Segmentary Damage Constitutive Model and Evolution Law of Rock under Water-Force Coupling Action of Pumped Storage in Deep Mine

Abstract

The deformation and failure of surrounding rock mass under different water environments is a basic mechanical problem encountered in the safe operation of ground pumped storage power station and abandoned mine pumped storage power station. According to the influence of different water environments on the failure characteristics of deep surrounding rock mass, it is necessary to summarize the damage evolution law of deep rock mass under different water environments and construct the constitutive model. In this paper, the loading mechanical test is carried out after the natural immersion of the rock in different water environments. The influence of the change of the geological water environment on the damage evolution characteristics of the rock is analyzed from the perspective of the deterioration of the mechanical parameters. On this basis, the damage statistical constitutive model is constructed, and the damage evolution analysis is carried out. The results show that the degradation degree of mechanical parameters such as compressive strength and elastic modulus of sandstone is in the order of distilled water immersion, simulated groundwater immersion and natural state. The damage evolution of sandstone under water–rock interaction is divided into four stages: no damage, rapid damage, deceleration damage and failure. The theoretical curve of the model is in good agreement with the uniaxial test curve of rock under different water environments. The segmented damage constitutive model based on the long compaction stage of sandstone under water–rock interaction reasonably reflects the change of stress–strain relationship of damage failure, and the physical meaning of parameters is clear.

1. Introduction

With the large-scale integration of clean energy such as wind energy and nuclear energy into the national power system, the uncertainty of the power system increases. It is difficult to accurately grasp the load demand of users by thermal power units alone, resulting in unnecessary waste of power resources [1]. Pumped storage power station is an ideal peaking power supply for power grids because of its two-way rapid response, but the establishment of a large number of reservoirs also has a serious impact on natural ecology [2,3]. In order to respond to the national “double carbon” goal, combined with the fact that a large number of abandoned mine underground resources have not been effectively utilized, turning waste into treasure and forming an energy technology system based on pumped storage has been proposed [4,5]. In the process of site selection, underground space reconstruction and support and operation safety of the storage power station, in view of the difference between underground and surface hydrological conditions, the surrounding rock is prone to dissolution and disintegration under the action of water and rock, endangering the circulation and integrity of the storage space and water circulation channel. Therefore, the study of the mechanical damage characteristics and constitutive model of sandstone after long-term immersion in distilled water and groundwater is conducive to the effective and reasonable safety evaluation of the operation process of pumped storage power station.

At present, a variety of related studies have been carried out on the deformation characteristics and damage and failure characteristics of rock under water–rock interaction by using the new technology of rock laboratory test, and many research results have been obtained. It mainly focuses on the use of water–rock interaction damage and failure of rock to carry out a variety of indoor mechanical tests, combined with the acoustic characteristic statistics and internal pore structure observation in the process of failure to study the damage mechanism or to describe the mathematical expression of rock constitutive relationship based on the analysis of its damage evolution characteristics [6,7]. Luo et al. [8] summarized the mechanical parameters and crack evolution of sandstone after soaking in aqueous solution with different PH values and summarized the degree of deterioration of mechanical properties of sandstone by aqueous solution. Li et al. [9] summarized the trend relationship between mechanical parameters and sandstone water content by analyzing the microscopic void characteristics of sandstone under different water contents and uniaxial compression tests. Zhang et al. [10] established a quantitative evaluation index of rock damage degree by using the energy evolution law of sandstone soaked in aqueous solution with different PH. Li et al. [11] studied the mechanical degradation characteristics of long-term soaking and dry-wet cycle sandstone, revealing that the first cycle soaking damage is the largest. Wang et al. [12] analyzed the deterioration of rock mechanical parameters under different times of water–rock cycle and concluded that the change of rock mechanical parameters tended to be stable after 16 times of water–rock cycle. Qiao et al. [13] constructed the damage variable by separating the secondary porosity generated by water chemistry from the total secondary porosity and verified the applicability of the damage variable through the CT test results.

