## 3 Results and Discussion  
### Analysis of Local Flow Patterns  
When two immiscible fluids compete for displacement in a porous media, their flow patterns are influenced by numerous factors, including the properties of the fluids (viscosity, density, wettability, etc.) and the dimensions of the porous media (particle size, throat size, surface roughness, etc.). When two immiscible fluids flow in a pore structure dominated by cylindrical particles, their flow mode is primarily crescent motion. [PERSON] and [PERSON] [39,40] analyzed and defined three basic forms of crescent motion: burst, touch, and overlap. As shown in Figure 2, the circular arcs denote stable menisci, menisci at corresponding maximal capillary pressure, and advanced menisci for the respective instability modes, respectively. When the invading phase contacts the defending phase during an invasion, crescent-shaped interfaces are formed, varying with pore size. The curvature of the crescent-shaped interfaces formed by the invading phase is the main determinant of these three forms. The burst mode (Figure 2a) typically appears when the curvature of the crescent-shaped interface begins to decrease gradually, causing a local decrease in capillary pressure, leading the invading phase to invade adjacent pores. The touch mode (Figure 2b) follows the burst mode. In the touch mode, the curvature of the crescent-shaped interface continuously increases, causing an increase in capillary pressure. The advancement of the crescent-shaped interface towards the cylindrical particles is accelerated. Then the crescent-shaped interface and the cylindrical particles contact. The original crescent is disrupted and bifurcated. The overlap mode (Figure 2c,d) requires two adjacent crescents. After the touch mode, the two branches formed by the split invading fluid meet and overlap at the three-phase contact line or fluid-fluid interface due to local geometric factors, forming a new main path. The burst and touch modes are influenced simply by the local geometry of a single channel and its adjacent pores, rather than the filling state of adjacent channels. Unlike the instability of the burst and touch modes, the overlap mode only occurs when there is at least one crescent-shaped interface between adjacent channels, leading to the penetration of the interface. The basic interface advancement mode is influenced by the contact angle and local geometry and is closely related to capillary pressure. Capillary pressure arises from the surface tension effects at the liquid-gas interface, which provides the main driving force for the invading fluid. Derived from the two-dimensional Young-Laplace equation:  
\[P_{c}=\sigma\bigg{(}\frac{1}{R_{1}}+\frac{1}{R_{2}}\bigg{)} \tag{13}\]  
where \(P_{c}\) is the capillary pressure, \(\sigma\) is the interfacial tension, and \(R_{1}\) and \(R_{2}\) are the radii of curvature of the invading curved lunar surface. Figure 1: Schematic diagram of the microscopic (The structure in the red circle is zoomed in on the right side). [PERSON] et al. [41; 42] analyzed the triggering process of pore filling in disordered porous media using geometric analysis based on the three basic forms of crescent motion and the size of the curvature radius. To more intuitively study the macroscopic displacement mechanisms of capillary fingering and viscous fingering, magnified local images of the simulation results are used to observe the evolution of the two-phase interface. When \(lgCa=-4.25\) and \(lgM=-0.52\) (Figure 3), the gas invasion phase exhibited an atypical invasion mode evolving from capillary fingering to viscous fingering during the breakthrough water displacement process. As shown in Figure 3a, when the curvature of the flow interface of the invading phase in pore throat A exceeds the critical curvature, the fluid continuously invades forward under the influence of capillary pressure, resulting in capillary fingering.