## 3 Results and Discussion
### Effect of Viscosity (M) and Capillary Force (Ca) on Flow in Porous Media
[PERSON] and [PERSON] and [PERSON] and [PERSON] (2019); [PERSON] and [PERSON] (2019) roughly defined capillary fingering and viscous fingering in porous media through extensive experiments. Capillary fingering typically occurs in small pores. Dominated by surface tension, elongated finger shapes are easily formed in porous media. The width of these channels is generally 1-3 pore widths, as shown in Figure 4(a1). It can be observed that three main capillary fingering channels appear due to capillary forces being the primary driving force in Figure 4(a1). As the capillary number \(Ca\) increases, the capillary force in the porous media gradually decreases while the viscous force increases. Viscous fingering typically occurs in large pores and high-viscosity conditions. The flow mode transitions from capillary fingering to viscous fingering, as shown in Figure 4(a2,a3). When viscous fingering occurs, the leading edge of the displacement front becomes unstable and irregular, forming multiple channels wider than 3 pore body widths, as shown in Figure 4(b3). At high \(Ca\) and \(M\) values, the CH\({}_{4}\) saturation is high. The capillary number \(lgCa=-4.07\) is the boundary where viscous fingering transitions to stable displacement. The displacement efficiency of CH\({}_{4}\) increases with the capillary number, transitioning the flow mode to stable displacement. The two-phase flow interface is stable, manifesting as stable displacement, as shown in Figure 4(c3,d3). The magnitude of saturation can be used to quantitatively describe the effects of \(Ca\) and \(M\) on two-phase flow. Figure 5a,b plot the functions of non-wetting phase (CH\({}_{4}\)) saturation versus \(Ca\) and \(M\) during CH\({}_{4}\) breakthrough and at stability, respectively. As shown in Figure 5a, the saturation of the non-wetting phase generally increases with \(Ca\) at different \(M\) values. However, an exceptional case occurs at \(lgM=-1.95\), where many interrupted fingering channels appear as shown in Figure 4(a2), resulting in a decrease in non-wetting phase saturation compared to Figure 4(a1). As the capillary number increases, the flow paths of the water phase in the porous media become more complex. Due to the viscosity of water molecules and the constraints of pore geometry, some branch channels may be occupied and blocked by the water phase, preventing CH\({}_{4}\) from continuing to advance in these channels, leading to interrupted fingering phenomena, and ultimately affecting the displacement efficiency of CH\({}_{4}\). Figure 5b shows the saturation of the non-wetting phase at stability after CH\({}_{4}\) breakthrough, which is almost consistent with the breakthrough saturation. However, at \(lgM=-0.52\), a special case occurs where the non-wetting phase saturation first increases, then decreases, and then increases again with increasing \(Ca\). The specific image can be seen in Figure 4 at \(lgM=-0.52\). When \(lgCa=-4.37\), the invading phase CH\({}_{4}\) gradually forms five capillary channels during the invasion breakthrough process and converges into one main flow channel after a period, rapidly displacing the defending water phase. The main cause of this phenomenon may be the changes in seepage conditions at the moment of breakthrough, when multiple capillary channels compete to absorb the invading CH\({}_{4}\). However, with changes in seepage conditions and adjustments in pore structure at stability, a more favorable seepage path with higher adsorption capacity may emerge. It causes more fluid flow to concentrate in this main channel. The cases of Figure 4(c2,c3), where CH\({}_{4}\) saturation decreases from breakthrough to stability, may indicate the differences in seepage velocity at different locations in homogeneous porous media. When the seepage velocity is higher at certain locations and lower at others, some fluid backflow may occur.