1 Introduction

Geothermal energy is, in principle, a decentralized, renewable, and limitless energy resource that is typically free from CO\(_2\) emissions during operations. Geothermal energy can serve as a baseload energy source (i.e., is available at all hours of the day and throughout the year) or as a dispatch-capable energy source (i.e., supporting other, only intermittently available, energy resources, such as wind and solar energy) (Fleming et al. 2022; Van Brummen et al. 2022). Four deep geothermal energy extraction and utilization technologies exist to date, namely: (1) Conventional Hydrothermal Systems (Eyerer et al. 2020; Menberg et al. 2021), (2) Enhanced Geothermal Systems (EGS) (Tester et al. 2006; Amann et al. 2018; Gischig et al. 2020), (3) Advanced Geothermal Systems (AGS) (Malek et al. 2022; Beckers et al. 2022), and (4) CO\(_2\)-Plume Geothermal (CPG) systems (Randolph et al. 2011; Adams et al. 2015; Ezekiel et al. 2020, 2022). Generally, these deep geothermal systems require deep wellbores, with large-diameter wells, to maximize energy throughput (Beckers and Young 2018; Adams et al. 2021; Ezekiel et al. 2021). However, aside from not requiring hydraulic stimulation, which can induce (micro) seismicity (Beckers et al. 2022; Ge and Saar 2022), AGS have the advantage that they can be deployed largely independently from geologic conditions. In particular, rock permeability is a notoriously difficult parameter to predict deep underground, yet the product of formation permeability and thickness, referred to as transmissivity, largely determines geothermal energy extraction rates. In AGS, geology plays a role only insofar as the drilling of wells deep underground is concerned.

In fact, AGS requires drilling several kilometers deep and many kilometers (sub)horizontally, which is extremely expensive when employing state-of-the-art rotary drilling technologies. To be cost-competitive, AGS typically require drilling deeper than \(\sim\) 5 km into hard rocks that are often crystalline and hotter than 150 \(^\circ\)C, at drilling costs that are much lower than what is currently possible with rotary drilling (Malek et al. 2022; Beckers et al. 2022). For instance, Malek et al. (2022) suggested that employing state-of-the-art mechanical rotary drilling technologies to drill 8 km deep, at a rate of penetration of 15 m/h and a bit lifetime of 50 h, will result in Specific Capital Costs (SpCC) of 145 $/W\(_e\), where $ refers to 2019 US Dollars. Such costs then need to be compared to a target SpCC of 2.0\(-\)5.0 $/\(W_e\) (Taylor et al. 2015) for renewable energies. While this acceptable cost range of 2.0\(-\)5.0 2019$/\(W_e\) from the year 2015 (Taylor et al. 2015) is likely too low, given the energy cost developments over the past years, the current SpCC estimate for AGS likely remains about an order of magnitude too high nonetheless, at least when rotary drilling is employed. Therefore, AGS utilization requires the development of alternative drilling technologies that are significantly cheaper than rotary drilling. One can reduce drilling costs through several drilling technology improvements, including increasing the penetration rate, decreasing the frequency and duration of tripping times (by prolonging drill bit lifetime), and reducing the likelihood of drill rig mechanical failure, to name the most significant ones. To this end, “contactless” or “abrasionless” drilling technologies, including Plasma-Pulse Geo-drilling (PPGD) (Schiegg et al. 2015; Ushakov et al. 2019; Walsh et al. 2020; Vogler et al. 2020; Ezzat et al. 2021, 2022; Zhu et al. 2022), laser drilling (Buckstegge et al. 2016), thermal spallation (Vogler et al. 2020b; Rossi et al. 2018, 2020), and microwave drilling (Woskov et al. 2014; Zheng et al. 2020; Zou et al. 2023), have the potential to decrease drilling costs significantly.

Plasma-Pulse Geo-Drilling (PPGD) is a relatively new drilling technology that fractures the rock by inducing a plasma inside the rock pores. The plasma is generated by high-voltage electric pulses (i.e., voltage peaks of \(~\sim\)200 kV and more) with extremely short pulse rise times \(<0.5~\upmu\)s, rather than by mechanical abrasion (Vorob’ev et al. 1961, as cited in Ushakov et al. 2019). In fact, several experimental and numerical studies, considering ambient pressure and temperature conditions, have suggested that PPGD has the potential to significantly reduce drilling costs, compared to mechanical drilling (see Sect. 2, for further details and citations). However, despite these recent advances in studying PPGD under ambient-air conditions, the effects of deep wellbore conditions (i.e., to the depths of 5 km and more) on PPGD performance have not been systematically studied to this date. Under deep wellbore conditions of \(\sim\)5 km, for example, PPGD often encounters hard crystalline rocks at elevated temperatures of 150 \(^\circ\)C and more, high lithostatic pressures of up to 150 MPa, and hydrostatic pressures of up to 50 MPa. Such investigations of the impact of deep wellbore conditions on the performance of PPGD are, however, necessary to enable estimating the potential drilling (and well completion) cost reductions that may be possible when employing the fully developed PPGD technology. This, in turn, would enable cost-effective deployment of geothermal power plants in general and of AGS in particular.

