# Research on Energy and Instability Criterion of Sandstone Hydraulic Fracturing in the Three Gorges Reservoir Area, Chongqing, China

Authors: Xiaojun Tang; Hongwei Yang
Journal: Geofluids
 (2023)

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## Abstract

At present, the understanding of structural failure and energy analysis of sandstone under hydrofracture is still insufficient. This time, we selected the sandstone in Chongqing Three Gorges Reservoir Area as the main analysis object to discuss and compare the characteristics of conventional uniaxial (triaxial) compression strength test and uniaxial (triaxial) compression hydraulic fracturing strength test and conduct instability analysis from the perspective of energy. Based on the mechanical characteristics and parameters about the uniaxial (triaxial) compressive strength test and uniaxial (triaxial) compressive hydrofracture strength test, we focus on the analysis of the evolution rules of the hydraulic pressure-strain curve, analyzing the criteria of crack expanding and instability by using energy principle. According to the viewpoint of fracture mechanics, the fracture morphology and failure type of sandstone under hydrofracture were discussed. The test results show that the deformation evolution rules of rock hydrofracture can be divided into four stages, including the characteristics of pore fissure water injection stage (OA), the elastic deformation stage (AB), the volume expansion stage (BC), and global rupture stage (CD). Using the P/C modulus (the ratio of hydraulic pressure and cohesive force, abbreviated as P/C), the ability of hydraulic pressure overcoming cohesive force can be evaluated during hydrofracture. Using energy variables (expressed byEk) about crack expanding from beginning to end, the unexpanded state, critical state, unstable state, and unstable failure of crack can be estimated. There is an interaction between tensile deformation and shear deformation from the crack initial stage to crack expanding stage.


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## Body

## 1. Introduction

The process of hydrofracture can lead to serious deformation and failure of the geo-framework during long operation. Many studies about hydrofracture and crack expanding rules have been carried out at home and abroad.Triaxial test of oil-bearing unconsolidated sandstone in Athabasca area shows that there may be four interaction modes between particles [1]. Alsayed studied the failure of sandstone under different axial pressure, confining pressure, and internal hydraulic pressure [2]. Daneshy found that under the action of hydrofracture, a series of changes will occur along the direction of the minimum resistance, and the shear stress may lead to invalidly shear and form a discontinuous rough fracture surface [3]. Haimson conducted a theoretical and experimental study on the influence of hydrofracture fracture morphology under the action of stress and natural fracture [4]. In terms of large-scale true triaxial simulation test system and test research, Yang et al. conducted hydrofracture tests on shale under different geostress and different displacement conditions, and the fracture propagation rules and final shape were studied by using acoustic emission and CT technology. At the same time, through the development of hydraulic sand fracturing test devices, red sandstone was used to study the migration law of proppants in the process of unconventional natural gas production [5–7]. Using rock samples, Zhao et al. carried out the simulation tests about fracture generation and fracture propagation of hydrofracture, and the geometric shape of the fracture after compression and the change rule of pressure with time during fracturing were obtained [8]. By studying the energy mechanism and the principle of energy dissipation and release in the process of rock fracture, Xie et al. analyzed rock strength and failure criteria from the damage theory [9, 10]. Through the hydrofracture of loose sandstone, Zhang et al. analyzed the physical mechanism of fracture formation, mechanical expansion behavior, expansion mechanism, and deflection mechanism [11].The research on hydrofracture mostly focuses on theoretical analysis, numerical simulation, model, material fracturing, and other aspects, while the research on mechanical mechanism and energy is relatively few; especially in the energy analysis of rock elastoplastic mechanical characteristics on crack initiation and propagation, the research on microfracture mechanism and energy analysis of hydrofracture failure strength and failure criteria require more research. In order to explore the issues of future work, this paper reviews previous studies related to the hydrofracture process related to experiments, the mechanical characteristics, energy changes, and instability criteria of the strength test results of uniaxial (triaxial) compression hydrofracture that are studied.


## 2. Samples and Testing

### 2.1. Rock Sample Preparation and Scheme Design

The sandstone was extracted from the Upper Triassic Xujiahe Formation in the Three Gorges Reservoir area of the Yangtze River. It is a terrigenous fine-grained clastic sedimentary rock with a grain size of 0.1 mm~0.5 mm. It is mainly composed of quartz, feldspar, flint, and muscovite [12]. The size of processed rock sample shall meet the rock mechanics test standard, and it shall be polished into a cylinder Φ50mm×100mm; the height of the test piece and the flatness of the and face meet the requirements. After screening, drill holes have a diameter of Φ6mm and a hole depth of 15 mm (Figures 1(b) and 1(c)).Figure 1
Schematic diagram of sandstone hydrofracture. (a) The RLW-2000M rock mechanical testing system and deformeter. (b) Uniaxial hydraulic failure. (c) Triaxial water pressure failure.
(a)(b)(c)The test device is RLW-2000M microcomputer-controlled coal rock rheometer (Figure1(a)). In order to conduct the hydrofracture test, a set of hydraulic hole sealing pressure plate is designed and manufactured (Figure 1(a)).The test mainly includes hydrofracture test and mechanical property analysis of sandstone under uniaxial (triaxial) compression. The conventional uniaxial and triaxial compression test results of sandstone are used as the comparison basis (Figure2), and the test plan and corresponding parameters are formulated (Table 1).Figure 2
Stress-strain curve. (a) Uniaxial stress-strain curve. (b) Triaxial stress-strain curve.
(a)(b)Table 1
Test programs and parameters.
Test classificationσ1σ2PStrength percentageEμ(MPa)(MPa)(MPa)(%)(GPa)Conventional uniaxial50.9800.0060.07.080.14Conventional triaxial94.34100.0032.411.590.11Uniaxial water pressure30.58012.2060.010.130.17Triaxial water pressure30.551018.8232.412.970.11The results of conventional uniaxial (triaxial) axial failure strength, the axial stress loading size, elastic modulus, and Poisson’s ratio of hydrofracture test under the same strength percentage value were analyzed. In the hydrofracture test, a certain axial constant stress was loaded first, and then hydrofracture was carried out to obtain the water pressure peak failure strength and strain evolution curve.


