## 4 Conclusions  
In this paper, a 2D numerical model considering the nucleation characteristics based on VOF method and Lee model is developed to predict the propagation of multiphase flow accompanied by solidification phase change during cryogenic fracturing. The formation and propagation of ice on the wall boundary along with multiphase flow around ice was studied. Conclusions are drawn as follows:  
1. For the classical Stefan problem, the max relative difference between the convergent numerical results and the theoretical solutions are about 6.58%, which can validate the feasibility and accuracy of the newly proposed method.
2. Through the simulation on the process of the ice generation and growth in Case 1, the improved Lee model in the proposed numerical method could consider the effect of low-temperature boundary on the freezing. It also could well reflect the different characteristic of the homogeneous nucleation and the heterogeneous nucleation on various kinds of boundaries.
3. By comparing the numerical solutions of Case 1 with that of Case 2, the concept of shape coefficient of nucleation in the proposed numerical method has the ability to distinguish the phase-change processes between the ice generation on the wall and the ice growth on the frozen ice, which could restore the practical situation.
4. The numerical results of Case 3-6 indicate that the sponge-layer absorbing technique is capable of handling the frozen ice and simulating its blocking effect on fluid flow.  
In future researches, the model experiments should be carried out to further validate the newly proposed numerical method. Furthermore, the method can be developed further on the following aspects: Considering the effects of temperature and pressure on the thermo-physical properties (thermal conductivity, density, viscosity, specific heat capacity, etc.) to meet the practical engineering requirements; extending the present method to three dimensions for more extensive application about the practical engineering problems; further performing the complete uncertainty analysis based on the experimental results; and further introducing more interactive effects among the multiphase fluids (such as mass transfer).  
**Author Contributions:** [PERSON] organized the research topic and draft the basic structure of the manuscript. [PERSON] draft the second chapter of the manuscript and helped edit the whole manuscript. [PERSON] sponsored the research project and revised the manuscript. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by The National Natural Science Foundation of China, grant number 51776031 and The Key Project of Natural Science Foundation of Liaoning Province of China, grant number 20170540182 and The National Innovation Team Foundation, grant number 50921001.  
**Conflicts of Interest:** The authors declare no conflict of interest.  
**Nomenclature**  
\begin{tabular}{l l} \(C_{p}\) & Specific heat capacity [J kg\({}^{-1}\) K\({}^{-1}\)] \\ \(H\) & Specific enthalpy [J kg\({}^{-1}\)] \\ \(H_{L}\) & Latent heat of fusion per mass [J kg\({}^{-1}\)] \\ \(H_{LV}\) & Latent heat of fusion per volume [J m\({}^{-3}\)] \\ \(h\) & Planck constant, 6.62607015 \(\times\) 10\({}^{-34}\) J s\({}^{-1}\) \\ \(\bar{I}\) & Unit tensor \\ \(J\) & Nucleation rate [m\({}^{-3}\) s\({}^{-1}\)] \\ \(K\) & Interfacial drag coefficient [kg m\({}^{-3}\) s\({}^{-1}\)] \\ \(k\) & Thermal conductivity [W m\({}^{-1}\) K\({}^{-1}\)] \\ \(k_{B}\) & Boltzmann constant, 1.380649 \(\times\) 10\({}^{-23}\) J K\({}^{-1}\) \\ \(m\) & Rate of mass transfer [kg m\({}^{-3}\) s\({}^{-1}\)] \\ \(n_{L}\) & Number of water molecule per water volume, 3.34 \(\times\) 10\({}^{28}\) [m\({}^{-3}\)] \\ \(p\) & Pressure [Pa] \\ \end{tabular}