Effects of Liquid Nitrogen on Mechanical Deterioration and Fracturing Efficiency in Hot Dry Rock

Abstract

Geothermal energy can be obtained from hot dry rock (HDR). The target temperatures for heat extraction from HDR range from 100 to 400 °C. Artificial fracturing is employed to stimulate HDR and create a network of fractures for geothermal resource extraction. Liquid nitrogen (LN₂) is environmentally friendly and shows better performance in reservoir stimulation than does conventional fracturing. In this study, triaxial compression experiments and acoustic emission location techniques were used to evaluate the impacts of temperatures and confining pressures on the mechanical property deterioration caused by LN₂ cooling. The numerical simulation of LN₂ fracturing was performed, and the results were compared with those for water and nitrogen fracturing. The results demonstrate that the confining pressure mitigated the deterioration effect of LN₂ on the crack initiation stress, crack damage stress, and peak stress. From 20 to 60 MPa, LN₂-induced reductions in these three stress parameters ranged between 7.73–18.51%, 3.46–12.15%, and 2.51–8.50%, respectively. Cryogenic LN₂ increased the number and complexity of cracks generated during rock failure, further enhancing the fracture performance. Compared with those for water and nitrogen fracturing, the initiation pressures of LN₂ fracturing decreased by 61.54% and 68.75%, and the instability pressures of LN₂ fracturing decreased by 20.00% and 29.41%, respectively. These results contribute to the theoretical foundation for LN₂ fracturing in HDR.

1. Introduction

Geothermal energy, as a low-carbon and stable source of energy, stands out as a highly promising option among sustainable energy alternatives for achieving low-carbon energy transitions [1,2,3,4,5]. Geothermal energy can be obtained from hot dry rock (HDR) that has a temperature of 180–650 °C and is located at a depth of more than 3 km [4,5,6]. The target temperatures for heat extraction from HDR are in the range of 100–400 °C [4,5,7]. The abundance of energy in HDR provides a sustainable solution to energy supply [4,5,8,9]. Artificial fracturing is used to stimulate HDR and form a network of fractures, thereby establishing a circulation system between injection and production wells that allows for the economical extraction of deep geothermal energy from HDR [10,11]. The utilization of liquid nitrogen (LN₂) has been developed for various new fracturing techniques that have good prospects for fracturing performance enhancement [12,13,14,15].

Unlike normal rock, the temperature of HDR can promote the deteriorating effects of LN₂ on rock properties. Wu et al. [16], Kang et al. [17], and Sha et al. [18] evaluated the influences of air, water, and LN₂ cooling on granite mechanical performance at different temperatures and found that LN₂ cooling can significantly damage granites. Wu et al. [19,20] investigated the LN₂ effects on the wave velocity, permeability, and compression strength of rocks (granite, shale, and sandstone) at different temperatures (25, 150, and 260 °C) and found that increasing the rock temperature can strengthen the LN₂’s thermal effect. Hou et al. [21] and Su et al. [22] analyzed the LN₂ effect on the mechanical performance of marble at different temperatures (25–300 °C) and also found temperature enhancement. Kang et al. [17] and Sha et al. [18] found that the mechanical property deterioration is more prominent above 400 °C. Wu et al. [23] and Rong et al. [24] found that increasing the cycle number of LN₂ treatments caused the continuous deterioration of rocks. Besides focusing on the mechanical parameters, Sha et al. [18], Ge et al. [25], and Rong et al. [24] analyzed the impact of LN₂ treatment on the acoustic emission (AE) parameters during rock failure. In addition, Wu et al. [20] found that confining pressure weakens the LN₂ effect. Most studies have failed to considered the confining pressure effect. For HDR under high temperatures and high in-situ stress, one must consider the confining pressure to reveal LN₂’s deterioration effect. Further research on fracturing performance is needed for LN₂ to be applied to the fracturing of HDR reservoirs. Laboratory-conducted LN₂ fracturing is carried out under low confining pressures [26]. Experiments with high confining pressures are difficult to implement. Numerical simulations can solve this difficulty by setting up conditions closer to site status. Therefore, this study investigated the mechanical performance of granite after LN₂ cooling under various temperatures and confining pressures via experiments and the fracturing performance of LN₂ via numerical simulations.

