## 2 Materials and Methods
### The Phase Equilibrium Calculation Theory Considers the Capillary Force Effect  
The PR-EOS is modified to incorporate capillary pressure, which is influenced by pore geometry and the wettability of the rock surface. In the calculation of capillary pressure between oil and gas, the oil phase is regarded as the wetting phase relative to the gas phase. Under this assumption, the gas phase pressure can be expressed as:  
\[p_{v}=p_{l}+p_{c} \tag{1}\]  
where \(p_{v}\) is the gas phase pressure (Pa), \(p_{l}\) is the liquid phase pressure (Pa), \(p_{c}\) is the capillary pressure (Pa), which is determined by the geometry of the pores and the wettability of the rock surface. Mercury intrusion experiments quantify the relationship between capillary pressure and mercury saturation (i.e., the pore volume occupied by mercury). Volume averaging of pore radius and interfacial tension allows derivation of this relationship from the Young-Laplace equation, a result validated in multiphase flow simulations [48]. This study investigates the impact of nanopore capillary pressure on phase equilibrium. Consequently, phase behavior calculations incorporate these effects as described by [PERSON] et al. [35] the calculation of capillary force in Equation (1) refers to the Young-Laplace equation  
\[p_{c}=\frac{2\sigma\cos\theta}{r} \tag{2}\]  
where \(\sigma\) represents the interfacial tension (N/m); \(\theta\) represents the contact angle (For gas-liquid phase equilibrium in reservoirs, the gas phase is regarded as non-wetting, while the liquid phase is considered wetting, assuming a contact angle of 45\({}^{\circ}\)[49] in this study; \(r\) represents the pore radius (nm). The method for calculating the interfacial tension \(\sigma\) in Equation (2) refers to the work of [PERSON] and [PERSON] [50; 51]  
\[ \begin{split}&\sigma=(\Omega_{i}\overline{p}_{l}-\Omega_{\circ} \overline{p}_{v})^{\circ}\\ &\Omega_{l}=\sum\limits_{i=1}^{N_{c}}x_{i}\Omega_{i}\\ &\Omega_{v}=\sum\limits_{i=1}^{N_{c}}y_{i}\Omega_{i}\end{split}  \tag{3}\]  
where \(v\) is the scaling exponent, \(v=3.6\); \(\overline{p}_{l/v}\) is the average molar densities of the liquid and gas phases in the pore (mol/cm\({}^{3}\)); \(x_{i}\) and \(y_{i}\) are the molar fractions of component \(i\) in the liquid and gas phases, respectively; \(\Omega_{i}\) is the isobaric molar volumes of the liquid and gas phases, respectively [50; 51]  
\[\Omega_{i}=(8.21307+1.97473\omega_{i})T_{ci}^{1.03406}P_{ci}^{-0.82636} \tag{4}\]  
where \(\omega_{i}\) is the acentric factor; \(T_{ci}\) is the critical temperature of component \(i\) (K); \(P_{ci}\) is the critical pressure of component \(i\) (bar). Equations (1)-(3) provide a method for integrating microscopic mechanisms into phase behavior calculations. Since this study focuses on the impact of capillary pressure in nanopores on phase behavior, capillary pressure is computed using an average pore radius in subsequent calculations. However, the option to incorporate a pore size distribution remains. The proposed model can be extended to account for the actual pore radius distribution of a reservoir, based on geological pore size data. In the PR-EOS, capillary pressure is not accounted for in the gas and liquid phases. Instead, Equations (1) and (2) compute capillary pressure and phase pressures for different pore sizes, subsequently affecting the compressibility factor and fugacity coefficient. As noted above, the original PR-EOS neglects capillary forces, resulting in no distinction between phase pressures. Consequently, the formulations for fugacity and compressibility factor must be modified to incorporate capillary effects and differentiate phase pressures. Referring to PR-EOS [12], the compressibility factor equation incorporating capillary forces is given as:  
\[Z_{\alpha}^{3}-(1-B_{\alpha})Z_{\alpha}^{2}+(A_{\alpha}-3B_{\alpha}^{2}-2B_{ \alpha})Z_{\alpha}-(A_{\alpha}B_{\alpha}-B_{\alpha}^{2}-B_{\alpha}^{3})=0 \tag{5}\]  
where \(\alpha\) denotes the phase, with \(l\) representing the gas phase and \(v\) the liquid phase.