## 3 Sample and Method
### Fractal Model  
Fractal theory has been applied to the quantitative study of sandstone reservoir pore structures since the 1980s. The microarchitectural topology of sedimentary pore spaces typically exhibits non-Euclidean geometric properties, manifesting as fractional scaling behaviors with optimal fractal scaling parameters (D\({}_{I}\)) within the 2.0-3.0 interval. Mathematical modeling reveals an inverse proportionality between D\({}_{I}\) values approaching theoretical minima (-2.0) and spatial uniformity of hydraulic conduit radii (\(\sigma_{\text{d}}<1\)\(\mu\)m\({}^{2}\)), whereas D\({}_{I}\) trending toward the upper asymptotes (-3.0) directly correlates with an exponential enhancement of microstructural anisotropy (\(\beta>0.85\)) quantified through capillary pressure hysteresis analysis. Therefore, the larger the fractal dimension, the more complex the pore-throat structure of the reservoir, and the stronger the heterogeneity of the reservoir. When the pore-throat radius of tight sandstone exhibits a fractal structure, the number of pore throats with a radius greater than \(r\) is denoted as N [35]. The relationship between N and \(r\) can be expressed as follows:  
\[\text{N}(>r)=\int_{r}^{r_{\text{max}}}\text{F}(r)\text{d}r=\text{a}r^{D} \tag{2}\]  
In the equation, \(r_{\text{max}}\) is the maximum pore-throat radius (\(\mu\)m); F(\(r\)) is the pore-throat radius distribution density function; a is the proportional constant [36]. Since the minimum pore-throat radius \(r_{\text{min}}\) is much smaller than \(r_{\text{max}}\), the following can be obtained:  
\[\text{S}=\big{(}\frac{r}{r_{\text{min}}}\big{)}^{3-D} \tag{3}\]  
In the equation, S represents the saturation of the wetted phase corresponding to capillary pressure P\({}_{\text{c}}\) (%). Assuming that the contact angle (\(\theta\)) is unaffected by pore-throat size, the fractal equation for the pore-throat radius distribution can be derived as follows:  
\[\text{S}=\big{(}\frac{\text{P}_{\text{c}}}{\text{P}_{\text{min}}}\big{)}^{D-3} \tag{4}\]  
In the equation, P\({}_{\text{min}}\) is the capillary pressure corresponding to r\({}_{\text{max}}\) (MPa). Taking the logarithm of the formula gives the following [37]:  
\[\text{lg}(\text{S})=(\text{D}-3)\text{lg}(\text{P}_{\text{c}})-(\text{D}-3) \text{lg}(\text{P}_{\text{min}}) \tag{5}\]  
Based on capillary pressure experimental data, the relationship between lg(S) and lg(P\({}_{\text{c}}\)) can be determined. In this experiment, air serves as the wetting phase, while mercury acts as the non-wetting phase. Therefore, lg(S) is converted to lg(1 - S\({}_{\text{Hg}}\)), where S\({}_{\text{Hg}}\) is the mercury saturation. Then, the slope of the relationship trend line of lg(1 - S\({}_{\text{Hg}}\)) and lg(P\({}_{\text{c}}\)) is analyzed, which represents the Hurst exponent H. Finally, the fractal dimension D is calculated as follows:  
\[\text{D}=3+\text{H} \tag{6}\]  
According to fractal theory, pore-throat structure fractals can be categorized into overall fractals and segmented fractals. When the relationship trend line between lg(1 - S\({}_{\text{Hg}}\)) and lg(P\({}_{\text{c}}\)) forms a straight line, it indicates a self-similar fractal distribution. The structures of large and small pore throats are similar, and their fractal dimensions are close [38]. When the relationship trend line of lg(1 - S\({}_{\text{Hg}}\)) and lg(P\({}_{\text{c}}\)) is not a straight line and there is a distinct inflection point in the middle, the line is divided into several straight lines based on the inflection point. The fractal dimensions of pore throats of different sizes are then calculated, and the total fractal dimension of the entire pore-throat structure is obtained by performing a weighted average calculation based on the pore-throat porosityof different sizes. This is used to determine the fractal characteristics of the pore size distribution and pore-throat structure.