Strength Parameters and Failure Criterion of Granite After High-Temperature and Water-Cooling Treatment

Abstract

Granite is the main rock type in hot dry rock reservoirs, and hydraulic fracturing is always required during the process of geothermal production. It is necessary to understand the strength parameters and failure criterion of granite after high-temperature and water-cooling treatment. In this paper, laboratory uniaxial and triaxial compression experiments are carried out on granite samples after high-temperature and water-cooling treatment. Combined with some experimental data collected from pre-existing studies, the variation behaviors of cohesion (c), the internal friction angle ($\phi$) and tensile strength $\left({\sigma }_{\mathrm{t}}\right)$ are systematically studied considering the heating and cooling treatment. It is found that c and $\phi$ generally show two different types of variation behaviors with the increasing heating temperature. Tensile strength decreases in a similar way for the different granite samples with the increasing treatment temperature. Empirical equations are provided to describe these strength parameters. Finally, a modified Mohr–Coulomb failure criterion with a “tension cut-off” is established for the granite samples, considering the effects of high-temperature and water-cooling treatment. This study should be helpful for understanding the mechanical behavior of hot dry rock during hydraulic fracturing in geothermal production, and the proposed failure criterion can be applied for the numerical modeling of reservoirs.

1. Introduction

As a main rock type in hot dry rock reservoirs, granite is an intrusive rock among acidic igneous rocks, and its main minerals are quartz, potassium feldspar, acidic plagioclase, etc., which are thought to be products of the partial melting of middle–lower crustal protoliths, during orogenesis and crustal thickening or thinning (I-type, S-type, A-type, M-type) [1,2], and it is always characterized by high strength, compactness and low permeability. The temperature of dry hot rock can reach 150–650 °C [3], and the exploitation technique is mainly based on the enhanced geothermal system. During the production process, the stability of wellbore in a high-temperature reservoir may be affected by the low-temperature drilling fluid [3,4]. In addition, the hydraulic fracturing process is also influenced by the mechanical properties of dry hot rock cooled by the low-temperature water or other fluids [5,6,7,8,9,10,11]. During the long-term geological disposal of high-level nuclear waste buried in a repository [12,13,14], it should also be taken into account that the surrounding rock mass heated by nuclear waste may be cooled by groundwater. Under the geological conditions of high temperature and high pressure, the above-mentioned engineering rock mass is subject to the complex coupling effect of temperature field, seepage field and stress field when in contact with low-temperature fluid. In order to determine the mechanical properties of rock mass under this condition, it is necessary to study certain strength parameters (cohesion and the internal friction angle) and the failure criterion of water-cooled granite under high temperature. This would be helpful for understanding the mechanism of hydraulic fracturing and wellbore stability problems during water injection in hot dry rock reservoirs.

In recent years, some experimental studies have been conducted on the physical and mechanical properties of granite samples after high-temperature and water-cooling treatment. The main research results are as follows:

(1) Physical properties: It has been found that the mass, volume and density of granite samples decrease with an increase in heating temperature [15,16,17]. In addition, it has been observed in some studies that the P-wave velocity of granite does not change obviously when the heating temperature is lower than 200 °C, while it decreases significantly with an increase in temperature from 200 °C to 800 °C [15,16,17,18], but some other studies have shown that the P-wave velocity decreases monotonically with increasing temperature [19,20]. There are also some studies on the porosity and permeability of granite after high-temperature and water-cooling treatment [21], and they find that porosity or permeability does not change significantly when the treatment temperature is lower than a threshold value (400–500 °C), while it increases obviously when the temperature is higher than this threshold [15,19,22,23].

