## Appendix A Detailed Derivation of Fractal Geometry Models for Porous Media
The value of \(D_{f}\) is calculated by the following equation [25]:  
\[D_{f}=d-\frac{\ln\emptyset}{\ln\frac{\lambda_{min}}{\lambda_{max}}}\] (A8)  
where \(\emptyset\) is the porosity and \(d\) is 2 in two dimensions and 3 in three dimensions. When a gas flows in the pores of a porous medium, its flow path may be curved, and [PERSON] [26] obtained the fractal scalar relationship for the characteristics of a tortuous flow pipe in a porous medium, which is given by:  
\[L_{t}=\lambda_{min}^{1-D_{\rm{T}}}L^{D_{\rm{T}}}\] (A9)  
\[\tau=\frac{L_{t}}{L}=\left(\frac{L}{\lambda_{min}}\right)^{D_{\rm{T}}-1}\] (A10)  
In the equations, \(D_{T}\) represents the tortuosity fractal dimension \(1<D_{T}<2\), \(L\) is the characteristic length of the channel, and \(L_{t}\) denotes the length of the capillary channel with diameter \(\lambda\). Generally, \(L_{t}>L\). Equations (A9) and (A10) characterize the curvature of capillary fluid flow paths in porous media. When \(D_{T}=1\), the fluid flows along a straight path. A higher \(D_{\rm{T}}\) value indicates stronger streamline curvature, and at the maximum limit (\(D_{\rm{T}}=2\)), the streamlines fully occupy the two-dimensional plane. In Equation (A10), \(\lambda_{\rm{min}}\) represents the minimum pore (capillary) size, which is a finite value. Therefore, the ratio \(Lt/L\) is also finite. Theory of fractal geometry porous media  
Viscous Damping Dissipation:  
The fractal model of porous media capillaries involves two structural parameters: the pore fractal dimension \(D_{f}\) and the tortuosity fractal dimension \(D_{T}\). The total flow rate \(Q\) through a unit volume of porous media is derived by conceptualizing fluid flow as occurring through tortuous capillary channels whose size distribution follows a fractal scaling law. Based on the modified Hagen-Poiseuille equation [23], the flow rate through a single tortuous capillary can be expressed as:  
\[N(L\geq\lambda)=\left(\frac{\lambda_{max}}{\lambda}\right)^{D_{f}}\]0 (A11)  
In the equation, \(\Delta p_{1}\) represents the pressure variation (unit: Pa); \(\lambda\) denotes the pore diameter (unit: mm); \(l_{t}\) indicates the total length of the tortuous capillary (unit: mm); and \(\mu\) stands for the fluid viscosity coefficient (unit: Pa-s). Equation (A11) is derived by approximating the capillary as circular. For cross-sectional shapes other than circular, they can be converted into a mean hydraulic radius model, yielding equations of identical form. Since the pore channel sizes follow a fractal distribution, the total flow rate \(Q\) can be obtained by integrating the flow rate in a single tube from the minimum pore diameter \(\lambda_{min}\) to the maximum pore diameter \(\lambda_{max}\):  
\[N(L\geq\lambda)=\left(\frac{\lambda_{max}}{\lambda}\right)^{D_{f}}\]1  
\[N(L\geq\lambda)=\left(\frac{\lambda_{max}}{\lambda}\right)^{D_{f}}\]2 (A12)  
In the equation, \(D_{f}\) represents the fractal dimension of the two-dimensional medium; \(D_{T}\) denotes the tortuosity fractal dimension; \(\lambda_{max}\) is the maximum pore diameter (unit: mm); and \(\lambda_{min}\) is the minimum pore diameter (unit: mm). Since the tortuosity satisfies \(1<D_{T}<2\) and the fractal dimension in two dimensions satisfies \(1<D_{f}<2\), the exponent \(3+D_{T}-2D_{f}>0\). Furthermore, because \(0<\left(\frac{\lambda_{min}}{\lambda_{max}}\right)^{3+D_{T}-2D_{f}}<1\), Based on [PERSON] and [PERSON]'s judgment (\(\lambda_{min}/\lambda_{max}<10^{-2}\)), Then, \(\left(\frac{\lambda_{min}}{\lambda_{max}}\right)^{D_{f}}\left(\frac{\lambda_{ min}}{\lambda_{max}}\right)^{3+D_{T}-2D_{f}}\approx 0\) It can be neglected.