Experimental Study on the Time-Dependent Gas Permeability of Fractures in Shales

Abstract

The gas permeabilities of shale fractures provide a critical basis for deeply understanding the subsurface fluid flow processes in many underground engineering activities in shales. However, the time-dependent behavior of shale permeability under formation stress has been rarely reported. In this study, two artificially fractured shale cores were used to experimentally investigate the time dependence of the fracture gas permeability and underlying mechanisms. Daily measurements of permeabilities were conducted at various gas pressures under multilevel confining stresses, where the confining stress was incrementally changed from 10 to 25 MPa and then reverted to 10 MPa. Numerical calculations were performed to determine the fracture apertures. The experimental results reveal a notable decline in gas permeability and aperture over time under each confining stress during the loading phase that proceeds from a decelerating decline stage to a steady decline stage. The gas permeability can be overestimated by at least a factor of 2 due to fracture creep. The magnitude of time-dependent permeability reduction is related to the fracture geometry, contact area, spatial distribution of apertures, and fracture stiffness. The observed significant permeability loss and limited time-dependent permeability recovery during the unloading stage indicate an irrecoverable process of creep-induced permeability reduction. The gas pressure-dependent permeability suggests notable gas slippage in fractures, which is influenced by creep and exhibits a power function decay with time. Considering the coupling effect of stress creep and gas slippage, a time-dependent gas permeability model is developed and validated using experimental data. This model is helpful to effectively predict the variation trend of fracture permeability during the implementation of underground engineering.

Highlights

• Time-dependent variations in gas permeability of shale fractures were measured and analyzed under multilevel confining stresses.

• Gas slippage phenomenon was observed in the fracture and decayed as a power function with time.

• A gas permeability model of fracture that considers the coupled effects of creep compaction and gas slippage was proposed and validated against experimental data.

1 Introduction

Shales are generally characterized by low porosity, extremely low permeability, and excellent sealing ability, which make them ideal caprocks for gas geological disposal. Due to their abundant hydrocarbon content, shales also serve as unconventional reservoir rocks (Bachu 2000; Cheng and Yu 2019). Many underground engineering practices in shales, such as hydraulic fracturing or massive-scale gas injection in underground storage projects, induce numerous microfractures or reactivate preexisting microfractures. Such fractures provide the main transport pathways for fluids in shale (Watanabe et al. 2008). The fluid flow process in fractures is critical in many engineering activities, including the production of oil and gas reservoirs (Cho et al. 2013; Deng et al. 2014; Zhang et al. 2021), effective recovery of hydrocarbons or geothermal fluids (Bandara et al. 2021; Cheng et al. 2021), evaluation of caprock sealing performance in CO₂ sequestration (Busch et al. 2008; Cheng and Yu 2022), and long-term storage of radioactive waste (Berkowitz 2002; Chen et al. 2022). Shales containing clay and soft materials exhibit time-dependent deformation characteristics under formation stress (i.e., creep), which results in variations in the aperture structure and long-term transport capacity of fractures (Liang et al. 2020; Matsuki et al. 2001). The altered aperture structures (the aperture magnitude and distribution, the contact area, etc.) and the resulting transport properties of fractures can be characterized by their time-dependent permeabilities, so that the evolution of the resulting fluid flow processes can be determined from permeability measurements under varied experimental conditions. Therefore, a comprehensive investigation of the time-dependent permeability of shale fractures is required to deeply understand the long-term safety and stability of the aforementioned underground projects.

The time-dependent behavior of the permeability induced by creep in porous rock under stress conditions has been extensively studied in many laboratory measurements (An et al. 2021; Guo et al. 2018; Liu et al. 2016a; Van Noort and Yarushina 2018). Steady-state permeability measurements on tight rocks under constant confining stress have shown that permeability declines significantly over time when all other variables remain constant (Chhatre et al. 2014; Sinha et al. 2013). Liu et al. (2016b) divided the evolution process of the gas permeability of low-permeable claystone with time into four stages and described the relation between the permeability variation and volumetric strain. The corresponding mechanisms were explained as clay packing, grain rearrangement, and progressive collapse of the pore spaces (Chang and Zoback 2009; Sone and Zoback 2014). For fractures, the widely accepted reasons for the time-dependent variation in permeability include mechanical creep, clay swelling, and pressure solution (Elkhoury et al. 2015; Polak et al. 2003; Yasuhara et al. 2006; Zhang and Talandier 2022). Mechanical creep can affect the fracture void space and aperture distribution, which alters the hydraulic properties of the fracture. Clay swelling on the fracture surface can cause the aperture to decrease and thus change the permeability. Pressure solution, a chemo-mechanical process, is one of the important processes leading to diagenetic compaction and deformation in rocks (Kamali-Asl et al. 2018; Yasuhara 2004). The contacting asperities dissolve under high local stress, and the dissolved minerals diffuse from the high-stress regions to the less-stress regions based on chemical potential and then precipitate on the free faces of the fracture surfaces. Among these three mechanisms, the latter two are applicable for water-saturated rocks where gas–water two-phase flow exists. Zhang (2013) conducted a series of laboratory flow-through experiments on fractured Callovo-Oxfordian argillite and Opalinus claystone, which can undergo significant self-sealing over time due to the combined effects of mechanical creep compaction and water-induced clay swelling, resulting in substantial permeability loss with time. Bandara et al. (2021) performed experimental investigations on propped siltstone fractures and found that creep-induced proppant embedment significantly affected fracture permeabilities, with a permeability reduction of up to 92% within 144 h, which can be roughly expressed as an exponential function. In addition to experimental studies, some permeability creep models have been established and developed to describe the time-dependent variations in permeability. Danesh et al. (2016) combined the Nishihara model and stress‒strain constitutive equation to derive a new permeability model that captures the effect of inelastic deformation on coal permeability. Based on the creep strain equation and stress-dependent permeability model, An et al. (2021) proposed an improved permeability model considering the creep effects and validated it with experimental data from Chhatre et al. (2014). Kamali-Asl et al. (2020) used a three-element rheological model to calculate the variation in fracture aperture and predict the time-dependent evolution of fracture permeability. The permeability prediction model was valid during the test period for both the loading and unloading paths. Most existing permeability models tend to focus only on the creep effect, which works well, but is difficult to apply to the calculation of time-dependent gas permeability because of the gas slippage phenomenon.

Gas slippage is a common occurrence during gas transport in porous media, and it significantly affects the permeability of rocks (Gao et al. 2017; Li et al. 2018). When the pore radius for gas flow approaches the mean free flow path of gas molecules, collisions between the gas molecules and pore walls become more dominant than molecule-to-molecule collisions. At this time, gas molecules “slip” near the wall surfaces, promoting gas transport and producing additional fluxes (Chen et al. 2020; Klinkenberg 1941). Our previous studies have shown that gas flow in microscale fractures cannot be entirely described by Darcy flow, and gas slippage has a significant impact on gas flow (Cheng and Yu 2019, 2022). In recent years, despite extensive research on gas slippage, the understanding of how it varies with the time-dependent deformation of the fracture remains unclear. This makes it difficult to interpret time-dependent gas transport processes in fractured rocks. Additionally, most gas permeability models incorporate gas slippage contributions but do not consider the creep effect, which may lead to incorrect rock parameters. To our knowledge, the time-dependent variation in the gas flow process in fractures has rarely been reported, and related experimental data are extremely limited. Considering the wide range of variation in the formation stress in long-term underground projects, it is critical to investigate the time-dependent behavior of gas permeability in fractures under multilevel confining stresses. This will help us better understand the gas transport processes in fractured shales. An effective permeability model that considers the coupling effect of creep and gas slippage is also required to predict the time-dependent evolution of fracture permeability for the long-term safety and stability of subsurface engineering.

