Environmental Earth Sciences (2018) 77:245 https://doi.org/10.1007/s12665-018-7423-5

ORIGINAL ARTICLE

Laboratory investigations of inert gas flow behaviors in compact sandstone Chaojun Jia1,2   · Weiya Xu1 · Huanling Wang1,3 · Wei Wang1 · Jun Yu1 · Zhinan Lin1 Received: 6 October 2017 / Accepted: 16 March 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract The seepage evolution behavior of compact rock is significant for the stability and safety of many engineering applications. In this research, both hydrostatic and triaxial compression tests were conducted on compact sandstone using an inert gas, namely argon. A triaxial compression test with a water permeability measurement was carried out to study the difference between the gas permeability and water permeability evolutions during the complete stress–strain process. Based on the experimental data, the hydrostatic stress-dependent gas permeability was discussed firstly. A second-order function was proposed to predict and explain the gas slippage effect. The mechanical properties and crack development of the sandstone samples were discussed to better understand the permeability evolution with crack growth during the complete stress–strain process. The results show that the gas permeability evolution can be divided into five stages according to the different crack growth stages. Then, the permeability changes in the crack closure stress 𝜎cc , crack initiation stress 𝜎ci , crack damage stress 𝜎cd and peak stress 𝜎p with confining pressures were analyzed. Finally, we found that the difference between the corrected gas permeability and water permeability can be attributed to the interaction between the water and sandstone grains. Keywords  Compact sandstone · Inert gas · Permeability · Slippage effect · Triaxial compaction · Hydrostatic compression

Introduction

* Chaojun Jia jia‑chao‑[email protected] Weiya Xu [email protected] Huanling Wang [email protected] Wei Wang [email protected] Jun Yu [email protected] Zhinan Lin [email protected] 1



Research Institute of Geotechnical Engineering, Hohai University, Nanjing 210098, China

2



Department of Civil Engineering, University of Toronto, Toronto M5S 1W8, Canada

3

Key Laboratory of Coastal Disaster and Defense, Ministry of Education, Hohai University, Nanjing 210098, China



Natural compact rock with a low seepage property is an ideal material for the storage of nuclear waste, oil and gas (Birgersson et al. 2008; Dana and Skoczylas 1999); Liu et al. (2014). Moreover, for huge hydraulic engineering projects, such as the construction of an underground power station, compact rock is also the perfect host rock (Popp et al. 2001). Natural rock is always subject to hydrostatic loading and deviatoric loading conditions. Therefore, it is of significant importance to study the permeability property of compact rock during a complex stress state to maintain the stability and safety of large-scale rock engineering (Yang et al. 2015). The permeability evolution of rock materials during a hydrostatic compression condition has been widely investigated by using a number of methods. David et al. (1994) investigated the dependence of the permeability on pressure in five types of sandstone with porosities ranging from 14 to 35%. The experimental results suggested that the data can often be approximated by an exponential function for a given compaction mechanism. Ding et al. (2016) tested the permeability of fractured sandstone after high-temperature treatment. The results indicated that the permeability

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slowly decreased when the temperature was below 400 °C. However, the permeability increased rapidly when the temperature increased from 400 to 600 °C. Tanikawa and Shimamoto (2009) measured the intrinsic permeability of sedimentary rocks from the western foothills of Taiwan by using nitrogen gas and distilled water. The results showed that the permeability of low permeability porous media to nitrogen gas is 2–10 times that to water and the difference can be partly explained by the Klinkenberg effect. During the last few decades, the permeability of rock materials under triaxial compression condition has also been extensively studied in laboratory experiments (Farquharson et al. 2016; Indraratna and Haque 1999; Jobmann et al. 2010; Naumann et al. 2007; Renner et al. 2000; Uehara and Shimamoto 2004; Wang et al. 2014). Zhu and Wong (1997) conducted triaxial compression experiments on five types of sandstone with porosities ranging from 15 to 35%. They found that a drastic decrease in the permeability was triggered by the onset of shear-enhanced compaction caused by grain crushing and pore collapse. In the brittle faulting regime, the permeability evolution relates to the onset of shear-induced dilation. Baud et al. (2012) reported results from an experimental study regarding the influence of preexisting structural anisotropy on the permeability evolution during the triaxial compaction of a porous sandstone. The results indicated that the decrease in the permeability can reach approximately one order of magnitude and is far more abrupt in samples deformed normal to bedding. Previous studies on the permeability evolution during triaxial deformation provide an understanding the fluid behavior of sandstone and other rock materials. However, many of them mainly focused on the fluid flow behavior of brittle rocks during a prefailure phase. The permeability characteristic during the entire deformation and failure process has not yet been intensively investigated, and to date, only very limited data are available. Both water and gas are used as flow media to measure the permeability of rock materials in the laboratory. Owing to its compressibility and viscosity, gas is less sensitive to temperature changes (Tanikawa and Shimamoto 2009). Therefore, gases such as ­CO2, ­N2 and ­CH4 have been widely investigated (Harpalani and Schraufnagel 1990; Somerton et al. 1975). However, rock swelling and the adsorption/desorption of gases may occur when using these gas flow types through a rock matrix (Jasinge et al. 2011; Levine 1996). Previous studies indicated that gas flow in compact rock, which differs from that in high-permeability materials, may be affected by a slippage effect (Moghadam and Chalaturnyk 2014). The slippage effect is defined as a phenomenon in which the velocity of gas molecules on the pore walls is no longer equal to zero (Anez et al. 2014). Therefore, the measured gas permeability is higher than the water permeability. However, the experimental data regarding the difference between gas

