## 2 Experimental Methodology
### Permeability Calculation
Figure 4: Relationship between mean free path and pressure at 22 \({}^{\circ}\)C.  
Since the flow regime mainly belongs to Darcy flow, the permeability of the tested siltstone for gas injection under various steady-state conditions can be calculated using the traditional Darcy equation (Equation (4)), based on the assumption that gas volume changes inversely proportionally with gas pressure at constant temperature [37]. \[k_{g}=\frac{2Q_{o}P_{o}\mu L}{A\left(P_{i}^{2}-P_{o}^{2}\right)} \tag{4}\]  
where, \(k_{g}\) is the permeability for CO\({}_{2}\), \(Q_{o}\) is CO\({}_{2}\) flow rate at downstream, \(\mu\) is its viscosity, \(L\) and \(A\) are the length and cross-sectional area of the specimen, \(P_{o}\) and \(P_{i}\) are downstream pressure and upstream pressure. However, when the injection pressure is higher than 6 MPa, the gaseous CO\({}_{2}\) and liquid CO\({}_{2}\) co-exist in the sample because CO\({}_{2}\) transfers into its liquid state at 6 MPa (data from NIST) [38], and the Boyle's Law that gas volume is proportional to the gas pressure is assumed to be applicable for the gaseous state for simplicity, but not for the liquid state, because this phase transition causes sudden enhancement of density [39]. For example, when CO\({}_{2}\) is in its liquid state, its density increases slightly from 760 kg/m\({}^{3}\) at 6.01 MPa to 901 kg/m\({}^{3}\) at 15 MPa, which is much higher than 210 kg/m\({}^{3}\) at 6.00 MPa. Therefore, the [PERSON] equation for compressed gas is not suitable for this case, and in consideration of the slight difference of density in the liquid state, the permeability for the liquid section can be calculated using the method proposed by [PERSON] et al. [24], utilizing Equations (5) and (6), and the ratio between the length of gaseous CO\({}_{2}\) section to sample length is inversely proportional to the ratio between the flow rate at an injection pressure of 6 MPa and that at each higher injection pressure. \[L_{l}=L-L_{g}=L-\frac{Q_{6}L}{Q_{o}} \tag{5}\]  
\[k_{l}=\frac{Q_{l}\mu_{l}L_{l}}{A(P_{i}-P_{t})} \tag{6}\]  
where, \(L_{l}\) is the length of liquid CO\({}_{2}\) in the sample, \(L_{g}\) is the length of gas CO\({}_{2}\) in the sample, \(Q_{6}\) is the flow rate at 6 MPa injection pressure, \(k_{l}\) is the permeability for liquid CO\({}_{2}\) at a certain injection pressure (\(>\)6 MPa), \(Q_{l}\) is the liquid CO\({}_{2}\) flow rate at transition point, \(\mu_{l}\) is the viscosity of liquid CO\({}_{2}\) at room temperature, \(P_{t}\) is pressure at transition point and here it equals 6 MPa, and \(P_{i}\) is upstream pressure. Figure 5: Dimensionless Knudsen number of CO\({}_{2}\) vs. pore pressure for different pore sizes at 22 \({}^{\circ}\)C.  
For transient conditions, the permeability at specific pressure was calculated using pulse-decay permeability measurement by the following equations (Equations (7) and (8)) [3]:  
\[\Delta P(t)=\Delta P_{0}e^{-\alpha t} \tag{7}\]  
\[\alpha=\frac{kA}{\beta V_{down}L\mu} \tag{8}\]  
where, \(\alpha\) is the decay exponent, \(\Delta P_{0}\) is the pressure difference between upstream and downstream at time \(t=0\), \(\Delta P(t)\) is the pressure difference between upstream and downstream with time, \(k\) is the permeability for CO\({}_{2}\), \(\beta\) is the CO\({}_{2}\) compressibility, \(V_{down}\) is the volume of the downstream reservoir, \(\mu\) is the CO\({}_{2}\) viscosity, and \(L\) and \(A\) are the length and cross-section area of the sample, respectively.