Introduction

Surrounding rock masses experience complex stress paths affected by engineering activities such as underground tunnel excavation, coal mining, and deeply buried radioactive waste disposal (Chen et al. 2004; Kim et al. 2019; Xue et al. 2017a). The theoretical research results of unloading rock mass mechanics indicate that the mechanical behavior of rock masses under loading and unloading conditions is essentially different (Yang et al. 2011; Chen et al. 2020; Li et al. 2010). Because the single loading path (the confining pressure of which is often constant) differs from the excavation process of rock masses (usually accompanied by a decrease in confining pressure), the mechanical parameters measured in the laboratory under loading conditions are not entirely accurate. It is more meaningful to perform confining pressure unloading tests and use the stress path and unloading rates to reflect the unloading characteristics of the excavation (Qiu et al. 2014; Huang et al. 2017).

Many scholars have carried out rock mass unloading tests and theoretical research to provide a reference for rock failure theory and stability analysis under excavation unloading. Zhou (2005) proposed an elastic-brittle material model based on micromechanics through theoretical analysis and investigated rock micromechanical damage mechanisms while axial stress was constant and lateral confinement was reduced. Qiu et al. (2014) performed a cyclic loading‒unloading pressure test on marble and revealed the influence of unloading damage on the evolution of volumetric dilation and elastic parameters. Huang and Huang (2014) conducted tests on artificial rock specimens under two conditions: unloading confining pressure with axial loading and unloading confining pressure with constant axial pressure. They analyzed the influence of prefabricated fractures on rock strength and fracture development. Huang et al. (2017) carried out triaxial unloading confining pressure tests and triaxial unloading creep tests on sandy mudstone and studied the influence of excavation method on the dilatancy and fracturing behavior of soft rock. Xu et al. (2019) presented minimum principal stress unloading tests of marble and analyzed the effects of initial stress and unloading rates on the bearing capacity and the deformation and failure characteristics. Using energy conversion, Wang et al. (2019) and Yang et al. (2019) analyzed the deformation and failure mechanisms of dacite and sandstone in the unloading confining pressure process. They found that the dissipated strain energy plays a significant role. Chen et al. (2020) discussed the evolution of crack strain during conventional loading and confining pressure unloading conditions and studied the effect of the unloading rates on the crack strain.

Scholars have also found that rock engineering instability and failure are closely related to seepage in surrounding rock masses. Thus, the changes in permeability caused by rock excavation and unloading must be considered (Alalade et al. 2018; Xue et al. 2017a; Wang et al. 2018, 2020b). For example, Yin et al. (2015) and Xue et al. (2017b) studied the effects of the unloading rate on the mechanical behavior and permeability to prevent coal ruptures and gas outburst hazards. Pardoen et al. (2016) numerically simulated the excavation of an underground gallery and reproduced a significant permeability increase in the excavation unloading damage. Chen et al. (2018) monitored the unloading process with acoustic emission technology and noted a positive correlation between the permeability and volumetric strain. Meng et al. (2012) discussed the phyllite failure mechanics under THM coupling and unloading and analyzed the effect of temperature and water pressure on penetrability.

Affected by the sedimentary environment, climate and other factors, Cretaceous sandstones in many parts of the world are poorly cemented with the characteristics of low density, strong water absorption capacity, and low strength (Vavro et al. 2019; Wang et al. 2020a), which greatly differ from other rocks. However, the unloading damage under hydromechanical coupling and the characteristics of permeability evolution of weakly cemented Cretaceous sandstone have rarely been studied. Cretaceous sandstone in the Ordos Basin, China, is an essential aquifer due to its late diagenetic period, shallow burial, and high porosity. Large-scale coal mining and underground construction in this area are bound to cause unloading damage to the weakly cemented Cretaceous sandstone, influencing the regional ecological environment and production safety. In this paper, confining pressure unloading-permeability tests of weakly cemented sandstone under different unloading paths were carried out by using the rock multi coupling test system, and the effects of the initial confining pressure and unloading rate on the unloading deformation and permeability evolution were studied. The test results can provide a basis for the stability analysis of surrounding rock and water disaster prevention of underground engineering excavation where such weakly cemented sandstone is distributed.

