Study on the Law of Rock Drillability under High-Temperature and High-Pressure Conditions in the Western Margin of the Ordos Basin

: The deep strata in the Ordos Basin exhibit characteristics of high temperature and high stress. Conventional methods for assessing drillability (normal temperature and pressure) fail to accurately understand the drilling resistance characteristics of deep rocks in this region, leading to improper guidance for selecting formation drilling tools and prolonging drilling cycles. This study employs physical experiments and numerical simulations to conduct drillability tests on core samples taken from the region under high-temperature and high-pressure conditions, simultaneously simulating the rock breaking process under different temperature and pressure conditions. The study investigates the variation patterns of rock drillability grade values and von Mises stress values during rock breaking under single-factor and multi-factor analyses of temperature and pressure conditions. Combining these variation patterns, an optimization analysis of the back rake angle of PDC drill bits used in drillability experiments is conducted to guide the selection of drill bits on site. The results indicate that the variation patterns of von Mises values from finite element simulations are consistent with the drillability grade values under high-temperature and high-pressure conditions. Under single-factor conditions, von Mises stress and drillability grade values generally increase with rising temperature before decreasing, while they increase with increasing confining pressure. Under multi-factor conditions, confining pressure is the primary influencing factor within the range of 0 to 50 MPa, while the influence of temperature becomes prominent between 180 ◦ C to 200 ◦ C, with a weakening effect of confining pressure. Model application: Selecting a back rake angle of 30 ◦ for PDC drill bits yields optimal rock-breaking results. The research findings hold significant implications for understanding the low rock-breaking efficiency of deep strata, optimizing drill bit parameters, and enhancing drilling efficiency.


Introduction
Rock drillability refers to the rock's ability to resist fragmentation [1][2][3] and is an important indicator of the ease of rock breaking during the rock-breaking process [4][5][6].In 1983, Miranda, A. [3] conducted experiments on rock drilling capacity, establishing the concept of drillability.Based on this concept, the national standards for drilling engineering drillability testing were formulated [7,8].Scholars [5] found that conducting experiments only under normal temperature and pressure conditions fails to reflect the true conditions of deep strata and cannot accurately assess the characteristics of deep rocks.Consequently, scholars at home and abroad have conducted research on high-temperature and high-pressure strata.Fu [9], and others have explored the impact of temperature and pressure on granite drillability through experiments, revealing the significant importance of granite properties for the development of dry-hot rocks.Wei et al. [10] combined experiments with finite element models and verified the differences between the rock drillability grade values and experiments under the condition of confining pressure variation, without considering the influence of temperature on drillability.Ngwu G. I. et al. [11] combined experiments with software to determine the influence of high temperature and high pressure on wellbore stress and concluded that drillability is affected by both temperature and pressure.Gao et al. [12] combined experiments with statistical methods, indicating that the analysis of rock characteristics under the coupled effects of temperature and confining pressure in granite is more accurate and pointing out the rock fragmentation mechanisms in different temperature and pressure regions.Zhou et al. [13] exclusively investigated the rock-breaking conditions of granite in deep and ultra-deep layers using indoor experimental methods, analyzing drillability and indicating that the drillability characteristics of deep rocks are influenced by geological environments and different rock types.Howarth, Rowlands [14], and Hoseinie, S.H. [15] believe that micro-cracks affect the mechanical properties and drillability of rocks.Abu Bakar et al. [16] and Paul, S. et al. [17] suggest that under temperature and pressure conditions, granite may develop erosive cracks, leading to fractured rocks affecting drillability; meanwhile, as temperature and pressure increase, the density of cracks in granite also increases.Ji et al. [5] conducted laboratory experiments to determine the distribution patterns of drillability under temperature and pressure conditions in sandstone-mudstone formations and discovered the presence of thermal cracks in sandstone-mudstone.
The geological structure of the western margin of the Ordos Basin is complex, and the development of oil and gas layers is shifting toward deeper layers.The high compaction degree of deep sandstone-mudstone layers, coupled with high stress and temperature [18,19], leads to poor drillability and low mechanical drilling speed during the drilling process.The average mechanical drilling speed of well Z 1 in zone Z on the western margin decreases from 30 m/h to 6 m/h with increasing depth; meanwhile, the extracted muddy cores are prone to fracturing, making effective experiments impossible.Therefore, this study utilizes the methods of physical experiments and numerical simulations on sandstone-mudstone, based on the research of scholars [9][10][11][12][13] on granite, to effectively utilize core samples and verify whether there is correlation and regularity between the experimental results of physical and numerical models, providing a new approach for regional drillability testing, reducing losses of underground cores under experimental conditions, accurately understanding the anti-drilling characteristics of deep strata rocks, and improving regional drilling speed to accelerate drilling efficiency [20][21][22][23].

