the dissolution of the The original glauberite was intact and hard; however, as the dissolvable components were dissolved dissolvable components. The original glauberite was intact and hard; however, as the dissolvable with high infiltration pressure, even though this dissolution may be trivial, the glauberite gradually components were dissolved with high infiltration pressure, even though this dissolution may be became porous and soft with cohesion decreased. The original solid and stable structure quickly trivial, the glauberite gradually became porous and soft with cohesion decreased. The original solid collapsed when this softening accumulated to a certain level. The picture of specimen 3 in Figure1b and stable structure quickly collapsed when this softening accumulated to a certain level. The picture after infiltration shows that the deformation mainly occurred at the bottom part where the dissolution of specimen 3 in Figure 1b after infiltration shows that the deformation mainly occurred at the bottom and softening effects are stronger. The water dissolution and softening effects vary along the specimen part where the dissolution and softening effects are stronger. The water dissolution and softening during the infiltration and dissolution process; these effects are obviously stronger at the bottom than at effects vary along the specimen during the infiltration and dissolution process; these effects are the top of the specimen even at a 3 MPa infiltration pressure. This result indicates that high infiltration obviously stronger at the bottom than at the top of the specimen even at a 3 MPa infiltration pressure. pressures and long dissolution times are needed to increase the recovery of the mineral components This result indicates that high infiltration pressures and long dissolution times are needed to increase from glauberite salt rock. the recovery of the mineral components from glauberite salt rock. For specimen 2, the infiltration pressure was 1 MPa lower than that of specimen 3. For specimen 2, the infiltration pressure was 1 MPa lower than that of specimen 3. The time- The time-dependent deformation strain of specimen 2 gradually increased to 0.127% in 137.5 h, dependent deformation strain of specimen 2 gradually increased to 0.127% in 137.5 h, and it sharply and it sharply increased to 0.137% before the specimen was hydraulically connected. Compared increased to 0.137% before the specimen was hydraulically connected. Compared with the 0.135% with the 0.135% strain of specimen 3, the strain of specimen 2 was approximately the same as that of strain of specimen 3, the strain of specimen 2 was approximately the same as that of specimen 1, but specimen 1, but the time needed for specimen 2 was 53 h more than that for specimen 3. This difference the time needed for specimen 2 was 53 h more than that for specimen 3. This difference in deformation in deformation between the two specimens reflects the effect of the infiltration pressure on the behavior between the two specimens reflects the effect of the infiltration pressure on the behavior of glauberite. of glauberite. This effect of the infiltration pressure is also shown in the time required to become This effect of the infiltration pressure is also shown in the time required to become hydraulically hydraulically connected for the two specimens. At a 3 MPa infiltration pressure, specimen 3 became connected for the two specimens. At a 3 MPa infiltration pressure, specimen 3 became hydraulically hydraulically connected after 84.5 h; at a 2 MPa infiltration pressure, specimen 2 required an additional connected after 84.5 h; at a 2 MPa infiltration pressure, specimen 2 required an additional 53 h. 53 h. The infiltration pressure for specimen 1 was 1 MPa, which was lower than that of both specimens The infiltration pressure for specimen 1 was 1 MPa, which was lower than that of both specimens 2 and 3. At this low pressure, specimen 1 was not hydraulically connected during the 500 h at the 2 and 3. At this low pressure, specimen 1 was not hydraulically connected during the 500 h at the design tri-axial stress and infiltration pressure. The strain gradually increased to 0.108% in 350 h and design tri-axial stress and infiltration pressure. The strain gradually increased to 0.108% in 350 h then remained approximately constant from 350 to 500 h (Figure 6b). Compared with the strains of andspecimens then remained 2 and 3, approximatelythe strains of specimens constant from 1 and 350 0 were to 500 much h (Figure smaller6b). in Compared the samewith timethe (in strains140 h). ofThus, specimens the infiltration 2 and 3, pressure the strains and of dissolution specimens ti 1me and greatly 0 were affect much the smaller water seepage in the same and timemineral (in 140dissolution h). Thus, inside the infiltrationthe glauberite. pressure Higher and injection dissolution and timeinfiltration greatly pressures affect the seem water favorable seepage andfor hydraulic coalescence of glauberite in a larger domain during solution recovery.

Energies 2020, 13, 1797 8 of 20 mineral dissolution inside the glauberite. Higher injection and infiltration pressures seem favorable Energiesfor hydraulic 2020, 13, x coalescence FOR PEER REVIEW of glauberite in a larger domain during solution recovery. 8 of 20

(a)

(b)

FigureFigure 6. 6. StrainsStrains of of specimens specimens 3, 3, 2, 2, 1, 1, and and 0 0 over over time time with with different different infiltration infiltration pressure pressure (enlarged (enlarged sectionsection of of Figure Figure 55).). ( (aa) )Strains Strains of of specimens specimens 2 2 and and 3 3over over time time for for 3.0 3.0 and and 2.0 2.0 MPa MPa infiltration infiltration pressure, pressure, respectively.respectively. In In this this case, case, specim specimensens 2 2and and 3 3became became hydraulically hydraulically connected connected after after 84.5 84.5 h and h and 137.5 137.5 h, respectively.h, respectively. (b) (Strainsb) Strains of ofspecimens specimens 1 and 1 and 0 over 0 over time time with with 1.0 1.0 and and 0 0MPa MPa infiltration infiltration pressure, pressure, respectively.respectively. In In this this case, case, specimen specimen 1 1 was was no nott hydraulically hydraulically connected connected during during the the 500 500 h h test.

