1 Introduction

As an abundant, widely distributed, stable, and clean renewable energy, geothermal energy is increasingly considered by many to offer major contribution to future energy structure, especially when low carbon and environmentally friendly practices are prevailing globally (Hu et al. 2021; Kumari and Ranjith 2019). Among various types of geothermal energy, extracting heat from the hot dry rock (HDR) is found to show the most potential as an important type of renewable energy in the future. The HDR geothermal energy is approximately over 300 times of the total fossil energy including oil and natural gas (Olasolo et al. 2016; Hu et al. 2018). Therefore, it is of great importance to develop improved reservoir stimulation techniques for more efficient geothermal extraction to mitigate the current energy shortage.

In the past decades, significant progress is made for hydraulic fracturing (HF) techniques applied in extracting oil and gas in low-permeability reservoirs (Zhuang and Zang 2021; Li et al. 2021). Different from conventional oil and gas reservoirs, however, the deep geothermal resources are often under complex stress conditions in high-strength rocks, resulting in great challenges for directly applying hydraulic fracturing in extraction of geothermal energy. Previous studies showed that during reservoir stimulation for the HDR, the breakdown pressure is often high and more prone to induce seismicity, which can pose potential risks when extracting HDR geothermal energy (Tomac and Sauter 2018; Cornet 2021). Additionally, groundwater may be contaminated by the fracturing fluid during the injection and production periods when using the hydraulic fracturing techniques (Jackson et al. 2013). Moreover, the patterns of the fracture network generated by adopting the hydraulic fracturing are relatively uniform, leading to unsatisfactory heat transfer efficiency (Blöcher et al. 2016).

Due to various drawbacks of hydraulic fracturing techniques in extracting geothermal energy, developing new stimulation methods specifically for HDR reservoirs has received massive attention. Fei et al. (2018) showed that using foam for fracturing can generate more complex fracture network with reduced breakdown pressure and seismicity. Similarly enhanced fracturing performance was also found by Middleton et al. (2015) using super-critical CO2 as the fracturing fluid. These studies demonstrated the advantages of employing less viscous fluids (e.g., N2, CO2, and foam) for fracturing, including creating more complex fracture networks for improved heat extraction performances (Ma et al. 2022). In recent years, many studies have focused on exploring different gases (low viscosity with high flowability and diffusivity) as the fracturing fluid to develop new fracturing techniques, which called gas fracturing techniques (GF) (Xie et al. 2020; Hou et al. 2017a, b). Compared to water, gas can better diffuse into micro-fractures in the rocks to generate more complex fractures with less breakdown pressure (Xue et al. 2022; Zhao et al. 2019). Lu et al. (2020) compared the fracturing performances of hydraulic fracturing and gas fracturing under true triaxial stress conditions. It was found that more micro-fractures were generated using gas fracturing with a reduced breakdown pressure by 30% compared to hydraulic fracturing. Yang et al. (2021) examined the failure behaviors of granite using both water and N2 as the fracturing fluids under high-temperature conditions. Their results showed that the generated fractures demonstrated more complex patterns with higher conductivity when N2 was used, resulting in reduced breakdown pressure and faster fracture propagation. Hou et al. (2017ab) found similar results in their experiments using N2 to fracture coal, sandstone, and shale, especially for low-permeability rocks. It was suspected that low-viscosity gas can alter pore pressure transfer and distribution, which assisted in enhancing fracture propagation. Hou et al. (2018) confirmed this suspicion in their subsequent analyses on fracturing behaviors of sandstone, and showed that gas fracturing can cause higher pore pressure which further promoted fracture development. Ougier-Simonin and Zhu (2015) assessed the effect of increasing pore pressure on the failure behavior of high-permeability sandstone under different confining pressures. They showed that higher pore pressure can directly cause failure of the stable fractures to increase internal fracture propagation speed. In the experiments examining hydraulic fracturing and gas fracturing with red sandstone, Gao et al. (2018) found that gas fracturing was not as sensitive to different injection rates as hydraulic fracturing. For gas fracturing, increasing injection rate only reduced the time for breakdown, whereas the breakdown pressure remained almost unchanged.

The key factors affecting the performance of gas fracturing were further explored in previous research regarding the breakdown pressure and fracture pattern of gas fracturing and hydraulic fracturing (Li et al. 2020; Zhao et al. 2021). He et al. (2020) studied the effects of bedding orientation and deviatoric stress on the gas fracturing performances. The results showed that the induced fractures generally propagated along the bedding orientation, although the fractures may deviate locally during the propagation process. The shale breakdown pressure decreased drastically as the bedding orientation and deviatoric stress increased. Jia et al. (2021) demonstrated that higher confining pressure and injection rate can result in higher breakdown pressure for shale, which was consistent with the results from hydraulic fracturing experiments (Li et al. 2016). Overall, the breakdown pressure was found lower when using the less viscous super-critical CO2 than N2. Xue et al. (2021) assessed the differences in the physical properties and sorption behaviors between the super-critical CO2 and N2. The use of super-critical CO2 led to higher breakdown pressure than N2, due to different stress intensity factors (SIF) at the tip of the fractures. Liang et al. (2021) systematically examined the effect of the fracture fractal dimension on the breakdown pressure in gas fracturing. It was shown that the difference in the breakdown pressure between using N2 and water was closely related to the varying SIF of the induced fractal fractures. Therefore, to further analyze the failure behaviors in gas fracturing, it is vital to understand the fracture propagation mechanism and establish effective theoretical models for the breakdown pressure. Since the famous elastic criteria proposed by Hubbert and Willis (1957) (H-W criteria), many predictive models were subsequently developed for the breakdown pressure, including the porous elastic criteria and point-stress criteria. However, Zhang et al. (2018) pointed out that the conventional models are essentially only for local fracture initiation, that is, the models only predict local pressure for fracture initiation rather than for the breakdown of the rock. Hou et al. (2017ab) provided a new approach for predicting breakdown pressure, by replacing the initial fracture length with the characteristic length to establish new breakdown pressure criteria based on the SIF of Model I tensile fractures. Inspired by the nonlocal assumption of the average stress criteria, Wang et al. (2021) used the equivalent fracture with a characteristic length l to describe the fracture initiation and stabilizing processes, where the rock only fails when the tensile stress along l reaches the tensile strength of the rock. Based on the above study, Yu et al. (2022) studied the breakdown pressure of rocks under stepwise pressurization and stepwise pulse pressurization by further incorporating two parameters to quantify the effect of fluid flow. Lu et al. (2020) suggested that under the true triaxial stress condition, initiation of the vertical fractures was controlled by the maximum shear stress of the well wall, whereas initiation of the horizontal fractures was determined by the maximum axial stress. Based on these results, a predictive model was proposed for the breakdown pressure, and the predictions were found consistent with those from numerical simulations and experiments.

