# Creep Behavior and Its Prediction of Saturated Weakly Cemented Medium-Grained Sandstone under Multiaxial Loadings

Authors: Qingheng Gu; Guangming Zhao; Xiangrui Meng; Jian Sun; Xiang Cheng; Wensong Xu; Ruofei Zhang
Journal: Geofluids
 (2023)

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## Abstract

Weakly cemented medium-grained sandstone in Ordos mining area of China is widely distributed under phreatic water. The creep behavior of saturated rock is important for the stability of water-resisting strata. In this paper, creep mechanical properties of saturated medium-grained sandstone were studied by triaxial rheological test. The results showed that medium-grained sandstone only shows attenuation creep and stable creep under low stress level, and accelerated creep occurs when axial load reaches about 75% of instantaneous compressive strength. The rock samples exhibit volume dilatancy under most loading levels and finally shear fracture. The failure mechanism is the dislocation separation of mineral particles. Based on the energy dissipation theory, the damage evolution equation was modified and introduced into the Burgers model, which can accurately describe the rheological behavior of saturated medium-grained sandstone. The modified model was used to predict the crack initiation time under different deviatoric stresses, which can provide guidance for early warning of water-resisting rock stability.


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## Body

## 1. Introduction

A special rock material, medium-grained sandstone, widely exists under phreatic water in Ordos mining area of Western China. This rock material is of late diagenesis, which results in the poor cementation of rock particles and low strength of rock [1–3]. The climate in this area is dry, and there is precious phreatic water above the weakly cemented rock stratum [4, 5]. Therefore, when mining coal seams under weakly cemented water-resisting strata, not only the long-term stability of medium-grained sandstone affected by groundwater should be paid attention to, but also the early warning method of rock fracture instability needs to be studied.Groundwater has a great influence on mechanical properties of soft rocks [6–17]. For example, Yao et al. [6] carried out uniaxial compression and acoustic emission monitoring tests of coal with different water contents. The results showed that the strength and deformation modulus of coal after water immersion decrease with the increase of water content, and the fracture evolution of coal can be predicted by acoustic emission signal. Fabre and Pellet [7] studied the creep characteristics of three types of sedimentary rocks by using uniaxial compression creep test and pointed out that isokinetic creep and accelerated creep would occur only when the stress exceeded a certain value; otherwise, the rock would only show deceleration creep. Cao et al. [8] studied the creep characteristics of anorthosite under water-saturated and air-dried conditions through uniaxial compression creep tests and pointed out that it takes longer for saturated rocks to achieve stable creep than air-dried rocks. Yan et al. [9] analyzed the creep experimental results of granite under different stresses and osmotic pressures and modified the Nishihara model. The results showed that the improved Nishihara model could better describe the whole process and nonlinear characteristics of rock creep. In a word, water will weaken the mechanical properties of rocks. With the increase of water content, the strength and deformation resistance of rock gradually decrease, and the creep rate increases under long-term load, and the duration of stable creep extends. Therefore, it is impossible to find a unified law of the influence of water on rock creep behavior with different water-weakening properties. The creep properties of weakly cemented medium-grained sandstone under the influence of water need to be studied specifically.In addition, the stability evaluation of water-resisting strata is important to ensure the safety of mining. The signs of rock failure can be obtained by abnormal deformation or external monitoring [18–22], but they are not suitable for failure prediction considering rock anisotropy. The creep model can be used to describe the relationship between rock deformation and time, and rock deformation is related to the development of cracks [23]. So the failure behavior of rock can be predicted by using creep model. Up to now, most creep models have been studied to accurately describe the existing test or field behavior [24–28]. However, the accuracy of using creep model to make further prediction remains to be demonstrated, especially to link fracture prediction with creep deformation.In this paper, the weakly cemented water-resisting medium-grained sandstone was taken as the research object. The water-resisting rock mass in daily water environment can be considered as saturated. The creep mechanical properties of saturated rock samples under different confining pressures were experimentally studied to obtain its unique rheological properties. The damage evolution equation and creep model based on energy dissipation were improved, and a prediction method of rock cracking initiation point in creep process was proposed.