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permeability and water permeability evolutions during triaxial compression are rare. To characterize the inert gas flow behaviors in compact sandstone during complex stress conditions, hydrostatic compression and conventional compression tests along with permeability measurements were carried out. The influence of a confining pressure and the Klinkenberg effect on the test gas permeability were discussed. The mechanical properties and the crack development of this compact sandstone were studied to better understand the permeability evolution with crack growth during triaxial compression. In addition, the difference between corrected gas permeability and water permeability evolutions during triaxial compression was discussed.

Experimental methods Sample characterization and preparation The materials used in this study were compact sandstone that was collected from a hydropower station in southwest China. The sandstone was an off-white fine grain rock material with a bulk density of 2608 kg/m3. The porosity was measured in two steps. Five samples were first dried until a constant mass was attained. Then, the samples were saturated with distilled water under vacuum for 24 h and the weight was measured. The determined connected porosity was 2.21% according to a method published in the literature (Dana and Skoczylas 1999). X-ray diffraction (XRD) experiments indicated that the main mineral was composed of quartz (92%), feldspar (2%), rock debris (3%) and other minerals (3%). Studies have shown that pore size distributions vary significantly with clay content in rock. Consequently, the effect of clay content on the permeability of a rock material is significant, particularly in a rock with a small porosity (Dewhurst et al. 1998). Therefore, the clay content was examined in detail. The results showed that clay was not found in this sandstone. Moreover, the sandstone was homogeneous without visible macrocracks and fractures. The microstructure of the test sandstone was examined using scanning electron microscope (SEM), and the results are shown in Fig. 1. It can be seen that the sandstone has granular structures. The mineral grains are uniformly mixed together tightly, which results in small pores. This sandstone is so compressed that the widths of the microcracks are extremely narrow. The observed small pores and microcracks are the main seepage channels for the fluids. Intact blocks were cored perpendicular to their sedimentary bedding and ground into cylindrical samples with a diameter of 50 mm and a length 100 mm, according to the International Society for Rock Mechanics (ISRM). The basic parameters of the compact sandstone and test conditions

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Fig. 1  SEM micrographs of sandstone: a magnified 500 times; b magnified 1000 times; c magnified 4000 times; d magnified 8000 times

are provided in Table 1. Note that the saturated condition influences the permeability greatly. For this purpose, the samples were treated under 105 °C until a stable mass was reached before testing the gas permeability of the sample. No significant effect on the permeability and porosity has been observed at this dry temperature (Chen et al. 2013). In addition, one sample was saturated with water under vacuum conditions before testing the permeability with water.

Experimental apparatus The apparatus used in this experiment is shown schematically in Fig. 2. This servo-controlled experimental system comprises a cylindrical cell, three high-pressure pumps and a data monitoring system. A confining pressure up to 60  MPa was applied by a chamber pressure pump. The deviatoric stress can be carried out on samples in the load Table 1  Basic parameters of the compact sandstone and test conditions

or displacement control mode by a deviatoric loading pump. The axial strain ε1 was measured using two linear variable displacement transducers (LVDT) with a resolution of ± 1 μm, while the radial deformation ε3 is obtained through a ring radial displacement transducer chain mounted around the middle of the samples. Based on the measured axial and radial strains, the volumetric strain can be calculated from 𝜀v = 𝜀1 + 2 × 𝜀3 . The samples were isolated from the confining oil by a rubber jacket. The upper and lower reservoirs were installed in the upper and lower sides of the sample, respectively. This hydromechanical coupling system is capable of accepting gas and water as seepage materials. When testing the permeability with water, only the flow channel of the apparatus needs to switch to water. Note that a both steady-state method and a transient pulse method can be adopted to test the permeability of rock in this apparatus. Because of a high-precision pressure gauge, the water and

Samples Diameter (mm) Length (mm) Mass (g) Density (g/cm3) Pc (MPa) Pp (MPa) Fluid type SY-1 SY-2 SY-3 SY-4 SY-5 SY-6

49.92 49.82 49.98 49.82 49.90 49.85

101.38 101.32 101.23 101.40 101.59 101.31

514.56 519.03 518.61 510.54 519.76 516.64

2.595 2.629 2.613 2.584 2.617 2.614

3–25 5 10 10 15 20

0.2–2 1 1 1 1 1

Ar Ar Ar H 2O Ar Ar

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Fig. 2  Schematic diagram of the permeability measurement apparatus. Both water and gas can be conducted on this system

gas permeabilities down to 1­ 0−22 and 1­ 0−24 m2 were measured, respectively.

