# Hydromechanical Coupling Characteristics of the Fractured Sandstone under Cyclic Loading-Unloading

Authors: Tong Zhang; Yang Liu; Ke Yang; Ming Tang; Xiang Yu; Fei Yu
Journal: Geofluids
 (2020)

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## Abstract

The mechanical and hydraulic properties of rock mass play a crucial role in underground engineering. To study the effect of hydraulic pressure, confining pressure, and axial cyclic loading-unloading on variation of the deformation and permeability in fractured rock mass, the coupling triaxial experiment of sandstone was conducted. The concept of permeability recovery rate (PRR) and permeability enhancement reduction rate (PERR) was proposed to characterize the change in permeability. The results show that the permeability of fractured sandstone quadratically varies with the change of hydraulic pressure and confining stress. In detail, the permeability decreases with the decrease of hydraulic pressure and increases with the decrease of confining stress, respectively. Compared with the single-fracture permeability, the double-fracture permeability is more sensitive to the change of hydraulic pressure. Furthermore, the permeability of fractured sandstone is more dependent on the hydraulic pressure than the confining stress. With the performance of axial cyclic loading-unloading, the permeability spirals down, and both the axial and radial residual strains quadratically evolve. Following the first axial cyclic loading-unloading, an obvious deformation memory phenomenon characterized by a parallelogram shape in axial stress-strain curves was observed for the sandstone. The cumulative PRR of 85%-95% was maintained in double-fracture sandstone. On the contrary, a fluctuation of cumulative PRR characterized by “V shape” was observed for single-fracture sandstone. The enhancement effect of axial cyclic loading on the permeability was characterized by the decrease of PERR for double-fracture sandstone and increase of PERR with a greater gradient for single-fracture sandstone.


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## Body

## 1. Introduction

Understanding the mechanical and hydraulic properties of rock mass is critical for the safety of underground engineering, such as deep tunnel engineering [1], geothermal extraction [2–5], radioactive waste treatment [6–8], CO2 geological storage [9–11], deep coal mining [12, 13], and underground reservoir [14]. The fluid flow mainly occurred in the strata comprised of plenty of natural and human activity-induced fractures and was easily influenced by the variable high stress [15]. To meet the widespread utilization of energy resources, underground mining has exploited into the deeper crust [16]. Therefore, recent research in geo-mechanical and hydromechanical mechanism of fractured rock mass extensively studied in the deep underground activity.The hydraulic conductivity as a vital index for fractured rock mass with the hydromechanical coupling effect has been extensively investigated using theoretical analysis, field and laboratory measurements, and numerical simulation [17]. In fractured rock mass, the deformation in the rock matrix and fracture caused by the effect of hydromechanical coupling also significantly affect the seepage and diffusion of fluid. Due to the high strength of the rock matrix, the deformation of fracture is more sensitive than that of rock matrix for stress-dependent deformities [18]. The equivalent continuum and discrete element algorithms have been employed in the characterization of the response of fractured rock mass to changes in stress and hydraulic pressure [19–21]. Considerable efforts have been focused on the flow behaviors, including Darcy and non-Darcy flow, of fractured rock mass with the effect of hydromechanical coupling [22–25]. Furthermore, the study on the variation of permeability in deformable rough-walled fractures subjected to the change of fracture geometric is also performed [26, 27].The rock mass, consist of matrix and fractures, is often subjected to mining stresses (cyclic or dynamic loads) in deep coal mining. It is important to study the effect of cyclic mining stresses on rock stratum, which would be beneficial to predict dynamic hazards in coal mines. Hence, the mechanical and seepage properties of different rocks under complicated stress conditions, especially under cyclic loading and unloading, have received extensive attention [28, 29]. For constant-amplitude cyclic loading-unloading, Chen et al. [30] investigated the deformation modulus of sandstone under different cyclic loading and found that the tangent modulus and Poisson ratio show a shape of asymmetric “X,” and the mean of unloading modulus is larger than the loading modulus under the sine wave cycle load. Fuenkajorn and Phueakphum [31] experimentally studied the deformation parameters and uniaxial compressive strength of salt rock under the effect of cycle loading and unloading. Liu and He [32, 33] researched the residual axial and volumetric strain characteristics with variable confining stress and frequency and described the degradation process of sandstone with damage variable under cycle loading-unloading. Liu et al. [34] experimentally investigated the permeability variation of fractured sandstone under confining stress cyclic loading. For cycle loading-unloading at different stress levels or tiered cyclic loading-unloading, Liu et al. [35] carried out the different stress level cyclic loading experiments to realize the damage evolution of salt rock and established a formula to describe the evolution of damage. Zhao et al. [36] studied the deformation and permeability of sandstone with cycle loading and unloading of different unloading rates. With the increase of cycle times, the shape of permeability curves is ∞ type. The relationship between the variation of axial strain, unloading rate, and loading stress can be described with a power function. Jiang et al. [37] experimentally studied the evolution of permeability, acoustic emission, and energy dissipation of gas-containing coal under tiered cyclic loading, described the relative process by defining permeability recovery rate, damping ratio, acoustic emission energy rate and ring count rate, and developing a coal damage variable equation. Duan et al. [38] carried out the hydromechanical experiments to analyze the inherent relationship between the residual strain, permeability, acoustic emission, and energy dissipation of gas-bearing coal under the confining stress cyclic unloading-loading. The existing studies are mainly focused on intact rock samples, while there are rare reports on the evolution of deformation and permeability of fractured rocks under cyclic loading-unloading.In this study, the hydromechanical experiment of fractured sandstone comprised of single-fracture and double-fracture is carried out. The deformation and permeability of fractured sandstone subjected to the change of hydraulic pressure, confining stress, and axial cycle loading-unloading are investigated and quantitatively analyzed based on the permeability recovery rate (PRR) and permeability enhancement reduction rate (PERR).