Methodology

Specimen preparation

The weakly cemented sandstone used in this study was fine sandstone from the Cretaceous Luohe Formation and was collected from an exploration borehole of the Shilawusu coal mine in the Ordos Basin, China. The drilling cores had roughly the same burial depth (approximately 300 m) to avoid the discrepancies induced by rock heterogeneity. The X-ray diffraction test results suggest that the main mineral components of this sandstone are quartz (33.32%), albite (18.14%), orthoclase (23.01%), calcite (5.26%), montmorillonite (3.2%), dolomite (4.56%), and analcime (12.51%). The sandstone is purplish-red, and the physical properties are shown in Table 1. Through drilling, cutting, and grinding, the sandstone was processed into standard cylindrical specimens with a diameter of 50 mm and a height of 100 mm, and the unevenness of the upper and lower planes of the specimens was less than 0.02 mm, as suggested by International Society for Rock Mechanics (2007). Before the experiment, ultrasonic testing was carried out to ensure the homogeneity and similarity of the specimens. Then, the specimens were fully saturated with water by a vacuum water saturation machine.

Table 1 Basic physical properties of sandstone specimen
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Experimental equipment

The experiments were carried out at the China University of Mining and Technology using Rock 1000–50 VHT multi-coupling test system (Fig. 1). The system consists of six parts, namely, the control system, triaxial pressure chamber, axial pressure system, confining pressure system, seepage system, and temperature system, as well as special high-precision sensors (displacement, pressure, and temperature). The test system can provide a maximum axial force of 2000 kN and a maximum confining pressure of 100 MPa. The maximum available pressure of the seepage system is 60 MPa upstream and 50 MPa downstream. The pressure control accuracy of the whole test system is ± 0.01 MPa and the resolution ratio is 0.0001 MPa. Two LVDTs were fixed on both sides of the specimen to measure the axial strain, and an annular LVDT was fixed on the middle of the specimen to obtain the lateral strain with a measurement range of ± 5 mm.

Fig. 1
figure 1

Experimental equipment

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Experimental scheme

The stress state of the surrounding rock could be changed during excavation for underground engineering. In this paper, the axial and confining pressure were set to represent the vertical and horizontal stresses, respectively, and the deformation and seepage of weakly cemented sandstone caused by underground engineering excavation were studied. First, the conventional triaxial test was carried out to obtain the variation characteristics of the strength, deformation parameters, and permeability of the sandstone during loading, which provided the basis for the unloading confining pressure test. During the test, three different confining pressures of 7.5 MPa, 10 MPa, and 12.5 MPa were selected to simulate the in situ stress at burial depths of 300 m, 400 m, and 500 m, respectively. According to 60–80% of the triaxial compressive strength commonly used as the unloading starting point in the unloading confining pressure test (Huang et al. 2017; Li et al. 2017; Qiu et al. 2014), 80% of the triaxial compressive strength under each confining pressure was selected as the unloading starting point of the corresponding confining pressure in this study. Moreover, three confining pressure unloading rates of 0.2 MPa/min, 0.5 MPa/min, and 1 MPa/min were selected to simulate different excavation rates. The specific test conditions and basic information of specimens are listed in Table 2.

Table 2 Basic information about the sandstone specimens
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The steps of the conventional triaxial compression test were as follows: (1) Increase the axial pressure and confining pressure of the specimen to the hydrostatic pressure \({\sigma }_{3}^{0}\) at a loading rate of 5 MPa/min (OA stage in Fig. 2). (2) Apply the permeation pressure of 2 MPa and keep it constant until a stable unidirectional flow is formed in the specimen. Then, keep the stress state remained unchanged and measure the initial permeability. (3) Increase the axial stress at a 2 MPa/min loading rate and maintain the confining pressure. Once the specimen was damaged, change the axial stress control mode to strain control immediately to obtain the residual strength of the specimen (AC stage in Fig. 2). (4) Test the permeability at a 0.8% interval of axial strain.

Fig. 2
figure 2

Sketch map of stress path in different experiment schemes

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The unloading confining pressure test was carried out as follows: (1) Increase the confining pressure and axial pressure to a designed value \({\sigma }_{3}^{0}\) at the same time at a loading rate of 2 MPa/min (OA stage in Fig. 2). (2) Increase the permeation pressure of the lower part of the specimen to 2 MPa and keep it constant, and then measure the initial permeability. (3) Maintain the confining pressure σ3 and increase the axial stress to the unloading starting point load (point B in Fig. 2) at a 2 MPa/min loading rate. In this process, measure the permeability every 0.06% of axial strain. (4) Unload the confining pressure with the given unloading rate (BD1, BD2, and BD3 in Fig. 2, representing different unloading rates) until the specimen fails. Since the time required for the permeability test is significantly longer than the time used in stress unloading, the permeability of too many points will affect the deformation of the specimen during unloading. Measure the permeability every 1.5–2 MPa reduction of the confining pressure during the unloading to reduce this effect. (5) Once the specimen failed, the unloading confining pressure was stopped immediately, and the permeability was measured after failure.

