J. Cent. South Univ. (2017) 24: 2396-2405

DOI: https://doi.org/10.1007/s11771-017-3651-1

Stress dependent permeability and porosity of low-permeability rock

JIA Chao-jun(贾朝军)^{1, 2}, XU Wei-ya(徐卫亚)¹, WANG Huan-ling(王环玲)^{1, 3},

WANG Ru-bin(王如宾)¹, YU Jun(俞隽)¹, YAN Long(闫龙)¹

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

Central South University Press and Springer-Verlag GmbH Germany 2017

Abstract: The seepage property of low-permeability rock is of significant importance for the design and safety analysis of underground cavities. By using a self-developed test system, both permeability and porosity of granite from an underground oil storage depot were measured. In order to study the influence of rock types on permeability, a tight sandstone was selected as a contrast. The experimental results suggested that the porosity of this granite is less than 5% and permeability is low to 10^{–20 m2} within the range of effective stress. During the loading process, both exponential relationship and power law can be utilized to describe the relationship between effective stress and permeability. However, power law matches the experimental data better during the unloading condition. The stress dependent porosity of granite during loading process can be described via an exponential relationship while the match between the model and experimental data can be improved by a power law in unloading paths. The correlation of permeability and porosity can be described in a power law form. Besides, granite shows great different evolution rules in permeability and porosity from sandstone. It is inferred that this difference can be attributed to the preparing of samples and different movements of microstructures subjected to effective stress.

Key words: permeability; porosity; effective stress; steady-state method; transient pulse method; low-permeability

1 Introduction

The permeability of low-permeability rock is very important for many engineering applications, such as the development of unconventional gas reservoir, the storage of gas and oil, the construction of hydropower station, the reserve of nuclear waste. Permeability is one of the most important parameters which is employed in the evaluation, site selection, exploration and development of large-scale civil engineerings. Compared with common rock projects, low-permeability rocks usually have lower porosity (less than 15%) and less permeability (less than 10^–15m2=1 mD). The testing of permeability and porosity of such low-permeability rocks becomes a crucial problem which restricts the development of projects.

The two most widely methods to evaluate permeability are the steady-state method and the transient pulse method. The steady-state method applies a constant pressure gradient between two ends of rock sample. Permeability is calculated through Darcy’s law after measuring the steady flow rate. After reaching the same value of pressure during the two ends of the rock sample, a sudden pressure gradient is employed to the upstream side. This technique is called transient pulse method. Permeability is back analyzed from the kinetics of the decay of pressure. The steady-state method which is easy to operate is a fundamental method to evaluate permeability. This method is particularly efficient for relatively high-permeability rocks. The transient pulse method, pioneered by BRACE et al [1], can greatly shorten the test time. Furthermore, the beam bending method [2], thermal expansion kinetics [3, 4] and dynamic pressurization [5] are also available to measure permeability of geomaterials.

Even though the stress dependent permeability and porosity were well documented, they are still complicated. Based on laboratory work, a great number of empirical models that can describe the relationship between permeability/porosity and effective stress have been proposed. In general, these models can be separated into exponential and power law forms. The exponential relationship for describing stress-dependent permeability or porosity takes the form of

                       (1)

where Y is permeability or porosity under effective stress P_e; Y₀ denotes permeability or porosity under atmospheric pressure P₀ (0.1 MPa); ξ is a material constant.

DAVID et al [6] estimated permeability of 5 types of sandstones using this exponential relationship. The minimum value of fitting initial permeability K₀ is 14.8×10^{–15 m2}. EVANS et al [7] proposed the stress- dependent permeability of intact core, fault core and the damage zone of granite. The results indicated that the material constant ξ of intact core is much higher than the other two kinds of granites. Exponential relationship is also utilized to estimate porosity of shale [8] and carbonate [9]. Moreover, St. Peter and Mount Simon sandstones with a porosity of 8%–18% are also best described by an exponential equation [10]. One issue for the exponential relationship is that it yielded poor fittings in the low effective stress ranges [11]. RUTQVIST et al [12] suggested a modified exponential equation:

                    (2)

where φ_r is residual porosity under high effective stress; φ₀ is porosity at zero stress and α is a material constant. Equation (2) can better estimate porosity in low effective stress ranges if the stress varies in a wide range.