A couple of studies investigated the combined effect of elevated temperature and hydrostatic pressure on PPGD productivity (Vazhov et al. 2010; Anders et al. 2015). Vazhov et al. (2010) showed that increasing both hydrostatic pressure (up to 35 MPa) and temperature (up to 120 \(^\circ\)C) reduces the productivity (excavated rock volume per pulse) of electropulse drilling (i.e., PPGD) in sandstone, limestone, and granite, by a factor of 5–11, with the most significant reduction occurring in the hydrostatic pressure range of 0.1–5 MPa. These reductions apparently occurred even when an oil-based drilling fluid was used during the experiments. In contrast, the results of Anders et al. (2017) suggest that increasing temperature (up to 90 \(^\circ\)C) and hydrostatic pressure (up to 50 MPa) has no impact on the specific energy required for electropulse drilling with oil-based muds, while a linear increase in the specific energy was observed when deionized water was used. Thus, these two studies show significant discrepancies and leave many questions unanswered. It is difficult to identify the cause of these discrepancies, because neither study provides all the details regarding the employed operating conditions, the characteristics of the drilling fluids used, the rise time of the pulses, or the electrode configurations.

Therefore, we develop here a standard experiment protocol that we suggest to be followed when investigating the effects of temperature, lithostatic pressure, and hydrostatic pressure on the performance (i.e., the productivity and the specific energy) of PPGD and related methods. This paper then experimentally investigates the effect of temperature (up to a maximum of 80 \(^\circ\)C, due to material limitations) on the PPGD process, to be followed by separate studies on the effect of lithostatic and hydrostatic pressures. In addition to the direct benefits of the experiment results on comparing PPGD with other drilling technologies under deep wellbore conditions, these experiment results can be used to expand the capabilities of the available numerical PPGD simulators (Kuznetsov et al. 2011; Cho et al. 2016; Li et al. 2019; Walsh and Vogler 2020; Vogler et al. 2020; Zhu et al. 2021, 2022; Li et al. 2021; Ezzat et al. 2022; Duan et al. 2022; Rao et al. 2022; Kang et al. 2023; He et al. 2023) to include deeper wellbore conditions.

2 Background on Plasma-Pulse Geo-Drilling (PPGD)

During PPGD, or electropulse drilling, electrodes transmit a high-voltage pulse to the rock surface. Such high-voltage pulses induce electric discharges (i.e., plasma formation) inside the rock, increasing the tensile pressure and, ultimately, fracturing the rock by tension (Lisitsyn et al. 1998; Ezzat et al. 2021, 2022; Liu et al. 2022). Under drilling conditions, the drilling fluid is above the rock surface, on which the electrodes rest. Thus, the electrical discharge is expected to occur within the drilling fluid, as the drilling fluid tends to have a dielectric strength that is lower than that of the rock, containing pores filled with a native fluid. The dielectric strength of any material (e.g., drilling fluid and rock) is the maximum voltage gradient that the material withstands before the electric breakdown occurs. However, Vorob’ev et al. (1961, as cited in Ushakov et al. 2019) found that if the rise time of the electric pulse is short enough, the dielectric strength of the rock becomes lower than that of the drilling fluid, so that the electric discharge occurs inside the rock, enabling the electro- or plasma-pulse drilling process. The critical maximum rise time is a function of the drilling fluid and the rock type. For instance, considering mineral transformer oil as the drilling fluid results in the following critical maximum rise times: 10 \(\upmu\)s for sandstone, 8 \(\upmu\)s for marble, and 0.9 \(\upmu\)s for quartz (Ushakov et al. 2019). When using deionized water, shorter critical maximum rise times are required than those needed for the mineral transformer oil, namely: 0.1 \(\upmu\)s for sandstone, 0.8 \(\upmu\)s for marble and 0.5 \(\upmu\)s for quartz (Ushakov et al. 2019).

