# Modeling the Submergence Depth of Oil Well States and Its Applications
## 2 Materials and Methods
### Submergence Depth Model Analysis
#### 2.2.2 Pumping State Transition  
During the pumping process, the oil well of the full-pumping type is in the full-pumping state, and the oil well of the nonfull-pumping type is in the nonfull-pumping state. However, for the oil well of the transition type, the state of the oil well may change during the pumping process, from the full-pumping state to the nonfull-pumping state or from the nonfull-pumping state to the full-pumping state, depending on the well's comprehensive seepage parameter \(C_{f}\) and pumping speed \(v\).  
Figure 4 shows a schematic diagram of the transition of an oil well submergence depth from the full-pumping state to the nonfull-pumping state. Combining Formulas (5) and (9), it can be found that this state transition should satisfy the following requirements:  
\[\left\{\begin{array}{l}h_{E}-\frac{v}{C_{f}\cdot S}<h_{p}\leq h_{0}\\ t>-\frac{1}{C_{f}}ln\left(\frac{C_{f}\cdot S\left(h_{E}-h_{p}\right)-v}{C_{f} \cdot S\left(h_{E}-h_{0}\right)-v}\right)\end{array}\right.. \tag{11}\]  
The denominator of Formula (11) is \(C_{f}\cdot S(h_{E}-h_{0})-v\
eq 0\); otherwise, the submergence depth of the full-pumping state model in Formula (5) will have a constant value of \(h_{0}\), and the pumping state will not change.  
It is assumed that \(v_{c}=1+v/\left(C_{f}\cdot S\cdot h_{p}\right)\); then, an oil well transitioning from the nonfull-pumping state to the full-pumping state should satisfy the following conditions:  
\[\left\{\begin{array}{l}\frac{h_{E}}{v_{c}}\geq h_{p}>h_{0}\\ t\geq-\frac{1}{C_{f}\cdot v_{c}}ln\left(\frac{h_{E}-h_{p}\cdot v_{c}}{h_{E}-h_{0 }\cdot v_{c}}\right)\end{array}\right.. \tag{12}\]  
Similarly, the denominator is \(h_{E}-h_{0}\cdot v_{c}\
eq 0\); otherwise, the submergence depth of the nonfull-pumping state model in Formula (8) will have a constant value of \(h_{0}\), and the pumping state of the oil well will not change. The submergence depth of the oil well from the nonfull-pumping state to the full-pumping state is shown in Figure 5.  
Figure 4: Schematic diagram of transition from the full-pumping state to the nonfull-pumping state.  
Regarding the parameters \(h_{E}\) and \(C_{f}\) of the oil well model in the full-pumping state (Formula (5)), when the oil well transitions into the nonfull-pumping state, the current submergence depth satisfies \(h<h_{p}\); the parameters \(h_{E}\) and \(C_{f}\) do not change, and at this time, the initial submergence depth \(h_{0}\) is replaced by \(h_{p}\). The state model Formula (5) is replaced by Formula (8), and the time restarts. The transition of an oil well from the nonfull-pumping state to the full-pumping state is similar to the transition process described above.  
The models (Formulas (5) and (8)) of the oil well's pumping submergence depth can also be applied to include only the submergence depth variation of pure oil storage or the pure oil pumping process. In fact, the pure oil storage submergence depth model and the pure oil pumping submergence depth model are special cases of oil well pumping submergence depth models. According to the oil well pumping submergence depth constructed above, the oil well pure storage model and the oil well pure pumping model can be obtained:  
(1) When the oil well stops pumping; that is, \(v=0\), the pure oil storage model [12] is as follows:  
\[h=h_{0}+(h_{E}-h_{0})\Big{(}1-e^{-C_{f}t}\Big{)}. \tag{13}\]  
Obviously, Formula (13) is a special case of Formulas (5) and (8) in the case of \(v=0\).  
(2) Under the assumption that the permeability coefficient of the oil well is \(K=0\), the pure oil pumping model [16] is as follows:  
\[h=\left\{\begin{array}{cc}h_{0}-\frac{v}{5}t,&h_{0}\geq h_{p},&t\leq\frac{S }{v}\ \left(h_{0}-h_{p}\right)\\ h_{0}e^{-\frac{S}{h_{p}}t},&h_{0}<h_{p}\end{array}\right.. \tag{14}\]  
Similarly, Formula (14) is a special case of Formulas (4) and (7) in the case of \(K=0\). In reality, oil wells that satisfy Formula (14) hardly exist, and they are suitable for the liquid pumping process of a straight vessel.