Hydrostatic and triaxial compression procedure Two types of tests were carried out in this study: hydrostatic compression tests and conventional triaxial compression tests. All the tests were performed under constant temperature (24 °C) conditions to reduce the influence of thermal variation. Argon was selected as the seepage media because of its chemical neutrality (Meziani and Skoczylas 1999). An inert gas is widely used in underground oil storage (Goodall et al. 1988). In addition, it is much easier to flow argon with small molecules through a porous network, so that the measurement can be rapid. To ensure that the observed reduction in the pressure across samples corresponded to the flow through the sample, rather than create a flow between the sample and the rubber jacket, fluid pressure stepping experiments were performed on a steel sample with same jacket and method as that of the permeability tests. The experimental results indicated that the flow between the sample and jacket can be neglected when the confining pressure exceeds the fluid pressure of 0.5 MPa. The hydrostatic compression test was carried out on sample SY-1 with different confining pressures and fluid

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pressures. Classically, the testing procedure was as follows: first, the sample was hooped to the axial platens and isolated from confining oil by a rubber jacket. Then, a confining pressure was applied to the desired value at a rate of 0.5 MPa/s. Next, gas was injected to the upstream of the sample with the desired pressure. The downstream of the sample was opened to atmosphere. The variations in the confining pressure were in the following increments: 3–5–10–15–20–25 MPa. During each step of constant confining pressure, the permeability was measured as the fluid pressure changed from 0.2 to 2 MPa. Conventional triaxial compression along with the permeability measurement tests can be described as follows. First, the samples were hooped to the axial platens and isolated from the confining oil by a rubber jacket. Then, the confining pressure was applied to the desired value at a rate of 0.5 MPa/s. The upper fluid pressure was applied later. Afterward, the deviatoric stress was increased stepwise under an axial strain control model at a rate of 2 × 10−4/min until sample failure. Such strain rate is slow enough to ensure steady-state flow to determine the permeability while continuously deforming the sandstone samples. During the entire process, the confining pressure and upper pressure were kept constant. After each step of deviatoric stress was applied, the variation in the lower pore pressure transducer was recorded during ∆t by closing the

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Permeability measurement method There are many methods that can be applied to test the permeability of rock materials. Generally, two different techniques are widely used in the laboratory according to the permeability range. These two methods are the steady-state flow method (Davy et al. 2007) and pulse test method (Brace et al. 1968). When the permeability of a test material is higher than ­10−19 m2, the steady-state flow method is a good choice because Darcy’s law can be applied directly. In the case of sandstone studied in this paper, its initial permeability was estimated to be on the order of 1­ 0−17 m2; therefore, the steadystate flow method was adopted. According to the law of conservation of mass, the lower mean flow rate Qdn is

Vdn ΔP Qdn = ( ) P0 + ΔP∕2 Δt

(1)

2𝜇g LQdn P0 ) ( A P21 − P20

(2)

where Vdn is the volume of the lower reservoir, ∆P is the pressure variation in the lower pore pressure transducer, P0 is the atmospheric pressure (0.1 MPa), and ∆t is the recording time. Based on the conservation of mass and Darcy’s law, the gas permeability can be calculated from

K=

where 𝜇g is the viscosity of argon, L is the length of the sample, A is the cross-sectional area of the sample, and P1 is pressure recorded by the upper pore pressure transducer. The water permeability was determined from sample SY-4 with a confining pressure of 10 MPa and a pore pressure of 1.0 MPa. The water permeability can be calculated from Darcy’s law:

K=−

𝜇w LQ AΔP

(3)

where 𝜇w is the viscosity of the distilled water, ∆P is pressure gradient cross the samples, and Q is the flow rate that is determined from the change in the pump position during the time change ∆t.

Experimental results Hydrostatic compression test results Effect of confining pressure on the permeability The relationship between the apparent gas permeability and the confining pressure of sample SY-1 is plotted in Fig. 3. The apparent permeability is obtained with different pore pressures. It can be seen that the gas permeability of sandstone decreases gradually with an increase in the confining pressure during the hydrostatic compression tests. In addition, the reduction in the permeability is nonlinear. When the confining pressure is low, the permeability decreases rapidly, and the rate of decrease becomes flat under higher confining pressures. A sample at a pore pressure of 2 MPa was utilized for analysis. The permeability decreases from 3.90 × 10−18 to 3.00 × 10−18 m2 when the confining pressure decreases from 3 to 5 MPa, respectively, which is a decreases of almost 23% from the initial permeability. For comparison, the gas permeability only decreases approximately 7.2% when the confining pressure increases from 20 to 25 MPa. In fact, the rock material is a porous medium that contains a large amount of microcracks, pores and other preexisting defects. The connected defects form flow channels for the transport media. An increase in the confining pressure closes or even shuts down the flow channels. The permeability eventually decreases. However, the resistance of samples to deformation becomes more difficult with high confining pressures. Thus, a reduction in the permeability becomes slow. Note that the influence of pore pressure on the permeability is also significant. The higher the pore pressure is, the lower the permeability is. Additionally, the change in the permeability with pore pressure is nonlinear. For instance, the

7.0x10-18

2MPa 1.5MPa 1.0MPa 0.5MPa 0.2MPa

6.0x10-18

Permeability/m2

valve next to the lower reservoir. This operation was performed when a steady flow condition was achieved. In this way, the evolution curves of the permeability during the entire loading process were obtained.