Experimental Equipment
Drillability determination experiments under high-temperature and high-pressure conditions were conducted using fully automated rock drillability testing equipment, as illustrated in Figure 1 (Shandong Zhongshi Dashiyi Technology Co., Ltd., Qingdao, China).The equipment comprises a fully automated high-pressure pump, a drillability tester, a rock core holder, and a computer control system.Temperature and pressure conditions are regulated by the holder on the right side.The required specifications for the rock samples by the holder are cylindrical specimens with a diameter of Φ100 mm and a height of 80 mm.

Rock Drillability Experiments under Different Temperature and Pressure Conditions
Experimental samples were obtained from the Z Block of the Ordos Basin, at a depth of 4800-5000 m.The experimental procedure is depicted in Figure 2. Upon completion of the experiment, the data processing interface was accessed to read experimental data such as drilling pressure, torque, and drillability grade.A total of 24 experiments were conducted on 12 rock cores, as outlined in Table 1, along with the experimental plan and results.

Rock Drillability Experiments under Different Temperature and Pressure Conditions
Experimental samples were obtained from the Z Block of the Ordos Basin, at a depth of 4800-5000 m.The experimental procedure is depicted in Figure 2. Upon completion of the experiment, the data processing interface was accessed to read experimental data such as drilling pressure, torque, and drillability grade.A total of 24 experiments were conducted on 12 rock cores, as outlined in Table 1, along with the experimental plan and results.

Rock Drillability Experiments under Different Temperature and Pressure Conditions
Experimental samples were obtained from the Z Block of the Ordos Basin, at a depth of 4800-5000 m.The experimental procedure is depicted in Figure 2. Upon completion of the experiment, the data processing interface was accessed to read experimental data such as drilling pressure, torque, and drillability grade.A total of 24 experiments were conducted on 12 rock cores, as outlined in Table 1, along with the experimental plan and results.   1 indicates that under the condition of maintaining a temperature of room temperature (20 • C), the drillability grade of the rock increases with the increase in confining pressure.When the confining pressure reaches 40 MPa, the rock has the longest drillability test time.Due to the inherent conditions of the rock in the formation, the radial compaction degree of the rock is optimal, leading to an increase in the drillability grade.As the confining pressure continues to rise, the cracks inside the rock samples are re-compacted, but small cracks still exist, resulting in a decrease in drillability grade.
At a confining pressure of 0 MPa, the drillability grade shows a decreasing trend with the increase in temperature.Some scholars have shown [5,17,24] that the critical temperature causing thermal cracks in sandstone and mudstone is around 180 • C. When below this temperature, the properties of the rock change little with temperature; when above this temperature, thermal cracks form in the rock, Young's modulus decreases significantly, the rock strength decreases, and the drillability grade decreases.
Under the combined effect of temperature and confining pressure, the drillability grade of rock changes more significantly with confining pressure.At a confining pressure of 40 MPa and a temperature of 180 • C, the impact of thermal cracks generated by temperature is more evident, significantly affecting the physical properties of the rock and resulting in a significant reduction in the drillability grade.When the temperature exceeds the critical value, the results are consistent with those of the single-temperature experiment.As the confining pressure increases, the impact of thermal cracks is evident, leading to a decrease in drillability grade.
After the experiments, thin section analysis was conducted on some of the rock samples, revealing prominent cracks under a microscope with a resolution of 300 µm (see Figure 3, the red line indicates the case of a crack.),indicating that as temperature and confining pressure increase, internal cracking occurs within the rock.
Processes 2024, 12, x FOR PEER REVIEW 5 of 17 experiment.As the confining pressure increases, the impact of thermal cracks is evident, leading to a decrease in drillability grade.
After the experiments, thin section analysis was conducted on some of the rock samples, revealing prominent cracks under a microscope with a resolution of 300 µm (see Figure 3, the red line indicates the case of a crack.),indicating that as temperature and confining pressure increase, internal cracking occurs within the rock.

