Influences of confining pressure and injection rate on breakdown pressure and permeability in granite hydraulic fracturing

Hot dry rock (HDR) geothermal resource is a renewable green energy source with great exploitation potential. The burial depth of HDR generally exceeds 2500 m and is typically under high in situ stress conditions, resulting in an ultralow permeability of the rock formations. To enhance the permeability of these formations, hydraulic fracturing is widely used as a reservoir stimulation technique in HDR geothermal resource exploitation. The differences in burial depth, in situ stress, and geological environment require different engineering designs when implementing hydraulic fracturing. Therefore, the confining pressure and injection rate play significant roles in determining the effectiveness of hydraulic fracturing, as they affect the propagation and distribution of fractures in the rock formation. To quantify the impact of these factors on the effectiveness of hydraulic fracturing, simulation experiments, and permeability tests were conducted using granite specimens under various confining pressure and injection rate conditions. The results of these experiments revealed the relationships among the confining pressure, injection rate, breakdown pressure, and permeability enhancement of the granite. The breakdown pressure of granite increased with the confining pressure, while the injection rate had little effect on the breakdown pressure. The hydraulic fractured sample produced new penetrating fractures, which increased the reservoir permeability, and owing to the higher complexity of hydraulic fractures under low confining pressure, the increase of permeability is correspondingly higher. The research results can provide an important reference for efficient stimulation development technology of deep HDR geothermal resources.


| INTRODUCTION
Hot dry rock (HDR) geothermal energy emerges as a green renewable resource with significant implications for the global energy landscape. With ample geothermal resources within its borders, China has the potential to tap into this clean energy source, thereby improving its energy structure and enhancing its energy security. Given the strategic significance of HDR geothermal energy, its development, and utilization are crucial to the country's future energy needs.
The permeability of granite has been extensively studied by scholars in the field. Wang et al. 1 measured the permeability and porosity of granite and found that its initial permeability ranged from 10 −17 to 10 −19 m 2 , with an initial porosity of 4.05% to 7.09%. Wu et al. 2 used a rock triaxial permeameter to study the permeability of granite under varying confining pressures, observing that as the confining pressure increased from 10 to 60 MPa, the permeability of granite decreased from 10 −18 to 10 −19 m 2 . Some experimental studies found the initial permeability of granites to be in the order of 10 −19 m 2 , and with an increase in confining pressure the permeability of granite decreases by one to two orders of magnitude. [3][4][5] The studies in question emphasize the inherent low permeability of HDRs, presenting a significant challenge in directly extracting stored heat energy. To overcome this obstacle, the application of hydraulic fracturing as a means to enhance permeability in the reservoir has been suggested. The application of hydraulic fracturing results in the creation of new fractures that intersect with preexisting ones within the rock formation, effectively forming an artificial heat storage with a defined permeability and scale.
The impact of hydraulic fracturing on permeability enhancement has been the subject of extensive research by numerous scholars. Zhang et al. 6 applied hydraulic fracturing to tight sandstone and found that the permeability increased by a factor of up to 2.9 × 10 6 . Compared to intact sandstone, single-fracture sandstone, and double-fracture sandstone have been found to have significantly greater permeability. This was determined in a study where the permeability of these three types of sandstone was compared. 7 Zhao et al. 8 conducted research on the seepage behavior of fracture networks after hydraulic fracturing of shale and concluded that the seepage channels formed during the process can significantly enhance the permeability of shale. In addition, hydraulic fracturing is also used in the commercial development of oil and gas wells to increase production. [9][10][11] These findings demonstrate that hydraulic fracturing has a substantial impact on rock permeability.
Most of the studies mentioned have focused on oil and gas resources, examining rocks such as sandstone and shale. Moreover, some researchers have investigated the permeability-enhancing effects of hydraulic fracturing on granite. Examination of the impact of hydraulic fracturing on the permeability of granite fractures revealed that the permeability was significantly increased after the fractures were subjected to hydraulic fracturing. 12 Watanabe et al. 13 carried out hydraulic fracturing on granite at temperatures ranging from 200°C to 450°C and observed that the permeability of the granite was greatly improved at varying temperatures. However, these experiments were limited to a single low confining pressure and fixed injection rate and did not take into account the effects of different confining pressures and injection rates on granite permeability. In addition, the depth of geothermal resources in HDRs and the ground stresses they endure differ, 14 and the injection rate in complex and practical engineering projects is subject to change in response to actual changes in circumstances. 15 Therefore, our study aimed to investigate the effect of hydraulic fracturing on granite permeability under varying confining pressures and injection rates. The permeability of the samples was measured both before and after hydraulic fracturing. The comparison of the prefracturing and postfracturing permeability enabled us to uncover the law of permeability enhancement in granite after hydraulic fracturing under different conditions. This study provides valuable data support for addressing complex and practical engineering challenges.