245

5.0x10-18 4.0x10-18 3.0x10-18 2.0x10-18 1.0x10-18 0.0

0

5

10

15

20

25

30

Confining pressure/MPa

Fig. 3  The relationship between permeability and confining pressure of sample SY-1

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permeability rises from 3.90 × 10−18 to 4.09 × 10−18 m2 when pore pressure changes from 2 to 1.5 MPa, respectively, under a confining pressure of 3 MPa. The variation in the permeability is 0.79 × 10−18 m2 when the pore pressure changes from 0.5 to 0.2 MPa, which is greater than 4 times the variation in the permeability between a pore pressure of 2 and 1.5 MPa. Similar results are reported by Wang et al. (2016) who conducted permeability measurements on the late Proterozoic granite gneiss from an underground oil and gas storage depot in China. The variation in the of test permeability with pore pressure can be attribute to the Klinkenberg effect. The stress-dependent permeability of sandstone has been well documented (David et al. 1994; Dong et al. 2010; Mohiuddin et al. 2000). Nevertheless, both the exponential and power law forms were used to describe the permeability changes with effect stress according to the laboratory results. The major problem of the exponential law is that it is very difficult to fit all the data when the stress spans large ranges. The power law, however, will yield an infinite permeability at zero stress (Zheng et al. 2015). Generally, the exponential equation and the power law equation that describe the stressdependent permeability can be expressed as: (4)

K = K0 exp (− 𝛾𝜎)

K = 𝛽𝜎 −𝜉 (5) where K0 is the initial permeability, σ is the effective stress that is defined as the difference between the confining pressure and the pore pressure, and γ, β and ξ are the material constants. Based on the experimental results, the stressdependent permeability fitting results are listed in Table 2. It can be seen that both the exponential and power equations can be used to describe the stress-dependent permeability. The determined K0 and β are reduced with increasing pore pressure. The determined γ range from 0.062/MPa to 0.081/MPa was much higher than the results published by David et al. (1994) on sandstones with higher permeabilities. Therefore, we conclude that the permeability of compact sandstone corresponds to a higher stress sensitivity. This result can be attributed to the small pores and the presence of microcrack networks in the compact sandstone. Applying stress to small-radius pores can decrease their flow capacity proportionally more than large pores (Vairogs et al. 1971). Table 2  The fitting results of the sample SY-1

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In addition, microcrack closure plays a significant role in the permeability reduction (Kwon et al. 2004). The determined ξ changes from 0.626 to 0.829, and the fitting R2 is slight lower than the fitting results of the exponential equation. Note here that the “stress” is the preferred confining pressure because a higher effective stress corresponds to a larger permeability if the confining pressure is constant. Influence of the Klinkenberg effect on the permeability Ideally, the permeability is a property of rock that depends on the structure of the porous medium. In this paper, the pore pressure changes from 0.2 to 2 MPa while the confining pressure ranges from 3 to 25 MPa. Therefore, the effect of pore pressure on the effective stress can be roughly ignored. Furthermore, if the pore pressure is considered, then the permeability at a higher effective stress should be smaller. However, the laboratory results show the opposite phenomena. Consequently, the effect of the Klinkenberg effect on this sandstone is significant. In addition, the relationship between the gas flow rate and the difference in the squared pressures is presented in Fig. 4. According to Darcy’s law, the gas flow rate Qdn should be linearly proportional to the difference in the squared pressures ( P21 − P20 ). As shown in 5x10-8

3 MPa 5 MPa 10 MPa 15 MPa 20 MPa 25 MPa

4x10-8

Qdn(m3/s)

245  

3x10-8

2x10-8

1x10-8

0

0

1

2

3

4

( P12 − P02 )( MPa 2 )

5

Fig. 4  The relationship between gas flow rate and the difference of square pressures

Pore pressure (MPa)

K = K0 exp (− 𝛾𝜎)

K0 ­(10−18 m2)

γ (1/MPa)

R2

β ­(10−18 m2/MPa)

ξ

R2

0.2 0.5 1.0 1.5 2

7.14 6.28 5.46 4.83 4.61

0.062 0.072 0.080 0.079 0.081

0.96 0.98 0.99 0.99 0.99

13.61 13.52 12.77 11.29 11.04

0.626 0.736 0.813 0.806 0.829

0.89 0.94 0.95 0.96 0.96

K = 𝛽𝜎 −𝜉

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Fig. 4, the relationship between the gas flow rate and the difference in the squared pressures is not linear, as described in Eq. (2). The curve is concave for a constant confining pressure. This phenomenon is more obvious under lower confining pressures. These problems might be related to the pore pressure-dependent gas permeability known as the Klinkenberg effect. When the pore throat dimension is within two orders of magnitude of the mean free path of the gas molecules, the gas molecules will slip on the surface of the pores. Therefore, the apparent permeability is higher than the real value, and the test results should be corrected. Klinkenberg (1941) proved that the relationship between the apparent permeability Kg and the intrinsic permeability K∞ can be expressed as: ) ( b Kg = K∞ 1 + (6) P̄