Due to the low porosity of the sandstone specimen, the permeability was measured using a transient method (Brace et al. 1968), which is widely used in testing the permeability of tight rocks and coal (Bhandari et al. 2015; Zhou et al. 2019). In this method, the water pressure decay curve passing through the sandstone specimen can be described as follows:

$$\left({P}_{1}-{P}_{2}\right)=\Delta P\left[{V}_{2}/\left({V}_{1}+{V}_{2}\right)\right]{e}^{-\alpha t}$$
(1)
$$\alpha =\left(\frac{kA}{\mu L}\right)\left(\frac{1}{{V}_{1}}+\frac{1}{{V}_{2}}\right)$$
(2)

where P1 is the upstream pressure (MPa); P2 is the downstream pressure (MPa); ΔP is the pressure difference between the upstream and downstream of the specimen in the initial stage, which is 2 MPa in this study; V1 and V2 are the reference volumes of the upstream and downstream, respectively; t is the time spent on this test(s); μ is the water viscous coefficient (Pa s); A is the cross-sectional area of the specimen (m2); L is the length of the specimen (m); and k is the permeability (m2). In the actual study, Eq. (1) is deformed, and α is obtained from the pressure difference-time semilogarithmic curve (see Fig. 3). Then, the permeability under different stress states can be calculated through Eq. (2).

Fig. 3
figure 3

The curve of semilogarithmic pressure difference vs. time

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Results

Stress and deformation evolution characteristics of sandstone

Stress‒strain characteristics under conventional compression

The variation in strain with deviatoric stress of the sandstone specimens during triaxial compression is shown in Fig. 4. As shown in Fig. 4, the stress‒strain curves can be divided into three stages: linear elastic stage, yield stage, and post-peak failure stage. When the specimen was loaded to the hydrostatic pressure state, the natural cracks were gradually closed, resulting in the axial strain (\({\varepsilon }_{1}\)) increasing linearly with the deviatoric stress at the first loading stage. In this stage, the radial strain (\({\varepsilon }_{3}\)) increased only slightly so that the specimen was in a compressed state overall. With increasing deviatoric stress, the specimen entered the yield stage. The rate increase in stress slowed, the radial strain increased rapidly, and the cracks in the specimen occurred and gradually expanded. The specimen entered the volume expansion state. In the third stage, the crack in the specimen continued to develop. Finally, it formed into a macrofracture zone, which made the specimen lose its bearing capacity, and the stress decreased rapidly. The volumetric strain (\({\varepsilon }_{\text{v}}\)) continued to increase as axial compression was applied. Table 3 shows that the elastic modulus and peak strength increased with increasing initial confining pressure.

Fig. 4
figure 4

Variation of the deviatoric stress–strain curve under different confining pressures

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Table 3 Mechanical parameters of sandstone specimens under conventional compression
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Stress‒strain characteristics under unloading confining pressure

Taking the condition of a confining pressure unloading rate of 0.2 MPa/min as an example, Fig. 5 plots the variation in the strain with deviatoric stress under different initial confining pressures. It can be seen from Fig. 5 that the stress‒strain curve under the condition of unloading confining pressure is similar to that of conventional triaxial compression, which can also be divided into three stages. The first stage corresponds to the axial stress loading process (AB stage in Fig. 2), so the curve changes the same as that in the linear elastic stage of conventional compression. When the confining pressure began to unload, it entered the yield stage, the radial strain began to increase rapidly, and cracks appeared in the specimen. In the post-peak failure stage, the radial expansion rate greatly decreased because the confining pressure unloading was stopped as soon as the specimen was damaged, and the confining pressure still constrained the specimen.