3.3.3.3. Time-Dependent Time-Dependent DeformationDeformation at at the the Saturated-Water Saturated-Water Stage Stage SpecimensSpecimens 2 2 and and 3 were exposed toto infiltrationinfiltration pressurespressures of of 2 2 and and 3 3 MPa, MPa, respectively, respectively, resulting resulting in insmall small eff effectiveective stresses stresses on on the the solid solid skeleton skeleton of the of the two two specimens. specimens. For For specimen specimen 3 with 3 with a uniform a uniform pore poredistribution, distribution, the e fftheective effective stress st onress the on solid the skeleton solid skeleton was 1–2 was MPa 1– when2 MPa the when specimen the specimen was subjected was subjectedto a pore pressureto a pore of pressure 3 MPa. Underof 3 MPa. the Under tri-axial the stress tri-axial state, stress little deformationstate, little deformation and failure occurredand failure for occurredspecimen for 3. Asspecimen a result, 3. the As axiala result, strain the gradually axial strain increased gradually to 0.137%increased from to 0.169%0.137%over from 414 0.169% h, and over the 414strain h, and increment the strain was increment 0.032%, as was shown 0.032%, in Figure as shown7. As in previously Figure 7. mentioned,As previously a distinct mentioned, di fference a distinct was differenceshown in thewas degree shown of in softening the degree and of the softening deformation and the of deformation the specimen of as the a result specimen of the as di aff erenceresult of in thethe difference dissolution in time. the dissolution Since the bottom time. Since of the the specimen bottom of experiences the specimen dissolution experiences for adissolution longer time, for the a longerpores weretime, relatively the pores better were developedrelatively better in this developed area, and the in degreethis area, of softeningand the degree by dissolution of softening reaction by dissolutionwas greater. reaction In contrast, was at greater. the top ofIn acontrast, specimen, at the the pores top wereof a poorlyspecimen, developed the pores due towere the poorly shorter developeddissolution due time, to andthe theshorter degree disso oflution softening time, due and to the dissolutiondegree of softening reaction wasdue weaker.to the dissolution Therefore, reaction was weaker. Therefore, the difference in porosity caused by water dissolution and erosion directly led to differences in the effective stress on the solid skeleton of the glauberite. The top part of the specimens had undeveloped pores and small or nonexistent pore pressure; in addition, the effective stress on the solid skeleton was smaller. Consequently, the combined effect of the effective stress and the solid skeleton characteristics was prone to producing deformation or shear failure.

Energies 2020, 13, x FOR PEER REVIEW 9 of 20

The time-dependent deformation strain of specimen 2 increased from 0.137% to 0.179% in 361 h; the increment was 0.042%, which was larger than the strain of 0.032% for specimen 3. The infiltration pressureEnergies 2020 on, 13 specimen, 1797 2 was 1 MPa less than that on specimen 1; therefore, at the bottom of specimen9 of 20 2, the pores caused by water dissolution and erosion were smaller, and the degree of internal deterioration was relatively low. However, the effective stress on specimen 2 was larger than that on specimenthe difference 3, resulting in porosity in the caused closure by of waterpores inside dissolution the specimen and erosion and deformation directly led toof ditheff erencessolid skeleton. in the Consequently,effective stress onas theshown solid in skeleton Figure of6, thethe glauberite.strain of specimen The top part 2 over of the time specimens clearly reflects had undeveloped the time- dependentpores and small deformation or nonexistent characteristic. pore pressure; This figure in addition, shows that the etheffective deformations stress on theof the solid two skeleton specimens was atsmaller. this stage Consequently, are by the theaxial combined effective estressffect of on the the e ffsolidective skeleton stress and of glauberite the solid skeleton specimens, characteristics as well as thewas dissolution prone to producing and softening deformation effects. or shear failure.

Figure 7. Strain of specimens 2 and 3 over time at the saturated-water stage (enlarged section of Figure5 7.). Strain of specimens 2 and 3 over time at the saturated-water stage (enlarged section of Figure 5). The time-dependent deformation strain of specimen 2 increased from 0.137% to 0.179% in 361 h; 3.4.the incrementTime-Dependent was 0.042%, Deformation which was at the larger Drainage than the Stage strain of 0.032% for specimen 3. The infiltration pressureFigure on 8 specimen shows the 2 wasstrain 1 MPa of specimens less than that1, 2, on3 resp specimenectively, 1; over therefore, time at the bottomdesign tri-axial of specimen stress 2, afterthe pores unloading caused bythe water infiltration dissolution pressure. and erosion The ti wereme-dependent smaller, and deformation the degree of strain internal of deteriorationspecimen 1 increasedwas relatively from low. 0.108% However, to 0.118%, the effective and stressits stra onin specimen increment 2was was larger 0.01%; than the that time-dependent on specimen 3, deformationresulting in the strain closure of ofspecimen pores inside 2 increased the specimen from and 0.179% deformation to 0.206%, of the and solid its skeleton. strain increment Consequently, was 0.026%,as shown which in Figure is 260%6, the of strain the strain of specimen increment 2 over of sp timeecimen clearly 1. reflectsThe time-dependent the time-dependent deformation deformation strain ofcharacteristic. specimen 3 increased This figure from shows 0.169% that to the 0.218%, deformations and its ofstrain the twoincrement specimens was 0.048%, at this stage which are is by 480% the ofaxial that e ffofective specimen stress 1. on Contrasting the solid skeletonthe strain of increments glauberite of specimens, specimens as1, well2, and as 3 theat this dissolution stage clearly and reflectssoftening the e ffimportantects. influence of the loading effect at the previous stage on the internal structures of the three specimens. After removing the infiltration pressure, the pore water pressure inside 3.4. Time-Dependent Deformation at the Drainage Stage specimens 1, 2, and 3 decreased to 0 MPa, the axial effective stress increased to 5 MPa, and the effects of theFigure pore 8water shows pressure the strain and of dissolution specimens and 1, 2, er 3osion respectively, on the specimen over time vanished. at the design Therefore, tri-axial the threestress specimens after unloading changed the infiltrationfrom a state pressure. of tri-axial The st time-dependentress and infiltration deformation pressure strain to a tri-axial of specimen stress 1 stateincreased only. from 0.108% to 0.118%, and its strain increment was 0.01%; the time-dependent deformation strain of specimen 2 increased from 0.179% to 0.206%, and its strain increment was 0.026%, which is 260% of the strain increment of specimen 1. The time-dependent deformation strain of specimen 3 increased from 0.169% to 0.218%, and its strain increment was 0.048%, which is 480% of that of specimen 1. Contrasting the strain increments of specimens 1, 2, and 3 at this stage clearly reflects the important influence of the loading effect at the previous stage on the internal structures of the three specimens. After removing the infiltration pressure, the pore water pressure inside specimens 1, 2, and 3 decreased to 0 MPa, the axial effective stress increased to 5 MPa, and the effects of the pore water pressure and dissolution and erosion on the specimen vanished. Therefore, the three specimens changed from a state of tri-axial stress and infiltration pressure to a tri-axial stress state only.

