1 Introduction

Plenty of fossil energy such as coal, oil and natural gas has been continuously consumed to support the rapid economic development. As a consequence, a mass of greenhouse gas is released into the atmosphere, exacerbating environmental deterioration. This is at odds with the current world's consensus on reducing carbon emissions. As a clean, efficient, widely distributed, abundant, safe and high-quality renewable resource, geothermal resources have become a hot spot for resource development in recent years (Liu et al. 2018). Geothermal resources can be used not only for power generation, but also for heating, planting, breeding, etc. without any gas emission or environmental pollution (Moya et al. 2018; Wang et al. 2018). Geothermal resources can be divided into shallow geothermal type, hydrothermal type and hot dry rock (HDR) type, and more than 90% of the available geothermal resources exist in HDR. HDR is a kind of rock distributed 3 ~ 10 km below the surface, with a temperature above 180 °C, containing no or only a small amount of fluid (Kubik 2006). It is estimated that thermal energy contained in HDR is equivalent to 30 times of the energy contained in all oil, natural gas and coal in the world. (Tianfu et al. 2012).

However, it is always a great challenge for exploration and exploitation of HDR, as it has the characteristics of high hardness, strong abrasiveness, and poor drill-ability. Currently, surface and HDR reservoirs are connected mainly through traditional rotary drilling, which breaks rock to form a well by applying weight to a rotating bit. However, it is usually difficult to break HDR efficiently with traditional drilling tools and conventional drilling methods (Zhang et al. 2019). Low rate of penetration (ROP) and high drilling cost are the key problems that encountered in HDR geothermal resources exploitation and utilization. (Zeng 2015; Tester et al. 2007). In order to improve efficiency and reduce cost, many scholars have conducted abundant experiments based on HDR. The temperature difference between the HDR and the fluid can cause thermal shock resulting in fragmentation (Dai et al. 2021). Cold shock has an impact on the strength and elastic modulus of HDR, which will induce thermal stress and produce a lot of damage (Huang et al. 2019). While increasing the density of crown cutter can effectively improve the drilling efficiency of PDC bit in HDR (Zhu and Dan 2019). Furthermore, gas drilling (Zhukov et al. 2018), laser drilling (Wang et al. 2020), thermal–mechanical combined drilling (Rossi et al. 2020) and other novel technologies that aim at HDR formations are constantly appearing. However, although these novel technologies have made some breakthroughs, most of them still regard the hard interaction between the bit and rock as the main and basic driving force for fragmentation, and others are used as supplements. It will inevitably lead to serious wear of the bit. Hence, it is urgent to explore an efficient technology used for HDR fragmentation to realize the low-cost exploitation and utilization.

Plasma Channel Drilling (PCD) is a new method (Timoshkin et al. 2004; Inoue et al. 2000), which has been widely used in mining and rock excavation (Yan et al. 2016), mineral decomposition (Andres et al. 2001; Huang and Shi 2020) and recycling of valuable resources (Duan et al. 2015), etc. PCD has the significant advantages of high drilling efficiency and low cost (Biela et al. 2009; Li et al. 2019; Zhang et al. 2017). In conventional drilling, based on hard interaction with the rock, the bit relies on its own scraping and shearing to cause damage to rock. It makes the bit vulnerable to severe wear, tooth breakage, and tooth loss, etc., especially in HDR formations. Instead, during the rock fragmentation process of PCD, the electrode bit is in soft contact with rock all the time. Rock at the bottom of the well is broken by the continuous electric discharge of bit. Therefore, there is almost no mechanical wear when using the electrode bit based on PCD. (Lehmann et al. 2017; Yudin et al. 2019). The PCD can efficaciously improve the efficiency of rock fragmentation in high hardness and strong abrasiveness strata such as HDR, thus reducing the cost and shortening the cycle of drilling.

In this paper, to explore the fragmentation mechanism of PCD, granite is selected as the experimental object, which is the most common rock type in HDR (Guo and Wong 2020; Zhu et al. 2021a, b; Guo et al. 2021). Then the numerical model of heterogeneous granite on the basis of the extend Grain-based Model is established. Subsequently, the generation of plasma channel trajectory in heterogeneous granite is explored the by using the probabilistic evolution model. Furthermore, based on the plasma channel generated when electrode gap is 15 mm, the fragments and cracks of granite under different confining pressures are studied. This study also includes an experimental program to examine the granite fragmentation and borehole generation caused by the electrode bit that based on PCD. The research results can provide new ideas and methods for improving the drilling efficiency in HDR, and help to design the structural of drill bits based on PCD technology.