Fig. 5
figure 5

Variation of the deviatoric stress–strain curve with the unloading confining pressures rate of 0.2 MPa/min under different initial confining pressure

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From the starting unloading confining pressure point to the specimen failure point, the change in axial strain under different confining pressures exhibited little difference and was 0.197%, 0.201%, and 0.198% at 7.5 MPa, 10 MPa, and 12.5 MPa, respectively. However, the confining pressure unloading greatly affected the radial deformation. The changes in radial strain were − 0.389%, − 0.297%, and − 0.262% at 7.5 MPa, 10 MPa, and 12.5 MPa, respectively, from the starting unloading point to the failure point. The greater the initial confining pressure was, the smaller the change in radial strain, so the smaller the change in volumetric strain.

Permeability evolution characteristics of sandstone

Permeability evolution under conventional compression

The permeability–axial strain curves of sandstone under constant confining pressure are presented in Fig. 6. The permeability of sandstone changes little before failure because the natural fractures in the specimen are closed under high confining pressure. When the deviatoric stress reaches the yield strength point, the sandstone produces macrocracks, the seepage channel gradually forms, and the permeability increases rapidly. However, the peak permeability lags behind the peak strength. Moreover, the magnitude of the peak permeability of the sandstone is dependent on the confining stress. At a high confining stress, the initiation, growth, and propagation of cracks was suppressed; therefore, the linear elastic stage before the onset of dilatancy lasted longer, which delayed the beginning of fracture seepage. The magnification of permeability after failure under high confining pressure was also smaller due to the limitation of confining pressure.

Fig. 6
figure 6

Permeability evolution curves of sandstone under conventional compression

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Permeability evolution under the unloading confining pressure

The variation trend of the permeability–axial strain curve during unloading confining pressure is similar to that in conventional compression, so this subsection focuses on the variation in sandstone permeability with volumetric strain as plotted in Fig. 7, to help analyze the influence of unloading rates and initial confining pressure on permeability. It can be seen from Fig. 7 that in the axial loading stage, the specimen was gradually compressed, and the volumetric strain increased so that the permeability decreased slightly. When the specimen reached the expansion point, it entered the expansion state from the compression state, and the permeability increased approximately linearly.

Fig. 7
figure 7

Permeability evolution with the volumetric strain of sandstone in unloading confining pressure

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Notably, the initial confining pressure significantly had a significant effect on permeability. As the initial confining pressure increased, the initial permeability and post-peak permeability of the sandstone specimen decreased. The unloading rate mainly influenced the evolution process of permeability. With increasing unloading rate, the rate of increase in permeability decreased gradually, and the volumetric strain-permeability curve exhibited inflection points, and thus was not as smooth as the curve under slow unloading. The smaller the unloading rate was, the greater the change in volumetric strain and the higher the post-peak permeability. Taking the test with an initial confining pressure of 10 MPa as an example, the unloading amount of the confining pressure at failure was 54.3%, 35.5%, and 23.3% at unloading rates of 0.2 MPa/min, 0.5 MPa/min, and 1 MPa/min, respectively. This indicated that under the same confining pressure, the lower the unloading rate was, the lower the confining pressure when peak failure occurred. Table 4 shows the ratio of post-peak permeability to initial permeability. Under an initial confining pressure of 10 MPa, when the specimen was unloaded at a rate of 0.2 MPa/min, 0.5 MPa/min, and 1 MPa/min, the permeability ratio was 6.37, 4.12, and 3.17, respectively. The permeability difference is because the cracks in the specimen were more developed while unloading slowly, and the confining pressure was lower when the specimen failed, which was conducive to the migration of fluid in the sandstone.

Table 4 Mechanical parameters of sandstone specimen under unloading confining pressure experiments
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Discussion

Effect of unloading rate on deformation and strength

The stress‒strain curves of sandstone specimens under different initial confining pressures and unloading rates are shown in Fig. 8. Before the start of unloading, the specimen was under axial loading, and the stress‒strain curves were similar to those under conventional triaxial compression, which suggests that specimen dispersion was small. Martin (1993) described that keeping the axial pressure constant and unloading the confining pressure is a process of increasing deviatoric stress (σ1–σ3), which corresponds to a monotonic effective loading path. It can be seen from Fig. 8 that when unloading at different rates under the same initial confining pressure, the axial deformation process of the specimen was basically the same, but the radial deformation was very different. From the beginning of unloading, the axial strain increased with the deviatoric stress, and its change was similar to the triaxial compression process (see Fig. 4), while the change in radial strain was related to the unloading rate. When the confining pressure was unloaded at a lower rate, with the increase in deviatoric stress, the radial strain first increased slowly, and this process was very short. After reaching the volume expansion point of the specimen, the radial strain began to increase rapidly, the curve presented an obvious “small slope,” and the volume expansion of the specimen was more significant. The deviatoric stress strain changed similarly while unloading at the different initial confining pressures with the same unloading rates. Notably, the curve shape changed with the increasing initial confining pressure under the larger unloading rate of 1 MPa/min. The deviatoric stress-radial strain curve was sharp under a lower initial pressure of 7.5 MPa and 10 MPa, and there was a “small slope” similar to that during unloading at slow rates under a pressure of 12.5 MPa. The variation in the stress‒strain curves of the sandstone during unloading was similar to the experimental results of Chen et al. (2020), Guo et al. (2021), and Fu et al. (2023).