Energies 2020, 13, x FOR PEER REVIEW 10 of 20

Energies 2020, 13, 1797 10 of 20 Energies 2020, 13, x FOR PEER REVIEW 10 of 20

Figure 8. Strain curves of specimens 1, 2, and 3 over time at the drainage stage (enlarged section of Figure 5).

3.5. Analysis of the Fractures before and after the Experiment Figure 9 shows the 750th layer’s CT scanned cross-section graph of the specimens 1, 2, and 3 before and after the experiment. The gray levels represent materials with different densities in these images,FigureFigure and 8. 8. StraintheStrain black curves curves parts of of specimens specimensare the pores1, 1, 2, 2, and andand 3 3 overfractures. over time time at atIt the thecan drainage drainage be clearly stage stage seen (enlarged (enlarged that sectionthe section connected of of fracturesFigureFigure of 5).5 ).all the specimens after the experiment all increase compared to that before the experiment, 3.5.which Analysis indicates of the that Fractures the glauberite before and salt after rock the gradua Experimentlly turns to a porous medium from an intact rock 3.5.under Analysis the condition of the Fractures of pore waterbefore pressure. and after Furthe the Experimentrmore, the lengths and apertures of the fractures Figure9 shows the 750th layer’s CT scanned cross-section graph of the specimens 1, 2, and 3 increaseFigure with 9 shows the increase the 750th of pore layer’s water CT pressure, scanned whichcross-section verifies graph that the of softeningthe specimens degree 1, caused 2, and by3 before and after the experiment. The gray levels represent materials with different densities in these beforethe dissolution and after increasesthe experiment. with larger The grayinfiltration levels repressure.present materials with different densities in these images, and the black parts are the pores and fractures. It can be clearly seen that the connected images,The and code the compiled black parts by MATLAB are the pores [31] was and used fractures. to reprocess It can thebe clearlyrebuilt CTseen images that the and connected obtain the fractures of all the specimens after the experiment all increase compared to that before the experiment, fracturesporosity. of Table all the 4 showsspecimens the porosityafter the experimentand weight aofll increaseglauberite compared specimens to thatafter before versus the before experiment, testing; which indicates that the glauberite salt rock gradually turns to a porous medium from an intact rock whichit can indicatesbe seen that that the the porosity glauberite of saltglauberite rock gradua specimenslly turns increase to a porous after testingmedium versus from beforean intact testing, rock under the condition of pore water pressure. Furthermore, the lengths and apertures of the fractures underwhile thethe conditionweight of ofglauberite pore water specimens pressure. decreases Furthermore, after testingthe lengths compared and ap toertures that before of the testing.fractures In increase with the increase of pore water pressure, which verifies that the softening degree caused by increaseaddition, with the theporosity increase and of weightpore water loss pressure,of glauberite which specimens verifies that with the higher softening infiltration degree pressurecaused by is the dissolution increases with larger infiltration pressure. thelarger. dissolution increases with larger infiltration pressure. The code compiled by MATLAB [31] was used to reprocess the rebuilt CT images and obtain the porosity. Table 4 shows the porosity and weight of glauberite specimens after versus before testing; it can be seen that the porosity of glauberite specimens increase after testing versus before testing, while the weight of glauberite specimens decreases after testing compared to that before testing. In addition, the porosity and weight loss of glauberite specimens with higher infiltration pressure is larger.

(a)

(a)

(b)

Figure 9. Cont.

(b)

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(c)

Figure 9. CT scan images (at 25 mm height for all the samples)samples) of glauberite specimens 1, 2, 3 before and after the experiment. ( a) specimen 1; ( b) specimen 2; (c) specimen 3.

The codeTable compiled 4. The by porosity MATLAB and mass [31] wasof glauberite used to specimen reprocess after the rebuiltversus before CT images testing. and obtain the porosity. Table4 shows the porosity and weight of glauberite specimens after versus before testing; it Porosity before Porosity after Increment of Mass before Mass after Mass canNumber be seen that the porosity of glauberite specimens increase after testing versus before testing, while Testing/% Testing/% Porosity/% Testing/g Testing/g Loss/g the weight1 of glauberite0.276 specimens 1.729 decreases after 1.453 testing compared199.45 to that before188.40 testing. In addition,11.05 the porosity2 and0.237 weight loss of glauberite 3.142 specimens 2.905 with higher198.26 infiltration pressure175.30 is larger.22.94 3 0.265 3.269 3.004 200.10 174.35 25.75 Table 4. The porosity and mass of glauberite specimen after versus before testing.