(2) Mechanical properties: Some studies [19,24,25,26] show that the uniaxial compressive strength and Young’s modulus of granite samples decrease monotonously with the treatment temperature. However, some other studies [17,18,27] report that the uniaxial compressive strength and Young’s modulus generally remain constant or increase slightly when the treatment temperature is lower than a threshold value (200–400 °C), while they decrease significantly when the temperature is higher than the threshold. In addition, triaxial compressive test results show that the compressive strength of the granite samples increases when the temperature increases from 25 °C to 300 °C, and it decreases when the temperature changes from 300 °C to 900 °C, and the range of strength variation decreases significantly with the enhancement of confining pressure [15]. Different experimental studies show that the tensile strength of granite samples decrease monotonously with increasing treatment temperature [17,25,26]. It can also be observed that the variation trend of Poisson’s ratio is not consistent with the increasing treatment temperature [15,17,24].

(3) The initiation and propagation of micro cracks: The scanning electron microscopy (SEM) technique is used on rock samples, and the propagation characteristics of inter-granular and trans-granular cracks related to the treatment temperatures have been studied [15,16,24]. The CT scanning of granite samples shows that water cooling significantly improves the pore space of granite and has an influence on micro and macro cracks [28]. Acoustic emission (AE) analyses are also applied in experiments, and the results suggest that heating temperature and cooling methods may have influence on crack propagation in the samples [22,24]. Dynamic SHPB experiment results show that the heating temperature and cooling methods have influence on the fracturing patterns of granite samples [19,21].

However, systematic studies on the evolution of strength parameters (cohesion and the internal friction angle) of granite samples after high-temperature and water-cooling treatment are still required, and it is necessary to create a failure criterion considering the influence of treatment temperature. Therefore, in Section 2, high-temperature heating–water cooling experiments on Fangshan granite samples are analyzed, and some more experimental results from the published researches are combined to study the evolution characteristics of the strength parameters. Based on the test results, a modified Mohr-Coulomb failure criterion with a “tension cut-off” has been proposed for the granite samples considering the treatment temperature in Section 3 and Section 4. Finally, Section 5 provides a discussion on the mechanism of the strength parameter variations.

2. Mohr-Coulomb Strength Parameters of Granite Samples After High-Temperature and Water-Cooling Treatment

2.1. Laboratory Experiments on Fangshan Granite Samples

2.1.1. Granite Samples and Experimental Methods

The granite samples were collected from a quarry in Fangshan, Beijing, and the cylindrical specimens with diameter of 25 mm and height of 50 mm (Figure 1) were well prepared for the experiments in this study. The samples are unweathered, with the color of gray white, and no obvious cracks can be observed by naked eyes. The average density is about 2.72 g/cm³. According to XRD analysis, the samples have the mineral composition of quartz (38%), plagioclase (47%), microcline (11%), and mica (4%) [29].

The main apparatus used in this experimental study includes: (1) SG-XL1200 furnace (developed by SIOM in Shanghai, China) with the maximum heating temperature of 1200 °C, and the heating rate can be controlled with a real-time temperature control accuracy of ±3 °C; (2) DHG-9053A drying oven (produced by Yiheng Equipment, Shanghai, China) with the temperature control accuracy of ±2 °C; (3) TAW-2000 servo-controlled triaxial testing system for rock specimens (produced by Chaoyang Test Instrument Co., Ltd., Changchun, China).

In this experiment, the Fangshan granite samples were first heated to the target temperatures (200 °C, 400 °C, 600 °C, 800 °C) at a constant rate of 3 °C/min, then the target temperature was maintained for each of the rock samples for 2 h. Thereafter, the rock samples were immediately put into water (initial water temperature was 20 °C, water volume was about 20 L) and cooled for 2 h in order to make the rock samples fully cooled. Finally, the rock samples were dried at 103 °C for 48 h, and a series of uniaxial and triaxial compression tests (σ₂ = σ₃ = 0, 10, 20, and 30 MPa) were carried out on these samples. For the uniaxial test, σ₁ increases continuously with σ₂ and σ₃ remaining zero until the failure of the sample occurs. For the triaxial test, the cylindrical sample is put into the loading cell firstly, and the confining pressure is applied with oil, until the initial stress state (σ₁ = σ₂ = σ₃) is reached, thereafter the confining pressures (σ₂ = σ₃) is kept constant, and the axial stress σ₁ is increased until the failure of the sample occurs.