The power law describes stress-dependent permeability/porosity as follows:

                               (3)

where ζ is a material constant.

SHI and WANG [13] proposed a power law to describe the stress-dependent permeability of clays provided by MORROW et al [14]. The results indicated that K₀ ranges from 10^–18 to 10^{–14 m2}; ζ ranges from 1.2 to 1.8. DONG et al [15] studied the sandstone and shale from TCDP Hole-A and pointed out that the permeability and porosity of sandstone are 10^–14–10^{–13 m2} and 15%–19%, respectively. The values for shale are 10^–20–10^{–15 m2} and 8%–14%, respectively. It was indicated that a power law is appropriate to describe the stress dependency of permeability and porosity of tested rocks.

Previous expressions indicated that the permeability and porosity changing with effective stress are not unique. The laboratory works about sandstone with a porosity higher than 10% are well documented. However, with the increasing of effective stress, low-permeability rocks undergo small changes of porosity. Permeability variation can range to several orders of magnitude. It is hard to provide a suitable model to explain hydraulic behavior of low-permeability rocks. Besides, the current model mainly focuses on hydraulic behavior during the loading process. A 300×10⁴ m³ underground cavity for the storage of oil is under construction in China at present. The surrounding rock is granite with a porosity less than 5% and permeability low to 10^{–20 m2}. It is a crucial issue to ensure the safety of the cavern during long time operation. In order to provide a suggestion to the better design and safety operate of this cavern, the objective of this work is to build the models to describe stress dependent permeability and porosity. et al [16] pointed out that porosity is a material property and only connected pores can provide the flow paths for the seepage fluid. Consequently, there is no universal permeability-porosity relationship. Based on experimental results, the relationship between permeability and porosity of granite was proposed. The comparison between granite and tight sandstone was analyzed. Finally, the influence of deformation mechanism on permeability was discussed.

2 Methodology

2.1 Description of rock samples

The granite from an underground water-sealed oil storage depot in China was collected for the experiments. It is in light fleshy red color, gneissic structure and tiny weathering appearance. The main mineral components are potassium feldspar, anorthose, quartz, hornblende and biotite. The tight sandstone is from a hydropower station in southwest China. The natural densities of sandstone and granite are 2.45 and 2.61 g/cm³. The initial porosities are 7.82% and 2.51%. Cylindrical samples with a diameter of 50 mm were drilled from rock blocks. In order to shrink the influence of anisotropy, the samples were cored perpendicularly to the bedding plane. The height was 30 mm and two ends of the samples were polished to make sure that the error was less than 0.3 mm. Only the intact samples were selected for the experiment. Before testing, samples were over heated under 105 °C for 3 d until constant mass.

2.2 Experimental procedures

Gas permeability and porosity were conducted on the apparatus which was developed by Hohai University and Lille University. This apparatus consists of a pressure chamber, a confining pressure loading system, a porosity test system and a gas control panel. Both steady-state and transient pulse methods can be applied in this apparatus. Meanwhile, porosity is measured with permeability step by step. Confining pressure ranges from 0 to 60 MPa, and the maximum upstream pressure reaches 10 MPa. This apparatus enables permeability of 10^{–24 m2} to be measured. For the permeability of sandstone, steady-state method is more appropriate to this relatively high permeability geomaterial. When permeability is less than 10^{–18 m2}, it costs several hours even days to reach the stable flow state. As a result, transient pulse method is employed to test permeability of granite. The schematic diagram of experimental apparatus is shown in Fig. 1.

Fig. 1 Schematic diagram of test apparatus for permeability (a) and porosity measurements (b)

When applying steady-state method, the one- dimensional Darcy’s law can be written as

                                (4)

where K, μ and P are gas permeability, viscosity and pressure, respectively; Q denotes steady flow rate; A is cross-sectional area of samples. The boundary conditions are as shown in Fig. 2. The pressure distribution in rock samples can be written as [17]

          (5)

where L is height of samples. The flow rate Q of gas inlet end can be calculated from Eq. (4) and Eq. (5).