| Experimental set-up
The experiment was conducted using the GCTS RTR-4600 triaxial testing system. The system is depicted in Figure 1. The GCTS test system is primarily used to evaluate the mechanical properties and seepage behavior of rock, concrete, and coal under complex stress conditions. The system boasts high precision and stable performance, with the ability to collect high and lowspeed data. It utilizes force, displacement, axial strain, and transverse strain control modes, with an axial maximum load capacity of 4600 kN, a confining pressure maximum of 140 MPa, a pore fluid pressure maximum of 140 MPa, and a highest temperature of 200°C. The system can simulate the stress, pore pressure, and temperature conditions found in deep underground rocks, and perform uniaxial compression, indirect tension, triaxial compression, fatigue, creep, and relaxation tests. In addition, it can perform hydraulic fracturing tests with an injection flow range of 0.1-30 mL/min, and the injection pressure can be monitored through the monitoring system.

| Sample preparation
Test cylindrical samples were collected from Macheng, Hubei Province, China, with a total of 12 samples divided into four groups. The samples were drilled vertically along the lamination direction with a diameter of 50 mm and a height of 100 mm. Figure 2 shows the samples utilized for hydraulic fracturing, and Table 1 summarizes the fundamental physical parameters of the rock. To facilitate the hydraulic fracturing experiment, a hole with a diameter of 6 mm and depth of 50 mm was drilled at the central axial position of each sample. To prevent the leakage of the fracturing fluid, the hole was sealed using rubber rings.
To facilitate subsequent analysis of the results from hydraulic fracturing experiments, we perform basic mechanical tests on the granite samples. 16

| Testing procedure
For each group of samples, permeability tests were conducted under confining pressures of 3, 5, 10, and 15 MPa, and an axial stress of 1 MPa. After the permeability tests were completed, the axial and confining stresses were increased to the target values, and hydraulic fracturing was performed. The experimental conditions are shown in Table 2. The aim of the study is to examine the impact of different confining pressures on the breakdown pressure, the direction of fracture development, and permeability enhancement of granite.
Within each group, having the same confining pressure and axial stress, a range of injection rates is applied to investigate the effects of different injection rates on the breakdown pressure, fracture development direction, and permeability enhancement of granite.

| Pulse decay method
In this experiment, the permeability of the rock samples was determined using the pulse decay method. This method involved the use of two gas tanks located upstream and downstream of the sample, respectively. The sample was subjected to the required confining pressure and helium gas was introduced into both ends to establish a balanced pressure. Next, the gas pressure in the downstream tank was reduced, while the gas pressure in the upstream tank was maintained constant. The switches from both tanks to the sample were then opened, and the time required for the pressure to reach balance was recorded. By inputting the gas pressure, pressure difference, and the time required for the reestablishment of balance into Equation (1), the permeability of the sample was calculated.
where k is the permeability. S is the slope of the line. μ g is gas viscosity. L and A are the height and cross-sectional area of the specimens. f z is the actual gas that deviates from the characteristic value of the ideal gas. P m is the 3 | EXPERIMENTAL RESULTS AND ANALYSIS