Assumptions of the Model
The simulation model for rock drilling is highly complex, but the focus of this study is on the motion and force of the drill bit during the rock-breaking process.To improve computational efficiency, minor factors are ignored, and the simulation model makes the following assumptions: (1) The rock is treated as a continuous isotropic medium, neglecting the influence of initial cracks and internal pore pressure in the rock.(2) The hardness of the composite drill bit is significantly greater than the strength of the rock; hence, the drill bit is assumed to be rigid.(3) Rock failure is modeled using the Drucker-Prager constitutive model, and the parameters of the constitutive model are not influenced by temperature and pressure.(4) The influence of temperature on the rock drilling process is disregarded.(5) In the event of rock element failure, failed elements are not considered, avoiding situations of repetitive cutting.

Model Establishment
The model of the PDC drill bit used in the drillability experiment was constructed using SolidWorks software.This complex model was then imported into ABAQUS

Assumptions of the Model
The simulation model for rock drilling is highly complex, but the focus of this study is on the motion and force of the drill bit during the rock-breaking process.To improve computational efficiency, minor factors are ignored, and the simulation model makes the following assumptions: (1) The rock is treated as a continuous isotropic medium, neglecting the influence of initial cracks and internal pore pressure in the rock.(2) The hardness of the composite drill bit is significantly greater than the strength of the rock; hence, the drill bit is assumed to be rigid.(3) Rock failure is modeled using the Drucker-Prager constitutive model, and the parameters of the constitutive model are not influenced by temperature and pressure.(4) The influence of temperature on the rock drilling process is disregarded.(5) In the event of rock element failure, failed elements are not considered, avoiding situations of repetitive cutting.

Model Establishment
The model of the PDC drill bit used in the drillability experiment was constructed using SolidWorks software.This complex model was then imported into ABAQUS (Abaqus/CAE 2020) simulation software to establish a simulation study of the drill bit and rock models, with specific conditions added as depicted in Figure 4, the red arrow indicate that the direction of applying confining pressure contains a circle.The study presents a sophisticated three-dimensional numerical model that accounts for the thermal interaction between PDC teeth and the formation.Geometric dimensions of the formation rock model are specified in millimeters Φ100 × H80 (mm).For thermal transfer analysis, the numerical model adopts the Standard-thermal transfer element library, employing linear geometric order C3D8R elements for enhanced accuracy and computational efficiency.The mesh of the PDC bit utilizes an approximate element size layout, featuring a 1.5 interval at the tooth ring and 2 outside the tooth ring.The global mesh size of the rock is partitioned into two parts, with intervals of 1 and 3 for the internal and external layouts, respectively.In the stress field analysis section, the rock model employs the standard-three-dimensional stress element library, utilizing linear geometric order C3D8R elements, while the PDC bit utilizes linear geometric order C3D10M elements for comprehensive stress analysis.
The mechanical parameters of the main materials in the finite element model (PDC bit and rock) are detailed in Table 2, with the rock parameters derived from geological mechanical experiments conducted at a depth of 5000 m in the Z area of the Ordos Basin (temperature 130 °C; pressure 40 MPa).The material properties were input into the property module of ABAQUS, with the incorporation of the Drucker-Prager (D-P) constitutive rock failure criterion and the establishment of the damage evolution coefficient.Temperature is modeled using steady-state heat transfer principles, which can be directly configured within the predefined field, while pressure is defined using the pressure setting feature of the load module.

Rock-Breaking Patterns in the Model
The parameters within the ABAQUS numerical simulation software are meticulously configured to maintain alignment between laboratory-calibrated parameters and model parameter settings.Employing the controlled variable method, both single-factor and multi-factor analyses are executed, systematically altering temperature and confining pressure conditions within the finite element model.Multiple finite element models are The study presents a sophisticated three-dimensional numerical model that accounts for the thermal interaction between PDC teeth and the formation.Geometric dimensions of the formation rock model are specified in millimeters Φ100 × H80 (mm).For thermal transfer analysis, the numerical model adopts the Standard-thermal transfer element library, employing linear geometric order C3D8R elements for enhanced accuracy and computational efficiency.The mesh of the PDC bit utilizes an approximate element size layout, featuring a 1.5 interval at the tooth ring and 2 outside the tooth ring.The global mesh size of the rock is partitioned into two parts, with intervals of 1 and 3 for the internal and external layouts, respectively.In the stress field analysis section, the rock model employs the standard-three-dimensional stress element library, utilizing linear geometric order C3D8R elements, while the PDC bit utilizes linear geometric order C3D10M elements for comprehensive stress analysis.
The mechanical parameters of the main materials in the finite element model (PDC bit and rock) are detailed in Table 2, with the rock parameters derived from geological mechanical experiments conducted at a depth of 5000 m in the Z area of the Ordos Basin (temperature 130 • C; pressure 40 MPa).The material properties were input into the property module of ABAQUS, with the incorporation of the Drucker-Prager (D-P) constitutive rock failure criterion and the establishment of the damage evolution coefficient.Temperature is modeled using steady-state heat transfer principles, which can be directly configured within the predefined field, while pressure is defined using the pressure setting feature of the load module.