2.1.2. Experimental Results

The experimental results of Fangshan granite samples after high-temperature and water-cooling treatment are presented in

Table 1. Experimental results of Fangshan granite samples after high-temperature and water-cooling treatment.

Confining Pressure (MPa)	Peak Deviatoric Stress (σ1–σ3) (MPa)
	20 °C 1	200 °C	400 °C	600 °C	800 °C
0	79.9	77.8	84.3	51.9	31.0
10	229.3	209.7	247.3	244.0	161.0
20	317.7	310.7	341.7	263.0	246.7
30	392.7	397.0	425.0	379.7	219.0

¹ 20 °C here means that these samples were tested under room temperature for a comparison study. The other temperatures (200~800 °C) are the treatment temperatures before water cooling.

. In order for a comparison study, the samples under natural condition were also applied in the experiments under different confining pressures, and the corresponding test results were recorded under the condition of 20 °C (room temperature). According to Mohr-Coulomb criterion shown in Equation (1), cohesion and the internal friction angle of the granite samples under different treatment conditions were calculated and shown in

Table 2. Cohesion and the internal friction angle of Fangshan granite samples after high-temperature and water-cooling treatment.

Heating Temperature (°C)	c (MPa)	φ (°)	R2
20	15.7	55.3	0.97
200	13.8	55.8	0.99
400	16.0	56.7	0.97
600	13.3	54.9	0.91
800	13.1	47.2	0.77

, and the results were plotted in Figure 2.

$${\sigma }_1={\sigma }_3{\displaystyle \frac{1+\mathrm{s}\mathrm{i}\mathrm{n}\phi }{1-\mathrm{s}\mathrm{i}\mathrm{n}\phi }}+{\displaystyle \frac{2c\mathrm{c}\mathrm{o}\mathrm{s}\phi }{1-\mathrm{s}\mathrm{i}\mathrm{n}\phi }}$$

(1)

where c and φ are cohesion and the internal friction angle of the samples, respectively.

As presented in Table 2 and Figure 1, after heating and water-cooling treatment, cohesion of Fangshan granite does not change significantly within the heating range from 20 °C (room temperature) to 400 °C, while it decreases by 15.3% and 16.6% under the conditions of 600 °C and 800 °C, respectively, compared with the value under room temperature. The internal friction angle almost remains constant when the temperature is not higher than 600 °C, and it decreases slightly at 800 °C, where it is 14.6% lower than that at room temperature.

2.2. Evolution Behaviors of Mohr-Coulomb Strength Parameters

2.2.1. Extended Data from Different Granite Samples

In order to obtain the evolution behaviors of strength parameters of granite after high-temperature and water-cooling treatment, more experimental data on different granite samples should be studied, in addition to the test results on Fangshan granite in this study. Therefore, experimental data from some previously published literatures are also included here.

Table 3. Summary of the samples and treatment conditions.

Samples	Sample Size (mm)	Mineral Contents 1	Heating Rate	Cooling Temprature 2	References
Fangshan granite	φ25 × 50	Qz (38%), Pl (47%), Mc (11%), and Mi (4%)	3 °C/min	20 °C	This study
Dabie granite	φ37 × 74	Qz (8.9%), Pf (45.1%), Ab (21.1%), Mi (23.2%)	5 °C/min	RM	[30]
Gonghe granite-1	50 × 50 × 50	Pl (40%~50%), Qz (20%~25%), Bt (5%~10%)	3~5 °C/h	20 °C	[31]
Gonghe granite-2				60 °C
Gonghe granite-3				100 °C
Sichuan granite	φ50 × 100	Fs (54.1%), Qz (40.2%), Bt (5.7%)	5 °C/min	RM	[32]
Wuyi granite	φ40 × 80	Qz (16%), Mc (51%), Bt (32%), Am (1%)	5 °C/min	25 °C	[33]
Rizhao granite-1	φ50 × 100	Qz (25%), Pl (39%), Pf (22%), Bt (14%)	2~4 °C/min	RM	[34]
Rizhao granite-2					[35]