                            (6)

Fig. 2 Test boundary condiontion of both steady state and transient pulse methods.

The upstream pressure variation during the time △t is defined as △P. As a result, the mean pressure of upstream can be written as P_m=P_u–△P/2. Gas mass conservation implies that the mass of gas flows from upstream end equals the value flows through rock samples:

                              (7)

where V_u is total volume of upstream pipes. According to Eq. (6) and Eq. (7), the permeability of steady-state method can be expressed as

                           (8)

Gas permeability and porosity measurement are conducted on the selected sandstone and granite. The test procedures are as follows: 1) Measure physical property of samples containing height, diameter, and mass. 2) Package the samples using a rubber wrap and place inside pressure chamber. Multi-pores shims are put in two ends of samples to ensure homogeneous gas flow. 3) Add hydraulic oil to pressure chamber and apply confining pressure to default value. 4) Open valve 1 and valve 2 to inject argon to pipes. The valve 5 is closed with opening to atmosphere of outlet end of samples during the same time. Close valve 1 when the upstream pressure is approximately above 1 MPa. 5) Record the variation of manometer 1 during the time Δt (usually to be 3 min) when the decreasing rate tends to be stable. The minimum recording spots are higher than 6. 6) Apply the next level of confining pressure. Meanwhile, pore pressure is applied as previous. 7) Repeat steps 5) and 6)

When using transient pulse method, the boundary condition is the same as that shown in Fig. 2. Firstly, the same pressure is applied to both sides of upstream and downstream (P_u=P_d). A sudden increasing pressure ΔP is applied to upstream side then. Knowing that the pulse pressure is negligible compared with initial stage, permeability is obtained by choosing the coefficient c [18]:

                  (9)

where P_f is the pressure at the end of the test; V₁ and V₂ are the volumes of gas reservoirs.

Compared with steady-state method, the transient pulse technique is carried out by the same apparatus which adds the imaginary line section in Fig. 1(a). Follow the first three steps as the steady-state method, 4) close valve 5 and open the other four valves to inject argon to samples from both ends. When the pressure is slight higher than 1 MPa, close valve 1 and valve 3. 5) Open valve 1 and apply a pulse pressure of 0.5 MPa when previous stable pressure is achieved. 6) Measure the variation of pulse pressure with time. The recording spots are over 12 points. 7) The next level of confining pressure is subjected to samples. Repeat steps 4)–6) until the end of test.

Porosity is conducted alternatively with permeability on same apparatus with the porosity part as shown in Fig. 1(b). Before measurement, a steel sample with the same dimension is inserted into the pressure chamber to calibrate the constant volumes of V_a and V_b of test system. Argon with pressure slight above 1 MPa is injected to test system by opening valve 1 firstly. The valve 2 and valve 3 are closed during the same time. The ultimate stable value of manometer P_a is then recorded after closing valve 1. After that, gas flows to samples from both ends by opening valves 2 and 3. The finial stable value of manometer P_b is recorded. The injected gas is argon of above 99% purity which can be consider a perfect gas. According to law of conservation of mass to perfect gas, we obtain

                    (10)

where V_v is pore volume of samples. In that way, porosity can be written as

                             (11)

The accuracy of porosity can reach to 0.01%.

3 Results

Figures 3 and 4 show variation of permeability and porosity of granite and sandstone with effective stress. The effective stress defines as the difference between confining pressure and pore pressure. It can be seen that an increase of hydrostatic load causes the permeability decreases to a relative low level, with porosity of both samples decreases during the same process. This can be explained by the closure of initial microcracks and pores subjected to confining pressure. During the unloading process, both permeability and porosity increase but the values are consistently lower than the loading process. Moreover, the gap between loading and unloading grows with the decrease of effective stress. This can be attributed to porous media which contains plenty of microcracks and pores. The decreasing of permeability relates to the compaction of pore system. This elastic deformation cannot be fully restored with the decrease of effective stress. The unrecoverable deformation shows accumulative effect on permeability which turns out to be an increasing discrepancy between the values of loading process.

