## 4 Conclusions  
This work proposed a method to reduce friction and improve the ROP of slide drilling. The method applies axial vibration on the drill-string at the surface. A mathematical vibration model was established to simulate the load transfer in the drill-string, and this paper discussed the advantages, influencing factors, and feasibility of the drill-string axial vibration at the surface technology.  
Numerical modeling results show that the motion of the drill-string in the conventional drilling process has stick-slip features, and the WOB is fluctuant with certain frequencies. If the friction force equals the weight of the upper drilling string, no weight can be transmitted to the bit. The conventional method of calibrating the WOB using a weight indicator gauge is not suitable when the ROP is slow. The proposed technology of applying axial vibration to the drill-string at the surface has been proven to be highly effective. This is supported by the significant increase in the WOB resulting from the release of friction after the application of the vibration. The amplitude and frequency of the exciting forces are the main influence factors of the effectiveness of the application of this technology and the safety of the drill-string. The horizontal well section exhibits a high RMS acceleration at low frequencies, whereas the vertical well section demonstrates a high RMS acceleration at high frequencies. There is an optimal frequency for a given case (10 Hz in this paper). This means that the conventional manual pick-up and slack-off at the surface with a high amplitude and low frequency has little friction reduction effect. Safety evaluation results indicate that the drill-string is in a safe state under most of the exciting parameters.  
Some simplification and issues should be considered in future research, such as:  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Exciting Force/kN** & **0** & **0.5** & **1** & **5** & **10** & **20** & **25** & **30** \\ \hline \multirow{4}{*}{100} & \multirow{4}{*}{0–4200} & \multirow{4}{*}{0–4200} & \multirow{4}{*}{0–1600} & \multirow{4}{*}{0–1750} & \multirow{4}{*}{0–4450} & \multirow{4}{*}{0–1920} \\  & & & & & & 1750 & \(\sim\)2450 & \(\sim\)4200 & \(\sim\)2330 \\ \cline{5-8}  & & & & & \(\sim\)3530 & \(\sim\)4100 & & \(\sim\)4170 \\ \cline{5-8}  & & & & & \(\sim\)4170 & \(\sim\)4200 & & \(\sim\)4200 \\ \hline \multirow{4}{*}{200} & \multirow{4}{*}{0–4200} & \multirow{4}{*}{0–1230} & \multirow{4}{*}{0–1520} & \multirow{4}{*}{0–1660} & \multirow{4}{*}{0–1920} & \multirow{4}{*}{0–1415} & \multirow{4}{*}{0–1865} & \multirow{4}{*}{0–1895} \\  & & & & & & & & \\  & & & &  
(a) Utilizing the soft-string model in the mathematical model accommodates accurate results for wells with less curvature on site. However, it may lead to significant discrepancies in high curvature intervals.  
(b) During the drilling process, the drill-string is composed of joints and centralizers. The borehole wall is not uniform in shape and the gap between the drill-string and the borehole wall cannot be disregarded. These factors will inevitably impact the principles of friction reduction and load transfer. Therefore, the actual conditions should be considered in future research.  
[PERSON] conceived and designed the program, and [PERSON] supervised the study and debugged the program. [PERSON] assisted in the development of the algorithm. [PERSON] analyzed the computed results and wrote the paper. All authors have read and agreed to the published version of the manuscript.  
This paper is sponsored by the Hubei Key Laboratory of Oil and Gas Drilling and Production Engineering (Yangtze University) (Grant No. YQZC202208) (X.W.), and Hubei Provincial Science and Technology Agency (Grant No. 2019 CFA093) (Z.F.).  
The data used to support the findings of this study are available from the corresponding author upon request.  
The authors also gratefully acknowledge the financial support of the CNPC Bohai Drilling Corporation.  
The authors declare no conflict of interest.