Rock-Breaking Patterns in the Model
The parameters within the ABAQUS numerical simulation software are meticulously configured to maintain alignment between laboratory-calibrated parameters and model parameter settings.Employing the controlled variable method, both single-factor and multifactor analyses are executed, systematically altering temperature and confining pressure conditions within the finite element model.Multiple finite element models are harnessed for pattern analysis.In single-factor analysis, temperatures are held constant at 20 • C while varying confining pressures at 20 MPa, 30 MPa, 40 MPa, and 50 MPa; likewise, with confining pressures set at 0, temperatures range from 120 • C to 200 • C. In multi-factor analysis, combinations of confining pressures and temperatures are explored, ranging from 20 MPa and 120 • C to 50 MPa and 200 • C.

Impact of Confining Pressure Variation on Rock Fracture
While controlling for other parameters' influence, the temperature was standardized at 20 • C, with confining pressures varying at 20 MPa, 30 MPa, 40 MPa, and 50 MPa, respectively.Extracted from both experimental data and the finite element simulation results, the drilling pressure versus depth curves and torque versus depth curves are displayed in Figures 5 and 6.Additionally, the finite element simulation employs an identical frame count to visualize the fragmentation of rocks, as shown in the von Mises stress contour plots in Figure 7.
From Figures 5 and 6, it is evident that in the drill pressure-depth curve, the fluctuation range of the drill pressure results falls within the error range.In the torque-depth curve, the average R 2 fitting value between the simulation and experimental conditions is 0.83, indicating good agreement between the simulation and experimental conditions.Figures 7 and 8 demonstrate that the actual von Mises stress initially increases and then decreases with increasing confining pressure.When the confining pressure reaches 40 MPa, the confining pressure of the rock gradually approaches the failure stress set by the rock parameters.The stress on the rock continues to increase, reaching the failure limit of the rock, resulting in a decrease in von Mises stress.While controlling for other parameters' influence, the temperature was standardized at 20 °C, with confining pressures varying at 20 MPa, 30 MPa, 40 MPa, and 50 MPa, respectively.Extracted from both experimental data and the finite element simulation results, the drilling pressure versus depth curves and torque versus depth curves are displayed in Figures 5 and 6.Additionally, the finite element simulation employs an identical frame count to visualize the fragmentation of rocks, as shown in the von Mises stress contour plots in Figure 7. From Figures 5 and 6, it is evident that in the drill pressure-depth curve, the fluctuation range of the drill pressure results falls within the error range.In the torque-depth curve, the average R 2 fitting value between the simulation and experimental conditions is 0.83, indicating good agreement between the simulation and experimental conditions.Fig-

Combined Effect of Temperature and Confining Pressure Variation on Rock Fracture
To control the influence of other parameters on the model, the pressure and temperature were systematically varied as follows: 20 MPa, 120 °C; 30 MPa, 150 °C; 40 MPa, 180 °C; and 50 MPa, 200 °C.The experimental and finite element simulation results of the drilling pressure versus depth curves, as well as the torque versus depth curves, were retrieved, and the output data are presented in Figures 13 and 14.The finite element simulation employs an identical frame count to illustrate the rock fragmentation scenario, while the von Mises stress cloud diagram is portrayed in Figure 15.C. The experimental and finite element simulation results of the drilling pressure versus depth curves, as well as the torque versus depth curves, were retrieved, and the output data are presented in Figures 13 and 14.The finite element simulation employs an identical frame count to illustrate the rock fragmentation scenario, while the von Mises stress cloud diagram is portrayed in Figure 15.
Figures 13 and 14 reveal that the drilling pressure results fluctuate within the error range in the drilling pressure-depth curve, while the correlation coefficient R 2 between the simulated and experimental torque-depth curves averages 0.95.As illustrated in Figures 15 and 16, the von Mises stress initially increases with rising temperature and pressure before declining.Analyzing the fractured rock conditions, it becomes apparent that at 20 MPa and 120 • C, as well as at 30 MPa and 150 • C, significant stress variations occur upon the initial contact of the rock with the drill bit, with stress values notably increasing as pressure rises.However, at 40 MPa and 180 • C, observations from the later frames (Figure 17) indicate that the drill bit fails to effectively engage with the rock, and no distinct stress concentration zones are evident during rock fracturing.This could be attributed to crack formation in the high-temperature zone under this condition, leading to stress dissipation.For the rock subjected to 50 MPa and 200 • C, cracks form in the high-temperature zone, yet due to pressure effects, these cracks are repeatedly compressed, resulting in reduced stress levels and diminished rock-breaking efficiency.

