1 Introduction

Hydrofracturing refers to injecting high-pressure fluid (water, gas, etc.) into a formation through drilling, which induces wellbore breakage and fracture propagation under the action of hydromechanical coupling. Finally, artificial fractures are formed in the formation (King 2012; Fjaer 2008). Hydraulic fracturing is the most widely used and mature hydrofracturing technology. It has been successfully applied to the development of shale oil and gas, tight oil and gas, coalbed methane and dry hot rock (Zhang and Yin 2014; Zhang et al. 2018; Li et al. 2018; Clarkson et al. 2016; Liu et al. 2017; Zhuang et al. 2016).

Laboratory physical model experiments are important methods for studying hydraulic fracturing. The true triaxial hydraulic fracturing experiment system is used to simulate hydraulic fracturing under in situ stress conditions (Zhao et al. 2019a, b; Zhang et al. 2019; Tan et al. 2017; Guo et al. 2018; Zhang and Fan 2014; Huang and Liu 2017). The initiation and propagation of hydraulic fractures are studied, as well as fracture morphology (Hou et al. 2018; Zhou et al. 2008; Li et al. 2016; Dehghan et al 2015; Lhomme et al. 2002). The effects of factors such as the principal stress difference, pum** rate, fracturing medium, natural fractures, reservoir lithology and mechanical properties on the breakdown pressure and fracture morphology of hydraulic fracturing are analyzed (Zhuang et al. 2019; Bennour et al. 2015; Zhao et al. 2019a, b; Liu et al. 2014). During the hydraulic fracturing of homogeneous rock, the initiation and propagation direction of the main hydraulic fractures are controlled by a three-dimensional in situ stress field, which initiates and propagates perpendicular to the direction of the minimum principal stress (Zhou et al. 2008; Zhuang et al. 2019). The three-dimensional hydraulic fractures are in the form of oblate ellipsoids (Zhang et al. 2019; Li et al. 2016; Zhao et al. 2019a, b). A smaller principal stress difference corresponds to an easier formation of the fracture network (Jiang et al. 2016). When hydraulic fractures propagate to the bedding plane, there are three propagation modes: propagation along the bedding plane, across the bedding plane, and alternate propagation along and across the bedding plane (Huang and Liu 2017). When hydraulic fracturing is performed in naturally fractured reservoirs, not only can new fractures be generated, but natural fractures can also be activated and extended (Mou et al. 2021; Cheng et al. 2015). A complex fracture network is easy to form after hydraulic fractures connect with natural fractures (Abe et al. 2021).

In oil and gas reservoirs, oil and gas are stored in the pores of the reservoir. The reservoir rock is in a fluid-saturated state and has a certain pore pressure, that is, the reservoir pressure (Peng et al. 2021). Crack initiation and propagation in fluid-saturated rock are controlled by the interaction between fluid flow and rock deformation. The description of hydraulic coupling is crucial for fracture experiments on fluid-saturated rock, and the related fluid mechanics model and constitutive theory are also well-developed (Zhang et al. 2021a, b; Smirnov et al. 2020; Kartseva et al. 2022). Hydraulic fracturing has a particularly significant effect on fluid-saturated rock damage. Specifically, the pore pressure in rock can promote the initiation and propagation of hydraulic fractures, and the distribution of the pore pressure gradient in porous rock can affect the propagation direction of hydraulic fractures (Geertsma 1973). Based on this, Laurent (Laurent et al. 1993) established a relationship between pore pressure and the mechanical properties of rock by using the effective stress coefficient method and verified its effectiveness in the study of the mechanical mechanism of fluid-saturated rock through various experiments. The study of fluid-saturated rock hydrostatic experiments includes various fields. The triaxial compression test of fluid-saturated rock for oil and gas reservoir exploitation shows that higher pore pressure can decrease the compressive strength of rock and reduce the compressibility of rock, thus improving the extraction efficiency (Wu et al. 2021; Li et al. 2021). The hydraulic fracturing of geothermal reservoirs also shows that the response of mechanical properties and fracture mechanism of fluid-saturated hot dry rock is the key to experimental research, and a higher degree of rock saturation correlates to lower mechanical parameters such as the crack damage stress ratio, fracture toughness and cohesion strength under different stress conditions; this trend confirms that the saturation of hot dry rock increases the pore pressure of rock and further promotes the expansion of cracks in hydraulic fracturing (Wang et al. 2021; Barbosa et al. 2019). In the plane strain compression experiment for evaluating microseismic activity and fracture in fluid-saturated rock, the occurrence of an inelastic response of fluid-saturated rock is consistent with the increase in the acoustic emission activity rate; thus, the crack initiation and propagation process of fluid-saturated rock can be monitored by acoustic emission location (Makhnenko et al. 2020; He et al. 2022). However, pore pressure was not considered in the previous laboratory hydraulic fracturing experiments. The fluid in the pores is lost or desorbed during sample preparation or processing (Zhou et al. 2017; Weijers and Pater 1992). Since the atmospheric pressure is not considered, there is no initial pore pressure in the pores of the sample before the experiment.