¹ Qz—quartz; Pl—plagioclase; Mc—microcline; Mi—mica; Pf—potassium feldspar; Ab—albite; Bt—biotite; Fs—feldspar. Am—amphibole ² RM—room temperature.

summarizes some basic information of the samples and the related treatment conditions, including the sample sizes, mineral contents, heating rates, and cooling water conditions.

It should be noted that the Rizhao granite-1 samples are cooled by liquid nitrogen after heating treatment [34]. This research is also presented here for a comparison with the samples suffering water-cooling treatment. In order to systematically compare the strength parameters of different groups of granite samples, the experimental data were normalized in this paper. The normalized cohesion $c_{\mathrm{N}}$ and normalized internal friction angle ${\phi }_{\mathrm{N}}$ are defined as follows:

$$c_{\mathrm{N}}=c_T/c_{\mathrm{R}}$$

(2)

$${\phi }_{\mathrm{N}}={\phi }_T/{\phi }_{\mathrm{R}}$$

(3)

where $c_T$ and ${\phi }_T$ are cohesion and the internal friction angle of the granite samples after heating temperature of T °C and water cooling treatment, respectively, while $c_{\mathrm{R}}$ and ${\phi }_{\mathrm{R}}$ are cohesion and the internal friction angle of the samples under room temperature, respectively. By this definition, the normalized cohesion and internal friction angle values obtained from the above-mentioned experiments are plotted in Figure 3.

The variation characteristics of cohesion and the internal friction angle of Fangshan granite have been introduced in Section 2.1. According to Figure 3b, cohesion of Dabie granite under the temperature of 200 °C is apparently higher than the condition of room temperature, and it does not show significant change from 200 °C to 500 °C, thereafter it decreases at 600 °C but it is still very close to the value under room temperature. It goes down further when the treatment temperature reaches 900 °C. However, the internal friction angle keeps almost constant in a very wide range of treatment temperature from 20 °C to 900 °C.

Figure 3c–e show the behaviors of cohesion and the internal friction angle of three groups of Gonghe granite samples. Cohesion is obviously higher at 250 °C than the condition of room temperature, while it decreases significantly with the increasing temperature, and the cohesion value is much lower at 600 °C than the condition of room temperature. At the meantime, the internal friction angle behaves an increasing trend at the range from 250 °C to 400 °C, thereafter it turns to be almost constant when the treatment temperature goes up further until 600 °C.

According to Figure 3f–i, cohesion of Sichuan granite, Wuyi granite and Rizhao granite—2 samples decrease continuously, while cohesion of Rizhao granite—1 sample increases to a peak value before it decreases with the growing heating temperature. On the other hand, the internal friction angle of these four samples do not have significant changes.

Consequently, it can be found that different granite samples exhibit quite different behaviors of cohesion and the internal friction angle after the high-temperature and water cooling treatment. For cohesion, the behaviors can be divided into two types as follows:

(1)

Type I: Cohesion keeps decreasing with the growing treatment temperature. For example, Fangshan granite, Sichuan granite, Wuyi granite, and Rizhao granite—2 samples.

(2)

Type II: Cohesion goes up to a peak value before going down with the growing treatment temperature. For example, Dabie granite, Gonghe granite, and Rizhao granite—1 samples.

For the internal friction angle, the behaviors of the granite samples can also be divided into two types as follows:

(1)

Type I: Internal friction angle remains almost constant under different treatment temperatures. For example, Fangshan granite, Dabie granite, Sichuan granite, Wuyi granite, and Rizhao granite—1 & 2 samples.