To explore the effectiveness of LN₂, triaxial compression experiments were performed on heated granite after air and LN₂ cooling. The impacts of different temperatures and confining pressures on mechanical property deterioration with LN₂ cooling were investigated. The numerical simulation of LN₂ fracturing was performed to analyze the fracturing performance enhancement.

2. Experiment and Simulation

2.1. Thermal Treatments and Mechanical Property Experiments

The granite was taken from an open-pit mining site in Rizhao, China. The mineral content was quartz (25%), plagioclase (39%), potassium feldspar (22%), and biotite (14%), based on the X-ray diffraction test results. The granite was prepared in 50 × 100 mm cylindrical samples (Figure 1a), according to the standards of the International Society for Rock Mechanics. By density and ultrasonic testing, samples with a low dispersion of test results were selected for the experiment. The samples were subjected to a thermal process (Figure 1b) before testing their mechanical properties. The heating temperatures were 25, 200, and 400 °C. The samples were first heated to the predetermined temperatures at a slow rate (2–4 °C/min) and then held for 2 h to ensure that the internal temperature of the sample was uniform. After the conclusion of the maintenance time, two cooling treatments were used, i.e., the air-cooling treatment and the LN₂-cooling treatment. The samples in the air-cooling group underwent natural cooling under ambient air conditions. The samples in the LN₂-cooling group were cooled by immersion in LN₂ tanks for 1 h. After the heating and cooling treatments, all samples were left under ambient air conditions for 24 h prior to mechanical testing, ensuring that the samples returned to room temperature.

For the samples after thermal treatment, triaxial compression experiments were performed using the TAW-2000 experimental system (Changchun Chaoyang Test Instrument Co., Ltd., Changchun, China) (Figure 1c). The confining pressure was initially applied at 0.05 MPa/s until it reached predetermined levels (0, 20, 40, and 60 MPa), after which axial displacement loading was initiated at 0.2 mm/min. Synchronized with the loading progress, the AE locations were collected via the PCI-2 AE test system (Figure 1d).

In the experiment, for the sample deformation process, the stress parameters for crack initiation and damage were identified using the stress–strain curve, according to the crack volume strain method [4,5] (Figure 1e). Crack initiation represents the stress level at which cracks begin and is marked according to the reversal points on the curve of crack volumetric strain [4,5]. Crack damage represents the stress level at which unstable crack growth occurs and is marked according to the reversal points on the curve of volumetric strain [4,5].

2.2. Numerical Simulation of Liquid Nitrogen Fracturing

The LN₂ fracturing for HDR reservoirs is a typical multi-physical field coupling problem. The cryogenic LN₂ injection process causes changes in the rock deformation, along with the damage, temperature, and fluid flow, and each physical field influences the other during the change process. Multi-physical field coupling problems can be solved by numerical methods. In this study, the fracturing process was simulated using COMSOL Multiphysics 5.6 software.

2.2.1. Governing Equations

If LN₂ is applied to fracturing for HDR reservoirs, one must consider the physical changes in the LN₂. The equation of state for LN₂ can be expressed as [27]

$${\displaystyle \frac{A\left(\rho ,T\right)}{RT}}=\alpha \left(\delta ,\tau \right)={\alpha }^0\left(\delta ,\tau \right)+{\alpha }^{\mathrm{r}}\left(\delta ,\tau \right)$$

(1)

where $A$ is the Helmholtz energy, $\alpha$ is the dimensionless Helmholtz energy, ${\alpha }^0$ is the ideal gas contribution to the Helmholtz energy [27], ${\alpha }^{\mathrm{r}}$ is the residual Helmholtz energy [27], $\rho$ is the density, $T$ is the temperature, R = 8.314510 J/(mol·K), $\delta =\rho /{\rho }_{\mathrm{c}}$, $\tau =T_{\mathrm{c}}/T$, ${\rho }_{\mathrm{c}}$ is 11.1839 mol/dm³, and $T_{\mathrm{c}}$ is 126.192 K.

The density and isobaric heat capacity equations of Span et al. [27], as well as the viscosity and thermal conductivity equations of Lemmon and Jacobsen [28], were adopted for LN₂.

