## 2 Mathematical Models
### Two-Phase Relative Permeability Hysteresis Model
Normalized water relative permeability and normalized water saturation reduction are shown by:  
\[S_{gr}=\frac{S_{gi}}{1+CS_{gi}} \tag{30}\]6  
Normalized oil relative permeability, normalized water saturation increase:  
\[S_{gr}=\frac{S_{gi}}{1+CS_{gi}} \tag{30}\]7  
where the normalized oil relative permeability is:  
\[S_{gr}=\frac{S_{gi}}{1+CS_{gi}} \tag{30}\]8  
Normalized oil relative permeability and decreased normalized water saturation are shown by:  
\[S_{gr}=\frac{S_{gi}}{1+CS_{gi}} \tag{30}\]9  
(5)  
Delshad Model [26]  
In 2003, [PERSON] obtained analytical expressions for water and oil relative permeabilities using [PERSON]'s relative permeability model and Brooks-Corey capillary pressure function for main drainage. The saturation \(M_{ow}\) characterizes the smallest pores where oil has displaced water within the required residence time, converting water-wet to oil-wet. \(M_{ow}\) is the initial water saturation before reservoir oil recovery. In many reservoirs, this is equal to the bound water saturation. To establish the relationship for calculating residual oil saturation, it is assumed that \(S_{or}\) has a maximum value when \(M_{ow}=S_{inv}\), and it equals zero when \(M_{ow}=1\). The relationship between \(S_{or}\) and \(M_{ow}\) is:  
\[C=\frac{1}{(S_{\text{gr}})_{max}}-1 \tag{31}\]0  
where \(S_{or}^{\max}\) is the residual oil saturation at \(M_{ow}=S_{inv}\):  
\[C=\frac{1}{(S_{\text{gr}})_{max}}-1 \tag{31}\]1  
\(\overline{M}_{ow}\) is used to distinguish between the saturation of pore throats that are water-wet from those that are oil-wet or intermediate wet. It is assumed that in mixed-wet reservoirs, the largest pores will be oil-wet or intermediate wet. The relative permeability expressions for water and oil are as follows. For \(\bar{S}_{w}\leq\bar{M}_{ow}\):  
\[C=\frac{1}{(S_{\text{gr}})_{max}}-1 \tag{31}\]2  
\[C=\frac{1}{(S_{\text{gr}})_{max}}-1 \tag{31}\]3For \(\bar{S}_{w}>\bar{M}_{ow}\):  
\[C=\frac{1}{(S_{\text{gr}})_{max}}-1 \tag{31}\]4  
\[C=\frac{1}{(S_{\text{gr}})_{max}}-1 \tag{31}\]5  
where \(\Omega=\bar{M}_{ow}+\bar{S}_{o}\) and \(\bar{S}_{o}=\frac{S_{o}-S_{w}}{1-S_{w}-S_{or}}\). \(\lambda\) is the pore size distribution index used in the Brooks\(-\)Corey model, and \(\Omega\) is the term in the integral of relative permeability. (6) Kjosavik Model [39]  
[PERSON] et al. established a two-phase model for mixed wettability relative permeability curves, which took into account the processes of primary drainage, spontaneous imbibition, and secondary drainage. They integrated two Corey-type relative permeability expressions and weighted them based on capillary pressure correlation, in order to comprehensively describe the characteristics of water-wet and oil-wet systems. Their model is based on the assumption that the relative permeability for each injection cycle should return to the starting point of the previous displacement and form a closed scanning loop. The capillary pressure equation for the proposed mixed-wet system is an extension of the Brooks-Corey capillary relationship, as shown below:  
\[C=\frac{1}{(S_{\text{gr}})_{max}}-1 \tag{31}\]6  
where \(1/a_{w}\) is the exponent of the pore size distribution, \(c_{w}\) is the inlet pressure, \(c_{ud}\) and \(a_{wd}\) were the displacement parameters in the water-wet system, and \(a_{od}\) and \(c_{od}\) were the displacement parameters in the oil-wet system, Replace the subscript \"d\" with \"i\" to make it applicable to the imbibition process. In Equation (35):  
\[C=\frac{1}{(S_{\text{gr}})_{max}}-1 \tag{31}\]7  
\[C=\frac{1}{(S_{\text{gr}})_{max}}-1 \tag{31}\]8  
\(P_{cw}\) and \(P_{co}\) represent the capillary pressures in the water and oil branches, respectively, \(a_{w}\) and \(a_{O}\) are constants.