During hydraulic fracturing, the pressure water penetrates into the rock along the fracture, creating pore pressure and generating a pore pressure gradient (Tang et al. 2002). The initiation and propagation of hydraulic fractures are the results of hydromechanical coupling. The initial pore pressure will affect the seepage process of pressure water from the borehole or fractures to the pores of the surrounding rock, which will further affect the initiation and propagation of hydraulic fractures.

In the past, the effect of pore pressure was mainly studied by theoretical analysis based on the effective stress principle of saturated soil mechanics (Hubbert and Willis 1957; Haimson and Fairhurst 1969). The effect of pore pressure and its gradient on hydraulic fracturing is still unclear (Ito 2008). The practice of hydraulic fracturing in gas-bearing coal seams in underground coal mines and preliminary experimental studies show that the breakdown pressure of hydraulic fracturing may increase with increasing pore pressure (Huang et al. 2018), which cannot be explained by traditional theory. Therefore, it is necessary to deeply understand the effect of pore pressure on rock hydraulic fracturing.

In this study, the problem of the disregarded initial pore pressure effect in traditional laboratory experiments was addressed; a hydraulic fracturing experimental system considering hydromechanical coupling under pseudotriaxial stress loading was utilized to perform rock hydraulic fracturing using an initial evenly distributed pore pressure. The influence of different initial evenly distributed pore pressures on the rock hydraulic fracturing process was analyzed. The experimental process was more similar to the actual hydraulic fracturing process of oil and gas reservoirs. These research results provide a more comprehensive theoretical basis for resolving technical problems of unconventional oil and gas resource exploitation.

2 The experimental principle of the effect of the initial evenly distributed pore pressure on sandstone hydraulic fracturing

The sandstone hydraulic fracturing experiment process under the initial evenly distributed pore pressure conditions includes two stages. The first stage is to preset the initial pore pressure (Fig. 1a), where the same pore pressure is applied to the borehole and to the top and bottom ends of the sample. The pressurized water continuously seeps into the sample, and finally, an initial evenly distributed pore pressure is formed inside the sample. The second stage is the hydraulic fracture stage (Fig. 1b), where hydraulic fracturing experiments are performed with a constant pum** rate after the internal seepage of the sample reaches a stable state. During hydraulic fracturing, the top and bottom ends of the sample are continuously injected with water to ensure that the initial pore pressure inside the sample remains constant during the hydraulic fracturing process.

Fig. 1
figure 1

The technological process of dense linear multidrilling hole hydraulic fracturing

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A key problem with this experiment is how to determine whether the seepage process reaches a steady state. The method used to judge the internal seepage balance in the sample is shown in Fig. 2. First, a pre-seepage experiment was performed to test the seepage equilibrium time. The same pore water pressure was applied to the top and bottom ends of the sample for seepage; at the same time, the borehole pipeline valve was closed to monitor the water pressure. The seepage time (h) was recorded when the water pressure of the borehole pipeline reached a stable level. Then, a designed pore water pressure was loaded at the borehole and on the top and bottom ends of the sample for seepage at the same time. After the seepage time reached h, the valve of the borehole pipeline was closed, and the water pressure gauge was observed. If the water pressure in the borehole did not fluctuate within half an hour, the seepage inside the sample was considered to have reached equilibrium. A hydraulic fracturing experiment with a constant pum** rate was carried out. If the water pressure in the borehole decreased, the seepage was continued until the water pressure became stable after the borehole pipeline valve was closed.