In general, the previous experimental studies on gas fracturing were mostly focused on the effects of injection rate and viscosity on the failure behaviors of rocks. For HDR reservoirs in deep underground, the in-situ stress conditions especially the confining pressure can evidently affect the gas fracturing performances (Li et al. 2023). However, comprehensive studies on how the confining pressure affects the gas fracturing behaviors are still lacking. Moreover, the rocks tested in previous studies were mainly sandstone, shale, and coal, which are different from the common rock type of the HDR, leading to different failure behaviors using gas fracturing (Li et al. 2017; Kong et al. 2023; Hu et al. 2022). In addition, it is important to effectively apply gas fracturing in the HDR applications is to develop a breakdown pressure predictive model that accounts for the effects of fracture propagation and fluid flow.

Therefore, this study aims to assess the fracturing behavior of granite outcrop under different confining pressures and injection rates and accordingly, to propose a predictive model for breakdown pressure considering the effect of confining pressure. First, fracturing tests were conducted on granite specimens using the self-remodeled gas fracturing apparatus under 5 different confining pressures and 2 injection rates to assess their effects on the breakdown pressure and fracture patterns. Then, variations of AE event count, AE energy, and b-value during the gas fracturing tests were analyzed in detail. Last, a novel predictive model was proposed for breakdown pressure considering the effect of confining pressure, which was found to agree well the experimental results. Findings of this work can provide valuable theoretical understanding of the gas fracturing process to improve the stimulation performance of HDR reservoirs for more effective extraction of deep geothermal resources.

2 Experimental specimen preparation and testing procedure

2.1 Testing material

The specimens used in this study were fabricated using outcrops of Yanshanian granite in the Pearl River Delta of the Guangdong-Hong Kong-Macao Greater Bay Area, and the typical mechanical properties of the granite can be found in Table 1. The average mineral contents of the granite were obtained via the X-ray diffraction (XRD), as shown in Fig. 1a. The tested granite consisted mainly of microcline, quartz, and albite, with the three taking up 81% of all minerals. Standard cylindrical specimens with a dimension of ϕ50 mm × H100 mm (diameter × height) were created from raw granite outcrops, and both ends of the specimen were grinded and polished to ensure an flatness accuracy of ± 0.02 mm. Finally, a circular hole with a dimension of ϕ10 mm×\(H\) 50 mm was drilled for each specimen, which is considered equivalent to the fracturing well in an actual geothermal application (Wang et al. 2021), as depicted in Fig. 2b.

Table 1 Typical mechanical parameters of granite
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Fig. 1
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Specimen preparation: a mineral contents, b cylindrical granite samples with a circular borehole

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Fig. 2
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Gas fracturing experimental system

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2.2 Experimental equipment

The fracturing tests were conducted using a self-remodeled gas fracturing (GF) apparatus based on the servo-controlled rock mechanics device, as shown in Fig. 2. The apparatus can enable a maximum axial load of 2000 kN and a maximum confining pressure of 80 MPa. The gas fracturing system mainly consisted of a gas cylinder, air compressor, air pressure controller, and gas tank. The maximum air pressure was 100 MPa with a rate between 0.001 and 10 MPa/s. Instant injection pressure and strain variation of the specimens can be accurately recorded during experiments. The PCI-2 AE monitoring device, equipped with the18 bit A/D converter, was used to receive AE signals. The maximum signal amplitude was 100 dB and the frequency ranged from 1 to 3 MHz. Three acoustic emission probes are arranged on the same cross section at three equally divided intervals, and the probes are spaced 60 mm apart along the axial direction of the rock sample. There are 6 probes in total and its arrangement is shown in Fig. 1b.

2.3 Experimental procedure

In the tests, nitrogen (N2) was used as the fracturing fluid. To assess the effects of both confining pressure and injection rate on the fracturing performance, 5 different confining pressures (0, 3, 6, 9, and 12 MPa) were adopted with 2 injection rates (0.2 and 0.4 MPa/s). Fracturing tests were conducted twice for each condition. The experimental procedure is as follows: (1) Loading axial and confining pressure: first, axial load was applied through stress control at a rate of 0.05 MPa/s until reaching 5 MPa to keep specimens well mounted; then, confining pressure was imposed at a rate of 0.01 MPa/s to the prescribed value; finally, the axial pressure was increased to 25 MPa at a rate of 0.1 MPa/s. (2) Gas fracturing: gas pressure was progressively increased at the designated rate, and the data such as the pressure–time variation and AE results were recorded during the tests. The test was ceased when the gas pressure dropped dramatically, indicating that breakdown of the specimen was achieved.