2 The plasma channel drilling

The PCD, also known as “Pulsed Power Drilling” (Inoue et al. 2000) and “Electro-Pulse-Boring” (Oschiegg et al. 2015), means to peel off the fragments from rock mass by high-voltage electric pulse (HVEP). The research on the rock-breaking mechanism of PCD began in Tomsk University in 1960s. Vorobiev et al. (2005) found that when a pair of positive and negative electrodes were placed at the solid–liquid interface, a visible discharge channel could form inside the solid by loading HVEP on the positive electrode. It is the basic principle of study on the fragmentation mechanism of PCD.

Through a good deal of laboratory experiments, Andres (1995) put forward the relationship between the breakdown electric field strength E0 of water, rock, air and oil and the rising edge time τ of pulse voltage, as shown in Fig. 1. The fragmentation mechanism of PCD has two types: Hydro-electric fragmentation (HEF) and Electric fragmentation (EF). It can be seen from Fig. 1 that the critical point of τ is 500 ns. When τ > 500 ns, the E0 of rock is greater than water, and the electrodes will discharge in the water to generate shock waves which will spread to the rock surface and cause fragmentation. The whole process is called HEF. The E0 of rock is less than water when τ < 500 ns, electric breakdown occurs preferentially in rock over water, HVEP will release energy directly inside rock. It is called the EF. Notably, the rock-breaking efficiency of EF is higher, and all the study on PCD in this paper is based on the whole process of EF.

Fig. 1
figure 1

The relationship between the breakdown strength of the electric field and the rise time of the pulse voltage of each phase medium

Full size image

On basis of the theories of the electric breakdown caused by HVEP in rocks, Li et al. (2020) proposed a typical structure of electrode bit, and plentiful research results show that the rock fragmentation of PCD can be exhaustively divided into four steps (Andres 2010; Boev et al. 1997; Cho et al. 2014; Lisitsyn et al. 1999): (I) Electric breakdown occurs preferentially in rock under the HVEP with τ < 500 ns. Plasma channel begins to form and expand inside the rock. (II) Plasma channel concatenates the high-voltage electrode and ground electrode to form a main discharge channel. Subsequently, the voltage on the high-voltage electrode decreases instantly, and the current in the loop increases rapidly. (III) The energy from the circuit release into the plasma channel instantaneously, and the sudden rise of the temperature makes the plasma channel produce a mass of thermal stress. (IV) Rock will be destroyed when the stress caused by hyperthermia exceeds its own strength, and fragments will be carried away by the flowing liquid. Figure 2 shows the typical structure of the electrode bit and four steps of the rock fragmentation of PCD.

Fig. 2
figure 2

Typical structure of the electrode bit and four steps of the rock fragmentation of PCD

Full size image

3 Plasma channel trajectory simulation in heterogenous granite

The time from initiation to complete formation of the plasma channel is actually extremely short, generally in nanoseconds to microseconds. (Otsuka et al. 2007; Zuo et al. 2018). On account of the complex structure of rock and the interference of strong electromagnetic and strong light, it is very difficult to monitor the rock-breaking process of PCD and obtain physical and mechanical information. It is comparatively tough to explore the fragmentation mechanism of PCD by means of continuum mechanics. Gratifyingly, the discrete element method (DEM) can reproduce the microscopic changes in rocks very well, and it is also convenient to observe and record relevant data. Rock in DEM is considered to be a system of numerous independent tiny rigid particles. The rigid particles are connected by a variety of alternative contact models, which allows the micro-structure of the rock to approximate the fact. Failure of the bond between particles represents micro-fracture of rock. All the micro-cracks together can reflect the macro-fractures. Due to DEM's unique advantages in rock micromechanics research, the numerical model based on DEM will be used to investigate the rock-breaking mechanism of PCD in this paper.

3.1 Heterogenous granite model establishment and calibration

3.1.1 The extend grain-based model

Particle Flow Code (PFC) is a software developed based on DEM, which is widely used in geotechnical engineering, mining engineering, etc. After years of improvement and innovation, varieties of mechanical models with different emphases have been proposed on the basis of PFC2D. (Peng et al. 2017, 2019). It is worth noting that Potyondy (2010) proposed a complex rock model containing deformable polygonal mineral grains, which is called the Grain-based Model (GBM). There are two kinds of mechanical contact models between rigid particles in GBM: parallel bond model (PBM) and smooth joint model (SJM). The contact model between particles in a single grain is PBM, while the contact model between different grains is SJM. Either the maximum normal stress or shear stress exceeds its corresponding strength, the parallel bond will break, and all mechanical properties associated with it will be removed simultaneously. The behavior of the SJM's bond is linear elastic. When the strength limit is exceeded, the bond breaks, particles separate.

In GBM, polygonal grains are bonded along adjacent sides. (Jian et al. 2019; Saadat et al. 2020). The boundaries of the grains are the edges of the polygon, which are smooth, regular straight lines. Practically, the grain boundaries in a rock mass are either parallel to each other, contain each other, mesh like cogs, or something else. The grain profile is rugged and complex. Hence, polygons are not excellent enough to reproduce the actual grain shape, which results in some deviation between the numerical model and rock. Therefore, if the polygon can be replaced by the practical grain boundary, the deviation between simulation and experiment can be reduced and the simulation results can be more convincing.