Fig. 2
figure 2

Determination of the condition of seepage equilibrium inside the sample

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3 The physical model experiment of the effect of the initial evenly distributed pore pressure on sandstone hydraulic fracturing

3.1 Experimental system and sample preparation

The hydraulic fracturing experimental system with hydromechanical coupling under triaxial stress loading was used to perform sandstone hydraulic fracturing experiments to determine the effect of the initial evenly distributed pore pressure (Fig. 3). The experimental system consists of the framework of the experimental platform, the loading system, and the monitoring system. This system can determine the integrated experimental function of triaxial loading, seepage, and hydraulic fracturing of cylinder samples with a diameter of 50 mm and a height of 100 mm.

Fig. 3
figure 3

The hydraulic fracturing experimental system considering hydromechanical coupling under triaxial stress loading

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The loading system includes four channels of an electrohydraulic servo control loading system and six channels of a hydraulic pressure control loading system. The four-channel electrohydraulic servo control loading system uses the MOOG valve to control the axial stress, confining stress and water pressure loading. The maximum output pressure of each channel is 31.5 MPa. Two channels are utilized for axial stress and confining stress loading. One channel controls the oil–water conversion supercharger to achieve water pressure control loading. The maximum water pressure loading can reach 63 MPa. The loading and unloading of the water pressure can be controlled by the L/min and MPa/min modes, and the loading path can be preset. Additionally, the pressure sensor and displacement sensor at the back end of the oil–water loading converter provide feedback signals to control the servo valve and achieve the dynamic control of water pressure loading. The loading of the top and bottom seepage pressure is controlled by the six-channel hydraulic pressure loading system. It can achieve 0 ~ 60 MPa water pressure loading, and the control accuracy is 0.5%. According to the loading mode of L/min or MPa/min, data such as loading pressure, displacement, and liquid injection volume are automatically collected. During the experiment, the change curve of the axial stress, confining stress, water pressure, displacement, and seepage pressure of the sample can be displayed in real time on the control software interface, and the data of the whole experimental process can be recorded. Moreover, an acoustic emission (AE) instrument with 8 channels was used to monitor the microfracturing and crack propagation process in real time.

Coarse sandstone samples with a diameter of 50 mm and a height of 100 mm were used for the experiment. A borehole with a depth of 65 mm and a diameter of 6 mm was drilled on the end surface of the cylindrical sandstone sample and was used as the injection hole for hydraulic fracturing. The stainless-steel tube was sealed to the borehole with epoxy-type reinforcement glue, and the depth of the sealing section was 35 mm. A naked borehole fracturing section with a length of 30 mm was left in the middle of the sample (Fig. 4).

Fig. 4
figure 4

Coarse sandstone samples

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To reduce the influence of the discreteness of the physical and mechanical properties of the samples on the experimental results, the samples were prepared by intensive sampling on the same sandstone block. At least 3 samples were tested for each initial pore pressure experiment. At the same time of sampling, a standard sample used for physical and mechanical parameter testing was sampled from the same sandstone block. The physical and mechanical properties are shown in Table 1.

Table 1 Physical and mechanical parameters of the samples
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3.2 The experimental scheme

The experimental scheme is shown in Table 2. The in situ stress condition of the reservoir with a burial depth of 480 m is simulated in the experiment. The lateral pressure coefficient is set as 0.67; thus, the axial stress is 12 MPa and the confining stress is 8 MPa. Tap water is used as the fracturing fluid. Five groups of hydraulic fracturing experiments under different initial evenly distributed pore pressures were carried out to compare and analyze the influence of the initial pore pressure on the hydraulic fracturing process. At least 3 samples were carried out for each group of experiments to eliminate the influence of the discreteness of samples on the experimental results (Table 2).