## 2. Experimental Method

### 2.1. Sample Preparation

The mineral components of the rock mainly include quartz, illite, chlorite, and albite, with the contents of 27.75%, 16.18%, 8.09%, and 47.98%, respectively. Illite and chlorite belong to clay minerals. That is to say, the clay mineral content of the medium-grained sandstone studied is about 24%.Medium-grained sandstone is mainly composed of mineral particles (MP), which are connected by clay minerals (Figure1). Most mineral particles are irregular aggregates, and some unconformity contact between particles forms original defects in the sample, which contributes to the time-dependent behavior of rocks and crack initiation [29].Figure 1
Photomicrographs of the internal structure of medium-grained sandstone.The saturated samples of medium-grained sandstone for the test were prepared according to the procedure outlined by ISRM [30]. The ratio of height to diameter of specimens is 2 : 1, and the diameter is about 50 mm. The height is about 100 mm, and the error of the surface roughness is less than 0.02 mm. The production process of saturated rock samples is to immerse the dry rock in clean water, weigh it every half an hour within the first 10 hours, and then weigh it every hour until its mass remains unchanged [12].


### 2.2. Axial Load Classification

The creep test adopts the graded loading method, and the compressive strength of the rock under the corresponding confining pressure needs to be tested before the classification of the graded loading creep level. The confining pressure (σ3) was designed as 1.5 MPa, 3.0 MPa, 4.5 MPa, and 6.0 MPa. The triaxial compression stress-strain relationship of rock under four confining pressures is shown in Figure 2. The compressive strength of saturated sample is 18.6 MPa, 23.8 MPa, 26.8 MPa, and 29.3 MPa with σ3 of 1.5 MPa, 3.0 MPa, 4.5 MPa, and 6.0 MPa.Figure 2
Stress-strain curves of saturated medium-grained sandstone under triaxial compression.The axial load of the test starts from about 50%-60% of the compressive strength. The difference between two adjacent loads is 2.0 MPa. The specific implementation scheme is shown in Table1.Table 1
Implementation scheme of creep test for medium-grained sandstone.
Rock sampleσ3 (MPa)σ1‐σ3 (MPa)Saturated medium-grained sandstone1.59.511.513.515.517.519.521.5…3.012141618202224…4.514.516.518.520.522.524.526.5…6.015171921232527…


### 2.3. Creep Test Equipment and Procedures

Triaxial creep test was carried out on RLJW-2000 servo-controlled testing machine [31, 32]. At the beginning of the test, the confining pressure is loaded to the set value at the rate of 0.05 MPa/s, and then, the axial deviating stress is loaded to the first level at the rate of 0.06 mm/min. The axial pressure is maintained for 6 hours before loading to the next level and so on until the specimen is destroyed.


## 2.1. Sample Preparation

The mineral components of the rock mainly include quartz, illite, chlorite, and albite, with the contents of 27.75%, 16.18%, 8.09%, and 47.98%, respectively. Illite and chlorite belong to clay minerals. That is to say, the clay mineral content of the medium-grained sandstone studied is about 24%.Medium-grained sandstone is mainly composed of mineral particles (MP), which are connected by clay minerals (Figure1). Most mineral particles are irregular aggregates, and some unconformity contact between particles forms original defects in the sample, which contributes to the time-dependent behavior of rocks and crack initiation [29].Figure 1
Photomicrographs of the internal structure of medium-grained sandstone.The saturated samples of medium-grained sandstone for the test were prepared according to the procedure outlined by ISRM [30]. The ratio of height to diameter of specimens is 2 : 1, and the diameter is about 50 mm. The height is about 100 mm, and the error of the surface roughness is less than 0.02 mm. The production process of saturated rock samples is to immerse the dry rock in clean water, weigh it every half an hour within the first 10 hours, and then weigh it every hour until its mass remains unchanged [12].


## 2.2. Axial Load Classification

The creep test adopts the graded loading method, and the compressive strength of the rock under the corresponding confining pressure needs to be tested before the classification of the graded loading creep level. The confining pressure (σ3) was designed as 1.5 MPa, 3.0 MPa, 4.5 MPa, and 6.0 MPa. The triaxial compression stress-strain relationship of rock under four confining pressures is shown in Figure 2. The compressive strength of saturated sample is 18.6 MPa, 23.8 MPa, 26.8 MPa, and 29.3 MPa with σ3 of 1.5 MPa, 3.0 MPa, 4.5 MPa, and 6.0 MPa.Figure 2
Stress-strain curves of saturated medium-grained sandstone under triaxial compression.The axial load of the test starts from about 50%-60% of the compressive strength. The difference between two adjacent loads is 2.0 MPa. The specific implementation scheme is shown in Table1.Table 1
Implementation scheme of creep test for medium-grained sandstone.
Rock sampleσ3 (MPa)σ1‐σ3 (MPa)Saturated medium-grained sandstone1.59.511.513.515.517.519.521.5…3.012141618202224…4.514.516.518.520.522.524.526.5…6.015171921232527…


## 2.3. Creep Test Equipment and Procedures

Triaxial creep test was carried out on RLJW-2000 servo-controlled testing machine [31, 32]. At the beginning of the test, the confining pressure is loaded to the set value at the rate of 0.05 MPa/s, and then, the axial deviating stress is loaded to the first level at the rate of 0.06 mm/min. The axial pressure is maintained for 6 hours before loading to the next level and so on until the specimen is destroyed.