The studies above have analyzed the damage characteristics and principles of rocks under water–rock interaction to a large extent. Based on the constitutive model, continuous damage mechanics and statistical theory are introduced to construct the damage stress–strain relationship. Deng et al. [14] verified the constitutive rationality through the indoor triaxial compression test data of sandstone soaking and air drying cycle. He et al. [15] constructed a rock damage constitutive model considering the influence of residual strength on the rock constitutive relationship to explore the influence of water pressure on the rock damage process. Zhao et al. [16] proposed a constitutive model of deep shale based on the D-P criterion and Weibull probability distribution and verified it theoretically in view of the research status of damage constitutive relationship of deep shale under hydration. Li et al. [17] used the characteristics of triaxial compression test curves of coal rocks with different water contents to derive a segmented damage constitutive model that conforms to the three stages of pre-peak, post-peak and residual stages.

In summary, previous scholars have made a lot of analysis and summary on the damage characteristics and constitutive relationship of rock under water–rock interaction. However, the relevant damage constitutive model needs to be further improved to make it more conducive to specific engineering applications. In this paper, based on the analysis of the characteristics that the compaction stage of uniaxial loading curve of sandstone gradually becomes longer under different water environments, a segmented damage constitutive model with Mohr–Coulomb criterion is constructed by means of continuous damage mechanics and Weibull probability statistical distribution to describe the stress–strain relationship of sandstone under different water–rock interactions. Combined with the summary of the weakening law of mechanical parameters and the analysis of damage evolution law in the loading process, it is expected to provide some theoretical support and reference for the maintenance and design of pumped storage power stations in abandoned mines. At the same time, it is hoped that it will be helpful to the subsequent research on the failure of rock mass with initial cracks caused by the secondary excavation of underground water storage space.

2. Materials and Methods

2.1. Laboratory Test Program

In order to simulate the real groundwater environment, according to the literature [10], the main cations of groundwater are Na⁺, K⁺, Ca²⁺, Mg²⁺, etc.; the anions are SO₄²⁻, Cl⁻, HCO₃⁻, etc.; and the PH is roughly 5–8. The above ions are added to the distilled water to simulate the groundwater composition. In this paper, the test materials involved are selected from deep sandstone. According to the standard specification of indoor rock test, sandstone is made into 6 standard rock size specimens with a diameter of 50 mm and a height of 100 mm by coring and grinding. The shape and size of sandstone used in this test are shown in Figure 1. The same amount was placed in three environments of natural state, distilled water and simulated groundwater, and the specimens were completely immersed in aqueous solution for 60 days. The rock soaking process is shown in Figure 2. After 60 days, the immersed specimens were taken out and placed in natural air for 3 days. The standard specimen was subjected to unconfined uniaxial compression test. The sandstone uniaxial compression tester is shown in Figure 3.

2.2. Constitutive Description Method

The damage constitutive model is constructed as an effective method to predict the mechanical behavior of rock. Firstly, according to the failure characteristics of the rock test curve, the curve is divided into two sections by the compaction stage. The compaction stage curve is described by empirical formula. The curve description after the compaction stage defines the damage variable based on some methods. Combined with probability statistics theory and existing failure criteria, the corresponding damage evolution equation is derived. Then, the constitutive relation which can describe the whole failure process is obtained.

3. Construction of Constitutive Model

3.1. Damage Variable Determination

According to the analysis of rock internal damage under different geological water environments, after soaking in aqueous solution, due to the dissolution or dissolution of internal debris and cements under hydraulic action, the porosity of sandstone increases continuously. Using the principle of effective stress, it is found that the effective bearing area of rock particles decreases under the same stress, and the effective stress and deformation of rock also increase. It is necessary to construct a damage factor to quantitatively describe the degree of water damage in rock under different geological environments. Through research, it is found that the water environment has a significant deterioration effect on the elastic modulus of rock, so the elastic modulus is used to describe the water damage $D_w$ [18].

$$D_w=1-\frac{E_w}{E_0}$$

(1)

where $E_w$ is the elastic modulus after soaking in aqueous solution, and $E_0$ is the elastic modulus in natural state.

