Effect of Intermediate Principal Stress on the Strength, Deformation, and Permeability of Sandstone

Abstract

Although the mechanical behaviors and flow aspects of sandstone have been previously investigated, studies of the effect of the intermediate principal stress (σ₂) on the strength, deformation, and permeability of sandstone are lacking. In this work, the mechanical behaviors and permeability of sandstone under true triaxial stress conditions were investigated using a newly developed true triaxial geophysical apparatus. The experimental results showed that with increasing σ₂, the peak strength, octahedral effective normal stress, and octahedral effective shear stress of the sandstone increased, and the rate of increase decreased. This is because a larger intermediate principal stress coefficient b has an inhibitory effect on rock strength. In our study, as the ratio of σ₂/σ₃ increased, the specimen entered compressive strain in the σ₂ direction during the first stress drop. The stress and strain path deviations occur during rock failure. The amount of deviation increased as the σ₂ increased before the peak stress. This phenomenon indicates that elastic mechanics are not suitable for understanding this sandstone rock during its failure. The permeability evolution of the sandstone under true triaxial stress conditions was measured and analyzed to investigate the effect of σ₂. During the complete true triaxial stress-strain experiments, the variation we found in gas seepage velocity could be divided into two stages. Before the first pressure drop, the gas seepage velocity was mainly affected by volume strain. After the first pressure drop, the seepage velocity was affected by the deviator strain, which can change the seepage channels.

1. Introduction

Minerals and energy are often found in deep rock formations and are generally in a complex, high-stress state. During the process of mineral and energy development, the stress state of the rock mass is changed, and fractures in the rock begin to germinate and develop. The fracture distribution under different stress states directly affects the mining process, particularly during the oil, gas, and geothermal recovery processes [1,2]. Triaxial experiments have been widely conducted to investigate the geomechanical and fluid flow behaviors of rocks under different stress conditions, which are of vital importance in evaluating the performance and efficiency of energy exploitation. Traditional triaxial experiments have been generally conducted under conventional triaxial stress (σ₂ = σ₃) conditions, ignoring the effects of σ₂ on the geomechanical properties and permeability of rocks.

Although many strength criteria for rocks have been proposed in previous research, most strength criteria have ignored the influence of the intermediate principal stress (σ₂) and cannot reliably predict the true strength characteristics of rocks [3,4]. Hence, studies have been conducted to investigate the effects of σ₂ during recent years. Colmenares et al. compared several failure criteria to publish true triaxial experimental data [5]. Haimson conducted several true triaxial experiments on crystalline rocks and summarized the effects of σ₂ on brittle rocks [6]. Yang conducted true triaxial experiments on concrete and found that its strength changed with increasing σ₂ [7]. Chang et al. proposed a criterion that involved fitting the curves of several true triaxial experimental data [8]. Mogi developed the first true triaxial apparatus for rock and proposed a complete true triaxial strength criterion based on his experimental data [9,10]. Following his pioneering work, other laboratories have developed true triaxial apparatuses and have proposed true triaxial strength criteria [11,12,13,14]. Meanwhile, researchers have also found that the peak strength, deformation, and ductility of rock are all affected by the intermediate principal [15,16,17]. Pan et al.’s research showed that a critical value of σ₂ exists. Below this value, in their experiments σ₂ had a positive effect on the peak strength, while above this critical stress, the peak strength was negatively affected [18]. Phueakphum et al. found that σ₂ significantly affects the tensile strength of rock [19]. Yin et al. used a self-developed true triaxial apparatus to investigate the rock failure mechanism along different stress paths [20]. The results showed that rock strength and deformation are highly dependent on the stress path. Haimson and Rudnicki found that the internal friction angles of rocks are not constant under different σ₂ values [21,22]. The internal friction angle of their samples increased with the increasing intermediate principal stress coefficient.

The permeability of rocks is also affected by σ₂ [23]. The directional transport characteristics and pore volume change in sandstone under true triaxial stress were studied by Al-Harthy et al. [24]. An advanced true triaxial apparatus with seismic and resistivity sensors was developed that can evaluate rock permeability properties and fracture delineations [25,26,27]. Li et al. developed a new geophysical type of equipment with which they investigated shale permeability under true triaxial stress states [28,29].

