## 2 Mathematical Models
The first and second terms represent the reduction in oil phase relative permeability due to the presence of water and gas, respectively, while the third term adjusts for the reduction caused by the interaction of water blocking oil and gas blocking oil. The resulting expression for oil phase relative permeability becomes:  
\[\kappa_{ro}=k_{row}+k_{rog}-\kappa_{ro}^{0}+(S_{w}+S_{o})\left[\kappa_{ro}^{0}-k_{ row}\right]\left[\kappa_{ro}^{0}-k_{rog}\right]/\kappa_{ro}^{0} \tag{6}\]  
In 1979, [PERSON] and [PERSON] [23] modified the Stone model to overcome the limitations of the original model, namely, that it could only be simplified to two-phase data when endpoint relative permeabilities were equal to 1. The modified Stone I model:  
\[k_{ro}=konor_{wgrowmax} \tag{7}\]  
\[\beta_{w}=\frac{k_{row}}{k(1-S_{wnor})_{rowmax}} \tag{8}\]  
\[\beta_{g}=\frac{k_{rog}}{k(1-S_{gor})_{rowmax}} \tag{9}\]  
The modified Stone II model:  
\[\mathrm{k_{ro}}=\frac{\left(\mathrm{S_{L}-S_{W}}\right)^{3}}{(1-\mathrm{S_{L }})^{4}}\left(\mathrm{S_{W}+S_{L}-2S_{L}}\right) \tag{1}\]0  
In 1984, [PERSON] [24] extended the statistical network model of capillary pressure and relative permeability under low capillary number displacement and percolation theory to gas, oil, and water three-phase flow, proposing a linear model. In this model, the permeability of oil can be approximated by a straight line connecting the two-phase relative permeabilities at connate water saturation. The relative permeability of oil in a three-phase system is directly proportional to the saturations of gas and water. The proportionate functions \(S_{g}\) (\(k_{ro}\)) and \(S_{w}\) (\(k_{ro}\)) are found through iteration. In 1988, [PERSON] [25] introduced a saturation-weighted interpolation model, where saturation weighting ensures that when the saturation of one phase approaches zero (or an irreducible saturation), the appropriate two-phase data will predominate. The calculation of oil-phase relative permeability is as follows:  
\[\mathrm{k_{ro}}=\frac{\left(\mathrm{S_{L}-S_{W}}\right)^{3}}{(1-\mathrm{S_{L }})^{4}}\left(\mathrm{S_{W}+S_{L}-2S_{L}}\right) \tag{1}\]1  
where: \(S_{wcrit}<S_{w}<1-S_{orw},0<S_{g}<\frac{(1-S_{orw}-S_{w})\left(1-S_{org}-S_{ wcrit}\right)}{1-S_{orw}-S_{wcrit}}\). In 1989, [PERSON] [26] and [PERSON] proposed a three-phase model independent of two-phase relative permeabilities, based on experimental data at two-phase limits. In this model, the residual oil saturation (Sor) in three-phase flow exhibits a linear relationship between the two limiting residual oil saturations (\(S_{orw}\), \(S_{org}\)):  
\[\mathrm{k_{ro}}=\frac{\left(\mathrm{S_{L}-S_{W}}\right)^{3}}{(1-\mathrm{S_{L }})^{4}}\left(\mathrm{S_{W}+S_{L}-2S_{L}}\right) \tag{1}\]2  
In 1989, [PERSON] and [PERSON] [27] discussed the Stone model modified by [PERSON] and [PERSON] [23] (1979). If \(k_{row}\) and \(k_{rog}\) do not equal the oil phase relative permeability at zero connate water saturation (oil-water) and gas saturation (oil-gas), then it is not possible to match relative permeabilities at the two-phase boundaries. To address the following two limitations:  
1. The measurement of two-phase oil-gas relative permeability data does not necessarily occur at connate water saturation. 2. Due to experimental errors, \(k_{row}\) often does not equal \(k_{rog}\). Introducing another normalization factor to correct the Stone I model:  
\[\mathrm{k_{ro}}=\frac{\left(\mathrm{S_{L}-S_{W}}\right)^{3}}{(1-\mathrm{S_{L }})^{4}}\left(\mathrm{S_{W}+S_{L}-2S_{L}}\right) \tag{1}\]3  
In 1997, [PERSON] [28] and others extended the research of [PERSON] (1993) by proposing a theoretical model. This model is based on a physically relevant description of phase distribution and flow mechanisms at the pore scale, considering only strong water-wet and extended conditions, but it can be expanded to oil-wet and even mixed-wet conditions.