Considering the internal inhomogeneity of rock and the continuity of loading, this paper assumes that the strength failure of rock microelement satisfies the Weibull probability density distribution function $R(\gamma )$.

$$R(\gamma )=\frac{m}{{\gamma }_0}{(\frac{\gamma }{{\gamma }_0})}^{m-1}\exp [-{(\frac{\gamma }{{\gamma }_0})}^m]$$

(2)

where $\gamma$ is the distribution variable of micro-unit strength failure, and ${\gamma }_0$ and $m$ are Weibull probability distribution parameters.

According to the continuous damage mechanics, when the number of rock micro-element damage continues to accumulate, the macroscopic strength of the rock continues to decay. Therefore, the ratio of the number of failure microelements to the total number of micro-elements under a certain microelement stress state is expressed as force damage $D_F$ [19].

$$D_F=\frac{N_p}{N}$$

(3)

where $N_p$ is the number of broken microelements, and $N$ is the total number of microelements.

The failure of rock microelement is similar to the macroscopic strength failure of rock. With the continuous expansion of rock cracks under mechanical loading, there is a critical threshold strength of rock micro-element. When the microelement reaches the threshold stress state, the failure occurs.

$$f({\sigma }^{*})=k_0$$

(4)

where $k_0$ is related to the internal friction angle and cohesion of rock [20].

The probability of both sides of the Equation (4) can be obtained.

$$p[f({\sigma }^{*})]=p[k_0]$$

(5)

From Equations Equations (4) and (5), the expression of the number of sandstone failure microelements $N_p$ can be obtained as follows:

$$N_p={\displaystyle {\int }_0^{k_0}N\cdot R(x)dx}$$

(6)

Substituting Equations (2) and (6) into Equation (3), the final expression of sandstone force damage factor can be obtained as

$$D_F=1-\exp [-{(\frac{\gamma }{{\gamma }_0})}^m]$$

(7)

3.2. Damage Constitutive Model after Water–Rock Interaction

The rock damage accumulates continuously; the fracture expands, and the effective stress between the particles in the rock leads to effective plastic deformation of the rock skeleton. According to the Lemaitre strain equivalence hypothesis and the effective stress principle, the effective stress expression is as follows [21].

$${\sigma }_i^{*}=\frac{{\sigma }_i}{1-D_F}(i=1,2,3)$$

(8)

where ${\sigma }_i^{*}$ is the effective stress, and ${\sigma }_i$ is the nominal stress.

It can be clearly seen from the uniaxial stress–strain relationship curve of sandstone that the rock deformation has an obvious elastic stage, and the rock stress–strain relationship at this stage satisfies the generalized Hooke law.

$${\epsilon }_l=\frac{1}{E}[{\sigma }_l^{*}-\nu ({\sigma }_m^{*}+{\sigma }_n^{*})](l,m,n=1,2,3)$$

(9)

where $E$ is elastic modulus, and $\nu$ is Poisson ratio.

According to the Equations (1) and (7)–(9), the stress–strain relationship of triaxial rock under different geological water environments can be expressed as

$${\sigma }_1=E_w{\epsilon }_1\exp [-{(\frac{\gamma }{{\gamma }_0})}^m]+\nu ({\sigma }_2+{\sigma }_3)$$

(10)

Since the Mohr–Coulomb strength criterion is based on rock triaxial test data, it is universal and reasonable for rock conventional compression process. Therefore, this paper assumes that rock microelement strength meets the Mohr–Coulomb strength criterion, and the expression is as follows:

$$\gamma ={\sigma }_1^{*}-\frac{1+\sin \phi }{1-\sin \phi }{\sigma }_3^{*}$$

(11)

where ${\sigma }_1^{*}$ and ${\sigma }_3^{*}$ are the effective first principal stress and the effective third principal stress after damage correction, and $\phi$ is the internal friction angle of sandstone.