Therefore, σ₂ plays an important role in rock strength, deformation, and permeability. This work presents strength, deformation, and permeability measurements of sandstone under true triaxial stress conditions using the newly-developed, multi-functional, true triaxial geophysical (TTG) apparatus [28]. The results were used to investigate the geomechanical and fluid flow behavior of the sandstone under true triaxial stress conditions.

2. Instruments Used and Methodology

2.1. Experimental Apparatus

A new multi-functional true triaxial geophysical (TTG) apparatus was self-developed by Chongqing University. Research on the mechanical properties and permeability of rock under a true triaxial stress state can be achieved using this apparatus.

Photographs and schematic diagrams of the apparatus are shown in Figure 1. There is a fluid channel in the loading plate that is in direct contact with the sample. To seal the gas fluid in the sample, a series of measures was taken. The four faces of the sample were covered with silicone rubber, and the sample was encased in heat-shrink tubing. A sealing oil with a pressure higher than that of the gas was also applied in order to contain the gas in the sample. The sealing structure is shown in Figure 1.

2.2. Experimental Scheme

The failure experiments were conducted under different stress conditions (see

Table 1. Stress conditions.

Serial Number	σ2/MPa	σ3/MPa	Displacement Rate
M-1	20	20	0.001 mm/s
M-2	30	20	0.001 mm/s
M-3	40	20	0.001 mm/s
M-4	50	20	0.001 mm/s
M-5	60	20	0.001 mm/s
M-6	30	30	0.001 mm/s
M-7	40	30	0.001 mm/s
M-8	50	30	0.001 mm/s
M-9	60	30	0.001 mm/s
M-10	40	40	0.001 mm/s
M-11	50	40	0.001 mm/s
M-12	60	40	0.001 mm/s
M-13	50	50	0.001 mm/s
M-14	60	50	0.001 mm/s
M-15	60	60	0.001 mm/s

). During the loading process, load σ₁ had a constant displacement rate of 0.001 mm/s, while σ₂ and σ₃ remained at a constant stress. The gas (CH₄) injection pressure = 1 MPa, and the gas outlet valve opened to the atmosphere. Three-dimensional (3-D) strain and the outlet flow rate were continuously recorded during the experiments.

2.3. Specimen Preparation for the True-Triaxial Experiment

The sandstone specimens used in the testing were collected from outcrops in the Chayuan district of Chongqing, in the Sichuan basin in southwest China. The bulk density of this rock is 2290 kg/m³. The Brunauer-Emmett-Teller (BET) surface area is the area per gram of the sandstone. It was measured by the ASAP2010 apparatus and showed a value of 3.05 m²/g. The uniaxial compression test showed that the rock strength was 56.43 MPa. The samples showed a Young’s modulus of 5.17 GPa and a Poisson’s ratio of 0.33. The sandstone samples were ground to 100 × 100 × 100 mm³ cubes using a programmable grinding machine. Each cube was ground with an accuracy of 5 μm flatness on each face and opposite side parallelism. Figure 2 shows photographs of the specimens and the computed tomography (CT) scan images. The photographs of the raw rock samples are on the left side of Figure 2 and the finished rock specimens are shown in the middle. The CT scan images of the finished samples are shown on the right. The CT images of the cubic (upper) and sliced (below) samples show the homogeneity of this rock. Figure 3 shows scanning electron microscope (SEM) images of the sandstone. It can be seen from the scanning electron microscope (SEM) images that the diameter of the particles is approximately 0.13 mm, that of a medium-fine sandstone, and there are many pores in the particles.

3. Results and Discussion

3.1. Effect of Intermediate Principal Stress on Rock Strength

The peak strengths of the sandstone are shown in Figure 4a; the strength of the sandstone increased with increasing σ₂ and σ₃ for all specimens. As seen in Figure 4b, σ₃ produced a greater increase in rock strength compared to σ₂. It can be concluded that σ₃ plays a more important role in enhancing rock strength than σ₂, indicating that if a constant mean stress state is applied, the rock has a lower peak strength when there is a greater ratio of σ₂ to σ₃.

