# Modeling the Submergence Depth of Oil Well States and Its Applications
## 3 Results
### Submergence Depth Models' Numerical Solutions
#### 3.1.1 Nonlinear-Interpolation Method  
(1) Full-pumping state  
It is considered that \(x=e^{-C_{f}}\in(0,1)\). For \(h\) in the rising phase, the sampling time points \(0=t_{0}<t_{1}<t_{2}\) and the corresponding submergence depth \(h_{p}<h_{0}<h_{1}<h_{2}\) satisfy the following:  
\[h_{i}=h_{0}+\left(h_{E}-\frac{v}{c_{f}\cdot S}-h_{0}\right)\left(1-x^{i} \right),\,(i=1,2). \tag{15}\]  
Thus, it holds that \((h_{1}-h_{0}){\sum_{i=0}^{t_{2}-1}x^{i}}=(h_{2}-h_{0}){\sum_{i=0}^{t_{1}-1}x^{ i}}\). The corresponding nonlinear-interpolation method's characteristic function of the full-pumping state is as follows:  
\[f(x)=(h_{1}-h_{0}){\sum_{i=0}^{t_{2}-1}x^{i}}-(h_{2}-h_{0}){\sum_{i=0}^{t_{1}- 1}x^{i}}. \tag{16}\]  
Thus, the following expressions hold:  
\[f(0)=h_{1}-h_{2}<0, \tag{17}\]  
\[ \begin{array}{ll}f(1)&=(h_{1}-h_{0})t_{2}-(h_{2}-h_{0})t_{1}\\ &=(h_{1}-h_{0})(t_{2}-t_{1})-(h_{2}-h_{1})(t_{1}-t_{0})\end{array} . \tag{18}\]  
In Formula (15), the submergence depth \(h_{i}\) and the solution \(x\) satisfy \(\frac{h_{2}-h_{1}}{h_{1}-h_{0}}=\frac{x^{t_{2}}-x^{t_{1}}}{x^{t_{1}}-x^{t_{0}}}\); therefore:  
\[f(1)=\frac{(h_{1}-h_{0})(t_{2}-t_{1})(t_{1}-t_{0})}{x^{t_{0}}-x^{t_{1}}}\bigg{(} \frac{x^{t_{2}}-x^{t_{1}}}{t_{2}-t_{1}}-\frac{x^{t_{1}}-x^{t_{0}}}{t_{1}-t_{0 }}\bigg{)}. \tag{19}\]  
It is assumed that \(F(t)=x^{t}\). Then, according to the Lagrange mean value theorem, there is a \(\tau_{1}\in(t_{0},t_{1})\) and a \(\tau_{2}\in(t_{1},t_{2})\) such that:  
\[f(1)=\frac{(h_{1}-h_{0})(t_{2}-t_{1})(t_{1}-t_{0})}{x^{t_{0}}-x^{t_{1}}}(Ft( \tau_{2})-Ft(\tau_{1})) \tag{20}\]  
Since \(Ft\prime(t)=x^{t}[\ln(x)]^{2}\) and \(\frac{(h_{1}-h_{0})(t_{2}-t_{1})(t_{1}-t_{0})}{x^{t_{0}}-x^{t_{1}}}>0\), for \(t\geq 0\), \(Ft(t)\) is a monotonically increasing function. Thus, \(f(1)>0\). Therefore, there is a solution to the characteristic equation \(f(x)=0\), \(x\in(0,1)\). The following proves that for \(\Delta x>0\), \(x\in(0,1)\), and \(x+\Delta x\in(0,1)\), when \(f(x)\geq 0\), the function is monotonically increasing. \[ \begin{array}{ll}f(x+\Delta x)&=(h_{1}-h_{0}){\sum_{i=t_{1}}^{t_{2}-1}}(x+ \Delta x)^{j}-(h_{2}-h_{1}){\sum_{j=0}^{t_{1}-1}}(x+\Delta x)^{j}\\ &>\left(1+\frac{\Delta x}{x}\right)^{t_{1}}\!\!\left[(h_{1}-h_{0}){\sum_{j=t_ {1}}^{t_{2}-1}x^{j}}-(h_{2}-h_{1}){\sum_{j=0}^{t_{1}-1}x^{j}}\right]\\ &=\left(1+\frac{\Delta x}{x}\right)^{t_{1}}\!\!f(x)\geq f(x)\end{array} . \tag{21}\]  
Therefore, \(f(x)\) crosses zero at a single point in \(x\in(0,1)\), so the corresponding characteristic equation \(f(x)=0\) has a unique solution \(x\in(0,1)\)[17]. Then,  
\[\left\{ \begin{array}{ll}C_{f}=-ln(x)\\ h_{E}=h_{0}+\frac{h_{1}-h_{0}}{1-x^{t_{1}}}+\frac{v}{C_{f}\cdot S},(1\leq i\leq 2 )\end{array} .\right. \tag{22}\]  
The solution given in Formula (22) is also suitable for \(h\) in the falling phase. (2) Nonfull-pumping stateSimilar to the full-pumping state, for \(x=e^{-C_{n}}\in(0,1)\) and \(h\) in the rising phase, the sampling time points \(0=t_{0}<t_{1}<t_{2}\) and the corresponding submergence depth \(0\leq h_{0}<h_{1}<h_{2}<h_{p}\) satisfy:  
\[h_{i}=h_{0}+\bigg{[}h_{E}\bigg{(}1-\frac{v}{C_{n}\cdot S\cdot h_{p}}\bigg{)}-h_{ 0}\bigg{]}\big{(}1-x^{-t_{i}}\big{)},(i=1,2). \tag{23}\]  
The corresponding nonfull-pumping state characteristic function is the same as the full-pumping state characteristic function, that is, Formula (16). Similarly, the solution \(x\in(0,1)\) of the corresponding characteristic equation leads to the following:  
\[\left\{ \begin{array}{l}C_{f}=-\big{(}ln(x)+\frac{v}{S\cdot h_{p}}\big{)}\\ h_{E}=\Big{(}h_{0}+\frac{h_{i}-h_{0}}{1-x^{t_{i}}}\Big{)}+\Big{(}1+\frac{v}{C _{f}\cdot S\cdot h_{p}}\Big{)},(1\leq i\leq 2)\end{array} \right.. \tag{24}\]  
The solution given in Formula (24) also applies to \(h\) in the falling phase. Regardless of a full-pumping state or the nonfull-pumping state, the submergence depth tester has a certain error (<1%). To avoid error amplification, the time between every two sampling points should not be too short.