The longer critical maximum rise times the transformer oil allows are easier to generate. However, transformer oils have two major disadvantages: (1) They are hazardous due to their low flammability temperature and (2) their biodegradability is low, raising environmental concerns if applied at relatively shallow depths, where freshwater aquifers are prevalent (Šárpataky et al. 2021). In contrast, deionized water can overcome these issues, which makes it more appropriate for conducting PPGD experiments and for PPGD deployment near freshwater aquifers, i.e., typically (but not exclusively) at relatively shallow depths. Although deionized water is convenient for conducting laboratory experiments, such experiments constitute simplified drilling environments, as real drilling conditions typically result in impure drilling fluids. Impure drilling fluids exhibit significantly decreased dielectric strengths, increasing the likelihood of the electric discharge occurring through the drilling fluid, not the rock. Nonetheless, deionized water has been used in several laboratory studies on electropulse drilling as a proxy for actual drilling fluids (e.g., Lisitsyn et al. 1998; Inoue et al. 1999). Thus, our use of deionized water also facilitates comparing our laboratory experiments with previous studies on electropulse drilling (i.e., PPGD).

Generally, the drilling-specific cost, \(C_\text {D}\) (USD/m), is given by the following equation:

$$\begin{aligned} C_\text {D} = \frac{C_\text {b} + C_\text {r}\left( t_\text {d} + t_\text {t} + t_\text {n}\right) }{D}, \end{aligned}$$
(1)

where \(C_\text {b}\) (USD) is the cost of the drill bits used to drill a depth, D (m), \(C_\text {r}\) (USD/h) is the rig operating cost per hour, \(t_d\) (h) is the drilling time (i.e., the time spent in rotation, achieving drilling progress), \(t_t\) (h) is the total tripping time (i.e., time spent for raising and lowering the drill string to replace the worn-out drill bit), and \(t_\text {n}\) (h) is the non-rotational time [i.e., the time spent in maintenance, fixing any failure of the drill string or bottom hole assembly (BHA)]. The drilling cost can be reduced by minimizing these three times as follows: The drilling time can be reduced by increasing the rate of penetration in various geological formations, the tripping time by increasing the bit’s lifetime, and the non-rotational time by reducing the mechanical moving parts, which are prone to failure. In addition, the specific energy (the energy required to drill a unit volume of rock) of the employed drilling method is an important factor, as it increases the operation cost rate, \(C_\text {r}\).

Laboratory experiments conducted over the past 3 decades suggest several advantages of PPGD, compared to mechanical rotary drilling, that are expected to reduce the three mentioned times in the drilling-specific cost Eq. (1) and the energy consumption. The latter will reduce the operational cost rate. Together, these cost reductions have the potential to reduce PPGD drilling costs to below the costs of mechanical rotary drilling (Ushakov et al. (2019), and the references therein). The following shows several arguments why PPGD has the potential to reduce drilling costs:

  1. 1.

    PPGD may significantly reduce drill-bit-related tripping times as the PPGD drill bit experiences no mechanical abrasion, resulting in longer bit lifetimes.

  2. 2.

    PPGD can operate at lower rotations per minute (RPM) and requires less torque (only for drilling mud circulation) by the drill string, reducing the total energy consumption of the drilling operation and reducing drill string wear. This low RPM and the associated torque are only for circulating the drilling fluid without contributing to the rock excavation process.

  3. 3.

    PPGD only has to overcome the rock’s low tensile strength, not its high compressive strength. Thus, much lower specific energy values are required during PPGD compared to mechanical rotary drilling, which has to work against the rock’s stronger compressive strength.

  4. 4.

    PPGD enables a low rate of decrease in the borehole diameter with depth, which means that extremely large-diameter boreholes are not needed at shallower depths for achieving the minimum required diameter at the bottom of the borehole, generally facilitating well completion by reducing the well emplacement costs. Lowering the rate “in principle” is possible when drilling with PPGD, as PPGD has a non-rotating drill string. This non-rotating drill string should result in sufficient space around it to mount equipment for the “casing-while-drilling” operation, reducing the necessary number of casing stages to satisfy the mud window constraints.

Thus, laboratory experiments under ambient-air conditions with oil-based drilling muds and comparative analyses (which assume drilling at a depth between 3.5 and 4.5 km), Anders et al. (2015, 2017) suggested that electropulse drilling (i.e., PPGD) may reduce drilling costs by 17–23%, compared to the cost of mechanical rotary drilling.