## 2. Experimental Methods

### 2.1. Experiment Principle

During the experiment process, we assume that (1) the permeable water is an incompressible fluid; (2) the steady seepage under constant pressure is regarded as continuous seepage; and (3) for low permeability fractured sandstone, the seepage obeys Darcy’s law during experiment process. The permeability formula is expressed as follows:(1)k=μLVAΔPΔt,where k is the permeability (m2); V is the inflow volume of the seepage fluid during the time Δt (m3); μ is the dynamic viscosity of water, μ=1×10−3 Pa∙s (T=20°C); L is the length of the rock sample (m); A is the cross-sectional area of the rock sample (m2); ΔP is the hydraulic pressure difference (Pa); and Δt is the increment of time (s).


### 2.2. Sample Preparation

The experimental samples were prepared with a dimension of50mm×100mm (diameter×length). The physical property of density and porosity is 2350 kg/m3 and 7.78%, respectively. The fractured rock mass with a single fracture of 100 degree inclined angle relative to the horizontal plane and two mutually perpendicular fracture of 100 degree inclined angle and 90 degree inclined angle relative to the horizontal plane were obtained through the Brazilian splitting test, as shown in Figure 1. The specific mechanical property of the intact sandstone under different confining stress is shown in Table 1.Figure 1
Fractured samples: (a) single-fracture; (b) double-fracture.
(a)(b)Table 1
Mechanical property of the sandstone.
Sample no.Confining pressure (MPa)Peak strength (MPa)Poisson’s ratioElastic modulus (GPa)Cohesion (MPa)Friction angle (°)15109.420.236.3514.7648.93215170.670.199.01325225.690.1813.37The hydromechanical tests were carried out by triaxial hydromechanical coupling experimental system, as shown in Figure2. The triaxial hydromechanical coupling experimental system includes a triaxial cell and a fluid injection pump. The scope of confining stress, axial stress, and hydraulic pressure are 0-60 MPa, 0-600 MPa, and 0-30 MPa, respectively. The specific experimental process is presented in Figure 3.Figure 2
Triaxial hydromechanical coupling experimental system.Figure 3
Triaxial hydromechanical coupling experiment process.


### 2.3. Testing Scheme

To investigate the evolution of deformation and fluid conductivity subjected to the change of hydraulic pressure, confining stress, and cycle loading-unloading, the fractured sandstones were loaded into the triaxial chamber subjected to a complicated triaxial loading path, which consists of three parts. In the first part, the axial and confining pressure were initially loaded to the hydrostatic pressure of 36 MPa with a rate of 0.1 MPa/s, and the water pressure was reduced from 4.3 to 0.4 MPa with a gradient of 0.86 MPa. In the second part, the axial and confining pressure decreased from 36 to 18.5 MPa with a gradient of 2.5 MPa, while maintaining the hydraulic pressure of 0.4 MPa. Finally, the axial cyclic loading-unloading stress was set to an increment of 21.6 MPa at each cycle. The specific loading path is shown in Figure4.Figure 4
The triaxial loading path for testing.


## 2.1. Experiment Principle

During the experiment process, we assume that (1) the permeable water is an incompressible fluid; (2) the steady seepage under constant pressure is regarded as continuous seepage; and (3) for low permeability fractured sandstone, the seepage obeys Darcy’s law during experiment process. The permeability formula is expressed as follows:(1)k=μLVAΔPΔt,where k is the permeability (m2); V is the inflow volume of the seepage fluid during the time Δt (m3); μ is the dynamic viscosity of water, μ=1×10−3 Pa∙s (T=20°C); L is the length of the rock sample (m); A is the cross-sectional area of the rock sample (m2); ΔP is the hydraulic pressure difference (Pa); and Δt is the increment of time (s).