4. Discussion Porosity Porosity Increment Mass before Mass after Number before after of Mass Loss/g Testing/g Testing/g 4.1. Time-DependentTesting Deformation/% Testing Mechanism/% Porosity of Glau/%berite Salt Rock under the Coupled Effects of Compression1 and Dissolution 0.276 1.729 1.453 199.45 188.40 11.05 2 0.237 3.142 2.905 198.26 175.30 22.94 For glauberite salt rock that is moderately soluble in water, the effect of water on rock should 3 0.265 3.269 3.004 200.10 174.35 25.75 consider not only the effective stress principle but also the chemical effects of dissolution, pressure solution, and hydration of the Ca component of glauberite to form gypsum. When eroded by water, 4. Discussion the internal structures of glauberite salt rock undergo a variety of changes, such as the disintegration 4.1.and Time-Dependentswelling of illite Deformation and montmorillonite Mechanism ofby Glauberite absorbing Salt water, Rock the under dissolution the Coupled of E theffects sodium of Compression sulfate, and Dissolutionthe hydration of calcium sulfate. Therefore, under the combined effect of these factors, the mechanical properties of glauberite salt rock decrease, and infiltration pressure reinforces this For glauberite salt rock that is moderately soluble in water, the effect of water on rock should decrease effect. When saturated by high infiltration pressure over a long period of time, the mutual consider not only the effective stress principle but also the chemical effects of dissolution, pressure reinforcing of dissolution and seepage cause all or some of the sodium sulfate to dissolve and a larger solution, and hydration of the Ca component of glauberite to form gypsum. When eroded by water, the number of pores to emerge between the solid matrix of calcium sulfate. In this way, glauberite salt internal structures of glauberite salt rock undergo a variety of changes, such as the disintegration and rock changes from a dense rock with low permeability to a porous medium with a main solid skeleton swelling of illite and montmorillonite by absorbing water, the dissolution of the sodium sulfate, and of a calcium sulfate matrix; in addition, its mineral components and structures vary, resulting in the hydration of calcium sulfate. Therefore, under the combined effect of these factors, the mechanical varying mechanical properties of glauberite salt rock [1]. Furthermore, compression of the solid properties of glauberite salt rock decrease, and infiltration pressure reinforces this decrease effect. skeleton in a tri-axial stress state causes closure of pores, decreasing the permeability [32]. As a When saturated by high infiltration pressure over a long period of time, the mutual reinforcing of consequence, the time-dependent deformations of the specimens are caused by the coupled effects of dissolution and seepage cause all or some of the sodium sulfate to dissolve and a larger number of compression and dissolution. pores to emerge between the solid matrix of calcium sulfate. In this way, glauberite salt rock changes The infiltration pressure has two important influences on the tri-axial time-dependent from a dense rock with low permeability to a porous medium with a main solid skeleton of a calcium deformation of glauberite salt rock. First, the infiltration pressure increases the speed of dissolution sulfate matrix; in addition, its mineral components and structures vary, resulting in varying mechanical and seepage, increasing the amount and size of pores and cracks in the specimen, greatly decreasing properties of glauberite salt rock [1]. Furthermore, compression of the solid skeleton in a tri-axial stress the mechanical properties, and increasing the deformation of glauberite salt rock. On the other hand, state causes closure of pores, decreasing the permeability [32]. As a consequence, the time-dependent increasing infiltration pressure constrains the deformation of glauberite salt rock, as the effective deformations of the specimens are caused by the coupled effects of compression and dissolution. stress on the specimen decreases. Therefore, the pore water pressure and deformation of glauberite salt rock not only enhance and accelerate each other but also inhibit each other. The time-dependent deformation mechanism of glauberite salt rock under tri-axial stress and infiltration pressure differs from that of rocks that are not soluble. The effective stress loaded on a specimen determines the axial deformation of glauberite salt rock. Varying infiltration pressure not only results in different effective stresses on the solid skeleton of glauberite rock but also causes distinctions in the internal components and the microstructure evolution law, such as pores and cracks, that can affect the time-dependent deformation of glauberite

