# 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  
According to Formulas (5) and (8) of the models, oil wells can be divided into three types: the full-pumping type, the transition type, and the nonfull-pumping type. For the maximum pumping speed \(v_{Max}\) of the pumping unit, after a long period of pumping, when \(h_{E}\geq h_{p}+v_{Max}/\big{(}C_{f}\cdot S\big{)}\), the oil well is the full-pumping type; when \(h_{E}<h_{p}\), the oil well is the nonfull-pumping type; otherwise, when \(h_{p}\leq h_{E}<h_{p}+v_{Max}/\big{(}C_{f}\cdot S\big{)}\), the oil well is the transition type. These three well types correspond to two well states: full-pumping and nonfull-pumping (see Table 1 and Figure 1). When the submergence depth \(h\geq h_{p}\), the oil pumping state of the oil well is the full-pumping state; otherwise, when the submergence depth \(h<h_{p}\) is the nonfull-pumping state. Among them, the oil well of the full-pumping type can only be in the full-pumping state, and the oil well of the nonfull-pumping type can only be in the nonfull-pumping state. In the transition-type oil well, the state of the oil well may be in the state of full pumping, or it may be in the state of nonfull pumping, and the well's state may also transition between the two, depending on the well's seepage and pumping conditions. \begin{table}
\begin{tabular}{c c c} \hline
**Oil Well Type** & **Condition** & **Pumping Model** \\ \hline Full-pumping type & \(h_{p}+\frac{v_{Max}}{C_{f}\cdot S}\leq h_{E}\) & Full-pumping state Model (5) \\  & \(h_{p}\leq h_{0}\) & \\ \hline Transition type & \(h_{p}\leq h_{E}<h_{p}+\frac{v_{Max}}{C_{f}\cdot S}\) & 1. If the size relationship between \(h\) and \(h_{p}\) remains unchanged, keep the original state model. \\  & & Otherwise, take \(h_{0}=h_{p}\) and restart the timing: \\  & & 2. When condition\(<h_{p}\) becomes \(h>h_{p}\), use Model (5); \\  & & 3. When condition \(h>h_{p}\) becomes \(h<h_{p}\), use Model (8). \\ \hline Nonfull-pumping type & \(h_{p}>h_{E}\) & Nonfull-pumping state Model (8) \\  & & \\ \end{tabular}
\end{table} 
Table 1: Relationship between discriminant conditions and pumping models. If the oil well is of the full-pumping type, for the full-pumping state model of Formula (5), \(h\geq h_{p}\). During the pumping process, when \(t=0\), the submergence depth is the initial submergence depth \(h_{0}\). At this time, \(h_{0}\geq h_{p}\). In the model, the submergence depth \(h\) changes with time \(t\), and the comprehensive seepage parameter \(C_{f}\) changes with the oil-storage condition of the reservoir during the oil-pumping process. Normally, \(C_{f}\) changes very slowly, and its change is negligible over a considerable period. Therefore, \(C_{f}\) can generally be treated as a constant. However, after a long period of pumping, when the remaining oil storage in the reservoir decreases by a certain amount and the variation in \(C_{f}\) reaches a certain level, it needs to be recalculated to match the current state of the oil well to the greatest extent. As the pumping process continues, the time \(t\) increases gradually. If the reservoir has sufficient oil storage, the submergence depth \(h\) will be higher than the pump stroke height \(h_{p}\); that is, \(h\geq h_{p}\). The oil well will be in the full-pumping state for a long time. If the oil well is of the nonfull-pumping type, for the nonfull-pumping state model of Formula (8), during the pumping process, the initial submergence depth of the oil well \(h_{0}<h_{p}\), and the change in the parameter \(C_{n}\) in the model is mainly related to the pumping speed \(v\); thus, the change in the submergence depth \(h\) will be more sensitive to \(v\). If the oil storage speed of the oil well is not less than the current oil pumping speed, after a long period, the oil pumping process reaches a certain time \(t\), and the submergence depth \(h\) will gradually increase but will not be greater than \(h_{p}\).