# 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 pressure difference across the static interface can be calculated based on the Young-Laplace equation and the surface energy equation and the relevant equations are given by  
\[\Delta p=\gamma_{\text{ow}}(k_{1}+k_{2}) \tag{14}\]  
\[\gamma_{\text{sep}}+\gamma_{\text{ow}}\cos\theta=\gamma_{\text{os}} \tag{15}\]  
where \(\Delta p\) is the pressure difference across the interface, Pa; \(\gamma_{\text{ow}}\) is the interfacial tension of the oil-water interface, N; \(k_{1}\) and \(k_{2}\) are components of the curvature of the oil-water interface in two perpendicular directions, m\({}^{-1}\); \(\gamma_{\text{sep}}\) is the interfacial tension of the water-solid interface, N; \(\gamma_{\text{os}}\) is the interfacial tension of the oil-solid interface, N.  
It is worth noting that the contact angle \(\theta\) is a constant during the process of oil-water two-phase flow. However, in order to derive expressions of capillary force in all the situations, the \(\theta\) adopted in the figure can be different. At the same stage, different symbols of \(\theta\) are used if there are different situations about the relation between contact angle \(\theta\) and opening angle \(\beta\). Otherwise, \(\theta\) is used. The advancing of the oil-water interface in the physical capillary barrier model can be divided into five stages. **Stage I**: The solid-liquid contact line advances along the wall of the capillary channel from the leftmost end to position \(a\), and the process is shown in Figure 4. The position of the interface at this stage satisfies the following condition:  
\[0\leq x<L-r\frac{1+\sin(\beta-\theta)}{\cos(\beta-\theta)} \tag{16}\]  
At stage I, with the continuous advancing of the interface from left to right, the radius of the capillary barrier tube gradually decreases, and the capillary force continues to increase. Here, the equivalent radius is the curvature radius of the oil-water interface. Thus, the equivalent capillary tube radius can be derived from the geometrical relationship of the capillary tube, which can be written as  
\[R=\frac{(L-x)\sin\beta+r\cos\beta}{\cos\theta+\sin\beta} \tag{17}\]  
Due to the centrosymmetric property of the capillary barrier tube, the radii of curvature in two perpendicular directions are identical and can be given by  
\[\kappa_{1}=\kappa_{2}=1/R \tag{18}\]  
The pressure difference across the interface can be calculated by  
\[\Delta p=\frac{2\gamma_{\text{lg}}}{R} \tag{19}\]  
Figure 4: The first stage during the process of interface advancing. **Stage II**: The solid-liquid contact line stays at position \(a\), but the contact angle initially formed by the liquid phase and the wall of the inlet section will gradually change to the one formed by the liquid phase and the wall of the throat section with the advance of the interface from left to right. Finally, a stable contact angle will be formed, and an immobile contact line will be observed, but the morphology of the interface is continuously changing; the initial state and the final state are shown in Figures 5 and 6, respectively. The position of the interface at stage II satisfies the following condition:  
\[L-r\frac{1+\sin(\beta-\theta_{2})}{\cos(\beta-\theta_{2})}\leq x<L-r\frac{1- \sin\theta_{2}}{\cos\theta_{2}} \tag{20}\]  
The equivalent capillary tube radius \(R\) at this stage can be written as  
\[R=\frac{\left((L-x)^{2}+r^{2}\right)}{2(L-x)} \tag{21}\]  
The change of the capillary pressure along the axis at this stage can be divided into two situations:  
**Situation 1**. When \(\beta>\theta_{2}\), the change of the capillary force is shown in Figure 7a.