Rejoicingly, owing to the advances in digital image processing technology, the factual grain boundaries of rock can be obtained easily now. By replacing the polygons in GBM with actual grain boundaries, an extend GBM that more closely resembles the practical rock structure can be built.

To obtain the actual grain boundaries, the prerequisite is to obtain a clear digital image that with a good quality of the granite. Firstly, placing granite and a ruler that with a factual scale in a bright environment. The initial image will be obtained by stable shooting with a portable high-resolution camera. The second step is to identify the grain boundaries of different minerals. The boundaries are obtained by changing the brightness and color channel of the initial image. Then the grain boundaries obtained will be divided into different layers according to mineral species. The final and most important step is that the extracted grain boundaries must be scaled to the actual size according to the scale of the ruler. Figure 3 shows two mechanical contact models between particles in the model and the establishment process of the extend GBM.

Fig. 3
figure 3

The establishment process of the extend GBM and the two contact models between particles

Full size image

Rock specimen used in this paper is the granite from Suizhou, Hubei Province, one of the five granite towns in China. According to the data of Wuhan Geological Exploration Institute, Suizhou granite reserves are 2 billion cubic meters, accounting for about 10% of the total granite reserves in China. The main mineral components of the granite are Plagioclase, Albite, Quartz, and Biotite. As the large grain size and distinct color difference of the granite, it is awfully conducive to image processing and computer analysis. Following the steps described above, the grain boundaries of the granite can be obtained effortlessly. Embedding the obtained grain boundaries that contain four layers into the PBM. Then particles will be divided into different minerals according to the layer. Then, the corresponding microscopic parameters are assigned to the particles.

3.1.2 Calibration of granite model

Calibration of microscopic parameters is the cornerstone of subsequent simulation. The coincidence between calibration and experimental results will directly affect the accuracy and rationality of subsequent experiments. Virtually, the calibration is to reproduce the macroscopic mechanical properties of rocks by adjusting micromechanical parameters constantly (Zhang et al. 2021). The main macroscopic mechanical parameters of granite are elastic modulus, uniaxial compressive strength (UCS) and Brazilian tensile strength (BTS). Elastic modulus and UCS are generally obtained from the uniaxial compressive test, and BTS is obtained from the Brazilian splitting test. Follow the steps in Fig. 3, an extend GBM of the granite is established for calibration. The model used for uniaxial compression test is a rectangle with a width of 25 mm and a height of 50 mm, containing 35,078 particles. The model used for the Brazilian split test is a circle with a diameter of 25 mm containing 13,644 particles. The upper and lower surfaces of the calibration models have a rigid wall, and the loading rate of which during simulation is constant 0.1 m/s.

The initial microscopic parameters of the particles should be given before the first calibration. Different PBM parameters are given within reasonable limits for particles representing different minerals. However, the granite model used in this paper includes six grain boundary types: “plagioclase-albite”, “plagioclase-quartz”, “plagioclase-biotite”, “albite-biotite”, and “quartz-biotite”. The rationality of boundary microscopic parameters is tough to verify and adjust. Therefore, to simplify the model and reduce the time cost of the calibration, all boundaries are given the same parameters. In a word, there are four sets of PBM parameters and only one set of SJM parameters in the model.

All microscopic parameters are adjusted and modified on the basis of the last calibration results until the macroscopic mechanical behavior of the model is consistent with the experimental results. After many attempts and constant revisions, the calibration result which is very close to the experiment is finally obtained: The UCS is 84.8MPa, elastic modulus is 10451.4MPa and BTS is 5.31MPa. UCS, elastic modulus and BTS obtained from experiment are 86.4 MPa, 10707.6 MPa and 4.85 MPa, respectively. The errors of the three parameters between calibration and experiment are all within 10%, see from Table 1. Fig. 4 shows the results of calibration and experiment. In the uniaxial compression test, a large number of fragments are generated in the simulation, which has a high coincidence with the experimental consequence. While there are obvious elongated cracks in the granite in both the results of simulation and experiment in the Brazilian splitting test. What’s more, the UCS-strain and BTS-strain curves obtained from calibration are intensely matched with the experimental results. Therefore, the microscopic parameters obtained from calibration are reasonable and can be used in subsequent simulations. Table 2 shows the microscopic parameters of the calibrated model in Fig. 4.

Table 1 The results of simulation and experiment
Full size table
Fig. 4
figure 4

The calibration results of the extend GBM

Full size image
Table 2 The calibrated micro-parameters of simulation model
Full size table

Follow the steps in Fig. 3, the model of granite based on the extend GBM can be established by using various parameters in Table 2. As shown in Fig. 5, the model is a rectangle with a width of 50mm and a height of 13mm, containing 18, 202 particles. It will be used for the electric breakdown experiment of the generation of plasma channel trajectory and the rock-breaking simulation of PCD, subsequently.