the porous media. For this purpose, to obtain the real value of the permeability of sandstone, the slippage factor b should be determined. Based on the test data of SY-1, the relationship between the test gas permeability and the reciprocal mean pressure is plotted in Fig. 5. It can be seen that the apparent permeability increases with an increase in the reciprocal mean pressure under a constant confining pressure. When the confining pressure is high, the curves are linear which agrees well with the Klinkenberg linear equation Eq. (2). However, when the confining pressure is low, i.e., 3 and 5 MPa, the curves deviate from the Klinkenberg straight line. Similar results were also reported by other authors (Dong et al. 2012; Li et al. 2009; Noman and Kalam 1990). Moghadam and Chalaturnyk (2014) suggested that the deviations stemmed from a departure from Warburg’s model (Kundt and Warburg 1875) or the assumption of a constant velocity gradient. Tang et al. (2005) suggested a higher-order equation to predict and explain the gas slippage effect, which can be written as: ( ) b a Kg = K∞ 1 + + (7) P̄ P̄ 2

where K∞ is the intrinsic permeability under a large pore pressure where the slippage effect can be neglected, P̄ is the mean pressure between the two ends of the sample, and b is the slippage factor that denotes the sensitivity of the apparent permeability to the pore pressure. The slippage factor depends on the temperature, gas type and pore structure of

6.0x10-18

Gas permeability Kg/m2

where a represents the a dynamic or secondary slippage factor. Thus, Eq. (7) is used to correct the apparent permeability of SY-1, and the results are listed in Table 3.

3 MPa 5 MPa 10 MPa 15 MPa 20 MPa 25 MPa

7.0x10-18

5.0x10-18

Triaxial compression test results Mechanical properties of compact sandstone For the triaxial compression test with a gas permeability measurement, four different values of the confining pressures were used, namely 5, 10, 15 and 20 MPa. The obtained stress–strain curves are plotted in Fig. 6a. Figure 6a shows that the effect of the confining pressure on the mechanical behavior of sandstone is significant. Table 4 compiles the strength and deformation parameters of compact sandstone under different confining pressures. In Table 4, E is the elastic modulus, ν is Poisson’s ratio, and 𝜎p represents the peak stress. Using the parameters listed in Table 4, the influence of confining pressure on the mechanical property

4.0x10-18 3.0x10-18 2.0x10-18 1.0x10-18 0

1

2

3

4

Reciprocal mean pressure

5

1/ P

6

7

8

(1/MPa)

Fig. 5  The relationship between test gas permeability and the reciprocal mean pressure Table 3  Fitting results of SY-1

245

Pc (MPa)

Equation

3

( Kg = K∞ 1 +

5 10 15 20 25

b P̄

+

a P̄ 2

)

K∞ ­(m2)

b (MPa)

a ­(MPa2)

R2

3.182 × 10−18

0.274

− 0.020

0.994

2.342 × 10−18 1.554 × 10−18 9.348 × 10−19 5.800 × 10−19 4.829 × 10−19

0.329 0.318 0.405 0.649 0.361

− 0.023 − 0.011 − 0.025 − 0.024 − 0.011

1.000 1.000 0.999 0.999 0.992

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Environmental Earth Sciences (2018) 77:245 300

(a)

(b)

σ1−σ3/MPa

280

Test results Mohr-Coulomb criterion

200

20MPa

20MPa

150 10MPa

100

10MPa

5MPa

5MPa

15MPa

-40

-30

-20

-10

0

ε1/10−3 0

10

20

240

y=149.57+5.7688*x R2=0.99

200

15MPa

50

ε3/10−3

(σ1−σ3)(MPa)

250

160

0

5

10

15

20

σ3(MPa)

30

(c)

SY-2

SY-3

SY-5

SY-6

Fig. 6  Triaxial compression test results. a Stress–strain curves; b is the relationship between peak stress and confining pressure and c is the failure model of sandstone Table 4  Strength and deformation parameters of the compact sandstone under different confining pressures Sample

Pc (MPa)

PP (MPa)

E (GPa)

ν

𝜎p (MPa)

SY-2 SY-3 SY-5 SY-6

5 10 15 20

1 1 1 1

15.52 16.53 15.98 17.15

0.14 0.15 0.16 0.16

175.68 210.99 236.42 263.47

and deformation behavior of the sandstone samples were investigated. It can be seen that the elastic modulus generally increases with increasing confining pressure. However, Poisson’s ratio remained almost constant. The peak stress values of the different samples of compact sandstone are plotted in Fig. 6b as a function of confining pressure. The relationship can be described by the Mohr–Coulomb criterion as follows: [ ( ( 𝜑 )] 𝜑) − 𝜎3 1 − tan2 45◦ + 𝜎1 − 𝜎3 = 2c tan 45◦ + 2 2 (8) where the concluded cohesion c is 31.1 MPa and the internal friction angle φ is 44.8°. By using the Mohr–Coulomb criterion, a shear plane angle β = 45°− φ/2 = 22.6, which is