## 2.2. Sample Preparation

The experimental samples were prepared with a dimension of50mm×100mm (diameter×length). The physical property of density and porosity is 2350 kg/m3 and 7.78%, respectively. The fractured rock mass with a single fracture of 100 degree inclined angle relative to the horizontal plane and two mutually perpendicular fracture of 100 degree inclined angle and 90 degree inclined angle relative to the horizontal plane were obtained through the Brazilian splitting test, as shown in Figure 1. The specific mechanical property of the intact sandstone under different confining stress is shown in Table 1.Figure 1
Fractured samples: (a) single-fracture; (b) double-fracture.
(a)(b)Table 1
Mechanical property of the sandstone.
Sample no.Confining pressure (MPa)Peak strength (MPa)Poisson’s ratioElastic modulus (GPa)Cohesion (MPa)Friction angle (°)15109.420.236.3514.7648.93215170.670.199.01325225.690.1813.37The hydromechanical tests were carried out by triaxial hydromechanical coupling experimental system, as shown in Figure2. The triaxial hydromechanical coupling experimental system includes a triaxial cell and a fluid injection pump. The scope of confining stress, axial stress, and hydraulic pressure are 0-60 MPa, 0-600 MPa, and 0-30 MPa, respectively. The specific experimental process is presented in Figure 3.Figure 2
Triaxial hydromechanical coupling experimental system.Figure 3
Triaxial hydromechanical coupling experiment process.


## 2.3. Testing Scheme

To investigate the evolution of deformation and fluid conductivity subjected to the change of hydraulic pressure, confining stress, and cycle loading-unloading, the fractured sandstones were loaded into the triaxial chamber subjected to a complicated triaxial loading path, which consists of three parts. In the first part, the axial and confining pressure were initially loaded to the hydrostatic pressure of 36 MPa with a rate of 0.1 MPa/s, and the water pressure was reduced from 4.3 to 0.4 MPa with a gradient of 0.86 MPa. In the second part, the axial and confining pressure decreased from 36 to 18.5 MPa with a gradient of 2.5 MPa, while maintaining the hydraulic pressure of 0.4 MPa. Finally, the axial cyclic loading-unloading stress was set to an increment of 21.6 MPa at each cycle. The specific loading path is shown in Figure4.Figure 4
The triaxial loading path for testing.


## 3. Experimental Results and Discussion

### 3.1. Effect of Hydraulic Pressure and Confining Stress

The deformations in the experiment procedure of 1st and 2nd are small, which are not the focus of this study. Therefore, the deformations of 1st and 2nd parts in this research are not analyzed in detail in Section3.1.


#### 3.1.1. Relationship between Permeability and Hydraulic Pressure

The relationship between permeability and hydraulic pressure is shown in Figure5. The permeability of double-fracture sandstone is almost three times that of single-fracture sandstone under the initial stress condition. With the decrease of hydraulic pressure, the permeability of both single- and double-fracture sample gradually decreased as a quadratic function. Compared with the single-fracture sandstone, the double-fracture sandstone is more sensitive to the change of the hydraulic pressure. This illustrates that seepage channels of the fractured sandstone are narrowed as the decrease of the hydraulic pressure, resulting in the decrease of the permeability. Furthermore, the decrease of double-fracture sandstone is larger than single-fracture sandstone, due to more fractures exiting in the double-fracture sandstone.Figure 5
Permeability evolution of fractured sandstones under different hydraulic pressures.


#### 3.1.2. Relationship between Permeability and Confining Stress

Figure6 depicts the change of permeability of fractured sandstone subjected to the variation of confining stress at a constant hydraulic pressure. With the decrease of confining stress, the permeability of both single- and double-fracture sandstones increases slowly as a quadratic function. And the permeability of double-fracture sandstone in each stage is 2.7 times that of single-fracture sandstone. It is indicating that the effect of confining stress for fractured sandstone is relatively limited.Figure 6
Permeability evolution of fractured sandstones under different confining stresses.


### 3.2. Effect of Axial Cyclic Loading-Unloading

#### 3.2.1. Evolution Characteristics of the Deformation and Permeability

Figure7 shows the relationship of the axial strain, axial stress, and permeability of the fractured sandstones in the process of axial cyclic loading and unloading. For single-fracture sandstone, with axial cyclic loading-unloading, both axial and radial strains gradually increase, and the correlative permeability spiralled down quickly in the compaction stage and slowly in the elastic and plastic stage. However, for double-fracture sandstone, the permeability slightly changes with axial cyclic loading and unloading. Furthermore, the axial stress-strain curves of both single- and double-fracture sandstones approximately present a parallelogram shape after the first cycle, showing obvious deformation memory characteristics.Figure 7
Stress-strain and permeability-axial strain curves for fractured sandstones under axial cyclic loading-unloading: (a) single-fracture sandstone; (b) double-fracture sandstone.
(a)(b)The deformation and permeability characteristics of each cycle stage are presented in Figures8 and 9. In the first cycle, the loading and unloading curves of both single-fracture sandstone and double-fracture sandstone did not form a closed loop, and a large amount of residual deformation was observed, resulting in nonlinear properties of rock materials and damages of artificial fractures [36]. The corresponding permeability decreased dramatically and did not completely recover with axial loading and unloading. For single-fracture sandstone, the crossing of permeability curves at the loading state and unloading state was observed at the second cycle, while it was not found in the double-fracture sandstone. This phenomenon indicates that fractures were majorly compressed in the second cycle. Consequently, the flow channels were narrowed, and permeability reduced step-wise. In the third cycle, the permeability curves of both single-fracture sandstone and double-fracture sandstone gradually decreased with the loading and unloading, indicating that the fractures and secondary cracks also compressed during the third cycle. After the third cycle, the reexpansion of existing fractures and the generation of secondary cracks dominated the loading and unloading process. The compressed flow channels were expanded under the action of hydraulic pressure. As a result, the corresponding permeability of single- and double-fracture sandstones gradually increased. Finally, the axial stress-strain and permeability-axial strain curves of both single- and double-fracture sandstone have similar trend at the fifth loading.Figure 8
Stress-strain and permeability-strain curves for single-fracture sandstone at different cycles: (a) the first cycle; (b) the second cycle; (c) the third cycle; (d) the fourth cycle; (e) the fifth loading.
(a)(b)(c)(d)(e)Figure 9
Stress-strain and permeability-strain curves for double-fracture sandstone at different cycles: (a) the first cycle; (b) the second cycle; (c) the third cycle; (d) the fourth cycle; (e) the fifth loading.
(a)(b)(c)(d)(e)


