## 2 Materials and Methods
Pathfinding was performed in three principal directions, \(x\), \(y\), and \(z\), by identifying whether a connected path exists from one face of the sample to the opposite face through pore voxels only. Once a path is detected, its total length is compared to the straight-line length of the sample in the corresponding direction, yielding the directional tortuosity. The function subsequently returns both the minimum and the average tortuosity across the three directions, thereby determining geometric tortuosity based on the shortest path method. Geometric tortuosity, \(\tau_{g}\), is defined as the ratio of the actual path length of the fluid, \(L_{min}\), to the straight line distance between the inlet and the outlet, \(L\), and is mathematically expressed as follows  
\[\tau_{g}=\frac{L_{min}}{L} \tag{11}\]  
Similar methodologies have been widely adopted in the literature, exemplified by [PERSON] et al. who utilized [PERSON]'s algorithm for pore microstructure characterization [21], and [PERSON] et al. who applied shortest-path analysis on 3D microstructural datasets of fuel cell materials [40]. These studies reinforce the validity and relevance of this approach in pore structure investigations. An empirical equation was subsequently employed to corroborate the accuracy of tortuosity estimations encompassing both streamline and geometric aspects. For instance, [PERSON] et al. calculated geometric tortuosity in 3D porous media using the A\({}^{\star}\) algorithm to identify the shortest efficient flow paths [30]. Furthermore, they developed an empirical model capable of predicting geometric tortuosity, \(\tau_{g}\), solely as a function of the effective porosity of the medium, \(\phi_{eff}\), which quantifies the fraction of void space available for fluid flow  
\[\tau_{g}=5.98e^{-7.666\phi_{eff}}+1.44e^{-0.3665\phi_{eff}} \tag{12}\]  
Additionally, [PERSON] et al. (2013) proposed a theoretical model for geometric tortuosity [41], expressed as follows  
\[\tau_{g}=\left[\frac{p-p_{c}+\left(\frac{c}{L_{s}}\right)^{\frac{1}{2}}}{1-p_{ c}}\right]^{p-p_{D_{opt}}} \tag{13}\]  
In percolation models of fluid flow in porous media, the occupation probability \(p\) represents the likelihood that a pore or throat is occupied by fluid, ranging from 0 to 1. The critical occupation probability, pc, marks the threshold for the formation of a continuous flow path. For 3D systems, the correlation length exponent \(\approx\) 0.88 and the fractal dimension of the optimal flow path is \(D_{opt}\approx\) 1.43. The linear system size is denoted by \(L_{s}\). [PERSON] et al. [32] showed that there is a linear relationship between the occupation probability \(p\) and the average pore coordination number \(Z\) in regular lattice structures, such as 2D square, hexagonal, and triangular lattices, and 3D simple cubic, body-centered cubic, and face-centered cubic lattices. This relationship underpins the connection between pore network topology and fluid transport properties. Building on this framework, [PERSON] et al. improved finite-size scaling by incorporating the coordination number \(Z\)[31] and replacing the occupation probability \(p\) and the critical threshold \(pc\) with the normalized ratio \(\frac{Z}{Z_{max}}\) and \(\frac{1.5}{Z_{max}}\), respectively, where \(Z_{max}\) is the maximum coordination number. The scaling constant \(C\) is defined as \(C=0.85l_{min}\), where \(l_{min}\) represents the minimum tortuous length, corresponding to the shortest pore throat path. This refined scaling approach enhances the accuracy of permeability predictions and improves the reliability of fluid flow modeling in complex porous media.