Table 2 Experimental scheme of hydraulic fracturing under different initial evenly distributed pore pressures
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In experiments 1 ~ 5, the hydraulic fracturing experiment was carried out by presetting an initial evenly distributed pore water pressure, and the pore pressure increased by 1.5 MPa between each group of experiments. Before the start of the formal experiment, a preseepage experiment was carried out to test the seepage equilibrium time. The borehole pipeline pressure reached a stable level after 8 h when the sample was loaded with a seepage pressure of 1.5 MPa on both ends under the abovementioned stress conditions. To ensure the even distribution of pore pressure inside the sample before hydraulic fracturing, the total seepage time set in the experiment was 10 h. After 9 h, the valve of the borehole pipeline was closed to observe whether the water pressure in the borehole was stable. If the water pressure was stable, then the hydraulic fracturing experiment was prepared. After 10 h, the hydraulic fracturing experiment was carried out with a pum** rate of 30 mL/min.

3.3 The experimental procedure

  1. (1)

    Sample installation: Photos of the sample were taken and then placed into the experimental loading device. The sealing performance of the system and the sealing effect of the samples with a hand pump were tested.

  2. (2)

    Seepage process: First, the axial stress and confining stress loading pipeline of the experimental loading frame were connected with the output oil circuit of the four-channel electrohydraulic servo control loading system. Then, the top and bottom seepage pipeline of the experimental loading frame and the borehole seepage pipeline were connected with the oil–water conversion supercharger. After the pipelines were connected and checked, the four-channel electrohydraulic servo control loading system was opened. After the loading target values of axial stress and confining stress were set, the system controlled the output of axial stress and confined stress to the set value and stabilized for five minutes. Finally, the six-channel hydraulic pressure stabilized control loading system was opened to set the loading target seepage pressure values for the borehole and the top and bottom of the sample. The system controlled the output seepage pressure to the set value.

  3. (3)

    Pressure stabilization of the borehole seepage pipeline: After the seepage time reached 9 h, the borehole seepage pipeline was closed to observe the change in pressure in the borehole. If the water pressure in the borehole did not change within half an hour, the inside of the sample was determined to reach a state of seepage equilibrium. The hydraulic fracturing pipeline of the oil–water conversion supercharger controlled by the servo control system was connected to the borehole pipeline through a three-way valve. The AE system was connected, and two AE probes were arranged outside the loading frame.

  4. (4)

    Hydraulic fracturing process: When the rock hydraulic fracturing experiment was performed, the rose-red dye used in the poster was added to the fracturing fluid. Then, the mixture was stirred evenly to ensure that the shape of the hydraulic fracture after the experiment was easily observed. The initial pore pressure target value was set and output through the servo control system. After the water pressure of the hydraulic fracturing pipeline reached the initial pore pressure target value, the valve connecting the hydraulic fracturing pipeline and the borehole pipeline was opened. The hydraulic fracturing experiment was carried out at a pum** rate of 30 mL/min after the pressure was stabilized for 5 min.

  5. (5)

    End of the experiment: The pump was turned off, and the experiment was stopped after 25 min.

4 Experimental results and analysis

4.1 Effects of the initial evenly distributed pore pressure on hydraulic fracture initiation

The water pressure and AE curves of sandstone hydraulic fracturing under different initial evenly distributed pore pressures are shown in Fig. 5. One test per group was selected for illustration. When the initial evenly distributed pore pressure is 1.5 MPa, the water pressure increases from 1.5 MPa to 13.465 MPa within 2.269 min after the start of pum**, and then the borehole wall ruptures. After hydraulic fracture initiation, the water pressure drops significantly to 3.029 MPa after 0.258 min. The rate of decrease is 40.450 MPa/min, and the reduced magnitude is 76.17%. Then, the water pressure slowly drops to 1.995 MPa and fluctuated at approximately 2 MPa (Fig. 5a). According to the water pressure curves, the sample rupture is relatively severe at the moment of fracture initiation. The hydraulic fractures instantly extended to the edge of the sample, leading to a sharp decrease in water pressure after fracture initiation. In the subsequent pum** process, the water pressure is always maintained at approximately 2 MPa. From the detected AE phenomenon, the AE count and energy abnormally increased at the moment of fracture initiation. The energy released at the moment of fracture initiation is large, and the rupturing is severe.