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oblique to the direction of the major principle stress σ1, can be estimated. The failure modes of the compact sandstone samples are presented in Fig. 6c. The results show a strong influence of the confining pressure on the failure model of the sandstone. Under a confining pressure of 5 MPa, the sample shows a mixed tension and shear fracture mode. Frictional sliding is along the direction of the shear fracture plane. The angle of the major shear fracture plane at a confining pressure of 5 MPa is approximately along the direction parallel to the σ1. When the confining pressure increases to 10 and 15 MPa, a typical shear fracture with local tensile cracks is observed in the samples. The shear frictional sliding is distinct, and the observed angles of the shear fracture planes are 17° and 22°, respectively. It can be seen that the angle of the shear fracture plane at a confining pressure of 15 MPa is equal to that obtained from the linear Mohr–Coulomb criterion. In addition, the fracture plane at a stress of 15 MPa extends across the two end surfaces. Under a confining pressure of 20 MPa, the sandstone sample SY-6 fails by a major shear fracture with some lateral tensile cracks. The shear fracture is not smooth compared with that at a lower confining pressure. In addition, the major plane is found to have ample powder, which is induced by the friction between the fracture surfaces. The angle of the major shear fracture

Environmental Earth Sciences (2018) 77:245

plane is approximately 24°, which is slightly higher than the determined angle using the linear Mohr–Coulomb criterion. Crack development in sandstone The failure process induced by stress is attributed to the initiation, propagation and coalescence of cracks. A number of techniques have been developed to investigate failure processes in rock by many researchers (Eberhardt et al. 1999; Hatzor and Palchik 1997; Pettitt et al. 1998; Wu et al. 2000). These researchers clearly show that the stress–strain behavior of brittle rock can be divided into several stages characterized by changes in the measured axial and lateral strains. Typical stress–strain curves of the sandstone sample SY-3 are shown in Fig. 7. When the stress is within the crack closure stress 𝜎cc , the preexisting microcracks in the sample close. Our experimental results showed that the closure of microcracks is related to the confining pressure. The closure of microcracks at a lower confining pressure is more distinct than that at a higher confining pressure (Fig. 6). After crack closure is attained, the elastic section is normally followed by a

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concave curve. The elastic modulus E and Poisson’s ratio ν can be determined from this portion. The crack initiation stress threshold 𝜎ci is defined as the point where the lateral strain curve departs from linearity. When the stress is higher than this threshold, stable growth occurs, which implies that additional stress should be loaded in order for the cracks to extend. Using a method suggested by Martin (1993), this stress threshold can be determined. The crack damage stress 𝜎cd marks the beginning of region IV and the onset of unstable crack growth. During this stage, crack coalescence will continue, even if the applied stress is kept constant. Based on the experimental results, the critical stress of sandstone samples under different confining pressures are listed in Table 5. It can be seen that the average values of 0.13 𝜎p for the crack closure stress threshold, and 0.36 𝜎p for the crack initiation stress threshold are determined. The average crack damage stress 𝜎cd is an amount of 0.89 of the peak stress 𝜎p. After the characteristic stress levels have been identified, the damage process of the sandstone can be characterized. The permeability that is associated with the development of cracks can be investigated.

Fig. 7  Typical stress–strain diagram showing the stages of crack development in sample SY-3

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Table 5  Critical stress of sandstone samples under different confining pressures

Sample

PC (MPa)

𝜎cc (MPa)

𝜎ci (MPa)

𝜎cd (MPa)

𝜎p (MPa)

SY-2 SY-3 SY-5 SY-6

5 10 15 20

18.10 (0.10σp) 30.11 (0.14σp) 33.17 (0.14σp) 37.06 (0.14σp)

68.81 (0.39σp) 74.03 (0.35σp) 85.98 (0.36σp) 90.39 (0.34σp)

154.47 (0.88σp) 195.55 (0.92σp) 193.21 (0.82σp) 245.07 (0.93σp)

175.68 210.99 236.42 263.47

Absolute permeability/m2

6.0x10-18

5.0x10

-18

4.0x10

-18

SY-2 SY-3 SY-5 SY-6

Crack growth and coalescence

3.0x10-18

σcc

σcd

σci

2.0x10-18

1.0x10-18

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

σcc

6.0x10-18

Absolute gas permeability/m2

245  

1.0

Normalized deviatoric stress

σci

5.0x10-18

σcd σp

4.0x10-18 3.0x10

-18

2.0x10

-18

1.0x10-18 0.0

y=2.582×10-18e-0.0765 R2=0.848 y=2.894×10-18e-0.0824 R2=0.995

y=1.177×10-17e-0.148 R2=0.906

y=2.234×10-18e-0.0829 R2=0.997 5

10

15

20

Confining pressure/MPa

Fig. 8  Relationship between absolute permeability and normalized deviatoric stress under different confining pressures

Fig. 9  Relationship between permeabilities at critical stresses and confining pressure

Evolution of the gas permeability with deviatoric stress

the stress is less than 𝜎cc because the flow channels become narrow. (2) If the stress is less than 𝜎ci , the permeability continues to decrease with the elastic compaction of the sandstone samples. (3) When the stress is higher than 𝜎ci , microcracks form or propagate mainly in the direction parallel to the axial stress. In addition, newly formed microcracks are not connected into the flow channels. Therefore, the change in the permeability is the limit. (4) The onset of microcrack coalescence starts at the crack damage stress 𝜎cd ; the gas can flow through the new channels; and the permeability increases rapidly during this stage. (5) During the post-peak stage, macrocracks or shear bands are formed. Therefore, the permeability increases to the peak value.

