Mechanical properties of sandstone under hydro-mechanical coupling

2.1 Preparation of water-saturated sandstone specimens

The sandstone used in the test was taken from the sandstone aquifer of the no.3 main well of Macheng Iron Mine in Hebei Province, China. The sandstone is yellowish-brown coarse sandstone, and its main component is quartz. The standard cylindrical specimen of ϕ50 mm × 100 mm was prepared from the collected sandstone, as shown in Figure 1. According to the SL264_2020 specifications for rock tests in water conservancy and hydroelectric engineering [21], the specimens were forced to be water-saturated by boiling method, and the coefficient of water saturation of sandstone specimens was measured to be about 6.56%. The specific steps are as follows:

1. Dry sandstone specimens are placed horizontally in the tank, and water is added to the tank to 1/4 of the height of the specimen. Then water was injected to 1/2 and 3/4 of the height of the specimen every 2 h. After 6 h, the specimens are completely immersed.

2. After the specimens are completely immersed for 48 h, the water inside the tank was boiled, the boiling time shall not be less than 6 h. After that, the specimens are taken out at regular intervals and its mass weighed with an electronic balance (the measurement accuracy is 0.01 g). When the mass of the specimens does not exceed 0.01 g after two consecutive measurements, it means that the specimen is saturated with water.

Figure 1

Sandstone specimens.

To further analyze the pore distribution in sandstone specimens, three water-saturated sandstone specimens were randomly selected and their porosity was measured by AniMR-150 nuclear magnetic resonance analyzer (Figure 2). The average porosity of the tested samples was 14.28%, the specific results are shown in Table 1, and the pore type was tubular pore. The specific results are shown in Figure 3. In Figure 3, the pore size of the specimen is distributed in 0.01–10 µm.

Figure 2

AniMR-150 nuclear magnetic resonance analyzer.

Figure 3

Experimental results of AniMR-150 nuclear magnetic resonance analyzer.

2.2 Test apparatus and schemes

Rock triaxial compression tests under different confining pressures and pore water pressures were carried out in rock mechanics laboratory of Hunan University of Science and Technology. The MTS815 rock mechanics test system was used as the power equipment in the test. The axial compression, confining pressure and pore water pressure of the test system are independent servo control systems (Figure 4(a)). The confining pressures of test are 10, 20 and 30 MPa. In each case, the pore water pressure P is always lower than the confining pressure and has four levels, namely 2, 4, 6 and 8 MPa. The specific steps of the test are as follows:

1. First, the water-saturated sandstone specimens are wrapped with heat shrink tube, and the permeable pressure plate is placed at both ends of the specimen. Then blow the heat shrink tube evenly with a hot air gun to keep it in close contact with the specimen and the upper and lower permeable pressure plate. In order to prevent additional damage caused by hydraulic oil entering into the specimen during the experiment, a wire is used to further fix the upper and lower permeable pressure plate and the heat shrink tube. Finally, the specimen is placed on the base of the test system and Axial Circ Extensometer, Axial Extensometer and aqueduct are installed (Figure 4(b)).

2. In the test process, the confining pressure σ ₃ and axial pressure σ ₁ were loaded to the set value at the speed of 2.0 MPa/min, and the axial pressure σ ₁ was kept consistent with the confining pressure σ ₃ to achieve the triaxial hydrostatic pressure state (σ ₁ = σ 2 = σ ₃). Then, the pore water pressure P was loaded to the specified value at the speed of 0.05 MPa/s.

3. The axial displacement loading rate of 0.1 mm/min was used for deviation stress loading until the specimen was destroyed. The test loading path and the stress state of sandstone specimens are shown in Figure 5.

Figure 4

(a) MTS815 rock mechanics test system and (b) diagram of specimen installation.

Figure 5

(a) Diagram of test loading paths and (b) diagram of rock specimen loading.

3.1 Deviation stress–strain curve

Figure 6 shows the deviation stress–strain curve obtained from triaxial compression tests of sandstone under pore water pressure. It can be seen from the figure that the deviation stress–axial strain experienced five stages [22–25], namely, the compression and closure stage of initial fissure, the linear elastic deformation stage, the stable development stage of crack, the unstable propagation stage of crack and the post-peak stage. The deviation stress–volumetric strain is mainly in the stage of volumetric compression and volumetric expansion. It can be seen from the pre-peak stage of the deviator stress–circumferential strain curve that the circumferential strain rate increases with the increase of the deviator stress. Under the same deviator stress, the circumferential strain of the specimen decreases with the increase of confining pressure, and increases with the increase of pore water pressure. This is because the confining pressure restricts the circumferential deformation of the specimen, and the pore water pressure will weaken the restraining effect of the confining pressure on the circumferential deformation.