### 2.2. Curve Analysis

#### 2.2.1. Stress-Strain Curve

According to the test curve (Figure2), the failure curve of conventional uniaxial (triaxial) compression strength conforms to the characteristics of the full stress-strain curve. The axial, transverse, and volumetric strain curves show four typical stages, namely, pore compaction stage, elastic deformation stage, unstable fracture stage, and postfracture stage. The characteristics of confining pressure show that the axial peak failure strength, axial strain, and transverse strain increase with the increase of confining pressure.


#### 2.2.2. Water Pressure-Strain Curve

According to the analysis method based on full stress-strain curve, under uniaxial (triaxial) compression condition, the whole process curve (stress and strain) of rock specimen deformation under water pressure was analyzed. The deformation curve of sandstone under hydrofracture can be divided into four stages (Figure3).
(1)
Water injection stage of pore and fissure (OA stage). The pores or microcracks existing in the rock specimen are filled under the action of low water pressure, forming early nonlinear deformation, and the curves of water pressure-strain and water pressure volume are concave. In this stage, the lateral expansion and volume deformation of the specimen are smaller(2)
Elastic deformation stage of pores and fractures (AB stage). In this stage, the water pressure-strain curve is approximately linear, reflecting the elastic deformation of the pores or fractures of the rock specimen. Under confining pressure, the changes of transverse strain and volumetric strain are also smaller(3)
Volume expansion stage of pore and fissure (BC stage). In this stage, due to the influence of water pressure, the deformation characteristics develop from elasticity to plasticity. The transverse deformation of pores or fissures increases, showing the characteristics of volume expansion, and the transverse strain and volumetric strain rate increase rapidly. Transverse strain and volume strain account for about 85-99%.There is a turning point in the process of extending from axial compression shear deformation to transverse tensile deformation, which can be called the shear tensile point. From the point of view of fracture mechanics, the development of cracks between particles is characterized by the extension of friction shear to tensile splitting failure(4)
Coalescence failure stage of pore and fissure (CD stage). In this stage, the peak value of water pressure decreases rapidly, the internal structure of the rock changes qualitatively, the pore and fissure runs through to failure, the lateral deformation and volume deformation increase rapidly, and the rock-bearing capacity decreases rapidly. The failure mode of rock is the joint failure of splitting and shearing along the macrofracture surface. From the perspective of fracture mechanics, the main role of water pressure in hydrofracture is to lubricate and overcome the friction between particles and play a role of tensile force on the tip of pore fractures. For the complexity and heterogeneity of rocks, the failure process includes the combined effects of shear and tensile deformationFigure 3
Comparison of the pore water pressure and strain curves of hydrofracture. (a) Uniaxial compression hydrofracture. (b) Triaxial compression hydrofracture.
(a)(b)


## 2.1. Rock Sample Preparation and Scheme Design

The sandstone was extracted from the Upper Triassic Xujiahe Formation in the Three Gorges Reservoir area of the Yangtze River. It is a terrigenous fine-grained clastic sedimentary rock with a grain size of 0.1 mm~0.5 mm. It is mainly composed of quartz, feldspar, flint, and muscovite [12]. The size of processed rock sample shall meet the rock mechanics test standard, and it shall be polished into a cylinder Φ50mm×100mm; the height of the test piece and the flatness of the and face meet the requirements. After screening, drill holes have a diameter of Φ6mm and a hole depth of 15 mm (Figures 1(b) and 1(c)).Figure 1
Schematic diagram of sandstone hydrofracture. (a) The RLW-2000M rock mechanical testing system and deformeter. (b) Uniaxial hydraulic failure. (c) Triaxial water pressure failure.
(a)(b)(c)The test device is RLW-2000M microcomputer-controlled coal rock rheometer (Figure1(a)). In order to conduct the hydrofracture test, a set of hydraulic hole sealing pressure plate is designed and manufactured (Figure 1(a)).The test mainly includes hydrofracture test and mechanical property analysis of sandstone under uniaxial (triaxial) compression. The conventional uniaxial and triaxial compression test results of sandstone are used as the comparison basis (Figure2), and the test plan and corresponding parameters are formulated (Table 1).Figure 2
Stress-strain curve. (a) Uniaxial stress-strain curve. (b) Triaxial stress-strain curve.
(a)(b)Table 1
Test programs and parameters.
Test classificationσ1σ2PStrength percentageEμ(MPa)(MPa)(MPa)(%)(GPa)Conventional uniaxial50.9800.0060.07.080.14Conventional triaxial94.34100.0032.411.590.11Uniaxial water pressure30.58012.2060.010.130.17Triaxial water pressure30.551018.8232.412.970.11The results of conventional uniaxial (triaxial) axial failure strength, the axial stress loading size, elastic modulus, and Poisson’s ratio of hydrofracture test under the same strength percentage value were analyzed. In the hydrofracture test, a certain axial constant stress was loaded first, and then hydrofracture was carried out to obtain the water pressure peak failure strength and strain evolution curve.