In this study, two borehole core samples of shale were retrieved to experimentally investigate the time-dependent gas permeability of shale fractures under multilevel confining stresses. Each shale core contained a single fracture along its axis. Prior to the experiments, the mineral composition, pore distribution characteristics, and matrix permeability of the shale samples were measured. The surface topographies of the two fracture surfaces were reconstructed, and the aperture distribution and roughness of the fracture were analyzed. Steady-state gas flow tests were conducted with different gas pressures to measure the gas permeabilities of the fractures under multilevel confining stresses, and the associated apertures were calculated numerically. Based on the experimental data, the time-dependent behavior of the fracture permeability was analyzed, and the factors influencing the evolution of the gas transport process were discussed. The variation in the gas slippage effect caused by creep was also evaluated and quantified by a power function. Finally, a time-dependent gas permeability model of the fracture considering the coupling effect of creep and gas slippage was proposed and validated by experimental data. The variations in permeability over the next 30 days were predicted based on this model.

2 Description of the Samples

2.1 Properties of the Shale Matrix

Two shale core samples were acquired from the Carboniferous formation of the eastern Qaidam Basin in China. These samples were drilled into cylindrical cores from core plugs of the ZK1-1 well at depths of 494 m and 518.6 m. The mineral compositions of the samples were measured by X-ray diffraction (XRD) on the shale powder, and the results are shown in Table 1. The main nonclay minerals are quartz and calcite. The contents of clay minerals for Samples 1 and 2 are 46.6% and 36%, respectively, and the dominant clay minerals are mixed layer illite–smectite, illite, and kaolinite. The total organic carbon content was measured by a carbon–sulfur analyzer, and the results are also listed in Table 1. The pore distribution of the matrix was determined by high-pressure mercury intrusion porosimetry (MIP) and low-pressure adsorption of N₂ and CO₂. The measurement results are displayed in Fig. 1 and Table 2. The pore structure analysis shows that the proportions of pore sizes less than 50 nm account for 90% of the total pore size. The clear peak values in Fig. 1 indicate that the most dominant pore sizes for the two samples are approximately 34.33 and 25.25 nm. In addition, the average pore size, porosity, and bulk density of the shale samples were acquired and listed in Table 2.

Table 1 Mineral composition and mechanical parameters of the shale cores

Fig. 1

Pore distribution of the shale matrix

Table 2 The dimensional specifications and elastic parameters of the samples

The matrix permeabilities of the dry cores were measured by the steady-state flow method at a range of methane gas pressures. The gas apparent permeability (k_app) of the core sample can be calculated according to the modified Darcy’s law:

$${k}_{app}=\frac{2{P}_{0}\mu QLZ}{A({P}_{1}^{2}-{P}_{2}^{2}){Z}_{a}},$$

(1)

where P₀ is atmospheric pressure (P₀ = 0.097 MPa); μ is the gas dynamic viscosity; Q is the gas flow rate; L is the sample length; Z and Z_a are gas compressibility factors under the experimental temperature and pressure and the experimental temperature and atmospheric pressure, respectively; A is the cross-sectional area of the core sample; and P₁ and P₂ are the inlet and outlet gas pressures. The intrinsic permeability (k_in) was then obtained by using Klinkenberg’s slip model to correct the gas apparent permeability (Klinkenberg 1941). The measured results are provided in Table 3.

Table 3 The gas flow test results of the intact samples before fracturing

2.2 Preparation and Microstructural Characteristic Measurement of the Fracture

The Brazilian fracturing method was used to artificially create a single rough fracture in each core. The corresponding details are described in Cheng and Yu (2019). The resulting fracture was appropriately located centrally and extended longitudinally along the axis throughout the core, as shown in Fig. 2. The elastic parameters of the fractured cores (the Young’s modulus and Poisson’s ratio) were measured by ultrasonic measurements, and the results are also presented in Table 2. Note that Sample 2 was used to conduct a series of gas percolation experiments, and the details are provided in Cheng and Yu (2022).

Fig. 2

Pictures of the fractured shale cores

Fracture topography was measured using a high-precision three-dimensional (3-D) laser scanner with a scanning interval of 30 μm and a resolution of 7 μm. The scanned data were applied to reconstruct the fracture surfaces and calculate the morphology parameters of the fractures via digital processing, with details described below. First, GOM Inspect (a software for detailed evaluations of 3-D data) was used to align the two fracture surfaces to a common base plane, and the data were exported in a stereolithography (STL) file. Subsequently, the obtained STL files were imported into Surfer software to reconstruct the 3-D surfaces, as shown in Fig. 3, and then the data were converted to a GRID file, which includes the XYZ coordinates of the two surfaces, to calculate the roughness parameters.

Fig. 3

Three-dimensional surface topography of the fracture

The roughness of each fracture was characterized by the mean asperity height (R_m), root mean square of the asperity height (R_rms), and joint roughness coefficient (JRC), which were calculated as follows (Isaka et al. 2020; Sausse 2002; Tse and Cruden 1979):

$${R}_{m}=\frac{1}{n}\sum_{i=1}^{n}\left|{z}_{i}-{z}_{a}\right|,$$

(2)

$${R}_{\mathrm{rms}}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}{({z}_{i}-{z}_{a})}^{2}},$$

(3)

$${Z}_{2}={\left[\frac{1}{L}\sum_{i=1}^{n-1}\frac{{({z}_{i+1}-{z}_{i})}^{2}}{\Delta x}\right]}^{0.5},$$

(4a)

$$\mathrm{JRC}=32.2+32.47\times {\mathrm{log}}_{10}{Z}_{2},$$

(4b)

where z_i is the asperity height at point i, z_a is the mean asperity height, and n is the number of total data points. Note that the 3-D profile of a fracture surface consists of multiple two-dimensional (2-D) profiles along the flow direction. These 2-D profiles were averaged to find the Z₂ and JRC values. The roughness parameters in Table 4 show that the values of R_m and R_rms for the two samples are approximately equal, but the JRC value of Sample 2 is larger than that of Sample 1, suggesting that the fracture surface of Sample 2 is rougher than that of Sample 1. According to the JRC value and the mining rock mass rating classification system proposed by Milne (1990), the fracture surfaces of the two samples are slightly rough and can be classified as curved rough.