Darcy’s law was adopted to calculate the flow of fracturing media in the HDR reservoirs. The governing equation of fluid flow during the fracturing process is derived using the work of Zhou et al. [29]. For slightly compressible fluids, the fluid flow equation can be expressed as [29]

$$\left(\varphi {\beta }_{\mathrm{f}}+{\displaystyle \frac{{\alpha }_{\mathrm{B}}-\varphi }{1+S}}{\displaystyle \frac{1}{K_{\mathrm{s}}}}\right){\displaystyle \frac{\partial p}{\partial t}}-\left(\varphi {\alpha }_{\mathrm{f}}+{\displaystyle \frac{{\alpha }_{\mathrm{B}}-\varphi }{1+S}}{\alpha }_T\right){\displaystyle \frac{\partial T}{\partial t}}-\nabla \cdot \left({\displaystyle \frac{k}{{\mu }_{\mathrm{f}}}}\nabla p\right)+{\displaystyle \frac{{\alpha }_{\mathrm{B}}-\varphi }{1+S}}{\displaystyle \frac{\partial {\epsilon }_{\mathrm{V}}}{\partial t}}=0$$

(2)

For compressible fluids, the fluid flow equation can be expressed as [29]

$$\begin{array}{l}\left(\varphi {\beta }_{\mathrm{f}}+{\displaystyle \frac{{\alpha }_{\mathrm{B}}-\varphi }{1+S}}{\displaystyle \frac{1}{K_{\mathrm{s}}}}\right){\displaystyle \frac{\partial p}{\partial t}}-\left(\varphi {\alpha }_{\mathrm{f}}+{\displaystyle \frac{{\alpha }_{\mathrm{B}}-\varphi }{1+S}}{\alpha }_T\right){\displaystyle \frac{\partial T}{\partial t}}\\ -\nabla \cdot \left({\displaystyle \frac{k}{{\mu }_{\mathrm{f}}}}\nabla p\right)-{\beta }_{\mathrm{f}}{\displaystyle \frac{k}{{\mu }_{\mathrm{f}}}}\nabla p\cdot \nabla p+{\alpha }_{\mathrm{f}}{\displaystyle \frac{k}{{\mu }_{\mathrm{f}}}}\nabla T\cdot \nabla p+{\displaystyle \frac{{\alpha }_{\mathrm{B}}-\varphi }{1+S}}{\displaystyle \frac{\partial {\epsilon }_{\mathrm{V}}}{\partial t}}=0\end{array}$$

(3)

where $\varphi$ is the porosity, ${\beta }_{\mathrm{f}}$ is the fluid compressibility coefficient, ${\alpha }_{\mathrm{B}}$ is Biot’s coefficient, $S={\epsilon }_{\mathrm{V}}+\left(p/K_{\mathrm{s}}\right)-{\alpha }_T\left(T-T_0\right)$, ${\epsilon }_{\mathrm{V}}$ is the volumetric strain, $p$ is the pore pressure, $K_{\mathrm{s}}$ is the modulus of the solid matrix, ${\alpha }_T$ is the volumetric thermal expansion coefficient, $T_0$ is the initial temperature, $t$ is the time, ${\alpha }_{\mathrm{f}}$ is the fluid thermal expansion coefficient, $k$ is the permeability, and ${\mu }_{\mathrm{f}}$ is the dynamic viscosity of a fluid.

The energy conservation equation for heat transfer is expressed as [30]

$${\left(\rho C_p\right)}_{\mathrm{eff}}{\displaystyle \frac{\partial T}{\partial t}}+{\rho }_{\mathrm{f}}{C}_{p,\mathrm{f}}\mathit{q}\cdot \nabla T-\nabla \cdot \left({\lambda }_{\mathrm{eff}}\nabla T\right)=Q_T-TK{\alpha }_T{\displaystyle \frac{\partial {\epsilon }_{\mathrm{V}}}{\partial t}}$$