(2)

Type II: Internal friction angle increases with the increasing treatment temperature. For example, Gonghe granite samples.

2.2.2. Empirical Formula for Cohesion and the Internal Friction Angle

It is necessary to provide an empirical formula to describe the behaviors of cohesion and the internal friction angle of different granite samples after high-temperature and water cooling treatment. As the different groups of experiments do not have the same maximum treatment temperature, considering that the temperature of 600 °C is high enough for most of the dry hot rock reservoirs, here in this study the experimental data in the range from room temperature to 600 °C are used for consistency to obtain the best fitted curves and equations. Although the granite samples show different behaviors, it has been found that unified empirical Equations (4) and (5) can be applied for the different types of cohesion and the internal friction angle. The fitting curves for the normalized cohesion and internal friction angle are presented in Figure 4 and Figure 5, respectively, and the fitting coefficients are listed in

Table 4. Fitting coefficients for different granite samples.

Samples	Parameters in Equation (4) for Cohesion		R2	Parameters in Equation (5) for the Internal Friction Angle			R2
	a	b		A	x0	p
Fangshan granite	0.0198	0.0008	0.19	1.02	0.24	8.92	0.07
Dabie granite	−0.2104	0.0351	0.61	0.99	0.87	54.19	0.11
Gonghe granite-1	−0.3681	0.1069	0.93	1.26	4.02	20.00	0.69
Gonghe granite-2	−0.2326	0.0750	0.52	1.28	3.79	35.00	0.43
Gonghe granite-3	−0.4848	0.1818	0.80	1.40	3.13	15.00	0.74
Sichuan granite	0.1012	−0.0069	0.87	1.06	0.47	198.90	0.62
Wuyi granite	0.0350	−0.0040	0.88	1.01	2.83	63.37	0.65
Rizhao granite-1	−0.1120	0.0460	0.99	0.99	0.10	0.65	0.38
Rizhao granite-2	0.1190	0.0130	0.99	1.02	0.50	13.57	0.93

.

$$c_{\mathrm{N}}={\displaystyle \frac{1}{1+a\left(\frac{T}{100}-0.2\right)+b{(\frac{T}{100}-0.2)}^2}}$$

(4)

$${\phi }_{\mathrm{N}}={\displaystyle \frac{1-A}{1+{((\frac{T}{100})/x_0)}^p}}+A$$

(5)

where a, b, A, x₀, P are the fitting coefficients.

The physical meanings of the fitting coefficients can be discussed by observing their influences on the fitting curves. Figure 6 shows the variation of normalized cohesion by the effects of coefficients a and b. It can be found that a dominates the two types of cohesion behaviors, i.e., cohesion shows type I behavior if a < 0, while it has type II behavior for a ≥ 0. According to Figure 6b, for both type I and II behaviors of cohesion, b value has an influence on the decreasing slope of the curve. The higher b value will result in the steeper decreasing trend.

Figure 7 illustrates the variation of the normalized internal friction angle by the effects of coefficients A, x₀ and p. It is apparent to see that A dominates the peak and residual value of the internal friction angle, while x₀ is the central point of the temperature range for the increasing phase of the internal friction angle, and p affects the range of the increasing stage as well as the increasing slope.

3. Tensile Strength of Granite Samples After High-Temperature and Water-Cooling Treatment

The tensile strength calculated by the Mohr-Coulomb model is unreasonably higher than the experimental result, and this discrepancy can usually be corrected by introducing the “tension cut-off” criterion [36]. Therefore, it is also required to study the tensile strength of granite samples after high-temperature and water cooling treatment.

The pre-existing experimental results have shown that high temperature heating and water-cooling treatment has obvious effects on the tensile strength of granite [17,23,25,32], however, it is still necessary to obtain the general evolution behaviors of tensile strength with different treatment situations.

Table 5. Samples and treatment conditions.