Fig. 5
figure 5

Water pressure and AE of hydraulic fracturing under different initial evenly distributed pore pressures

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When the initial evenly distributed pore pressure is 3.0 MPa, the water pressure increases from 3.0 MPa to 14.179 MPa within 2.557 min after the start of pum**, and the borehole wall rupture. Then, the water pressure decreases slightly and then decreases to 9.37 MPa after 0.32 min. The rate of decrease is 40.450 MPa/min, and the reduced magnitude is 76.17%. During fracture propagation, the water pressure begins to rise again and fluctuates at approximately 13 MPa (Fig. 5b).

When the initial evenly distributed pore pressure is 4.5 MPa, the water pressure increases from 4.5 MPa to 14.757 MPa within 2.897 min after the start of pum**, and the borehole wall ruptures. Then, the water pressure immediately drops to 8.999 MPa with a decrease in magnitude of 39.02%. Then, the water pressure begins to quickly rise to 13.955 MPa after 4.697 min. In the subsequent pum** process, the water pressure drops instantly to 6.487 MPa and stabilizes at approximately 6.5 MPa (Fig. 5c).

When the initial evenly distributed pore pressure is 6.0 MPa, the water pressure increases from 6.0 MPa to 15.646 MPa within 1.728 min after the start of pum**, and the borehole wall ruptures. Then, the water pressure decreases significantly to 7.205 MPa at 0.104 min later, with a decrease in magnitude of 53.95%. During fracture propagation, the water pressure fluctuates, slowly increases to 10 MPa and then fluctuates down to 7.5 MPa after 14 min. In the subsequent pum** process, the water pressure is stable at 7.5 MPa (Fig. 5d).

When the initial evenly distributed pore pressure is 7.5 MPa, the water pressure increases from 7.5 MPa to 16.262 MPa within 1.784 min after the start of pum**, and the borehole wall ruptures. Then, the water pressure decreases significantly to 7.095 MPa with a decrease of 56.37%. Finally, the water pressure stabilizes at 7 MPa (Fig. 5e).

The breakdown pressure of sandstone hydraulic fracturing under different initial evenly distributed pore pressures is shown in Table 3. With increasing pore pressure, the breakdown pressure of hydraulic fracturing also increases. By fitting the experimental results, the relationship between breakdown pressure and pore pressure follows a linear growth trend (Fig. 6) (Eq. 1), which indicates that the breakdown pressure has a linearly increasing relationship with the initial evenly distributed pore pressure.

$$P_{b} = 12.4906 + 0.5513P_{0}$$
(1)
Table 3 Fracture pressure, AE energy, and pressure drop after hydraulic fracturing under different initial evenly distributed pore pressure conditions
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Fig. 6
figure 6

Relationship between fracture pressure and initial pore pressure

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It should be noted that although the breakdown pressure of sandstone hydraulic fracturing increases with increasing initial pore pressure, the difference is not significant. There are two main reasons for this: (1) In the process of long-term seepage of sandstone samples, the pore pressure will cause deformation of the pores and weaken the samples. This causes the strength of the sample to decrease. In addition, the greater the pore pressure (seepage pressure) is, the more serious the damage to the strength of the samples during the seepage process. Therefore, the water pressure required to fracture the borehole wall decreases accordingly. Actually, the difference in the breakdown pressure for sandstone hydraulic fracturing under different initial evenly distributed pore pressures is greater than the experimental results. (2) As only the pore pressure is different in the comparison experiments, the other conditions remain the same. In addition, the difference in pore pressure in each group of comparison experiments is only 1.5 MPa, so the difference in breakdown pressure does not drastically differ.

The AE energy at the moment of borehole wall rupture and the reduced magnitude of water pressure after rupture are shown in Fig. 7. According to the AE energy monitored in the process of hydraulic fracturing, apart from the initial pore pressure of 1.5 MPa, the breakdown pressure increases with increasing initial pore pressure. The AE energy at the moment of borehole wall fracturing also increases accordingly. Additionally, with an increase in the initial pore pressure, the reduced magnitude of the water pressure after borehole wall fracturing correspondingly increases; therefore, with an increase in the initial pore pressure, the fracturing range of the borehole wall correspondingly increases.

