## 3 Permeability Calculation Method
### Permeability Calculation  
The pseudo-pressure method is used to calculate the specimen permeability. The gas seepage equation is based on the following assumptions: (1) the fluid is a one-component gas; (2) the flow is isothermal; (3) the medium is uniform and isotropic, and the porosity \(\phi\) is constant; (4) gravity can be neglected; and (5) flow is laminar so that [PERSON]'s law holds.  
Based on the above assumptions and conditions and combining continuity, momentum and state equations, the general form of the partial differential equation for gas seepage is derived [24]:  
\[\frac{K_{g}}{\phi}\
abla\cdot[\frac{p\
abla p}{\mu(p)Z(p)}]=\frac{\partial}{ \partial t}[\frac{p}{Z(p)}] \tag{11}\]  
Here, \(p\) is the gas pressure; \(Z\) is the deviation factor for correction of ideal gas; \(Z(p)\) is a function of the deviation factor with respect to \(p\); and \(\mu(p)\) is the gas viscosity function with respect to \(p\).  
The pseudo-pressure is defined as:  
\[m(p)=2\int_{p_{m}}^{p}\frac{p}{\mu Z}dp \tag{12}\]  
where \(m(p)\) is the pseudo-pressure; \(p_{m}\) is any reference pressure, which can be taken as 0 or 0.1 MPa.  
Equations (11) and (12) are used to obtain a partial differential equation, which describes the true gas percolation represented by the pseudo-pressure \(m\).  
\[\
abla^{2}m=\frac{\phi c_{g}(p)\mu(p)}{K_{g}}\cdot\frac{\partial m}{\partial t} \tag{13}\]  
where \(c_{g}=\frac{1}{p}-\frac{1}{Z(p)}(\frac{dZ(p)}{dp})_{T}\) is the isothermal compression coefficient of the real gas.  
For an isotropic homogeneous specimen of length \(L\) and cross-sectional area \(A\) with the pressures at the upper (\(x=0\)) and the lower (\(x=L\)) extremes delineated as \(p_{0}\) and \(p_{L}\), the seepage equation and the boundary conditions are as follows.  
\[\frac{d^{2}m}{dx^{2}}=0\hskip 56.905512 pt(0<x<L) \tag{14}\]  
\[m=m_{0}=\frac{p_{0}^{2}}{\mu_{0}Z_{0}}\hskip 56.905512 pt(x=0) \tag{15}\]  
\[m=m_{L}=\frac{p_{L}^{2}}{\mu_{L}Z_{L}}\hskip 56.905512 pt(x=L) \tag{16}\]  
where \(\mu_{0}\) is the gas viscosity at the top of the specimen of length \(L\); \(\mu_{L}\) is the gas viscosity at the bottom of the specimen; \(Z_{0}\) is the deviation factor at the top of the specimen; \(Z_{L}\) is the deviation factor at the bottom of the specimen.  
Linear interpolation of (14)-(16) gives:  
\[m=m_{0}-\frac{m_{0}-m_{L}}{L}x \tag{17}\]  
where \(x\) is the distance from the top of the specimen.  
Taking the average, \(\mu=\mu,Z=\overline{Z}\), we get \(m-m_{0}=\frac{p^{2}-p_{0}^{2}}{\mu Z}\). Taking Equation (12) into Equation (17), we can write:  
\[p^{2}=p_{0}^{2}-(\frac{\mu\overline{Z}}{\mu_{0}Z_{0}}p_{0}^{2}-\frac{\mu \overline{Z}}{\mu_{L}Z_{L}}\,p_{L}^{2})\frac{x}{L} \tag{18}\]For gas seepage, the volume flow varies with pressure. Using the mass flow \(F\), which is invariant, and assuming that the flow rate is not too large and follows [PERSON]'s law, [PERSON] [24] obtained the volume flow rate \(Q_{sc}\) as:  
\[Q_{sc}=\frac{F}{\rho_{sc}}=\frac{AK_{g}T_{sc}}{2 pL\mu ZT}(p_{0}^{2}-p_{L}^{2}) \tag{19}\]  
where \(Q_{sc}\) is the volume flow rate, \(\rho_{sc}\) is the gas density, \(T_{sc}\) is the temperature, all under standard conditions, and \(p\) is atmospheric pressure.  
Equation (19) can also be written as:  
\[K_{g}=\frac{2Q_{sc}pL\mu ZT}{AT_{sc}(p_{0}^{2}-p_{L}^{2})} \tag{20}\]  
During the test, we keep the outlet pressure constant, change the inlet pressure and get the gas flow rate \(Q_{sc}\) at different pressure differences. The gas permeability \(K_{g}\) is obtained according to Equation (20), and the Klinkenberg permeability \(K_{\infty}\) and slip factor \(b\) can be determined by fitting the permeability of gas obtained by multipoint measurements through Equation (6).