# Modeling the Submergence Depth of Oil Well States and Its Applications
## 2 Materials and Methods
### Submergence Depth Model Analysis
#### 2.2.1 Trends of Change in the Submergence Depth
If the oil well storage speed is always lower than the current pumping speed, the submergence depth \(h\) will gradually decrease with increasing time \(t\). When \(h\) is low to a certain extent, it is necessary to stop pumping. If the oil well is of the transition type, it may be in a full-pumping state or a nonfull-pumping state, which is determined by the oil well's permeability and pumping speed. The current state of the oil well needs to be determined according to the relationship between \(h\) and \(h_{p}\). During the pumping process, when \(h\geq h_{p}\), the oil well is in a full-pumping state, corresponding to the model of the full-pumping state in Formula (5). For the transition-type oil well in the full-pumping state, if the oil storage speed of the oil well is less than the current pumping speed, the submergence depth \(h\) will gradually decrease with increasing time \(t\). After reaching a certain time, \(h<h_{p}\), the oil well will transition from the full-pumping state to the nonfull-pumping state. At this time, the corresponding Formula (8) is the nonfull-pumping state model. For the transition-type oil well in the state of nonfull pumping, if the oil storage speed of the well is not less than the current pumping speed, after a long pumping time, the submergence depth \(h\) will gradually increase until \(h\geq h_{p}\), and the well will again transition from the nonfull-pumping state to the full-pumping state. When the pumping state changes, that is, from the full-pumping state to the nonfull-pumping state, or from the nonfull-pumping state to the full-pumping state, its initial submergence depth \(h_{0}\) is replaced by \(h_{p}\), and the time \(t\) starts counting from 0. Regardless of whether a well is in the full or nonfull-pumping state, for a given pumping speed, its submergence depth is in one of three phases: the rising phase, the  
Figure 1: Relationship between oil well types and pumping states. holding phase, or the falling phase. For the full-pumping model, these three phases (see Figure 2) can be expressed as follows:  
\[ \begin{array}{ll}\frac{dh(t)}{dt}&=C_{f}\Big{(}h_{E}-\frac{v}{C_{f}S}-h_{0} \Big{)}e^{-C_{f}t}\\ &=\left\{\begin{array}{c}>0\;v<C_{f}\cdot S(h_{E}-h_{0})\\ =0\;v=C_{f}\cdot S(h_{E}-h_{0})\\ <0\;v>C_{f}\cdot S(h_{E}-h_{0})\end{array}\right..\end{array}  \tag{9}\]  
When \(dh(t)/dt>0\), the submergence depth is in the rising phase, and when \(dh(t)/dt=0\), the submergence depth is in the holding phase; otherwise, the submergence depth is in the falling phase. Similarly, for the nonfull-pumping model, the three phases (see Figure 3) can be expressed as follows:  
\[ \begin{array}{ll}\frac{dh(t)}{dt}&=C_{n}\Big{(}h_{E}-h_{E}\frac{v}{C_{n}S \cdot S\cdot h_{p}}-h_{0}\Big{)}e^{-C_{n}t}\\ &=\left\{\begin{array}{c}>0\;v<C_{f}\cdot S(h_{E}-h_{0})\frac{h_{p}}{h_{0}} \\ =0\;v=C_{f}\cdot S(h_{E}-h_{0})\frac{h_{p}}{h_{0}}\\ <0\;v>C_{f}\cdot S(h_{E}-h_{0})\frac{h_{p}}{h_{0}}\end{array}\right..\end{array}  \tag{10}\]  
Figure 3: Schematic diagram of three-phase trends of the submergence depth in the nonfull-pumping model. Figure 2: Schematic diagram of three-phase trends of the submergence depth in the full-pumping model. The trend of change in the submergence depth is determined by the comprehensive seepage parameter \(C_{f}\) of the oil well and the pumping speed \(v\) of the pumping unit. Especially for the nonfull-pumping model, because the oil well has fewer liquid reserves, the variation trend of submergence depth highly depends on the pumping speed.