## 3. Results and Discussion

### 3.1. Time-Dependent Strain of Medium-Grained Sandstone

The variation of axial creep deformation of saturated medium-grained sandstone with time is shown in Figure3. The specimen produces obvious instantaneous deformation when each level of deviator stress is applied. The instantaneous strain of the specimen is the largest when the first loading level is applied, and the subsequent instantaneous deformation increases with the increase of the loading level. When the confining pressure is different, the application of the first-order load has little effect on the instantaneous strain, which is about 0.35%, indicating that the compression effect of about 60% compressive strength on saturated medium-grained sandstone is basically the same.Figure 3
Strain curves in a multiloading triaxial creep test.
(a)
σ3=1.5MPa(b)
σ3=3.0MPa(c)
σ3=4.5MPa(d)
σ3=6.0MPaWhen the deviator stress is less than 75% of the compressive strength, the rock sample only shows instantaneous deformation and decay creep, and the rock deformation reaches a stable state in a short time. When the deviator stress increases to a certain extent, the rock deformation shows obvious time correlation, and it takes a long time for the rock deformation to reach a stable state. With the increase of loading level, the steady-state creep rate of rock (slope of line segment in steady-state deformation stage) gradually increases until accelerated creep occurs. Macroscopic fracture instability occurs in a short time after the start of accelerated creep.


### 3.2. Strain Rate Analysis

The creep rate of rock can indirectly reflect its damage and fracture state. If the internal cracking of rock is serious, the creep rate of rock will increase obviously. The creep rate of rock samples under different deviatoric stresses and confining pressures is shown in Figure4. Under the same confining pressure, the stable creep rate of rock increases with the increase of deviatoric stress. Before the last stage of loading, the creep rate increases linearly with the deviatoric stress. At the last stage of loading, the creep rate increases rapidly, which indicates that the cracks in the rock expand, and the bearing capacity of the rock weakens.Figure 4
Relationship between steady strain rate and deviatoric stress.It can also be seen from Figure4 that the stable creep rate of rock with large lateral restraint is also large, but the difference of creep rate between two adjacent loading levels is slightly smaller. This reminds us that in practical engineering, support and reinforcement can effectively improve confining pressure and long-term stability of the project.


### 3.3. Fracture Characteristics

The damage and fracture of rock are interrelated. In fact, the damage process of rock is a process in which cracks gradually initiate, expand, and coalesce to form macrofractures. The damage in this process grows slowly first and then rapidly to 1 [31]. The fracture morphology of saturated medium-grained sandstone is shown in Figure 3. Shear failure occurs in the rocks under four confining pressures. The rock with high confining pressure has a small fracture angle; that is, when the lateral restraint is small, the rock shear angle is large, resulting in large shear deformation.To further understand the fracture mechanism of medium-grained sandstone, the fracture appearance of rock is scanned, and the scanning image is shown in Figure5. Compared with Figure 1, it can be seen that the fracture of rock is mainly caused by the dislocation of mineral particles, which eventually leads to the formation of macrofracture surface. This is mainly due to the small cementation between mineral particles, which is easy to fail under high pressure and long time. This is also the difference between weakly cemented rock and some macroscopic rock damage caused by mineral particle fracture.Figure 5
Photomicrographs of the fracture appearance of medium-grained sandstone.