(4)

where ${\left(\rho C_p\right)}_{\mathrm{eff}}=\varphi \left({\rho }_{\mathrm{f}}{C}_{p,\mathrm{f}}\right)+\left(1-\varphi \right)\left({\rho }_{\mathrm{s}}{C}_{p,\mathrm{s}}\right)$, ${\rho }_{\mathrm{f}}$ is the density of the fluids, $C_{p,\mathrm{f}}$ is the isobaric heat capacity of the fluids, ${\rho }_{\mathrm{s}}$ is the rock density, $C_{p,\mathrm{s}}$ is the isobaric heat capacity of the rock matrix, ${\lambda }_{\mathrm{eff}}=\varphi {\lambda }_{\mathrm{f}}+\left(1-\varphi \right){\lambda }_{\mathrm{s}}$, ${\lambda }_{\mathrm{f}}$ is the thermal conductivity of the fluids, ${\lambda }_{\mathrm{s}}$ is the thermal conductivity of the rock matrix, $Q_T$ is the energy sources, and $K$ is the bulk modulus.

The mechanical equilibrium equation can be given as follows (tensile is positive) [31]:

$$G{u}_{i,jj}+{\displaystyle \frac{G}{1-2\nu }}{u}_{j,ji}-{\alpha }_{\mathrm{B}}{p}_{,i}-K{\alpha }_T{T}_{,i}+f_i=0$$

(5)

where $G$ is the shear modulus, $u_i$ is the displacement component, $\nu$ is Poisson’s ratio, and $f_i$ is the body force component.

Two strength criteria were adopted using the following formula [32]:

$$\left\{\begin{array}{l}{F}_1={\sigma }_1-f_{\mathrm{t}}=0\\ F_2=-{\sigma }_3+{\sigma }_1{\displaystyle \frac{1+\sin \phi }{1-\sin \phi }}-f_{\mathrm{c}}=0\end{array}\right.$$

(6)

where ${\sigma }_1$ and ${\sigma }_3$ are the principal stresses, $f_{\mathrm{t}}$ is the uniaxial tensile strength, $f_{\mathrm{c}}$ is the uniaxial compressive strength, and $\phi$ is the internal frictional angle.

The damage variable D is defined by [32]

$$D=\left\{\begin{array}{cc}0& F_1<0,F_2<0\hfill \\ {\left|{\displaystyle \frac{{\epsilon }_{\mathrm{t}0}}{{\epsilon }_1}}\right|}^2-1& F_1=0,\mathrm{d}\left|{\epsilon }_1\right|>0\\ 1-{\left|{\displaystyle \frac{{\epsilon }_{\mathrm{c}0}}{{\epsilon }_3}}\right|}^2& F_2=0,\mathrm{d}\left|{\epsilon }_3\right|>0\end{array}\right.$$

(7)

where ${\epsilon }_{\mathrm{t}0}$ and ${\epsilon }_{\mathrm{c}0}$ are the maximum tensile and compressive principal strains when damage is initiated, respectively, and ${\epsilon }_1$ and ${\epsilon }_3$ are the principal strains. The tensile damage is the priority.

When a rock element is damaged, its elastic modulus, permeability, and thermal conductivity change accordingly, as follows [32,33]:

$$E=\left(1-\left|D\right|\right)E_0$$

(8)

$$k=k_0{\left({\displaystyle \frac{\varphi }{{\varphi }_0}}\right)}^3\exp \left({\alpha }_k\left|D\right|\right)$$

(9)

$${\lambda }_{\mathrm{s}}={\lambda }_0\exp \left({\alpha }_{\lambda }\left|D\right|\right)$$

(10)

where $E_0$ is the initial elastic modulus, $k_0$ is the initial permeability, ${\varphi }_0$ is the initial porosity, ${\lambda }_0$ is the initial thermal conductivity, and ${\alpha }_k$ and ${\alpha }_{\lambda }$ are the amplification factors.

To account for the inherent heterogeneity in the physical and mechanical properties of the rock constituents at the mesoscale, the Weibull distribution [34,35] was implemented to statistically characterize the probabilistic strength distribution across rock elements:

$$P\left(\chi \right)={\displaystyle \frac{m}{{\chi }_0}}{\left({\displaystyle \frac{\chi }{{\chi }_0}}\right)}^{m-1}\exp \left[-{\left({\displaystyle \frac{\chi }{{\chi }_0}}\right)}^m\right]$$

(11)

where $\chi$ is the parameter of mesoscopic elements, ${\chi }_0$ is the scale parameter, and $m$ is the heterogeneity coefficient.