Analysis of Experimental and Finite Element Method Results
The physical experiments and numerical simulations, as presented in Table 1 and  Figures 8, 12, and 16, reveal a consistent correlation and systematic relationship between the drillability grade and von Mises stress values under elevated temperature and pressure conditions.The insights derived from the numerical simulations hold significant

Analysis of Experimental and Finite Element Method Results
The physical experiments and numerical simulations, as presented in Table 1 and Figures 8, 12 and 16, reveal a consistent correlation and systematic relationship between the drillability grade and von Mises stress values under elevated temperature and pressure conditions.The insights derived from the numerical simulations hold significant value for conducting laboratory analyses on drillability under such extreme conditions, providing valuable insights into the intricate dynamics involved.

Model Application
The physical experiments and numerical simulation results demonstrate that the simulated scenarios can provide a reasonable reference for real-world applications.Currently, laboratory experiments utilize PDC drill bits with a cutting tooth back rake angle of 20 • .There is a need to explore the optimal back rake angle for the cutting teeth under specific temperature and pressure conditions, aiming to generate higher stress values at suitable angles.This optimization is crucial for enhancing drilling efficiency and providing guidance for field applications.
In response to the conditions of 4800 m-5000 m depth in the Ordos Basin Zone Z, with a temperature of 130 • C and a pressure of 40 MPa, finite element simulations were conducted to optimize the back rake angle of the cutting teeth.Maintaining consistent parameter settings and controlling for other factors, the PDC drill bit model was varied with cutting tooth angles of 20

Discussion
This study provides direction for researching drillability under different temperature and pressure conditions.However, there is still a lack of accurate consistency between the simulations and actual conditions.The simulation results show significant numerical differences from reality, and there is a lack of empirical formulas to accurately determine the drillability level through numerical simulation.Many domestic and international scholars have conducted numerous repeated indoor experiments and have gradually recognized

Discussion
This study provides direction for researching drillability under different temperature and pressure conditions.However, there is still a lack of accurate consistency between the simulations and actual conditions.The simulation results show significant numerical differences from reality, and there is a lack of empirical formulas to accurately determine the drillability level through numerical simulation.Many domestic and international scholars have conducted numerous repeated indoor experiments and have gradually recognized the patterns of different lithologies.However, progress in numerical simulation has reached a bottleneck.The broad prospects of numerical simulation require scholars to focus on the correlation between various parameters.By establishing related functions through multiple regression analysis of data, the reasonable application of numerical simulation can effectively improve the utilization rate of rock and further guide oil and gas extraction.

Conclusions
This study combines physical experimentation with finite element analysis to explore the consistency between laboratory methods and numerical modeling approaches.It investigates the variations in von Mises stress values and rock drillability grades under single and multiple factors in the high-temperature and high-pressure conditions of the Z block in the Ordos Basin.The following conclusions were drawn: (1) The experiments and simulations conducted under different temperature and pressure conditions indicate that, under different temperature conditions, the drillability level and von Mises stress value exhibit a pattern of initial increase followed by a decrease with rising temperature.Under pressure, the drillability level and von Mises stress value demonstrate a positive correlation.Under the combined influence of temperature and pressure, the impact of pressure becomes more pronounced, particularly when the temperature is confined within the range of 180-200 • C. Upon reaching the original temperature and pressure conditions of the formation, the drillability level of the rock undergoes significant changes.(2) The results from the physical experiments and finite element analysis exhibit high consistency, affirming the rationality of employing the finite element method for determining rock drillability under high temperature and pressure conditions.(3) To expand upon the research findings, an optimization simulation model for the back-tilt angle of PDC drill bits under different temperature and pressure conditions in the Ordos Basin was developed.The simulation results reveal that under the temperature and pressure conditions of the Ordos Basin, a back-tilt angle of 30 • achieves the highest efficiency in rock breaking, making it applicable to this region.