As mentioned before, when external parameters such as temperature and pore pressure are constant, the slippage effect is only affected by the pore structures. During the process of deformation induced by deviatoric stress, the pore structure changes under different stress states. Therefore, the influence of slippage on the apparent permeability is not always the same. However, from the study of SY-1, the slippage factor only varies from 0.174 to 0.377 MPa. The mean value of the slippage factor is equal to 0.27 MPa and is used to correct the gas permeability test during triaxial compression. Based on large amount of test data, the constant slippage factor b was used by Wang et al. (2013) to obtain the intrinsic permeability under triaxial compression conditions. Therefore, only the corrected permeability will be discussed in the next phase. To investigate the permeability evolution with crack development, the relationship between the intrinsic permeability and the normalized deviatoric stress is shown in Fig. 8, where the normalized deviatoric stress is defined as the ratio between the deviatoric stress and peak stress 𝜎p . It can be seen that the permeability evolution of sandstone during triaxial compression can be divided into five stages according to the critical stress thresholds. (1) With the closure of preexisting cracks, the permeability decreases when

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Evolution of the permeability with confining pressure Based on the experimental results, the relationship between the permeabilities at the critical stresses and confining pressure are plotted in Fig. 9 under a triaxial compression condition. The results show the strong influence of the confining pressure on the intrinsic permeability of sandstone. In particular, the permeability at peak stress 𝜎p changes greatly with the confining pressure. For instance, the permeability under a confining pressure of 5 MPa is 5.85 × 10−18 m2. Under a confining pressure of 10 MPa, the permeability is reduced to 1.91 × 10−18 m2, which is an almost a 67%

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decrease compared with the value under 5 MPa. Under a confining pressure of 20  MPa, the permeabilities at the crack closure stress 𝜎cc , crack initiation stress 𝜎ci and crack damage stress 𝜎cd decrease by 1.34 × 10−18, 1.05 × 10−18 and 1.05 × 10−18 m2, respectively. Compare with the peak stress, These values do not change significantly. Previous studies have shown that the relationship between the permeability and the effect stress can be described by an exponential function (David et al. 1994). Under hydrostatic stress conditions, the samples only contain initial defects. When the deviatoric stress is applied to a critical stress threshold ( 𝜎cc , 𝜎ci , 𝜎cd and 𝜎p ), the corresponding damage states are crack closure, crack initiation, stable crack growth and fracture formation, respectively. The exponential function is examined whether the permeability at the critical stress threshold is related to the confining pressure, similar to that under hydrostatic stress conditions: (9) where K ′ and 𝜂 are constants. The fitting lines using Eq. (9) are shown in Fig. 9 for the crack closure stress 𝜎cc , crack initiation stress 𝜎ci , crack damage stress 𝜎cd and peak stress 𝜎p . In Fig.  9, the constant K ′ values are 2.849 × 10−18, 2.234 × 10−18, 2.582 × 10−18 and 1.177 × 10−17 m2 for different critical stresses. However, the corresponding constant 𝜂 values are 0.0824/MPa, 0.0829/MPa, 0.0765/MPa and 0.148/MPa. It is found that the values of R2 are 0.995, 0.997, 0.848 and 0.906. Therefore, the permeabilities at the critical stress thresholds decrease nonlinearly with the confining pressures. Natural rock is always under a complex stress state. For example, the permeability property of rock in an excavation damaged zone (EDZ) is of great significant for the cavern stability. Knowledge of the permeability evolution with confining pressure can help us understand the weak point of a long cavern at different stress stages.

K = K � exp(−𝜂pc )

Evolution of the gas permeability with axial strain In the previous section, we discussed the stress-dependent permeability before the peak stress. Then, the relationship between the deviatoric stress and strain and the intrinsic permeability-axial strain during the completed process is plotted in Fig. 10. The intrinsic permeability is corrected from the apparent gas permeability. A series of platforms in the stress–strain curves are induced by the creep of sandstone under constant a deviatoric stress during gas permeability testing. According to the permeability evolution with crack development, the change in the permeability from crack closure stress to crack damage stress is the limit. Therefore, the permeability before peak stress can be divided into three regions: the initial defect closure stage (region I); the linear elastic deformation stage (region II), and the yield to peak strength stage (region III). The permeability of post-peak

200

245

2.5x10-17

σ1−σ3/MPa

Κ−ε1 2.0x10-17

150

1.5x10-17 100

σ−ε3 σ−ε1 50

-35

-30

-25

-20

-15

ε3/10−3

-10

-5

0

1.0x10-17

5.0x10-18

0

5

10

15

20

Absolute permeability/m2

Environmental Earth Sciences (2018) 77:245

0.0 25

ε1/10−3

Fig. 10  Relationship between absolute permeability and axial strain of compact sandstone sample SY-2

stress region can be divided into two regions: the post-peak stress softening (region IV) and the residual strength (region V). In region IV, the sandstone samples failed and more fractures are formed. The permeability increases to a peak value. For sample SY-2, the permeability varies from 5.44 × 10−18 to 2.29 × 10−17 m2. When the deviatoric stress reaches the residual value, the existing fractures remain stable. Therefore, the permeability is almost constant. The ultimate permeability of sample SY-2 remains at 3.31 × 10−17 m2.