## 2.2. Curve Analysis

### 2.2.1. Stress-Strain Curve

According to the test curve (Figure2), the failure curve of conventional uniaxial (triaxial) compression strength conforms to the characteristics of the full stress-strain curve. The axial, transverse, and volumetric strain curves show four typical stages, namely, pore compaction stage, elastic deformation stage, unstable fracture stage, and postfracture stage. The characteristics of confining pressure show that the axial peak failure strength, axial strain, and transverse strain increase with the increase of confining pressure.


### 2.2.2. Water Pressure-Strain Curve

According to the analysis method based on full stress-strain curve, under uniaxial (triaxial) compression condition, the whole process curve (stress and strain) of rock specimen deformation under water pressure was analyzed. The deformation curve of sandstone under hydrofracture can be divided into four stages (Figure3).
(1)
Water injection stage of pore and fissure (OA stage). The pores or microcracks existing in the rock specimen are filled under the action of low water pressure, forming early nonlinear deformation, and the curves of water pressure-strain and water pressure volume are concave. In this stage, the lateral expansion and volume deformation of the specimen are smaller(2)
Elastic deformation stage of pores and fractures (AB stage). In this stage, the water pressure-strain curve is approximately linear, reflecting the elastic deformation of the pores or fractures of the rock specimen. Under confining pressure, the changes of transverse strain and volumetric strain are also smaller(3)
Volume expansion stage of pore and fissure (BC stage). In this stage, due to the influence of water pressure, the deformation characteristics develop from elasticity to plasticity. The transverse deformation of pores or fissures increases, showing the characteristics of volume expansion, and the transverse strain and volumetric strain rate increase rapidly. Transverse strain and volume strain account for about 85-99%.There is a turning point in the process of extending from axial compression shear deformation to transverse tensile deformation, which can be called the shear tensile point. From the point of view of fracture mechanics, the development of cracks between particles is characterized by the extension of friction shear to tensile splitting failure(4)
Coalescence failure stage of pore and fissure (CD stage). In this stage, the peak value of water pressure decreases rapidly, the internal structure of the rock changes qualitatively, the pore and fissure runs through to failure, the lateral deformation and volume deformation increase rapidly, and the rock-bearing capacity decreases rapidly. The failure mode of rock is the joint failure of splitting and shearing along the macrofracture surface. From the perspective of fracture mechanics, the main role of water pressure in hydrofracture is to lubricate and overcome the friction between particles and play a role of tensile force on the tip of pore fractures. For the complexity and heterogeneity of rocks, the failure process includes the combined effects of shear and tensile deformationFigure 3
Comparison of the pore water pressure and strain curves of hydrofracture. (a) Uniaxial compression hydrofracture. (b) Triaxial compression hydrofracture.
(a)(b)


## 2.2.1. Stress-Strain Curve

According to the test curve (Figure2), the failure curve of conventional uniaxial (triaxial) compression strength conforms to the characteristics of the full stress-strain curve. The axial, transverse, and volumetric strain curves show four typical stages, namely, pore compaction stage, elastic deformation stage, unstable fracture stage, and postfracture stage. The characteristics of confining pressure show that the axial peak failure strength, axial strain, and transverse strain increase with the increase of confining pressure.


## 2.2.2. Water Pressure-Strain Curve

According to the analysis method based on full stress-strain curve, under uniaxial (triaxial) compression condition, the whole process curve (stress and strain) of rock specimen deformation under water pressure was analyzed. The deformation curve of sandstone under hydrofracture can be divided into four stages (Figure3).
(1)
Water injection stage of pore and fissure (OA stage). The pores or microcracks existing in the rock specimen are filled under the action of low water pressure, forming early nonlinear deformation, and the curves of water pressure-strain and water pressure volume are concave. In this stage, the lateral expansion and volume deformation of the specimen are smaller(2)
Elastic deformation stage of pores and fractures (AB stage). In this stage, the water pressure-strain curve is approximately linear, reflecting the elastic deformation of the pores or fractures of the rock specimen. Under confining pressure, the changes of transverse strain and volumetric strain are also smaller(3)
Volume expansion stage of pore and fissure (BC stage). In this stage, due to the influence of water pressure, the deformation characteristics develop from elasticity to plasticity. The transverse deformation of pores or fissures increases, showing the characteristics of volume expansion, and the transverse strain and volumetric strain rate increase rapidly. Transverse strain and volume strain account for about 85-99%.There is a turning point in the process of extending from axial compression shear deformation to transverse tensile deformation, which can be called the shear tensile point. From the point of view of fracture mechanics, the development of cracks between particles is characterized by the extension of friction shear to tensile splitting failure(4)
Coalescence failure stage of pore and fissure (CD stage). In this stage, the peak value of water pressure decreases rapidly, the internal structure of the rock changes qualitatively, the pore and fissure runs through to failure, the lateral deformation and volume deformation increase rapidly, and the rock-bearing capacity decreases rapidly. The failure mode of rock is the joint failure of splitting and shearing along the macrofracture surface. From the perspective of fracture mechanics, the main role of water pressure in hydrofracture is to lubricate and overcome the friction between particles and play a role of tensile force on the tip of pore fractures. For the complexity and heterogeneity of rocks, the failure process includes the combined effects of shear and tensile deformationFigure 3
Comparison of the pore water pressure and strain curves of hydrofracture. (a) Uniaxial compression hydrofracture. (b) Triaxial compression hydrofracture.
(a)(b)