Substituting Equation (11) into Equation (10), the constitutive equation of sandstone can be expressed as

$${\sigma }_1=E_w{\epsilon }_1\exp [-{(\frac{{\sigma }_1^{*}-\frac{1+\sin \phi }{1-\sin \phi }{\sigma }_3^{*}}{{\gamma }_0})}^m]+\nu ({\sigma }_2+{\sigma }_3)$$

(12)

According to the analysis of the uniaxial stress–strain curve of sandstone under three water environments, with the immersion of aqueous solution, the rock compaction stage is prolonged, and the stress–strain curve shows obvious concave function evolution law. Through the analysis of literature [14], the threshold strength ${\sigma }_b$ of rock compaction stage in this paper is 40% ${\sigma }_c$ (peak strength). In order to describe the constitutive relationship of sandstone at this stage, it can be set as

$${\sigma }_1={\sigma }_b{(\frac{{\epsilon }_1}{{\epsilon }_b})}^2$$

(13)

where ${\epsilon }_b$ is the strain value corresponding to the threshold strength in the compaction stage.

In summary, the constitutive model of sandstone is

$${\sigma }_1=\left\{\begin{array}{l}{\sigma }_b{(\frac{{\epsilon }_1}{{\epsilon }_b})}^2, \epsilon \le {\epsilon }_b\\ {\sigma }_b+E_w({\epsilon }_1-{\epsilon }_b)\exp [-{(\frac{\gamma }{{\gamma }_0})}^m]+2\nu {\sigma }_3, \epsilon \ge {\epsilon }_b\end{array}\right.$$

(14)

3.3. Parameter m and ${\gamma }_0$ Identification

The parameters of the damage statistical constitutive model are determined by linear fitting method, peak point method and curve fitting method [15]. As shown in the above triaxial constitutive model relationship, parameters such as ${\sigma }_b$, ${\sigma }_c$, $E_w$ and $\nu$ can be measured by rock indoor compression test. In order to obtain the probability distribution parameters ${\gamma }_0$ and m, the numerical calculation is carried out by using the two conditions of curve continuity and slope at the peak of the curve equal to 0.

$${\epsilon }_1={\epsilon }_c, {\sigma }_1={\sigma }_c$$

(15)

$${\epsilon }_1={\epsilon }_c, \frac{d{\sigma }_1}{d{\epsilon }_1}=0$$

(16)

where ${\epsilon }_c$ is the peak strain, and ${\sigma }_c$ is the peak stress.

Substituting Equation (15) into Equation (14), we obtain

$${(\frac{\gamma }{{\gamma }_0})}^m=\ln \frac{E({\epsilon }_c-{\epsilon }_b)}{{\sigma }_c-{\sigma }_b-2\nu {\sigma }_3}$$

(17)

Substituting Equation (16) into Equation (14), we obtain

$${\left.\frac{\partial {\sigma }_1}{\partial {\epsilon }_1}\right|}_{\epsilon ={\epsilon }_c,\sigma ={\sigma }_c}=E\exp [-{(\frac{\gamma }{{\gamma }_0})}^m]+E({\epsilon }_1-{\epsilon }_b)\exp [-{(\frac{\gamma }{{\gamma }_0})}^m](-\frac{m{\gamma }^{m-1}}{{\gamma }_0^m})\frac{\partial \gamma }{\partial {\epsilon }_1}=0$$

(18)

The expression $\gamma$ in Equation (18) can be obtained by substituting Equations (8) and (10) into Equation (11)

$$\gamma =\frac{E{\epsilon }_1({\sigma }_1-\frac{1+\sin \phi }{1-\sin \phi }{\sigma }_3)}{{\sigma }_1-2\nu {\sigma }_3}$$

(19)

From Equations (17)–(19), the expressions of $m$ and ${\gamma }_0$ are as follows:

$$m=\frac{\alpha }{({\epsilon }_c-{\epsilon }_b)\beta \ln \frac{E({\epsilon }_c-{\epsilon }_b)}{{\sigma }_c-{\sigma }_b-2\nu {\sigma }_3}}$$

(20)

$${\gamma }_0={[\frac{{\alpha }^m}{\ln \frac{E({\epsilon }_c-{\epsilon }_b)}{{\sigma }_c-{\sigma }_b-2\nu {\sigma }_3}}]}^{\frac{1}{m}}$$

(21)

where

$$\alpha =\frac{E{\epsilon }_c({\sigma }_c-\frac{1+\sin \phi }{1-\sin \phi }{\sigma }_3)}{{\sigma }_c-2\nu {\sigma }_3}$$

