## Appendix A Detailed Derivation of Fractal Geometry Models for Porous Media  
(1) Fractal Geometry Theory  
The interstitial spaces within brush seal bristle packs are intricate and microscale. The structural similarity between porous media and bristle packs has led to the widespread use of porous media models for simulating leakage flow in brush seals. However, the capillary diameter, tortuosity, and surface morphology of porous media pores exhibit inherent randomness and irregularity, making traditional theoretical methods inadequate for analyzing their seepage characteristics. Given the fractal nature of porous microstructures and pore size distributions [22; 23; 24], advancements in fractal geometry offer novel approaches to refine porous media models for brush seals. A fractal-geometry porous medium refers to a structure where the microarchitecture (e.g., pore sizes in the bristle pack) follows a fractal scaling law within a defined dimensional range [25]. Specifically, the cumulative number of pores with diameters greater than or equal to \(\lambda\) satisfies:  
\[M(L)\,\sim L^{D_{f}}\] (A1)where \(D_{f}\) is the fractal dimension, \(M(L)\) represents the structural characteristic measure of the medium, and \(L\) denotes the scale. When the measurement scale \(L\) changes, the measure \(M(L)\) of the fractal object varies, but the fractal dimension \(D_{f}\) remains constant. Taking the logarithm of both sides of Equation (A1) yields a straight line in a log-log coordinate system, where the slope corresponds to \(D_{f}\). [PERSON] [24] applied fractal geometry theory to porous media pores, demonstrating that the cumulative number of pores with diameters greater than or equal to \(\lambda\) satisfies:  
\[N(L\geq\lambda)=\left(\frac{\lambda_{max}}{\lambda}\right)^{D_{f}}\] (A2)  
where \(D_{f}\)is the fractal dimension, satisfying \(0<D_{f}<2\) in two-dimensional space and \(0<D_{f}<3\) in three-dimensional space; \(\lambda\) denotes the pore scale, and \(\lambda_{\text{max}}\) is the maximum pore scale. From Equation (A2), the total number of pores is derived as:  
\[N_{t}=\left(\frac{\lambda_{max}}{\lambda_{min}}\right)^{D_{f}}\] (A3)  
where \(\lambda_{\text{min}}\) represents the minimum pore scale. Equation (A3) quantifies the total pore count in a fractal-geometry porous medium and implies statistical self-similarity within the scale range \(\lambda_{\text{min}}\leq\lambda\leq\lambda_{\text{max}}\), where the fractal dimension \(D_{f}\) remains constant. Treating Equation (A2) as a continuous and differentiable function, differentiation with respect to \(\lambda\) yields:  
\[-dN=D_{f}\lambda_{max}^{D_{f}}\lambda^{-(D_{f}+1)}d\lambda\] (A4)  
This equation describes the number of pores within the interval [\(\lambda\), \(\lambda+d\lambda\)], where \(-dN>0\), indicating that the pore count decreases as the pore diameter increases. \[-\frac{dN}{N_{t}}=D_{f}\lambda_{min}^{D_{f}}\lambda^{-(D_{f}+1)}d\lambda=f( \lambda)d\lambda\] (A5)  
where \(f(\lambda)=D_{f}\lambda_{min}^{D_{f}}\lambda^{-(D_{f}+1)}\) is the probability density function of the pore distribution, which should satisfy the normalization condition:  
\[\int_{-\infty}^{+\infty}f(\lambda)d\lambda=\!\!\int_{\lambda_{min}}^{\lambda _{max}}f(\lambda)d\lambda=1-\left(\frac{\lambda_{min}}{\lambda_{max}}\right)^ {D_{f}}=1\] (A6)  
The condition for the above equation to hold is:  
\[\left(\frac{\lambda_{min}}{\lambda_{max}}\right)^{D_{f}}\approx 0\] (A7)  
This equation can be regarded as a criterion for determining whether fractal geometry theory and methods are applicable to porous media. In brush seals, \(\lambda_{min}/\lambda_{max}<10^{-2}\), so \((\lambda_{min}/\lambda_{max})^{D_{f}}\approx 0\), thereby approximately satisfying Equation (A7). Therefore, the porous medium in brush seals can be treated using fractal theory and methods.