## 3. Energy and Instability Criteria

### 3.1. Strength Failure Assessment Analysis

According to M-C criterion in rock mechanics [13–15], the peak strength, confining pressure, and fracture angle of uniaxial (triaxial) compression strength test were analyzed (Figure 4). The rock fracture pattern is dominated by single inclined plane shear failure, and there is a small amount of transverse tensile failure in the uniaxial compression fracture. According to the characteristics of the full stress-strain curve and the stress peak value (water pressure peak value) at failure, the basic mechanical parameters (Table 2) can be obtained. In this paper, the ratio of water pressure to bonding force is set as P/C coefficient for analysis, wherein the principal stress is expressed by Coulomb’s criterion as follows.
(1)σ1=1+sinφ1−sinφσ3+2c×cosφ1−sinφ.Figure 4
Uniaxial (triaxial) compression failure characteristics [12]. (a) Uniaxial -∠76°. (b) σ2=10MPa−∠67°.
(a)(b)Table 2
The Mechanical parameters of conventional uniaxial (triaxial) compression failure.
Test classificationFailure characteristicsFracture angleInternal friction angleAdhesive forceP/Cθ, °Φ, °C/MPaConventional uniaxialSingle bevel shearing76626.361.92Conventional triaxialSingle bevel shearing67448.242.28According to the above formula, we have the following.(2)c=1−sinφσ1−1+sinφσ32cosφ.Therefore, the P/C coefficient is(3)P/C=P/1−sinφσ1−1+sinφσ32cosφ=2Pcosφ1−sinφσ1−1+sinφσ3.Therefore, the P/C coefficient can be used to evaluate the ability of water pressure to overcome the bonding force in the process of hydrofracture. The larger the ratio, the more difficult the hydrofracture failure will be. This parameter is related to the axial stress and confining pressure. When the water pressure is constant, the larger the axial stress, the more conducive the hydrofracture is; the higher the confining pressure, the more detrimental it is to the hydrofracture. When the axial stress is constant, the greater the water pressure, the faster the failure; the smaller the confining pressure, the better the hydrofracture. When the axial stress and confining pressure are constant, the greater the water pressure is, the faster the damage is; the internal friction angle will also affect the hydraulic failure.


### 3.2. Analysis of Energy and Instability Criteria

#### 3.2.1. Energy Analysis

Heping et al. [16] studied the concepts of energy dissipation and damage, energy release, and overall failure in the process of deformation and failure of rock mass elements. According to the theory of elastic deformation energy in rock mass [17], the elastic strain energy (U), volume change energy (Uv), and distortion energy (Ud) of rock specimens under uniaxial (triaxial) compression were analyzed (Figure 5).Figure 5
Comparison of energy and axial stress curves. (a) Uniaxial compression failure energy curve. (b) Triaxial compression failure energy curve.
(a)(b)That is(4)U=Ud+Uv,where
(5)U=12Eσ12+σ22+σ32−2μσ1σ2+σ2σ3+σ3σ1,(6)Uv=1−2μ6Eσ1+σ2+σ32,(7)Ud=U−Uv=1+μ6Eσ1−σ22+σ2−σ32+σ3−σ12.It can be seen from the energy comparison curve in Figure5 the total energy provided by elastic deformation to the unclosed system during uniaxial (triaxial) compression failure. Most of the energy forms in the failure process are the distortion energy of shape change. At the same time, the reversible energy of volume change can be converted into surface energy. The elastic deformation energy in the process of uniaxial (triaxial) compression failure is the total energy provided by the unclosed system. In the process of failure, most of the energy forms are the distortion energy of shape change. At the same time, the volume change energy is a reversible energy, which can be converted into surface energy. The prestress energy in the process of uniaxial (triaxial) compression hydrofracture is mainly elastic deformation energy, which is the energy storage process of rock elastic deformation, and reversible energy.


