# Study of Rock Crack Extension under Liquid Nitrogen Low-Temperature Fracturing
## 4 Numerical Simulation of Liquid Nitrogen Low-Temperature Fracturing
### Effect of the Heat Transfer Coefficient on the Crack Propagation  
The monitoring of acoustic emissions in rock can reflect the changes in internal structure, help us to understand the time, location, and strength of microfractures, and further reveal the basic laws of rock crack initiation and propagation. Accordingly, the process is divided into various stages throughout the process of liquid nitrogen cold shock damage to the circular orifice plate model under different heat transfer coefficients, according to the changes in acoustic emission quantity and tensile stress: (1) the linear elastic deformation stage; (2) the microcrack initiation to propagation stage; (3) the instability fragmentation stage, or the stop expansion stage. Figure 11 shows the temporal variations in the number and cumulative number of acoustic emissions during liquid nitrogen cold shock, which is not described in the figure due to the rapidness of the first stage. In the second stage, the graded cracks around the wellbore are formed within 100 s. Thus, the second stage is analyzed based on 100 s.  
Figure 19 shows that the cumulative number of acoustic emissions increases with the cooling time, and the growth rate rapidly increases first and then fluctuates. Microcracks appeared in the rock almost within 0.2 s, which is very rapid. A large number of acoustic emission events mainly occur in the first 5 s. The reason is that the contact boundary is directly subjected to cold shock at this stage. The rapid increase in temperature gradient results in the instantaneous increase in tensile stress at the contact boundary, resulting in nearly a dozen microcracks rapidly initiating and propagating along the radial direction of the borehole (microcrack propagation period). After 5 s, some microcracks stopped propagating and gradually evolved into graded cracks. However, the number of AE events showed a certain fluctuation decrease at this stage, indicating that the crack growth rate slowly decreased over time. The higher number of acoustic emissions at a certain time was mainly due to the uneven distribution of the material properties caused by the heterogeneity of the rock materials, and more point damage occurred around the crack tip. Rocks with low heat transfer coefficients (H = 100 W\(\cdot\)m\({}^{-2}\)\(\cdot\)K\({}^{-1}\)) showed obvious hysteresis during a liquid nitrogen cold shock. Figure 17 demonstrates that the failure elements appeared only after 1 s. The number of these random and disordered point-like failure elements continuously increased after 5 s, turning to microcracks. This was slower than that of rock with a high heat transfer coefficient at 0.2 s. Hence, the number of acoustic emissions in the first few seconds of rock with a low heat transfer coefficient was smaller than that of the rock with a high heat transfer coefficient, showing a fluctuating upward trend. At 20 s, some microcracks stopped propagating and gradually evolved into graded structures. Figure 17: Temperature distribution and crack propagation in the circular hole plate models with different heat exchange coefficients (unit: \({}^{\circ}\)C). (a) H = 100 W \(\cdot\) m\({}^{-2}\)\(\cdot\) K\({}^{-1}\); (b) H = 500 W \(\cdot\) m\({}^{-2}\)\(\cdot\) K\({}^{-1}\); (c) H = 1000 W \(\cdot\) m\({}^{-2}\)\(\cdot\) K\({}^{-1}\). cracks. This phenomenon further shows that the larger the heat transfer coefficient, the faster the temperature gradient changes, and the faster the rock failure rate. After a long crack propagation period, the rock enters the third stage, namely, the unstable fracture stage, or the stop propagation stage. Given the high elastic modulus and strength of rock, the above model needs at least 1800 s to achieve this stage. Table 3 illustrates the statistical table of the failure rate of rocks with various heat transfer coefficients.