Energies 2020, 13, 1797 12 of 20

The infiltration pressure has two important influences on the tri-axial time-dependent deformation of glauberite salt rock. First, the infiltration pressure increases the speed of dissolution and seepage, increasing the amount and size of pores and cracks in the specimen, greatly decreasing the mechanical properties, and increasing the deformation of glauberite salt rock. On the other hand, increasing infiltration pressure constrains the deformation of glauberite salt rock, as the effective stress on the specimen decreases. Therefore, the pore water pressure and deformation of glauberite salt rock not only enhance and accelerate each other but also inhibit each other. The time-dependent deformation mechanism of glauberite salt rock under tri-axial stress and infiltration pressure differs from that of rocks that are not soluble. The effective stress loaded on a specimen determines the axial deformation of glauberite salt rock. Varying infiltration pressure not only results in different effective stresses on the solid skeleton of glauberite rock but also causes distinctions in the internal components and the microstructure evolution law, such as pores and cracks, that can affect the time-dependent deformation of glauberite rock. The effective stress mainly influences the time-dependent deformations in the following two aspects. First, at saturated pore pressure stage, for the same specimen, the effective stress varies at different stages, leading to ununiform deformations of glauberite salt rock. Thus, the time-dependent deformation behavior is more obvious with larger effective stress. Second, at the water-saturated stage, when the effective stress loaded on the specimen is sufficiently small, the deformation may be small or nonexistent; conversely, the deformation can increase over time with larger effective stress, which can overcome the confining pressure effect. This result is also shown in the deformation differences of specimens 2 and 3. In the solution mining process, initially, applied infiltration pressure results in the decrease of the effective stress of glauberite salt rock, and therefore, the ore body deformations are relatively small. However, ore body deformation increases sharply after the infiltration pressure returns to 0, and the effective stress increases greatly, causing ground surface subsidence. Furthermore, the loading history of the previous stage also has an important effect on the time-dependent deformation of glauberite salt rock [32]. The parameters of the previous stage, such as loading time, the internal structure differences under infiltration pressure, and the dissolution effect, greatly influence the experimental results of the next stage. Comparing the results at the drainage stage clearly shows that the degree of softening of the solid skeleton caused by the infiltration pressure loading history dominates the time-dependent deformation. Generally, the degree of weakening of the solid skeleton is more serious with larger dissolution and infiltration pressure; at this state, the time-dependent deformation strain and the average time-dependent deformation rate are relatively large at the same stress. The influence of the solution concentration on the time-dependent deformation properties was not considered in this experiment because the fresh water used for the infiltration pressure was changed regularly to ensure that the solution concentration did not reach the saturation point. However, because solute concentration gradients must have existed in the samples, the solution was likely saturated or near-saturated at many points in the sample. While only fresh water was injected, the concentration of dissolved fluid varied spatially and temporally within each sample, depending on fluid pressure gradient, flow rate, and damage development; therefore, in the experiments, the output concentration of the fluid during the saturation stage was measured by a sodium-specific ion meter, and the results are shown in Figure 10. From the figure, it can be seen that during the experiment the solute concentration did not reach the saturation point. Energies 2020, 13, x FOR PEER REVIEW 12 of 20 rock. The effective stress mainly influences the time-dependent deformations in the following two aspects. First, at saturated pore pressure stage, for the same specimen, the effective stress varies at different stages, leading to ununiform deformations of glauberite salt rock. Thus, the time-dependent deformation behavior is more obvious with larger effective stress. Second, at the water-saturated stage, when the effective stress loaded on the specimen is sufficiently small, the deformation may be small or nonexistent; conversely, the deformation can increase over time with larger effective stress, which can overcome the confining pressure effect. This result is also shown in the deformation differences of specimens 2 and 3. In the solution mining process, initially, applied infiltration pressure results in the decrease of the effective stress of glauberite salt rock, and therefore, the ore body deformations are relatively small. However, ore body deformation increases sharply after the infiltration pressure returns to 0, and the effective stress increases greatly, causing ground surface subsidence. Furthermore, the loading history of the previous stage also has an important effect on the time-dependent deformation of glauberite salt rock [32]. The parameters of the previous stage, such as loading time, the internal structure differences under infiltration pressure, and the dissolution effect, greatly influence the experimental results of the next stage. Comparing the results at the drainage stage clearly shows that the degree of softening of the solid skeleton caused by the infiltration pressure loading history dominates the time-dependent deformation. Generally, the degree of weakening of the solid skeleton is more serious with larger dissolution and infiltration pressure; at this state, the time-dependent deformation strain and the average time-dependent deformation rate are relatively large at the same stress. The influence of the solution concentration on the time-dependent deformation properties was not considered in this experiment because the fresh water used for the infiltration pressure was changed regularly to ensure that the solution concentration did not reach the saturation point. However, because solute concentration gradients must have existed in the samples, the solution was likely saturated or near-saturated at many points in the sample. While only fresh water was injected, the concentration of dissolved fluid varied spatially and temporally within each sample, depending on fluid pressure gradient, flow rate, and damage development; therefore, in the experiments, the output concentration of the fluid during the saturation stage was measured by a sodium-specific ion Energiesmeter,2020 and, 13 the, 1797 results are shown in Figure 10. From the figure, it can be seen that during13 ofthe 20 experiment the solute concentration did not reach the saturation point.

Figure 10. Solution concentration (Na SO ) at the water-saturated stage. Figure 10. Solution concentration (Na22SO44) at the water-saturated stage. 4.2. Constitutive Model Development and Parameter Identification 4.2. Constitutive Model Development and Parameter Identification The generated laboratory results make the development of a constitutive model for glauberite salt rockThe possible.generated Several laboratory conclusions results canmake be the obtained develo bypment analyzing of a constitutive the time-dependent model for deformation glauberite Energies 2020, 13, x FOR PEER REVIEW 13 of 20 curvessalt rock of possible. the different Several stages conclusions for glauberite can saltbe obtained rock. First, by theanalyzing model shouldthe time-dependent contain an elastic deformation element becausecurves of glauberite the different salt rock stages produces for glauberite transient salt elastic rock. strain First, after the loading; model then,should a viscouscontain elementan elastic is affectedelement bybecause the compression glauberite salt and rock dissolution produces showtransients obvious elastic viscoelastic strain after properties.loading; then, Because a viscous the needed for this model to account for the increasing strain over time. Glauberite salt rock affected by the generalizedelement is needed Kelvin formodel this canmodel describe to account the viscoelastic for the increasing characteristics strain over of rocks time. and Glauberite it only has salt three rock compression and dissolution shows obvious viscoelastic properties. Because the generalized Kelvin elements that can be easily obtained, this model is used to fit the strain curves of the different stages model can describe the viscoelastic characteristics of rocks and it only has three elements that can be for glauberite salt rock. easily obtained, this model is used to fit the strain curves of the different stages for glauberite salt rock. For a single contractive condition, the generalized Kelvin model is composed of one Hooke For a single contractive condition, the generalized Kelvin model is composed of one Hooke element and one Kelvin element, and the generalized Kelvin model is shown in Figure 11. element and one Kelvin element, and the generalized Kelvin model is shown in Figure 11.