Figure 6

Deviation stress–strain curve of water-saturated sandstone under pore water pressure: (a) σ ₃ = 10 MPa, (b) σ ₃ = 20 MPa and (c) σ ₃ = 30 MPa.

Under the action of pore water pressure (P = 2, 4, 6, 8 MPa), the peak deviation stress and residual deviation stress of sandstone specimens increase with the increase of confining pressure. This is because the confining pressure can restrain the circumferential deformation of the specimen. The higher the confining pressure, the greater the degree of restraint. And under constant confining pressure, with the increase of pore water pressure, the peak deviation stress and residual deviation stress of sandstone specimens decrease to varying degrees.

Table 2 shows the stress and strain of water-saturated sandstone under the combined action of confining pressure σ ₃ and pore water pressure P, where (σ ₁–σ ₃)_max is the peak deviation stress; ε _1max, ε _3max and ε _vmax are the axial strain, circumferential strain and volumetric strain corresponding to the peak deviation stress, respectively; (σ ₁–σ ₃)_d is the initiation deviation stress of dilation and (σ ₁–σ ₃)_r is the residual deviation stress. The correlation curve is shown in Figure 7.

Figure 7

(a) (σ ₁–σ ₃)_max – P, (b) (σ ₁–σ ₃)_d – P, (c) (σ ₁–σ ₃) _r – P, (d) ε _1max – P, (e) ε _3max – P and (f) ε _vmax – P.

Figure 7(a) shows that under the same confining pressure σ ₃, pore water pressure P has a significant effect on the peak strength (σ ₁–σ ₃)_max of water-saturated sandstone. Under constant confining pressure σ ₃, the initiation deviation stress of dilation (σ ₁–σ ₃)_d of water-saturated sandstone specimen is negatively correlated with pore water pressure P (Figure 7(b)), and its linear fitting formula is

where G and H are fitting parameters (Table 3). It can be found from the table that the R ² of (σ ₁ –σ ₃)_d – P fitting lines under different confining pressures are all above 0.95, indicating that the expression of the relationship between (σ ₁–σ ₃)_d and P by equation (1) is appropriate.

Figure 7(c) shows that the residual deviation stress (σ ₁–σ ₃)_r of sandstone specimens decreases with the increase of pore water pressure P. In the case of σ ₃ = 10 MPa, when the pore water pressure P increases from 2 to 8 MPa, the residual deviation stress drops from 26.73 to 8.40 MPa, a decrease of 68.57%. In the case of σ ₃ = 20 MPa, when the pore water pressure P increases from 2 to 8 MPa, the residual deviation stress decreases from 41.25 to 18.18 MPa, the decrease is 55.93%.

When the confining pressure is constant, ε _1max decreases linearly with the increase of P (Figure 7(d)). Linear fitting shows that ε _1max − P has the following relationship:

where I and L are fitting parameters (Table 4). It can be found from Table 4 that under different confining pressures, R ² of the fitting formula of ε _1max and P is above 0.92, indicating that the description of the relationship between ε _1max and P in equation (2) is reliable. In addition, it is found from Figure 7(e) and (f) that ε _3max and ε _vmax fluctuate up and down with the increase of P, and there is no obvious law between them.

3.2 Deformation characteristic analysis

In the triaxial compression test, the deformation modulus E ₅₀ and Poisson’s ratio μ of water-saturated sandstone specimens at 50% peak strength are, respectively [15,26]:

where σ 1 50 = 50 % peak axial stress, σ ₃ is the confining pressure acting on the specimen; ε 1 50 and ε 3 50 are the axial strain and circumferential strain corresponding to 50% peak strength, respectively.

The deformation modulus E ₅₀ and Poisson’s ratio μ of sandstone specimen can be calculated according to formulas (3)–(5). Figure 8(a) is the curve of E ₅₀ – P, Figure 8(b) is the curve of μ – P. From the graph, when σ ₃ is constant, E ₅₀ decreases with the increase of P. In the case of confining pressure σ ₃ = 20 MPa, when the pore water pressure P increased from 2 to 8 MPa, the deformation modulus E ₅₀ decreased by 27.39% from 10.55 GPa to 7.66 MPa. When the pore water pressure P remains constant, E ₅₀ increases with the increase of confining pressure σ ₃. Take the example of pore water pressure P = 4 MPa, when the confining pressure σ ₃ increased from 10 to 30 MPa, the deformation modulus E ₅₀ increased from 6.66 to 10.00 GPa, by up to 50.15%.

