# Modeling the Submergence Depth of Oil Well States and Its Applications
## 3 Results
### Error Analysis
#### 3.2.2 Application Analysis of the Nonlinear-Interpolation Method  
For \(h_{E}=1000\) (m), \(v/(C_{f}\cdot S)=12.5\) (m), and \(C_{f}=0.005\) (h\({}^{-1}\)), according to pumping Formula (5), the corresponding model is:  
\[h=100+887.5\Big{(}1-e^{-0.005t}\Big{)}. \tag{34}\]  
The sampling time nodes are \(t_{1}=24\) (h) and \(t_{2}=96\) (h), and the corresponding submergence depths are \(h_{1}=200.358\) (m) and \(h_{2}=438.330\) (m). The calculation results are shown in Table 2. In this table, \"\(\varepsilon\)\" is the specified error setting for the solution of Formula (16), and \"error \(C_{f}\)\" and \"error \(h_{E}\)\" are the absolute error values corresponding to \(\left|\overline{C_{f}}-C_{f}\right|\) and \(\left|\overline{h}_{E}-h_{E}\right|\), respectively. Since the value of \(\beta\) in Formula (32) is in the order of \(1.9\times 10^{-05}\), it can be seen in Formulas (27) and (28) that both \"error \(C_{f}\)\" and \"error \(h_{E}\)\" satisfy the set requirements. The solution error of the corresponding Formula (16) can be set as needed, but the measurement error cannot be set to meet a particular requirement. Table 3 shows the calculation results obtained with a solution error setting of \(\varepsilon=1\times 10^{-12}\) and a measurement error of \(1\%\). In Table 3, \"relative error \(C_{f}\)\" is \(\left|\overline{C_{f}}-C_{f}\right|\)/\(C_{f}\), and \"relative error \(h_{E}\)\" is \(\left|\overline{h}_{E}-h_{E}\right|\)/\(h_{E}\). For \(h_{i}\pm\Delta h_{i}(i=0,1,2)\), the symbol \"+\" or \"\(-\)\" indicates \(h_{i}+1\%\times h_{i}\) or \(h_{i}-1\%\times h_{i}\) respectively. For example, \"+\" in term \(h_{0}\pm\Delta h_{0}\) indicates \(h_{0}+1\%\times h_{0}=1.01\times h_{0}\), and similarly, \"\(-\)\" indicates \(0.99\times h_{0}\). \"0\" means no error. Formula (16) shows that in the solution equations for the model, when the errors of \(h_{i}(i=0,1,2)\) increase or decrease in the same value, the errors cancel each other out, and the result does not affect the solution \(x\). Therefore, the value of the parameter \(C_{f}\) is not affected, but the value of \(h_{E}\) is affected. Under the assumption that \(\overline{h}_{i}=(1+\delta)h_{i}\), \((i=0,1,2)\), where \begin{table}
\begin{tabular}{c c c} \hline \hline \(\varepsilon\) & **Error \(C_{f}\)** & **Error \(h_{E}\)** \\ \hline \(1.00\times 10^{-01}\) & \(2.31\times 10^{-03}\) & \(7.29\times 10^{+02}\) \\ \(1.00\times 10^{-02}\) & \(1.93\times 10^{-04}\) & \(3.16\times 10^{+01}\) \\ \(1.00\times 10^{-03}\) & \(3.64\times 10^{-05}\) & \(6.22\times 10^{+00}\) \\ \(1.00\times 10^{-04}\) & \(8.73\times 10^{-07}\) & \(1.48\times 10^{-01}\) \\ \(1.00\times 10^{-05}\) & \(2.02\times 10^{-07}\) & \(3.43\times 10^{-02}\) \\ \(1.00\times 10^{-06}\) & \(2.17\times 10^{-08}\) & \(3.68\times 10^{-03}\) \\ \(1.00\times 10^{-07}\) & \(2.08\times 10^{-09}\) & \(3.52\times 10^{-04}\) \\ \(1.00\times 10^{-08}\) & \(3.27\times 10^{-10}\) & \(5.54\times 10^{-05}\) \\ \(1.00\times 10^{-09}\) & \(2.03\times 10^{-11}\) & \(3.44\times 10^{-06}\) \\ \(1.00\times 10^{-10}\) & \(1.14\times 10^{-12}\) & \(2.00\times 10^{-07}\) \\ \(1.00\times 10^{-11}\) & \(1.15\times 10^{-13}\) & \(2.00\times 10^{-08}\) \\ \(1.00\times 10^{-12}\) & \(1.36\times 10^{-14}\) & \(2.30\times 10^{-09}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter error of solving equations. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(h_{0}\pm\Delta h_{0}\) & \(h_{1}\pm\Delta h_{1}\) & \(h_{2}\pm\Delta h_{2}\) & **Relative Error \(C_{f}\)** & **Relative Error \(h_{E}\)** \\ \hline + & + & + & \(2.71\times 10^{-12}\) & \(9.88\times 10^{-03}\) \\ \(-\) & \(-\) & \(2.71\times 10^{-12}\) & \(9.88\times 10^{-03}\) \\ \(0\) & \(+\) & \(-\) & \(2.05\times 10^{-01}\) & \(1.29\times 10^{-01}\) \\ \(0\) & \(-\) & \(+\) & \(2.02\times 10^{-01}\) & \(1.92\times 10^{-01}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Measurement error distribution and calculation results.