# Investigation of the Effect of Capillary Barrier on Water-Oil Movement in Water Flooding
## 3 Capillary Barrier Phenomenon
### Effect of Different Contact Angles and Opening Angles on the Capillary Force
#### 3.4.1 Water-Wet Conditions  
Figure 12 shows the change of the capillary force with the contact angle and the opening angle under water-wet conditions. As shown in Figure 12, the reverse of the oil-water interface is observed during the advancing process when \(\theta+\beta>90^{\circ}\), and the capillary force presents resistance characteristics, and there is a negative maximum value of \(p_{c}^{min}\). The increase and decrease of the driving force in the process can only result in the deformation of the interface. When the driving force of the fluid is sufficient to overcome the maximum resistance, the resistance value will decrease, and the interface will continue advancing even at a small driving pressure. Therefore, the driving pressure must be above a certain threshold to induce fluid motion, and once this threshold value is reached, the displacement force required for fluid motion will decrease. The following can be known from Figure 12: (1) Under water-wet conditions, the capillary force is positive in the process of the liquid entering the throat section; (2) When \(\theta>0^{\circ}\), there exist extrema of \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Wettability** & **Conditions** & **Maximal Value** & **Minimal Value** \\ \hline \(0<\theta<90^{\circ}\) (water-wet) & \(0<\beta+\theta<90^{\circ}\) & \(r\) (driving force) & \(r/\cos\theta\) (driving force) \\ \(90^{\circ}<\theta<180^{\circ}\) (oil-wet) & \(-90^{\circ}<\beta-\theta<0\) & \(r/\cos(\theta-\beta)\) (driving force) & \(r/\cos\theta\) (driving force) \\  & \(-180^{\circ}<\beta-\theta<0\) & \(r/\cos(\theta-\beta)\) (driving force, resistance) & \(r/\cos\theta\) (resistance) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Conditions of the extrema when oil-water interface entering the throat section. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Wettability** & **Conditions** & **Maximal Value** & **Minimal Value** \\ \hline \(0<\theta<90^{\circ}\) (water-wet) & \(0<\beta+\theta<180^{\circ}\) & \(r/\cos\theta\) (driving force) & \(r/\cos(\theta+\beta)\) (driving force, resistance) \\ \(90^{\circ}<\theta<180^{\circ}\) (oil-wet) & \(90^{\circ}<\beta+\theta<180^{\circ}\) & \(r/\cos\theta\) (resistance) & \(r/\cos(\theta+\beta)\) (resistance) \\  & \(180^{\circ}<\beta+\theta<270^{\circ}\) & \(r/\cos\theta\) (resistance) & \(-r\) (resistance) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Conditions of the extrema when oil-water interface leaving the throat section. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Wettability** & **Conditions** & **Maximal Value** & **Minimal Value** \\ \hline \(0<\theta<180^{\circ}\) & \(0<\beta+\theta<90^{\circ}\) & \(r/\cos(\theta+\beta)\) (driving force) & \((L\sin\beta+r\cos\beta)/(\cos\theta-\sin\beta)\) \\ \(90^{\circ}<\beta+\theta<270^{\circ}\) & \((L\sin\beta+r\cos\beta)/(\cos\theta-\sin\beta)\) & \(r/\cos(\theta+\beta)\) (resistance) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Conditions of the extrema when oil-water interface entering the outlet section. the capillary force, but the maximum capillary force (\(\Delta p_{max}=2\gamma_{lg}/r\)) appears when \(\beta>\theta\); (3) When \(\beta+\theta>90^{\circ}\), the capillary force is negative and the occurrence of the capillary barrier phenomenon can be observed in the process of the liquid leaving the throat section; (4) Define \(\Delta=\beta+\theta-90^{\circ}\), which can characterize the intensity of the capillary phenomenon; effect. The larger the value of \(\Delta\), the larger the extrema of the capillary resistance, indicating stronger capillary barrier effect. Figure 12: The influence of opening angles on the capillary force with different contact angles under water-wet conditions: (**a**) \(\theta=0^{\circ}\); (**b**) \(\theta=20^{\circ}\); (**c**) \(\theta=40^{\circ}\); (**d**) \(\theta=60^{\circ}\); (**e**) \(\theta=80^{\circ}\).