2.2.2. Model Descriptions

A plane strain model (Figure 2) of 1 × 1 m with a 0.1 m borehole diameter was adopted in this study. The mechanical parameters are presented in

Table 1. Mechanical parameters for the mesoscopic element.

Parameter	Value
$\mathrm{Average}\mathrm{value}\mathrm{of}\mathrm{initial}\mathrm{permeability}(m=20$) (m2)	4 × 10−19
Average value of the coefficient of thermal expansion (K−1)	8 × 10−6
Average value of modulus of elasticity (GPa)	45 *
Average compressive strength (MPa)	400 *
Average compressive strength/average tensile strength	16
Internal friction angle (°)	54
Heterogeneity coefficient (m)	4 *

* For heterogeneous models, the parameter values of mesoscopic elements are not equal to the macroscopic values and need to be determined via trial calculations.

. The initial temperature and initial pore pressure were 200 °C and 30 MPa, respectively. The initial values were maintained for both temperature and pore pressure at the outer boundary. The in-situ stresses were 75 MPa (for a 3000 m depth, according to 25 MPa/km [11]), applied to the outer boundaries. For LN₂ fracturing, the borehole boundary was pressurized at 0.5 MPa/s [26,36], and the time step was 20 s/step. The LN₂ temperature was 100 K [37]. For water or nitrogen fracturing, both water and nitrogen were injected at 25 °C. The pressurization rate was the same as that used for the liquid nitrogen fracturing.

3. Mechanical Property Deterioration

3.1. Crack Initiation Stress

Figure 3 shows the crack initiation stress of heated granite after air and LN₂ cooling under various temperatures and confining pressures. For all samples (Figure 3a), high temperatures led to a reduction in the crack initiation stress, especially above 200 °C, and the temperature sensitivity was weakened by the confining pressure. For instance, the crack initiation stress of the LN₂-cooled samples decreased by 46.11% when the sample temperature escalated from 25 to 400 °C under unconfined conditions (0 MPa); in contrast, the crack initiation stress decreased by 6.1% from 25 to 400 °C under the confining pressure of up to 60 MPa. The confining pressure caused the cracks to close and made crack initiation more difficult, thus mitigating the temperature-induced differences [20,38]. In addition, the crack initiation stresses of all samples significantly increased with the increase in confinement pressure, suggesting that the presence of confinement pressure makes rock cracking difficult. The crack initiation stress of the LN₂-cooled samples at 200 °C increased by 5.97 times when the confinement pressure was increased from 0 to 60 MPa. Moreover, the crack initiation stresses in the LN₂-cooled samples were lower than those of their air-cooled counterparts under different confinement pressures, highlighting the efficacy of LN₂ treatment in thermally induced mechanical weakening across varying confining pressures. Figure 3b shows the crack initiation stress reduction rate induced by LN₂. The reduction rate induced by LN₂ was the highest under unconfined conditions. The reduction rate significantly decreased when the pressure exceeded 20 MPa, indicating that the deterioration effect of LN₂ on crack initiation stress was weakened by confining pressure due to crack closure caused by the confining pressure. The reduction rate resulting from LN₂ was 7.73–18.51% in the range of 20 to 60 MPa, indicating that the LN₂ still had a weakening effect on the rock cracking threshold under high confining pressures.

3.2. Crack Damage Stress

Figure 4 shows the crack damage stress of heated granite after air and LN₂ cooling under various temperatures and confining pressures. The temperature effect on the crack damage stress was weakened by the confinement pressure. The crack damage stress of the LN₂-cooled samples from 25 to 400 °C decreased by 51.77% under unconfined conditions (Figure 4a) and by 5.33% under the confining pressure of 60 MPa. Under different conditions, the LN₂-cooled samples exhibited a reduction in crack damage stress compared to the results for their air-cooled counterparts, attributable to the preferential coalescence of cryogenically induced microcracks under loading conditions. Figure 4b shows the reduction rate of crack damage stress induced by LN₂. The reduction rate in crack damage stress caused by LN₂ cooling was significant under 0 MPa. The reduction rate caused by LN₂ cooling was lower when high confining pressures were applied, which indicates that the presence of confining pressure weakened the deterioration of crack damage stresses induced by LN₂. The reduction rate induced by LN₂ cooling was 3.46–12.15% when the confining pressure was in the range of 20 to 60 MPa, indicating that LN₂ remained effective in reducing the crack damage stress, while the confining pressure generally enhanced the resistance to cracking.