| Breakdown behavior
In this experiment, the term "breakdown pressure" refers to the critical fluid pressure at which the rock fractures. The results of the 12 samples, including the confining pressure, axial stress, injection rate, breakdown pressure, and breakdown time, are presented in Table 3. Figure 3 depicts the variation of fluid pressure with time under various confining pressure conditions. As the figure demonstrates, the breakdown process can be divided into four stages: (1) the filling stage, (2) the pressure-increase stage, (3) the fracturing stage, and (4) 19 Based on this information, the hydraulic breakdown was performed on different samples with the same confining pressure and varying injection rates. The experimental results are presented in Figure 4. As can be seen from the figure, despite an increase in the injection rate from 5 to 15 mL/min, the breakdown pressure of the sample did not exhibit a significant change. These results demonstrate that after applying confining pressure, the breakdown pressure of the sample remains largely unaffected by variations in the injection rate. Figures 3 and 4 depict the fluid pressure change trends of 12 samples, which are found to be consistent.
The hydraulic fracturing process commences as water fills the pores of the sample. After the pores have been saturated, the fluid pressure increases rapidly with the continued injection of water. Upon reaching the breakdown pressure, the fluid undergoes a sudden decrease in pressure, which is a clear indication of sample fracture resulting from the high pressure of the fluid. This leads to water flowing out of the sample and a reduction in pressure. The injection of water is then continued at a similar rate, and the fluid pressure remains stable as it is equal to the confining pressure. This is due to the hydraulic connection between the injected fluid and the silicon oil providing the confining pressure.

| Changes in permeability before and after hydraulic fracturing
Hydraulic fracturing tests were conducted on four sample groups at an axial stress of 40 MPa and a water injection rate of 15 mL/min, under confining pressures of 3, 5, 10, and 15 MPa. The permeability of the samples was assessed both before and following the hydraulic fracturing tests. These tests were conducted to uncover the relationship between the increase in granite permeability and different confining pressures during the hydraulic fracturing process. Figure 5 illustrates the change in permeability of the samples before and after hydraulic fracturing. Before hydraulic fracturing, the permeability of the sample is determined by its primary fractures. Natural granite has limited primary fractures, leading to low permeability, primarily ranging from 10 −3 to 10 −4 mD. 20 After hydraulic fracturing, the permeability of the sample shows a significant improvement, with an increase of one to two orders of magnitude, primarily ranging from 10 −1 to 10 −3 mD. This improvement is due to the creation of new fractures and the extension of existing ones during the hydraulic fracturing process, which results in the formation of a fracture network. This network interconnects the new and original fractures and leads to a substantial increase in permeability. 21 The experimental results indicate that the permeability of samples A 3-15 , A 5-15 , A [10][11][12][13][14][15] , and A  , under a confining pressure of 3 MPa, was 2.1 × 10 −3 , 6.3 × 10 −3 , 3.8 × 10 −3 , and 5.9 × 10 −3 mD, respectively, before hydraulic fracturing. After the process was conducted, the permeability of samples A 3-15 , A 5-15 , A [10][11][12][13][14][15] , and A  increased to 5.32 × 10 −2 , 13.82 × 10 −2 , 6.84 × 10 −2 , and 5.85 × 10 −2 mD, respectively. When compared to the values before hydraulic fracturing, the permeability growth multipliers were found to be 26.86, 22.94, 17.00, and 8.92, respectively. It is observed that the growth multiplier decreases as the confining pressure increases during hydraulic fracturing. Furthermore, as the confining pressure continues to increase to 15 MPa, the permeability of the samples decreases to 9.67 × 10 −3 , 7.87 × 10 −3 , 4.92 × 10 −3 , and 20.3 × 10 −3 mD, respectively. It is widely accepted that rock permeability decreases with increasing confining pressure. 22 The results of these tests are consistent with this general trend. Both before and after hydraulic fracturing, the permeability of granite demonstrates a marked decrease with increasing confining pressure.
After hydraulic fracturing, the confining pressure increased from 3 to 15 MPa, as shown in Figure 6. In this experiment, the confining pressure interval is small, so the downward trend of permeability shows linear characteristics. To better understand this relationship, we conducted a regression analysis of the permeability F I G U R E 5 Prefracturing and postfracturing permeability. data and the change in confining pressure and calculated the slope, whose absolute value represents the rate of permeability decline. Further details can be found in Table 4 (the confining pressure is represented by σ 3 ). The table shows that, as the confining pressure increased from 5 to 15 MPa during hydraulic fracturing, the rate of permeability decline decreased from 0.0108 to 0.0028 after hydraulic fracturing. The results indicate that with increasing confining pressure during hydraulic fracturing, the rate of permeability change in the sample decreases.