3 Experimental results and analysis

3.1 Evolution of injection pressure curves

The time-pressure curves of the granite specimens were obtained under different confining pressures and injection rate during the gas fracturing test, as shown in Fig. 3. Compared to the time-pressure curves from conventional hydraulic fracturing tests, different stages can be obviously observed from the time-pressure curves of gas fracturing (Liang et al. 2021). Based on the variation feature of injection pressure, the gas fracturing can be divided into three stages including the gas compaction stage, gas pressuring stage, and pressure relieving stage. Specifically, the gas compaction stage transitioned to the gas pressuring stage when the injection pressure first became stabilized and then increased drastically. In the gas compaction stage, N2 was rapidly compressed with a fast-growing injection pressure and as the gas was further compressed at high gas pressure, the injection pressure began to climb slowly until becoming stabilized for about 10 s. During the gas pressuring stage, the injection rate increased almost linearly with respect to time. As N2 intruded into the original fractures of the granite, irreversible damage was progressively caused by the increased pore pressure on the internal micro-fractures. As depicted in Fig. 3a–j, the average time in the gas pressuring stage changed slightly under same confining pressures regardless of the injection rate. The results indicated that under a fixed injection rate, the specimens experienced longer gas pressuring stages at higher confining pressure and became less prone to failure. This was due to the increased closure of micro-fractures as the confining pressure became higher to reduce pore aperture and porosity; therefore, the gas needed more time to intrude into the microfractures to cause internal damage for the specimens. In addition, as the rock strength also became higher at larger confining pressure, it required longer time for the fluid pressure in the injection hole to grow to cause rock breakdown (Deng et al. 2020). When the confining pressure was fixed, the time in the gas pressuring stage can be evidently reduced with the increase of injection rate to reach the breakdown pressure faster. The amounts of reduction were 10.42%, 0%, − 5.21%, 25%, and 10% as the confining pressures increases from 0 to 12 MPa. From Fig. 3e and h, the durations of the gas pressuring stage of GF-5a and GF-8a deviated largely from those of GF-5b and GF-8b, which led to slight fluctuations (70 and 50.5 s) when averaged to obtain the actual durations. The internal damage continued to accumulate as the gas pressure increased and when reaching its peak, the rock failed instantly with a sharp drop of the gas pressure to enter the pressure relieving stage.

Fig. 3
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Time-pressure curves of the fracturing tests under different injection rate ν and confining pressures σ3

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3.2 Evolution of breakdown pressure

To analyze the effects of different injection rates and confining pressures on the failure behaviors of granite using gas fracturing, the breakdown pressure was obtained with the increase of confining pressure for two different injection rates, as demonstrated in Fig. 4. At fixed injection rate, the increment of breakdown pressure decreased with the increase of confining pressure. The largest increment occurred when the confining pressure increased from 0 to 3 MPa at both injection rates, with an increment of 68.5% for the low injection rate and 53.6% for the high injection rate. This was because the granite strength is higher under the triaxial condition than uniaxial condition. Under the effect of confining pressure, the micro-fractures inside the granite began to close with reduced porosity and fracture aperture, which contained the gas from rapidly spreading to the internal fractures to cause rock breakdown. The granite was continuously compressed as the confining pressure increased. However, further compaction generally ceased when reaching certain confining pressure and therefore, the granite porosity showed insignificant drop as the confining pressure kept increasing. Resultantly, the increment of the breakdown pressure consistently decreased as the confining pressure increased. For example, the increment in the breakdown pressure was only 20.8% at low injection rate and 12.5% at high injection rate, which were only 30.4% and 23.3% of their peak values, respectively.

Fig. 4
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Breakdown pressure under different confining pressures and injection rates

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It was observed that when the confining pressure was 0 MPa, the breakdown pressure at high injection rate (0.4 MPa/s) was higher than that at low injection rate (0.2 MPa/s) with an increment of 7.69%, which is consistent with the results in hydraulic fracturing experiments (Zhuang et al. 2019). With the increase of confining pressure, however, the breakdown pressure at high injection rate was obviously lower than that at low injection rate, and the difference became more evident as the confining pressure increased. Compared to the case of low injection rate, the higher the confining pressure, the larger the breakdown pressure at high injection rate dropped. This was due to further closure of micro-fractures and porosity reduction induced by increasing the confining pressure to contain the flow of N2 and when higher injection rate was used, the gas pressure in the hole rapidly grew to promote gas intrusion in the granite and the rock became more prone to failure.

3.3 Evolution of AE counts and AE energy during gas fracturing

The AE parameters obtained during the rock failure were used to study the damage process of the granite during gas fracturing (Xie et al. 2023), as they can imply the process of fracture development and propagation and depict the internal damage evolution and failure features. Figure 5 shows the time-dependent variation of AE counts and energy (along with the injection pressure) in the gas fracturing tests under different confining pressures when the injection rate was 0.2 MPa/s. It can be found in Fig. 5a–e that the AE counts varied similarly as the AE energy, with larger AE counts at higher AE energy. Almost no AE activities were observed before reaching the breakdown pressure, suggesting only trivial fracture development and propagation during the gas compaction and gas pressuring stages. However, the AE counts and AE energy both increased dramatically when reaching the breakdown pressure, indicating massive internal damage in the rock specimen with major macro-fractures. As the macro-fractures further developed, the injection pressure quickly dropped with much weakened AE activities.