As is known, b is a parameter that can reflect the relationship of three principal stresses. It determines the direction of deviatoric stress in the cylindrical coordinate system. Thus, parameter b can affect the deviatoric strain and rock failure model. b decreases when σ₃ increases, and increases when σ₂ increases. Thus, considering the variation in b, the rock strength always increases under a smaller b stress state when the same increment of σ₂ or σ₃ is added. For example, in our experiment under the stress state of σ₃ = 40 MPa and σ₂ = 590 MPa, by increasing σ₃ from 40 MPa to 50 MPa, the rock strength increased by 6.2 MPa. At the same time, b was reduced from 0.06 to 0. When σ₂ increased from 40 MPa to 50 MPa, b increased from 0.06 to 0.12 and the increase in rock strength was only 1.9 MPa. Therefore, it is apparent that for the same amount of increase in σ₂ or σ₃, the difference in the impact on rock strength was mainly caused by the variation in b. Increasing σ₂ and σ₃ by the same amount had a greater influence on rock strength under a lesser b stress state than a greater one. Therefore, the influence of σ₂ and σ₃ on rock strength was mainly caused by b.

The octahedral shear stress ${\tau }_{oct}=\frac{1}{3}\sqrt{{({\sigma }_1-{\sigma }_2)}^2+{({\sigma }_1-{\sigma }_3)}^2+{({\sigma }_2-{\sigma }_3)}^2}$ and the octahedral effective normal stress ${\sigma }_{oct}=\frac{1}{3}({\sigma }_1+{\sigma }_2+{\sigma }_3)$ are important mechanical parameters for many materials. The octahedral shear stress and octahedral effective normal stress represent two basic parameters of the cylindrical coordinate system. The normal stress can only cause a volume change and shear stress can only cause a shape change. Thus, many scholars use these two parameters in their strength criteria, such as the Mogi criterion [30] and the Drucker and Prager (D-P) strength criterion [31]. Figure 5 shows that with increasing σ₂, the τ_oct and σ_oct of the sandstone increased at a reducing rate. In addition, the growth attenuation rate of τ_oct and σ_oct increased with increasing σ₂. The growth attenuation rate was smaller under σ₃ = 30 MPa than when σ₃ = 20 MPa. These phenomena were still caused by the intermediate principal stress coefficient b. As previously mentioned, when σ₃ is small, as σ₂ increases, the rate of the increase in b is more rapid, which inhibits the continued growth of σ₁. Therefore, the growth rate of the volume and deviatoric stresses was slow under a high b stress state. With increasing σ₂, the τ_oct and σ_oct of our samples increased at a reducing rate.

It can be seen that during the true triaxial experiment, the intermediate principal stress coefficient b played an important role in rock strength increase. At the same time, the increase in σ₂ not only caused an increase in the hydrostatic stress, but also a change in the intermediate principal stress coefficient b. Therefore, the change in rock strength caused by the change in σ₂ under different σ₃ values can be understood as the combinatory effect of the hydrostatic and intermediate principal stress coefficients. Thus, it was necessary to further study the influence of hydrostatic stress and b on rock strength under true triaxial stress conditions.

3.2. Effect of Intermediate Principal Stress on Rock Deformation

The post-peak deformation of rock is an important issue in understanding the mechanism of underground dynamic disasters. It shows how rock energy is released. We defined the deformation during the first stress drop as Δε, which is shown in Figure 6. ε₂ is defined as the strain in the σ₂ direction at the peak stress σ₁. It can be affected not only by the amount of σ₂, but also by the parameter b. It can act as a benchmark for studying the post-peak deformation in the σ₂ direction. The ratio of Δε and ε₂ is defined as ω. Figure 7 shows the relationship between ω and σ₂. As shown in Figure 7, when σ₂ was relatively low, the sandstone specimen was still expanding in the σ₂ direction during the first stress drop stage. As the ratio of σ₂/σ₃ increased, the specimen entered compressive strain in the σ₂ direction during the first stress drop, in which the compression exponentially increased with the increasing stress ratio.