Further reductions in PPGD drilling costs require determining the optimum operating conditions (e.g., pulse rise times, pulse application frequency, pulse shape, etc.) and electrode configurations for various rock types and drilling fluid types. Therefore, several experimental and modeling studies investigated the fundamental physics underlying the PPGD process. Here, we list several studies highlighting the impact of the rock pores and the pulse shape (we refer to the references in these studies for more details). In a laboratory experiment, Lisitsyn et al. (1998) found that during the PPGD process, the fracturing of the rock begins with the formation of the plasma in the rock’s gas-filled pores and pre-existing micro-fractures (referred to as local electric discharge). Boev et al. (1999) show that the dielectric strength of granite can be reduced to half of its original value by applying a square-like pulse shape (i.e., respecting the required rise time to be less than 500 ns) instead of a saw-tooth-like pulse shape. Experimentally, Li et al. (2022) found that during the PPGD process, chemical reactions (e.g., oxidization) and physical processes (e.g., dissociation of grains) occur at the microscale, which both promote rock fragmentation. Finally, Ezzat et al. (2022) and Liu et al. (2022) found that the pore fluid, as well as the pore shape and size, can affect the rock’s dielectric strength. The above studies investigated plasma formation under ambient conditions of 25 \(^\circ\)C, 0.1 MPa lithostatic pressure, and 0.1 MPa hydrostatic pressure. Therefore, the experimental work presented here investigates the effect of somewhat deep wellbore temperatures on the performance of PPGD (i.e., the productivity and the specific energy). However, the maximum temperature employed in our study is 80 \(^\circ\)C, due to experiment material limitations (discussed below).

3 Experiment Method

The primary objective of our experimental study is to identify the effect of elevated temperatures (up to 80 \(^\circ\)C, representing temperatures typically found at 2–3 km depth) on the PPGD process. We quantify the temperature effect with respect to PPGD productivity and specific energy exerted when drilling in granite. We quantify PPGD performance in terms of the excavated rock volume per electric pulse, Q (mm\(^3\)/pulse), and the specific energy, SE (J/mm\(^3\)), in terms of the amount of energy required to drill a unit volume of rock. The investigated temperature limit of 80 \(^\circ\)C is the maximum temperature that the used PEEK electrical insulator can withstand when being exposed to 200 kV pulses and hydrostatic pressures of up to 50 MPa. Higher temperatures could be studied using different electrical insulators (to be made of a material with higher dielectric strength and larger compressive strength at higher working temperatures), which, however, were unavailable to us when conducting the experiments. Table 1 lists our experiment parameters, with further explanations listed in the following experiment setup section.

Table 1 Experiment operating parameters and measured quantities
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3.1 Experiment Setup

Figure 1 shows the experiment setup, which includes the following three components: (1) the drilling autoclave with the heating system; (2) the Marx generator (i.e., high-voltage pulse generator); and (3) the electrical diagnostics system. To reproduce downhole drilling conditions, an autoclave is used that can generate a pressure of 50 MPa and 80 \(^\circ\)C ambient temperature. The granitic rock sample, the deionized water drilling fluid, and the electrode system, with the high-voltage insulation PEEK material, are placed inside the autoclave. The electrical heating jacket heats the autoclave and its contents to the desired temperature (max. 80 \(^\circ\)C in our case). The autoclave’s thermal insulation jacket facilitates keeping a constant temperature during the experiments, with a maximum deviation of ±2 \(^\circ\)C. The temperature is measured using an RS Pro 6070 digital thermometer.

A 12-stage Marx generator (PULSREX-20: manufactured by Kammermann GmbH and supplied by SwissGeoPower AG) is used to generate 200 kV pulses with a 0.1 \(\upmu\)s rise time, 210 J of energy, and a pulse repetition rate of 2 pulses/minute. This low repetition rate is sufficient to achieve the objectives of our study, as outlined above. For actual commercial PPGD operations, however, the pulse repetition rate needs to be significantly higher (on the order of 20 pulses/second). During our experiments, the maximum pulse voltage is limited to 200 kV, due to the following two constraints: (1) this is the limit of our electrical diagnostics system and (2) this is the maximum voltage, limited by the PEEK electrical isolation diameter that can be implemented inside the autoclave (i.e., the drilling cell). The generated pulses are applied via the high-voltage electrodes to the surface of the rock sample, placed inside the autoclave.

The high-voltage electrode and the grounded electrode are combined in a two-leg, co-axial cylindrical electrode configuration (i.e., a drill bit), as shown in Fig. 2. Duan et al. (2022) and He et al. (2023) used such a configuration in lab experiments and numerical simulations and showed that this co-axial configuration is more efficient (at the lab scale) than the other common configurations, namely, the multi-electrode pair configuration and the electrode cross configuration.

