## 2 Mathematical Models
### Two-Phase Relative Permeability Hysteresis Model  
The concept of incorporating the effects of relative permeability hysteresis on the basis of three-phase relative permeability models is referred to as the relative permeability hysteresis model. Currently, the mathematical models for relative permeability hysteresis have been extensively developed, which can be divided into two-phase relative permeability hysteresis models and three-phase relative permeability hysteresis models. The individual model could either consider the direction of saturation or both the direction of saturation and the saturation history. Different models vary in their scanning curves, among which the most widely used models include the two-phase hysteresis models such as the Killough model [34] and the Carlson model [35], and the three-phase hysteresis model by [PERSON] [36]. (1) Land Model [37]  
[PERSON] discovered an empirical relationship between the residual non-wetting phase saturation after imbibition and the initial non-wetting phase saturation. Based on this empirical relationship and the theory related to the distribution of relative permeability pore sizes, he established expressions for the trapping coefficient and the movable non-wetting phase saturation. By describing the trapping behavior experienced by the non-wetting phase and explaining its impact on relative permeability, he formally established the concept of hysteresis:  
\[\frac{1}{S_{\text{gr}}}-\frac{1}{S_{\text{gi}}}=\frac{1}{(S_{\text{gr}})_{max}} \tag{29}\]  
where:  
\[S_{gr}=\frac{S_{gi}}{1+CS_{gi}} \tag{30}\]  
\[C=\frac{1}{(S_{\text{gr}})_{max}}-1 \tag{31}\]  
where \(S_{\text{gi}}\) is the initial gas saturation; \(S_{\text{gr}}\) is the residual gas saturation after imbibition; \((S_{\text{gr}})_{max}\) is the maximum residual gas saturation when the initial gas saturation is consistent; and \(C\) is a constant, known as the Land constant. (2) Killough Model [34]  
Building on the Land model, [PERSON] assumed that no trapping occurs during the displacement process, and the relative permeability of the wetting phase does not exhibit hysteresis effects; the imbibition-relative permeability curve is reversible. Therefore, any imbibition curve, when reversed, will return precisely to its origin until we once again reach the main drainage curve. Assuming the first drainage process reverses at saturation \(S_{ni}\). At this time:  
\[K_{n}{}^{I}(S_{ni})=K_{n}{}^{D}(S_{ni}) \tag{32}\]  
where \(K_{n}^{I}(S_{ni})\) represents the permeability of the non-wetting phase during the imbibition stage at the reversal saturation \(S_{ni}\), and \(K_{n}^{D}(S_{ni})\) represents the permeability of the non-wetting phase during the displacement stage at the reversal saturation \(S_{ni}\). However, when the saturation reaches \(S_{nr}\), the new imbibition curve will reach \(K_{n}=0\), which is related to \(S_{ni}\) in Equation (33). \[K_{n}{}^{I}(S_{nr})=0 \tag{33}\]  
To predict the intermediate curve \(K_{n}{}^{I}\) located in (32) and (33), [PERSON] considered two methods: (a) parameter interpolation, and (b) normalization of experimental data. Specifically, the calculation of the non-wetting phase imbibition relative permeability is as follows:  
\[K_{rn}=\frac{K_{rnd}(S_{nh})}{K_{rnd}(S_{nmax})_{mi}(\overline{S_{nh}})} \tag{34}\]  
where:  
\[\overline{S}_{w}=\left(\frac{S_{n}-S_{nrn}}{S_{nh}-S_{nrn}}\right)(kurrm_{ nmax}) \tag{35}\]  
where \(K_{rnd}(S_{nh})\) represents the relative permeability during \(S_{nh}\) displacement, \(K_{rnd}(S_{nmax})\) represents the relative permeability during \(S_{nmax}\) displacement, and \(K_{rni}(\overline{S}_{nh})\) represents the relative permeability during \(\overline{S}_{nh}\) imbibition. (3) Carlson Model [35]  
[PERSON] focused on the study of the relative permeability of the non-wetting phase \(K_{n}\) in two-phase systems within the Killough model, and assumed that the relative permeability of the wetting phase \(K_{w}\) did not exhibit hysteresis behavior.