Figure 8

(a) E ₅₀ – P and (b) μ – P.

When σ ₃ is constant, μ keeps increasing with the increase of P, but its increasing speed is affected by σ ₃. The larger the σ ₃ is, the slower it increases. Taking P = 6 and P = 8 MPa for example, when σ ₃ = 10 MPa, μ increased from 0.4442 (P = 6 MPa) to 0.5133 (P = 8 MPa), an increase of 15.56%. When σ ₃ = 20 MPa, μ increased from 0.3969 (P = 6 MPa) to 0.4139 (P = 6 MPa), with an increase of 4.28%. When σ ₃ = 30 MPa μ increased from 0.3801 (P = 6 MPa) to 0.3949 (P = 8 MPa), with an increase of only 3.89% (Figure 8(b)).

3.3 Failure mode of water-saturated sandstone

Figures 9–11 show the failure of sandstone specimens with confining pressures of 10, 20 and 30 MPa under different pore water pressures P. When the confining pressure σ ₃ = 10 MPa (Figure 9), the X-shaped conjugate inclined plane shear failure mainly occurs in the specimens, and the distribution range of angle of rupture θ was 60°–76°. With the increase of pore water pressure P, the phenomenon of X-shaped conjugate failure is more obvious. In the triaxial compression test under hydro-mechanical coupling, the specimen is constrained by confining pressure and expanded by water pressure at the same time. Based on the principle of effective stress, the existence of pore water pressure will reduce the restraint effect of confining pressure, which will increase the circumferential deformation and brittleness of the rock specimen, resulting in an X-shaped shear failure of the specimen. Under the condition of large confining pressures (σ ₃ = 20 and 30 MPa), the sandstone specimen has shear failure of single inclined plane. Under constant confining pressure, the angle of rupture θ of sandstone specimen increases gradually with the increase of pore water pressure. When the pore water pressure P increases from 2 to 8 MPa, the angle of rupture θ increases from 66° to 71° at confining pressure of 20 MPa, and the angle of rupture θ increases from 45° to 70° at confining pressure of 30 MPa.

Figure 9

Failure diagram of water-saturated sandstone specimens under confining pressure of 10 MPa: (a) P = 2 MPa, (b) P = 4 MPa, (c) P = 6 MPa and (d) P = 8 MPa.

Figure 10

Failure diagram of water-saturated sandstone specimens under confining pressure of 20 MPa: (a) P = 2 MPa, (b) P = 4 MPa, (c) P = 6 MPa and (d) P = 8 MPa.

Figure 11

Failure diagram of water-saturated sandstone specimens under confining pressure of 30 MPa: (a) P = 2 MPa, (b) P = 4 MPa, (c) P = 6 MPa and (d) P = 8 MP.

4.1 Analysis of effective peak failure strength of water-saturated sandstone under hydro-mechanical coupling

In the triaxial compression test with pore water pressure, the pores between the solid skeleton and solid particles of sandstone will be affected by axial pressure, confining pressure and pore water pressure in the compression process. To further study the peak failure strength of sandstone solid skeleton under pore water pressure, the effective stress principle is used [27–29]. According to the peak failure strength σ _1max and confining pressure σ ₃ obtained by triaxial compression test under different pore water pressures P, the effective peak strength σ 1 max ′ and the effective minimum principal stress σ 3 ′ can be calculated by the following equation:

The calculation formula of effective initiation strength of dilation σ d ′ and effective residual strength σ r ′ of sandstone under hydro-mechanical coupling is as follows:

Figure 12(a) shows the relationship between effective peak failure strength σ 1 max ′ and effective minimum principal stress σ 3 ′ under hydro-mechanical coupling. Figure 12(b) shows the relationship between effective initiation strength of dilation σ d ′ and effective minimum principal stress σ 3 ′ under hydro-mechanical coupling. Figure 12(c) shows the relationship between effective residual strength σ r ′ and effective minimum principal stress σ 3 ′ under hydro-mechanical coupling. In general, the σ 1 max ′ of sandstone increases with the increase of σ 3 ′ , i.e., the degree σ 1 max ′ is positively correlated with σ 3 ′ . σ d ′ and σ r ′ are positively correlated with σ 3 ′ . The linear fitting of Figure 12 shows that

where k, k _d, k _r, λ, λ _d and λ _r are fitting parameters. The correlation coefficient R ² of the fitting formula is greater than 0.94. The expression of its effective strength is further obtained as:

Figure 12

(a) σ 1 max ′ − σ 3 ′ , (b) σ d ′ − σ 3 ′ and (c) σ r ′ − σ 3 ′ .

