## 2 Mathematical Models  
This Section will cover the theoretical frameworks used to describe the phenomenon of relative permeability hysteresis. This may include models based on physical principles, such as extensions of the Land model and capture models, which attempt to estimate  
Figure 2: Schematics of scanning curves. the increasing capture saturation of intermediate-wet and non-wet phases by introducing dynamic coefficients. Additionally, there are methods based on behaviors observed in experiments to update and optimize the model, which uses innovative techniques to develop new hysteresis models that predict the saturation of oil and water at the end of each injection cycle. The permeability model is the fundamental of the relative permeability hysteresis model. After the calculation of the scanning curve of from the hysteresis model, the permeability model will be used for the construction of the whole relative permeability curve. In 1956 [19], [PERSON] introduced a method to calculate the oil-water permeability of a three-phase system from the easily measurable gas relative permeability, related to saturation:  
\[\mathrm{k_{ro}}=\frac{\left(\mathrm{S_{L}-S_{W}}\right)^{3}}{(1-\mathrm{S_{L }})^{4}}\left(\mathrm{S_{W}+S_{L}-2S_{L}}\right) \tag{1}\]  
where \(S_{W}\) is the water saturation, \(S_{L}\) is the liquid saturation, and \(S_{L}\) is the residual liquid saturation. In 1970, [PERSON] [20] proposed a model based on channel flow theory, treating porous media as a combination of flow channels and assuming that only one type of flowing fluid exists in any given channel. Wettability determined the occupancy rate of channels of different sizes. The intermediate-wetting phase separates the wet phase in the smallest channels from the non-wet phase in the largest channels. Therefore, just like in two-phase systems, three-phase systems have the same saturation history, and the wet phase (water) saturation occupies the same flow channels. This also means that the water phase relative permeability is a function of its own saturation, the same in both two-phase and three-phase systems. The relationship for oil phase relative permeability:  
\[k_{ro}=S_{onor}\cdot\beta_{w}\cdot\beta_{g} \tag{2}\]  
\[S_{onor}=\frac{S_{o}-S_{on}}{1-S_{wc}-S_{om}} \tag{3}\]  
where \(S_{om}\) is the three-phase residual oil saturation, and \(S_{wc}\) is the connate water saturation. The function \(\beta_{w}\) depends only on water saturation, and \(\beta_{g}\) depends only on gas saturation. In 1973, [PERSON] [21] introduced another model (Stone II model), which uses the relationships of four two-phase relative permeabilities in oil/gas and oil/water systems to predict the three-phase oil phase relative permeability. The main difference between Stone I and Stone II is that Stone II does not require So for calculations. To solve for \(k_{ro}\), the following equation is used:  
\[k_{ro}=\left(k_{row}+k_{rwo}\right)\left(k_{rgg}+k_{rgo}\right)-\left(k_{rwo}+ k_{rgo}\right) \tag{4}\]  
In 1976, [PERSON] and [PERSON] [22] proposed that the oil-relative permeability would be a function of gas-water saturation, and assumed that water and gas could flow simultaneously with oil in the pore spaces. They considered the overall reduction in oil-phase relative permeability caused by water and gas, deriving an expression for three-phase oil-phase relative permeability. The reduction in oil-phase relative permeability is as follows:  
\[\left[\mathrm{k_{ro}^{0}-k_{row}}\right]+\left[\mathrm{k_{ro}^{0}-k_{rgg}} \right]-\left(S_{w}+S_{o}\right)\left[\mathrm{k_{ro}^{0}-k_{row}}\right]\left[ \mathrm{k_{ro}^{0}-k_{rog}}\right]/\mathrm{k_{ro}^{0}} \tag{5}\]  
where \(\mathrm{k_{ro}^{0}}\) is the oil phase relative permeability at irreducible water saturation and zero gas saturation, \(k_{rwa}\) is the oil phase relative permeability in oil-water two-phase flow, and \(k_{rog}\) is the oil phase relative permeability in oil-gas two-phase flow.