(a) (b)

Figure 11. (a) the generalized Kelvin model; (b) the modifiedmodified generalizedgeneralized KelvinKelvin model.model.

As mentionedmentioned before, during the dissolutiondissolution processprocess of thethe glauberiteglauberite salt rock,rock, thethe mechanicalmechanical parametersparameters weaken with with time time due due to to the the gradual gradual incr increaseease of the of the pores pores and and cracks cracks inside inside the salt the rock salt rock[1,33]. [1 ,Therefore,33]. Therefore, based based on the on damage the damage mechanics mechanics [34], [we34], can we propose can propose a damage a damage variable variable that thatcan candenote denote how how the time-dependent the time-dependent deformation deformation paramete parametersrs evolve evolve with time with during time during the coupled the coupled effect eofff ectcompression of compression and dissolution. and dissolution. Damage Damage variable variable D canD becan given be givenas [19] as [19]

−ααt D ==−1 e− t (1) D 1− e (1) where t is the time, and a is a constant that denotes how the damage law changes with time, and the where t is the time, and a is a constant that denotes how the damage law changes with time, and larger the damage function, the larger the α parameter. In order to take into account the influence of the larger the damage function, the larger the parameter. In order to take into account the influence of damage evolution on the creep behavior, the following expression of elastic modulus E1 (p,D) and E2 (p,D) and viscosity η(p,D) were proposed:

=−  E11(,pD ) E ()(1 p D )  =− E22(,pD ) E ()(1 p D ) (2)   ηη(,p DpD )=− ()(1 ) where E1(p) is the elastic modulus at the infiltration pressure p and E2(p) and η(p) are the elastic modulus and viscosity coefficient of the Kelvin model, respectively. To take into account of infiltration pressure effect on the transient creep strain, the constitutive equation of the Hooke model in one dimension is given by σσσ ε ==000= eαt 1 − (3) EpD11(, ) Ep ()(1 D ) Ep 1 () where is the initial stress. For the Kelvin model, the relation of stress and strain is

• σ Ep2 () 0 αt εε+ = e (4) 22ηη()pp ()

Energies 2020, 13, 1797 14 of 20

damage evolution on the creep behavior, the following expression of elastic modulus E1 (p,D) and E2 (p,D) and viscosity η(p,D) were proposed:   E1(p, D) = E1(p) (1 D)  · −  E2(p, D) = E2(p) (1 D) (2)  · −  η(p, D) = η(p) (1 D) · − where E1(p) is the elastic modulus at the infiltration pressure p and E2(p) and η(p) are the elastic modulus and viscosity coefficient of the Kelvin model, respectively. To take into account of infiltration pressure effect on the transient creep strain, the constitutive equation of the Hooke model in one dimension is given by σ σ σ ε = 0 = 0 = 0 eαt (3) 1 E (p, D) E (p) (1 D) E (p) 1 1 · − 1 where σ0 is the initial stress. For the Kelvin model, the relation of stress and strain is

E2(p) σ0 αt ε•2 + ε2 = e (4) η(p) η(p)

On the basis of the solution method of first-order nonlinear differential equations and the initial condition ε (t) =0 t= , the general solution of Equation (4) can be obtained as 2 | 0

E2 σ0 αt η t ε2 = (e e− ) (5) αη(p) + E2(p) −

The total strain is combined with the following two parts:

E2 σ0 αt σ0 αt η t ε(t) = ε1 + ε2 = e + (e e− ) (6) E1(p) αη(p) + E2(p) −

It should be noted that tri-axial stress state should be considered in true rock engineering, so we need get the creep equation in the three dimensions. In the three dimensional stress state, the stress tensor σij can be decomposed to deviatoric stress tensor Sij and spherical stress tensor σm, and the strain tensor can be decomposed to deviatoric strain tensor eij and spherical strain tensor em [35]. The relationship between these tensors can be written as ( σ = S + δ σ ij ij ij m (7) εij = eij + δijεm in which δij is Kronecker function, i, j = 1, 2, 3, and spherical stress tensor σm and spherical strain tensor em can be given by ( 1 1 σm = 3 (σ1 + σ2 + σ3) = 3 σii 1 1 (8) εm = 3 (ε1 + ε2 + ε3) = 3 εii Generally speaking, the spherical stress tensor can only lead to the variation of the bulk of the material, while the deviatoric stress tensor can only change its shape. According to the generalized Hooke law, we obtain the relationship between elastic mechanics parameters of the elastic element on the tri-axial stress state, ( S = 2Ge ij ij (9) σm = 3Kεm where G is shear modulus and K means bulk modulus. Energies 2020, 13, 1797 15 of 20

Substituting Equations (7)–(9) into Equation (6), the creep damage constitutive equation in the three dimensional stress state can be expressed as

G (p) Sij Sij 2 t σm αt αt η(p) εij(t) = + e + (e e− ) (10) 3K(p) 2G1(p) αη(p) + 2G2(p) − where G1 is the viscoelastic shear modulus of the Hooke model at the infiltration pressure p, and G2 and η are viscoelastic shear modulus and viscosity shear coefficient of the Kelvin model, respectively. In addition, effective stress is the important factor that actually determines the mechanical and hydraulic properties of porous rock and soil materials. According to the effective stress principle, the effective stress σ0 is equal to total stress σ minus pore water pressure p and can be written as [36]