#### 3.2.2. Relationship between Axial Residual Strain and Axial Cycle Stress

To quantitatively analyze the irrecoverable deformation of fractured sandstone subjected to the axial cyclic loading-unloading, the cumulative residual strain, comprised axial, and radial residual strains are defined. And the cumulative residual strainεjpi is expressed as [38]:
(2)εpij=εij−ε1j,where the superscripts j are 1 and 3 represent the axial and radial directions, respectively; εij is the strain at the ith cycle when the axial stress is 36 MPa, %; and ε1j is the strain for the initial axial stress of 36 MPa, %.Figure10 reflects the relationship between the cumulative residual strain and the cycle loading increment of the axial stress. Along the axial and radial direction, the cumulative residual strains increase in form of quadratic function as the increase in the loading increment of the axial stress. For cumulative axial residual strain and radial residual strain of double-fracture sandstone, the increment of residual strain increases remarkably in the first cycle and then increases slowly. Conversely, when the increment of axial stress is small, the cumulative radial residual strain of single-fracture is small with a slight increase gradient. The radial residual strain increases sharply after the second cycle. Generally, the change of residual strains can be divided into two stages: the sharp increase stage and slow increase stage.Figure 10
The residual strain at different cycle loading gradients: (a) cumulative axial residual strain; (b) cumulative radial residual strain.
(a)(b)


#### 3.2.3. Evolution of Permeability Recovery Rates (PRR) and Permeability Enhancement Reduction Rates (PERR)

To quantitatively characterize the permeability change of fractured rock caused by the axial cyclic loading-unloading and cyclic loading enhancement atσ1 =108 MPa, the concept of PRR and PERR was proposed, respectively. The PRR is defined as the ratio of the permeability in every cycle after unloading to the permeability at time of initial loading (cumulative PRR) or at the last cycle after unloading (relative PRR). Similarly, the PERR is defined as the ratio of the permeability in every cyclic loading at 108 MPa to the permeability at initial loading at 108 MPa (cumulative PERR) or at the last cyclic loading at 108 MPa (relative PERR). And the formulas for calculating the cumulative PPR (CPRR) and cumulative PERR (CPERR) λjc and relative PRR (RPRR) and relative PERR (RPERR) λjr are as follows:
(3)λcj=kik1×100%,(4)λrj=ki+1ki×100%,where the superscripts j=36 and 108 indicate the PPR and PERR, respectively, and ki is the permeability at the ith cyclic loading-unloading and loading when the axial stress is 36 MPa and 108 MPa, respectively, m2. k1 is the corresponding permeability when the initial axial stress is 36 MPa and 108 MPa for the first time, respectively, m2.Figure11 shows the relationship between PRR and axial cyclic loading and unloading. During the cyclic loading-unloading process, the CPRR of single-fracture sandstone gradually decreased. This change mainly caused by the continuous increment of load stress. With the continuously increasing of axial stress, the seepage channels, consisting of mainly compressed pores and fractures, were gradually narrowed. And the CPRR of double-fracture sandstone was stabled at 85%-95%, while the RPRR of both single- and double-fracture sandstones shows a certain fluctuation. This indicates that the stress sensitivity of permeability of double-fracture sandstone is lower than that of single-fracture sandstone, because it has more seepage channels.Figure 11
The permeability recovery at different cycles: (a) cumulative permeability recovery rate; (b) relative permeability recovery rate.
(a)(b)Figure12 presents the relationship between PERR and axial cyclic loading at the axial stress of 108 MPa. As the axial cyclic loading, the CPERR and RPERR of double-fracture sandstone decreased synchronously, presenting the enhancement effect of cyclic loading. The sandstone pore-fracture system was further compressed with increased cycle times, and then, the flow channels were narrowed. In contrast, the CPERR of single-fracture sandstone increased gradually with the first three axial loadings in the elastic stage, and the increasing degree was greater (about 2.4 times), which is attributed to the dislocation of artificial fracture surface. With axial cyclic loading time increase, the dislocated spaces increase, combing with fracture swelling and developing, the flow channels increase, and the permeability increases dramatically. Mine water inrush is one of the main disasters that restrict the safe and efficient production of coal mines. Combining Figure 12, when mining above or below the deep confined aquifer, especially the effect of repeated mining, even if the aquifuge is not failure, the displacement of roof and floor should be controlled reasonably to prevent the dislocation of vertical fractures, the increase of permeability of rock stratum, and water inrush disaster.Figure 12
The permeability enhancement reduction at different loading stages (at 108 MPa): (a) cumulative permeability enhancement reduction rate; (b) relative permeability enhancement reduction rate.
(a)(b)