Fig. 5
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AE counts and AE energy under different confining pressures at an injection rate of 0.2 MPa/s

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By examining the AE counts obtained from rock specimens at different confining pressures, the stress condition obviously demonstrated evident impact on the AE activities. Specifically, the highest AE counts was observed when the confining pressure was 0 MPa. Larger AE counts were observed during specimen breakdown as the confining pressure increased, indicating more active AE activities. The AE counts recorded near the breakdown pressure were found to significantly increase with increasing confining pressures. The breakdown pressure became higher as the confining pressure increased, leading to more secondary fractures generated during the fracturing process which enhanced AE activities.

As shown in Fig. 6, the variations of AE counts and AE energy in the gas fracturing tests with respect to time were obtained under different confining pressures when the injection rate was 0.4 MPa/s. Although both AE counts and AE energy showed similar time-dependent features as in the case of low injection rate (such as increased AE activities near the breakdown pressure), the recorded AE counts and AE energy were significantly higher. For example, the AE counts at higher injection rate increased by 160.4% when the confined pressured was 12 MPa. This indicated that during gas fracturing, a high injection rate can lead to more energy been released when the specimen failed, causing larger areas of internal damage, more developed secondary fractures, and stronger fracturing performances. In addition, the recorded AE counts increased with the increase of the confining pressure (expect for 0 MPa confining pressure) with over 10,000 AE counts at 12 MPa confining pressure, suggesting that the confining pressure can also affect the gas fracturing performance at high injection rate. The relatively high AE counts and AE energy observed at 0 MPa confining pressure were due probably to the injected gas fulfilling the relatively large amount of internal micro-fractures to promote fracture propagation, resulting in more energy been released along with abundant AE signals.

Fig. 6
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AE event and AE energy under different confining pressures at an injection rate of 0.4 MPa/s

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The instant energy release at the breakdown under 0.2 MPa/s injection rate showed a trend of first increasing and then decreasing with the continue increase of confining pressure. Similar results were also found in the case of 0.4 MPa/s injection rate. This was because the brittle behavior of rock was affected by the confining pressure, and the rock became less brittle as the confining pressure increased (Xie et al. 2021; Zhao 2020; Kuang et al. 2021). Under low confining pressure, the granite demonstrated typical brittle behaviors and the energy release became greater as the confining pressure increased. However, the rock transitioned from brittle to ductile with increased plasticity under high confining pressure, resulting in less energy release as the confining pressure continued to increase.

3.4 Evolution of AE b-value during gas fracturing

To demonstrate the internal fracture evolution for deepened understanding of the rock failure process, the b-value of AE was used to assess the distribution of the maximum AE event amplitude during the gas fracturing tests (Yang et al. 2022). Figure 7 shows the change of b-value of the specimens under different confining pressures and injection rates. Compared to other rock types, the b-value of granite was found relatively high in the gas fracturing experiments due to its distinct mineral content and internal structure. The tested granite was found to mainly contain feldspar, in which the microcline and albite are generally weak and more prone to induce micro-fractures under pressure (Shengxiang et al. 2021). Additionally, the natural fractures were compacted and the mineral particles were tightly attached under the confining pressure, which limited the fracture development and propagation to form macro-fractures, resulting in relatively high b-value (Chen et al. 2022).

Fig. 7
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Variation of AE b-value at different confining pressures

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At 0.2 MPa/s injection rate, the obtained b-value generally increased with some fluctuations. The b-value grew as the confining pressure increased with more small-amplitude AE events taking place. The high confining pressure can contain fracture initiation and hence, no macro-fractures can be formed in a short period of time. With the continuous injection of gas, the pore pressure became higher to create stress concentrations at the fracture tips, and the micro-fractures started to propagate under the internal stress. However, due to the high confining pressure, the extent of fracture propagation was rather limited. Most of the newly induced fractures were still at the micro scale and hence, more small-amplitude AE events were observed as the confining pressure increased, leading to an increase of the AE b-value.

When the injection rate was 0.4 MPa/s, the b-value varied less intensively with the increase of confining pressure, which was also observed in previous experimental studies (Dong et al. 2022). As granite generally possesses more extensive internal defects than other rock types, the extent of natural fracture development can vary among different granite specimens. The b-value results did not show a linear increase with the increase of the confining pressure. It can be seen that the small-amplitude AE event did not necessarily increase as the confining pressure became higher. Interestingly, it was shown that adopting high injection rate resulted in more evident fluctuations for b-value. This indicated that fracture development became more active at high injection rate, which induced more fractures at both micro and macro scales under the same confining pressure. The increase of injection rate can significantly raise the pressure within the fractures to enhance fracture generation and propagation. Adopting gas fracturing can result in more evident volume-fracturing for granite, which is beneficial for extracting geothermal energy.

