## 2 Mathematical Models
### Summary of the Model Study
#### 2.4.1 Two-Phase Relative Permeability Model
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Hysteresis Model**} & \multicolumn{3}{c}{**Application**} & \multirow{2}{*}{**Wettability**} & \multirow{2}{*}{**Assumed Conditions/Applicable Conditions**} \\ \cline{2-2} \cline{4-6}  & **Water** & & & & & \\ \hline [PERSON] et al. (2002) & & Yes & No & Mix-wet & \begin{tabular}{c} Assume that each injection cycle of kr should return to the starting point of \\ the previous displacement and form a closed scanning loop. \\ \end{tabular} \\ \hline [PERSON] et al. (2003) & Yes & Yes & No & Mix-wet & \begin{tabular}{c} Assume that Sor has a maximum value at \(M_{\text{cur}}=S_{\text{inv}}\) and is zero at \(M_{\text{cur}}=1\). \\ \end{tabular} \\ \hline [PERSON] et al. (2008) & Yes & Yes & No & Water-wet & \begin{tabular}{c} The captured non-wetting saturation does not always monotonically \\ change with the initial non-wetting saturation. \\ \end{tabular} \\ \hline \multicolumn{6}{c}{Three-phase hysteresis (related to saturation history: saturation direction and path)} \\ \hline [PERSON] and [PERSON] et al. (2004, 2011) (1987) & Yes & Yes & Both & \begin{tabular}{c} Consider the effect of pore blocking by gas trapped in water and oil phases, as well \\ as oil trapped in the water phase. \\ \end{tabular} \\ \hline [PERSON] and [PERSON] et al. (2016) & Yes & No & Yes & Water-wet & \begin{tabular}{c} Gas-water phase permeability is consistent during both two-phase and three-phase \\ flow processes; oil phase permeability in three-phase flow is calculated using the \\ Stone I model; there \\ is no hysteresis effect in the oil phase. \\ \end{tabular} \\ \hline [PERSON] et al. (2000) & Yes & Yes & Both & \begin{tabular}{c} C\({}_{\text{L}}\), C\({}_{\text{T}}\) and C\({}_{\text{U}}\) represent the Land, capture, and uncaptured constants, respectively. \\ \end{tabular} \\ \hline [PERSON] et al. (2000) & Yes & Yes & Mix-wet & \begin{tabular}{c} It is assumed that the hysteresis for each phase depends only on the saturation of \\ that phase. \\ Consideration of component effects. \\ \end{tabular} \\ \hline UTHYST Beygi et al. (2013) & Yes & Yes & Both & \begin{tabular}{c} Any relative permeability model containing capture saturation. \\ \end{tabular} \\ \hline [PERSON] et al. (2014) & Yes & Yes & Yes & Water-wet & \begin{tabular}{c} Oil remobilization induced by gas injection in water-wet media; a smooth transition \\ towards the layer-drainage regime for low oil saturations; \\ the consequent reduction of residual oil saturation in a three-phase system. \\ \end{tabular} \\ \hline WAG-HW & & & & & \\ [PERSON] and [PERSON] et al. (2015) & Yes & Yes & Both & \
\begin{tabular}{c} Application of hysteresis methods suitable for WAG simulations, based on \\ three-phase relative permeability data obtained from WAG experiments. \\ \end{tabular} \\ \hline [PERSON] et al. (2019) & Yes & No & Yes & Both &
\begin{tabular}{c} It is assumed that the relative permeability of gas and water depends only on \\ their saturation. Water saturation fluctuates within a certain range of saturations. \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._The Kjosavik model is a hysteresis model for mixed-wet rocks. It extends the Corey-Burdine relative permeability correlation developed for water-wet and oil-wet systems to mixed-wet rocks using a weighted approach with capillary pressure data. The model assumes that each cycle's relative permeability should return to the starting point of the previous displacement, forming a closed scanning loop. The Spiteri model overcomes limitations of the Land trapping saturation model. It characterizes the Spiteri trapping model in strongly water-wet systems with two parameters obtained by fitting results from pore network simulations.