Table 4 The results of the geometrical parameters of the fracture surfaces

The fracture aperture was defined as the small gap between the two surfaces, which can be obtained by subtracting the elevation levels of the asperities at a particular location on the two surfaces, mathematically represented as follows:

$${e}_{ij}={T}_{ij}-{B}_{ij},$$

(5a)

$${e}_{ij}\ge 0,$$

(5b)

where e_ij is the aperture at given coordinates and T_ij and B_ij are the elevation levels of the asperities at coordinates (i, j) for the top and bottom surfaces, respectively. Based on Eq. (5b), the overlapping points on the fracture surfaces were identified, and the corresponding apertures were assigned a value of 0, which represents a zero fracture aperture (i.e., the contacted asperities). The contact area of the two fracture surfaces was calculated by the percentage of zero aperture points.

The distribution histograms of the fracture apertures are plotted in Fig. 4, along with the variations in the probability density of the corresponding normal distribution functions. The average apertures and the standard deviations of the aperture distribution are listed in Table 4. The results show that more than 95% of the apertures of Samples 1 and 2 are in the ranges of 0.12–0.5 mm and 0–0.14 mm, respectively, and the e_ave of Sample 1 is higher than that of Sample 2.

Fig. 4

Frequency distribution histogram and corresponding normal distribution curve of the fracture aperture

3 Methodology

3.1 Experimental Procedure

Our main experimental goal is to monitor the time-dependent variation in fracture permeability under multilevel confining stresses. The fracture permeabilities were measured by the steady-state flow method using methane gas. Four different confining stresses of 10, 15, 20, and 25 MPa were applied in the experiments. Considering the safe operation of the experimental apparatus as well as actual formation temperature, all experimental temperatures were set to 40 ℃. This temperature also minimizes the influence of ambient temperature on the flow-through experiments.

A schematic of the experimental apparatus for the gas flow measurements is shown in Fig. 5. The core was wrapped in a fluororubber sleeve and placed into the core holder. To ensure that the core was evenly stressed, two cylindrical steel cores with diameters of 5 cm were placed at both ends of the shale core. A syringe pump with a maximum capacity of 50 MPa was used to apply the confining stress to the cores to simulate in situ formation conditions. The fluid injection system was connected to the core holder and could inject gas into the sample at a constant pressure. The inlet and outlet pressures at both ends of the sample were monitored by two high-precision pressure gauges (0–20 MPa, ± 0.1 kPa). A gas flow meter at the outlet was used to measure the gas volume flow rate. All experimental devices and pipelines were thermostatically controlled in a thermostat (± 0.05 ℃).

Fig. 5

Schematic diagram of the apparatus for the gas permeability measurements

The experimental procedure was as follows. Prior to testing, the samples were oven dried at 105 ℃ for 8 h. The dry core and two steel cores were placed into the core holder, a confining stress of 10 MPa was applied to the cores, and the entire system was vacuumed and leak-tested. Subsequently, methane was injected into the core holder. The inlet pressures, outlet pressures at both ends of the core, and gas volume flow rates were recorded and used to calculate the permeability. To improve the accuracy of the gas permeability, the gas flow rates were measured three times under each inlet pressure to ensure that steady-state flow was achieved. The gas permeabilities were measured every 24 h for 14 days. The confining stress was maintained at a constant 10 MPa during this period and then loaded to the next level. At later confining stress levels (15, 20, and 25 MPa), the duration of the permeability measurement was shortened from 14 days to 7 days. The confining stress was monotonically increased in steps of 5 MPa from 10 to 25 MPa and then unloaded in 5 MPa steps from 25 to 10 MPa at a rate of 0.2 MPa/min during all the loading and unloading processes. Note that each level of confining stress lasted only 2 days during unloading. The specific test path is illustrated in Fig. 6.

Fig. 6

Loading and unloading path in the experimental run

3.2 Determination of the Fracture Permeability and Aperture

Comparing the flow rates of the cores before and after fracturing under the same gas pressure, we found that the gas flow rates of the fractured cores were three orders of magnitude higher than those of the cores before fracturing. This suggests that the gas flow in the matrix is negligible compared to the total flow in the fractured cores and that all pressure and flow measurements directly reflected the fracture properties. Similar to the matrix, Darcy’s law (Eq. (1)) was used to calculate the apparent permeability of the fracture. Due to the high-velocity gas flow in the fracture, however, the applicability of Darcy’s law should first be demonstrated based on the Reynolds number (Rathnaweera et al. 2015), which is given by (Chen et al. 2015; Ranjith and Darlington 2007):

$$Re=\frac{\rho \nu {e}_{h}}{\mu }=\frac{\rho Q}{\mu w},$$

(6)

where ρ and ν are the density and flow velocity of the fluid, respectively, and e_h is the hydraulic aperture, which is defined as the aperture of a smooth fracture generating the same volumetric flow as the rough fracture. With the knowledge that the fluid flow rate of the fracture $Q={A}_{f}\nu$ and the cross-sectional area of the fracture ${A}_{f}={e}_{h}w$, Re can be rewritten as the second equation in Eq. (6), where w is the width of the fracture along the cross-section and is equal to the core’s diameter. The Reynolds number (Re) at all experimental conditions were calculated and listed in Table 5. The Re are in the range of 1.6–219.7, indicating that the methane flow behaves as a laminar flow, and Darcy’s law can be applied to calculate the apparent permeability for fractured shale samples. For fracture, Eq. (1) can thus be rewritten as:

Table 5 The measurement results of the gas permeability under experimental conditions

$$Q=\frac{{k}_{app}{A}_{f}({P}_{1}^{2}-{P}_{2}^{2}){Z}_{a}}{2\mu {P}_{o}LZ}=\frac{{k}_{app}{e}_{h}w({P}_{1}^{2}-{P}_{2}^{2}){Z}_{a}}{2\mu {P}_{o}LZ}.$$

(7)

According to the cubic law of gas flow in a smooth fracture, Q can be determined as follows (Witherspoon et al. 1980):

$$Q=\frac{{{e}_{h}}^{3}}{12\mu L}\frac{w({P}_{1}^{2}-{P}_{2}^{2}){Z}_{a}}{2{P}_{o}Z}.$$

(8)

Combining Eqs. (7) and (8), the apparent fracture permeability and hydraulic aperture can be derived as follows (Cheng and Yu 2022):

$${k}_{app}=\frac{1}{12}{(\frac{12\mu QZ}{w{Z}_{a}}\frac{2{P}_{0}L}{{P}_{1}^{2}-{P}_{2}^{2}})}^\frac{2}{3},$$

(9)

$${e}_{h}=\sqrt[3]{\frac{24\mu QZ{P}_{0}L}{{(P}_{1}^{2}-{P}_{2}^{2})w{Z}_{a}}}.$$

(10)