Fig. 7
figure 7

AE energy at the moment of borehole wall rupture and the reduced magnitude of water pressure after rupture

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According to the water pressure curve after fracture initiation, an increase in the initial pore pressure corresponds to an increase in the fracturing range of the borehole wall. When the initial pore pressure is 3.0 MPa, the water pressure drops sharply after the borehole wall ruptures and then increases to 13 MPa. During the subsequent pum** process, the water pressure maintains a fluctuation of approximately 13 MPa, indicating that the range of fracture initiation is smaller. The water pressure is always maintained at a higher pressure value after fracture initiation, which promotes the continuous propagation of hydraulic fractures (Fig. 5b). When the initial pore pressure is 4.5 MPa, the water pressure drops sharply after the borehole wall ruptures. Then, it begins to rise in a fluctuating manner and remains at a high-pressure value within 5 min. Next, the water pressure stabilizes at approximately 6.5 MPa. This indicates that the fracturing range of the borehole wall is larger than that when the initial pore pressure is 3.0 MPa, and the hydraulic fracture is connected with the sample surface after propagating for 5 min (Fig. 5c). When the initial pore pressure is 6.0 MPa, the tip of the hydraulic fracture is close to the sample surface after the initial borehole wall fracturing. The subsequent water pressure fluctuates between 7.5 MPa and 9 MPa and then remains at approximately 7.5 MPa (Fig. 5d). When the initial pore pressure is 7.5 MPa, the initial fracturing range of the borehole wall is larger. After the initial fracturing, the fracture is connected with the edge of the sample, resulting in a maintained water pressure of approximately 7 MPa (Fig. 5e).

4.2 Morphology of hydraulic fracture propagation under an initial evenly distributed pore pressure

The hydraulic fracture morphology of sandstone hydraulic fracturing under different initial evenly distributed pore pressures is shown in Fig. 8. Under the control of the loading stress field, the confining stress is the minimum principal stress, and the hydraulic fracture is an axial fracture. After each group of experiments, axially distributed hydraulic fractures can be observed on the surface of the samples.

Fig. 8
figure 8figure 8

Fracture morphology under different initial evenly distributed pore pressures

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When the initial pore pressure is 1.5 MPa, a hydraulic fracture can be observed at the upper part of the sample, and it propagates through to the upper surface of the sample (Fig. 8a). The lower part of the sample has no visible hydraulic fracture. Combined with the water pressure curve in the hydraulic fracturing process, it can be inferred that the initial rupture occurs mainly along the upper part of the sample. In addition, the rupture is more severe at the moment of fracture initiation. The hydraulic fracture directly penetrates to the upper surface of the sample, resulting in the water pressure drop** to 3.029 MPa instantaneously after the initial rupture. Then, the water pressure is maintained at approximately 2 MPa and does not rise again. Therefore, the hydraulic fractures do not further propagate to the lower part of the sample.

When the initial pore pressure is 3.0 MPa, a red watermark distributed along the axial direction is clearly observed in the middle of the sample. The red watermark is only in the middle of the sample and is not connected with the top and bottom surfaces of the sample. This indicates that microfractures or the permeation zone at the front of the hydraulic fracture tip have extended to the sample surface, and the borehole walls initially rupture along the middle of the fracturing section. Combined with the water pressure curve, it can be seen that the water pressure is always maintained at a high state in the fracture propagation process, which indicates that the hydraulic fracture is always in a state of propagation and extension (Fig. 8b).

When the initial pore pressure is 4.5 MPa, a connected axial hydraulic fracture can be observed on the surface of the sample (Fig. 8c), indicating that the hydraulic fracture is fully connected with the sample surface. Combined with the water pressure curve, it decreases slightly after the borehole wall initially ruptures. During fracture propagation, the water pressure continues to rise. When the hydraulic fracture is connected with the surface of the sample, the water pressure drops sharply again.

When the initial pore pressure is 6.0 MPa, a penetrating axial hydraulic fracture appears on the sample surface after the experiment. The fracture opening is significantly larger than that of the hydraulic fracture when the initial pore pressure is 4.5 MPa (Fig. 8d).