Fig. 8
figure 8

Stress–strain curves of sandstone under different confining pressure unloading rates

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Table 4 shows the deformation parameters of a sandstone specimen at peak failure. Under a greater unloading rate, the radial deformation and volume expansion of the specimen was smaller from the beginning of unloading to peak failure. For example, the radial strain at peak failure was − 0.287% under conventional compression with an initial confining pressure of 10 MPa (see Table 3), and it was − 0.341%, − 0.268%, and − 0.113% at an unloading rate of 0.2 MPa/min, 0.5 MPa/min, and 1 MPa/min, respectively (see Table 4), which changed significantly. The difference in volumetric strain between the loading and unloading experiments was more obvious. The volumetric strain was 0.283% in the conventional compression test, indicating that the specimen did not expand much. The volumetric strain was − 0.028% and 0.056% at unloading rates of 0.2 MPa/min and 0.5 MPa/min, respectively, indicating the specimen volume expanded significantly. The volumetric strain was 0.325% at an unloading rate of 1 MPa/min, which indicated that the expansion under fast unloading was not as apparent as that under compression.

The peak stress in the unloading confining pressure test was different from that in the uniaxial or triaxial compression tests due to the difference in stress paths: in the unloading confining pressure test, the bearing capacity of the specimen was reduced to the axial stress, resulting in specimen failure, while in the axial loading test, the axial stress was gradually increased to the bearing capacity of the specimen and failure (Huang et al. 2017; Qiu et al. 2014; You and Hua 1998). Therefore, Qiu et al. (2014) proposed a new concept of “limited bearing capacity” to refer to the peak stress of specimen failure during unloading confining pressure, which is different from the concept of “peak strength” in the axial loading test. Figure 9 plots the limited bearing capacity of the specimen under different unloading rates. The limited bearing capacity decreased gradually with the increasing unloading rate. This was because when unloading slowly, the stress in the specimen was adjusted gently, resulting in a series of microcracks and induced friction, which improved the bearing capacity of the specimen. During rapid unloading, the lateral restraint decreased rapidly, the local shear of the specimen intensified, internal failure occurred rapidly, and the bearing capacity decreased significantly.

Fig. 9
figure 9

Limited bearing strength under different unloading rates

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Typical failure characteristics of sandstone specimens under different unloading confining pressure rates are shown in Fig. 10. Under the unloading condition, axial tensile microcracks easily occurred to occur with lateral restraint reduction. As the confining pressure was released, the bearing capacity of the specimen decreased gradually, the specimen was damaged in shear, and a tensile shear composite fracture zone was formed during the final failure. As illustrated in Fig. 10a, when the unloading rate was low, the dense axial tension microcracks interpenetrated and formed a shear zone under the action of axial stress. The rock fragments in the shear zone squeezed and rubbed each other, resulting in secondary cracks along the shear zone and a thicker shear. As shown in Fig. 10b, when the unloading rate was large, the specimen suddenly experienced brittle failure when reaching the limited bearing capacity, and the friction effect was weak, so the shear failure zone was relatively thin and flat.

Fig. 10
figure 10

Typical failure characteristics of sandstones after unloading confining pressure tests

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Effect of the unloading rate on dilatancy characteristics

In rock mechanics theory, the dilatancy angle (ψ) is often used to describe the rock volume deformation characteristics, and it is also adopted to indicate the increase in cracks in the rock. To discuss the effect of the unloading confining pressure rate, the dilatancy angle was calculated using the following formula defined by Vermeer and Borst (1984):

$$\psi ={\text{arcsin}}\left(\frac{\Delta {\varepsilon }_{\text{v}}^{\text{c}}}{\Delta {\varepsilon }_{\text{v}}^{\text{c}}-2\Delta {\varepsilon }_{1}^{\text{c}}}\right)$$
(3)

where \(\Delta {\varepsilon }_{1}^{\text{c}}\) and \({\varepsilon }_{\text{v}}^{\text{c}}\) are the volumetric plastic strain increment and the axial plastic strain increment, respectively.