3.5 Analysis of macro-fracture propagation path

The macro-fracture patterns of the specimens under different confining pressures at 0.2 MPa/s injection rate are shown in Fig. 8, as indicated by red lines. It can be observed that macro-fractures occurred at both sides of the injection borehole extending from the borehole to the specimen surfaces, which were similar to those generated from hydraulic fracturing (Zhuang et al. 2020). The specimens showed the longest fractures extending nearly to the bottom of the specimen when the confining pressure was 0 MPa. Without confining pressure, microfractures were widely distributed in the granite specimen with high aperture. Therefore, the injected gas can easily penetrate through the micro-fractures under high pressure to expand the natural fractures, which generally resulted in macro-fractures extending from the top of the borehole to its bottom. The vertical fracture length decreased with the increase of the confining pressure, and the vertical fracture length under 12 MPa confining pressure was only 50% of that under 0 MPa confining pressure. This was also consistent with the change of AE counts discussed in Sect. 3.3 (Fig. 5a–e). The internal fractures of the specimen became closed under the confining pressure and therefore, it required higher pressure for the fluid to intrude into the fractures. The length and width of the induced macro-fractures under high confining pressure were obviously smaller than those under low confining pressure. In addition, more secondary fractures were generated under high confining pressure with more intense AE activities. For actual engineering applications, it suggested that as the horizontal ground stress increased, it became increasingly difficult for gas fracturing to fracture rock formations far from the fracturing well. The rock porosity decreased progressively as the confining pressure increased and as a result, the high-pressure gas became difficult to penetrate into the microfractures in the rock formations at the bottom of the well. Accordingly, the extent of volume-fracturing of the granite would be reduced using gas fracturing.

Fig. 8
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Fracturing patterns of the tested specimens using gas fracturing under different confining pressures with an injection rate of 0.2 MPa/s

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Figure 9 shows the macro-fracture patterns of the specimens under different confining pressures at 0.4 MPa/s injection rate. Similar to the case with low injection rate, the vertical fracture length also decreased as the confining pressure increased at high injection rate. This suggested that despite at large injection rate, fracture propagation was limited by higher confining pressure, leading to less volume-fracturing. In realistic conditions, the increase of the reservoir depth can cause major challenges for effective application of the gas fracturing technique to achieve desired volume-fracturing performance. In addition, the induced macro-fractures at high injection rate (0.4 MPa/s) were found to have greater length and width than those at low injection rate (0.2 MPa/s) under the same confining pressure, which also corresponded to more AE events observed at high injection rate in Sect. 3.2. This was because high gas pressure can be reached more promptly at high injection rate to accumulate more damage at the internal defects of the granite, resulting in greater energy release at the breakdown and higher volume-fracturing extent. Moreover, the length of the vertical fractures at high injection rate varied less significant than that at low injection rate. The vertical fracture length under 12 MPa confining pressure can still reach 80% of that under 0 MPa confining pressure. Recall the interesting results discussed in Sect. 3.4, where the variation of b-value under different confining pressures became more evident at high injection rate. These indicated that fracture development in the specimens was more active under high injection rate, and increasing confining pressure only cannot contain fracture propagation. Resultantly, this suggested that fracture networks may be greatly enhanced by increasing the injection rate of gas fracturing the more complex fracture network helps to improve the heat recovery efficiency in EGS and obtain more geothermal energy.

Fig. 9
figure 9

Fracturing patterns of the tested specimens using gas fracturing under different confining pressures with an injection rate of 0.4 MPa/s

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4 A new prediction model for breakdown pressure of gas fracturing

4.1 Basic mechanical model of gas fracturing process

The physical model of the gas fracturing process is illustrated in Fig. 10, where the borehole radius is a. The stress conditions were affected by the tectonic stresses σ2 and σ3, fluid pressure pw, and initial pore pressure p0. Considering stress superposition, the stress components (σrr, σθθ, and τrθ) near the borehole during gas fracturing can be expressed as:

$$\left\{ {\begin{array}{*{20}l} {\sigma_{rr} = \frac{1}{2}\left( {\sigma_{2} + \sigma_{3} } \right)\left( {1 - \frac{{a^{2} }}{{r^{2} }}} \right) + \frac{1}{2}\left( {\sigma_{2} - \sigma_{3} } \right)\left( {1 - \frac{{4a^{2} }}{{r^{2} }} + \frac{{3a^{4} }}{{r^{4} }}} \right)\cos 2\theta + p_{w} \frac{{a^{2} }}{{r^{2} }}} \hfill \\ {\sigma_{\theta \theta } = \frac{1}{2}\left( {\sigma_{2} + \sigma_{3} } \right)\left( {1 + \frac{{a^{2} }}{{r^{2} }}} \right) + \frac{1}{2}\left( {\sigma_{2} - \sigma_{3} } \right)\left( {1 + \frac{{3a^{4} }}{{r^{4} }}} \right)\cos 2\theta - p_{w} \frac{{a^{2} }}{{r^{2} }}} \hfill \\ {\tau_{r\theta } = \frac{1}{2}\left( {\sigma_{3} - \sigma_{2} } \right)\left( {1 + \frac{{2a^{2} }}{{r^{2} }} + \frac{{3a^{4} }}{{r^{4} }}} \right)\sin 2\theta } \hfill \\ \end{array} } \right.$$
(1)

where σrr, σθθ, and τrθ are the radial stress, circumferential stress, and shear stress of a point (r,θ), respectively; r is the distance between the point and the well center (r ≥ a); and θ is the orientation of the point. As σ3 is used as the confining pressure in this study with σ2 = σ3, Eq. (1) can be further given by:

$$\begin{array}{*{20}l} {\sigma_{rr} = \sigma_{3} \left( {1 - \frac{{a^{2} }}{{r^{2} }}} \right) + p_{w} \frac{{a^{2} }}{{r^{2} }}} \hfill \\ {\sigma_{\theta \theta } = \sigma_{3} \left( {1 + \frac{{a^{2} }}{{r^{2} }}} \right) - p_{w} \frac{{a^{2} }}{{r^{2} }}} \hfill \\ {\tau_{r\theta } = 0} \hfill \\ \end{array}$$
(2)
Fig. 10
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Schematic of stress condition around the borehole

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From Fig. 10, the total circumferential stress σθθ is given by superposing σθθ-1, σθθ-2, and σθθ-3 (Haimson and Fairhurst 1969; Ito and Hayashi 1991).

