# Investigation of the Effect of Capillary Barrier on Water-Oil Movement in Water Flooding
## 4 Effect of the Capillary Barrier on Oil-Water Two-Phase Flow in Porous Media Model
### Results and Discussion  
#### 4.3.1 Water-Wet Condition  
Better understanding of the characteristics of the oil-water two-phase flow pattern during waterflooding is necessary to elucidate the influence of the capillary barrier on the macroscopic immiscible displacement process. To this end, the dimensionless capillary number \(Ca\) is adopted to characterize the two-phase flow dynamics and is used to describe the relative importance of the viscous force to the capillary force, which can be defined as follows [36]:  
\[Ca=\frac{\mu_{water}V}{\sigma} \tag{32}\]  
where \(\mathbf{V}\) is the injection velocity of the water phase, m/s; \(\sigma\) is the interface tension of the oil phase and water phase, N; \(\mu_{water}\) is the dynamic viscosity of the water, Pa s.  
Figure 17 shows the change of pressure difference between the inlet and outlet and the change of saturation of oil phase in the porous media model under water-wet conditions. It can be seen from Figure 17a that there are large fluctuations of the pressure difference between the inlet and outlet before and after the breakthrough of the water phase at the low capillary number of 1.8 \(\times\)\(10^{-4}\) and there are negative values of the pressure difference. This dramatic fluctuation can be explained by the instability induced by the strong capillary effects during a capillary-dominated displacement process at a low capillary number [28]. With the advancing of the water phase, small fluctuations of the pressure difference are observed, and the pressure difference is positive. Although the pressure difference in the process of liquid leaving the throat is negative, the decrease amplitude of the capillary force is still larger than the increase amplitude of the resistance induced by the capillary barrier effect. Therefore, the fluctuations of the pressure difference weaken. The capillary force plays a dominant role at low injection velocity. At the intermediate capillary number of 1.8 \(\times\)\(10^{-3}\), the capillary force and the viscous force are at play simultaneously. There are obvious fluctuations of the pressure difference before and after the breakthrough of the water phase and the value of the pressure difference is always above zero. At the large capillary number of 1.8 \(\times\)\(10^{-2}\), viscous force plays a crucial role in the displacement process and there are little fluctuations of the pressure difference before the breakthrough of the water phase. This is because the frequency of Haines-jump events decreased with the increase of the capillary number and the displacement process is thus dominated by viscous fingering [37]. Large fluctuations of the pressure difference are observed after the breakthrough of the water phase. As can be seen from Figure 17b, the highest oil displacement efficiency under water-wet conditions is obtained at the intermediate capillary number of 1.8 \(\times\)\(10^{-3}\). The reason is that the sum of the contact angle and the opening angle satisfies the minimum conditions of the occurrence of the capillary barrier effect. Dramatic fluctuations of pressure difference between and the outlet under water-wet conditions (the contact angle is 60\({}^{\circ}\)) and at low capillary number of 1.8 \(\times\)\(10^{-4}\) are observed in Figure 18a. In order to clarify the underlying reasons behind the fluctuations, the waterflooding process is systematically investigated from the change of oil and water distribution, with time at capillary number of 1.8 \(\times\)\(10^{-4}\) shown in Figure 18, based on the aforementioned theoretical analysis of the capillary barrier phenomenon. Considering the \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Physical Quantiity**} & \multicolumn{3}{c}{**Boundaries**} \\ \cline{2-4}  & **Inlet** & **Outlet** & **Wall** \\ \hline velocity & fixed value & zero gradient & fixed value \\ pressure & zero gradient & fixed value & zero gradient \\ water volume fraction & fixed value & zero gradient & constant contact angle \\ \hline \hline \end{tabular}
\end{table}