Fig. 3 Permeability of sandstone (a) and granite versus effective stress (b)

Granite shows lower permeability and tighter porosity than sandstone. For instance, when the effective stress is 4.5 MPa, the permeability of granite is 12.43×10^–20 m² while the value of sandstone is 12.73× 10^{–17 m2}, which is almost three orders of magnitudes higher than granite. The porosities of these two types of rock under the same effective stress are 5.60% and 1.14%, respectively. The porosity of sandstone is about 5 times that of granite. It is notable that low permeability undergoes fairly small porosity changes with significant decrease in permeability. In fact, porosity as well as pore structures has great effect on permeability [19]. As a matter of comparison, granite shows more sensitive to stress path. For instance, the permeability of sandstone is about 61.18% of initial value while it is 23.38% for granite when the stress restores completely. For comparison, the porosity of sandstone can restore to 86.30% of initial porosity whereas 13.30% for granite.

Fig. 4 Porosity of sandstone (a) and granite (b) versus effective stress

3.1 Models for describing effective stress dependent permeability

Even though the relationship between effective stress and permeability is well documented, the models for describing the stress dependent permeability are also a controversial problem. Some researchers pointed out [6, 7, 9, 20, 21] that exponential relationship can be employed to describe their correlation. DAVID et al [6] found that the material constant γ=(6.21–18.1)×10^–3 MPa^–1 and initial permeability K₀=(14.8–2166)× 10^{–15 m2} for five types of sandstones. EVANS et al [7] suggested that γ=0.035–0.200 MPa^–1 and K₀=1.26×10^–14– 5.92×10^{–18 m2} for intact core, fault core and damage core of granite. The equation describing the relationship between effective stress and permeability can be written as

                     (12)

On the other hand, the other researchers [15, 22, 23] suggested that power law is superior to an exponential relationship for describing the stress dependent permeability:

                             (13)

DONG et al [15] studied sandstone from TCDP and concluded that β=0.057–0.303 and K₀=(1.14–9.07)×10^{–14 m2}. Due to the importance in safety analysis of large-scale civil engineerings, a rigid model to describe the stress dependent permeability must be put forward, especially for such compact granite. Even though the existed two types of models have been used widely, the differences are huge when the measured permeability spans a large ranges. The models, as a results, should be verified.

Based on laboratory works, both Eqs. (12) and (13) are used to estimate permeability. The fitting parameters are given in Table 1. During the loading and unloading process, the constant materials γ of sandstone are 0.053 and 0.033 MPa^–1, respectively. They are much higher than that reported by DAVID et al [6]. As a result, the sandstone test in my laboratory is more sensitive than previous. In addition, the sensitive of permeability to variation of effective stress during unloading process is less than loading process. The initial permeabilities are 16.60×10^{–17 m2} and 9.26×10^{–17 m2} for sandstone whereas K₀=(4.82–23.69)×10^{–20 m2} and γ=0.031–0.069 MPa^–1 for granite. It is notable that the initial permeability of granite is smaller and more sensitive to effective stress.

Power law fitting indicates that the material constants for sandstone are 0.466 and 0.333 with initial permeabilities of 71.24×10^–17 and 27.44×10^{–17 m2}. The fitting parameters for granite show the same rule compared with exponential relationship. For instance, the material constant for granite is 0.49 during the loading process which is higher than that of sandstone. During the unloading process, this parameter is 0.30, slight less than 0.333 of sandstone.

Through comparing two fitting methods, both power law and exponential relationship during the loading process fit the test results well. On the other hand, the unloading condition prefers to power law whose correlation coefficients are higher than 0.99. Even though its flaw is quite obvious, the initial permeability fitted by power law is also four times that of exponential relationship. This power law correlation is a relatively conservative estimate for engineering.