Since the mechanical aperture of the fracture cannot be directly measured during our experiments, a numerical method named the interpenetration model was performed based on the digital data of fracture surface topography to determine the variation in mechanical aperture associated with the time-dependent fracture permeability. The interpenetration model assumed a perfectly plastic rock response. All local apertures of the fracture were uniformly reduced based on the two surfaces being displaced relative to each other. The overlapping parts of the contacting asperities were removed, and the apertures at these points were set as 0.03 μm. Although this model is relatively simple, it is effective and agrees quite well with the results of flow tests (Cardona et al. 2021; Cheng and Milsch 2021; Deng et al. 2021; Watanabe et al. 2008). Meanwhile, a local cubic law (LCL)-based flow-through simulation was conducted under the same boundary conditions as the seepage experiment to model the single-phase gas flow in a 2-D field while the aperture distribution changed. In the flow simulation, the equation of continuity (Reynolds equation) for the steady-state laminar flow was used (Nemoto et al. 2009; Yeo et al. 1998):

$$\frac{\partial }{\partial x}\left(\rho \frac{{ek}_{g}}{\mu }\frac{\partial P}{\partial x}\right)+\frac{\partial }{\partial y}\left(\rho \frac{{ek}_{g}}{\mu }\frac{\partial P}{\partial y}\right)=0,$$

(11)

where P is the gas pressure, μ and ρ are the viscosity and density of the gas, which are functions of the gas pressure and can be acquired using REFPROP software, e is the local aperture, and k_g is the local gas permeability considering the slippage effect in the fracture, which can be expressed as follows (Cheng and Yu 2022; Wang et al. 2019; Zaouter et al. 2018):

$${k}_{g}=\frac{{e}^{2}}{12}\left(1+6\frac{\lambda \zeta }{e}\right),$$

(12)

where λ is the mean free path of the methane molecules ($\lambda =\frac{{k}_{B}T}{\sqrt{2}\pi {\delta }^{2}}$, where k_B is the Boltzmann constant and δ is the collision diameter of the gas molecule) and ζ is the slip factor ($\zeta =\frac{2-{\sigma }_{v}}{{\sigma }_{v}}$, where σ_v is the tangential momentum accommodation coefficient and assumed to be 0.75 in this study). Note that the permeability is invalid and the gas pressure cannot be defined at the location where the fracture aperture is zero (Watanabe et al. 2008). Considering that the most dominant pore size of our two samples is approximately 0.03 μm (Fig. 1), we set the apertures at the contacting asperities to 0.03 μm for this analysis.

Constant gas pressures were applied at the boundary perpendicular to the direction of macroscopic flow. The boundary parallel to the macroscopic flow direction was assigned a no-flow boundary. By solving the finite difference form of Eq. (11), the velocity field was determined. The outlet flow rate can thus be calculated as follows:

$$Q={\int }_{0}^{w}eVdx,$$

(13)

where e and V are the aperture and flow velocity in each grid at the outlet boundary, respectively, and w is the length of the outlet boundary (i.e., fracture width). By comparing and matching the experimentally and numerically derived flow rates, the mean mechanical aperture e_m can be obtained for prescribed confining stresses and times and expressed as follows:

$${e}_{m}=\frac{1}{wL}\iint edxdy.$$

(14)

4 Experimental Results

The experimental duration for the two samples was 41 days. Table 5 presents the apparent gas permeabilities k_app at different gas pressures and multilevel confining stresses P_c over the experimental period. The results show that k_app decreases significantly in Samples 1 and 2, up to 3.8 and 5.2 times, respectively, within the experimental gas pressure and confining stress ranges due to the elastic deformation and creep of the fracture. For simplicity, the experimental results at a gas pressure difference of 1.297 MPa are taken as a representative for analysis.

4.1 Time-Dependent Variation in the Fracture Permeability

The variations in apparent gas permeabilities of fractures under multilevel confining stresses are presented in Fig. 7a and b. To quantify the reduction in k_app, the permeability variations relative to the initial permeability are calculated and presented in Fig. 7c. Under a constant P_c of 10 MPa, k_app initially experiences a notable decrease over time. After four days, significant reductions of 26% and 42% are observed in the k_app of Samples 1 and 2, respectively. With increasing testing time, the rate of permeability reduction decreases. As P_c increases, k_app exhibits an instantaneous drop and then manifests the same time-dependent reduction as that at the previous confining stress, but the variation amplitude is smaller than that in the previous stress stage. The time-dependent k_app variation at each stress level can be roughly divided into two stages: the decelerating decline stage with a rapid rate and the steady decline stage with a slow rate. The rapid and decelerating decline stage mainly lasts 4–5 days when the samples are initially subjected to a confining stress of 10 MPa. With increasing P_c, the duration of the first stage shortens to 1–2 days within each subsequent stress level. The shorter duration of the rapid and decelerating decline stage at higher confining stresses can be attributed to changes in the mechanical properties and microstructure of the fracture.

Fig. 7

Time-dependent fracture gas permeability of (a) Sample 1 and (b) Sample 2 at multilevel confining stresses and the cumulative reduction in gas permeability with (c) time and (d) confining stress

Figure 7d presents that the cumulative reduction in k_app caused by creep rises with increasing P_c, and at P_c values of 10, 15, 20, and 25 MPa, it reaches 30.90%, 36.47%, 41.11%, and 44.61% in Sample 1 and 45.36%, 49.18%, 51.87%, and 53.82% in Sample 2, respectively, but the increment of change decreases with increasing P_c. For clarity, the amounts of permeability reduction (Δk) caused by creep (Δk_c) and elastic deformation (Δk_i) during each confining stress stage are calculated separately and compared in Fig. 8. Notably, Δk_c at 10 MPa is significantly higher than all other values. As P_c increases from 10 to 25 MPa, both Δk_c and Δk_i gradually decrease, particularly Δk_i. However, the ratio of Δk_c to Δk_i rises with increasing P_c by up to 63.33% and 44.01% in Samples 1 and 2, respectively. In addition, the variations in the fracture permeability during the unloading phase are also monitored at the different confining stresses. It is observed in Fig. 7 that there is a slight recovery and irreversible reduction in permeability during the unloading process, and the permeability loss can be up to 71%. The recovery of permeability appears to increase with increasing stress drop. However, such a recovery process lasts only 1–2 days, and the magnitude and rate of permeability variations during this recovery process are much less than those monitored in the loading steps. Combined with Fig. 8, the permeability recovery mainly relies on the elastic recovery, while the time-dependent variation in permeability during this process is negligible.

Fig. 8

The amount of permeability reduction caused by elastic deformation and creep during each confining stress stage

4.2 Changes in Fracture Aperture

Note that mechanical apertures were not calculated for the entire duration of the experiment due to slight changes in permeability at the end of each loading stage and the negligible role of time-dependent permeability variation during the unloading phases. Only the evolution of mechanical apertures and hydraulic apertures of Samples 1 and 2 with time during the loading process is presented in Fig. 9. The initial mechanical apertures of Samples 1 and 2 under unconfined conditions are 249.9 and 58.2 μm, respectively. After being subjected to a confining stress of 10 MPa, the mechanical and hydraulic aperture of Samples 1 and 2 are reduced to 46.9 and 6.9 μm and 20.9 and 4.2 μm, respectively. Under the constant confining stress, the fracture aperture initially changes drastically with time and then starts converging. Larger aperture variations are observed at the initial loading (10 MPa), which decreases at higher confining stresses. At the end of the loading phase, the mechanical and hydraulic apertures are reduced to 42.3 and 3.7 μm for Sample 1, and to 17.2 and 1.8 μm for Sample 2, and fractures are still not fully closed. The comparison of aperture changes in Fig. 9c show that the change magnitudes of the mechanical aperture are larger than that of the hydraulic aperture, especially at high confining stress levels.

