# Modeling the Submergence Depth of Oil Well States and Its Applications
## 4 Conclusions  
(1) In accordance with the fluid-balance equation, a full-pumping state model and a nonfull-pumping state model are constructed. The dimensionality of the parameters other than time and the oil well submergence depth is reduced using the correlation characteristics of the seepage parameters, which effectively reduces the number of parameters in the models of submergence depth for oil wells during pumping and enhances the availability of this model.  
(2) The relationship between the three types of oil wells and the two states are combed, the trends of the changes in the submergence depth in three phases (rising phase, holding phase, and falling phase) for the models of an oil well in the full and nonfull-pumping states are analyzed, and the judgment criteria for the corresponding submergence depth and the law governing the change in the pumping speed are given. On this basis, the criteria for transitioning an oil well from the full-pumping state to the nonfull-pumping state and from the nonfull-pumping state to the full-pumping state are given, which provides a basis for practical applications.  
(3) For the nonlinear-interpolation method and the least squares curve-fitting method, the corresponding univariate characteristic equations are derived, and the unique existence of the solution of the characteristic equations in the (0, 1) interval is proven. The errors of the two numerical solution methods are estimated, the iteration times of the dichotomy are given, the relevant numerical solution tests and result analysis are carried out, and the possible adverse conditions of the sampling data are evaluated to provide a reference for avoiding such conditions.  
(4) For the numerical-solution method of least squares curve-fitting, the computing environment was an Intel(r) CoreTM i7-3770 CPU @ 3.40 GHz. When the number of sampling points was 10, 50, 100, 200, and 500, the time consumption was no more than 0.02, 0.6, 2, 7, and 45 s, respectively.  
Conceptualization, T.L.; methodology, T.L., X.T., L.L. and X.G.; software, X.T. and Y.Z.; validation, X.T. and Y.Z.; formal analysis, L.Z. and X.S.; investigation, X.G. and X.S.; resources, X.G., L.Z. and X.S.; data curation, X.G., L.Z. and X.S.; writing--original draft preparation, T.L. and L.L.; writing--review and editing, L.L. and X.G.; supervision, T.L.; project administration, L.L.; funding acquisition, T.L. and X.S. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by the National Natural Science Foundation of China, grant number 52104032; the Natural Science Basic Research Plan in Shaanxi Province of China, grant number 2020 JQ-787; the Natural Science Basic Research Plan in Shaanxi Province of China, grant number 2022 JQ-671; the 68 th Batch of China Postdoctoral Foundation, grant number 2020M683597; and the Natural Science Basic Research Plan in Shaanxi Province of China, grant number 2019 JM-174.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Conflicts of Interest:** The authors declare no conflict of interest.