σ0 = σ p (11) − Substituting Equations (7) and (8) and (11) into Equation (10), we obtain the tri-axial creep damage constitutive equation considering the coupled effect of compression and dissolution

G (p) σm δijp Sij Sij 2 t αt αt η(p) εij(t) = − + e + (e e− ) (12) 3K(p) 2G1(p) αη(p) + 2G2(p) −

In this paper, based on the experimental condition for the cylindrical specimen, the minor principle stress and intermediate stress has the same value

σ2 = σ3 (13)

The axial creep damage constitutive equation can be simplified to

G (p) σ + 2σ 3p 2(σ σ ) 2(σ σ ) 2 t 1 3 1 3 1 αt 1 3 1 αt η(p) ε1(t) = − + − e + − (e e− ) (14) 9K(p) 3 2G1(p) 3 αη(p) + 2G2(p) −

In this test, the elastic deformation was not taken into account in order to precisely simulate the deformation of glauberite deposits during solution mining, so we have

ε (t) =0 t= (15) 1 | 0 Therefore, in this paper, the axial damage constitutive equation can be converted to

G (p) 2(σ σ ) 2 t 1 3 1 αt η(p) ε1(t) = − (e e− ) (16) 3 αη(p) + 2G2(p) −

In this model, not only was the effective stress principle taken into consideration, but the dissolution effect caused by the infiltration pressure was also added. The time-dependent deformation damage constitutive model was adopted to fit the time-dependent deformation strain curves, and the parameters of this model were obtained using the Levenberg–Marquardt algorithm [37]. It should be noted that in this part, we only used our model to fit experimental data at the hydraulic connection stage and saturated-water stage, not including the drainage stage. The model parameters are shown in Table5, and the experimental data and time-dependent deformation damage constitutive model for the strain-time curves are shown in Figure 12. The results show that the correlation coefficients of the two results are greater than 0.95, illustrating good fit. Therefore, the time-dependent deformation damage constitutive model can depict the time-dependent deformation characteristics of glauberite salt rock under the coupled effects of compression and dissolution. Energies 2020, 13, 1797 16 of 20

Table 5. Parameters of creep damage constitutive model.

Sequence Number of Correlation Creep Stage α/1 G2/MPa η/(MPa h) Specimen · Coefficient/R2 0 – 653.84 27,127.6 0.95 1 0.00012 332.25 12,999.26 0.98 a and b 2 0.00045 215.97 12,000.01 0.96 3 0.00060 251.53 4483.82 0.96 Energies 2020, 13, x FOR PEER REVIEW 16 of 20

(a) (b)

(c) (d)

FigureFigure 12. 12. ComparisonComparison of of predicted predicted curves curves of of the the creep creep damage damage model model and and test test data data under under different different infiltrationinfiltration pressure pressure levels. levels. ( (aa)) specimen specimen 0; 0; ( (bb)) specimen specimen 1; 1; ( (cc)) specimen specimen 2; 2; ( (dd)) specimen specimen 3. 3.

TableTable 55 showsshows thatthat viscoelasticviscoelastic shearshear modulusmodulus GG2 andand viscosity viscosity shear shear coefficient coefficient ηη ofof glauberite glauberite decreasedecrease over over increasing increasing infiltration infiltration pressure pressure when when the the specimens specimens are are affected affected by by the the dissolution dissolution and and erosionerosion effects, effects, while while the the parameter parameter a a increases increases with with increasing increasing infiltration infiltration pressure, pressure, indicating indicating more more damagedamage ofof mechanicalmechanical properties properties of glauberiteof glauberite with with higher higher infiltration infiltration pressure. pressure. The parameter The parameter changes changesfurther demonstrate further demonstrate the damage the degree damage of the degree solid skeleton of the solid mechanical skeleton properties mechanical of glauberite properties caused of glauberiteby infiltration caused pressure by infiltration and dissolution pressure e ffandect. di Throughssolution the effect. careful Through analysis, the thecareful variation analysis, process the variationof these threeprocess parameters of these three over parameters infiltration over pressure infiltration can be pressure described can respectively, be described with respectively, Equations with(17)–(19). Equations (17)–(19). p/0.66 2 G2 = 423.09e− + 231.50 (R = 0.98) (17) − p/0.66 2 Ge=+423.09 231.500.53p 2 ( R = 0.98) (17) 2 η = 26474.01e− (R = 0.91) (18) α = 0.0005244 ln(p + 0.2919)(R2 = 0.95) (19) −0.53p 2 η ==26474.01eR ( 0.91) (18)

2 α =+0.0005244ln(pR 0.2919) ( = 0.95) (19) where G2 and p are in the unit of MPa, and η is in the unit of MPa*h, and α is a dimensionless number. Figure 13 shows the comparison of fitting curve and inversion data of corresponding parameters; it can be seen that the equation basically reflects the evolution of parameters α, G2, and η with infiltration pressure. Substituting Equations (17)–(19) into Equation (16), we can finally

Energies 2020, 13, x FOR PEER REVIEW 17 of 20 establish the time-dependent deformation model for the glauberite, with its axial time-dependent deformation equation as follows