Figure 1 .
Figure 1.Cross-section of the fully automated rock drillability testing equipment and the rock core holder.

Figure 1 .
Figure 1.Cross-section of the fully automated rock drillability testing equipment and the rock core holder.

Figure 1 .
Figure 1.Cross-section of the fully automated rock drillability testing equipment and the rock core holder.

Figure 2 .
Figure 2. Flowchart of the rock drillability testing procedure.

Figure 3 .
Figure 3. Thin section identification of rock samples with 3% sandstone and 97% mud (a) and 100% mud (b) under polarized light conditions.

Figure 3 .
Figure 3. Thin section identification of rock samples with 3% sandstone and 97% mud (a) and 100% mud (b) under polarized light conditions.

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12,  x FOR PEER REVIEW 6 of Abaqus/CAE 2020) simulation software to establish a simulation study of the drill bit and rock models, with specific conditions added as depicted in Figure4, the red arrow indicate that the direction of applying confining pressure contains a circle.

Figure 4 .
Figure 4. Model diagram with applied conditions in ABAQUS Software.

Figure 4 .
Figure 4. Model diagram with applied conditions in ABAQUS Software.

Figure 5 .
Figure 5. Drilling pressure-depth curve under a single confining pressure condition.

Figure 5 .
Figure 5. Drilling pressure-depth curve under a single confining pressure condition.

Figure 6 .
Figure 6.Torque-depth curve under a single confining pressure condition.Figure 6. Torque-depth curve under a single confining pressure condition.

Figure 6 . 17 Figure 6 .
Figure 6.Torque-depth curve under a single confining pressure condition.Figure 6. Torque-depth curve under a single confining pressure condition.

Figure 7 .
Figure 7. Von Mises stress cloud diagram from the finite element simulation under a single confining pressure condition.

Figure 7 .
Figure 7. Von Mises stress cloud diagram from the finite element simulation under a single confining pressure condition.

Figure 8 .
Figure 8.Comparison between experimental and numerical von Mises stress values under a single confining pressure condition.3.3.2.Impact of Temperature Variation on Rock Fracture We controlled the other parameters' influence on the model, with confining pressure set to zero, and temperatures set at 120 °C, 150 °C, 180 °C, and 200 °C, respectively.The experimental and finite element simulation results of the drill pressure-depth curve and torque-depth curve output are shown in Figures 9 and 10.The finite element simulation illustrates the rock fragmentation using the same frame rate, with the von Mises stress cloud map depicted in Figure 11.
. The finite element simulation illustrates the rock fragmentation using the same frame rate, with the von Mises stress cloud map depicted in Figure11.

Figure 8 .
Figure 8.Comparison between experimental and numerical von Mises stress values under a single confining pressure condition.3.3.2.Impact of Temperature Variation on Rock Fracture We controlled the other parameters' influence on the model, with confining pressure set to zero, and temperatures set at 120 • C, 150 • C, 180 • C, and 200 • C, respectively.The experimental and finite element simulation results of the drill pressure-depth curve and torque-depth curve output are shown in Figures 9 and 10.The finite element simulation illustrates the rock fragmentation using the same frame rate, with the von Mises stress cloud map depicted in Figure 11.
. The finite element simulation illustrates the rock fragmentation using the same frame rate, with the von Mises stress cloud map depicted in Figure11.

Figure 7 .
Figure 7. Von Mises stress cloud diagram from the finite element simulation under a single confining pressure condition.

Figure 8 .
Figure 8.Comparison between experimental and numerical von Mises stress values under a single confining pressure condition.3.3.2.Impact of Temperature Variation on Rock Fracture We controlled the other parameters' influence on the model, with confining pressure set to zero, and temperatures set at 120 °C, 150 °C, 180 °C, and 200 °C, respectively.The experimental and finite element simulation results of the drill pressure-depth curve and torque-depth curve output are shown in Figures 9 and 10.The finite element simulation illustrates the rock fragmentation using the same frame rate, with the von Mises stress cloud map depicted in Figure 11.
. The finite element simulation illustrates the rock fragmentation using the same frame rate, with the von Mises stress cloud map depicted in Figure11.

Figure 9 .
Figure 9. Drilling pressure versus depth curve under single-temperature conditions.

Figure 9 .
Figure 9. Drilling pressure versus depth curve under single-temperature conditions.

