## 2 Mathematical Models
### Summary of the Model Study
#### 2.4.2 Three-Phase Relative Permeability Model  
The Parker and Lenhard model predicts arbitrary phase scanning curves using empirical interpolation schemes and scaling procedures. It uses two-phase wetting-saturation-capillary force and phase-permeability-saturation-capillary force models for incremental and decremental processes, defining saturation paths with three parameters. The Larsen and Skauge model depends on the initial saturation at the start of the given process. It proposes an algorithm to implement a non-wetting phase hysteresis model in numerical simulators using experimental wetting and non-wetting relative permeability as input.  
The Eggemann model analyzes hysteresis in gas-water two-phase relative permeability developed through continuous gas and water injection experiments. It identifies cycle hysteresis on the relative permeability based on historical matches and applies different Land formula constants considering displacement and history.  
The [PERSON]. [PERSON] model is used for mixed-wet systems and accounts for oil-gas composition changes. It uses Baker saturation-weighted interpolation for three-phase relative permeability, modifying the oil phase by rescaling saturation. The partitioning of the mixed-wet pore space is derived from the cross-saturation of water/oil two phases.  
The UTHYST model proposes new three-phase relative permeability and three-phase hysteresis models using multi-cycle three-phase water-alternating-gas experiments on non-water-wet rocks. It includes saturation history and composition effects, estimating three-phase parameters based on two-phase parameter saturation weighting interpolation. The hysteresis model is an extension of the Land trapping model, but introduces dynamic Land coefficients. The trapping model estimates continuously increasing capture saturations for intermediate wetting and non-wetting phases.  
The WAG-HW model is applied in WAG experiments on water-wet cores to obtain three-phase relative permeability values for oil, water, and natural gas. It observes hysteresis during WAG injection cycles of water and gas, where oil, water, and gas relative permeability exhibit significant and irreversible hysteresis. The model combines hysteresis effects for simulating WAG injection-related relative permeability.  
The Aghabozorgi model updates formulas for estimating hysteresis in water and gas phase relative permeability data to capture behavior observed in WAG experiments. It develops new hysteresis models based on innovative techniques to predict oil and water saturation at the end of each injection cycle. The main assumption of the model is that water saturation fluctuates within a certain saturation range with known limits (water saturation at the start and end of the water absorption cycles).  
To calculate saturation during the WAG process using three-phase relative permeability models, it is necessary to determine relative permeability data under different saturation histories. Due to the complexity of flow paths in three-phase flow, it is challenging to measure three-phase permeability experimentally. [PERSON]'s [44] research found that the Stone model overestimates the relative permeability in low oil saturation areas during the WAG process. [PERSON]'s [50] research identified several significant flaws in the WAG hysteresis model. Additionally, the Stone-I model exhibits certain errors when calculating the oil-phase relative permeability.