### 3.4. Creep Behavior Prediction

#### 3.4.1. Creep Model Analysis

The constitutive model can describe the deformation and failure process of rock. For medium-grained sandstone, the rock presents deceleration creep and steady creep under deviating stress, i.e., linear creep. This creep behavior can be well described by the Burgers model [26]. However, the development of rock failure mainly occurs in the accelerated creep stage, so a Burgers model considering damage can be used to describe the fracture process, as shown in Figure 6.Figure 6
Burgers model considering damage.As we all know, rock fracture propagation is a process of energy accumulation and dissipation [33–38]. When the energy accumulated at the crack tip reaches the energy required for its expansion, the rock fracture will expand. Xie [39] regards the microcracks and microvoids in rock as initial damage and puts forward the rock damage evolution equation based on energy expression.
(1)D=1−exp−αUd−U0dβ,where α and β are the parameters related to the material properties, U0d is the energy dissipation value corresponding to the initial defect, and Ud is the energy dissipation in the process of crack propagation.In the initial stage of loading, the dissipated energyUd of rock is less than U0d, which makes the base part of exponential function Ud−U0d in equation (1) less than zero. That is, the exponential function is not tenable. Therefore, the ratio of Ud to U0d is compared with 1 to judge whether the exponential function is tenable or not. And then, the square of Ud/U0d−1 is taken as the base number of the exponential function, which not only eliminates the meaningless situation of the exponential function but also makes the formula de unit. Thus, the damage evolution equation based on formula (1) is obtained.
(2)D=1−exp−αUdU0d−12β.Under constant axial load, the dissipation energyUd of rock can be expressed as follows [32]:
(3)Ud=21+v3σ1−σ3ε,where σ1 and σ3 are axial pressure and confining pressure, υ is the Poisson’s ratio of rock, and ε is the strain.Therefore, the constitutive equation of the Burgers model considering damage can be expressed as follows [40]:
(4)ε=σ−Dσcr1−D1Em0+1ηm0+1Ek01−e−Ek0/ηk0t,where t is time; Em0, ηm0, Ek0, and ηk0 are viscoelasticity coefficients of the Burgers model; and σcr is the residual strength.The initial damage of rock is closely related to the compaction stage of prepeak stress-strain curve. Because the creep behavior of rock in this study was constrained by confining pressure, the primary void of rock decreases under confining pressure, and the prepeak compaction stage of rock was not obvious. That is, the initial damage decreases. Under uniaxial or very small confining pressure, the initial damage of rock is obvious. The application effect of the modified model in this paper will be better.


#### 3.4.2. Creep Behavior Prediction

It can be seen from Figure3 that the saturated medium-grained sandstone with confining pressure of 3.0 MPa and 6.0 MPa has obvious accelerated creep phenomenon. We use the modified creep model to fit the test data; the related parameters and fitting parameters of the Burgers model with damage are shown in Table 2. The fitting results are shown in Figure 7. The model was found to be suitable to describe the experimental data with good accuracy (R2>0.99). This shows that the modified model can accurately describe the creep behavior of medium-grained sandstone, especially the accelerated creep behavior.Table 2
Creep model parameters of saturated medium-grained sandstone.
σ1‐σ3 (MPa)σ3 (MPa)υσcr (MPa)Em0 (GPa)ηm0 (GPa∗h)E0 k (GPa)ηk0 (GPa∗h)αβU0d (MJ/m3)22.03.00.4010.54.022.42122.14265.496.35∗10−267.28508.7127.06.00.3418.35.213.01151.45636.109.26∗10−478.14120.54Figure 7
Comparison between damage creep model curve with the experimental result of saturated medium-grained sandstone.
(a)
σ3=3.0MPa(b)
σ3=6.0MPaTo explore the evolution law of rock fracture in the process of rheology, the parameters ofU0d, α, and β are introduced into formula (2), and the damage evolution curve under the failure load level can be obtained, as shown in Figure 7. It shows that the damage variable D almost remains unchanged at the stage of deceleration creep and stable creep, which is approximately equal to 0. The damage variable D increases exponentially with the strain in the accelerated creep stage. This indicates that the rock crack almost does not expand in the decay and stable creep stages but expands exponentially in the accelerated creep stage.To study the damage evolution of rock under different loading levels, the creep characteristics of medium-grained sandstone under different deviator stresses are predicted based on the creep constitutive equation (formula (2)-formula (4)). The creep and damage evolution curves are shown in Figures 8 and 9. When the axial loading is not large enough, the creep rate of rock is low, and the damage variable is close to 0. The microcracks will not expand. The accelerated creep phenomenon of rock gradually appears with the increase of deviator stress. The greater the deviator stress is, the greater the steady creep rate is and the earlier the exponential accelerated creep occurs. The point at which the rock enters into the accelerated creep or the point at which the damage begins to increase can be regarded as the point of crack initiation. For example, when the confining pressure is 3.0 MPa and the deviator stress is 23 MPa, the accelerated creep behavior, i.e., the rock begins to fracture, occurs at t=2.85h, and the crack initiation point occurs at t=0.36h when the deviatoric stress is 27 MPa. The creep curve of rock under constant load of 23 MPa is close to that of test under 22 MPa, and the time of steady creep rate increasing slightly and entering accelerated creep is slightly earlier. This is consistent with the objective law, which shows that this method is reliable for predicting the initiation or acceleration creep time of rock materials. We can give early warning of rock and engineering based on this moment.Figure 8
Damage creep model curves of saturated medium-grained sandstone under different deviatoric stress conditions (σ3=3.0MPa).
(a)
Creep curve(b)
Damage evolution curveFigure 9
Damage creep model curves of saturated medium-grained sandstone under different deviatoric stress conditions (σ3=6.0MPa).
(a)
Creep curve(b)
Damage evolution curveBecause we cannot get the internal structure of rock in real time during the experiment, we can only show the failure process of rock visually by virtue of the damage evolution law of rock. Therefore, we simplified the creep fracture process of water-saturated sandstone, as shown in Figure10. Under the low axial loading, the rock only shows the phenomena of attenuation creep and stable creep. The micropores and microcracks of rock are gradually compressed, and no new fracture occurs. When the axial load reaches about 80% of the instantaneous compressive strength, the rock begins to appear accelerated creep. The rock fracture develops exponentially, and finally, macroshear failure occurs.Figure 10
Fracture development process of saturated medium-grained sandstone under multiaxial loadings.