When the initial pore pressure is 7.5 MPa, two connected and parallel hydraulic fractures appear on the surface of the sample. The spacing between the two hydraulic fractures is very small. However, according to the water pressure curve, no secondary rupture occurs in the borehole wall. Therefore, the double fractures might be caused by bifurcations in the process of fracture propagation to the sample surface (Fig. 8e). Bifurcations occurred in two tests with 7.5 MPa pore pressure, which are shown in Fig. 8e. An anomaly occurred in the third test, where rapid rupture of the sample occurred, the test duration was very short, and only a single hydraulic fracture was observed. The high pore pressure led to an increase in the pure water pressure in the fracture tip. The randomness of the crack direction may increase, which leads to bifurcation of the cracks.

According to the hydraulic fracture morphology on the sample surface after the experiment and the water pressure curve, it can be concluded that a larger initial pore pressure correlates to a larger range of the initial rupture. During fracturing propagation, the propagation range and opening of the fracture also increase as the initial pore pressure increases within the same pum** time.

5 Discussion

5.1 The comparison of the experimental results with classic models

The breakdown pressure is an important parameter in the initiation of hydraulic fractures, which can reflect the fracturing mechanism of rock during hydraulic fracturing. The most widely used formulas of traditional theory for calculating the breakdown pressure of rock hydraulic fracturing are the H-W formula proposed by Hubbret M.K. and Willis D.G. in 1957 and the H-F formula proposed by Bezalel Haimson and Charles Fairhurst in 1967 (Hubbert and Willis 1957; Haimson and Fairhurst 1969). The comparison of experimental results with classic models is shown in Fig. 9. For the axial fractures, the experimental results are opposite to the variation trend predicted by the classic model.

Fig. 9
figure 9

Comparison of the experimental results with the classic models

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Sandstone is a permeable porous medium composed of mineral particles and inter-particle pores. The pore pressure effect lead to the pores of the sandstone expands and deforms. The mutual squeezing force between mineral particles increases.

5.2 The effect of pore pressure on the initiation of the hydraulic fracture tip

Sandstone is a permeable porous medium composed of mineral particles and interparticle pores (Fig. 10a). Based on the mesostructure of sandstone, the particle structure model of the hydraulic fracture tip is established (Fig. 10b). The mesoscopic initiation process of the hydraulic fracture tip under the action of pore pressure is analyzed.

Fig. 10
figure 10

The effect of pore pressure on the initiation of the hydraulic fracture tip

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In the same pore, the pore pressure is equal and uniformly acting on the surface of the particle (Fig. 10c). In addition, the direction of the force points to the center of the particle. The pore pressure acts on the surface of the particles, which manifests as a compressive stress perpendicular to the contact surface. The uniform load acting on the particle surface by the pore pressure is converted into a concentrated force using the following equation:

$$f_{P} = \smallint_{ - \beta }^{\beta } P\cos \theta r{\text{d}}\theta$$
(2)

where \(\beta\) is half of the inner angle of the pore pressure area, and the integral of Eq. (2) can be obtained as follows:

$$f_{P} = 2Pr\sin \beta$$
(3)

where \(r\) is the radius of the bonded particle.

The pore pressure on both sides of the bonding plane after seepage is defined as \(P_{1}\), \(P_{2}\), then the concentrated forces on the particles by the pore pressure on both sides of the bonding plane are as follows:

$$f_{{P_{1} }} = 2P_{1} r\sin \beta$$
(4)
$$f_{{P_{2} }} = 2P_{2} r\sin \beta$$
(5)

The concentrated forces \(f_{{P_{1} }}\) and \(f_{{P_{2} }}\) acting on the particles by the pore pressure on both sides of the bonding plane are decomposed along the normal and tangential directions of the bonding surface.