Samples	Sample Size (mm)	Mineral Contents 1	Heating Rate	Cooling Water 2	References
Gonghe granite-1	φ50 × 25	Pl (40%~50%), Qz (20%~25%), Bt (5%~10%)	3~5 °C/h	20 °C	[31]
Gonghe granite-2				60 °C
Gonghe granite-3				100 °C
Songliao granite	φ50 × 25	Qz (26.1%), Mi (8.9%), PF (36%), Pl (25.2%)	30 °C/h	20 °C	[17,37]
Rizhao granite	φ50 × 25	Pl (35%), PF (40–45%), Qz (20–25%), Bt (3–5%)	3 °C/h	RM	[23]
Pingyi granite	φ50 × 25	Il (25%), Qz (28%), Fs (43%)	N/A	RM	[25,38]

¹ Qz—quartz; Pl—plagioclase; Mc—microcline; Mi—mica; PF—Potassium feldspar; Ab—albite; Bt—biotite; Il—illite; Fs—feldspar. ² RM—room temperature.

lists the basic information of the samples and treatment conditions for studying the tensile strength variation characteristics.

In order to systematically compare the tensile strength of different granite samples, the normalized tensile strength ${\sigma }_{\mathrm{t}\mathrm{N}}$ is defined as shown in Equation (6).

$${\sigma }_{\mathrm{t}\mathrm{N}}={\sigma }_{\mathrm{t}T}/{\sigma }_{\mathrm{t}\mathrm{R}}$$

(6)

where ${\sigma }_{\mathrm{t}T}$ is the tensile strength of granite after the heating treatment with temperature of T °C and water cooling, and ${\sigma }_{\mathrm{t}\mathrm{R}}$ is the tensile strength of granite under room temperature. The normalized tensile strength ${\sigma }_{\mathrm{t}\mathrm{N}}$ of different granite samples are plotted in Figure 8.

It is observed that the normalized tensile strength ${\sigma }_{\mathrm{t}\mathrm{N}}$ of these six groups of granite samples decrease generally linearly with the increasing heating temperature, with the similar slope, so a unified empirical Equation (7) can be fitted to describe the tensile strength behaviors of different granite samples. The normalized tensile strength and the linear fitting curve are shown in Figure 8.

$${\sigma }_{\mathrm{t}\mathrm{N}}=mT+n$$

(7)

where m and n are both fitting coefficients. Figure 8 also presents the linear fitting curve of the normalized tensile strength ${\sigma }_{\mathrm{t}\mathrm{N}}$ with increasing heating temperature T. It can be obtained that m = −0.0013, and n = 1.03, with R² = 0.94 for these groups of experimental data.

4. Failure Criterion of Granite Samples After High-Temperature and Water-Cooling Treatment

The above-mentioned studies have provided the variation behaviors of cohesion, the internal friction angle and tensile strength of granite samples after different high-temperature and water-cooling treatments, and the empirical equations have been proposed based on laboratory experimental results. Accordingly, a modified Mohr-Coulomb criterion with a tension cut-off can be built for granite considering the effect by different high-temperature and water-cooling treatment. The modified Mohr-Coulomb criterion is shown in Equation (8):

$$\tau ={\sigma }_{\mathrm{n}}\tan {\phi }_T+c_T,$$

(8)

where ${\sigma }_{\mathrm{n}}$ and $\tau$ are the normal stress and shear stress, respectively. $c_T$ and ${\phi }_T$ are cohesion and the internal friction angle dependent on treatment temperature, respectively. This equation can be converted to the formula expressed with principal stresses ${\sigma }_1$ and ${\sigma }_3$ as shown in Equation (9):

$${\sigma }_1={\sigma }_3{\displaystyle \frac{1+\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_T}{1-\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_T}}+2c_T{\displaystyle \frac{\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_T}{1-\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_T}}$$

(9)

In addition, Equation (10) is required for the “tension cut-off” as follows:

$${\sigma }_{\mathrm{t}}={\sigma }_{\mathrm{t}T}$$

(10)