## 3.1. Effect of Hydraulic Pressure and Confining Stress

The deformations in the experiment procedure of 1st and 2nd are small, which are not the focus of this study. Therefore, the deformations of 1st and 2nd parts in this research are not analyzed in detail in Section3.1.


### 3.1.1. Relationship between Permeability and Hydraulic Pressure

The relationship between permeability and hydraulic pressure is shown in Figure5. The permeability of double-fracture sandstone is almost three times that of single-fracture sandstone under the initial stress condition. With the decrease of hydraulic pressure, the permeability of both single- and double-fracture sample gradually decreased as a quadratic function. Compared with the single-fracture sandstone, the double-fracture sandstone is more sensitive to the change of the hydraulic pressure. This illustrates that seepage channels of the fractured sandstone are narrowed as the decrease of the hydraulic pressure, resulting in the decrease of the permeability. Furthermore, the decrease of double-fracture sandstone is larger than single-fracture sandstone, due to more fractures exiting in the double-fracture sandstone.Figure 5
Permeability evolution of fractured sandstones under different hydraulic pressures.


### 3.1.2. Relationship between Permeability and Confining Stress

Figure6 depicts the change of permeability of fractured sandstone subjected to the variation of confining stress at a constant hydraulic pressure. With the decrease of confining stress, the permeability of both single- and double-fracture sandstones increases slowly as a quadratic function. And the permeability of double-fracture sandstone in each stage is 2.7 times that of single-fracture sandstone. It is indicating that the effect of confining stress for fractured sandstone is relatively limited.Figure 6
Permeability evolution of fractured sandstones under different confining stresses.


## 3.1.1. Relationship between Permeability and Hydraulic Pressure

The relationship between permeability and hydraulic pressure is shown in Figure5. The permeability of double-fracture sandstone is almost three times that of single-fracture sandstone under the initial stress condition. With the decrease of hydraulic pressure, the permeability of both single- and double-fracture sample gradually decreased as a quadratic function. Compared with the single-fracture sandstone, the double-fracture sandstone is more sensitive to the change of the hydraulic pressure. This illustrates that seepage channels of the fractured sandstone are narrowed as the decrease of the hydraulic pressure, resulting in the decrease of the permeability. Furthermore, the decrease of double-fracture sandstone is larger than single-fracture sandstone, due to more fractures exiting in the double-fracture sandstone.Figure 5
Permeability evolution of fractured sandstones under different hydraulic pressures.


## 3.1.2. Relationship between Permeability and Confining Stress

Figure6 depicts the change of permeability of fractured sandstone subjected to the variation of confining stress at a constant hydraulic pressure. With the decrease of confining stress, the permeability of both single- and double-fracture sandstones increases slowly as a quadratic function. And the permeability of double-fracture sandstone in each stage is 2.7 times that of single-fracture sandstone. It is indicating that the effect of confining stress for fractured sandstone is relatively limited.Figure 6
Permeability evolution of fractured sandstones under different confining stresses.