## 3.1. Time-Dependent Strain of Medium-Grained Sandstone

The variation of axial creep deformation of saturated medium-grained sandstone with time is shown in Figure3. The specimen produces obvious instantaneous deformation when each level of deviator stress is applied. The instantaneous strain of the specimen is the largest when the first loading level is applied, and the subsequent instantaneous deformation increases with the increase of the loading level. When the confining pressure is different, the application of the first-order load has little effect on the instantaneous strain, which is about 0.35%, indicating that the compression effect of about 60% compressive strength on saturated medium-grained sandstone is basically the same.Figure 3
Strain curves in a multiloading triaxial creep test.
(a)
σ3=1.5MPa(b)
σ3=3.0MPa(c)
σ3=4.5MPa(d)
σ3=6.0MPaWhen the deviator stress is less than 75% of the compressive strength, the rock sample only shows instantaneous deformation and decay creep, and the rock deformation reaches a stable state in a short time. When the deviator stress increases to a certain extent, the rock deformation shows obvious time correlation, and it takes a long time for the rock deformation to reach a stable state. With the increase of loading level, the steady-state creep rate of rock (slope of line segment in steady-state deformation stage) gradually increases until accelerated creep occurs. Macroscopic fracture instability occurs in a short time after the start of accelerated creep.


## 3.2. Strain Rate Analysis

The creep rate of rock can indirectly reflect its damage and fracture state. If the internal cracking of rock is serious, the creep rate of rock will increase obviously. The creep rate of rock samples under different deviatoric stresses and confining pressures is shown in Figure4. Under the same confining pressure, the stable creep rate of rock increases with the increase of deviatoric stress. Before the last stage of loading, the creep rate increases linearly with the deviatoric stress. At the last stage of loading, the creep rate increases rapidly, which indicates that the cracks in the rock expand, and the bearing capacity of the rock weakens.Figure 4
Relationship between steady strain rate and deviatoric stress.It can also be seen from Figure4 that the stable creep rate of rock with large lateral restraint is also large, but the difference of creep rate between two adjacent loading levels is slightly smaller. This reminds us that in practical engineering, support and reinforcement can effectively improve confining pressure and long-term stability of the project.


## 3.3. Fracture Characteristics

The damage and fracture of rock are interrelated. In fact, the damage process of rock is a process in which cracks gradually initiate, expand, and coalesce to form macrofractures. The damage in this process grows slowly first and then rapidly to 1 [31]. The fracture morphology of saturated medium-grained sandstone is shown in Figure 3. Shear failure occurs in the rocks under four confining pressures. The rock with high confining pressure has a small fracture angle; that is, when the lateral restraint is small, the rock shear angle is large, resulting in large shear deformation.To further understand the fracture mechanism of medium-grained sandstone, the fracture appearance of rock is scanned, and the scanning image is shown in Figure5. Compared with Figure 1, it can be seen that the fracture of rock is mainly caused by the dislocation of mineral particles, which eventually leads to the formation of macrofracture surface. This is mainly due to the small cementation between mineral particles, which is easy to fail under high pressure and long time. This is also the difference between weakly cemented rock and some macroscopic rock damage caused by mineral particle fracture.Figure 5
Photomicrographs of the fracture appearance of medium-grained sandstone.