Fig. 5
figure 5

The model of granite based on the extend GBM

Full size image

3.2 Generation of plasma channel

3.2.1 The probabilistic evolution model

To study the whole rock-breaking process of PCD, the evolution of plasma channel generation should be clarified first, which is the step (I) and step (II) in Fig. 2b. As the time consumption of the plasma channel generation is fantastically short, there is no other obvious crushing inside rock at the moment of plasma channel generation. As a rule of thumb, the whole process could be recorded with a high-speed camera. Then a clear trajectory of the plasma channel can be obtained by screening the frames. Frustratingly, this method requires that the rock or the viewing surface be transparent. Another approach is to slice the rock after HVEP has pierced. The plasma channel on the flakes of the rock can be obtained. However, this method requires ensuring that only the electrical breakdown occurs inside the rock and HVEP does not lead to further fragmentation. The above two methods to obtain the plasma channel trajectory have great difficulties in practice. As a consequence, it is an alternative choice to get the plasma channel trajectory through numerical simulation, which not only reduces the consumption of materials and manpower, but also improves the work efficiency.

In 1980s, Daccord et al. (1985) proposed the fractal dielectric breakdown model (FDBM). In FDBM, a simple random model is used to describe the evolution of discharge channel, and the breakdown probability at any point is proportional to the strength of electric field. On the basis of FDBM, Wieseman and Zeller (1986) proposed a model with a stable pre-breakdown structure in solid medium. The model describes discharge channel as a tree, and introduces two new parameters: critical breakdown field strength Ec and voltage drop field strength Ed. The basic expression of the model is,

$$ p_{i} = \frac{{(\left| {E_{i} } \right| - E_{c} )^{\eta } }}{{\sum {\left( {\left| {E_{i} } \right| - E_{c} } \right)^{\eta } } }} $$
(1)

where the Ei is the local field strength, V/m; η is the probability exponent; pi is the development probability of each ramification to be breakdown, which only depends on Ei, %.

The experiment results that Zhang et al. (2012) conducted show the pi is not only related to Ec, but also the complete breakdown field strength Ew. When Ei exceeds Ew, the rock will be broken down with a probability of 100% as long as the voltage is applied.

On the strength of theories of predecessors, Zhu et al. (2020) proposed the probabilistic evolution model (PEM) of plasma channel trajectory. The model takes the evolution of a single plasma channel's ramification as the basic unit, and its basic expression is,

$$ p_{i} = {\text{ }}\left\{ {\begin{array}{*{20}l} 1 \hfill & {\left| {E_{i} } \right| \ge E_{w} } \hfill \\ {\left[ {\frac{{(\left| {E_{i} } \right| - E_{c} )}}{{E_{w} - E_{c} }}} \right]^{\eta } {\text{ }}} \hfill & {E_{c} < \left| {E_{i} } \right| < E_{w} ,\eta > 0} \hfill \\ 0 \hfill & {\left| {E_{i} } \right| \ge E_{c} } \hfill \\ \end{array} } \right.$$
(2)

Similar to Noskov's approach (Noskov et al. 1995), the "physical time" is also introduced into the PEM, which is called generation time in this paper. The generation time τt of ramification has been derived based on Poisson's statistical processing theory. The equation for τt is,

$$ \tau_{t} = \frac{{ - \ln (1 - \xi_{t} )}}{{kp_{i} }} $$
(3)

where ξt is the evolution probability of ramification, %; k is a constant related to the magnitude of τt. Regarding the evolution of the channel as a quasi-static process, and selecting Poisson equation as the governing equation of the spatial electric field distribution in the non-channel region inside the rock, then Eq. (4) can be obtained.

$$ - \varepsilon_{r} \varepsilon_{0} \nabla^{2} u = \rho_{q} $$
(4)

where ε0 is the vacuum dielectric constant, which is 8.854 × 10–12 F/m, and εr is the relative dielectric constant of medium (rock, electrode and insulating liquid); u is electric potential, V; ρq is the charge density, C/m3.

The relationship between charge density and current density is,

$$ \frac{{\partial \rho_{q} }}{\partial t} = \nabla {\varvec{H}} $$
(5)

where H is the current density vector.

PEM can generate one single ramification of the plasma channel after a calculation cycle. The loop does not stop until a preset boundary condition is reached. It must be noted that the premise of using PEM to obtain the trajectory of plasma channel is to obtain a point model of rock. A complete calculation cycle of PEM is:

  1. (a)

    Taking rock properties and HVEP into consideration, give the initial parameters, including initial potential boundary matrix u0, iterative electric field error ε, probability index η, critical breakdown field strength Ec and voltage drop field strength Ed.