Table 1 Fitting parameters for permeability

Fig. 5 Comparison between models adopting an exponential relationship and a power law for permeability of sandstone (a) and granite (b)

3.2 Models for describing effective stress dependent porosity

The stress dependent porosity has been studied extensively due to their importance in engineering applications. In general, this model can be divided into exponential relationship and power law. Exponential relationship has been used to describe the stress dependent porosity of sandstone [10], shale [24] and carbonate [9]:

                     (14)

where φ₀ is porosity under atmospheric pressure; α is material constant which denotes compressibility of pores. Equation (14) is employed to fit porosity of sandstone and granite. The fitting parameters are given in Table 2. During the loading process, φ₀=1.80% and α=0.071 MPa^–1 for granite. However, the fitting parameters for sandstone are φ₀=6.40% and α=0.014 MPa^–1. It is obvious that the granite used in our experiment is tighter and more compressible than sandstone. During the unloading process, both φ₀ and α decrease for sandstone and granite.

A power law with the form:

                              (15)

is used to describe the stress dependent porosity, where ω is a material constant. Using Eq. (15), the stress dependent porosity can be estimated. The determined parameters in the power law for sandstone are φ₀=8.14%–10.87% and ω=0.11–0.16 whereas for granite are φ₀=0.82–5.44% and ω=0.44–0.45. These determined φ₀ values are quite higher than that of exponential equation. Furthermore, there is no significant difference between fitting parameters for different stress paths of sandstone and granite. For sandstone, the experiment results indicate that the power law is superior to the exponential relationship for describing stress dependent porosity under loading and unloading conditions. Stress path shows great influence on porosity of granite. During the loading process, exponential relationship is better to estimate experimental data. On the contrary, the match between the models and the test results can be greatly improved by using power law during unloading process.

3.3 Relationship between permeability and porosity

The compaction of the flow paths induced by applying of effective stress is the main factor in causing the decreasing of permeability. As a result, porosity is the bridge which connects the permeability changes and effective stress variation. However, porosity is dominated by the volume of pores, while the volume as well as pore structures influences permeability. Pore structures are determined by rock types. Therefore, there is no universal relationship between permeability and porosity. Previous studies suggested that the following power law can be used to describe the permeability– porosity relationship induced by hydrostatics compaction:

                              (16)

where m denotes porosity sensitive coefficient. DAVID et al [6] proposed that m ranges greatly for different geomatierials. Equation (16) is employed to fit with sandstone and granite, as shown in Fig. 7. The determined parameters m for sandstone are 2.64 and 2.84 which fall within the range of the parameters reported by DAVID et al [6]. A much lower m is determined for granite, 0.64–0.98 for loading and unloading conditions which indicates lower porosity sensitive compared with sandstone.

Table 2 Fitting parameters for porosity

Fig. 6 Comparison between models adopting an exponential relationship and a power law for porosity of sandstone (a) and granite (b)

4 Discussion

Sandstone is a typical sedimentary rock. The permeability parallel to bedding plane is much higher than perpendicular direction [25]. In order to reduce the effect of anisotropic, the samples are cored perpendicularly to the bedding plane. In this way, the test results can only reflect permeability evolution under typical condition. From previous study, the stress dependent of permeability/porosity shows much difference from that of granite. Both permeability and porosity of granite are much lower than sandstone. In addition, the uniaxial compressive strength as well as brittleness is higher than sandstone. The damage to samples is inevitable during the coring and drilling. Therefore, the additional damage to granite affects the permeability evolution during the initial low stress ranges.

The scanning electron microscopy (SEM) images of sandstone and granite are given in Fig. 8. A is microcracks and B is pores which build up the flow paths for gas. The sandstone with short microcracks is dominated by pores while the granite is opposite. Compared with pores, microcracks easily deformed which means sharply reduction of permeability of small increase of effective stress in the low stress ranges. KLINKENBERG [26] suggested that the permeability of low permeability rock measured by gas is different from that measured by water. The relationship between gas permeability and water permeability can be expressed by equation as follows:

                           (17)

where K_g is gas permeability; K_∞ is water permeability.

