## 1 Introduction  
Relative permeability is one of the key properties of rocks, closely related to production. Relative permeability can be used to estimate production capacity, injection capability, hydrocarbon reserves, breakthrough times, and ultimate recovery rates [1, 2]. The relative permeability curve depends on the direction of saturation changes and the maximum and minimum saturations reached [3, 4]. It is generally recognized that the relative permeability increases with the increase in saturation and decreases with the decrease, which is a constant curve. In fact, the relative permeability changes with the displacement fluid and the cycle, and the relative permeability curve has a lag phenomenon between different saturation cycles. The irreversibility or path dependence of the physical properties is called \"hysteresis\", so when observed in the relative permeability data, it is called \"phase permeability hysteresis\", and the degree of this irreversibility depends on the fluid type [5, 6, 7], pore structure [8], and wettability [5, 9]. The hysteresis effect of relative permeability in porous media has formed a relatively unified conclusion through years of experimental research. Ignorance of the relative permeability hysteresis effect of gas water alternation will misinterpret the production/experimental result [10]. Not considering hysteresis of relative permeability will underestimate the recovery factor and storage capacity in WAG-related CCUS application [11]. There are two main causes for the occurrence of hysteresis in relative permeability [12; 13; 14]. First, wettability hysteresis: the advancing contact angle (the contact angle as the wetting phase displaces the non-wetting phase) is greater than the receding contact angle (the contact angle as the wetting phase retreats due to the intrusion of the non-wetting phase), closely related to surface chemical properties, roughness, and changes caused by adsorbed solution particles or solvents. Second, trapping of the non-wetting phase: during imbibition, part of the non-wetting phase breaks off in clumps or ganglia and becomes immobile (trapped). Since the non-wetting phase primarily occupies the centers of pores and throats in the porous medium, even a small amount of trapping can significantly affect the flow of fluids and the values of relative permeability. The trapping of residual oil varies in different wettability systems. In water-wet systems, the mechanism of disconnection is the main reason for oil trapping. During the water injection process, due to the bridging of two water layers on either side of the throat, the water in the wetting phase will fill the smaller pores and disconnect the crude oil between two pores. Whereas in oil-wet systems, due to the difference in capillary forces caused by pore size differences, the smaller capillaries in interconnected capillaries will become the main sites for trapping residual oil. There are two types of relative permeability [7; 15] hysteresis, as shown in Figure 1: displacement-direction-dependent and saturation-history-dependent hysteresis [12; 14; 16]. Displacement direction-dependent hysteresis refers to the difference between drainage and imbibition permeability curve. Saturation-history-dependent means that the relative permeability curves of different cycles of continuous displacement in the same phase are different. In water-wet systems, it is generally believed that the relative permeability of the water phase increases with increasing water saturation, while the relative permeability of the non-wetting phase decreases with increasing water saturation. The value of the non-wetting phase relative permeability depends on the water saturation and its direction of movement; the process of increasing water saturation, known as waterflooding, is called the imbibition process, while the process of decreasing water saturation is called the drainage process. Figure 1: Different relative permeability hysteresis. (**a**) Displacement direction dependent. (**b**) Saturation history dependent.