3.3. Peak Stress

Figure 5 presents the peak stress of heated granite after air and LN₂ cooling under various conditions. Similar to the crack initiation and damage stresses, the peak stress tended to decrease with temperature, and the confining pressure suppressed the temperature effect. The peak stress of the LN₂-cooled samples from 25 to 400 °C decreased by 30.52% under 0 MPa and by 5.5% under 60 MPa (Figure 5a). Additionally, the LN₂-cooled samples exhibited reduced peak stress magnitudes compared to those of their air-cooled counterparts, with this weakening effect attributed to thermally induced microcracks and localized stress concentration under cryogenic treatment [22,39]. Figure 5b shows the reduction rate in the peak stress induced by LN₂. The reduction rate in the peak stress caused by LN₂ cooling was the highest under unconfined conditions and decreased when it exceeded 20 MPa, which indicated that the presence of confinement pressure weakened the deterioration effect of LN₂. When the rock is unconstrained, LN₂ cooling can cause more stress reduction due to the thermal-induced cracks. The confining pressure limited this thermal-induced damage by restricting the expansion of cracks and maintaining the structural integrity of the material, thus weakening the overall degrading effect of LN₂ at higher stress levels. In addition, the weakening of the confinement pressure became stable when the confinement pressure exceeded 20 MPa. In the range of 20–60 MPa, the reduction rate in the peak stress caused by LN₂ cooling was in the range of 2.51–8.50%, showing that LN₂ cooling could decrease the peak stress under high confining pressures. This indicates that the thermal cracking due to LN₂ remained effective in reducing strength, even though the confining pressure led to crack closure and rock densification.

4. Acoustic Emission Event Statistics

In the triaxial compression experiments, the AE events during sample failure were collected using an AE testing system. The AE event distribution can reflect the crack distribution during sample failure. The LN₂ effect on the crack distribution of heated granite during sample failure was evaluated in terms of the statistical characteristics of the AE events.

4.1. Event Count

Figure 6 presents the count of AE events of heated granite after air and LN₂ cooling under various conditions. Each datum in Figure 6a is the average value of four scenarios (0, 20, 40, and 60 MPa) at the same temperature. Each datum in Figure 6b is the average value of three scenarios (25, 200, and 400 °C) at the same confining pressure.

For all samples, the count of AE events showed an ascending trend with the sample temperature (Figure 6a) and a descending trend with the confining pressure (Figure 6b). The LN₂-cooled samples consistently generated a higher count of AE events compared to their air-cooled counterparts across varying temperature and confining pressure regimes. At 25, 200, and 400 °C, the AE event counts after LN₂ cooling increased by 19.47%, 129.87%, and 25.66%, respectively, compared with the counts after air cooling. Under 0 and 60 MPa, the AE event counts after LN₂ cooling increased by 29.38% and 18.98%, respectively, compared with the counts after air cooling. The AE event counts were an indicator of the number of cracks formed during sample failure. These results revealed that the cryogenic effect of LN₂ contributed to an increase in crack formation across various conditions. The cryogenic LN₂ promoted crack initiation and propagation, leading to a higher number of cracks during failure and enhancing the rock’s susceptibility to cracking. This behavior indicated the role of cryogenic treatments in altering the rock’s mechanical response, weakening its structural integrity under specific conditions.

4.2. Fractal Dimension

Figure 7 presents the fractal dimension (box counting) of AE events in heated granite after air cooling and LN₂ cooling under various conditions. Each datum in Figure 7a is the average of four scenarios (0, 20, 40, and 60 MPa) at the same temperature. Each datum in Figure 7b is the average of three scenarios (25, 200, and 400 °C) at the same confining pressure. The fractal dimension of all samples tended to increase with the sample temperature (Figure 7a) and decrease with confining pressure (Figure 7b). The fractal dimension of the LN₂-cooled samples exceeded that of their air-cooled counterparts across varying temperature and confining pressure regimes, indicating that the cryogenic LN₂ enhanced crack complexity under diverse conditions. The higher fractal dimension indicates a more intricate crack network, resulting from the LN₂-induced thermal damage. This increased crack complexity under LN₂ cooling suggested that the cryogenic effect not only influenced the initiation of cracks but also their distribution.

