## 2 Mathematical Models
The porous medium is described as a set of fractal pores, with their linear fractal dimension and size distribution derived from the mercury intrusion capillary pressure curve. The following relative permeability for the three-phase oil phase was proposed:  
\[\mathrm{k_{ro}}=\frac{\left(\mathrm{S_{L}-S_{W}}\right)^{3}}{(1-\mathrm{S_{L }})^{4}}\left(\mathrm{S_{W}+S_{L}-2S_{L}}\right) \tag{1}\]4  
The authors extended the model to oil-wet and intermediate-wet conditions ([PERSON] et al., 1999 [29]) using the wettability index WI (where a water-wet medium equals 1, and an oil-wet medium equals \(-1\)):  
\[\mathrm{k_{ro}}=\frac{\left(\mathrm{S_{L}-S_{W}}\right)^{3}}{(1-\mathrm{S_{L }})^{4}}\left(\mathrm{S_{W}+S_{L}-2S_{L}}\right) \tag{1}\]5  
\[\mathrm{k_{ro}}=\frac{\left(\mathrm{S_{L}-S_{W}}\right)^{3}}{(1-\mathrm{S_{L }})^{4}}\left(\mathrm{S_{W}+S_{L}-2S_{L}}\right) \tag{1}\]6  
In 1999 [30], [PERSON] introduced an empirical model for three-phase relative permeability that accounts for changes in hydrocarbon composition, different saturation paths, and the trapping of oil, water, and gas. The model is based on saturation-weighted interpolation, using flowing saturation as a parameter for two-phase permeability functions. The relative permeability for the three-phase oil phase is as follows:  
\[ \begin{array}{c}k_{ro}(S_{o})=\alpha\frac{\big{\{}\big{(}S_{w}-S_{ww}\big{)} \big{[}a_{o}K_{row(w)}\big{(}S_{off}\big{)}\big{]}+\big{(}S_{g}-S_{gr}\big{)} \big{[}a_{o}k_{obs}\big{(}S_{off}\big{)}+b_{o}k_{rg(i)}\big{(}S_{off}\big{)} \big{]}\big{\}}}{\big{(}S_{w}-S_{ww}\big{)}+\big{(}S_{g}-S_{gr}\big{)}}\\ +\alpha k_{ol}(S_{ol})+\beta\frac{S_{o}}{S_{o}+S_{g}}\Big{[}a_{o}k_{ro(w)} \Big{(}S_{hf}\Big{)}+b_{o}k_{rg(w)}\Big{(}S_{hf}\Big{)}\Big{]}\end{array}  \tag{17}\]  
\(\alpha\), \(\beta\), \(a_{o}\), and \(b_{o}\) are functions of interfacial tension and fluid density when considering component consistency, that is, changes in fluid properties due to variations in pressure and components. If the changes are minimal, these function values are specified as: 1, 0, 1, and 0. In 2008 [31], [PERSON] and [PERSON] introduced a new empirical model for three-phase oil relative permeability to provide flexibility in adjustments for improving simulator predictions. The new model is based on a saturation-weighted interpolation between two-phase values, similar to the [PERSON] model, but with simple tuning coefficients, allowing for adjustments to match experimental measurements. \[k_{ro}=\frac{(S_{w}-S_{wc})k_{row}+S_{on}^{\ 2}\alpha\big{(}S_{g}-S_{gr}\big{)}k_{ rog}}{(S_{w}-S_{wc})+S_{on}^{\ 2}\alpha\big{(}S_{g}-S_{gr}\big{)}} \tag{18}\]  
\[S_{on}=\frac{(S_{o}-S_{om})}{\big{(}1-S_{wc}-S_{g}-S_{om}\big{)}} \tag{19}\]  
\[S_{om}=\frac{S_{w}S_{orw}+S_{g}S_{org}}{S_{w}+S_{g}} \tag{20}\]  
In 2010, [PERSON] and [PERSON] [32] proposed a model based on three sets of two-phase data, appropriately considering six types of saturation, namely, two-phase saturation,residual saturation, or critical saturation. They used the following interpolation to plot permeability maps of the reservoir:  
\[k_{ro}(S_{omnx})=\frac{S_{w}}{S_{w}+S_{g}}\cdot k_{row}(S_{omnx})+\frac{S_{g}}{S_{ w}+S_{g}}k_{rog}(S_{omnx}) \tag{21}\]  
In 2019, [PERSON] and [PERSON] [33] modified the Stone I model, taking into account the flow in three-phase systems during immiscible and near-miscible gas-water alternating injection experiments to correct oil phase relative permeability. The results indicated that the Stone model overestimated oil phase relative permeability data in regions of low oil saturation. Additionally, it was found that [PERSON]'s exponential model could not simulate the gradual decrease in oil phase relative permeability data. To improve the results, a constant parameter called the oil reduction factor (\(\beta_{o}\)) was introduced into the Stone I model at low oil saturations, which remained constant throughout the entire gas-water alternating experiment to consider the effect of separated oil clusters.