Some scholars (Alejano and Alonso 2005; Yuan and Harrison 2004; Zhao and Cai 2010) have proposed that the dilatancy angle is a function of the confining pressure and plastic parameters, which usually refers to the plastic shear strain \({\gamma }_{p}\) (Qiu et al. 2010). It can be calculated as follows:

$${\gamma }_{p}={\varepsilon }_{1}^{\text{p}}-{\varepsilon }_{3}^{\text{p}}$$
(4)

Figure 11 shows the evolution process of the dilatancy angle with plastic shear strain under different confining pressures and unloading rates from the beginning of unloading to when the specimen reached its limited bearing strength. To facilitate comparison, the plastic shear strain is normalized: the normalized value 0 indicates the beginning of unloading, and the normalized value 1 indicates the specimen reaching the ultimate bearing strength.

Fig. 11
figure 11

Change of dilatancy angle under confining pressure unloading tests

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It can be seen from Fig. 11 that the unloading rate greatly influenced the dilatancy angle with the same initial confining pressure. With an increase in the unloading rate, it was easier to reach the peak dilatancy angle, which was smaller. When unloading at a low speed of 0.2 MPa/min, the dilatancy angle first increased slowly to a certain level, and then remained almost unchanged during unloading and decreased rapidly when approaching the ultimate bearing strength. When unloading at a rate of 1 MPa/min, the dilatancy angle quickly reached a maximum and then decreased slowly with the increasing plastic shear strain. The reason for this difference is that when the unloading rate was low, the cracks in the specimen more slowly formed and connected with the decrease of confining pressure, which made the damage degree of the rock increase slowly and the shear expansion angle gradually increase. When the unloading rate was large, the rapid reduction in confining pressure significantly reduced the lateral restraint of the specimen, causing the rapid release of rock strain energy and the rapid formation of cracks. In this process, the radial plastic deformation increased rapidly, and the change in axial plastic deformation lagged behind it, which resulted in a rapid increase in the shear expansion angle. With the expansion and penetration of the crack, the strain energy was released gradually; the damage in the specimen slowed down, and the dilatancy angle began to decrease gradually. In addition, the initial confining pressure affected the dilatancy angle. The higher the initial confining pressure is, the smaller the dilatancy angle at the beginning of unloading and at the peak stress, which indicates that the confining pressure had an inhibitory effect on the whole dilatancy process.

Crack propagation behaviors of sandstone

Crack evolution under conventional compression

The progressive failure of rock includes the process from “elasticity” to “damage” and then to “fracture,” and the evolution of cracks in the rock during the failure process has a significant influence on the permeability. To study the evolution of cracks in rock during loading, Martin (1993) introduced the concept of crack strain to analyze the number of cracks during rock failure quantitatively. It is defined as the axial and radial deformation caused by crack initiation, propagation, penetration, and new crack initiation in the rock under loading. Therefore, the crack strain can be obtained by subtracting the elastic strain from the measured total strain, that is, the plastic strain. In the conventional triaxial test (σ2 = σ3), the elastic strain can be calculated by the following formula:

$$\left\{\begin{array}{l}{\varepsilon }_{1}^{\text{e}}=\frac{1}{E}\left({\sigma }_{1}-2\nu {\sigma }_{3}\right)\\ {\varepsilon }_{3}^{\text{e}}=\frac{1}{E}\left[\left(1-\nu \right){\sigma }_{3}-\nu {\sigma }_{1}\right]\\ {\varepsilon }_{\text{v}}^{\text{e}}=\frac{1-2\nu }{E}\left({\sigma }_{1}+2{\sigma }_{3}\right)\end{array}\right.$$
(5)

where \({\varepsilon }_{1}^{\text{e}}\), \({\varepsilon }_{3}^{\text{e}}\), and \({\varepsilon }_{\text{v}}^{\text{e}}\) are the elastic axial strain, elastic radial strain, and elastic volumetric strain, respectively. σ1 and σ3 are the axial stress and confining pressure, respectively. E is the elastic modulus, and ν is Poisson’s ratio.

