## 4 Results
### Fractal Dimensions  
At present, several methods to calculate the fractal dimension based on the gas adsorption method have been proposed, including the BET model method, the PSD model method, and the Frenkel-Halsey-Hill (FHH) model method [64; 65], but the FHH model qA the most widely used and the most effective method [39; 65; 66; 67]. In this study, the fractal dimensions of the 16 shale samples of K\({}_{2}\)n\({}^{1+2}\) in the Songliao Basin were calculated by the FHH model. The calculation formula (Equation (3)) is as follows:  
\[ln\left(\frac{v_{n}}{v_{n_{0}}}\right)=A\left[\ln\left(\ln\left(\frac{P_{0}}{ P}\right)\right)\right]+\beta \tag{3}\] \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Sample**} & **Average Pore** & **BET Surface** & **Total Pore Volume** & **Hysteresis** \\  & **Diameter (nm)** & **Area (m\({}^{2}\)/g)** & **(cm\({}^{3}\)/g)** & **Loop Type** \\ \hline SK1-S-1 & 8.73 & 44 & 0.0750 & H3 \\ SK1-S-2 & 9.80 & 44 & 0.0881 & H3 \\ SK1-S-3 & 9.58 & 31 & 0.0756 & H3 \\ SK1-S-4 & 9.21 & 40 & 0.0883 & H3 \\ SK1-S-5 & 10.01 & 31 & 0.0798 & H3 \\ SK1-S-6 & 8.74 & 38 & 0.0776 & H3 \\ SK1-S-7 & 9.62 & 42 & 0.0935 & H3 \\ SK1-S-8 & 9.77 & 32 & 0.0781 & H3 \\ SK1-S-9 & 8.84 & 35 & 0.0743 & H3 \\ SK1-S-10 & 10.52 & 22 & 0.0636 & H3 \\ SK1-S-11 & 14.61 & 10 & 0.0444 & H3 \\ SK1-S-12 & 8.26 & 50 & 0.0944 & H3 \\ SK1-S-13 & 8.22 & 66 & 0.1187 & H3 \\ SK1-S-14 & 8.51 & 41 & 0.0752 & H3 \\ SK1-S-15 & 15.03 & 10 & 0.0450 & H3 \\ SK1-S-16 & 12.25 & 16 & 0.0580 & H3 \\ \hline \end{tabular}
\end{table} 
Table 3: Pore parameters of K\({}_{2}\)n\({}^{1+2}\) by low-pressure N\({}_{2}\) adsorption. where \(v_{n}\) is the volume of adsorbed gas at pressure equilibrium \(P\); \(v_{n_{0}}\) is the volume of the saturated adsorption of the monolayer; \(A\) is a power-law exponent related to the fractal dimension; \(\beta\) is the intercept; \(P_{0}\) is the saturated vapor pressure of the gas. The value \(A\) can be calculated by plotting the \(ln\left(\frac{v_{n}}{v_{n_{0}}}\right)\) and \(\ln\left(\ln\left(\frac{P_{0}}{P}\right)\right)\) of the gas adsorption isotherm data, and the slope of the straight line is equal to \(A\). The method for calculating the fractal dimension \(D_{n}\) (Equation (4)) is as follows:  
\[D_{n}=A+3 \tag{4}\]  
Figure 5 shows the FHH diagram of the sample SK1-S-2. At the relative pressures \(\frac{P}{P_{0}}\) of 0-0.5 (R\({}^{2}\) = 0.9999) and 0.5-1 (R\({}^{2}\) = 0.9971), there are two different linear segments because the gas adsorption mechanism was different at different relative pressures (Appendix A) [63; 72; 73; 74; 75; 76; 77]. The characteristics of the pores were also different. Considering the difference in the gas adsorption behavior between \(\frac{P}{P_{0}}\) < 0.5 and \(\frac{P}{P_{0}}\) > 0.5, the fractal dimensions of the samples were calculated in these two regions, which are defined as D1 and D2. D1 is monolayer-multilayer adsorption controlled by van der Waals forces, and D2 represents the adsorption behavior controlled by capillary condensation at a higher \(\frac{P}{P_{0}}\)[73; 74; 75; 76; 77]. The general fractal dimension was between 2 and 3, which was determined by the geometric irregularity and roughness of the surface. The larger value of D represents a more complex and irregular surface (with a value close to 2, the surface is smoother and flatter; with a value close to 3, the surface is rougher and the heterogeneity is stronger) [35; 36; 37]. The fractal dimension D1 indicates the irregularity of the sample surface (the larger the fractal dimension D1, the more irregular and rougher surface) [65]. The fractal dimension D2 indicates the irregularity of the pore structure (the larger the fractal dimension D2, the higher the heterogeneity of the pore distribution, the smaller the pore size, the higher the liquid/gas surface tension, and the lower the gas adsorption capacity) [65]. Fractal dimensions D1 and D2 represent the pore surface fractal dimensions and pore structure fractal dimensions, respectively (Figure 5, Appendix A).