## 3.4. Creep Behavior Prediction

### 3.4.1. Creep Model Analysis

The constitutive model can describe the deformation and failure process of rock. For medium-grained sandstone, the rock presents deceleration creep and steady creep under deviating stress, i.e., linear creep. This creep behavior can be well described by the Burgers model [26]. However, the development of rock failure mainly occurs in the accelerated creep stage, so a Burgers model considering damage can be used to describe the fracture process, as shown in Figure 6.Figure 6
Burgers model considering damage.As we all know, rock fracture propagation is a process of energy accumulation and dissipation [33–38]. When the energy accumulated at the crack tip reaches the energy required for its expansion, the rock fracture will expand. Xie [39] regards the microcracks and microvoids in rock as initial damage and puts forward the rock damage evolution equation based on energy expression.
(1)D=1−exp−αUd−U0dβ,where α and β are the parameters related to the material properties, U0d is the energy dissipation value corresponding to the initial defect, and Ud is the energy dissipation in the process of crack propagation.In the initial stage of loading, the dissipated energyUd of rock is less than U0d, which makes the base part of exponential function Ud−U0d in equation (1) less than zero. That is, the exponential function is not tenable. Therefore, the ratio of Ud to U0d is compared with 1 to judge whether the exponential function is tenable or not. And then, the square of Ud/U0d−1 is taken as the base number of the exponential function, which not only eliminates the meaningless situation of the exponential function but also makes the formula de unit. Thus, the damage evolution equation based on formula (1) is obtained.
(2)D=1−exp−αUdU0d−12β.Under constant axial load, the dissipation energyUd of rock can be expressed as follows [32]:
(3)Ud=21+v3σ1−σ3ε,where σ1 and σ3 are axial pressure and confining pressure, υ is the Poisson’s ratio of rock, and ε is the strain.Therefore, the constitutive equation of the Burgers model considering damage can be expressed as follows [40]:
(4)ε=σ−Dσcr1−D1Em0+1ηm0+1Ek01−e−Ek0/ηk0t,where t is time; Em0, ηm0, Ek0, and ηk0 are viscoelasticity coefficients of the Burgers model; and σcr is the residual strength.The initial damage of rock is closely related to the compaction stage of prepeak stress-strain curve. Because the creep behavior of rock in this study was constrained by confining pressure, the primary void of rock decreases under confining pressure, and the prepeak compaction stage of rock was not obvious. That is, the initial damage decreases. Under uniaxial or very small confining pressure, the initial damage of rock is obvious. The application effect of the modified model in this paper will be better.


### 3.4.2. Creep Behavior Prediction

It can be seen from Figure3 that the saturated medium-grained sandstone with confining pressure of 3.0 MPa and 6.0 MPa has obvious accelerated creep phenomenon. We use the modified creep model to fit the test data; the related parameters and fitting parameters of the Burgers model with damage are shown in Table 2. The fitting results are shown in Figure 7. The model was found to be suitable to describe the experimental data with good accuracy (R2>0.99). This shows that the modified model can accurately describe the creep behavior of medium-grained sandstone, especially the accelerated creep behavior.Table 2
Creep model parameters of saturated medium-grained sandstone.
σ1‐σ3 (MPa)σ3 (MPa)υσcr (MPa)Em0 (GPa)ηm0 (GPa∗h)E0 k (GPa)ηk0 (GPa∗h)αβU0d (MJ/m3)22.03.00.4010.54.022.42122.14265.496.35∗10−267.28508.7127.06.00.3418.35.213.01151.45636.109.26∗10−478.14120.54Figure 7
Comparison between damage creep model curve with the experimental result of saturated medium-grained sandstone.
(a)
σ3=3.0MPa(b)
σ3=6.0MPaTo explore the evolution law of rock fracture in the process of rheology, the parameters ofU0d, α, and β are introduced into formula (2), and the damage evolution curve under the failure load level can be obtained, as shown in Figure 7. It shows that the damage variable D almost remains unchanged at the stage of deceleration creep and stable creep, which is approximately equal to 0. The damage variable D increases exponentially with the strain in the accelerated creep stage. This indicates that the rock crack almost does not expand in the decay and stable creep stages but expands exponentially in the accelerated creep stage.To study the damage evolution of rock under different loading levels, the creep characteristics of medium-grained sandstone under different deviator stresses are predicted based on the creep constitutive equation (formula (2)-formula (4)). The creep and damage evolution curves are shown in Figures 8 and 9. When the axial loading is not large enough, the creep rate of rock is low, and the damage variable is close to 0. The microcracks will not expand. The accelerated creep phenomenon of rock gradually appears with the increase of deviator stress. The greater the deviator stress is, the greater the steady creep rate is and the earlier the exponential accelerated creep occurs. The point at which the rock enters into the accelerated creep or the point at which the damage begins to increase can be regarded as the point of crack initiation. For example, when the confining pressure is 3.0 MPa and the deviator stress is 23 MPa, the accelerated creep behavior, i.e., the rock begins to fracture, occurs at t=2.85h, and the crack initiation point occurs at t=0.36h when the deviatoric stress is 27 MPa. The creep curve of rock under constant load of 23 MPa is close to that of test under 22 MPa, and the time of steady creep rate increasing slightly and entering accelerated creep is slightly earlier. This is consistent with the objective law, which shows that this method is reliable for predicting the initiation or acceleration creep time of rock materials. We can give early warning of rock and engineering based on this moment.Figure 8
Damage creep model curves of saturated medium-grained sandstone under different deviatoric stress conditions (σ3=3.0MPa).
(a)
Creep curve(b)
Damage evolution curveFigure 9
Damage creep model curves of saturated medium-grained sandstone under different deviatoric stress conditions (σ3=6.0MPa).
(a)
Creep curve(b)
Damage evolution curveBecause we cannot get the internal structure of rock in real time during the experiment, we can only show the failure process of rock visually by virtue of the damage evolution law of rock. Therefore, we simplified the creep fracture process of water-saturated sandstone, as shown in Figure10. Under the low axial loading, the rock only shows the phenomena of attenuation creep and stable creep. The micropores and microcracks of rock are gradually compressed, and no new fracture occurs. When the axial load reaches about 80% of the instantaneous compressive strength, the rock begins to appear accelerated creep. The rock fracture develops exponentially, and finally, macroshear failure occurs.Figure 10
Fracture development process of saturated medium-grained sandstone under multiaxial loadings.


