## 3 Results and Discussion
### Effect of Effective Stress Law on Permeability Behavior of CO\({}_{2}\)  
According to [PERSON] [47], rock deformation is governed by the difference between confining pressure and pore pressure, as in Equation (11), which reveals the opposite effects of confining pressure and pore pressure on the volumetric strain and pore structure. For consolidated rock, the sensitivity of rock deformation to confining pressure and pore pressure is different, and similarly, rock permeability, which is closely related to the pore structure of rock, is affected differently by confining pressure and pore pressure [48]. To describe how permeability varies with confining pressure and pore pressure, the effective stress law is applied, considering an effective stress coefficient (Equation (12)). \[\sigma_{Terzaghi}=C_{p}-P_{p} \tag{11}\]  
\[\sigma_{eff}=C_{p}-\chi P_{p} \tag{12}\]  
where, \(C_{p}\) is confining pressure, \(P_{p}\) is the pore pressure, \(\chi\) is the effective stress coefficient for permeability, and \(\sigma_{eff}\) is the effective stress for permeability. The effective stress coefficient for permeability \(\chi\) indicates the different effects of confining pressure and pore pressure on permeability, and the permeability can be related to single variable parameter \(\sigma_{eff}\), effective pressure [43]. The effective stress coefficient for permeability can be calculated using the ratio between the sensitivity of permeability to change in pore pressure and confining pressure (see Equation (13)). \[\chi=-\frac{\
icefrac{{\partial k}}{{\partial k}}/\
icefrac{{\partial P_{p}}} {{\partial k}}}{\
icefrac{{\partial C_{p}}}{{\partial k}}} \tag{13}\]  
where, \(k\) is the permeability of the sample, and \(\chi\) indicates the ratio between the relative sensitivity of permeability to changes in confining pressure and pore pressure. Considering the effect of slip-flow on the difference of permeability between gaseous and liquid CO\({}_{2}\), only the permeability for liquid CO\({}_{2}\) was analyzed for the effective stress coefficient \(\chi\). According to Figure 24, there is a linear relationship between permeability and effective stress under steady-state conditions, and \(\chi\) equals 3.91 in liquid CO\({}_{2}\) state, indicating that the permeability of the siltstone sample is more sensitive to change in pore pressure than confining pressure. This result is consistent with many other studies where clay mineral-abundant samples have higher \(\chi\) values (>1) [49; 50; 51], and a similar \(\chi\) value (2.96) was also calculated for liquid CO\({}_{2}\) under transient conditions (see Figure 25). This can be explained by a simplified conceptual model of pore structure proposed by [PERSON] and [PERSON] [49]. The very tiny pore channel for CO\({}_{2}\) flow is surrounded by mineral particles, and the quartz particles (accounting for 42% in weight) with a larger diameter and low compressibility make up the skeleton, while many tiny and soft clay particles (accounting for 40% in weight) are present between the quartz particles and tiny pore channels (see Figure 26). The simple conceptual model can be verified by SEM images (see Figure 8). In this model, the outer ring is a rigid framework composed of quartz particles and the inner ring is composed of soft clay minerals. When the confining pressure and pore pressure are applied to the model, the rigid framework incurs a small strain which has limited impact on the thickness of the CO\({}_{2}\) flow channel. The inner clay mineral layer is more sensitive to pore pressure because the confining pressure is mainly supported by quartz particles, and pore pressure can produce higher strain on clay minerals compared with the same confining pressure, and the clay mineral strain can have a considerable impact on the thickness of the flow channels. Therefore, pore pressure has a greater effect than confining pressure on the flow behavior of CO\({}_{2}\) in this clay-abundant sample [50].