Fig. 9

The variations in the mechanical and hydraulic apertures of (a) Sample 1 and (b) Sample 2 over time and (c) the comparison of the variations in mechanical and hydraulic apertures

The variations in the contact area during the loading process are also presented in Fig. 10. The initial unstressed contact areas of Samples 1 and 2 are 1.16% and 10.3%, respectively. After being subjected to the confining stress, the contact area of Samples 1 and 2 increases drastically to 39.7% and 44.1%, respectively, reaching 44.1% and 50.2% at the end of the loading phase, respectively. The contact area first rises and then stabilizes with time at each confining stress. The overall variation in the contact area of Sample 2 is greater than that of Sample 1 due to the difference in aperture distribution.

Fig. 10

The time-dependent variation in contact area of two samples during the loading

5 Discussion

5.1 Time-Dependent Behavior of the Fracture Permeability

5.1.1 Creep-Induced Variation in Fracture Permeability and Aperture

The time-dependent behavior of the fracture permeability and aperture observed in many experiments can be attributed to various mechanisms, including mechanical creep, pressure solution, stress corrosion, dissolution/precipitation, and fines migration (Bandara et al. 2021; Chen et al. 2022; Guo et al. 2018; Kamali-Asl et al. 2018; Polak et al. 2003; Yasuhara and Elsworth 2008). Considering single-phase gas flow in our experiment, the dominant mechanism is likely mechanical creep or fines migration. Fines migration is a process in which crushed fine grains migrate with the gas flow to fill or block small apertures and thus reduce permeability. However, we did not observe particles on the fracture surfaces after the experiment, which indicates that particle clogging is not the primary mechanism but mechanical creep compaction. Creep refers to the deformation process of the rock with time under a constant normal stress (Heap et al. 2009). As shown in Figs. 7 and 9, the time-dependent variation in k_app and aperture under each confining stress can be roughly divided into two stages. Such behavior is analogous to the primary and secondary creep stages observed in deformation experiments (Chang and Zoback 2009; Hamza and Stace 2018; Rassouli and Zoback 2018). This suggests that the time-dependent reductions in fracture permeability and aperture are closely related to the creep deformation process of the fracture. Creep-induced fracture deformation changes the geometry of the contacting asperities and void spaces adjacent to these contacting asperities, as well as the contact area of the fracture surface, resulting in the alteration of the hydraulic properties of the fracture (Pyrak-Nolte and Morris 2000). In our experiments, after the application of confining stress (10 MPa), most of the asperities of the two fracture surfaces come into contact, inducing high-stress concentrations (Cheng et al. 2021). Due to the contact, sliding, rearrangement, and crushing of asperities under a constant confining stress, the fracture apertures are squeezed, and the resulting contact areas are gradually expanded, causing a notable reduction in gas flow pathways and an increase in flow tortuosity, thereby leading to a significant decline in gas permeability. As the confining stress increases to a further stage, more asperities come into contact, and the contact area further increases, further reducing the fracture aperture and permeability.

Our experimental results show that the permeability and the associated aperture and contact area change mainly in the elastic deformation stage and the first stage of time-dependent deformation (i.e., the decelerating decline stage). Over time, the two fracture surfaces come closer to each other, contact occurs at the high asperities, small apertures preferentially close and large apertures remain open. Meanwhile, some isolated apertures are created due to the asperities contact and heterogeneous fracture structures. The gas flow paths are basically determined at the end of this period. As time goes on, subsequent fracture deformation depends almost entirely on the deformability of contacted asperities (Bandis et al. 1983). Creep-induced fracture closure gradually stabilizes in the steady decline stage, where the slight compression of large apertures and the closure of isolated apertures mainly occur, so that the preferential flow paths no longer change, and only minor reductions in fracture permeability and hydraulic aperture are observed. The mechanical aperture changes are always greater than the hydraulic aperture changes because some isolated apertures gradually close during the elastic and creep deformation, and the closure of these isolated apertures has almost no impact on the seepage capacity of the fracture, thus not affecting the change in permeability and hydraulic aperture.

During the loading process, more asperities will come into contact at higher confining stress. Deformation of these contacted asperities is the main source of creep, so the cumulative creep-induced k_app reduction increases with confining stress (Fig. 7d). This experimental observation suggests that the time-dependent creep deformation is expected to exert significant effects on the permeability at high formation stresses. Even at the confining stress of 10 MPa in our experiments, a comparison of k_app on the first and last day revealed almost a twofold overestimation of permeability if creep effects are not considered. Similar observations have been made by other scholars. Chhatre et al. (2014) stated that the initial permeability of the intact Eagle Ford shale is approximately four times the steady permeability at 2500 psi confining stress due to the creep effect. Bandara et al. (2021) found that for propped sandstone fractures, creep-induced permeability was overestimated by a factor of 1.2–13 under 20 MPa confining stress. If a higher confining stress is applied to our samples in the first loading stage, we may see more pronounced time-dependent variations in fracture permeability. Therefore, it is important to consider the time-dependent behavior of fracture permeability in numerical simulations or reservoir engineering calculations of underground projects. In addition, Figs. 7 and 8 show an irreversible reduction in permeability and slight permeability recovery in the unloading process. This is attributed to the fact that the creep compaction of shale fractures is nearly irrecoverable deformation and characterized as a combination of viscoelastic and viscoplastic behavior (Chang and Zoback 2009; Sone and Zoback 2014). The rearrangement and crushing of asperities in fracture surfaces are irreversible, and the reduced apertures are difficult to recover. The variations in fracture permeability during unloading are insignificant under each confining stress, and creep plays little role in this process.

5.1.2 Time-Dependent Variation in Gas Slippage

The experimental results in Table 5 show that the gas permeabilities of the two samples decrease with increasing gas pressure under all confining stresses. k_app declines up to 1.4 times as the gas pressure increases from 0.6 to 1.3 MPa in Sample 1, and it declines up to 1.2 times as the gas pressure increases from 0.6 to 1.5 MPa in Sample 2. This gas pressure-dependent behavior of gas permeability is due to the gas slippage effect in the fracture (Klinkenberg 1941). According to the theory of Klinkenberg (1941), the mean free path of gas molecules varies with gas pressure, and when the mean free flow path of the methane molecules approaches the aperture size, the collisions between gas molecules and fracture walls become dominant, inducing additional gas fluxes near the fracture surfaces. Similar phenomena have been reported by many scholars in fractures of different rock types, including siltstone (Wanniarachchi et al. 2018), sandstone (Bandara et al. 2021), and shale samples (Cheng and Yu 2022). Figure 11 indicates that k_app is inversely proportional to the average gas pressure, and the linear relationship can be expressed as follows (Klinkenberg 1941):

$${k}_{\mathrm{app}}={k}_{\mathrm{in}}\left(1+\frac{b}{{P}_{g}}\right),$$

(15)

where k_in is the intrinsic permeability; P_g is the average gas pressure, which can be calculated as P_g = (P₁ + P₂)/2; and b is the slippage factor, which indicates the influence degree of gas slippage on the total gas flow. The values of k_in and b under different confining stresses are determined by fitting the experimental results based on Eq. (15). The results show that the contribution of slip flow can reach 40% in the experimental ranges of gas pressure.