## 3.4.1. Creep Model Analysis

The constitutive model can describe the deformation and failure process of rock. For medium-grained sandstone, the rock presents deceleration creep and steady creep under deviating stress, i.e., linear creep. This creep behavior can be well described by the Burgers model [26]. However, the development of rock failure mainly occurs in the accelerated creep stage, so a Burgers model considering damage can be used to describe the fracture process, as shown in Figure 6.Figure 6
Burgers model considering damage.As we all know, rock fracture propagation is a process of energy accumulation and dissipation [33–38]. When the energy accumulated at the crack tip reaches the energy required for its expansion, the rock fracture will expand. Xie [39] regards the microcracks and microvoids in rock as initial damage and puts forward the rock damage evolution equation based on energy expression.
(1)D=1−exp−αUd−U0dβ,where α and β are the parameters related to the material properties, U0d is the energy dissipation value corresponding to the initial defect, and Ud is the energy dissipation in the process of crack propagation.In the initial stage of loading, the dissipated energyUd of rock is less than U0d, which makes the base part of exponential function Ud−U0d in equation (1) less than zero. That is, the exponential function is not tenable. Therefore, the ratio of Ud to U0d is compared with 1 to judge whether the exponential function is tenable or not. And then, the square of Ud/U0d−1 is taken as the base number of the exponential function, which not only eliminates the meaningless situation of the exponential function but also makes the formula de unit. Thus, the damage evolution equation based on formula (1) is obtained.
(2)D=1−exp−αUdU0d−12β.Under constant axial load, the dissipation energyUd of rock can be expressed as follows [32]:
(3)Ud=21+v3σ1−σ3ε,where σ1 and σ3 are axial pressure and confining pressure, υ is the Poisson’s ratio of rock, and ε is the strain.Therefore, the constitutive equation of the Burgers model considering damage can be expressed as follows [40]:
(4)ε=σ−Dσcr1−D1Em0+1ηm0+1Ek01−e−Ek0/ηk0t,where t is time; Em0, ηm0, Ek0, and ηk0 are viscoelasticity coefficients of the Burgers model; and σcr is the residual strength.The initial damage of rock is closely related to the compaction stage of prepeak stress-strain curve. Because the creep behavior of rock in this study was constrained by confining pressure, the primary void of rock decreases under confining pressure, and the prepeak compaction stage of rock was not obvious. That is, the initial damage decreases. Under uniaxial or very small confining pressure, the initial damage of rock is obvious. The application effect of the modified model in this paper will be better.