$$f_{{P_{1} }}^{n} = 2P_{1} r\sin \beta \cos \beta = P_{1} r\sin 2\beta$$
(6)
$$f_{{P_{1} }}^{s} = 2P_{1} r\sin \beta \sin \beta = 2P_{1} r\sin^{2} \beta$$
(7)
$$f_{{P_{2} }}^{n} = 2P_{2} r\sin \beta \cos \beta = P_{2} r\sin 2\beta$$
(8)
$$f_{{P_{2} }}^{s} = 2P_{2} r\sin \beta \sin \beta = 2P_{2} r\sin^{2} \beta$$
(9)

Among them, the component force along the normal direction produces a moment of force on the particles, which causes the particles to rotate with the bonding point as the fulcrum.

$$M_{{P_{1} }} = \frac{{f_{{P_{1} }}^{n} L_{p} }}{2} = \frac{1}{2}P_{1} r\sin 2\beta L_{p}$$
(10)
$$M_{{P_{2} }} = \frac{{f_{{P_{2} }}^{n} L_{p} }}{2} = \frac{1}{2}P_{2} r\sin 2\beta L_{p}$$
(11)

where \(L_{P}\) is the length of the seepage channel of the bonding plane.

The two moments act in opposite directions, and the resultant moment is as follows:

$$M_{P} = M_{{P_{1} }} - M_{{P_{2} }} = \frac{1}{2}\Delta Pr\sin 2\beta L_{p}$$
(12)

The normal tensile stress generated by the moment on the bonding plane is as follows:

$$\sigma_{P} = \frac{{M_{P} L_{p} }}{2I} = \frac{{\Delta Pr\sin 2\beta L_{P}^{2} }}{4I}$$
(13)

The rupture of the mineral particle bonding surface during rock hydraulic fracturing is the result of fluid pressure overcoming the mineral particle bonding force and in situ stress. From Eq. (13), the tensile stress induced by the fluid acting on the mineral particle bonding surface during hydraulic fracturing is determined by the pore pressure gradient. For a given stress condition and rock strength, the pore pressure gradient that induces rock rupture during hydraulic fracturing is constant. Therefore, the greater the pore pressure is, the greater the breakdown pressure.

5.3 Mesoscopic initiation process of the hydraulic fracture tip

The pore pressure effect led to the pores of the sandstone expands and deforms. The mutual squeezing force between mineral particles increases. The pore pressure effect causes the pores in sandstone to expand and deform. The mutual squeezing force between mineral particles increases. Because the sandstone is in hydrostatic equilibrium, with increasing pore pressure, the stability of the porous media system and the overall strength of resistance to destruction increase (Fig. 11a). In the hydraulic fracturing stage, the injection of pressurized water breaks the equilibrium state of the porous media system. The pore pressure gradient is formed on both sides of the particle bonding surface, resulting in tensile stress or shear stress on the particle bonding surface (Fig. 11b). As the water injection pressure continues to increase, the pore pressure gradient on both sides of the particle bonding surface continues to increase. When the tensile stress or shear stress induced by the pore pressure gradient reaches the ultimate strength of the particle bonding surface, the particle bonding surface will break and open (Fig. 11c).

Fig. 11
figure 11

Forced state of sandstone mineral particles before and after hydraulic fracturing considering the initial pore pressure and its gradient effect

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6 Conclusions

  1. (1)

    The hydraulic fracturing experimental system with hydromechanical coupling under triaxial stress loading was used to perform sandstone hydraulic fracturing experiments to determine the effect of the initial evenly distributed pore pressure. The experimental results show that as the initial pore pressure increases, the breakdown pressure of hydraulic fracturing also increases. The data fitting results show that the breakdown pressure and initial pore pressure follow a linear growth trend.

  2. (2)

    With an increase in the initial pore pressure, the AE energy at the moment of borehole wall fracturing correspondingly increases. After borehole wall fracturing, the reduced magnitude of the pum** pressure correspondingly increases, indicating that a greater initial pore pressure correlated to a greater range of initial rupture.

  3. (3)

    During fracturing propagation, the propagation range and opening of the fracture increase as the initial pore pressure increases within the same pum** time.

  4. (4)

    During hydraulic fracturing, a pore pressure gradient is generated on both sides of the mineral particles. When the tensile stress or shear stress induced by the pore pressure gradient reaches the ultimate strength of the mineral particle bonding surface, the particle bonding surface breaks and opens.