Fig. 11

Relationship between the apparent gas permeability and reciprocal of the gas pressure

As shown in Fig. 11, at a constant 10 MPa confining stress, the slope of both plots decreases with time, indicating that the gas slippage effect weakens over time. We then calculated the time-dependent variation in slippage factor b using data collected under 10 MPa confining stress as an example. Figure 12 shows that b presents a gradual decline with time. Fitting the experimental data indicates that b decays as a power function with time, which can be expressed as follows:

$$b={b}_{0}-m\times \left(1-{n}^{t}\right),$$

(16)

where b₀ is the initial slippage factor at 0 days and m and n are fitting parameters. Equation (16) is also applicable to other confining stresses, and the fitting parameters are listed in Table 6.

Fig. 12

Relationship between the slippage factor and time at a confining stress of 10 MPa

Table 6 Parameters of the permeability model at different confining stresses

Note that the trend of b can be divided into two stages, which is consistent with the variation in permeability, with a high decrease rate in the primary stage and then a slight decrease rate during the secondary stage. Theoretically, the slippage factor can be expressed as (Cheng and Yu 2022; Zaouter et al. 2018):

$$b=\frac{6{K}_{B}T\zeta }{\sqrt{2}\pi {\delta }^{2}e}.$$

(17)

Equation (17) suggests that the slippage factor is inversely proportional to the fracture aperture and that the gas slippage effect is significant at small apertures. Some scholars have stated that b increases with effective stress because the increasing stress constricts pore channels (Chen et al. 2021; Meng et al. 2021). One might expect that the fracture void space would be compressed under a constant confining stress and the gas slippage would be enhanced with time. However, the opposite phenomenon was observed in our experiments. As mentioned in Sect. 5.1.1, small apertures gradually close due to creep in the primary stage, while most of the large apertures remain open. The decreasing number of apertures most responsible for the gas slippage leads to the reduction in b over time. During the secondary stage, the aperture distribution is stable, and fluid pathways are formed, so the gas slippage barely changes.

5.1.3 Factors Influence the Time-Dependent Fracture Permeability

The experimental results show that although the permeability presents a similar variation trend at each confining stress stage, the magnitude of variation gradually decreases with increasing stress. This behavior probably results from the change in fracture normal stiffness (Bandis et al. 1983; Cammarata et al. 2006; Li et al. 2021; Liu et al. 2012). Figure 13a shows the fracture stiffness calculated from the confining stress and aperture variation of the two fractured samples during the loading phase. The fracture normal stiffness is related to the mechanical properties of the contacted asperities and depends mainly on the amount and distribution of the fracture contact area (Brown and Scholz 1985; Cheng and Milsch 2021; Pyrak-Nolte and Morris 2000). To investigate the interrelationship between fracture stiffness and time-dependent permeability, the variation in permeability (percentage) during each confining stress stage is displayed in Fig. 13b. When the sample is first subjected to confining stress (10 MPa), the contact area and fracture stiffness are low, and the fracture is easily compressed, resulting in a major time-dependent variation in fracture permeability. As the confining stress rises, more contact asperities are formed. The increasing contact area can withstand higher stresses, and the fracture stiffness gradually increases (Zhao et al. 2017). Only slight fracture compaction and permeability reduction occur at higher confining stresses. Therefore, as shown in Fig. 13, the noteworthy difference in the magnitude of reduction in permeability at the first two confining stress levels (10 and 15 MPa) is ascribed to the dramatic increase in fracture stiffness. Under the subsequent confining stresses (15, 20, and 25 MPa), the difference in the time-dependent permeability reduction between each stage is not significant, but it still decreases with increasing fracture stiffness. This may be because the fracture stiffness is already large and the deformability of the fracture is very poor. The continuous increase in fracture stiffness does not have a noticeable effect on fracture deformation (i.e., permeability reduction) as it did initially. Moreover, combining Figs. 8 and 13, it can be inferred that in our multilevel confining stress experiments, the fracture stiffness increases after each stress level, which has a greater impact on the instantaneous reduction in fracture permeability than the time-dependent reduction in permeability. The decline in Δk_i is apparently greater than that in Δk_c, and thus the ratio of Δk_c to Δk_i increases during loading. This suggests that the time-dependent creep compaction exhibits an increasingly important role in dominating the fracture closure and transport capacity with increasing confining stress. Chang and Zoback (2009) also observed the stiffening effect in Gulf of Mexico shale at higher stresses. They found that the creep strain was larger than the elastic strain when the stress increased above 20 MPa.

Fig. 13

a The variation in fracture stiffness with confining stress and b the magnitude of the time-dependent reduction in permeability at each confining stress

Figure 13b reveals a different degree of the time-dependent reduction in permeability for the two samples under a constant P_c, which may be closely related to the fracture surface morphology, aperture distribution, mechanical properties, and mineral composition (An et al. 2021; Kamali-Asl et al. 2018; Zhang et al. 2021). The 3-D surface topography and the aperture measurements show that Sample 2 is rougher than Sample 1, and the average aperture of Sample 1 is approximately four times larger than that of Sample 2, which corresponds to the lower fracture permeability of Sample 2 than Sample 1. As shown in Fig. 6, the aperture distribution range of Sample 1 is wider than that of Sample 2. Considering that the narrow apertures will be closed preferentially under stress, more void spaces remain in Sample 1, while more asperities on the fracture surfaces in Sample 2 will be in contact. This could explain why the permeability of Sample 2, which has a higher fracture stiffness, shows a greater reduction magnitude than that of Sample 1 at a 10 MPa confining stress and why the permeability of Sample 1 is consistently higher than that of Sample 2. If so, Sample 2 will have more deformed asperities than Sample 1, thereby resulting in a larger irreversible amount of gas permeability in Sample 2. At the following higher confining stresses, the reduction in permeability of Sample 1 is slightly larger than that of Sample 2, which is associated with the difference in fracture stiffness between the two samples. Fracture stiffness is directly related to elastic modulus, and Sample 1 with less fracture stiffness means it is easier to deform than Sample 2. In addition, previous studies on fractures indicate that the fracture stiffness is proportional to the JRC value (fracture geometry) and contact area (Bandis et al. 1983; Pyrak-Nolte and Morris 2000), and fracture stiffness is implicitly related to the rock mineral composition through the mechanical properties of contact asperities. The presence of soft material such as clay and total organic carbon (TOC) in fractured rocks decreases the fracture stiffness, and the amount of creep deformation is correlated to the clay and TOC contents (Rassouli and Zoback 2018; Sone and Zoback 2013). Considering that the clay and TOC contents of Sample 1 are higher than those of Sample 2 and the elastic modulus, JRC value, and contact area of Sample 1 are smaller than those of Sample 2, Sample 1 thus exhibits more time-dependent reduction in permeability than Sample 2. In summary, the size and spatial distribution of the apertures mainly control the time-dependent permeability variation during the first loading stage, and the dominant influence factor of time-dependent permeability converts to fracture stiffness at higher confining stress levels.