#### 3.2.2. Analysis of Instability Criteria

In order to further analyze the energy change law of hydrofracture process, based on Griffith’s fracture strength theory [18, 19], the energy balance equation between the former total energy (E1) and the total energy at a certain instant (E2) of the nonenergy closed system is as follows:
(8)E2=E1−We+Ws+Wi.Here, the rock is assumed to be linear elastic [20–22], We is the energy released by the rock when the crack appears or expands and can be defined as We=1/2σeεe [23], where σe is the effective stress during crack propagation and εe is the effective strain of crack propagation; Ws is the surface energy required for the formation of new crack surface, which can be defined as Ws=1/2σsεs, where σs is the effective stress when the new surface of the crack propagates and εe is the effective strain of crack new surface propagation; Wi is the energy of the whole system supplemented by the outside during crack growth and can be defined as Wi=σiεi, where σi is constant stress and εi is the instantaneous strain is the instantaneous strain.By comparing the energiesWe, Ws, and Wi (Figure 6), the results show that in the stage of elastic deformation of pores and fractures, the energies Wi and Ws increase linearly, We has little change, and the energy is mainly expressed as the storage of elastic energy of Wi. At the stage of volume expansion of pores and fractures, energy We increases nonlinearly. The main form of energy is the dissipated energy We released by the rock when cracks appear or grow.Figure 6
Comparison of the energy curves of hydrofracture. (a) Energy comparison curve of uniaxial compression hydrofracture. (b) Energy comparison curve of triaxial compression hydrofracture.
(a)(b)If the energy change of the whole system before and after crack propagation is set, which is the kinetic energy that promotes crack growth, the equation is as follows [24–28]:
(9)Ek=E1−E2=We−Ws−Wi.The physical meaning of the equation is the energy instability failure criterion of rock during crack propagation. WhenEk<0, the whole system is in the process of energy storage, and the crack is in the state of nonpropagation. When Ek=0, the change rate of kinetic energy is equal to zero, and the crack is in the critical state of growth. When Ek>0, the change rate of kinetic energy is greater than zero, the system is in the process of energy release, and the crack propagation is in an unstable state, which is easy to cause crack instability and damage.In order to discuss the law of hydrofracture energy change, the comparison curve of water pressureP, energy change Ek, and water pressure volume Vp was analyzed. It can be seen from Figure 7(a) that the energy change before and after crack propagation in the process of uniaxial compression hydrofracture is less than 0 Ek<0 in OA and AB stages, which shows that the energy of crack propagation at this stage is small, and the energy provided by the outside is converted into internal energy (stored energy). At the transition point B Ek=0 between the elastic deformation stage and the volume expansion stage of the pore fissure, this point indicates that the crack is in the critical state of expansion, which shows that the energy change before and after the crack growth is balanced with the energy provided by the outside world and the surface energy required for the formation of the new surface of the crack. The change rate of the kinetic energy is equal to zero, and the water volume accounts for 74.9% of the whole volume. In the expansion stage of pore fissure volume, when Ek>0, the change rate of kinetic energy is greater than zero, and the system is in the process of energy release. The internal energy between particles releases energy to the crack propagation. At this stage, the energy is unstable, and the crack grows rapidly, which is easy to be unstable and destroyed.Figure 7
Comparison of the pore water pressure and energy and water volume curves of hydrofracture. (a) Uniaxial compression hydrofracture curve. (b) Triaxial compression hydrofracture curve.
(a)(b)As shown in Figure7(b), the evolution law of energy change before and after crack growth in the triaxial compression hydrofracture process is similar to that in the uniaxial compression hydrofracture process. Due to the influence of confining pressure, the volume of pore fracture is in the expansion stage when the crack is at the critical point of expansion Ek=0, and the water volume is 81.5% of the total volume.Comparatively speaking, the energy change before and after crack propagation in the triaxial compression hydrofracture process is larger than that in the uniaxial compression hydrofracture process.


## 3.1. Strength Failure Assessment Analysis

According to M-C criterion in rock mechanics [13–15], the peak strength, confining pressure, and fracture angle of uniaxial (triaxial) compression strength test were analyzed (Figure 4). The rock fracture pattern is dominated by single inclined plane shear failure, and there is a small amount of transverse tensile failure in the uniaxial compression fracture. According to the characteristics of the full stress-strain curve and the stress peak value (water pressure peak value) at failure, the basic mechanical parameters (Table 2) can be obtained. In this paper, the ratio of water pressure to bonding force is set as P/C coefficient for analysis, wherein the principal stress is expressed by Coulomb’s criterion as follows.
(1)σ1=1+sinφ1−sinφσ3+2c×cosφ1−sinφ.Figure 4
Uniaxial (triaxial) compression failure characteristics [12]. (a) Uniaxial -∠76°. (b) σ2=10MPa−∠67°.
(a)(b)Table 2
The Mechanical parameters of conventional uniaxial (triaxial) compression failure.
Test classificationFailure characteristicsFracture angleInternal friction angleAdhesive forceP/Cθ, °Φ, °C/MPaConventional uniaxialSingle bevel shearing76626.361.92Conventional triaxialSingle bevel shearing67448.242.28According to the above formula, we have the following.(2)c=1−sinφσ1−1+sinφσ32cosφ.Therefore, the P/C coefficient is(3)P/C=P/1−sinφσ1−1+sinφσ32cosφ=2Pcosφ1−sinφσ1−1+sinφσ3.Therefore, the P/C coefficient can be used to evaluate the ability of water pressure to overcome the bonding force in the process of hydrofracture. The larger the ratio, the more difficult the hydrofracture failure will be. This parameter is related to the axial stress and confining pressure. When the water pressure is constant, the larger the axial stress, the more conducive the hydrofracture is; the higher the confining pressure, the more detrimental it is to the hydrofracture. When the axial stress is constant, the greater the water pressure, the faster the failure; the smaller the confining pressure, the better the hydrofracture. When the axial stress and confining pressure are constant, the greater the water pressure is, the faster the damage is; the internal friction angle will also affect the hydraulic failure.


## 3.2. Analysis of Energy and Instability Criteria

### 3.2.1. Energy Analysis

Heping et al. [16] studied the concepts of energy dissipation and damage, energy release, and overall failure in the process of deformation and failure of rock mass elements. According to the theory of elastic deformation energy in rock mass [17], the elastic strain energy (U), volume change energy (Uv), and distortion energy (Ud) of rock specimens under uniaxial (triaxial) compression were analyzed (Figure 5).Figure 5
Comparison of energy and axial stress curves. (a) Uniaxial compression failure energy curve. (b) Triaxial compression failure energy curve.
(a)(b)That is(4)U=Ud+Uv,where
(5)U=12Eσ12+σ22+σ32−2μσ1σ2+σ2σ3+σ3σ1,(6)Uv=1−2μ6Eσ1+σ2+σ32,(7)Ud=U−Uv=1+μ6Eσ1−σ22+σ2−σ32+σ3−σ12.It can be seen from the energy comparison curve in Figure5 the total energy provided by elastic deformation to the unclosed system during uniaxial (triaxial) compression failure. Most of the energy forms in the failure process are the distortion energy of shape change. At the same time, the reversible energy of volume change can be converted into surface energy. The elastic deformation energy in the process of uniaxial (triaxial) compression failure is the total energy provided by the unclosed system. In the process of failure, most of the energy forms are the distortion energy of shape change. At the same time, the volume change energy is a reversible energy, which can be converted into surface energy. The prestress energy in the process of uniaxial (triaxial) compression hydrofracture is mainly elastic deformation energy, which is the energy storage process of rock elastic deformation, and reversible energy.