  2. (b)

    After several times iterations, according to the Laplace equation, calculating the electric field matrix Ψ0.

  3. (c)

    Determining the set of latent breakdown points B0. It is selected from the points that set of breakdown points J0 may spread to in the growth direction.

  4. (d)

    Through Eq. (2), a new set of latent breakdown points B1 can be obtained from B0.

  5. (e)

    Through Eq. (3), the point with the smallest growth time in set B1 will be selected as the new breakdown point.

  6. (f)

    Determining whether the conditions for stopping the calculation have been reached. If the condition is reached, the cycle will be terminated; Otherwise, the potential of the new branch will be recalculated by Ed, and then the potential of the new ramification will be counted into Ψ0, and the process returns to step (b).

Figure 6 shows the specific process of the step (b) to (e). In Fig. 6, black hollow dots are on behalf of initial grid points in the model; Green dots represent all breakdown points that have evolved into plasma channel; The black dots are the elements belong to B0, and the red hollow dots pertain to set B1. Firstly, the set of breakdown points J0 is determined from the existing plasma channel trajectory, that is, all the green dots in Fig. 6a. Then, as shown in Fig. 6b, the set of latent breakdown points B0 whose elements may become new breakdown points, is selected from J0 according to the feasible direction. The quantity of feasible direction of the channel is 8 (Zhu et al. 2020). According to the pi, that is, Eq. (2), eliminate the elements with Ei less than Ec at both ends of the branch in B0, and a new set of latent breakdown points B1 will be obtained. Finally, through Eq. (3), calculating the τt of of all elements in B1. The ramification with the shortest generation time will be selected as a new ramification. And, the new breakdown point will be added into the j0 to form a new J0, as shown in Fig. 6d.

Fig. 6
figure 6

Expansion process of plasma channel trajectory based on PEM

Full size image

3.2.2 Generation of plasma channel trajectory

Figure 7 shows the steps of the generation of plasma channel trajectory with PEM. Firstly, meshing the granite model in Fig. 5, and the spacing between meshes in X and Y directions both are 2.5 × 10−5 m, which is the largest diameter of particles. Intersections of the grid lines will become new element points of the granite. Next, identifying mineral properties of each intersection according to the initial model. Meanwhile, the electrical parameters of minerals are assigned to the corresponding intersection points. Then the electric breakdown model (EBM) can be established by importing the PEM into point model of granite that with property assigned. Eventually, the trajectories of plasma channel can be obtained by changing the parameters of EBM. Table 3 shows the basic parameters of EBM, and Table 4 shows the electrical parameters of each mineral component. Le is the gap between high-voltage and ground electrode in EBM. It ranges from 5 to 20 mm with the interval of 5 mm.

Fig. 7
figure 7

The steps of obtaining the trajectory of the plasma channel

Full size image
Table 3 The parameters in the electric breakdown model
Full size table
Table 4 The electrical parameters of each mineral component
Full size table

There are two modes of electrical breakdown in rock under a single HVEP (Zhu et al. 2021a, b). When the complete electrical breakdown (CEB) occurs, a complete plasma channel that penetrates the rock completely and connects the high-voltage electrode tip with the ground electrode tip will be generated. On the contrary, the plasma channel only extends a small distance forward from the tip of the high-voltage electrode and cannot successfully penetrate the rock when the partial electrical breakdown (PEB) occurs. Through multiple simulations in EBM, the minimum breakdown voltage U0 at different Le can be obtained. As shown in Fig. 8, when the voltage is less than U0, complete plasma channel cannot form in rock, that is, PEB occurs. Once the voltage reaches U0, a complete plasma channel will generate in rock.

Fig. 8
figure 8

The plasma channel trajectory at different Le

Full size image

What is noteworthy is, although a complete channel cannot generate when PEB occurs, the maximum depth of the channels generated is larger at the same electrode gap. What's more, the ends of these channels seemingly tend to continue to grow deeper into rock. It can reduce the energy requirement of the subsequent rock-breaking process. This phenomenon shows that breaking rock with HVEP at low voltage and high frequency may achieve the same effect when providing the same energy.

3.2.3 Analysis of electric field

It is bound to be accompanied by the change of electric potential in process of electric breakdown by HVEP in rock. Figures 9 and 10 show the variation of the electric potential when PEB and CEB occur. Upon the solid–liquid interface, the electric potential always decreases uniformly from the high-voltage electrode to the ground electrode at the initial state in any case, and the electric potential decreases from maximum to zero. However, the isoline is obviously deflected underneath the solid–liquid interface, as the red dot box shown in Figs. 9 and 10. Since electrical properties between different minerals are distinct, a deflection may occur on the isolines along grain boundaries or through mineral grains, as the red dot circle shown in Figs. 9 and 10. This indicates that mineral grains in heterogeneous rocks may disturb the distribution of initial electric potential.