Fig. 7 Relationship between permeability and porosity for sandstone and granite under loading and unloading processes:

Fig. 8 SEM images of sandstone (a) and granite (b) (A: microcracks; B: pores)

P_av is mean pore pressure and b_g is slippage factor which reflects the influence of slippage effect on measured gas permeability. The slippage factor b_g depends on pore structures, temperature, gas types and so on. The influence of slippage effect on sandstone and granite is different. Therefore, these two types of rock show greatly different in permeability variation.

In this work, the effective stress is defined as the difference between confining pressure and pore pressure. For more general condition, the effective stress can be defined as [27–29]:

                               (18)

where λ is an effective stress coefficient which reflects the influence of pore pressure on permeability. Previous studies indicate that λ is a constant which relates on rock types, fracture shapes and connection, porosity, pore geometries and tortuosity. Generally speaking, λ is less than 1.0. AL-WARDY and ZIMMERMAN [19] pointed out that effective stress coefficients on clay-rich sandstone are greater than 1.0. The effective stress coefficient by Walls varies from 1.2 for clean sandstone to 7.1 for sandstone contains 20% clay. The experimental results also indicated that the effective stress coefficient increases with clay content [23]. In order to determine the value of effective stress coefficient, experiments should be run at different confining pressures and pore pressures. Even though the measurement was conducted on signal pore pressure in this work, we can estimate crudely how λ varies as a function of rock types. The SEM image (Fig. 8) of sandstone shows that the main mineral component is quartz, trace amounts of clay particles clog the pore throat, therefore the sensitivity of permeability to pore pressure is slighter higher than a rock consisting of single mineral. Thus, the value of would be slight higher than 1.0. BERRYMAN [30] pointed out that λ for a clay-free rock should not exceed 1. Compared with the result shown in Fig. 3 and that published by WALSH [29], we infer that it is higher density of fracture that makes fluid transfer effectively. As a result,is quiet close to 1.0; otherwise, it is small. Even though a large number of microcraks distribute in granite SEM image, there are no visible fractures in macrostructure. Therefore, λ of clay-free granite is much less than 1.0.

5 Conclusions

1) During the loading process, both permeability and porosity decrease with the increasing of effective stress and then turn to restore due to the reduction of effective stress. But the values are consistently lower than those in the loading conditions. Moreover, the gap gets wider when the stress is smaller. Compared with tight sandstone, porosity of granite is lower and permeability can decline up to 10^{–20 m2}. Almost three orders of magnitude drop of permeability is observed for granite and accompanied with only five times of porosity reduction compared with sandstone. Based on the experimental results, it can be inferred that permeability is determined by porosity as well as pore structures.

2) During the loading process, both exponential relationship and power law can be utilized to describe the relationship between effective stress and permeability of granite. Notably, the determined initial permeability K₀ is overestimated when the power law is adopted. Power law is superior to exponential relationship for describing the stress dependent permeability of granite with unloading condition. Stress paths also have great effect on porosity. The stress dependent porosity of granite during loading process can be described via an exponential relationship while the match between the model and experimental data can be improved by a power law in unloading path. The correlation of permeability and porosity can be described in a power law form. The determined porosity coefficient m for sandstone is 2.64–2.84 whereas m=0.64–0.98 for granite.

3) The relationship to describe stress dependent permeability/porosity various from different rock types. In addition, significant difference exists in the determined parameters for different types of rock.

4) The observed difference of permeability in different types of rock can be illustrated through different degree of damage of samples when coring and drilling. It is also postulated that the microstructures which contain microcracks and pores contribute to the permeability variation. Based on experimental results, effective stress coefficient for sandstone is slighter than 1.0 and for granite is much lower than 1.0.