3.3. Effect of Intermediate Principal Stress on the Transport Properties of the Sandstone

Figure 8 shows the relationship between permeability, σ₁, and ε₁. Considering the differing initial permeability of the sandstone, a uniform permeability was used to analyze the influence of σ₂ on the permeability of the sandstone.

The permeability at first gradually decreased with increasing σ₁. The permeability reached a minimum value when the rock reached the turning point from the elastic stage to the plastic stage. During the first stage, the main factor affecting the gas flow velocity was the frictional resistance. Therefore, as the rock volume stress increased, the internal fractures and pores continuously reduced. As a result, the rock penetration rate continuously reduced. During the plastic stage, the permeability slowly increased with increasing σ₁ because of the micro-cracks produced in the internal rock structure. During the later plastic stage, the permeability rapidly increased because of the micro-cracks connecting and gradually forming macro-cracks. After the sandstone reached its peak strength, fractures occurred in the sandstone and the permeability entered into a linear increasing stage, related to the first stress drop (see Figure 6). After the first stress drop, the permeability entered another linearly increasing stage. During the first stress drop, crack propagation processes formed many new seepage channels in the sandstone, and hence the permeability steadily increased. As shown in Figure 9a, during the first pressure drop, the volume strain of the rock expanded under all stress states. However, the expansion decreased with the increase in the intermediate principal stress, which was 0.12, 0.07, 0.06 and 0.02, respectively. Stress and strain deviation occurred obviously during this stage. This is among the reasons why the fluid flow velocity changed with the variation in σ₂ during this stage.

After the first stress drop, the permeability increased more slowly. A friction effect on the rupture surface might explain this difference. The fracture surface was polished in the following loading procedures, which hindered the permeability increase. At the same time, the rock continued to be compressed after the first pressure drop. However, there was no obvious correlation between the amount of compression and seepage velocity. During this stage, the flow of the fluid was mainly controlled by the deviatoric strain. The deformation of the rock was mainly due to the deviatoric strain. As shown in Figure 9a, there were one or two pressure drops after the first, and each pressure drop was accompanied by a change in the direction of the deviatoric strain and the characteristics of the seepage velocity. As shown in Figure 9b, the direction of the strain at an intermediate principal stress of 40 MPa and 50 MPa was very similar. In addition, the pattern of the flow rate changes was also very similar, as shown in Figure 8. The change in the partial strain caused a change in the internal fissure channel of the rock, which in turn caused a change in the permeation rate.

After the first pressure drop, the flow inside the rock was more inclined to fissure flow. During this process, the main factors affecting gas flow were inertial resistance and frictional resistance. Therefore, the seepage rate after the peak showed a certain volatility. Thus, gas permeation in sandstone rock can be divided into two periods during the rock damage process. Before the first pressure drop, the gas seepage velocity is mainly affected by volume strain. Rock bulk expansion always leads to an increasing gas seepage velocity. After the first pressure drop, gas flows through cracks in the rocks. Thus, the seepage velocity is affected by the deviator strain, which can change the seepage channels.

Thus, in the geothermal mining process, under the same fluid pressure conditions, the flow velocity of the fluid will be affected by the rock failure morphology and strain path.

4. Conclusions

A series of experiments was conducted to investigate the effect of σ₂ on the strength, deformation, and permeability of sandstone. From the analysis of the extensive experimental data, we can draw the following conclusions:

(1)

The peak strength, octahedral shear stress, and octahedral effective normal stress of the sandstone increased with increasing σ₂. However, the rate of increase decreased with increasing σ₂. This is because the relatively high intermediate principal stress coefficient b has an inhibitory effect on rock strength.

(2)

As the ratio of σ₂/σ₃ increased, the specimen entered compressive strain in the σ₂ direction during the first stress drop, when the compression exponentially increased with the increasing stress ratio.

(3)

Stress and strain deviation occurred during the loading period. This phenomenon indicates that elastic mechanics are not suitable for analyzing this sandstone rock from a macro view.

(4)

During the complete true triaxial stress–strain experiments, the variation in gas seepage velocity could be divided into two stages. Before the first pressure drop, the gas seepage velocity was mainly affected by volume strain. After the first pressure drop, the seepage velocity was affected by the deviator strain, which can change the seepage channels.