Fig. 9
figure 9

The isoline of the electric potential when PEB occurs

Full size image
Fig. 10
figure 10

The isoline of the electric potential when CEB occurs

Full size image

The isoline of the electric potential along the plasma channel path is also changing with the continuous extension and expansion of the channel. Compared with the initial state, the "bulge" on isolines will appear. The direction of the bulge always towards the tip of the ground electrode. In other words, the electric potential of each point on the channel is higher than the initial state. What's apparent is that the "bulge" is the strongest at the extreme end of the plasma channel, as the black dot box shown in Figs. 9 and 10, especially when CEB occurs. The density of isolines at the tip of the ground electrode become much denser than before when a complete channel generated. Instead, when PEB occurs, the potential around the ground electrode tip does not alter. It shows that the closer the channel is to the tip of the ground electrode, the more obvious the potential of the rock around the tip of the ground electrode will rise.

3.3 Rock-breaking simulation of PCD in PFC2D

3.3.1 Thermal–mechanical coupling model

As shown in step (III) and step (IV) in Fig. 2b, after the plasma channel generation, the energy from the circuit will release into the plasma channel instantaneously, and the sudden rise of the temperature makes the plasma channel produce a mass of thermal stress. Rock will be destroyed when the stress caused by hyperthermia exceeds its own strength. To study the evolution of granite in the process above, the thermal contact model is added to the model shown in Fig. 5, and the model turns into a thermo-mechanical coupling model.

As shown in Fig. 11, pressure is applied on the thermo-mechanical coupling model. The pressure is applied by the rigid walls on both sides and the particles of the flexible boundary on the upper surface. The pressure exerted by the rigid wall represents the formation pressure PF, and the pressure exerted by the flexible boundary represents the hydraulic pressure PL. The formation pressure and the liquid column pressure take the same value in this paper. Hence, the PF and PL are collectively referred to as confining pressure PC. PC ranges from 0 to 20 MPa, with the increment of 5 MPa. The rigid wall at the bottom of the model is fixed all the time.

Fig. 11
figure 11

The thermal–mechanical coupling model

Full size image

As the the time consumption of the plasma channel generation is extremely short, it can be considered that the micro-structure of rock does not change during the formation of the plasma channel. Therefore, it is logical to import the plasma channel trajectory directly into the thermo-mechanical coupling model without considering other changes. The particles in contact with the channel trajectory will be identified as the thermal-loaded particle. The hyperpyrexia will be applied to the plasma channel trajectory, that is, the hyperpyrexia is given to the thermal-loaded particles. Subsequently, under the action of the instantaneous influx of high temperature, the granite will be broken.

The initial temperature T0 is set at 20 °C without considering the influence of the high temperature of granite itself, as the change of temperature in the range of 0 ~ 300 °C will not significantly change the microstructure of granite (Zhang et al. 2018). The temperature on the channel can exceed 8000 °C instantly with incoming energies. And the heat source roughly conforms to the Gaussian distribution. Thus the hyperthermia Tt loaded on the plasma channel instantaneously is 2000 °C, 4000 °C, 6000 °C and 8000 °C. Table 5 shows the remaining other parameters in detail.

Table 5 The parameters of the thermal–mechanical coupling
Full size table

3.3.2 Results of the thermo-mechanical coupling experiment

3.3.2.1 Effect of temperature on plasma channel

To explore the mechanical behavior of the granite under the action of thermal–mechanical coupling, the trajectory of plasma channel when CEB occurs in granite with electrode gap of 15 mm obtained in Sect. 3.2.2 is introduced into the thermo-mechanical coupling model. Then, PC is set at 5 MPa, 10 MPa, 15 MPa and 20 MPa to discuss the effect of temperature on the plasma channel.

From Fig. 12, there are always four types of micro-cracks in the granite: Intragranular tensile crack, intragranular shear crack, intergranular tensile crack and intergranular shear crack. Intergranular shear crack is caused by the relative displacement of two particles belonging to different grains along the tangential direction of the contact surface. And intergranular tensile crack is caused by the relative displacement of two particles belonging to different grains along the normal direction of the contact surface. Analogously, intragranular shear crack is caused by the relative displacement of two particles belonging to the same kind of grain along the tangential direction of the contact surface. Intragranular tensile crack is caused by the relative displacement of two particles belonging to the same grain along the normal direction of the contact surface. It can be seen from the simulation results that the quantity of almost all kinds of cracks increases linearly with increasing temperature. The slope of the quantity curve of intragranular shear crack is the largest, in other words, its number is most affected by the change of the temperature on plasma channel. Moreover, the count of intragranular shear cracks is always the largest. While there is a weeny difference between the variation rate of the quantity curves of intergranular tensile crack and intragranular tensile crack, and the former consistently outnumber the latter. However, the initiation of intergranular shear cracks is rare or non-existent.