Nomenclature

P_e Effective stress (MPa)

P₀ Atmospheric pressure (0.1 MPa)

Y Permeability (K) or porosity (φ) under effective stress

Y₀ Permeability (K₀) or porosity (φ₀) under atmospheric pressure

ξ, ζ, ω, γ,

α, β, m

Material constants

φ_r Residual porosity under high effective stress (%)

Q Steady flow rate (m³/s)

μ Viscosity of Argon (Pa·s)

P  Pressure (MPa)

A  Cross sectional area of rock samples (m²)

L  Length of rock samples (m)

P_u Pressure of upstream side of the samples (MPa)

P_d Pressure of downstream side of the samples (MPa)

P_m Mean pressure of upstream side of the samples (MPa)

△P Pressure gradient (MPa)

△t Time variation (s)

V_u Total volume of upstream pipes (m³)

P_f Pressure at the end of the permeability test (MPa)

V₁, V₂ Volumes of gas reservoirs (m³)

V_a, V_b Constant Volumes of test system (m³)

V_v  Pore volume of rock samples (m³)

P_a, P_b Stable values of manometer (MPa)

K_g  Gas permeability (m²)

K_∞ Water permeability (absolutely permeability) (m²)

P_av Mean pore pressure (MPa)

b_g  Gas slippage factor (MPa)

λ Effective stress coefficient

References

[1] BRACE W F, WALSH J B, FRANGOS W T. Permeability of granite under high pressure [J]. Journal of Geophysical Research, 1968, 73(6): 2225–2236.

[2] SCHERER G W. Bending of gel beams: method for characterizing elastic properties and permeability [J]. Journal of Non-crystalline Solids, 1992, 142: 18–35.

[3] SCHERER G W. Thermal expansion kinetics: Method to measure permeability of cementitious materials: I, theory [J]. Journal of the American Ceramic Society, 2000, 83(11): 2753–2761.

[4] AI H, YOUNG J F, SCHERER G W. Thermal expansion kinetics: Method to measure permeability of cementitious materials: II, Application to hardened cement pastes [J]. Journal of the American Ceramic Society, 2001, 84(2): 385–391.

[5] GRASLEY Z C, SCHERER G W, LANGE D A, VALENZA J J. Dynamic pressurization method for measuring permeability and modulus: II. Cementitious materials [J]. Mater Struct, 2007, 40(7): 711–721.

[6] DAVID C, WONG T F, ZHU W, ZHANG J. Laboratory measurement of compaction-induced permeability change in porous rocks: Implications for the generation and maintenance of pore pressure excess in the crust [J]. Pure and Applied Geophysics, 1994, 143(1–3): 425–456.

[7] EVANS J P, FORSTER C B, GODDARD J V. Permeability of fault- related rocks, and implications for hydraulic structure of fault zones [J]. Journal of Structural Geology, 1997, 19(11): 1393–1404.

[8] DICKINSON G. Geological aspects of abnormal reservoir pressures in Gulf Coast Louisiana [J]. AAPG Bulletin, 1953, 37(2): 410–432.

[9] SCHMOKER J W, HALLEY R B. Carbonate porosity versus depth: A predictable relation for south Florida [J]. AAPG Bulletin, 1982, 66(12): 2561–2570.

[10] HOHOLICK J D, METARKO T, POTTER P E. Regional variations of porosity and cement: St. Peter and Mount Simon sandstones in Illinois Basin [J]. AAPG Bulletin, 1984, 68(6): 753–764.

[11] ZHENG J, ZHENG L, LIU H H, JU Y. Relationships between permeability, porosity and effective stress for low-permeability sedimentary rock [J]. International Journal of Rock Mechanics and Mining Sciences, 2015, 78: 304–318.

[12] RUTQVIST J, WU Y S, TSANG C F, BODVARSSON G. A modeling approach for analysis of coupled multiphase fluid flow, heat transfer, and deformation in fractured porous rock [J]. International Journal of Rock Mechanics and Mining Sciences, 2002, 39(4): 429–442.

[13] SHI Y, WANG C Y. Pore pressure generation in sedimentary basins: Overloading versus aquathermal [J]. Journal of Geophysical Research: Solid Earth (1978–2012), 1986, 91(B2): 2153–2162.

[14] MORROW C, SHI L, BYERLEE J. Permeability of fault gouge under confining pressure and shear stress [J]. Journal of Geophysical Research: Solid Earth (1978–2012), 1984, 89(B5): 3193–3200.