### 3.2.2. Analysis of Instability Criteria

In order to further analyze the energy change law of hydrofracture process, based on Griffith’s fracture strength theory [18, 19], the energy balance equation between the former total energy (E1) and the total energy at a certain instant (E2) of the nonenergy closed system is as follows:
(8)E2=E1−We+Ws+Wi.Here, the rock is assumed to be linear elastic [20–22], We is the energy released by the rock when the crack appears or expands and can be defined as We=1/2σeεe [23], where σe is the effective stress during crack propagation and εe is the effective strain of crack propagation; Ws is the surface energy required for the formation of new crack surface, which can be defined as Ws=1/2σsεs, where σs is the effective stress when the new surface of the crack propagates and εe is the effective strain of crack new surface propagation; Wi is the energy of the whole system supplemented by the outside during crack growth and can be defined as Wi=σiεi, where σi is constant stress and εi is the instantaneous strain is the instantaneous strain.By comparing the energiesWe, Ws, and Wi (Figure 6), the results show that in the stage of elastic deformation of pores and fractures, the energies Wi and Ws increase linearly, We has little change, and the energy is mainly expressed as the storage of elastic energy of Wi. At the stage of volume expansion of pores and fractures, energy We increases nonlinearly. The main form of energy is the dissipated energy We released by the rock when cracks appear or grow.Figure 6
Comparison of the energy curves of hydrofracture. (a) Energy comparison curve of uniaxial compression hydrofracture. (b) Energy comparison curve of triaxial compression hydrofracture.
(a)(b)If the energy change of the whole system before and after crack propagation is set, which is the kinetic energy that promotes crack growth, the equation is as follows [24–28]:
(9)Ek=E1−E2=We−Ws−Wi.The physical meaning of the equation is the energy instability failure criterion of rock during crack propagation. WhenEk<0, the whole system is in the process of energy storage, and the crack is in the state of nonpropagation. When Ek=0, the change rate of kinetic energy is equal to zero, and the crack is in the critical state of growth. When Ek>0, the change rate of kinetic energy is greater than zero, the system is in the process of energy release, and the crack propagation is in an unstable state, which is easy to cause crack instability and damage.In order to discuss the law of hydrofracture energy change, the comparison curve of water pressureP, energy change Ek, and water pressure volume Vp was analyzed. It can be seen from Figure 7(a) that the energy change before and after crack propagation in the process of uniaxial compression hydrofracture is less than 0 Ek<0 in OA and AB stages, which shows that the energy of crack propagation at this stage is small, and the energy provided by the outside is converted into internal energy (stored energy). At the transition point B Ek=0 between the elastic deformation stage and the volume expansion stage of the pore fissure, this point indicates that the crack is in the critical state of expansion, which shows that the energy change before and after the crack growth is balanced with the energy provided by the outside world and the surface energy required for the formation of the new surface of the crack. The change rate of the kinetic energy is equal to zero, and the water volume accounts for 74.9% of the whole volume. In the expansion stage of pore fissure volume, when Ek>0, the change rate of kinetic energy is greater than zero, and the system is in the process of energy release. The internal energy between particles releases energy to the crack propagation. At this stage, the energy is unstable, and the crack grows rapidly, which is easy to be unstable and destroyed.Figure 7
Comparison of the pore water pressure and energy and water volume curves of hydrofracture. (a) Uniaxial compression hydrofracture curve. (b) Triaxial compression hydrofracture curve.
(a)(b)As shown in Figure7(b), the evolution law of energy change before and after crack growth in the triaxial compression hydrofracture process is similar to that in the uniaxial compression hydrofracture process. Due to the influence of confining pressure, the volume of pore fracture is in the expansion stage when the crack is at the critical point of expansion Ek=0, and the water volume is 81.5% of the total volume.Comparatively speaking, the energy change before and after crack propagation in the triaxial compression hydrofracture process is larger than that in the uniaxial compression hydrofracture process.


## 3.2.1. Energy Analysis

Heping et al. [16] studied the concepts of energy dissipation and damage, energy release, and overall failure in the process of deformation and failure of rock mass elements. According to the theory of elastic deformation energy in rock mass [17], the elastic strain energy (U), volume change energy (Uv), and distortion energy (Ud) of rock specimens under uniaxial (triaxial) compression were analyzed (Figure 5).Figure 5
Comparison of energy and axial stress curves. (a) Uniaxial compression failure energy curve. (b) Triaxial compression failure energy curve.
(a)(b)That is(4)U=Ud+Uv,where
(5)U=12Eσ12+σ22+σ32−2μσ1σ2+σ2σ3+σ3σ1,(6)Uv=1−2μ6Eσ1+σ2+σ32,(7)Ud=U−Uv=1+μ6Eσ1−σ22+σ2−σ32+σ3−σ12.It can be seen from the energy comparison curve in Figure5 the total energy provided by elastic deformation to the unclosed system during uniaxial (triaxial) compression failure. Most of the energy forms in the failure process are the distortion energy of shape change. At the same time, the reversible energy of volume change can be converted into surface energy. The elastic deformation energy in the process of uniaxial (triaxial) compression failure is the total energy provided by the unclosed system. In the process of failure, most of the energy forms are the distortion energy of shape change. At the same time, the volume change energy is a reversible energy, which can be converted into surface energy. The prestress energy in the process of uniaxial (triaxial) compression hydrofracture is mainly elastic deformation energy, which is the energy storage process of rock elastic deformation, and reversible energy.