Fig. 12
figure 12

Effect of temperature on plasma channel

Full size image

From the distribution of cracks in granite under different PC and temperature when ts = 5 μs, it can be seen that the cracks generated at higher temperature seem to be the result of propagation on the basis of the cracks generated at the previous temperature gradient. Hence, it can be inferred that the increase of temperature will not alter the trend of initiation and propagation, but simply reduces the time consumption of initiation of various cracks.

3.3.2.2 The initiation and propagation of cracks

Initiation and propagation of various cracks in the rock-breaking process of PCD is worthwhile to study, because it will directly affect the efficiency of rock-breaking. It is most reasonable to study propagation and initiation of cracks when the temperature on the channel is 8000 °C, as the quantity of cracks is the largest in the case.

When PC = 0, as shown in Fig. 13, a large amount of intragranular shear cracks will appear first in the initial stage (ts ≤ 0.01 μs). With time increasing, the number of intragranular shear cracks will continue to increase, accompanied by the initiation of intergranular tensile cracks and a small number of intragranular shear cracks. At the final stage, the number of intragranular shear cracks is the largest, and they are mainly distributed around the channel. Only a few intergranular shear cracks were generated. The quantity curves of the cracks show that the cracks are mainly initiated in the period when ts < 1 μs. Almost no new cracks are initiated when ts exceed 1 μs. This phenomenon is in line with the reality that HVEP causes damage to rocks on the order of nanoseconds to microseconds.

Fig. 13
figure 13

The broken image and the change of cracks of granite when PC = 0

Full size image

As the broken image of granite shows, all cracks generated in the early stage (ts ≤ 0.1 μs) are around the plasma channel. Later, cracks initiated in the areas out of the plasma channel, as the highlighted hollow ellipse shown in Fig. 12a. The cracks are tensile cracks on grain boundaries of the fragments that have been peeled off from granite, and intergranular tensile cracks that extend deeper into granite at the lower part of the plasma channel. It can be predicted that the cracks will further increase and the fragments will be further broken into smaller under the continuous action of HVEP. It indicates that the damage of electric pulse to the granite when PC = 0 is quite thorough. The intergranular tensile crack has spread to the granite in the unbroken area, which is helpful to reduce the energy demand for subsequent rock breaking and thus improve the rock breaking efficiency.

According to Fig. 14 there are also four kinds of micro-cracks in granite with confining pressure. When PC ≠ 0, there is little difference in the quantity of cracks under different PC. The distribution of cracks under different PC is not much different from the case when PC = 0, but the quantity is less. However, it's not difficult to find that the initiation and propagation of the intergranular tensile cracks in the areas out of the plasma channel are obviously inhibited. This is because when PC gradually increases in a certain range, the micropores in granite gradually close, the grains are pressed more tightly under pressure, and the boundaries between grains are strengthened, thus inhibiting the initiation and propagation of intergranular tensile cracks. The influence of PC on the number of intragranular shear cracks around the plasma channel is smaller, because the mechanical contact between particles of the same grain is stronger than that between particles of different grains, but the extrusion of PC still can make the particles in the same grain tighter. Figure 15 shows the of variation of the quantity of cracks more intuitively. It can be found that when PC changes from 0 to 5 MPa, all kinds of cracks reduce greatly. What's more, when PC ≠ 0, the number of intragranular shear crack, intragranular tensile crack and intragranular tensile crack tends to decrease with increasing PC, and this trend is most obvious in total number of cracks.

Fig. 14
figure 14

The broken image the change of cracks of granite when PC ≠ 0

Full size image
Fig. 15
figure 15

The quantity of cracks under different Pc

Full size image

4 Laboratory experiment

4.1 Experimental program

Figure 16 shows the devices of the experiment and the granite specimen. The electrode bit used in the laboratory experiment is made up of nylon rod, high-voltage electrode and ground electrode. Nylon rod has a through hole in the center and a conical surface at one end. The ground electrode is a hollow cylinder. The high-voltage electrode is a rod-shaped structure with a small disk at the end. Ground electrode is installed at the end of the conical surface of the nylon rod, and the high-voltage electrode passes through the hole of the nylon rod. The end face of the small disk of the high-voltage electrode and the end face of the ground electrode are on a same plane. The electrodes are all made of stainless steel, and the maximum outer diameter of the bit is 60 mm. The output voltage of the pulsed power supply is 60 kV and the discharge frequency is 2 Hz. The liquid medium used in the experiment is tap water, and the rock is still the granite from Suizhou, with a size of 300 mm × 300 mm × 200 mm.

