## 4 Results and Discussion
### The Viscosity Ratio Effect  
The viscosity ratio is one of the most important dimensionless numbers in two-phase displacement through a porous media, and its effect was studied for a constant contact angle of 120\({}^{\circ}\), corresponding to a weak wetting property of the displacing fluid. The inlet pressure is also the same for different viscosity ratios, i.e., \(p_{in}=(p_{out}+0.0125)\)  mu\(\cdot\)lu\({}^{-1}\cdot\)ts\({}^{-2}\). Figure 4 shows the fluid distributions inside the pore network when breakthrough of the displacing fluid occurs for viscosity ratios of 1, 2, 5, and 10. It can be seen that for all the viscosity ratios considered, two or three preferential paths are generally formed. As the viscosity ratio increases, it becomes harder for the finger to flow into the large pore as highlighted by the black elliptical circles. The front of the bottom finger is finally halted at the position near the inlet for the viscosity ratio of 10. Also, the breakthrough finger switches to a lower position at the outlet when the viscosity ratio increases to 10.  
Figure 4: Fluid distributions at breakthrough for a wetting fluid invading a porous medium initially saturated with a non-wetting fluid. The wetting fluid (indicated in blue) is injected from the left side, and the viscosity ratio \(\lambda\) equals 1, 2, 5, and 10, respectively.  
Figure 3: Porosity as a function of the domain size. L is the length of the entire micromodel, while H is its width; x refers to the distance to the left boundary, and y is the distance to the bottom boundary. For each point shown above, the same scaling was applied to x and y, i.e., x/L = y/H.  
Figure 5 shows the time evolution of the displacing (wetting) fluid saturation for a viscosity ratio changing from 1 to 30. It can be clearly seen that a higher viscosity ratio results in a lower fluid saturation which is less favourable for the recovery of displaced fluid. This is consistent with previous experimental [44] and numerical results [45]. At the viscosity ratio of unity, the highest displacing fluid saturation (around 0.37) is observed, and the displacing fluid saturation keeps increasing linearly with time. Although the displacing fluid saturation reduces with \(\lambda\), the reduction becomes less significant at larger \(\lambda\) values.  
We also tested whether the above observations are valid for varying wettability and interfacial tension. Four pairs of contact angle and interfacial tension were considered (see Table 1), and the simulation results are shown in Figure 6. It is clear that the displacing fluid saturation drops rapidly when the viscosity ratio increases from 1 to 20 and then stays almost unchanged for all the cases considered. For each viscosity ratio, increasing contact angle or interfacial tension always leads to a higher displacing fluid saturation or displacement efficiency. In addition, the limitation of the current model was tested by decreasing the viscosity of the displacing fluid, and it was found that the maximum viscosity ratio that it can successfully access is as high as 50.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & \(\Delta\)**P** & \(\sigma\) & \(\cos\theta\) \\ \hline I & 0.0125 & 0.015 & \(-\)0.5 \\ II & 0.0125 & 0.0005 & \(-\)0.5 \\ III & 0.0125 & 0.0005 & 0.5 \\ IV & 0.0125 & 0.015 & 0.5 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters used in four typical cases of immiscible displacement.  
Figure 5: Evolution of the displacing fluid saturation for the viscosity ratio \(\lambda\) increased from 1 to 30 at \(\sigma=0.015\) mu\(\cdot\)ts\({}^{-2}\) and \(\theta=120^{\circ}\) (imbibition). Note that the time is normalized by \(t_{b}\), defined as the time at which the breakthrough occurs.