2(σσ− ) 1 ε ()=t 13 Energies 2020, 13, 17971 +×+−−0.53pp /0.66 17 of 20 3 0.1388ln(pe 0.2919) 846.18 e ) (20) −0.985 p ×−(e)e0.0005244ln(pt+− 0.2919) 0.032 et where G2 and p are in the unit of MPa, and η is in the unit of MPa*h, and α is a dimensionless number. Figure 13 shows the comparison of fitting curve and inversion data of corresponding parameters; The damage constitutive model takes into account the damage process of time-dependent (creep) it can be seen that the equation basically reflects the evolution of parameters α, G , and η with mechanical parameters of glauberite caused by infiltration pressure, which can provide2 a reference infiltration pressure. Substituting Equations (17)–(19) into Equation (16), we can finally establish the for predicting the time-dependent deformation of glauberite ore during the mining of rough solution. time-dependent deformation model for the glauberite, with its axial time-dependent deformation It should be noted that the constitutive model is a fit to experimental data obtained under specific equation as follows boundary conditions of fluid injection (0, 1, 2, 3 MPa) and applied stress (σ1 = 5 MPa, σ3 = 4 MPa). 2(σ1 σ3) 1 ε1(t) = − 0.53p p/0.66 Moreover, there are some limitations3 to0.1388 this ln constitutive(p+0.2919) e− model.+846.18 eA− truly) general model should × 0.985p (20) account for the full coupling of fluid(e0.0005244 flow, lnmineral(p+0.2919 dissolution,)t e 0.032e− diffusiont) and precipitation, porosity × − − evolution, and compression.

(a) (b)

(c)

Figure 13. VariationVariation processes of creep paramete parametersrs over infiltration infiltration pressure. ( (aa)) parameter parameter αα;;( (bb))

parameter G2;;( (c)) parameter parameter η.

5. ConclusionsThe damage constitutive model takes into account the damage process of time-dependent (creep) mechanical parameters of glauberite caused by infiltration pressure, which can provide a reference for Cylindrical samples of glauberite salt rock were deformed in 30 days under the conditions of an predicting the time-dependent deformation of glauberite ore during the mining of rough solution. axial pressure of 5 MPa and a confining pressure of 4 MPa and infiltration pressures in the range of It should be noted that the constitutive model is a fit to experimental data obtained under specific 0–3 MPa. The aim of the experiments was to investigate the time-dependent deformation of boundary conditions of fluid injection (0, 1, 2, 3 MPa) and applied stress (σ = 5 MPa, σ = 4 MPa). glauberite salt rock within three stages, hydraulic connection stage, water-saturated1 3stage, and Moreover, there are some limitations to this constitutive model. A truly general model should account drainage stage, and this study is expected to provide some guidance on the prediction and for the full coupling of fluid flow, mineral dissolution, diffusion and precipitation, porosity evolution, and compression.

5. Conclusions Cylindrical samples of glauberite salt rock were deformed in 30 days under the conditions of an axial pressure of 5 MPa and a confining pressure of 4 MPa and infiltration pressures in the range of Energies 2020, 13, 1797 18 of 20

0–3 MPa. The aim of the experiments was to investigate the time-dependent deformation of glauberite salt rock within three stages, hydraulic connection stage, water-saturated stage, and drainage stage, and this study is expected to provide some guidance on the prediction and assessment of surface subsidence during the process of in-situ solution mining for glauberite ore. Some conclusions can be made based on the results presented in this paper. During the hydraulic connection stage, the level of infiltration pressure greatly affects differences in the hydraulic connection of glauberite salt rock, and the higher infiltration pressure gives rise to the shorter hydraulic connection time. For specimen 3 to which an infiltration pressure of 3 MPa was applied, it took 84.5 h to become hydraulically connected, which was 53 h shorter than specimen 2 with an infiltration pressure of 2 MPa. With an infiltration pressure of 1 MPa, specimen 1 was not hydraulically connected in 500 h. During the hydraulic connection stage and water-saturated stage, it is mainly the effective stress and the degree of weakening (caused by the dissolution and erosion effects) that determine the time-dependent deformation of glauberite salt rock. At the hydraulic connection stage, it seems that the degree of weakening of glauberite salt rock dominates the time-dependent deformation. Conversely, for the water-saturated stage, the higher effective stress loaded on the solid skeleton of the glauberite salt rock brings larger time-dependent deformation. At the drainage stage, the loading history (time, infiltration pressure, etc.) of the previous stage impacts the time-dependent deformation of glauberite salt rock. The larger the infiltration pressure applied on the specimen was, the more pores and fractures developed in the glauberite salt rock, which brings larger time-dependent deformation. For the specimen loaded with higher infiltration pressure, the lengths and apertures of the fractures and porosity of glauberite salt rock were larger, which further verifies that the softening degree caused by the dissolution increases with larger infiltration pressure. Considering damage law of creep mechanical parameters induced by the infiltration pressure, the damage constitutive model was proposed and used to fit the time-dependent deformation strain curves, and this model can describe the time-dependent deformation of glauberite salt rock under the coupled effects of compression and infiltration pressure.

Author Contributions: Conceptualization, M.C.; methodology, M.C.; software, M.C.; validation, M.C.; formal analysis, M.C.; investigation, M.C.; resources, M.C.; data curation, M.C.; writing—original draft preparation, M.C.; writing—review and editing, S.Y. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Science Fund for Distinguished Young Scholars, grant number 51225404 and the National Natural Science Foundation of China, grant number 51504159 and 51904196. Acknowledgments: This research has been financed by the National Science Fund for Distinguished Young Scholars (no.51225404) and the National Natural Science Foundation of China (no.51504159 and no. 51904196), which are greatly appreciated. Conflicts of Interest: The authors declared that they have no conflicts of interest related to this work.

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