## 3.2.2. Analysis of Instability Criteria

In order to further analyze the energy change law of hydrofracture process, based on Griffith’s fracture strength theory [18, 19], the energy balance equation between the former total energy (E1) and the total energy at a certain instant (E2) of the nonenergy closed system is as follows:
(8)E2=E1−We+Ws+Wi.Here, the rock is assumed to be linear elastic [20–22], We is the energy released by the rock when the crack appears or expands and can be defined as We=1/2σeεe [23], where σe is the effective stress during crack propagation and εe is the effective strain of crack propagation; Ws is the surface energy required for the formation of new crack surface, which can be defined as Ws=1/2σsεs, where σs is the effective stress when the new surface of the crack propagates and εe is the effective strain of crack new surface propagation; Wi is the energy of the whole system supplemented by the outside during crack growth and can be defined as Wi=σiεi, where σi is constant stress and εi is the instantaneous strain is the instantaneous strain.By comparing the energiesWe, Ws, and Wi (Figure 6), the results show that in the stage of elastic deformation of pores and fractures, the energies Wi and Ws increase linearly, We has little change, and the energy is mainly expressed as the storage of elastic energy of Wi. At the stage of volume expansion of pores and fractures, energy We increases nonlinearly. The main form of energy is the dissipated energy We released by the rock when cracks appear or grow.Figure 6
Comparison of the energy curves of hydrofracture. (a) Energy comparison curve of uniaxial compression hydrofracture. (b) Energy comparison curve of triaxial compression hydrofracture.
(a)(b)If the energy change of the whole system before and after crack propagation is set, which is the kinetic energy that promotes crack growth, the equation is as follows [24–28]:
(9)Ek=E1−E2=We−Ws−Wi.The physical meaning of the equation is the energy instability failure criterion of rock during crack propagation. WhenEk<0, the whole system is in the process of energy storage, and the crack is in the state of nonpropagation. When Ek=0, the change rate of kinetic energy is equal to zero, and the crack is in the critical state of growth. When Ek>0, the change rate of kinetic energy is greater than zero, the system is in the process of energy release, and the crack propagation is in an unstable state, which is easy to cause crack instability and damage.In order to discuss the law of hydrofracture energy change, the comparison curve of water pressureP, energy change Ek, and water pressure volume Vp was analyzed. It can be seen from Figure 7(a) that the energy change before and after crack propagation in the process of uniaxial compression hydrofracture is less than 0 Ek<0 in OA and AB stages, which shows that the energy of crack propagation at this stage is small, and the energy provided by the outside is converted into internal energy (stored energy). At the transition point B Ek=0 between the elastic deformation stage and the volume expansion stage of the pore fissure, this point indicates that the crack is in the critical state of expansion, which shows that the energy change before and after the crack growth is balanced with the energy provided by the outside world and the surface energy required for the formation of the new surface of the crack. The change rate of the kinetic energy is equal to zero, and the water volume accounts for 74.9% of the whole volume. In the expansion stage of pore fissure volume, when Ek>0, the change rate of kinetic energy is greater than zero, and the system is in the process of energy release. The internal energy between particles releases energy to the crack propagation. At this stage, the energy is unstable, and the crack grows rapidly, which is easy to be unstable and destroyed.Figure 7
Comparison of the pore water pressure and energy and water volume curves of hydrofracture. (a) Uniaxial compression hydrofracture curve. (b) Triaxial compression hydrofracture curve.
(a)(b)As shown in Figure7(b), the evolution law of energy change before and after crack growth in the triaxial compression hydrofracture process is similar to that in the uniaxial compression hydrofracture process. Due to the influence of confining pressure, the volume of pore fracture is in the expansion stage when the crack is at the critical point of expansion Ek=0, and the water volume is 81.5% of the total volume.Comparatively speaking, the energy change before and after crack propagation in the triaxial compression hydrofracture process is larger than that in the uniaxial compression hydrofracture process.


## 4. Analysis of Hydrofracture Failure Characteristics

The mechanism of hydrofracture includes fracture initiation and fracture propagation [29–31]. By studying the corresponding relationship between water pressure and displacement field, the failure conditions of rock hydrofracture are analyzed. The hydrofracture strength of rock is much lower than the peak strength. According to the crack effect in rock fracture mechanics, the hydrofracture belongs to “low stress fracture” [32–34].