(22)

$$\beta =\frac{E({\sigma }_c-\frac{1+\sin \phi }{1-\sin \phi }{\sigma }_3)}{{\sigma }_c-2\nu {\sigma }_3}$$

(23)

4. Model Validation and Parameter Study

4.1. Test Verification

In this paper, the rationality of the constitutive model is verified by comparing the data curve of sandstone unconfined uniaxial compression test under three water environments with the derived constitutive model relationship curve. The stress–strain curves of sandstone under three water environments after uniaxial compression are shown in Figure 4. When the fitting degree of the two curves is high, the model is verified. The verification results are shown in Figure 5, and the selection of theoretical curve parameters in the verification process is shown in

Table 1. Parameter selection of theoretical curve.

Rock State	Elastic Modulus E/GPa	Threshold Strain in Compaction Stage ${\mathit{\epsilon }}_{\mathit{b}}\mathbf{/}\mathbf{\%}$	Threshold Stress in Compaction Stage ${\mathit{\sigma }}_{\mathit{b}}\mathbf{/}\mathbf{MPa}$	$\mathbf{Peak}\mathbf{Strain}{\mathit{\epsilon }}_{\mathit{c}}\mathbf{/}\mathbf{\%}$	$\mathbf{Peak}\mathbf{Stress}{\mathit{\sigma }}_{\mathit{c}}\mathbf{/}\mathbf{MPa}$	$\mathbf{Model}\mathbf{Parameter}\mathit{m}$	$\mathbf{Model}\mathbf{Parameter}{\mathit{\gamma }}_0$
Distilled water immersion	6.5	0.538	13.75	0.894	34.37	22.83	65.22
Simulated groundwater immersion	6.7	0.516	14.21	0.875	35.50	20.31	63.78
Natural state	8.9	0.465	19.38	0.829	48.46	20.70	80.17

. According to the four stages of rock curve characteristic compaction, elasticity, damage evolution and failure damage, the theoretical curve is well described, which reflects the characteristics of sandstone long strain value compaction stage under three different water environments and verifies the applicability of the constitutive model.

4.2. Comparison of Theoretical Curves in Literature

Through the comparison between the theoretical curve and the experimental data above, it is found that the uniaxial constitutive model established by the Equation (14) has a good description of the uniaxial stress–strain relationship of sandstone under different water–rock interactions, and through the comparison of the laws, it is found that the stress–strain relationship of the three compaction stages shows obvious concave characteristics. The constitutive relation uses the above characteristics to establish a two-stage constitutive relation expression with the threshold strain of the compaction section as the segmentation point. By comparing the theoretical curve in Figure 6 with the constitutive relation curve of conventional sandstone in the literature [22], it is found that the initial stage description in the model established for conventional sandstone is usually straight or convex, which is difficult to reasonably express as the constitutive relation of sandstone under water–rock interaction. However, after the initial compaction stage of rock, the similar shape of the two curves further indicates that the constitutive model established in this paper is reasonable.

4.3. Parameter Study

The model parameters $E$, ${\epsilon }_c$, ${\epsilon }_b$ and other parameters in the above model have clear physical significance. In order to strengthen the application scope of the constitutive model, this study adopts the control variable method to analyze the physical significance of Weibull distribution parameters $m$ and ${\gamma }_0$. Taking sandstone under natural conditions as an example, the fixed parameter ${\gamma }_0$ is 80.17, and a series of theoretical curves are obtained by selecting $m$ = 15, 20, 25 and 30, respectively. As shown in Figure 7, the compaction stage and elastic stage of the curve family are basically the same. For the damage evolution stage and failure stage, the damage evolution development and failure rate of the rock increase with the increase in m value, and the rock shows obvious brittle characteristics. That is, the parameter m is closely related to the internal damage accumulation rate and brittleness of rock. Similarly, when the fixed parameter m is 20.70, a series of theoretical curves are obtained by selecting ${\gamma }_0$ = 75, 80, 85 and 90, respectively, as shown in Figure 8. As the ${\gamma }_0$ value increases, the peak strain and strength of the rock continue to move in the direction of strain increase—that is, the ${\gamma }_0$ value controls the macroscopic strength of the sandstone.