The high voltage is transferred via a co-axial capacitive voltage divider for measuring the voltage pulse characteristics, i.e., the peak value and the rise time, which are both critical during the PPGD process. The voltage divider is a prototype, designed by the HPE lab at ETH Zurich, and has been “calibrated” against an industrial high-voltage divider (from HAEFELY). The voltage divider has a division ratio of 1:1050, as shown by Hobejogi and Biela (2011). A DSOX3034T oscilloscope (from Keysight with a 2 GSample/second sampling rate and 70 MHz bandwidth) records the low-voltage signal from the voltage divider via a 2x attenuator (10 dB/each) probe, which is equivalent to a 1:20 division ratio. The two division ratios and the low-voltage measurement of the oscilloscope are combined to reconstruct the actual high-voltage pulse profile (see the example shown in Fig. 1).

Fig. 1
figure 1

The PPGD autoclave experiment setup at Fraunhofer IEG, Germany. For the system description, see the main text. Color figure online

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3.2 Sample Preparation

In the experiments, we investigate granite samples, taken from Chüenzentennlen, close to the Grimsel Pass in the Central Alps of Switzerland (Rossi et al. 2018). This granite consists of 46% quartz, 31% feldspar, 20% plagioclase, 3% biotite, and minor amounts of chloride. The granite has a bulk density of 2.61 g/cm\(^3\), a porosity of 1.37 ± 0.52%, and a grain size ranged from a few micrometers to \(\sim\) 4 mm. This granite has a uniaxial compressive strength (UCS) of 116.9 ± 47.9 MPa, a tensile strength of 9.54 ± 2.17 MPa, and a Youngs modulus of 47.3 ± 15.4 MPa (Keusen et al. 1989). A coring bit is used to obtain cylindrical samples with a diameter of 53 mm out of a larger granite block. The granite cores are then cut to a length of 100 mm, which is roughly twice the diameter. Each cylinder side is ground to yield a flat contact surface for the high-voltage and the ground electrodes (Figs. 2, 3a).

Fig. 2
figure 2

The two-leg co-axial cylindrical electrode configuration used in the experiment. a Top view. b Side view

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3.3 Experiment Procedure

The following five steps are necessary to perform an experiment. First, the granite rock sample is placed in the autoclave. Second, to simulate the drilling fluid, the autoclave is filled with deionized water with an electrical conductivity of \(\sigma _w < 5\,\upmu\)S/m (produced by the ion exchanger MBK with 20 l/min rate) to minimize plasma arcing within the drilling fluid, maximizing the discharging energy injection into the granite sample. The deionized water and the rock sample are under a constant and identical lithostatic and hydrostatic pressure of 0.1 MPa. The sample is considered to be dry as the sample is immersed in water for only 15 min during an experiment run, whereas reaching the nominal liquid saturation of granite requires at least 100 h (Lisitsyn et al. 1998). Third, the top lid of the autoclave is closed, and the heater is switched on until the desired temperature is reached. Then, the heater is switched off. The thermal insulation jacket keeps the temperature constant during the experiment duration of approximately 5 min. Fourth, each rock sample is subjected to ten 200 kV pulses, provided by the Marx generator, at a fixed temperature. The number of applied pulses is ten, to ensure that the damage volume is large enough to compare the results for different temperatures. In particular, Lisitsyn et al. (1998) showed that at least 3–4 pulses are necessary in granite to induce minimally visible damage. Another consideration is that it becomes impractical to exceed about ten pulses, as otherwise the excavated volume outside the sample surface area (i.e., 10\(\times\)10 cm\(^2\)) becomes too large, preventing quantifying the damaged volume. Finally, the granite sample is taken out of the cell and investigated using light microscopy to measure the excavated volume, as shown in Fig. 3, thereby calculating the productivity and the specific energy.

Fig. 3
figure 3

a Top view of the granite sample GG-R-01, tested at 0.1 MPa and 20 \(^\circ\)C. The high-voltage electrode position is marked with a red dot. The grounded electrode can be seen as the two legs of the aluminum flange. The yellow-dashed rectangle marks the damage area, which has been scanned by light microscopy. b The 3D-constructed damage profile using light microscopy (Color figure online)

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4 Results

In our experiments, we investigate the impact of temperature (up to 80 \(^\circ\)C, due to material limitations as discussed above) on the PPGD process, including the following three characteristics and quantities:

  1. 1.

    the high-voltage pulse profile characteristics (i.e., peak, rise time, and damping factor) and how these characteristics are influenced by the discharge medium (i.e., deionized water or granite);

  2. 2.

    the PPGD productivity (i.e., the excavated rock volume per pulse); and

  3. 3.

    the specific energy of PPGD (i.e., the amount of energy required for PPGD to excavate a unit volume of granite).