| Rock fracture mode
We will analyze the permeability variation of the samples in this section by combining the fracture morphology of the samples.
With the continuous infusion of water, fluid pressure increases and instigates the formation of fractures within the rock. Once the fractures have formed, water permeates the interior of the rock, leading to the further development of the fractures. With the sustained increase in fluid pressure, the fractures extend to the perimeter of the sample, ultimately causing the sample to break apart and creating a network of fractures. This marks the completion of the fracking process. Previous studies have shown that the direction of fracture extension is perpendicular to the direction of the minimum principal pressure. 23 In this paper, the minimum principal pressure was the confining pressure, and the fracture that was generated by the specimen was situated on a plane perpendicular to the axial direction, which concurs with existing cognition. As depicted in Figures 7 and 8.
The sample was subjected to hydraulic fracturing under various conditions, as outlined in Table 5. As shown in Figure 7, under the influence of fluid pressure, the specimen experienced the formation of two newly generated macrofractures. These fractures originated from the holes and expanded radially. The trajectory of fracture expansion was generally linear, however, localized bending along the fracture trace was observed in some regions, which can be attributed to the influence of mineral form and distribution.  Figure 8 illustrates the formation of a dominant principal fracture as a result of hydraulic fracturing. The length of the principal fracture is summarized in Table 5. It can be observed that with the increase of confining pressure during hydraulic fracturing, there was a reduction in sample fracture length, from 100.0 to 80.8 mm. After hydraulic fracturing, under 3 MPa confining conditions, the growth rate of permeability in samples A 3-15 , A 5-15 , A [10][11][12][13][14][15] , and A 15-15 decreased with the reduction in the length of the principal fracture. This reduction can be attributed to the fact that, as confining pressure decreases during hydraulic fracturing, the sample is less affected by confining pressure, leading to the formation of a more complex network of fractures. With an increase in confining pressure, the fractures formed by hydraulic fracturing tend to gradually close, leading to a decrease in the permeability of the sample. However, due to the rough and irregular surface of the fractures, which contain protuberances serving as proppants, the fractures will not close completely. As a result, even under higher confining pressure, the permeability of the sample after hydraulic fracturing remains significantly higher compared to its permeability before hydraulic fracturing.

| CONCLUSION
Hydraulic fracturing has been extensively studied. 24,25 On the basis of previous research, to evaluate the effects of hydraulic fracturing on enhancing rock permeability, and the influence of confining pressure and water injection rate on hydraulic fracturing effectiveness, we conducted hydraulic fracturing and permeability testing. Through the results of these experiments, we have reached the following conclusions.
The breakdown pressure of granite increases with increasing confining pressure. After applying confining pressure, the breakdown pressure is basically not affected by the injection rate. And it suggests that the injection rate may not be a critical parameter for controlling the fracturing process in HDR mining. After hydraulic fracturing, two newly formed macrofractures penetrate the specimen. The fractures originate from the injection points and radiate outward. The expansion trajectory of the fractures is generally linear. And with increasing confining pressure during hydraulic fracturing, the length of the fractures decreases.
After hydraulic fracturing, the permeability of the sample is expected to increase by one to two orders of magnitude compared to its pre-fracturing state. When the confining pressure is low, the greater the decrease in confining pressure during hydraulic fracturing, the greater the increase in granite permeability. And after hydraulic fracturing, the permeability of the sample decreases linearly with increasing confining pressure. The change rate in permeability decreases as the confining pressure during hydraulic fracturing increases. This information can be used to predict the performance of an HDR geothermal reservoir over confining pressure and optimize its operation.