5. Damage Evolution Analysis

5.1. Damage Evolution Analysis of Mechanical Parameters

As a parameter index of rock mechanics, elastic modulus highlights the characteristics of rock stiffness. In this paper, the linear section of the uniaxial stress–strain curve of sandstone is fitted by linear function, and the slope value of the fitting curve is the elastic modulus. According to the mechanical parameters of sandstone in different water environments in Figure 9, with the soaking of aqueous solution, the elastic modulus and compressive strength of sandstone are obviously weakened. In distilled water and groundwater soaking, the elastic modulus of sandstone decreases by 27% and 25%, respectively, and the compressive strength decreases by 29% and 27%, respectively. According to [13], the content of Na⁺, K⁺, Ca²⁺ and Mg²⁺ increased due to the dissolution or dissolution of feldspar and other mineral components in sandstone after soaking in aqueous solution. Since Na⁺, K⁺ and other related cations were added to distilled water to simulate the real groundwater environment, the ion precipitation caused by the dissolution or dissolution of water osmotic pressure and mineral composition feldspar was reduced, and the expansion pressure of sandstone was effectively reduced. After air drying, the salt-out particles filled the pores, and the mechanical strength of rock structural plane increased. The macroscopic mechanical properties such as elastic modulus and compressive strength of sandstone in simulated groundwater are higher than those in distilled water.

5.2. Constitutive Damage Evolution Analysis

In order to further intuitively reflect the constitutive damage evolution process of rock after water–rock interaction, combined with the expression of total damage variable by Xu et al. [23], the total damage variable $D$ evolution equation and damage evolution rate equation considering the initial damage and force damage of water–rock are obtained.

$$D=D_F+D_w-D_F{D}_W$$

(24)

$$\begin{array}{ll}D& =1-\frac{E_w}{E_0}\exp [-{(\frac{\gamma }{{\gamma }_0})}^m]\\ & =1-\frac{E_w}{E_0}\exp [-{(\frac{{\epsilon }_1}{{\epsilon }_c})}^{m}G\ln \frac{E({\epsilon }_c-{\epsilon }_b)}{{\sigma }_c-{\sigma }_b-2\nu {\sigma }_3}]\end{array}$$

(25)

where

$$G=\frac{{\sigma }_c-2\nu {\sigma }_3}{{\sigma }_c-\frac{1+\sin \phi }{1-\sin \phi }{\sigma }_3}\frac{{\sigma }_1-\frac{1+\sin \phi }{1-\sin \phi }{\sigma }_3}{{\sigma }_1-2\nu {\sigma }_3}$$

The expression of the damage evolution rate equation is obtained by using the Equation (25).

$$\frac{\partial D}{\partial {\epsilon }_1}=\frac{E_w}{E_0}\frac{m}{{\gamma }_0}{(\frac{\gamma }{{\gamma }_0})}^{m-1}\exp [-{(\frac{\gamma }{{\gamma }_0})}^m]$$

(26)

According to Equation (25) and the experimental data, the rock damage evolution curves under the three water environments are obtained as shown in Figure 4. The damage evolution curves of the three are all S-shaped curves. With the continuous increase in strain, the internal damage of the rock is continuously accumulated, and the crack is continuously expanded. The damage reaches the maximum value 1 when the rock is broken. Through curve comparison, it is found that the initial damage of rock under water–rock interaction increases significantly. The rock damage values after distilled water and groundwater immersion increased by 25% and 23%, respectively, compared with the natural control group. According to the damage evolution and test curve analysis, as shown in Figure 10, the rock damage evolution process can be roughly divided into four stages: no damage stage, rapid damage stage, deceleration damage stage and failure stage. In the non-damage stage, the initial voids in the rock are compacted and the deformation is within the recoverable range. With the increase in external force loading, the internal particle skeleton of rock produces irreversible damage deformation; the internal micro-cracks continue to produce and expand, and the damage value increases rapidly. At the same time, as the crack continues to expand to the inner core of the rock, due to the compact particles of the rock core skeleton and the large friction resistance between the particles, the rock damage value appears at the stage of deceleration development until the rock forms a critical crack and is destroyed, and the rock damage value reaches the maximum value of 1.