Consequently, the modified Mohr-Coulomb criterion with a tension cut-off consists of Equations (9) and (10), and it can be applied for granite considering the effect by different high-temperature and water-cooling treatment. In these two equations, ${\sigma }_{\mathrm{t}T}$ can be obtained according to Equations (6) and (7). $c_T$ and ${\phi }_T$ can be determined by Equations (11) and (12) as follows:

$$c_T=c_{\mathrm{R}}{c}_{\mathrm{N}}={\displaystyle \frac{c_{\mathrm{R}}}{1+a\left(\frac{T}{100}-0.2\right)+b{(\frac{T}{100}-0.2)}^2}}$$

(11)

$${\phi }_T={\phi }_{\mathrm{R}}{\phi }_{\mathrm{N}}={\phi }_{\mathrm{R}}({\displaystyle \frac{1-A}{1+{(\left(\frac{T}{100}\right)/x_0)}^p}}+A)$$

(12)

The fitting coefficients in these equations have been obtained in Table 4. The values of $c_{\mathrm{R}}$, ${\phi }_{\mathrm{R}}$ and ${\sigma }_{\mathrm{t}\mathrm{R}}$ can also be determined according to the test results under room temperature and presented in

Table 6. The values of $c_{\mathrm{R}}$, ${\phi }_{\mathrm{R}}$ and ${\sigma }_{\mathrm{t}\mathrm{R}}$ under room temperature.

Samples	${\mathit{c}}_{\mathbf{R}}$ (MPa)	${\mathit{\phi }}_{\mathbf{R}}$ (°)	${\mathit{\sigma }}_{\mathbf{t}\mathbf{R}}$ (MPa)
Fangshan granite	15.74	55.34	2.46 1
Dabie granite	19.03	65.35	5.30 1
Gonghe granite	65.50	27.50	10.68
Sichuan granite	32.97	58.71	8.44 1
Wuyi granite	54.43	51.08	10.5
Rizhao granite-1	20.93	53.44	1.73 1
Rizhao granite-2	20.29	54.06	1.67 1

¹ The tensile strength of Fangshan granite, Dabie granite, Sichuan granite and Rizhao granite samples were not measured during the experiment, and their values were estimated by the Hoek-Brown strength envelope of triaxial test results, i.e., σ₃ is obtained when σ₁ = 0.

. It should be noted that the tensile strength of Fangshan granite, Dabie granite, Sichuan granite and Rizhao granite samples were not measured in the laboratory experiments, and their corresponding values in Table 6 are estimated by the Hoek-Brown strength envelope of tri-axial compression test results (i.e., σ₃ is obtained when σ₁ = 0).

Substituting the fitting coefficients in Table 4 and the values in Table 6 into the above equations, the strength envelops based on the new established criterion for each of the above-mentioned groups of granite samples studied in Section 2 have been obtained and plotted in Figure 9. The test results are also presented in Figure 8 for comparison. It should be noted that the experimental data of Gonghe granite samples in Figure 9c–e are based on direct shear tests instead of tri-axial compression tests, and their shear strength values at various normal stresses have been converted to the form of principal stresses (${\sigma }_1$,${\sigma }_3$). Comparing with the experimental results, it can be found that this proposed criterion can describe the failure strength of different granite samples well considering the effect of high-temperature and water-cooling treatment.

5. Discussion

Based on the experimental results in this study as well as the data collected from the references, it has been found that the variation behaviors of cohesion and the internal friction angle have two different types for the granite samples after high-temperature and water-cooling treatment, while the tensile strength changes in the similar way. It is necessary to understand the mechanism of these behaviors. As shown in some studies, this mechanism should be closely related to the mineral composition in the samples [39,40,41,42].

