# Modeling the Submergence Depth of Oil Well States and Its Applications
## 2 Materials and Methods
### Submergence Depth Model Construction
#### 2.1.1 Oil Well Full-Pumping State Model  
In the full-pumping state, the pumping speed corresponding to the pumping unit speed is \(v\) (m\({}^{3}\)/s), according to fluid balance Formula (1):  
\[\frac{KA}{\mu L}(P_{E}-P)-v=5\frac{dh}{dt}, \tag{2}\]  
where \(K\) is the permeability coefficient (m\({}^{2}\)), \(A\) is the cross-sectional area of the liquid passing through the rock (m\({}^{2}\)), \(\mu\) is the viscosity of the liquid (Pa-s), \(L\) is the length of the rock (m), \(P_{E}\) is the reservoir pressure (Pa), \(P\) is the oil well pressure (Pa), and \(S\) is the cross-sectional area of the oil well (m\({}^{2}\)).  
The density of the oil is denoted by \(\rho\) (kg/m\({}^{3}\)), and the acceleration due to gravity is denoted by \(g\) (N/kg). According to the liquid pressure formula \(P=\rho gh\), Formula (2) can be transformed into  
\[\frac{KA\rho g}{\mu L}(h_{E}-h)-v=S\frac{dh}{dt}, \tag{3}\]  
where \(h_{E}\) is the reservoir height (m) of the oil well corresponding to the reservoir pressure \(P_{E}\), hereafter referred to as the maximum submergence depth; and \(h\) represents the current submergence depth of the oil well (m).  
Taking the comprehensive seepage parameter of the oil well \(C_{f}=\frac{KA\rho g}{\mu LS}\), Formula (3) can be simplified as follows:  
\[\frac{dh}{dt}+C_{f}\cdot h=C_{f}\cdot h_{E}-\frac{v}{S}. \tag{4}\]  
For a given rotational speed of the pumping unit, solving the differential equation in Formula (4) to obtain the model of the oil well in the full-pumping state yields the following:  
\[h=h_{0}+\left(h_{E}-\frac{v}{C_{f}\cdot S}-h_{0}\right)\left(1-e^{-C_{f}t} \right), \tag{5}\]  
where \(h_{0}\) is the initial submergence depth (m); that is, the submergence depth of the oil well when \(t=0\).  
The term \(h_{p}\) denotes the pump stroke height (m). Since Formula (5) (also called full-pumping state Model (5)) applies to the full-pumping state, for any time \(t\), it meets the following requirements: \(h\geq h_{p}\). Then, the conditions \(h_{E}\geq h_{p}\) and \(h_{0}\geq h_{p}\) are met. The submergence depth \(h\) in the model changes with time. In addition to the gravity acceleration \(g\) and the cross-sectional area \(S\) of the oil well, other parameters will change with the pumping time of the oil well pump. It is not difficult to see that the comprehensive seepage parameter \(C_{f}\) in the model will change with the other parameters. Defined and used in this way, parameter \(C_{f}\) also includes other factors that affect trace amounts that are not reflected in the formula. Therefore, in the pumping process, the parameter \(C_{f}\) will also change. However, under normal circumstances, the change in parameter \(C_{f}\) is relatively slow, and the amount of change is small, which indicates that a certain law occurs. The degree of its change mainly depends on the remaining oil reserves of the reservoir and the oil-pumping capacity of the oil well. The greater the ratio of the reservoir's residual volume to pumping volume, the slower the change speed of parameter \(C_{f}\). After pumping for a long time, the value of parameter \(C_{f}\) will inevitably change. Therefore, when the change in \(C_{f}\) reaches a certain degree, its value must be obtained again through measurement or calculation in practical applications.  
According to Formula (5), when the oil well is in the full-pumping state and starts pumping, the corresponding submergence depth is the initial submergence depth \(h_{0}\) when the time \(t\) is 0. Assuming that the pumping speed \(v=0\), when the time \(t\) approaches infinity, the submergence depth tends to be the largest; that is, the submergence depth \(h\) tends to \(h_{E}\). Assuming that the pumping speed \(v>0\), when the time \(t\) is great enough, \(h_{E}\) will drop to \(h_{p}\). If pumping is continued, the oil well will enter the nonfull-pumping state.