## 4 Conclusions  
The current lack of understanding of CO\({}_{2}\) flow behavior in deep shale layers, particularly with its expected phase transition and using different measurement techniques, adds many complications to the prediction of the effectiveness of the fracturing process. The intention of the present study was therefore to determine the flow behavior of CO\({}_{2}\) in shale formations and the influence of phase transition on it. Further, the influence of utilizing laboratory techniques (steady-state and transient tests) on permeability calculation was also examined. The test results enable the following conclusions to be drawn:  
1. Much lower permeability values for liquid CO\({}_{2}\) compared to gaseous CO\({}_{2}\) were observed under transient conditions, due to the reduced slip-flow effect. Considering the significant difference of density between two phases, the mass flow rate of liquid CO\({}_{2}\) is much higher than that of gaseous CO\({}_{2}\). 2. Under steady-state conditions, the heat exchange with the decrease of pore pressure can greatly decrease the flow temperature due to the unique thermodynamic properties of CO\({}_{2}\), which can liquefy the gaseous CO\({}_{2}\) and extend the length of liquid CO\({}_{2}\) along the sample, and much higher density of liquid CO\({}_{2}\) can lead to higher flow rate recorded by flowmeter at room temperature. Faster temperature variation at higher injection pressure than 9 MPa can results in the presence of small drikold particles, which is responsible for the unstable flow rate at higher injection pressure. 3. Permeability calculated under transient conditions is much lower than that under steady-state conditions, except the reason that the slip-flow effect under steady-state conditions increased the flow behavior of CO\({}_{2}\) through the tested sample with ultralow permeability, the influence of temperature variation can be excluded under transient conditions due to its much lower flow rate. During the field fracturing process, liquid CO\({}_{2}\) is injected into the formation matrix at high pore pressure, and the conditions are closer to transient laboratory conditions. Therefore, the transient permeability calculation approach is more reliable for the shale fracturing process. 4. The variation of apparent permeability with injection pressure is affected by the combined effects of slip-flow effect, temperature variation and effective stress. Under transient conditions without any temperature variation, increasing the injection pressure caused an opposite trend of apparent permeability for gaseous and liquid CO\({}_{2}\), because the slip-flow effect dominates the downward of permeability for gaseous CO\({}_{2}\) and the effect of effective stress dominates the upward of permeability for liquid CO\({}_{2}\), respectively. However, under steady-state conditions, although liquid CO\({}_{2}\) showed a similar upward trend to that under transient conditions, gaseous CO\({}_{2}\) permeability also exhibited an upward trend with the increase of injection pressure, which is  
Figure 26: Conceptual model of flow channel structure in clay-abundant sample [49]. opposite to that under transient conditions, because the influence of temperature variation becomes stronger with the increase of flow rate and increases the apparent permeability with the increase of injection pressure. 5. Increasing the confining pressure causes the internal natural micro-fractures or preferential flow pathways in the rock matrix to close, which significantly reduces its flow capacity or permeability. A greater than 1 effective stress coefficient (\(\chi\)) for permeability confirms the greater sensitivity of CO\({}_{2}\) to pore pressure compared to confining pressure due to the abundance of clay minerals. [PERSON] and [PERSON] conceived and designed the experiments; [PERSON] performed the experiments; [PERSON] and [PERSON] analyzed the data and wrote the paper; [PERSON] revised the paper according to reviewers' comments. This research received no external funding.