[15] DONG J J, HSU J Y, WU W J, SHIMAMOTO T, HUNG J H, YEH E C, WU Y H, SONE A. Stress-dependence of the permeability and porosity of sandstone and shale from TCDP Hole-A [J]. International Journal of Rock Mechanics and Mining Sciences, 2010, 47(7): 1141–1157.

[16] BERNABE Y, MOK U, EVANS B. Permeability-porosity relationships in rocks subjected to various evolution processes [J]. Pure and Applied Geophysics, 2003, 160(5, 6): 937–960.

[17] DANA E, SKOCZYLAS F. Gas relative permeability and pore structure of sandstones [J]. International Journal of Rock Mechanics and Mining Sciences, 1999, 36(5): 613–625.

[18] BURLION N, SKOCZYLAS F, DUBOIS T. Induced anisotropic permeability due to drying of concrete [J]. Cement and Concrete Research, 2003, 33(5): 679–687.

[19] AL-WARDY W, ZIMMERMAN R W. Effective stress law for the permeability of clay-rich sandstones [J]. Journal of Geophysical Research: Solid Earth (1978–2012), 2004, 109(B4): 1029–1039.

[20] MCKEE C, BUMB A, KOENIG R. Stress-dependent permeability and porosity of coal and other geologic formations [J]. SPE Formation Evaluation, 1988, 3(01): 81–91.

[21] WANG H, XU W, ZUO J. Compact rock material gas permeability properties [J]. Physica B: Condensed Matter, 2014, 449: 10–18.

[22] SHI Y, WANG C Y. Generation of high pore pressures in accretionary prisms: Inferences from the Barbados Subduction Complex [J]. Journal of Geophysical Research, 1988, 93(B8): 8893–8910.

[23] GHABEZLOO S, SULEM J, GU DON S, MARTINEAU F. Effective stress law for the permeability of a limestone [J]. International Journal of Rock Mechanics and Mining Sciences, 2009, 46(2): 297–306.

[24] ATHY L F. Density, porosity, and compaction of sedimentary rocks [J]. AAPG Bulletin, 1930, 14(1): 1–24.

[25] RENNER J, HETTKAMP T, RUMMEL F. Rock mechanical characterization of an argillaceous host rock of a potential radioactive waste repository [J]. Rock Mechanics and Rock Engineering, 2000, 33(3): 153–178.

[26] KLINKENBERG L. The permeability of porous media to liquids and gases [C]// Proceedings of the Drilling and Production Practice, F, American Petroleum Institute, 1941.

[27] NUR A, BYERLEE J D. An exact effective stress law for elastic deformation of rock with fluids [J]. Journal of Geophysical Research, 1971, 76(26): 6414–6419.

[28] ZOBACK M D, BYERLEE J. Permeability and effective stress: Geologic notes [J]. AAPG Bulletin, 1975, 59(1): 154–158.

[29] WALSH J. Effect of pore pressure and confining pressure on fracture permeability [C]// Proceedings of the Int J Rock Mech Min, F, 1981 Pergamon: Elsevier, 18: 429–435.

[30] BERRYMAN J G. Effective stress for transport properties of inhomogeneous porous rock [J]. Journal of Geophysical Research: Solid Earth, 1992, 97(B12): 17409–17424.

(Edited by HE Yun-bin)

Cite this article as: JIA Chao-jun, XU Wei-ya, WANG Huan-ling, WANG Ru-bin, YU Jun, YAN Long. Stress dependent permeability and porosity of low-permeability rock [J]. Journal of Central South University, 2017, 24(10): 2396–2405. DOI:https://doi.org/10.1007/s11771-017-3651-1.

Foundation item: Projects(11172090, 51479049, 11272113, 11572110, 51209075) supported by the National Natural Science Foundation of China; Project(BK2012809) supported by the Natural Science Foundation of Jiangsu Province, China; Project(201406710042) supported by China Scholarship Council

Received date: 2015-11-11; Accepted date: 2016-03-17

Corresponding author: JIA Chao-jun, PhD candidate; Tel: +86–18066109363; E-mail: jia-chao-jun@163.com