According to Mohr–Coulomb criterion, the relationship between effective peak strength σ 1 max ′ and effective minimum principal stress σ 3 ′ can be further expressed as:

In addition, by defining effective residual cohesion c ′ ¯ and effective residual angle of internal friction φ ′ ¯ of sandstone at the residual stage, the effective residual strength can be expressed as:

Combining equations (9), (10) and (11), it can be obtained that under the action of pore water pressure, the effective cohesion c′ = 11.49 MPa and effective angle of internal friction φ′ = 38.32° of sandstone, and the effective residual cohesion c ′ ¯ = 3.13 MPa, the effective residual angle of internal friction φ ′ ¯ = 25.32°.

4.2 Criteria for relative strength of sandstone under hydro-mechanical coupling

Although based on the principle of effective stress, the effective strength and effective minimum principal stress of sandstone can be linearly fitted, which conforms to the Mohr–Coulomb criterion. However, according to the research of Mitchell and Skempton [30–32], for saturated materials, the effective stress principle when considering shear strength should be expressed as:

where Ψ is the true friction angle of the solid phase and ϕ′ is the effective friction angle. When α _c ≠ 0, equations (6) and (7) are not applicable. When considering the volumetric compression deformation of saturated materials, the effective stress principle can be expressed as:

where C is the compressibility of the material system (soil skeleton, rock mass and concrete body) and C _S is the compressibility of solid minerals. Table 5 shows the compressibility ratio of various soils, rocks and concrete.

It can be seen from Table 5 that for rock materials, the ratio of compressibility coefficient is large, and the effective stress principle in the form of equations (6) and (7) is not applicable. When we analyze the Mohr–Coulomb criterion of rock under hydro-mechanical coupling, equations (6) and (7) are often used [33,34,35,36]. Therefore, it is inevitable that there will be large errors. When analyzing the test data, it is found that there is a certain relationship between the strength change of sandstone and P/σ ₃. In order to further explore the mystery of this relationship, the conventional triaxial compression test of water-saturated sandstone was carried out using the same experimental device and loading rate. The peak strength σ _1,P, residual strength σ _r,P, initiation strength of dilation σ _d,P, Poisson’s ratio μ and deformation modulus E ₅₀ obtained from the tests are shown in Table 6. It can be seen from the table that σ _1,P, σ _r,P, σ _d,P, μ and E ₅₀ of water-saturated sandstone increase gradually with the increase of σ ₃.

Figure 13 shows the σ _1,Pi/σ _1,P – P/σ ₃, σ _r,Pi/σ _r,P – P/σ ₃ and σ _d,Pi/σ _d,P – P/σ ₃ relationships, respectively, where σ _1,Pi, σ _r,Pi and σ _d,Pi are the peak strength, residual strength and initiation strength of dilation of sandstone specimens under pore water pressure, respectively. In general, the σ _1,Pi/σ _1,P, σ _r,Pi/σ _r,P and σ _d,Pi/σ _d,P of water-saturated sandstone are negatively correlated with P/σ ₃, and the linear fitting of these shows the following relationship

where k ₁, k ₂, k ₃ are fitting parameters, and k ₁ = −0.71, k ₂ = −0.75, k ₃ = −0.99. The correlation coefficient of equation (12) is 0.995, which is greater than that of equation (9) (R ² = 0.94), indicating that it is reasonable and reliable to use it to describe the relationship between the ratio of strength of saturated sandstone and P/σ ₃. Therefore, the criterion of relative strength of sandstone under hydro-mechanical coupling is established, i.e.:

where σ _Pi is the strength of sandstone under hydro-mechanical coupling, σ _P is the strength under the independent action of external stress; γ is the relationship coefficient (γ = −0.71 for peak strength, γ = −0.75 for residual strength and γ = −0.99 for initiation strength of dilation).

Figure 13

(a) σ _1,Pi/σ _1,P – P/σ ₃, (b) σ _r,Pi/σ _r,P – P/σ ₃ and (c) σ _d,Pi/σ _d,P – P/σ ₃.

This article studies the mechanical properties and deformation characteristics of sandstone under hydro-mechanical coupling through series of triaxial compression tests. It is of great significance to reveal the mechanism of mine water inrush, water inrush in shaft excavation and tunnel water gushing. According to the ratio relationship between water pore pressure and confining pressure, establishing the relative strength criterion of sandstone under the action of water-force coupling. It provides a theoretical basis for the design of surrounding rock stability control scheme. Since the relative strength coefficient γ has not been further discussed, the triaxial compression test of different rock samples can be carried out under the same test plan to study the influencing factors of the relative strength coefficient γ.