Stress state I σθθ-1 is affected by the ground stress (i.e., confining pressure σ3) and can therefore be given by:

$$\sigma_{\theta \theta - 1}^{{}} = \sigma_{3} \left( {1 + \frac{{a^{2} }}{{r^{2} }}} \right)$$
(3)

Stress state II σθθ-2 is controlled by the total fluid pressure pw (which equals the sum of the initial pore pressure p0 and the atmospheric pressure pg), and it can be expressed as:

$$\sigma_{\theta \theta - 2}^{{}} = - p_{w} \frac{{a^{2} }}{{r^{2} }} = - \left( {p_{0} + p_{g} } \right)\frac{{a^{2} }}{{r^{2} }}$$
(4)

Stress state III σθθ-3 equals to the additional circumferential stress caused by the fluid pressure difference between the well and the rock formations (i.e., pw-p0). Based on the thermal-elastic theory of porous media, σθθ-3 can be given by Haimson and Fairhurst (1969), Ito and Hayashi (1991):

$$\sigma_{\theta \theta - 3}^{{}} = \alpha \frac{1 - 2\nu }{{1 - \nu }}\left[ {\left[ {p\left( {r,t} \right) - p_{0} } \right] - \frac{1}{{r^{2} }}\int\limits_{a}^{r} {\left[ {p\left( {r,t} \right) - p_{0} } \right]rdr} } \right]$$
(5)

where α is the effective stress coefficient α = 1 − K/Ks, K is the bulk modulus of granite, Ks is the bulk modulus of the rock matrix of granite, ν is the Poisson’s ratio, p(r,t) represents the pore pressure distribution during gas fracturing under a constant loading rate C, as given by:

$$p\left( {r,t} \right) = C\int\limits_{0}^{t} {f\left( {r,s} \right)} ds + p_{0}$$
(6)

where f(r,s) is derived following the work by Ito (2008). When C is approaching 0, p(r,t) approaches pw and Eq. (5) becomes:

$$\sigma_{\theta \theta - 3}^{{}} = \frac{\alpha }{2}\frac{1 - 2\nu }{{1 - \nu }}\left( {1 + \frac{{a^{2} }}{{r^{2} }}} \right)\left( {p_{w} - p_{0} } \right)$$
(7)

The conventional failure theories believed that tensile fractures would take place within the rock when the minimum circumferential stress of the well surpasses the tensile strength of the rock (Xie et al. 2020). Accordingly, the total circumferential stress on the well wall is given by:

$$\begin{aligned} \sigma_{\theta \theta } & = \sigma_{\theta \theta - 1} + \sigma_{\theta \theta - 2} + \sigma_{\theta \theta - 3} \\ & = \sigma_{3} \left( {1 + \frac{{a^{2} }}{{r^{2} }}} \right) - p_{w} \frac{{a^{2} }}{{r^{2} }} + \frac{\alpha }{2}\frac{1 - 2\nu }{{1 - \nu }}\left( {1 + \frac{{a^{2} }}{{r^{2} }}} \right)\left( {p_{w} - p_{0} } \right) \\ \end{aligned}$$
(8)

Based on the conventional breakdown pressure models, many subsequent classic predictive models were proposed for the breakdown pressure, such as the H-W criteria (Hubbert and Willis 1957), H-F criteria (Haimson and Fairhurst 1969), and I-H criteria (Ito and Hayashi 1991). These models suggested that tensile fractures would occur near the borehole when the effective circumferential stress σ′θθ reaches the tensile strength of the rock σt, where σ′θθ can be given by:

$$\sigma_{\theta \theta }^{\prime } = \sigma_{\theta \theta } - p_{0} = \sigma_{\theta \theta - 1} + \sigma_{\theta \theta - 2} + \sigma_{\theta \theta - 3} - p_{0} = - \sigma_{t}$$
(9)

The following expression can be derived by substituting Eq. (8) into Eq. (9):

$$\sigma_{\theta \theta }^{\prime } = \sigma_{3} \left( {1 + \frac{{a^{2} }}{{r^{2} }}} \right) - p_{w} \frac{{a^{2} }}{{r^{2} }} + \frac{\alpha }{2}\frac{1 - 2\nu }{{1 - \nu }}\left( {1 + \frac{{a^{2} }}{{r^{2} }}} \right)\left( {p_{w} - p_{0} } \right) - p_{0} = - \sigma_{t}$$
(10)

4.2 Establishment of new breakdown pressure prediction model of gas fracturing

The study by Wang et al. (2021) found that the above predictive models for breakdown pressure, based on the tensile stress criteria, are still essentially local stress criteria which normally give predictions lower than the actual breakdown pressures. Owing to the strain softening or fracturing transitioning regions, the rock would not actually break although the effective tensile stress exceeds the rock tensile strength at certain local point. Ito (2008) suggested that the occurrence of fracture development lagged behind the point failure at the well wall, and the induced fractures originated from inside the rock rather than from the borehole surface. Similar conclusions were also derived from experimental results by Zhang et al. (2018), who stated that the conventional point stress-based models can only be used to obtain the fracture initiation pressure instead of the breakdown pressure of the rock. Subsequently, Wang et al. (2021) proposed a new conceptual model for rock breakdown based on the results of Zhang et al. (2018), as demonstrated in Fig. 11. Tensile failure can occur at the fracture tip when the fluid pressure keeps increasing until reaching the fracture initiation pressure. The fracture length is increased to l0 under the fracture initiation pressure, and higher pressure is required for the fracture length to reach l to cause breakdown of the entire rock.