4.1 Pulse Voltage Profile

Figure 4 shows four different pulse voltage profiles. The first profile (dotted-blue) represents an undesired electric discharge occurring within the deionized water, i.e., the drilling fluid (essentially a short circuit). The other three profiles represent discharges within the granite at different temperatures, namely at 20 \(^\circ\)C (dotted-dashed-black), 50 \(^\circ\)C (dashed-red), and 80 \(^\circ\)C (solid-orange). The deionized water pulse profile (discharge took place within deionized water) was measured without a (granite) sample being placed in the autoclave. The pulse voltage profiles in granite have a higher damping factor than the deionized water profile (Fig. 4), which means that the amplitudes of the secondary oscillations (i.e., the ones after the first and the main peak) approach zero voltage faster in granite than in deionized water.

Figure 4 also shows that the temperature affects neither the pulse voltage profile, the rise time, nor the damping factor. Even though the Marx generator is set to deliver 200 kV pulses, the measured pulse voltage peaks do not exceed 150 kV, as shown in Fig. 4. The difference between the charging voltage and the measured pulse voltage is caused by the internal impedance (i.e., resistance, inductance, and capacitance) of the Marx generator (Adhikary et al. 2011). This internal impedance, in combination with the impedance of the arc through the load (i.e., the rock sample), basically forms a voltage divider and, therefore, only 75% of the voltage, stored in the stages of the Marx generator, is delivered to the load (i.e., the rock sample). The rest of the voltage drops across the internal impedance of the Marx generator. The Marx generator output voltage could be increased by reducing the internal impedance of the Marx generator by a more compact design with lower parasitic inductances.

Fig. 4
figure 4

Typical pulse voltage profiles exerted during the experiment. Granite profiles refer to the measured pulse profile when the discharge occurs within the granite sample at temperatures of 20 \(^\circ\)C (dotted-dashed-black), 50 \(^\circ\)C (dashed-red), and 80 \(^\circ\)C (solid-orange). The deionized water profile refers to the case, where the discharge occurs within the deionized water in the absence of the (granite) sample (Color figure online)

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4.2 PPGD Productivity

Figure 5 shows the productivity, Q (mm\(^3\)/pulse), versus the rock temperature, T (\(^\circ\)C). The right y-axis in the figure shows the productivity normalized to the baseline productivity value, labeled by a red circle. The baseline condition is a temperature of 20 ± 2 \(^\circ\)C, a hydrostatic pressure of 0.1 MPa, and a lithostatic pressure of 0.1 MPa.

Fig. 5
figure 5

PPGD productivity, Q, as a function of temperature, T

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The results show that the PPGD productivity decreases linearly when temperature increases to the maximum applied temperature of \(80\,^\circ\)C. For instance, at 80 \(^\circ\)C, the PPGD productivity has decreased by 40% below the productivity observed under the baseline condition of 20 \(^\circ\)C. A least-square linear fit yields an \(R^\text {2}\) value of 93%, given by

$$\begin{aligned} Q\left( T\right) = 116.99 - 0.78\ T, \end{aligned}$$
(2)

where T is given in \(^\circ\)C and Q in mm\(^3\)/pulse.

4.3 Specific Energy

Figure 6 shows the specific energy (SE) (kJ/cm\(^3\)), as a function of temperature, T (\(^\circ\)C). In each experiment, we apply ten electric pulses, meaning that the total amount of energy implemented in the SE calculation is ten times the amount of electric energy released per pulse. We calculate this amount of energy per pulse (i.e., the amount of energy delivered to the rock surface and the losses through the high-voltage electrode), \(E_\text {P}\), by multiplying the number of stages, \(n_\text {s}\), by the electrostatic energy stored per stage, according to the following equation:

$$\begin{aligned} E_\text {P} = n_\text {s} \times \frac{1}{2} \times C_\text {s} \times V_\text {s}^2, \end{aligned}$$
(3)

where \(C_\text {s}\) is the capacitance per stage, \(V_\text {s}\) is the voltage per stage (\(V_\text {P}\)/\(n_\text {s}\)), and \(V_\text {P}\) is the pulse voltage, which we set at the Marx generator control panel to a constant value of 200 kV in all experiments. By substituting the values of the \(n_s = 12\) stages with the capacitance per stage, \(C_\text {s} = 140\) nF, Eq. 3 can be written in a compact form as a function of the total pulse voltage as follows:

$$\begin{aligned} E_\text {P} = 35/6 \times 10^{-3} \times V_\text {P}^2, \end{aligned}$$
(4)

where \(E_\text {P}\) is given in Joule and \(V_\text {P}\) in kV.

