## 2 Mathematical Models
### Two-Phase Relative Permeability Hysteresis Model
By integrating the water portion relationship in the capillary (Equation (35)), the following relative permeability equations for the water-wet system were obtained, applicable for drainage and imbibition:  
\[C=\frac{1}{(S_{\text{gr}})_{max}}-1 \tag{31}\]9  
\[K_{n}{}^{I}(S_{ni})=K_{n}{}^{D}(S_{ni}) \tag{32}\]0  
\[K_{n}{}^{I}(S_{ni})=K_{n}{}^{D}(S_{ni}) \tag{32}\]1  
\[K_{n}{}^{I}(S_{ni})=K_{n}{}^{D}(S_{ni}) \tag{32}\]2  
\(k_{rw.awd}\) and \(k_{ro.awd}\) are the relative permeabilities of water and oil during the drainage process in the water-wet system. Similarly, the relative permeabilities of water and oil during the imbibition process in the oil-wet system:  
\[K_{n}{}^{I}(S_{ni})=K_{n}{}^{D}(S_{ni}) \tag{32}\]3  
\[K_{n}{}^{I}(S_{ni})=K_{n}{}^{D}(S_{ni}) \tag{32}\]4  
\[K_{n}{}^{I}(S_{ni})=K_{n}{}^{D}(S_{ni}) \tag{32}\]5\[K_{n}{}^{I}(S_{ni})=K_{n}{}^{D}(S_{ni}) \tag{32}\]6  
The parameter a has the same value in both the capillary pressure and relative permeability equations, while the curvature exponent (m) is only used in the relative permeability relationships. The definition of normalized saturation used in the relative permeability equations is as follows:  
\[K_{n}{}^{I}(S_{ni})=K_{n}{}^{D}(S_{ni}) \tag{32}\]7  
\[K_{n}{}^{I}(S_{ni})=K_{n}{}^{D}(S_{ni}) \tag{32}\]8  
They argue that when \(S_{wR}\) is small, entering water-wet pores, fluid behavior can be predicted using the water-wet equation. When water saturation reaches (\(1-S_{oR}\)) the oil-wet equation should be employed:  
\[K_{n}{}^{I}(S_{ni})=K_{n}{}^{D}(S_{ni}) \tag{32}\]9  
\[K_{n}{}^{I}(S_{nr})=0 \tag{33}\]0  
\(k_{rud}\) and \(k_{rod}\) are the relative permeabilities of water and oil, respectively, in the mixed-wet system during the drainage process. Substituting the subscript \"d\" with \"i\" in the equation yields the relationship for the imbibition relative permeability. Once boundary drainage experiences its first reversal, the Land closure model is used to compute the residual oil saturation. The normalized saturation \(S_{oR}\) should be adjusted using the computed value, and the oil branch (from Equation (37)) should also be adjusted according to the new residual saturation \(\left(p_{cui}^{1}\right)\).The water branch remains unchanged. Therefore, the relative permeability of \(S_{oR}\) will be oil-wet at the initial saturation of the first imbibition, obtained through the following equation for the first imbibition:  
\[K_{n}{}^{I}(S_{nr})=0 \tag{33}\]1  
\[K_{n}{}^{I}(S_{nr})=0 \tag{33}\]2  
Their hysteresis model is based on the assumption that each scanning curve should move towards the initial point of the previous displacement and form a closed loop. Therefore, by having the relative permeability values of water at the initial and final saturations with water and putting them into the equation, \(k_{rwi}^{0}[j]\) and \(k_{rwi}^{1}[j]\) are obtained. The relative permeability of the first imbibition water can be calculated using Equation (52). The same procedure can be used to obtain the relative permeability of oil:  
\[K_{n}{}^{I}(S_{nr})=0 \tag{33}\]3  
\[K_{n}{}^{I}(S_{nr})=0 \tag{33}\]4  
Spiteri Model [10]  
The obtained non-wetting saturation does not always change monotonically with the initial non-wetting saturation. Therefore, [PERSON] et al. proposed the following equation to overcome the limitations of the Land trapping saturation model (only applicable to strong water-wet systems). The Spiteri capture model is characterized by two parameters, \(\alpha\) and \(\beta\), which are obtained by fitting the results of pore network simulations. Parameters \(\alpha\) and \(\beta\) represent the initial slope and curvature of the curve, respectively. \[K_{n}{}^{I}(S_{nr})=0 \tag{33}\]5  
where \(S_{ot}\) is the captured oil saturation, and \(S_{oi}\) is the initial oil saturation. The constraints of the [PERSON] model parameters are as follows:  
\[K_{n}{}^{I}(S_{nr})=0 \tag{33}\]6  
[PERSON]'s hysteresis model is an improvement over the Land and Carlson models for predicting relative permeability.