5. Comparison of Liquid Nitrogen Fracturing and Conventional Fracturing

Experimental analysis showed that LN₂ increased the number and complexity of cracks and decreased the mechanical strength of high-temperature granite. LN₂ fracturing was compared with water and nitrogen fracturing via numerical simulation to reveal the advantages of LN₂ applied to fracturing for HDR reservoirs. Figure 8 displays the evolution of the damage distribution during water, nitrogen, and LN₂ fracturing, with positive values representing shear damage and negative values representing tensile damage. In the three fracturing models, the damage first appeared around the borehole and gradually extended outward, which is consistent with the fracture propagation during the fracturing process. When the fracturing continued, the fracture zones formed by LN₂ fracturing were more uniformly distributed in the radial directions, and the number of fracture zones was more than that of the other two modes. These results indicate that LN₂ fracturing contributed to the formation of multi-fractures, which is in agreement with the findings of Yang et al. [6] and proves the model’s correctness.

Figure 9 presents the cumulative number of damage elements and the fractal dimension (box counting) of the damage distribution during water, nitrogen, and LN₂ fracturing. The number and fractal dimension of damage due to LN₂ fracturing were higher than those of the other two methods, indicating that LN₂ fracturing promoted fracture complexity. The temperature difference between the reservoir and LN₂ led to significant thermal stresses, combined with the injection pressure, resulting in the formation of a greater number of more complex fractures. There was zero damage at a lower injection pressure. The number of damages increased with the increase in the injection pressure, and the corresponding injection pressure was determined as the initiation pressure [40]. The number and fractal dimension of damages rapidly increased as the injection pressure further increased, indicating that unstable damage occurred. The corresponding injection pressure was determined as the instability pressure [40].

Figure 10 presents the initiation pressure and instability pressure of the water, nitrogen, and liquid nitrogen fracturing models. The initiation and instability pressures of nitrogen fracturing exhibited high values, indicating that a higher breakdown pressure was needed for deep geothermal reservoirs with a high temperature and in-situ stress via mere injection pressure. This also reflected the fracturing difficulty regarding deep geothermal reservoirs. The initiation and instability pressures of water fracturing were lower than those of nitrogen fracturing because thermal stresses were generated to reduce breakdown pressures. However, the breakdown pressures of water fracturing still displayed high values under the effect of thermal stress, which was one of the limiting factors for hydraulic stimulation in deep geothermal reservoirs. Compared with water and nitrogen fracturing, the initiation pressures of LN₂ fracturing decreased by 61.54% and 68.75%, and the instability pressures of LN₂ fracturing decreased by 20% and 29.41%, respectively. These results suggest that LN₂ fracturing offers advantages by reducing the breakdown pressures for the geothermal reservoirs under high temperatures and high in-situ stress conditions.

6. Conclusions

The stress parameters and AE event statistics of heated granite after air cooling and LN₂ cooling were compared. Additionally, numerical simulation of LN₂ fracturing was performed and compared with the results for water and nitrogen fracturing. The following conclusions are summarized:

(1)

Confining pressure mitigated the effect of temperature on crack initiation stress, crack damage stress, and peak stress. It also reduced the deterioration effect of LN₂ cooling on these stress parameters.

(2)

LN₂ still reduced stress parameters under high confinement pressures. From 20 to 60 MPa, the reduction rates of crack initiation stress, crack damage stress, and peak stress induced by LN₂ cooling were 7.73–18.51%, 3.46–12.15%, and 2.51–8.50%, respectively.

(3)

Cryogenic LN₂ increased the counts and fractal dimensions of AE events during sample failure under different conditions.

(4)

LN₂ fracturing led to a greater number and complexity of fractures and reduced the breakdown pressures. Compared with water and nitrogen fracturing, the initiation pressures of LN₂ fracturing decreased by 61.54% and 68.75%, while the instability pressures of LN₂ fracturing decreased by 20.00% and 29.41%, respectively.

This study investigated the impacts of LN₂ on the mechanical degradation of granite. Future work will systematically explore the influence of LN₂ cycling on the rock’s mechanical behavior and its application in enhancing fracturing efficiency.