The right y-axis in the figure shows the specific energy, normalized to the baseline value, labeled as a red circle. The baseline condition refers to a temperature of 20 ± 2 \(^\circ\)C, a hydrostatic pressure of 0.1 MPa, and a lithostatic pressure of 0.1 MPa.

Fig. 6
figure 6

The PPGD specific energy (SE) as a function of temperature, T

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These results show that the specific energy for rock excavation during PPGD increases linearly with increasing temperature, up to the maximum applied temperature of 80 \(^\circ\)C. A least-squares linear fit yields an \(R^\text {2}\) value of 88%, given by

$$\begin{aligned} \text {SE}\left( T\right) = 1.57 + 3T/100, \end{aligned}$$
(5)

where T is given in \(^\circ\)C and SE in kJ/cm\(^\text {3}\).

The data point at 40 \(^\circ\)C is likely an outlier as it shows a much lower value of specific energy, which is inconsistent with the trend of the results. We thus exclude this data point from the curve fitting. The outlier could be caused by an unfavorable positioning of the electrode tip at the grain boundary or a pre-existing natural micro-crack at the rock surface, thereby increasing the damaged rock volume and the associated productivity. Nonetheless, further investigations of the impact of temperature between 20 and 55 \(^\circ\)C should provide better insights into whether this data point is indeed an outlier.

4.4 Error Analysis

To calculate the error bars shown in Figs. 5 and 6, we conduct an error analysis. Table 2 summarizes the different error sources when determining temperature, drilling productivity, and specific energy.

Table 2 Sources of measurement errors
Full size table

For the temperature axes (x-axes) in both Figs. 5 and 6, the error bars represent the systematic error in the temperature measurement of ±2 \(^\circ\)C, i.e., an absolute error. This uncertainty is caused by switching off the heater after reaching the desired temperature, i.e., during the five minutes of running the experiment. Nonetheless, the heating jacket that surrounds the experiment keeps the temperature constant with an uncertainty of ±2 \(^\circ\)C.

For the productivity axis (y-axis) in Fig. 5, the error bars include both systematic and random errors. The systematic error of 3% is the same for all data points, as it represents the accuracy of the 3D reconstruction model used to calculate the volume of the damage profile, scanned using light microscopy. The random relative error of 7% is estimated by repeating the baseline experiments (20 \(^\circ\)C) twice and then calculating the standard deviation from the mean value. Due to the project constraints, i.e., the time and the budget, only the baseline experiment could be repeated twice, whereas the rest of the data points represent only one set of experiments. Therefore, the calculated random relative error of 7% is used for the baseline experiment to estimate the absolute “random” error for the rest of the data points.

For the specific energy (y-axis) in Fig. 6, the error bars also include both systematic and random errors. The systematic error of 7% is the same for all data points, which includes the following: (1) A 3% uncertainty in the 3D reconstruction of the damage profile, scanned by light microscopy. (2) A 4% uncertainty in the evaluated amount of released energy by the Marx generator, which represents ±1 Joule per pulse as an absolute error. The random relative error of 7% is estimated by repeating the baseline experiments (at 20 \(^\circ\)C) twice and then calculating the standard deviation from the mean value.

5 Discussion

Figure 4 shows four pulse profiles. The first profile is for discharging through deionized water (i.e., without a rock sample), whereas the other three pulses represent discharges through granite at three different temperatures. The granite pulses yield higher damping factors than the pulse going through the deionized water. When also the granite sample is included, the total electrical resistivity of the load (deionized water and the rock sample) decreases as both are connected in parallel. The decreased overall electrical resistivity results in a higher damping factor of the considered RLC (defined below) parallel resonant circuit, which is used to model the electric breakdown process, i.e., the electric breakdown in the PPGD process (Burkin et al. 2009; Zhu et al. 2022; Rao et al. 2022). An RLC parallel circuit is a circuit with three components, namely a resistor with resistance R, an inductor with inductance L, and a capacitor with capacitance, C, which are connected in parallel (further details are reported by Ouyang et al. (2023)and the references therein). Therefore, the discharge pulses, which travel through the granite, have a higher damping factor than the ones that travel through the deionized water. This is consistent with the findings by Anders et al. (2015), who have shown higher damping factors when electric discharges occur through rocks than through deionized water, i.e., the drilling fluid. We can thus use the characteristics of the damping of the voltage pulse profile to evaluate whether the discharge goes through the rock sample, the deionized water, or the surrounding air. In the experiments, this approach is used to exclude the pulses going through the surrounding air from counting as part of the 10 pulses applied per experiment.

