## 3 Results and Discussion
### Effect of Viscosity (M) and Capillary Force (Ca) on Flow in Porous Media  
Capillary forces and viscous forces have a significant correlation in a two-phase flow, affecting the morphology, distribution, and flow characteristics of the fluids. For a specific two-phase flow system, it is necessary to comprehensively consider the influence of these two forces to deeply understand and describe fluid behavior. The impact of viscous forces and capillary forces on interface instability and fluid saturation patterns can be judged from the interface flow state, which is classified into capillary fingering, viscous fingering, and stable displacement. Capillary forces and viscous forces can be characterized by two dimensionless numbers: capillary number (\(Ca\)) and viscosity ratio (\(M\)). The capillary number (\(Ca\)) is related to capillary forces, and the viscosity ratio (\(M\)) is the ratio of the viscosity of the invading non-wetting fluid to the viscosity of the displacing wetting fluid. They are defined as:  
\[Ca=\frac{\mu_{i}\,U_{i}}{\sigma} \tag{14}\]  
\[M=\frac{\mu_{i}}{\mu_{d}} \tag{15}\]  
where \(\mu\) is the dynamic viscosity, \(U\) is the characteristic velocity, and \(\sigma\) is the interfacial tension between the two phases. The subscripts \(i\) and \(d\) denote the \"invading phase\" and \"defending phase\", respectively. In this simulation, the invading phase is CH\({}_{4}\) and the defending phase is water. Under the conditions of \(\sigma\) is 70 mN/m and wetting angle \(\theta_{c}\) is 90\({}^{\circ}\), displacement is carried out at different constant velocities to explore the effects of capillary forces and viscous forces on the CH\({}_{4}\)-water two-phase flow characteristics. The specific data are shown in Table 1[13; 35]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Invading Fluid** & **Defending Fluid** & \(\sigma\) **(mN/m)** & **Contact Angle** & **Flow Rate \(U\) (m/s)** & _IgM_ & _IgCa_ \\ \hline \multirow{3}{*}{CH\({}_{4}\)} & \multirow{3}{*}{water} & \multirow{3}{*}{70} & \multirow{3}{*}{90\({}^{\circ}\)} & \(-1.95\) & \(-6.09\), \(-5.79\), \(-5.49\), \(-5.09\) \\  & & & & \(-1.22\) & \(-5.37\), \(-5.07\), \(-4.77\), \(-4.37\) \\ \cline{1-1}  & & & & \(0.05\) & \(-0.52\) & \(-4.67\), \(-4.37\), \(-4.07\), \(-3.67\) \\ \cline{1-1}  & & & & \(0\) & \(-4.15\), \(-3.85\), \(-3.55\), \(-3.15\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Effect of flow rate and capillary number on fluid properties. Figure 3: Evolution of the pore-scale CH\({}_{4}\)–water two-phase interface when \(lgCa=-4.25\), \(lgM=-0.52\) (The green arrow shows the direction of flow): (**a**) Burst mode in simulation; (**b**) transformation to touch mode in simulation after burst mode; (**c**) schematic of transformation of touch mode to overlapping mode in simulation; (**d**) transformation of capillary fingering to viscous fingering after overlapping of multiple overlapping modes. Figure 4 shows the snapshots of the simulation for different values of \(lgCa\) and \(lgM\) at the breakthrough moment. The viscosity ratio gradually increases from left to right. The results shown in Figure 4 qualitatively illustrate the three modes of immiscible displacement: capillary fingering, viscous fingering, and stable displacement. It is clearly observed that the change in displacement mode depends on the \(Ca\) and \(M\) values. At lower capillary numbers and lower viscosity ratios, i.e., \(lgCa<-4.37\) and \(lgM<-0.52\), the fingering phenomenon is quite pronounced, with the two-phase flow interface becoming unstable and multiple elongated flow channels appearing. However, the fingering phenomenon  
Figure 4: Repulsion patterns at the breakthrough moment for different capillary numbers and viscosity ratios (red and blue regions represent CH4 and water, respectively). results from the individual or combined action of capillary force and viscosity ratio, making it difficult to distinguish the type of fingering that occurs.