## 1 Introduction
In this work, displacement flow of the residual water in the stratum by the injected cryogenic fracturing fluid is considered. In this process, the residual water in the reservoir is constantly cooled down by the cryogenic fracturing fluid while flowing in the stratum pore channel, which further leads the residual water to freeze. The newly-formed ice is supposed to occupy the pore channel partly or completely, affecting the subsequent state of flow. In this case, the cold energy transferred in the stratum during fracturing would also be influenced. Additionally, large-scale ice blockage can result in local pressure rise in the reservoir, which could enhance the initiation and propagation of fractures. According to the above analysis, it is noted that the displacement flow of residual water by the cryogenic fracturing fluid would have profound influence on the fracturing process. The process is quite complicated in practice as the following physical processes are involved and worked together including miscible/immiscible displacement flow (depends on the type of fracturing fluid (liquid carbon dioxide, liquid nitrogen)), liquid-gas phase transition (vaporization of the fracturing fluid), liquid-solid phase transition (solidification of residual water), pore-scale flow. It is indicated by the literature survey that nearly no research has focused on such a displacement process by far. Accordingly, a primary study on this freezing-coupling displacement process is carried out which introduces a newly developed numerical model to simulate the freezing process during the displacement flow. Due to the complexity of the actual process, simplifications as illustrated in Figure 1 were made as follows. 1. A 2D model is developed. The main purpose of this research is to qualitatively reflect the freezing process during the displacement process and the influence of the formed ice on the flow status in a numerical method. The 2D method is considered sufficient for achieving this goal. Moreover, compared with 3D simulation, 2D simulation could significantly reduce the time cost of calculation. 2. The effect of porous structure is neglected and immiscible displacement flow in a single passage is considered. The \"single passage\" simplification is made referring to the research works on pure displacement process [22; 23; 24; 25; 26; 27; 28], in which the essential feature of the displacement process could be reflected. The \"immiscible displacement\" simplification comes from the fact that as a candidate of the cryogenic fracturing fluid, liquid nitrogen (LN\({}_{2}\)) is hardly soluble in water. 3. The influences due to vaporizing of the fracturing fluid and pore scale are neglected. This simplification is made in consideration of the complexity of the vaporization process and the capillary effect. Research works on freezing soil [29] were also referred to, where the influence of vapor component is neglected. Based on the above simplifications, the major effort is spent on simulating the growth of ice phase and the influence of ice phase on the flow status during the displacement process (Figure 1c). Research on this process is supposed to provide a method to predict the ice blockage pattern in the reservoir properly, which is necessary for a further prediction of the fracture development, pressure distribution, permeability evolution, thermal shock position, and other factors during fracturing. An improvement in the fracturing efficiency can then be expected.