Fig. 11
figure 11

Conceptual breakdown pressure prediction model based on the average tensile stress criteria (Wang et al. 2021; Zhang et al. 2018)

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Based on the line stress criteria, Wang et al. (2021) proposed a predictive model for the breakdown pressure. By assuming that the fracture initiation and development can be considered equivalent to a fracture with a characteristic length l, the final breakdown occurs when the average tensile stress is reached for the equivalent fracture. This is different from the case where breakdown takes place when the effective stress at a certain point reaches the rock tensile strength. The average tensile stress predictive model can be given by:

$$\frac{1}{l}\int\limits_{l}^{l + a} {\theta_{\theta \theta }^{\prime } dr} = \frac{1}{l}\int\limits_{l}^{l + a} {\left( {\sigma_{\theta \theta } - p_{0} } \right)dr} = - \sigma_{t}$$
(11)

Substituting Eq. (10) into Eq. (11), one can obtain:

$$\frac{1}{l}\int\limits_{l}^{l + a} {\left[ {\sigma_{3} \left( {1 + \frac{{a^{2} }}{{r^{2} }}} \right) - p_{w} \frac{{a^{2} }}{{r^{2} }} + \frac{\alpha }{2}\frac{1 - 2\nu }{{1 - \nu }}\left( {1 + \frac{{a^{2} }}{{r^{2} }}} \right)\left( {p_{w} - p_{0} } \right) - p_{0} } \right]dr} = - \sigma_{t}$$
(12)

where l is the characteristic length of the equivalent fracture. Considering λ = l/a and η = α(1 − 2ν)/(1 − ν), the expression for the breakdown pressure can be obtained by integrating both sides of Eq. (12) after simplification as:

$$p_{w} = \frac{{2(2 + \lambda )\sigma_{3} + 2(1 + \lambda )\sigma_{t} - \eta (2 + \lambda )p_{0} }}{(2 + \lambda )(2 - \eta )}$$
(13)

From the analysis in Sect. 3.1, higher fluid pressure was required to enable fracture initiation and development, as the fluid became more difficult to penetrate the compacted natural fractures under the confining pressure. Resultantly, this would affect the circumferential stress induced by the total fluid pressure and the process of fluid penetrating from the injection borehole wall to the inside of the granite, as demonstrated in Fig. 12. Hence, modifications are needed for σθθ-2 and σθθ-3, i.e., Eqs. (4) and (7) (Wang et al. 2021; Yu et al. 2022). Considering the effect of confining pressure on the gas fracturing process from fracture initiation to breakdown, we incorporated both ξ1 and ξ2 to account for the influences of confining pressure on gas flow. Accordingly, Eqs. (4) and (7) become:

$$\sigma_{\theta \theta - 2} = - \xi_{1} p_{w} \frac{{a^{2} }}{{r^{2} }} = - \xi_{1} \left( {p_{0} + p_{g} } \right)\frac{{a^{2} }}{{r^{2} }}$$
(14)
$$\sigma_{\theta \theta - 3} = \frac{\alpha }{2}\frac{1 - 2\nu }{{1 - \nu }}\left( {1 + \frac{{a^{2} }}{{r^{2} }}} \right)\left( {\xi_{2} p_{w} - p_{0} } \right)$$
(15)
Fig. 12
figure 12

Schematic of fracture compaction under the confining pressure

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Substituting Eqs. (13) and (14) into Eq. (10), one may obtain:

$$\begin{aligned} & \frac{1}{l}\int\limits_{l}^{l + a} {\theta_{\theta \theta }^{\prime } dr} \\ & = \frac{1}{l}\int\limits_{l}^{l + a} {\left[ {\sigma_{3} \left( {1 + \frac{{a^{2} }}{{r^{2} }}} \right) - \xi_{1} p_{w} \frac{{a^{2} }}{{r^{2} }} + \frac{\alpha }{2}\frac{1 - 2\nu }{{1 - \nu }}\left( {1 + \frac{{a^{2} }}{{r^{2} }}} \right)\left( {\xi_{2} p_{w} - p_{0} } \right) - p_{0} } \right]dr} = - \sigma_{t} \\ \end{aligned}$$
(16)

Similarly, by setting λ = l/a and η = α(1 − 2ν)/(1 − ν), the breakdown pressure predictive model can be derived by integrating both sides of Eq. (16) after simplification as:

$$p_{w} = \frac{{2(\lambda^{2} - 1)\sigma_{3} + 2\lambda^{2} \sigma_{t} - \left[ {\left( {\eta - 2} \right)\lambda^{2} - \eta } \right]p_{0} }}{{\xi_{2} \eta - 2\xi_{1} - \eta \lambda^{2} }}$$
(17)

In addition, higher confining pressure leads to significant increase in the breakdown pressure, whereas λ is derived from the elastic solution for isotropic infinite plane under plane stress, which is different from the actual conditions. Therefore, a coefficient ζ was introduced to modify σ3 in Eq. (17), and the predictive model for the breakdown pressure with consideration of the effect of confining pressure can be expressed as:

$$p_{w} = \frac{{2\zeta (\lambda^{2} - 1)\sigma_{3} + 2\lambda^{2} \sigma_{t} - \left[ {\left( {\eta - 2} \right)\lambda^{2} - \eta } \right]p_{0} }}{{\xi_{2} \eta - 2\xi_{1} - \eta \lambda^{2} }}$$
(18)

To examine the validity of the proposed predictive model for breakdown pressure, the breakdown pressure of the granite was assessed under two different injection rates, and the associated parameters were given as follows: the tensile strength σt = 12.34 MPa, Poisson’s ratio ν = 0.20, and effective stress coefficient α = 0.12. The effect of initial pore pressure p0 can be neglected during the gas fracturing process as the granite has quite low permeability and porosity (Yu et al. 2022).