The elastic modulus and Poisson’s ratio of the specimen are calculated in the elastic deformation section of the specimen during the loading part before unloading, and it is assumed that they remain unchanged during the unloading process. The crack strain is calculated as follows (summarized by Martin and Chandler 1994):

$$\left\{\begin{array}{l}{\varepsilon }_{1}^{\text{c}}={\varepsilon }_{1}-\frac{1}{E}\left({\sigma }_{1}-2\nu {\sigma }_{3}\right)\\ {\varepsilon }_{3}^{\text{c}}={\varepsilon }_{3}-\frac{1}{E}\left[\left(1-\nu \right){\sigma }_{3}-\nu {\sigma }_{1}\right]\\ {\varepsilon }_{\text{v}}^{\text{c}}={\varepsilon }_{v}-\frac{1-2\nu }{E}\left({\sigma }_{1}+2{\sigma }_{3}\right)\end{array}\right.$$
(6)

where \({\varepsilon }_{1}^{\text{c}}\), \({\varepsilon }_{3}^{\text{c}}\), and \({\varepsilon }_{\text{v}}^{\text{c}}\) are the crack axial strain, crack radial strain, and crack volumetric strain, respectively. \({\varepsilon }_{1}\), \({\varepsilon }_{3}\), and \({\varepsilon }_{v}\) indicate the measured axial, radial, and volumetric strain, respectively.

The evolution of crack strain during conventional triaxial loading is illustrated in Fig. 12a. There was an initial crack volumetric strain after the confining pressure was applied, which reflected the closure effect of the confining pressure on the original crack of the specimen in the hydrostatic pressure stage. As the axial stress increased, the crack volumetric strain increased gradually. When the crack volumetric strain remained unchanged as the axial stress increased, the original crack in the sandstone specimen was closed entirely, and this axial stress was the crack closure stress (σcc). Then, the specimen entered a short stage in which the axial stress continued to increase vertically, and the crack volumetric strain basically remained unchanged, indicating that no new crack occurred in the specimen in this stage. With the continuous increase in axial stress, the tip of the original crack in the specimen experienced stress concentration, crack initiation occurred, and the crack volumetric strain started to increase. The corresponding stress was the crack initiation stress (σci), which indicates that the specimen entered the crack stable propagation stage. Then, as the axial stress increased, the specimen state gradually transitioned from the compressed state to the expanded state, and the stress-volumetric strain curve shows a turning point (expansion point). Generally, the stress corresponding to the expansion point is called the crack damage stress σcd and indicates that the specimen entered the unstable crack growth stage after this point. In this study, due to the weak cementation and low strength of Cretaceous sandstone, it was noticed that the crack initiation stress (σci) and the crack damage stress (σcd) are at the same point (i.e., the expansion point of the specimen), indicating that once the specimen started to crack, the crack grew rapidly. This explains why the permeability increased rapidly after the volume dilatancy point (see Fig. 6). The specimen was fractured when the peak deviatoric stress (σf) was reached, and the crack strain at this time is called the crack volumetric propagation strain. Then, the specimen entered the post-peak stage, and the crack continued to expand. It can be seen from Fig. 12b that the larger the initial confining pressure was, the larger the initial crack volumetric strain, indicating a higher initial crack closure level, and lower the initial permeability (shown in Table 3).

Fig. 12
figure 12

The crack evolution behaviors of sandstone under conventional compression

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Crack evolution under unloading confining pressure

In the unloading test, the axial pressure remains unchanged, and the confining pressure is reduced, which is different from the condition that the axial pressure is increased when Martin proposed the concept of crack volume strain. However, both are paths of increased deviatoric stress, so the crack volumetric strain also can be calculated by the deviatoric stress in the confining pressure unloading test. As shown in Fig. 13a, in the axial stress loading stage of the test, the crack strain of the specimen increased first. Then, it remained unchanged when the deviatoric stress increased to the crack closure stress (σcc), similar to the conventional compression test. As the deviator stress continued to increase to the crack initiation stress (σci), cracks occurred in the specimen, and the crack volumetric strain started to decrease. Notably, this stress was equal to the stress at the beginning of unloading. This indicates that when the confining pressure started unloading, the crack in the specimen began to sprout immediately, which explained the apparent increase in the specimen permeability. With a further decrease in the confining pressure, the specimen rapidly entered the expansion state from the compressed state, indicating that the deviator stress increased to the crack damage stress (σcd), the crack in the specimen started to propagate, and the permeability increased rapidly. The difference between the crack initiation stress and the crack damage stress was very small. After the deviator stress reached the limited bearing strength, the specimen was damaged and entered the post-peak crack propagation stage, and the rate of increase in the crack strain decreased.

