# Investigation of the Effect of Capillary Barrier on Water-Oil Movement in Water Flooding
## 3 Capillary Barrier Phenomenon
### Variation of the Capillary Force along the Central Axis
#### 3.3.1 Analysis of the Capillary Force at Different Stages
The equivalent capillary tube radius \(R\) changes from \(R_{1}\) to \(r\), and then from \(r\) to \(R_{2}\); thus, the capillary force gradually increases and finally, it will reach the maximum value, which can be expressed as  
\[\gamma_{\text{sep}}+\gamma_{\text{ow}}\cos\theta=\gamma_{\text{os}} \tag{15}\]0  
\[\gamma_{\text{sep}}+\gamma_{\text{ow}}\cos\theta=\gamma_{\text{os}} \tag{15}\]1 Oil–water interface morphology when the static contact angle is formed in equilibrium with the wall of the inlet part. \[\gamma_{\text{sep}}+\gamma_{\text{ow}}\cos\theta=\gamma_{\text{os}} \tag{15}\]2 Oil–water interface morphology when the static contact angle is formed in equilibrium with the wall of the throat. Then, the capillary force will gradually decrease to a value when the stable contact angle is formed, and the value is given by  
\[\gamma_{\text{sep}}+\gamma_{\text{ow}}\cos\theta=\gamma_{\text{os}} \tag{15}\]3  
**Situation 2**. When \(\beta\leq\theta_{2}\), the change of the capillary force is shown in Figure 7b. The equivalent capillary tube radius \(R\) changes from \(R_{3}\) to \(R_{4}\); thus, the capillary force is gradually decreasing along the axis. **Stage III**: At this stage, the contact line moves in the throat section and the interface advances steadily from position \(a\) to position \(b\). The initial state and the final state are illustrated in Figures 6 and 8, respectively. The position of the interface at this stage satisfies the following condition:  
\[\gamma_{\text{sep}}+\gamma_{\text{ow}}\cos\theta=\gamma_{\text{os}} \tag{15}\]4  
The equivalent capillary tube radius \(R\) at this stage can be written as  
\[\gamma_{\text{sep}}+\gamma_{\text{ow}}\cos\theta=\gamma_{\text{os}} \tag{15}\]5  
The capillary force at this stage is constant (\(r/\text{cos}\theta\)) and is closely related to the contact angle and the radius of the capillary tube. **Stage IV:** At this stage, the contact line stops at junction \(b\) of the throat section and the outlet section. The contact angle will gradually change to the one formed by the liquid phase and the outlet section, and the contact line still stays immobile. The morphology of the interface is determined based on the contact angle \(\theta\) and the opening angle \(\beta\) and can be divided into two situations:  
\[\gamma_{\text{sep}}+\gamma_{\text{ow}}\cos\theta=\gamma_{\text{os}} \tag{15}\]6 The third stage of interface advancing. \[\gamma_{\text{sep}}+\gamma_{\text{ow}}\cos\theta=\gamma_{\text{os}} \tag{15}\]7 The change of the capillary force along the axis at the second stage of interface advancing: (**a**) situation 1; (**b**) situation 2. **Situation 1**. When \(0<\theta_{4}+\beta<90^{\circ}\), the interface advances from the position shown in Figure 8 to the position shown in Figure 9a. The capillary force at this stage gradually decreases along the axis, but the value is always positive. Thus, no capillary barrier phenomenon is observed. **Situation 2**. When \(\theta_{4}+\beta\geq 90^{\circ}\), the interface advances from the position shown in Figure 8 to the position shown in Figure 9b. The capillary force along the axis gradually decreases to zero and the value is negative. The capillary barrier is observed. The position of the interface at this stage satisfies the following condition:  
\[\gamma_{\text{sep}}+\gamma_{\text{ow}}\cos\theta=\gamma_{\text{os}} \tag{15}\]8  
The equivalent capillary tube radius \(R\) at this stage can be written as  
\[\gamma_{\text{sep}}+\gamma_{\text{ow}}\cos\theta=\gamma_{\text{os}} \tag{15}\]9  
The capillary barrier phenomenon occurs when \(\theta_{4}+\beta\geq 90^{\circ}\) and the change of the equivalent capillary tube radius \(R\) is shown in Figure 4.0 The equivalent capillary tube radius gradually increases from \(R_{5}\) to \(+\infty\), then gradually decreases to \(R_{6}\). The negative values of the capillary force indicate the occurrence of the capillary barrier phenomenon. The plus or minus value of the capillary force is determined by the contact angle \(\theta\) and the opening angle \(\beta\).