### 4.1. Fracture Morphology of Uniaxial Compression Hydrofracture

When the rock is under uniaxial compression, the axial stress is 60% of the peak strength, which is lower than the yield stress and belongs to the elastic deformation stage. When water pressure is applied, the transverse deformation is changed from shear deformation to tensile deformation under the action of water pressure. Figure8(a) shows that in the failure mode, the end face cracks symmetrically and parallel outward along the borehole, there are branch cracks, and the cracks crack with dislocation; the outer surface of the specimen has a semipenetrating vertical crack, cracking from the end face to the lower part of the specimen, and the crack has a rotating phenomenon. It can be judged that the fracture form of the crack is type I, namely, open type.Figure 8
Fracture morphology of sandstone hydrofracture. (a) Hydrofracture with uniaxial compression. (b) Diagram of triaxial compression hydrofracture.
(a)(b)


### 4.2. Fracture Morphology of Triaxial Compression Hydrofracture

When the rock is in the triaxial compression state, the axial stress is 32.4% of the peak strength, which is far lower than the yield stress. The confining pressure is 10 MPa, which belongs to the elastic deformation stage. When the applied water pressure is lower than the confining pressure, the lateral effective stress is the pressure effect, which shows that the lateral deformation and volume deformation are small, the stress state is compressive elastic deformation, and the rock shows mainly shear deformation; when the applied water pressure is greater than the confining pressure, the transverse effective stress is tension, which shows that the transverse deformation and volume deformation gradually increase, and the rock changes from shear deformation to tensile deformation.Therefore, Figure8(b) shows that there are cracks with an included angle of 90° on the end face outside the borehole in the failure mode, and the cracks are staggered. The test piece has fully penetrated the nonlinear crack, and the crack initiation point may start from the bottom of the borehole, showing a vertical upward and downward tensile cracking. At the same time, the upper and lower cracks showed obvious dislocation during the cracking process, and particle friction traces (particle powder) were observed at the dislocation position. The upper crack is affected by the end drilling and stress transition state, and the crack cracking angle is relatively large. The cracking angle of the lower crack in the particle is relatively small. It can be judged that the fracture form of the crack is I and II compounds, that is, open and staggered.


## 4.1. Fracture Morphology of Uniaxial Compression Hydrofracture

When the rock is under uniaxial compression, the axial stress is 60% of the peak strength, which is lower than the yield stress and belongs to the elastic deformation stage. When water pressure is applied, the transverse deformation is changed from shear deformation to tensile deformation under the action of water pressure. Figure8(a) shows that in the failure mode, the end face cracks symmetrically and parallel outward along the borehole, there are branch cracks, and the cracks crack with dislocation; the outer surface of the specimen has a semipenetrating vertical crack, cracking from the end face to the lower part of the specimen, and the crack has a rotating phenomenon. It can be judged that the fracture form of the crack is type I, namely, open type.Figure 8
Fracture morphology of sandstone hydrofracture. (a) Hydrofracture with uniaxial compression. (b) Diagram of triaxial compression hydrofracture.
(a)(b)


## 4.2. Fracture Morphology of Triaxial Compression Hydrofracture

When the rock is in the triaxial compression state, the axial stress is 32.4% of the peak strength, which is far lower than the yield stress. The confining pressure is 10 MPa, which belongs to the elastic deformation stage. When the applied water pressure is lower than the confining pressure, the lateral effective stress is the pressure effect, which shows that the lateral deformation and volume deformation are small, the stress state is compressive elastic deformation, and the rock shows mainly shear deformation; when the applied water pressure is greater than the confining pressure, the transverse effective stress is tension, which shows that the transverse deformation and volume deformation gradually increase, and the rock changes from shear deformation to tensile deformation.Therefore, Figure8(b) shows that there are cracks with an included angle of 90° on the end face outside the borehole in the failure mode, and the cracks are staggered. The test piece has fully penetrated the nonlinear crack, and the crack initiation point may start from the bottom of the borehole, showing a vertical upward and downward tensile cracking. At the same time, the upper and lower cracks showed obvious dislocation during the cracking process, and particle friction traces (particle powder) were observed at the dislocation position. The upper crack is affected by the end drilling and stress transition state, and the crack cracking angle is relatively large. The cracking angle of the lower crack in the particle is relatively small. It can be judged that the fracture form of the crack is I and II compounds, that is, open and staggered.


## 5. Conclusions

(1)
According to the analysis method of full stress-strain curve, the deformation evolution law of rock during hydraulic fracturing can be divided into four stages (pore fissure water injection stage (OA stage); elastic deformation stage of pore fissure (AB stage); pore fissure volume expansion stage (BC stage); and pore fracture through fracture stage (CD stage)). Through analysis, the shear tension point is obtained at the volume expansion stage (BC segment) of the pore fracture, that is, the interparticle or fracture expands from frictional shear to tensile splitting failure(2)
According to the M-C criterion in rock mechanics, the P/C coefficient (ratio of water pressure to cohesive force) is obtained through analysis, which can evaluate the ability of water pressure to overcome the cohesive force during hydraulic fracturing. The larger the ratio, the more difficult the hydraulic fracturing is to be damaged. This parameter is mainly related to the axial stress and confining pressure(3)
According to theP−Ek−Vp relation curve, the instability failure criterion of rock during crack propagation is discussed. The following relationships exist. When Ek<0 (energy change Ek), the whole system is in the energy storage stage and the crack does not propagate. When Ek=0, the change rate of kinetic energy is equal to zero, and the crack is in the critical state of propagation. When Ek>0, the change rate of kinetic energy is greater than zero, the system is in the process of energy release, and the crack propagation is in an unstable state, which is easy to cause crack instability and damage(4)
The fracture morphology of rock hydraulic fracturing is analyzed. The fracture form of uniaxial compression hydraulic fracturing is type I, that is, open type. The fracture forms of triaxial compression hydraulic fracturing are I and II composites, that is, open and staggered. It shows that there is an interaction between tensile deformation and shear deformation in the process of fracture initiation and fracture extension of sandstone hydraulic fracturing



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*Source: 9985452-2023-01-25.xml*