## 3.4.2. Creep Behavior Prediction

It can be seen from Figure3 that the saturated medium-grained sandstone with confining pressure of 3.0 MPa and 6.0 MPa has obvious accelerated creep phenomenon. We use the modified creep model to fit the test data; the related parameters and fitting parameters of the Burgers model with damage are shown in Table 2. The fitting results are shown in Figure 7. The model was found to be suitable to describe the experimental data with good accuracy (R2>0.99). This shows that the modified model can accurately describe the creep behavior of medium-grained sandstone, especially the accelerated creep behavior.Table 2
Creep model parameters of saturated medium-grained sandstone.
σ1‐σ3 (MPa)σ3 (MPa)υσcr (MPa)Em0 (GPa)ηm0 (GPa∗h)E0 k (GPa)ηk0 (GPa∗h)αβU0d (MJ/m3)22.03.00.4010.54.022.42122.14265.496.35∗10−267.28508.7127.06.00.3418.35.213.01151.45636.109.26∗10−478.14120.54Figure 7
Comparison between damage creep model curve with the experimental result of saturated medium-grained sandstone.
(a)
σ3=3.0MPa(b)
σ3=6.0MPaTo explore the evolution law of rock fracture in the process of rheology, the parameters ofU0d, α, and β are introduced into formula (2), and the damage evolution curve under the failure load level can be obtained, as shown in Figure 7. It shows that the damage variable D almost remains unchanged at the stage of deceleration creep and stable creep, which is approximately equal to 0. The damage variable D increases exponentially with the strain in the accelerated creep stage. This indicates that the rock crack almost does not expand in the decay and stable creep stages but expands exponentially in the accelerated creep stage.To study the damage evolution of rock under different loading levels, the creep characteristics of medium-grained sandstone under different deviator stresses are predicted based on the creep constitutive equation (formula (2)-formula (4)). The creep and damage evolution curves are shown in Figures 8 and 9. When the axial loading is not large enough, the creep rate of rock is low, and the damage variable is close to 0. The microcracks will not expand. The accelerated creep phenomenon of rock gradually appears with the increase of deviator stress. The greater the deviator stress is, the greater the steady creep rate is and the earlier the exponential accelerated creep occurs. The point at which the rock enters into the accelerated creep or the point at which the damage begins to increase can be regarded as the point of crack initiation. For example, when the confining pressure is 3.0 MPa and the deviator stress is 23 MPa, the accelerated creep behavior, i.e., the rock begins to fracture, occurs at t=2.85h, and the crack initiation point occurs at t=0.36h when the deviatoric stress is 27 MPa. The creep curve of rock under constant load of 23 MPa is close to that of test under 22 MPa, and the time of steady creep rate increasing slightly and entering accelerated creep is slightly earlier. This is consistent with the objective law, which shows that this method is reliable for predicting the initiation or acceleration creep time of rock materials. We can give early warning of rock and engineering based on this moment.Figure 8
Damage creep model curves of saturated medium-grained sandstone under different deviatoric stress conditions (σ3=3.0MPa).
(a)
Creep curve(b)
Damage evolution curveFigure 9
Damage creep model curves of saturated medium-grained sandstone under different deviatoric stress conditions (σ3=6.0MPa).
(a)
Creep curve(b)
Damage evolution curveBecause we cannot get the internal structure of rock in real time during the experiment, we can only show the failure process of rock visually by virtue of the damage evolution law of rock. Therefore, we simplified the creep fracture process of water-saturated sandstone, as shown in Figure10. Under the low axial loading, the rock only shows the phenomena of attenuation creep and stable creep. The micropores and microcracks of rock are gradually compressed, and no new fracture occurs. When the axial load reaches about 80% of the instantaneous compressive strength, the rock begins to appear accelerated creep. The rock fracture develops exponentially, and finally, macroshear failure occurs.Figure 10
Fracture development process of saturated medium-grained sandstone under multiaxial loadings.


## 4. Conclusions

The main purpose of this study is to investigate the triaxial creep behavior of saturated medium-grained sandstone, which represents the extreme state of water-resisting rock mass. The damage variable and creep model were improved to describe the creep behavior of rock and determine the initiation time of rock fracture.(1)
The first-loading instantaneous deformation of saturated rock samples under different confining pressures is similar, and then, the instantaneous deformation increases with the increase of deviator stress. Under low stress level, medium-grained sandstone only shows attenuation creep and stable creep, and accelerated creep occurs when axial load reaches about 75% of instantaneous compressive strength(2)
Shear failure of rock samples occurs under different confining pressures, and the failure mechanism is dislocation separation between mineral particles(3)
The creep rate increases rapidly at the loading level of fracture. Based on the energy dissipation, the damage evolution equation was modified and introduced into the Burgers model, which can accurately describe the rheological behavior of saturated medium-grained sandstone. The modified model was used to predict the crack initiation time under different deviatoric stresses, which can provide guidance for early warning of rock stability(4)
The creep damage model modified in this paper is theoretically suitable for describing other rocks, especially the rocks with rich internal voids. Next, we will focus on the description effect of this model on the creep behavior of other rocks and further modify it to adapt to more rocks



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*Source: 3747027-2023-02-06.xml*