The test results were fitted using Eq. (16) under the two injection rates, as shown in Fig. 13. The equivalent characteristic fracture length l was 5.379 mm at 0.2 MPa/s injection rate and 5.254 mm at 0.4 MPa/s injection rate, indicating that l can be evidently reduced at high injection rate. Smaller l suggests shorter distance for stable fracture development and lower fracture initiation length, resulting in more significant unstable fracture development and greater extent of progressive damage. Consequently, the injected gas can more easily cause fracture propagation, which leads to reduced breakdown pressure that is consistent with the observations in the experiments.

Fig. 13
figure 13

Comparison between the proposed model prediction results and experiments

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The breakdown pressure with the increase of confining pressure is shown in Fig. 14 considering various equivalent characteristic length l (ζ, ξ1, and ξ2 were determined from the fitting results at 0.2 MPa/s injection rate). From the results, the effect of confining pressure demonstrated important effect on the gas fracturing process, where the breakdown pressure increased with the increase of confining pressure, which is consistent with the experimental results. Additionally, the increment of the breakdown pressure under an increasing confining pressure became more evident with larger equivalent characteristic length l. Larger equivalent characteristic length l generally suggested longer distance of stable fracture propagation and hence, higher gas pressure was needed to fully penetrate the fractures. Therefore, it became more difficult for the fractures to propagate from the fracture tip till breakdown, resulting in higher breakdown pressure. Under the effect of confining pressure, the fracture aperture was reduced under compression with less fracture void space. However, the stable fracture propagation length became larger in this condition, which led to the increase of equivalent characteristic length l and greater increment of granite breakdown pressure. Higher confining pressure resulted in larger breakdown pressure required to achieve the same length for the stable fractures. The new model can well describe the fracture behavior of granite under confining pressure and provide a theoretical basis for the development of geothermal energy in hot dry rock.

Fig. 14
figure 14

Change of breakdown pressure for different characteristic fracture length l and confining pressure

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5 Conclusions

In this study, the mechanism and influencing factors of gas fracturing were assessed to explore its application in extracting geothermal energy. The effect of the confining pressure on the gas fracturing performances was analyzed for the first time. Gas fracturing tests were conducted with Yanshanian granite outcrops from the Pearl River Delta of the Guangdong-Hong Kong-Macao Greater Bay Area, under varying confining pressure, axial pressure, and injection rate. The breakdown pressure–time relationships, AE characteristics, and macro-fracture patterns were examined to assess the combined effect of confining pressure and injection rate on gas fracturing performances. In addition, a novel theoretical model was presented by incorporating the effect of confining pressure to study the fracture propagation during gas fracturing. Results from the proposed model were found to closely match the test results, suggesting its robustness in describing the gas fracturing mechanism under the effect of confining pressure. The main conclusions of this work are summarized as follows:

  1. (1)

    From the gas breakdown pressure–time curves, the gas fracturing process can be divided into three stages including the gas compaction stage, gas pressuring stage, and pressure relieving stage. When at the fixed injection rate, longer gas compaction stage was observed as the confining pressure became higher, with larger breakdown pressure but reduced breakdown pressure increment. Under fixed confining pressures, the breakdown pressure was evidently reduced with the increase of injection rate, and this difference became increasingly large as the confining pressure increased.

  2. (2)

    Both higher confining pressure and injection rate can lead to the increase of AE counts. More secondary fractures appeared under high confining pressure to cause more internal damage and greater extent of volume-fracturing. At high injection rate (0.4 MPa/s), more energy was instantly released when the specimens failed, generating a wider range of internal damage for more developed secondary fractures and higher extent of volume-fracturing.

  3. (3)

    At low injection rate (0.2 MPa/s), the b-value of AE generally showed a rising trend as the confining pressure increased with more small-amplitude AE event occurred. However, the b-value fluctuated significantly as the confining pressure increased at high injection rate (0.4 MPa/s), and demonstrated large deviations from that at low injection rate under all confining pressures. This was because the granite often consists of large mineral particles, which results in varying internal defects for different specimens. The number of micro-fractures and macro-fractures was significantly increased when adopting a higher injection rate, which led to more evident fluctuation of the b-value.

  4. (4)

    The length of the vertical macro-fractures generated on both sides of the injection borehole decreased as the confining pressure increased from the gas fracturing experiments, indicating that the specimen became less prone to breakdown under high confining pressure. The extent of volume-fracturing was evidently enhanced with the increase of injection rate. Both the length and width of the generated vertical fractures were obviously larger at high injection rate than at low injection rate.

  5. (5)

    By incorporating the equivalent characteristic fracture length l and coefficients ζ, ξ1, and ξ2 for the effect of confining pressure on gas fracturing, a novel predictive model was proposed for the breakdown pressure of granite during gas fracturing based on the average tensile stress criteria. The developed model can provide accurate estimate of the breakdown pressure, and can be used to effectively analyze the relations between the breakdown pressure, confining pressure, and equivalent characteristic fracture length l.