Our results show that increasing the temperature of the granite linearly decreases the PPGD productivity (Fig. 5) and linearly increases the specific energy required for PPGD (Fig. 6). Such an effect may mostly be attributed to the different increase rates in the electric conductivity of the deionized water versus the granite. Usually, heating the material increases its electric conductivity; however, different increase rates are recorded for each material (granite, deionized water). For instance, from 20 \(^\circ\)C and higher, deionized water has a higher increase rate in electric conductivity, with increasing temperature, than that of granite (Light et al. 2004; Olhoeft 1979). This difference in the electric conductivity increase rates results in a higher electric conductivity of the deionized water than that of the granite. The less the resistance (i.e., the higher the conductivity) of the deionized water, the higher is the electric current passing through the water (i.e., current leakage) and the energy loss, which reduces the probability of arcing through the granite, which would decrease the PPGD performance (Boev et al. 1999). While Vazhov et al. (2014) did not observe a dependence in PPGD performance on temperature, their results are to be expected as they used oil-based drilling muds, which have much higher dielectric strengths than rocks. In practice, using often poorly biodegradable oil-based drilling muds can be problematic, at least at relatively shallow drilling depths of perhaps 1 km, where freshwater aquifers may be expected and could be contaminated.

Even though we show here that PPGD performance decreases with temperature increasing, up to 80 \(^\circ\)C (corresponding to a maximum depth of \(\sim\) 3 km), when deionized water is employed, the effect of lithostatic pressure on PPGD performance is expected to counteract this negative trend, according to a preliminary analysis of our ongoing investigations of the impact of the lithostatic pressure. Nonetheless, this study and its results have a couple of implications for PPGD research:

  1. 1.

    It sets a standard protocol to examine the effect of each deep wellbore condition (temperature, lithostatic pressure, and hydrostatic pressure) separately, which can be utilized by the PPGD community to facilitate comparing research results.

  2. 2.

    It will allow the PPGD modeling community to expand their numerical models from simulating PPGD under ambient conditions to deep wellbore conditions, starting here with elevated temperatures.

Ultimately, the goal is to investigate the impact of deep drilling conditions on the PPGD process by examining the combined effect of elevated temperature, lithostatic pressure, and hydrostatic pressure on PPGD performance and its implications for drilling costs and associated AGS power plant costs.

6 Conclusions

This paper introduces a standardized experiment setup to investigate the effect of elevated temperature under deep wellbore conditions (down to 3 km depth) on Plasma-Pulse Geo-drilling (PPGD) performance, using deionized water as the drilling fluid. The effect of temperatures up to 80 \(^\circ\)C on the PPGD productivity and specific energy is investigated, from which we draw the following conclusions.

  1. 1.

    High-voltage pulse profiles exhibit a higher damping factor in granite than in deionized water. This enables distinguishing between whether the pulse discharged in the water (outside the granite) or within the granite sample. Furthermore, increased temperatures have shown no effect on the pulse profiles in granite.

  2. 2.

    Temperature elevation decreases the PPGD productivity and increases the specific energy requirement when deionized water is used as the drilling fluid. This dependency is linear and can be observed up to the maximum tested temperature of 80 \(^\circ\)C (i.e., \(\sim\) 3 km drilling depth conditions).

  3. 3.

    Decreasing productivity, observed with increasing temperature, can likely be attributed to the stronger increase of electric conductivity with temperature in the deionized water, compared to granite. This higher electric conductivity in deionized water, when compared to granite, holds true for temperatures larger than 20 \(^\circ\)C. The difference in the electric conductivity yields a higher probability of plasma formation in the deionized water, which prevents plasma formation, and ultimately fracturing, within the granite rock.

  4. 4.

    The effect of increasing temperature on the PPGD drilling performance depends highly on the employed drilling fluid. Our results show that temperature affects the PPGD drilling performance negatively when deionized water is used, while Anders et al. (2017) have shown that such temperature impacts on drilling performance are negligible when (often undesirable) oil-based drilling muds are used.

  5. 5.

    Based on preliminary results from our companion study (Ezzat et al. 2024), we hypothesize that the negative trend of decreasing PPGD performance with increasing temperature is likely counteracted when elevated lithostatic, and possibly hydrostatic, pressures are applied.