## 4 Applications
### The Impact of Relative Permeability Hysteresis on Recovery
When using non-condensable gas for injection, water saturation can be preferentially reduced by selectively altering the relative permeability of water rather than oil, thus prioritizing the reduction in water saturation. A similar effect can be achieved by directly injecting immiscible, non-condensable gas into the water-saturated zone. This will establish a region of high-capillary gas saturation, significantly reducing the characteristics affecting water influx, depending on rock geometry and relative availability features. Establishing trapped gas saturation in oil-bearing formations is evidently undesirable and should be avoided, as this could severely damage oil-phase permeability. **Negative impacts:**  
(1) Decrease in injection capacity [72; 76; 77]  
The hysteresis effect of phase permeability can lead to an increase in injection well pressure, especially during water injection after gas displacement. Once the pressure reaches a certain level, it may result in the inability to inject water and gas. In fact, instances of decreased injection capacity have been observed in real oilfield development scenarios, attributed to injection losses caused by the hysteresis effect. The potential loss of injection capacity can significantly impact the economic benefits of projects aimed at enhancing gas recovery. (2) Decrease in recovery [11]  
Several numerical simulation studies have found that models considering hysteresis in relative permeability provide predictions of recovery rates closer to actual conditions, with hysteresis in simulations leading to a decrease in recovery rates. Reasonably modeling hysteresis phenomena in the alternating gas-water process can significantly enhance the stability and accuracy of production outcomes. Failing to account for hysteresis effects may result in errors in predicting gas-water phase flow and distribution, leading to inaccurate recovery rate forecasts and irrational development scheme formulations. (3) Phase trapping [11]  
In many reservoir applications, phase trapping has been demonstrated in the literature as a significant factor for reducing productivity. The hysteresis effect of relative permeability has been found to cause trapping of the wetting phase by the non-wetting phase. (4) Early breakthrough of gas and water [11]  
Numerous experts have studied the impact of hysteresis on breakthrough times, with research indicating that considering hysteresis effects results in earlier breakthrough times for gas and water compared to conventional models that do not consider hysteresis effects. Failing to consider hysteresis effects may lead to erroneous estimates of the time for water breakthrough, preventing timely water shutoff measures and thereby hindering the maximization of economic benefits. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Author** & **Application** & \begin{tabular}{c} **Two Phase or** \\ **Three Phase** \\ \end{tabular} & \begin{tabular}{c} **Injection** \\ **Method** \\ \end{tabular} & **Gas** & \
\begin{tabular}{c} **Phenomena and Conclusions** \\ \end{tabular} \\ \hline [PERSON] et al. [78] (1998) & Reservoir & Two-phase & WAG & Natural gas &
\begin{tabular}{c} WAG injection has better accessibility and lower \\ residual oil saturation than water injection. \\ \end{tabular} \\ \hline [PERSON] et al. [53] (2003) & Core & & & & \\
[53] (2003) & experiment & Three-phase & WAG & N2 &
\begin{tabular}{c} The hysteresis cycle is irreversible. Reduce \\ residual oil saturation by capturing gas with water. \\ \end{tabular} \\ \hline [PERSON] & & & & & \\ [PERSON] et al. [79] & Core & & & & \\
[2011] & Core & & & & \\
[2011] & Experiment & Three-phase & WAG &
\begin{tabular}{c} The three-phase kr values of water and gas \\ depend on two independent saturation levels. The \\ kr values of three-phase water, oil, and gas are \\ significantly lower than their corresponding \\ two-phase values. \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The practical application of the Relative Permeability Hysteresis model.