## 3.2. Effect of Axial Cyclic Loading-Unloading

### 3.2.1. Evolution Characteristics of the Deformation and Permeability

Figure7 shows the relationship of the axial strain, axial stress, and permeability of the fractured sandstones in the process of axial cyclic loading and unloading. For single-fracture sandstone, with axial cyclic loading-unloading, both axial and radial strains gradually increase, and the correlative permeability spiralled down quickly in the compaction stage and slowly in the elastic and plastic stage. However, for double-fracture sandstone, the permeability slightly changes with axial cyclic loading and unloading. Furthermore, the axial stress-strain curves of both single- and double-fracture sandstones approximately present a parallelogram shape after the first cycle, showing obvious deformation memory characteristics.Figure 7
Stress-strain and permeability-axial strain curves for fractured sandstones under axial cyclic loading-unloading: (a) single-fracture sandstone; (b) double-fracture sandstone.
(a)(b)The deformation and permeability characteristics of each cycle stage are presented in Figures8 and 9. In the first cycle, the loading and unloading curves of both single-fracture sandstone and double-fracture sandstone did not form a closed loop, and a large amount of residual deformation was observed, resulting in nonlinear properties of rock materials and damages of artificial fractures [36]. The corresponding permeability decreased dramatically and did not completely recover with axial loading and unloading. For single-fracture sandstone, the crossing of permeability curves at the loading state and unloading state was observed at the second cycle, while it was not found in the double-fracture sandstone. This phenomenon indicates that fractures were majorly compressed in the second cycle. Consequently, the flow channels were narrowed, and permeability reduced step-wise. In the third cycle, the permeability curves of both single-fracture sandstone and double-fracture sandstone gradually decreased with the loading and unloading, indicating that the fractures and secondary cracks also compressed during the third cycle. After the third cycle, the reexpansion of existing fractures and the generation of secondary cracks dominated the loading and unloading process. The compressed flow channels were expanded under the action of hydraulic pressure. As a result, the corresponding permeability of single- and double-fracture sandstones gradually increased. Finally, the axial stress-strain and permeability-axial strain curves of both single- and double-fracture sandstone have similar trend at the fifth loading.Figure 8
Stress-strain and permeability-strain curves for single-fracture sandstone at different cycles: (a) the first cycle; (b) the second cycle; (c) the third cycle; (d) the fourth cycle; (e) the fifth loading.
(a)(b)(c)(d)(e)Figure 9
Stress-strain and permeability-strain curves for double-fracture sandstone at different cycles: (a) the first cycle; (b) the second cycle; (c) the third cycle; (d) the fourth cycle; (e) the fifth loading.
(a)(b)(c)(d)(e)


### 3.2.2. Relationship between Axial Residual Strain and Axial Cycle Stress

To quantitatively analyze the irrecoverable deformation of fractured sandstone subjected to the axial cyclic loading-unloading, the cumulative residual strain, comprised axial, and radial residual strains are defined. And the cumulative residual strainεjpi is expressed as [38]:
(2)εpij=εij−ε1j,where the superscripts j are 1 and 3 represent the axial and radial directions, respectively; εij is the strain at the ith cycle when the axial stress is 36 MPa, %; and ε1j is the strain for the initial axial stress of 36 MPa, %.Figure10 reflects the relationship between the cumulative residual strain and the cycle loading increment of the axial stress. Along the axial and radial direction, the cumulative residual strains increase in form of quadratic function as the increase in the loading increment of the axial stress. For cumulative axial residual strain and radial residual strain of double-fracture sandstone, the increment of residual strain increases remarkably in the first cycle and then increases slowly. Conversely, when the increment of axial stress is small, the cumulative radial residual strain of single-fracture is small with a slight increase gradient. The radial residual strain increases sharply after the second cycle. Generally, the change of residual strains can be divided into two stages: the sharp increase stage and slow increase stage.Figure 10
The residual strain at different cycle loading gradients: (a) cumulative axial residual strain; (b) cumulative radial residual strain.
(a)(b)


### 3.2.3. Evolution of Permeability Recovery Rates (PRR) and Permeability Enhancement Reduction Rates (PERR)

To quantitatively characterize the permeability change of fractured rock caused by the axial cyclic loading-unloading and cyclic loading enhancement atσ1 =108 MPa, the concept of PRR and PERR was proposed, respectively. The PRR is defined as the ratio of the permeability in every cycle after unloading to the permeability at time of initial loading (cumulative PRR) or at the last cycle after unloading (relative PRR). Similarly, the PERR is defined as the ratio of the permeability in every cyclic loading at 108 MPa to the permeability at initial loading at 108 MPa (cumulative PERR) or at the last cyclic loading at 108 MPa (relative PERR). And the formulas for calculating the cumulative PPR (CPRR) and cumulative PERR (CPERR) λjc and relative PRR (RPRR) and relative PERR (RPERR) λjr are as follows:
(3)λcj=kik1×100%,(4)λrj=ki+1ki×100%,where the superscripts j=36 and 108 indicate the PPR and PERR, respectively, and ki is the permeability at the ith cyclic loading-unloading and loading when the axial stress is 36 MPa and 108 MPa, respectively, m2. k1 is the corresponding permeability when the initial axial stress is 36 MPa and 108 MPa for the first time, respectively, m2.Figure11 shows the relationship between PRR and axial cyclic loading and unloading. During the cyclic loading-unloading process, the CPRR of single-fracture sandstone gradually decreased. This change mainly caused by the continuous increment of load stress. With the continuously increasing of axial stress, the seepage channels, consisting of mainly compressed pores and fractures, were gradually narrowed. And the CPRR of double-fracture sandstone was stabled at 85%-95%, while the RPRR of both single- and double-fracture sandstones shows a certain fluctuation. This indicates that the stress sensitivity of permeability of double-fracture sandstone is lower than that of single-fracture sandstone, because it has more seepage channels.Figure 11
The permeability recovery at different cycles: (a) cumulative permeability recovery rate; (b) relative permeability recovery rate.
(a)(b)Figure12 presents the relationship between PERR and axial cyclic loading at the axial stress of 108 MPa. As the axial cyclic loading, the CPERR and RPERR of double-fracture sandstone decreased synchronously, presenting the enhancement effect of cyclic loading. The sandstone pore-fracture system was further compressed with increased cycle times, and then, the flow channels were narrowed. In contrast, the CPERR of single-fracture sandstone increased gradually with the first three axial loadings in the elastic stage, and the increasing degree was greater (about 2.4 times), which is attributed to the dislocation of artificial fracture surface. With axial cyclic loading time increase, the dislocated spaces increase, combing with fracture swelling and developing, the flow channels increase, and the permeability increases dramatically. Mine water inrush is one of the main disasters that restrict the safe and efficient production of coal mines. Combining Figure 12, when mining above or below the deep confined aquifer, especially the effect of repeated mining, even if the aquifuge is not failure, the displacement of roof and floor should be controlled reasonably to prevent the dislocation of vertical fractures, the increase of permeability of rock stratum, and water inrush disaster.Figure 12
The permeability enhancement reduction at different loading stages (at 108 MPa): (a) cumulative permeability enhancement reduction rate; (b) relative permeability enhancement reduction rate.
(a)(b)