5.2 Time-Dependent Gas Permeability Model of the Fracture

A mathematical model to quantitatively describe the time-dependent behavior of the gas permeability in a fracture under formation stress is critical for predicting the long-term response of fractures associated with geotechnical performance and engineering design under reservoir conditions (An et al. 2021; Bandara et al. 2021; Kamali-Asl et al. 2020). In this study, we propose a gas permeability model that considers the coupled effects of gas slippage and long-term creep of the fracture.

Burgers and Power-law models have been widely used to model and predict the creep behavior of rock (Hamza and Stace 2018; Kaiser and Morgenstern 1982; Kamali-Asl et al. 2021; Sone and Zoback 2014). The Burgers model, which can describe elastic, primary, and steady-state creep deformation, is composed of a Kelvin model (a spring and a dashpot connected in series) and a Maxwell model (a spring and a dashpot connected in parallel) in series (Nomikos et al. 2011; Parsons and Hedley 1966), and it is expressed as follows:

$$\varepsilon =\sigma \left[\frac{1}{{E}_{1}}+\frac{t}{{\eta }_{1}}+\frac{1}{{E}_{2}}\left(1-{e}^{-\frac{{E}_{2}t}{{\eta }_{2}}}\right)\right],$$

(18)

where ɛ denotes strain; σ represents the applied stress on the fracture; E₁ and η₁ are the spring modulus and dashpot viscosity for the Maxwell unit; and E₂ and η₂ are the spring modulus and dashpot viscosity for the Kelvin unit. The Power-law model can fit the creep data well, which is simpler and safer when extrapolated (Rassouli and Zoback 2018; Trzeciak et al. 2018). It can be expressed as (Sone and Zoback 2014):

$$\varepsilon =\frac{\sigma }{E}+\sigma B{t}^{D},$$

(19)

where E is Young’s modulus and B and D are empirical parameters.

In a fracture, ɛ can be quantified by the aperture variation, defined as the ratio of the aperture variation (Δe) to the initial aperture (e₀). The first term in both equations represents the elastic strain, and only the creep strain induced by the confining stress P_c is considered in this study. Equations (18) and (19) are thus transformed into the following form:

$$\frac{\Delta e}{{e}_{0}}={P}_{c}\left[\frac{t}{{\eta }_{1}}+\frac{1}{{E}_{2}}\left(1-{e}^{-\frac{{E}_{2}t}{{\eta }_{2}}}\right)\right],$$

(20)

$$\frac{\Delta e}{{e}_{0}}={P}_{c}B{t}^{D}.$$

(21)

According to the cubic law, the fracture permeability (k) is correlated to the aperture variation Δe, and thus k can be rewritten as follows:

$$k={k}_{0}{(1-\frac{\Delta e}{{e}_{0}})}^{2},$$

(22)

where k₀ is the initial permeability value. According to Eqs. (20)–(22), the time-dependent permeability of the fracture can be determined as follows:

$$k={k}_{0}{\left[1-{P}_{c}\left[\frac{t}{{\eta }_{1}}+\frac{1}{{E}_{2}}\left(1-{e}^{-\frac{{E}_{2}t}{{\eta }_{2}}}\right)\right]\right]}^{2},$$

(23)

$$k={k}_{0}{\left(1-{P}_{c}B{t}^{D}\right)}^{2}.$$

(24)

Notably, k in Eqs. (23) and (24) is the intrinsic permeability reflecting the intrinsic properties of the rock but not the apparent gas permeability (k_app). Equations (23) and (24) cannot be directly used to model our experimental results due to the gas slippage effect.

Combining Eqs. (15) and (16), Eqs. (23) and (24) can be transformed as follows:

$${k}_{app}={k}_{app0}{\left[1-{P}_{c}\left[\frac{t}{{\eta }_{1}}+\frac{1}{{E}_{2}}\left(1-{e}^{-\frac{{E}_{2}t}{{\eta }_{2}}}\right)\right]\right]}^{2}\left[1-\frac{m(1-{n}^{t})}{{P}_{g}+{b}_{0}}\right],$$

(25)

$${k}_{app}={k}_{app0}{\left(1-{P}_{c}B{t}^{D}\right)}^{2}\left[1-\frac{m(1-{n}^{t})}{{P}_{g}+{b}_{0}}\right].$$

(26)

We then verified the gas permeability models based on the experimental results. The experimental data (data for 7 days within each confining stress level) are used to fit the two models and obtain the best-fit model parameters, including η₁, E₂, η₂, B, and D. Figure 14 shows that both Eqs. (25) and (26) can match the experimental data well, indicating that the two models are suitable for describing the time-dependent variation in fracture permeability during flow-through experiments with good accuracy (R² = 0.97–0.99). The associated parameters are determined at confining stresses of 10, 15, 20, and 25 MPa and displayed in Table 6. The parameters E₂ and η₂ reflect the property of primary creep stage and the parameter η₁ exhibits the property of secondary creep stage (Hamza and Stace 2018). The lower parameter values mean a greater decrease in the fracture permeability during the corresponding creep stage. These three parameters are related to the fracture stiffness and contact area. The fracture stiffness of the sample is very low when it is first stressed. With increasing confining stress P_c and loading time, the contact area between the two fracture surfaces also increases, increasing the fracture stiffness. The lower fracture stiffness suggests that the fractured sample is prone to exert deformation, thereby resulting in much more permeability reduction under P_c of 10 MPa compared to that under the other higher stresses, which corresponds to the smallest values of these three parameters. An increase in E₂ and η₂ with increasing P_c indicates that the duration and magnitude of permeability reduction in the primary stage diminish at higher P_c. The η₁ increases with P_c means that the decline rate of permeability in the steady decline stage decreases. These behaviors are consistent with our experimental observations and can be attributed to the fact that increasing fracture stiffness with increasing P_c and loading time results in a weakening of the fracture deformation capacity. In the Power-law model, B is roughly inversely proportional to the elastic Young’s modulus. Parameter D determines the relative contribution of the time-dependent strain to the total strain (Sone and Zoback 2014). The decrease of B value with increasing P_c (especially 10 to 15 MPa) can be attributed to the increasing fracture stiffness after subjected each stress step. Larger values of D at higher P_c indicate that time-dependent permeability variations become more important. Since the increase in the fracture stiffness affects Δk_i more than Δk_c, the ratio of Δk_c to Δk_i rises progressively (Fig. 8), which is consistent with the increase in D value.