Comparing the damage evolution curves under three water environments, the non-damage threshold strain corresponding to the rock in the simulated groundwater state is 0.0055, which is 17% higher than the natural state threshold strain. The non-damage threshold strain in distilled water is 0.0059, which is 7.3% and 26% higher than that in simulated groundwater state and natural state, respectively. In the long-term coexistence of water and rock, the two have a profound impact on the internal results of rock due to various complex weakening effects, including physical, chemical and mechanical aspects [13].

(1)

Physical effect

With the long-term immersion, the aqueous solution continues to enter the gap between the rock particles, which has a good lubrication and migration effect on the particles.

(2)

Chemical effect

The chemical reaction and ion exchange of the aqueous solution to the rock cement and some minerals lead to dissolution; the connection between the rock particles weakens, and holes appear.

(3)

Mechanical effect

The seepage of aqueous solution leads to the pressure on the pores inside the rock.

The non-damage stage of the rock damage evolution curve under the three states is delayed due to the water–rock weakening effect. The non-damage threshold strain corresponding to the larger initial internal void and damage also increases. After entering the stage of rapid damage growth, the three curves intersect, showing that the rock strain is the largest after long-term immersion in distilled water under the same damage value, which corresponds to the larger damage deformation generated in the compression process on the macro level.

In order to further discuss the influence of different water–rock interaction on rock damage evolution process, the damage evolution rate under different water environments is analyzed. The results are shown in Figure 11. The physical meaning corresponding to the damage evolution rate is the slope of each point of the damage evolution curve, and its size corresponds to the speed of damage evolution. The faster the damage evolution, the greater the brittleness of the rock, and the slower the damage evolution, the higher the ductility of the rock [24]. Through the comparison of Figure 12, it is found that the damage evolution rate of rock in natural state is the largest, followed by distilled water state and simulated groundwater state. Thus, groundwater can effectively enhance the ductility of rock.

In summary, the constitutive model has two parameters $m$ and ${\gamma }_0$ and gives the corresponding expression. In practical engineering applications, combined with the selection of mechanical parameters such as elastic modulus and internal friction angle in the engineering site, the overall stress–strain evolution relationship of rock can be better predicted. According to the deformation of surrounding rock, the internal damage degree of pumped storage power station under long-term immersion will be better predicted. In the future, with the help of certain model software, it can be well applied to the site selection, secondary excavation and repeated operation of pumped storage power stations.

6. Conclusions

This study mainly constructs a rock segment damage constitutive model that is more in line with the characteristics of the compaction stage under water–rock interaction. The applicability of the constitutive model was verified by uniaxial loading tests of sandstone soaked in three geological water environments. Through the study of mechanical parameters and constitutive damage evolution law, the damage weakening law of sandstone is summarized. It can provide reference for the safety support and operation of pumped storage power station in deep mine. The main conclusions are as follows:

(1)

Compared with sandstone in natural state, the mechanical parameters such as compressive strength and elastic modulus of rock are obviously weakened after soaking in aqueous solution, and the weakening effect in distilled water is greater than that in simulated groundwater.

(2)

The failure process of sandstone is a continuous damage accumulation process. Similar to the uniaxial loading process, the damage evolution process of sandstone can be roughly divided into four stages: no damage, rapid damage, deceleration damage and failure. Simulated groundwater can effectively enhance the ductility of rock.

(3)

According to the curve of long strain compaction section, a piecewise damage statistical constitutive model based on Mohr–Coulomb criterion is constructed, which can describe the stress–strain relationship of rock loading failure under three water environments.

The distribution parameters $m$ and ${\gamma }_0$ have clear physical significance. m reflects the internal damage accumulation rate and brittleness of rock, and ${\gamma }_0$ affects the macroscopic strength of rock.

The study highlights the importance of understanding the behavior of rock masses under water-force coupling actions in pumped storage power stations and suggests that further research in this area could lead to more effective risk mitigation strategies and improved safety outcomes.