## 3.2.1. Evolution Characteristics of the Deformation and Permeability

Figure7 shows the relationship of the axial strain, axial stress, and permeability of the fractured sandstones in the process of axial cyclic loading and unloading. For single-fracture sandstone, with axial cyclic loading-unloading, both axial and radial strains gradually increase, and the correlative permeability spiralled down quickly in the compaction stage and slowly in the elastic and plastic stage. However, for double-fracture sandstone, the permeability slightly changes with axial cyclic loading and unloading. Furthermore, the axial stress-strain curves of both single- and double-fracture sandstones approximately present a parallelogram shape after the first cycle, showing obvious deformation memory characteristics.Figure 7
Stress-strain and permeability-axial strain curves for fractured sandstones under axial cyclic loading-unloading: (a) single-fracture sandstone; (b) double-fracture sandstone.
(a)(b)The deformation and permeability characteristics of each cycle stage are presented in Figures8 and 9. In the first cycle, the loading and unloading curves of both single-fracture sandstone and double-fracture sandstone did not form a closed loop, and a large amount of residual deformation was observed, resulting in nonlinear properties of rock materials and damages of artificial fractures [36]. The corresponding permeability decreased dramatically and did not completely recover with axial loading and unloading. For single-fracture sandstone, the crossing of permeability curves at the loading state and unloading state was observed at the second cycle, while it was not found in the double-fracture sandstone. This phenomenon indicates that fractures were majorly compressed in the second cycle. Consequently, the flow channels were narrowed, and permeability reduced step-wise. In the third cycle, the permeability curves of both single-fracture sandstone and double-fracture sandstone gradually decreased with the loading and unloading, indicating that the fractures and secondary cracks also compressed during the third cycle. After the third cycle, the reexpansion of existing fractures and the generation of secondary cracks dominated the loading and unloading process. The compressed flow channels were expanded under the action of hydraulic pressure. As a result, the corresponding permeability of single- and double-fracture sandstones gradually increased. Finally, the axial stress-strain and permeability-axial strain curves of both single- and double-fracture sandstone have similar trend at the fifth loading.Figure 8
Stress-strain and permeability-strain curves for single-fracture sandstone at different cycles: (a) the first cycle; (b) the second cycle; (c) the third cycle; (d) the fourth cycle; (e) the fifth loading.
(a)(b)(c)(d)(e)Figure 9
Stress-strain and permeability-strain curves for double-fracture sandstone at different cycles: (a) the first cycle; (b) the second cycle; (c) the third cycle; (d) the fourth cycle; (e) the fifth loading.
(a)(b)(c)(d)(e)


## 3.2.2. Relationship between Axial Residual Strain and Axial Cycle Stress

To quantitatively analyze the irrecoverable deformation of fractured sandstone subjected to the axial cyclic loading-unloading, the cumulative residual strain, comprised axial, and radial residual strains are defined. And the cumulative residual strainεjpi is expressed as [38]:
(2)εpij=εij−ε1j,where the superscripts j are 1 and 3 represent the axial and radial directions, respectively; εij is the strain at the ith cycle when the axial stress is 36 MPa, %; and ε1j is the strain for the initial axial stress of 36 MPa, %.Figure10 reflects the relationship between the cumulative residual strain and the cycle loading increment of the axial stress. Along the axial and radial direction, the cumulative residual strains increase in form of quadratic function as the increase in the loading increment of the axial stress. For cumulative axial residual strain and radial residual strain of double-fracture sandstone, the increment of residual strain increases remarkably in the first cycle and then increases slowly. Conversely, when the increment of axial stress is small, the cumulative radial residual strain of single-fracture is small with a slight increase gradient. The radial residual strain increases sharply after the second cycle. Generally, the change of residual strains can be divided into two stages: the sharp increase stage and slow increase stage.Figure 10
The residual strain at different cycle loading gradients: (a) cumulative axial residual strain; (b) cumulative radial residual strain.
(a)(b)


