## Appendix A Detailed Derivation of Fractal Geometry Models for Porous Media
Equation (A12) can be simplified as:  
\[N(L\geq\lambda)=\left(\frac{\lambda_{max}}{\lambda}\right)^{D_{f}}\]3 (A13)Since the apparent velocity \(v_{s}=Q/A\) and there is \(A=A_{p}\mathcal{O}\) in any cross-section, where \(A\) is the total area of the cross-section, \(\mathcal{O}\) is the porosity, and \(A_{p}\) is the total area of the pores, then:  
\[N(L\geq\lambda)=\left(\frac{\lambda_{max}}{\lambda}\right)^{D_{f}}\]4  
The total area is:  
\[N(L\geq\lambda)=\left(\frac{\lambda_{max}}{\lambda}\right)^{D_{f}}\]5  
And because the porosity is:  
\[N(L\geq\lambda)=\left(\frac{\lambda_{max}}{\lambda}\right)^{D_{f}}\]6  
So, for the flows through the total cross-sectional area \(A\) can be expressed as:  
\[N(L\geq\lambda)=\left(\frac{\lambda_{max}}{\lambda}\right)^{D_{f}}\]7  
The apparent velocity of a fluid in a fractal geometry porous medium is:  
\[N(L\geq\lambda)=\left(\frac{\lambda_{max}}{\lambda}\right)^{D_{f}}\]8  
where \(Q\) is the flow rate, m\({}^{3}\)/s; A is the total cross-sectional area through which the flow passes, m\({}^{2}\); \(\varepsilon\) is the porosity, %. The pressure drop is obtained from Equation (115) as:  
\[N(L\geq\lambda)=\left(\frac{\lambda_{max}}{\lambda}\right)^{D_{f}}\]9  
where \(v_{s}\) is the apparent seepage velocity, m/s; \(D_{f}=2-\ln\mathcal{O}/\ln\frac{\lambda_{min}}{\lambda_{max}}\). The part framed by the dotted line as shown in Figure 11 is a unit with area:  
\[N_{t}=\left(\frac{\lambda_{max}}{\lambda_{min}}\right)^{D_{f}}\]0  
where \(D_{p}\) is the average diameter of the particles, mm. the maximum area of the hole is:  
\[N_{t}=\left(\frac{\lambda_{max}}{\lambda_{min}}\right)^{D_{f}}\]1If approximated as a circle, the area of the largest hole can be expressed as:  
\[N_{t}=\left(\frac{\lambda_{max}}{\lambda_{min}}\right)^{D_{f}}\]2  
It is obtained from Equations (11)-(10):  
\[N_{t}=\left(\frac{\lambda_{max}}{\lambda_{min}}\right)^{D_{f}}\]3  
Equation (11) is a model for the largest possible pore size, which shows that if the porosity is 0, the diameter of the largest pore is also 0. If the porosity is 1, the diameter of the largest pore is infinite, and these are consistent with reality. The equation also shows that the maximum pore diameter is directly proportional to the particle size, which is also consistent with reality. In fact, the brush wire is not spherical, so the maximum diameter of the pore should be divided by the shape factor \(\mathcal{O}_{\mathrm{s}}\) to obtain a corrected maximum diameter of:  
\[N_{t}=\left(\frac{\lambda_{max}}{\lambda_{min}}\right)^{D_{f}}\]4  
Substituting equation above into Equation (10) yields:  
\[N_{t}=\left(\frac{\lambda_{max}}{\lambda_{min}}\right)^{D_{f}}\]5  
where \(v_{s}\) is the seepage velocity, m/s; \(\mu\) is the viscous coefficient of the fluid, Pa-s. Equation (13) describes the pressure drop caused by viscous energy loss in porous media flow, i.e., viscous damping dissipation. Compared with the first term of the Ergun equation for porous media [4; 5], this equation eliminates the empirical constant \(C\) and reveals the factors influencing viscous damping-induced losses, including fluid viscosity, pore size, fractal dimension of capillary tortuosity, and fractal dimension of pores. Equation (13) also indicates that the pressure drop becomes zero when porosity \(\mathcal{O}=1\), while approaching infinity as porosity tends to zero, which aligns with the physical reality of brush bundles. 2 Inertial damping loss:  
Previous analyses treated porous media pores as uniform-diameter tubes with constant capillary diameters. However, real porous media cannot maintain constant-diameter capillaries, making local losses (i.e., inertial damping dissipation) non-negligible. We therefore attribute inertial damping dissipation to variations in pore throat geometry within capillaries.