Fig. 13
figure 13

The crack evolution behaviors of sandstone under unloading confining pressure

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Figure 13b shows that the change in crack strain was closely related to the unloading rate of the confining pressure. The lower the unloading rate was, the larger the crack volumetric propagation strain, which means that there were more cracks in the specimen when the specimen reached peak failure. This was because when the unloading rate was low, the internal stress of the specimen was fully adjusted during the unloading process, and the cracks continued to initiate, propagate, and coalesce slowly. Therefore, the smaller the unloading rate was, the greater the increase in permeability when the specimen reached peak failure.

Figure 13c shows the crack volumetric propagation strain of the sandstone specimen under different initial confining pressures. The confining pressure had an apparent inhibitory effect on the development of internal cracks in the specimen, and the crack volumetric propagation strain was smaller under larger confining pressures with the same unloading rate. This was because when unloading the confining pressure, small tensile cracks were formed in the specimen with the continuous adjustment of the stress, and a larger confining pressure inhibited the inoculation and propagation of tensile cracks, resulting in a small crack volumetric propagation strain and a smaller increase in permeability when the rock failed. At the same time, the influence of the unloading rate on the crack volumetric propagation strain decreased with increasing initial confining pressure. When the initial confining pressure was 7.5 MPa, the crack volumetric propagation strain with an unloading rate of 0.2 MPa/min and 1 MPa/min was 0.9% and 0.1%, respectively, and the crack volumetric propagation strain of the lower speed was nine times that of the higher speed. However, when the confining pressure increased to 12.5 MPa, the crack volumetric propagation strain with an unloaded rate of 0.2 MPa/min and 1 MPa/min was 0.25% and 0.1%, respectively, and the low speed was only 2.5 times that of the high speed. This further affected the evolution of permeability. Table 4 shows that with increasing confining pressure, the difference in the increase in permeability during slow unloading and rapid unloading failure was significantly smaller than that under a low initial confining pressure state.

Conclusion

Laboratory triaxial unloading confining pressure tests were carried out on weakly cemented sandstone. Based on the test results, the effects of the unloading rate and initial confining pressure on the deformation and permeability behaviors are studied, and the rock dilatancy and crack propagation behaviors are revealed. The main conclusions are as follows.

  1. (1)

    Under the condition of an unloading confining pressure and a constant axial pressure, the volumetric strain of weakly cemented sandstone is mainly related to radial deformation. The smaller the unloading rate is, the greater the radial strain and the more obvious the volume expansion. When unloading slowly, many secondary tensile cracks are generated in the specimen, inducing friction and secondary splitting along the shear zone, and the limited bearing strength of the specimen is improved. The shear zone is smoother and flatter under faster unloading.

  2. (2)

    Under the same unloading rate, the greater the confining pressure is, the smaller the initial permeability, and the smaller the increase in permeability after peak failure, the confining pressure has an apparent inhibitory effect on permeability. Under the same initial confining pressure, as the unloading rate decreases, the unloading degree of the confining pressure after peak failure increases, indicating a lower confining pressure of the specimen, so the permeability after failure is greater.

  3. (3)

    The larger the initial confining pressure is, the smaller the dilatancy angle at the beginning of unloading, and the smaller the dilatancy angle during unloading overall. Under fast unloading, the specimen cracks rapidly at the initial stage, reaches the peak dilatancy angle quickly, and then decreases significantly. However, under a low unloading rate, the damage process is slow, and the dilatancy angle first increases slowly and then remains unchanged.

  4. (4)

    Crack propagation behaviors could explain the effects of the confining pressure and unloading rate on the permeability. The smaller the unloading rate is, the greater the crack volumetric propagation strain, which means there are more cracks in the specimen when damaged, so the higher the permeability after failure. The larger the confining pressure is, the smaller the crack volumetric propagation strain and the smaller the specimen permeability.

  5. (5)

    In actual coal mining and underground construction, the excavation speed should be increased appropriately to control the increase in permeability, to reduce the risk of water inrush disaster from the weakly cemented Cretaceous sandstone aquifer during unloading, especially in places where the Cretaceous stratum is thin and shallow.