Discussions Owing to the determined intrinsic permeability, the difference between the corrected gas permeability and water permeability evolution during the triaxial compression tests is discussed. Overall, the mechanical properties for the gas and water seepage experiments are qualitatively the same. The evolution in the volumetric strain and permeability with deviatoric stress for samples SY-3 and SY-4 are plotted in Fig. 11. At same confining pressure and pore pressure, the peak stress of the sandstone sample is 189.31 MPa, which is approximately 89% that of sample SY-4. The result shows a strong weakening of the strength of sandstone due to water. Aqueous solutions can weaken the deformation of sandstone in many ways. Baud et al. (2000) conducted triaxial compression experiments on four different types of sandstone with porosities ranging from 11 to 35%. The results indicated that the weakening effect of water on sandstone during the brittle faulting regime can be attributed to a reduction in the specific surface energy as the result of the absorption of pore fluid onto the internal pore surfaces. Additionally, water weakens rocks by promoting subcritical crack growth and reducing the friction between the fractured surfaces. As an inert gas, the transport medium argon does not interact with the rock matrix. Note that the

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Environmental Earth Sciences (2018) 77:245

50 K-σ (SY-3) εv-σ (SY-3)

40

K-σ (SY-4)

Dilatancy

εv(10-3)

30 20

1.4x10-17 1.2x10-17 1.0x10-17 8.0x10-18

Compression

6.0x10-18

10

4.0x10-18

0

2.0x10-18

-10 0

50

100

150

(σ1−σ3)(MPa)

200

Absoulte permeability(m2)

εv-σ (SY-3)

0.0 250

Fig. 11  Evolution of volumetric strain and permeability with deviatoric stress for sample SY-3 and SY-4

seepage media is the only difference between the tests. Thus, we assume that the different evolutions of the corrected gas and water permeability with stress can be mainly attributed to the weakening effect of water on sandstone. Under a triaxial compression condition, water acts to differentiate the water permeability from the gas permeability via effecting the subcritical crack growth (Heap et al. 2009). Compared with argon, water, which has a higher viscosity, can draw and connect the grains together. Moreover, the very small pore radii of water (Fig. 1) make the molecules easier to absorb onto the internal pore surfaces. Therefore, the deformation of sample SY-4 is smaller than that of the SY-3. For this reason, the water permeability is lower than the gas permeability during the crack closure stage. Figure 11 shows that there is a good correlation between the permeability and the volumetric strain. The results indicate that the evolution of the permeability can be attributed to the initiation, propagation and coalescence of microcracks. However, the reactions that occur preferentially between water and the strained atomic bonds close to crack tips accelerate cracks growth. Water is a lubricant that can reduce the connection and frictional force between grains. In addition, if two grains in the sandstone are close to each other, the grains will draw the water molecules into the surrounding area, and fill in the gap between the grains. Therefore, the water permeability increases quickly with crack growth and is higher than the gas permeability. When the samples fail, the grain powder in the shear fractures flow with the water. Finally, the ultimate permeability of water is higher than gas due to wider flow paths.

Conclusions In this paper, both hydrostatic compression and conventional triaxial compression experiments were conducted on compact sandstone to investigate the inert gas flow behavior.

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Based on the experimental results, the following conclusions can be drawn. During the hydrostatic compression tests, the permeability decreases nonlinearly with the confining pressure. When the confining pressure is constant, the permeability decreases with the pore pressure. Both the exponential and power law equations can be used to describe the stressdependent permeability of compact sandstone. The slippage effect influences the gas permeability significantly. Based on the experimental results, a second-order equation is put forward to determine the intrinsic permeability of sandstone and predict and explain the gas slippage effect. During the triaxial compression test, the failure process of sandstone can be divided into the crack closure stage, linear elastic deformation stage, stable crack growth stage, unstable crack growth stage and post-peak stage. The critical stress thresholds corresponding to different stages are determined. The critical stress thresholds reflect the crack development in the samples. Therefore, the permeability at different stages are different from each other. The relationship between permeabilities at the critical stress thresholds and the confining pressure can be described by an exponential function. The variation in the intrinsic gas permeability is significantly different than the variation in the water permeability during the triaxial compression tests. These test results indicate that the water permeability is lower at lower stress values and higher at higher stress values. This phenomenon can be attributed to the interaction between the water and sandstone grains. The ultimate water permeability is higher than the gas permeability because the grain powder in the shear fractures flow with water. Acknowledgements  This work presented in this paper was financially supported by the National Natural Science Foundation of China (Grant Nos. 11172090, 11572110, 11672343), and the Natural Science Foundation of Jiangsu Province (Grant No. BK2012809). We thank AJE (www.aje.com) for its linguistic assistance during the preparation of this manuscript.

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