Fig. 16
figure 16

The device of the experiment and the granite specimen

Full size image

Before experiment, putting the granite into an insulated container, and putting the electrode drill on the upper surface of the rock. Connecting the grounding electrode of the electrode drill with the output negative pole of the high-voltage pulse power supply, and the high-voltage electrode is connected with the output positive pole. Subsequently, adding tap water to the container until the granite and exposed electrodes are completely submerged. The start and end of the experiment are controlled by the switch on and off of the high-voltage pulse power supply. The start time of the rock-breaking process is the time when the switch is closed, and the end time is when the switch is off. The operations are controlled by computer.

4.2 Analysis of experimental results

Six groups of experiments are carried out. Each group of experiments is carried out on a new surface of the granite at room temperature and pressure. The maximum radius rm and the maximum breaking depth hm of the broken area is recorded after each rock-breaking process. Figure 17 shows the method of measurement and the broken pits of the granite, and Table 6 shows rm, hm, and discharge time t0 in detail. From the results, the broken pits are usually distributed within the maximum outer diameter of the electrode bit, that is, the electrode bit only causes damage to the rock between the high-voltage electrode and ground electrode. Under the continuous discharge of the electrode bit, the shapes and distribution of pits formed on the granite surface are random.

Fig. 17
figure 17

The method of measurement and the broken pits

Full size image
Table 6 The results in the process of rock breaking
Full size table

The debris produced in each experiment are collected, as shown in Fig. 18. The maximum length of a few debris exceeds 5 mm, but most are below 2 mm. In this experiment, there is no device or mechanism used to scour or discharging debris, so that the debris that has already been stripped will be repeatedly broken by HVEP. It is a vital cause why the sizes of debris are generally small.

Fig. 18
figure 18

The debris produced by granite in 6 groups of experiments

Full size image

Other attempts have been made on the basis of the above experiments. Replacing the tap water in the container by oil, and the granite by denser gray-white granite. The granite can be punched with holes that through the whole rock sample with the parameters of the HVEP unchanged. This is because the breakdown voltage of oil is usually much higher than that of granite, as shown in Fig. 1. Electrode bit only discharge in granite, which reduces the energy loss. Figure 19 shows the holes in granite after drilling. However, different from the regular holes made by conventional bits, the hole formed by the electrode bit is unsmooth, and there are many pits of different sizes on the wall. This shows once again that the damage caused by PCD to granite is random. The problem with this is that the stability of borehole wall can not be guaranteed. To solve the problem, it may be effective to use a bit that composited by a conventional bit and an electrode bit to drill. Conventional bit can ensure drilling direction and trim borehole wall, and electrode bit can improve drilling efficiency. But it is of vital importance to handle the relationship between electricity and conductors well.

Fig. 19
figure 19

The destruction of granite when the liquid medium is oil

Full size image

5 Conclusions and discussion

The main objective of this paper is to investigate the fragmentation mechanism of granite by HVEP, for this purpose, the probabilistic evolution model (PEM) is used to explore the generation of plasma channel trajectory in heterogeneous granite. Besides, the thermal–mechanical coupling model is carried out to reveal the formation of debris and cracks under different confining pressures. This research also includes an experimental test to examine the granite fragmentation and borehole caused by an electrode bit. The following conclusions can be summarized,

  1. 1.

    The electric potential along the plasma channel path will changing with the continuous extension and expansion of the channel. The potential of each point on the channel is higher than the initial state. And the closer to the end of the plasma channel, the more drastic this variation will be.

  2. 2.

    The maximum depth of the channels generated in PEB is larger at the same electrode gap and the ends of these channels seemingly tend to continue to grow deeper into rock. It can reduce the energy requirement of the subsequent rock-breaking process.

  3. 3.

    The increase of temperature on the plasma channel will not alter the trend of initiation and propagation of cracks, but simply reduces the time consumption of crack initiation. The quantity of intragranular shear cracks is most affected by the change of the temperature on plasma channel.

  4. 4.

    In the rock fragmentation process of PCD, intragranular shear cracks are always the first to initiate. Besides, the quantity of intragranular shear cracks is always the largest, and they are mainly distributed around the channel. There are only a few intergranular shear cracks generate in granite. And the initiation and propagation of the intergranular tensile cracks in the areas out of the plasma channel will obviously inhibited by confining pressure.

  5. 5.

    The hole caused by only an electrode bit is exceeding unsmooth, and the hole wall is full of pits with different shapes. It may be effective to use a bit that composited by a conventional bit and an electrode bit to solve the problem. Conventional bit can ensure drilling direction and trim borehole wall, and electrode bit can improve drilling. Also noteworthy is that the appropriate voltage, electrode gap and liquid medium play an important role in the rock fragmentation of PCD. And the effect of the liquid medium seems to be the greatest, as it determines whether the energy can fully release into rock.

The fly in the ointment is that all the rock-breaking experiments in this paper have not considered the role of drilling fluid which can erode and clean debris. The author is making further efforts to consider the role of drilling fluid and further optimize the model.