Fig. 14

Comparison between experimental data and two permeability models at each confining stress

The two permeability models are valid in the experimental time (seven days), but the capability of the two models to predict variations in fracture permeability in long-term subsurface engineering deserves further examination. We used the model parameters obtained by fitting the 7 days of data (parameters in Table 6) to calculate the permeability changes within 14 days under 10 MPa confining stress and compared them with the experimental values to verify the practicability of the model. Figure 15a shows that the theoretical permeability calculated by the Burgers-permeability model deviates from the experimental data with time, while the Power-law-permeability model can better predict the trend of the decrease in permeability. In addition, to further evaluate the usefulness of the Power-law-permeability model, it is examined whether this model can successfully fit experimental results from other sources. Bandara et al. (2021) measured fracture gas permeability for saturated siltstone, and the creep-induced permeability variations at 20 MPa confining stress were observed. The experimental data are extracted and plotted in Fig. 15b, where the measurement time is 144 h. The gas slippage effect in those samples is considered and calculated from the flow-through test data. In Fig. 15b, the permeability model is well matched with the experimental data. Therefore, the Power-law-permeability model is effective, and by providing appropriate rock parameters and associated coefficients, it can be used to predict the time-dependent permeability variations in large-scale underground projects.

Fig. 15

a Comparison of the capacity of two models to predict permeability variation and b verification of the applicability of the Power-law-permeability model by other published results (Bandara et al. 2021)

As shown in Fig. 16, the permeability variations over the next 30 days are predicted by the permeability model. Although the permeability changes slowly, further permeability reductions of 6.6% and 8.0% are produced in Samples 1 and 2, which account for 15.7% and 9.8% of the permeability decline in the first 7 days, respectively. This implies that most of the time-dependent permeability reduction in the fractured sample under constant stress occurs over the first week. Based on the trends of permeability variation predicted by the model, a certain amount of permeability loss may be expected during the long-term compaction phase of subsurface fractures over decades or centuries. This time-dependent compaction is likely to be an action mechanism that induces the self-sealing of the fracture. However, fractures are difficult to close completely within a short time due to the interaction of the asperities on the two opposing fracture surfaces (Zhao et al. 2017; Zhou et al. 2019). Under long-term formation stress, the rock fractures gradually stiffen so that the permeability reductions in the later phases are extremely low. The validity of this permeability model may need to be further verified by actual data on a larger time scale.

Fig. 16

The predicted permeability reduction over the next 30 days using the time-dependent permeability model at a confining stress of 10 MPa

5.3 Implications

Fracture permeability measurements of shale cores are routinely conducted in the laboratory at approximate in situ stress conditions, and the experimental conclusions can be extended to the field scale for efficient implementation of underground engineering. The study of creep-induced time-dependent permeability evolution in fractured shale is important but challenging because it takes a long time. Our experiments show that fracture permeability would be significantly overestimated if the creep effect is not taken into account. Gas slippage, as a gas-specific property that influences gas permeability (Chen et al. 2020; Cheng and Yu 2022), is also affected by creep, exhibiting a power function decay with time. The rate of permeability change decreases continuously and then stabilizes over time under the constant confining stress. After approximately five days, the permeability gradually enters a quasi-stable stage. Therefore, to accurately measure rock permeability, one can either apply a certain confining stress to the rock sample in the core holder for at least a week before permeability measurements or successively measure the permeability until it reaches a quasi-stable value. Using the proposed permeability model to calculate the time-dependent variation caused by creep is also a way to roughly eliminate measurement errors and improve accuracy. Note that our permeability model is relatively simple, and determining the right model parameters is necessary. More data from flow-through experiments and deformation experiments are required to strengthen the model practicability.

Shale-gas/oil development relies on multistage fracturing and horizontal drilling technology (An et al. 2021; Zhou et al. 2019). These advanced engineering activities create numerous fractures. Correct evaluation of shale fracture permeability is crucial to understanding and predicting reservoir production performance. Our experimental results indicate that creep compaction greatly reduces the transport capacity of shale fractures during the first loading stage, while this effect is weakened at higher confining stresses due to the increase in fracture stiffness. This means that newly generated self-propping fractures will exhibit a certain time-dependent permeability reduction under formation stress, and as the effective stress rises (for example, during resource depletion), those fractures will only be slightly compacted and may persist for a long time. Shales are also considered as caprocks or host rocks for CO₂ geological sequestration and radioactive waste geological disposal (Cheng and Yu 2019; Tsang et al. 2012). Fractures, as the most probable leakage pathways, determine the success or failure of these underground projects (Zaouter et al. 2018). An effective prediction of the seepage characteristics of subsurface fluids during the disposal period is necessary for the long-term safety of geological disposal repositories. Our data show that the fracture permeability changes slightly at the end of each confining stress stage. Dry fractures are difficult to close completely under creep. Previous studies have demonstrated that moisture can promote mechanical creep, clay swelling, or even complete self-sealing (Sone and Zoback 2014; Wang et al. 2022; Wenning et al. 2021; Zhang and Talandier 2022). Therefore, introducing some aqueous fluids to accelerate fracture creep (i.e., enhance the time-dependent reduction of fracture permeability) is required to prevent shallow groundwater contamination caused by leakage of greenhouse gases and radioactive waste. Compared with dry fractures, the evolution of the seepage characteristics over time of water-saturated fractures under hydro-mechanical coupling effect is more complex and thus needs to be further investigated and elucidated in future work.

6 Conclusions

A series of gas permeability measurements and associated aperture calculations were conducted on shale microfractures under multilevel confining stresses to experimentally investigate the time-dependent behavior of the gas permeability and underlying mechanisms. The following conclusions were reached.

The observed permeability reduction of up to 3.8–5.2 times during confining stress loading from 10 to 25 MPa is a result of the elastic deformation and creep of the fracture. The gas permeability and aperture of the fracture significantly decrease with time due to creep at a constant confining stress, with a rapid and decelerating decline stage and then a slow and steady decline stage. Neglecting creep effects can lead to an overestimation of fracture permeability by at least two times under 10 MPa confining stress. Considering the time-dependent creep behavior of fractures is essential in calculating fracture permeability for numerical simulations or reservoir engineering calculations of underground projects.

Increasing confining stress results in a decrease in the effective aperture and an increase in the contact area of the fracture, thereby further decreasing the gas permeability, but the permeability reductions caused by both elastic deformation (Δk_i) and creep (Δk_c) are weakened. This behavior is related to the increase in fracture stiffness. The considerable decline in the magnitude of permeability reduction is due to the dramatically increase in fracture stiffness. The increase in fracture stiffness with confining stress has a greater effect on Δk_i than on Δk_c; thus, the ratio of Δk_c to Δk_i also increases with confining stress. During the unloading phase, permanent permeability loss and limited permeability recovery over time are observed, indicating that the creep-induced permeability reduction is an irrecoverable process. Our results also show that two fractured samples exhibit different time-dependent permeability reductions, which are attributed to differences in aperture distributions and fracture stiffness.

The dependence of permeability on gas pressure indicates the presence of the gas slippage phenomenon in the fractures. The contribution of slip flow can be up to 40%, even in a relatively small gas pressure range. Under a constant confining stress, gas slippage manifests a power function decay over time with the change of aperture distribution caused by creep. Considering the coupling effect of creep compaction and gas slippage, the time-dependent gas permeability model is developed and verified by experimental data. Subsequently, the permeability reductions over the next 30 days are predicted by this model. The predicted results suggest that the gas permeability of the two samples decreases further during this period, but to a small extent.