For each single group of the experiments, the samples have almost the same mineral components and micro-structures, and the treatment method is also the same, so the variation of the strength parameters should be mainly dependent on the damage evolution or crack propagation owing to the heating and cooling effects. Reference [43] explained that the behaviors of cohesion weakening and friction strengthening are resulted from the crack propagation during the damage process of the rock. Therefore, it is reasonable to explain the two types of cohesion and the internal friction angle variation behaviors with different cracking process during the heating and cooling treatment.

For the continuous decreasing behavior of cohesion (type I), it should be resulted from the growing cracks in the granite samples with increasing treatment temperature before the water cooling. However, cohesion of some samples increases when the treatment temperature is not very high (type II). This behavior should be related to the expansion of the minerals and closure of the cracks owing to the heating treatment before the water cooling. References [44,45] also reported that the strength values of granite samples have an increase with the increasing temperature within a certain range, and reference [44] presented the decreasing width of cracks with increasing temperatures ranging from 65 °C to 160 °C in SEM images.

Reference [43] shows that the growth of cracks may lead to the increase of internal friction angle, therefore the increasing behavior of the internal friction angle (type II) should be due to the growing cracks with increasing treatment temperature before the water cooling. For type I behavior of the internal friction angle, it does not change significantly. This should be owing to the limited crack number in the samples (e.g., Fangshan granite, Sichuan granite, Wuyi granite, and Rizhao granite—1 & 2 samples), as the decreasing of cohesion is not very significant for these type of samples, either.

However, it should also be noted that although the variations of cohesion and the internal friction angle show different behaviors, their tensile strength values drop in the similar way with the increasing treatment temperature. Bieniawski (1967) observed that the crack initiation and propagation stresses of norite under uniaxial tension are 95% and 97% of the peak tensile strength, respectively [46]. Tham et al. found that AE (acoustic emission) initiation stress level of medium grained granite in direct tensile tests with a single-notched plate is 98% of the peak tensile strength [47]. Cai also concluded that under overall tensile loading, the stress levels of crack initiation and crack propagation are very close to the peak strength [48]. This means that the tensile strength is more sensitive to the induced cracks. Consequently, for the type I granite samples, even very limited amounts of induced cracks may lead to a decrease of the tensile strength.

6. Conclusions

In this paper, based on the high-temperature and water-cooling experiments and some other experimental data collected from the references, the variation behaviors of cohesion, the internal friction angle and tensile strength have been systematically studied, and the failure criterion considering the effect of heating and cooling treatment have been proposed. Some conclusions can be drawn as follows:

(1)

It is found that there are two types of variation behaviors of cohesion and the internal friction angle for the granite samples after heating and cooling treatment.

(2)

Unified empirical equations have been provided to describe the different types of variation behaviors of cohesion and the internal friction angle for the granite samples after heating and cooling treatment. The sensitivity analyses show that the fitting coefficients have quite clear physical meanings.

(3)

The tensile strength decreases in a similar way for the different granite samples with the increasing treatment temperature. A linear empirical equation has been given to describe this behavior.

(4)

A modified Mohr-Coulomb failure criterion with a “tension cut-off” has been established for the granite samples considering the effects of high-temperature and water-cooling treatment. The criterion has been validated by well describing the different types of granite behaviors after heating and cooling treatment.

(5)

A discussion has been supplied on the mechanism of the different granite behaviors. It should be closely related to the damage and cracking process during the heating and cooling treatment on the granite samples.

The variation behaviors of cohesion, the internal friction angle, and tensile strength should be helpful for understanding the mechanical behavior of hot dry rock during the hydraulic fracturing in geothermal production. As a main rock type for the hot dry rock reservoirs, the mechanical properties of granite under high temperature will be affected by the injected water, which may have significant influences on the fracturing characteristics. The proposed failure criterion can be applied for numerical modelling on the reservoir. In addition, the strength parameters and failure criterion of granite after high-temperature and water-cooling treatment in this paper should also be helpful for understanding wellbore stability problems.

In the future studies, more systematic experiments should be carried out for a better understanding on the mechanism of the different behaviors, and more experimental data are still required for improving the failure criterion.