# Study of Rock Crack Extension under Liquid Nitrogen Low-Temperature Fracturing
## 4 Numerical Simulation of Liquid Nitrogen Low-Temperature Fracturing  
### Numerical Model and Boundary Conditions of the Frost Heave  
The last section shows that low temperatures can cause the pore water inside rocks to undergo a phase transition and solidify into ice. In this process, the volume expansion of water is about 9% [29]. Research shows that the force generated by the transformation of the pore water phase into ice exceeds 65 MPa [30], which is sufficient to cause pore damage. The liquid nitrogen temperature required for this experiment was much lower (about \(-195.8\)\({}^{\circ}\)C) than the \(-20\)\({}^{\circ}\)C temperature load simulated in the previous section. When the rock is frozen by liquid nitrogen, the freezing rate inside the rock is faster, the water transfer flux is smaller, and the frost heaving force is larger, which causes the easy breakage of the rocks.  
In addition to liquid nitrogen freezing damage, liquid nitrogen cold shock is also one of the main mechanisms of liquid nitrogen low-temperature fracturing. Under reservoir conditions, the heat of the rock near the contact boundary dissipates to the low-temperature fluid under the action of cold shock and quickly cools and contracts when the low-temperature fluid makes contact with the reservoir rock. Meanwhile, cracks perpendicular to the contact boundary will be formed when the tensile stress exceeds the tensile strength of the rock [16] (Figure 14).  
Given the influence of liquid nitrogen characteristics, flow velocity, and rock properties, a certain time process is necessary to reduce the temperature of the contact boundary to the same temperature as liquid nitrogen, and the temperature cracks are mainly concentrated in a certain range from the contact boundary. Most current studies directly assume that the contact boundary temperature is the liquid nitrogen temperature, which is inconsistent with the actual situation. Consequently, this work sets the contact boundary as the third boundary condition: a temperature difference exists between the rock and the liquid nitrogen at the contact boundary. The contact boundary to a lower temperature (\(-160\)-\(170\)\({}^{\circ}\)C) not only needs to consider the time factor but also sets the heat transfer coefficient. In this  
Figure 12: Comparative analysis of the minimum principal stress numerical and analytical solutions at \(\xi=0.06\).  
Figure 13: Comparison between the numerical and analytical solutions of frost heave pressure.  
section, the effect of the heat transfer coefficient on the temperature and stress field during liquid nitrogen cold shock is studied by taking the liquid nitrogen freezing test under the real triaxial confining pressure used by [PERSON] et al. [16] as an example. The required heat transfer coefficient is calculated according to the cracking effect, which provides basic data for the numerical simulation experiment considering in situ stress in the next section.  
In the field test, [PERSON] et al. [17] injected liquid nitrogen into a hole with a center diameter of 1 inch in the shale cube (8 in \(\times\) 8 in \(\times\) 8 in), and cracks were generated in the boundary of the hole to the rock interior under cold shock for nearly 40 min. Considering that the hole depth was much larger than its diameter, the specimen can be regarded as the plane strain state and simplified as the circular hole square plate model. The size of the model is shown in Figure 15. Quadrilateral elements were used, the number of elements was 609 \(\times\) 609, and the initial temperature of the model was 20 \({}^{\circ}\)C. The values of the elastic modulus and Poisson's ratio were: \(E_{s}\) = 49.3 GPa and \(V_{s}\) = 0.268. Furthermore, the outer boundary of the model was not constrained, the in situ stress \(\sigma\) = 0, and the other parameters were the same as those in Table 2.