Figure 10 .
Figure 10.Torque versus depth curve under single-temperature conditions.

Figures 9
Figures 9 and 10 reveal that in the drilling pressure versus depth curves, the drilling pressure results fluctuate within the range of error, while in the torque versus depth curves, the average R 2 value of the simulated data compared to the experimental data is 0.79.Figures11 and 12illustrate that the actual von Mises stress initially increases with temperature before decreasing.Initially, temperature exerts a limited influence; however, upon reaching 180 °C, the rock is significantly affected by the temperature field, leading to thermal diffusion toward the drill bit and the formation of a high-temperature zone within the rock.Consequently, this induces rock fracturing, resulting in a substantial decrease in the von Mises stress values and a diminished rock-breaking effect.With a temperature of 200 °C, the pronounced high-temperature phenomenon further exacerbates the reduction in stress values.

Figure 11 .
Figure 11.Von Mises stress distribution in the finite element simulation under single-temperature conditions.

Figures 9
Figures 9 and 10 reveal that in the drilling pressure versus depth curves, the drilling pressure results fluctuate within the range of error, while in the torque versus depth curves, the average R 2 value of the simulated data compared to the experimental data is 0.79.Figures11 and 12illustrate that the actual von Mises stress initially increases with temperature before decreasing.Initially, temperature exerts a limited influence; however, upon reaching 180 • C, the rock is significantly affected by the temperature field, leading to thermal diffusion toward the drill bit and the formation of a high-temperature zone within the rock.Consequently, this induces rock fracturing, resulting in a substantial decrease in the von Mises stress values and a diminished rock-breaking effect.With a temperature of 200 • C, the pronounced high-temperature phenomenon further exacerbates the reduction in stress values.

Figure 11 .
Figure 11.Von Mises stress distribution in the finite element simulation under single-temperature conditions.

Figure 12 .
Figure 12.Comparison of the experimental data and von Mises stress values under single-temperature conditions.

Figure 12 .
Figure 12.Comparison of the experimental data and von Mises stress values under single-temperature conditions.3.3.3.Combined Effect of Temperature and Confining Pressure Variation on Rock Fracture To control the influence of other parameters on the model, the pressure and temperature were systematically varied as follows: 20 MPa, 120 • C; 30 MPa, 150 • C; 40 MPa, 180 • C; and 50 MPa, 200 • C. The experimental and finite element simulation results of the drilling pressure versus depth curves, as well as the torque versus depth curves, were retrieved, and the output data are presented in Figures 13 and 14.The finite element simulation employs an identical frame count to illustrate the rock fragmentation scenario, while the von Mises stress cloud diagram is portrayed in Figure 15.Figures13 and 14reveal that the drilling pressure results fluctuate within the error range in the drilling pressure-depth curve, while the correlation coefficient R 2 between the simulated and experimental torque-depth curves averages 0.95.As illustrated in Figures15 and 16, the von Mises stress initially increases with rising temperature and pressure before declining.Analyzing the fractured rock conditions, it becomes apparent that at 20 MPa and 120 • C, as well as at 30 MPa and 150 • C, significant stress variations occur upon the initial contact of the rock with the drill bit, with stress values notably increasing as pressure rises.However, at 40 MPa and 180 • C, observations from the later frames (Figure17) indicate that the drill bit fails to effectively engage with the rock, and no distinct stress concentration zones are evident during rock fracturing.This could be attributed to crack formation in the high-temperature zone under this condition, leading to stress dissipation.For the rock subjected to 50 MPa and 200 • C, cracks form in the high-temperature zone, yet due to pressure effects, these cracks are repeatedly compressed, resulting in reduced stress levels and diminished rock-breaking efficiency.

Figure 13 .
Figure 13.Illustration of the drilling pressure-depth curve under varying temperature and pressure conditions.

Figure 14 .
Figure 14.Depiction of the torque-depth curve under changing temperature and pressure conditions.

Figure 13 . 17 Figure 13 .
Figure 13.Illustration of the drilling pressure-depth curve under varying temperature and pressure conditions.

Figure 14 .
Figure 14.Depiction of the torque-depth curve under changing temperature and pressure conditions.

Figure 14 .
Figure 14.Depiction of the torque-depth curve under changing temperature and pressure conditions.

Figure 15 .
Figure 15.Displayed is the von Mises stress cloud diagram from the finite element simulation under the influence of temperature and pressure.