## 3.2.3. Evolution of Permeability Recovery Rates (PRR) and Permeability Enhancement Reduction Rates (PERR)

To quantitatively characterize the permeability change of fractured rock caused by the axial cyclic loading-unloading and cyclic loading enhancement atσ1 =108 MPa, the concept of PRR and PERR was proposed, respectively. The PRR is defined as the ratio of the permeability in every cycle after unloading to the permeability at time of initial loading (cumulative PRR) or at the last cycle after unloading (relative PRR). Similarly, the PERR is defined as the ratio of the permeability in every cyclic loading at 108 MPa to the permeability at initial loading at 108 MPa (cumulative PERR) or at the last cyclic loading at 108 MPa (relative PERR). And the formulas for calculating the cumulative PPR (CPRR) and cumulative PERR (CPERR) λjc and relative PRR (RPRR) and relative PERR (RPERR) λjr are as follows:
(3)λcj=kik1×100%,(4)λrj=ki+1ki×100%,where the superscripts j=36 and 108 indicate the PPR and PERR, respectively, and ki is the permeability at the ith cyclic loading-unloading and loading when the axial stress is 36 MPa and 108 MPa, respectively, m2. k1 is the corresponding permeability when the initial axial stress is 36 MPa and 108 MPa for the first time, respectively, m2.Figure11 shows the relationship between PRR and axial cyclic loading and unloading. During the cyclic loading-unloading process, the CPRR of single-fracture sandstone gradually decreased. This change mainly caused by the continuous increment of load stress. With the continuously increasing of axial stress, the seepage channels, consisting of mainly compressed pores and fractures, were gradually narrowed. And the CPRR of double-fracture sandstone was stabled at 85%-95%, while the RPRR of both single- and double-fracture sandstones shows a certain fluctuation. This indicates that the stress sensitivity of permeability of double-fracture sandstone is lower than that of single-fracture sandstone, because it has more seepage channels.Figure 11
The permeability recovery at different cycles: (a) cumulative permeability recovery rate; (b) relative permeability recovery rate.
(a)(b)Figure12 presents the relationship between PERR and axial cyclic loading at the axial stress of 108 MPa. As the axial cyclic loading, the CPERR and RPERR of double-fracture sandstone decreased synchronously, presenting the enhancement effect of cyclic loading. The sandstone pore-fracture system was further compressed with increased cycle times, and then, the flow channels were narrowed. In contrast, the CPERR of single-fracture sandstone increased gradually with the first three axial loadings in the elastic stage, and the increasing degree was greater (about 2.4 times), which is attributed to the dislocation of artificial fracture surface. With axial cyclic loading time increase, the dislocated spaces increase, combing with fracture swelling and developing, the flow channels increase, and the permeability increases dramatically. Mine water inrush is one of the main disasters that restrict the safe and efficient production of coal mines. Combining Figure 12, when mining above or below the deep confined aquifer, especially the effect of repeated mining, even if the aquifuge is not failure, the displacement of roof and floor should be controlled reasonably to prevent the dislocation of vertical fractures, the increase of permeability of rock stratum, and water inrush disaster.Figure 12
The permeability enhancement reduction at different loading stages (at 108 MPa): (a) cumulative permeability enhancement reduction rate; (b) relative permeability enhancement reduction rate.
(a)(b)


## 4. Conclusions

Based on the hydromechanical experiments of triaxial loading, the change of the axial and radial strain and permeability subjected to the effect of the hydraulic pressure, confining stress, and cyclic loading-unloading was observed. And cumulative residual strain, PRR, and PERR were proposed to quantitatively analyze these changes. The main conclusions are as follows:(1)
The relative permeability of both single- and double-fracture sandstone varies as a quadratic function of hydraulic and confining stress. The permeability of fractured sandstone is positively related to the hydraulic pressure, and double-fracture sandstone is more sensitive to the hydraulic pressure than the single-fracture sandstone. The permeability of fractured sandstone is negatively related to the confining stress. However, the effect of confining stress on the strain is more obvious than the permeability(2)
As the advance of axial cyclic loading-unloading, the axial and radial strains of fractured sandstone increase, and the permeability of single-fracture sandstone spirals down. However, for double-fracture sandstone, the permeability slightly changes with more seepage channels. Furthermore, the axial stress-strain curves approximately present a parallelogram shape after the first cycle, showing obvious deformation memory characteristics. The CPRR of single-fracture sandstone decreases first and then increases. The CPRR of double-fracture sandstone maintains 85%-95% with a fluctuated RPRR(3)
With axial cycle enhancement loading, the PERR of double-fracture sandstone decreases gradually; in contrast, the PERR of single-fracture sandstone dramatically increases. For the single-fracture sandstone, the dislocation of fracture surface was presented and developed with the increase of the axial loading time, resulting in the increase of the seepage channels characterized by the increase of the permeability



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*Source: 8811003-2020-10-31.xml*