Figures 13 and 14
Figures 13 and 14 reveal that the drilling pressure results fluctuate within the error range in the drilling pressure-depth curve, while the correlation coefficient R 2 between the simulated and experimental torque-depth curves averages 0.95.As illustrated in Figures 15 and 16, the von Mises stress initially increases with rising temperature and pressure before declining.Analyzing the fractured rock conditions, it becomes apparent that at 20 MPa and 120 °C, as well as at 30 MPa and 150 °C, significant stress variations occur upon the initial contact of the rock with the drill bit, with stress values notably increasing as pressure rises.However, at 40 MPa and 180 °C, observations from the later frames (Figure17) indicate that the drill bit fails to effectively engage with the rock, and no distinct stress concentration zones are evident during rock fracturing.This could be attributed to crack formation in the high-temperature zone under this condition, leading to stress dissipation.For the rock subjected to 50 MPa and 200 °C, cracks form in the high-temperature zone, yet due to pressure effects, these cracks are repeatedly compressed, resulting in reduced stress levels and diminished rock-breaking efficiency.

Figure 15 .
Figure 15.Displayed is the von Mises stress cloud diagram from the finite element simulation under the influence of temperature and pressure.Processes 2024, 12, x FOR PEER REVIEW 14 of 17

Figure 16 .
Figure 16.The figure presents the experimental data alongside the von Mises stress values affected by temperature and pressure variations.

Figure 17 .
Figure 17.The figure illustrates variations in the simulation of rock fracturing under identical numbers of later frames for different temperature and pressure conditions.

Figure 16 .
Figure 16.The figure presents the experimental data alongside the von Mises stress values affected by temperature and pressure variations.

Figure 16 .
Figure 16.The figure presents the experimental data alongside the von Mises stress values affected by temperature and pressure variations.

Figure 17 .
Figure 17.The figure illustrates variations in the simulation of rock fracturing under identical numbers of later frames for different temperature and pressure conditions.

Figure 17 .
Figure 17.The figure illustrates variations in the simulation of rock fracturing under identical numbers of later frames for different temperature and pressure conditions.
, 25 • , 30 • , and 35 • .The results of the finite element simulations are illustrated in the following figure.Observation of Figures 18 and 19 reveals that the von Mises stress values for cutting teeth ranging from 20 • to 35 • are 1332.0MPa, 1365.8MPa, 1416.8MPa, and 910.4 MPa, respectively.The rock's stress condition was assessed at the same frame number when the drill bit made contact with the rock.From the perspective of fractured rocks, the range of von Mises stress for rock breaking lies between the maximum and minimum values.Under the mechanical parameters of the Z block rocks, when the cutting tooth angle is 30 • , the von Mises stress value is maximized, indicating the highest level of rock damage intensity and consequently, relatively higher rock-breaking efficiency.Processes 2024, 12, x FOR PEER REVIEW 15 of 17 conducted to optimize the back rake angle of the cutting teeth.Maintaining consistent parameter settings and controlling for other factors, the PDC drill bit model was varied with cutting tooth angles of 20°, 25°, 30°, and 35°.The results of the finite element simulations are illustrated in the following figure.Observation of Figures 18 and 19 reveals that the von Mises stress values for cutting teeth ranging from 20° to 35° are 1332.0MPa, 1365.8MPa, 1416.8MPa, and 910.4 MPa,respectively.The rock's stress condition was assessed at the same frame number when the drill bit made contact with the rock.From the perspective of fractured rocks, the range of von Mises stress for rock breaking lies between the maximum and minimum values.Under the mechanical parameters of the Z block rocks, when the cutting tooth angle is 30°, the von Mises stress value is maximized, indicating the highest level of rock damage intensity and consequently, relatively higher rock-breaking efficiency.

Table 1 .
Rock drillability test results under different temperature and pressure conditions in Block Z of the Ordos Basin.

Table 2 .
Mechanical parameters of the main materials in finite element models.

Table 2 .
Mechanical parameters of the main materials in finite element models.
Processes 2024, 12, x FOR PEER REVIEW 7 of 17 harnessed for pattern analysis.In single-factor analysis, temperatures are held constant at 20 °C while varying confining pressures at 20 MPa, 30 MPa, 40 MPa, and 50 MPa; likewise, with confining pressures set at 0, temperatures range from 120 °C to 200 °C.In multi-factor analysis, combinations of confining pressures and temperatures are explored, ranging from 20 MPa and 120 °C to 50 MPa and 200 °C.