## 4 Conclusions  
Excessive friction, fluid-structure interaction vibrations, and energy consumption in deep shale gas horizontal wells severely restrict their efficient and sustainable develop  
Figure 38: Distribution patterns of annular velocity under varying friction coefficients: (**a**) \(\mu\) = 0.07; (**b**) \(\mu\) = 0.15; (**c**) \(\mu\) = 0.3; (**d**) \(\mu\) = 0.5.  
ment. To address these challenges, this study developed and experimentally validated a bidirectional FSI model for a 76 mm slim-hole drill string-drilling fluid system. Compared to explicit dynamic analysis, the recursive algorithm developed herein reduces the computation time by 59.3% while maintaining an equivalent accuracy. The key findings are summarized as follows:  
1. Drilling fluid lowers the drill string natural frequency by 20-25%, with low-order modes showing lateral coupled vibrations. An increased pump pressure (0.5-8 MPa) further reduces the second-order frequency by up to 46.5%, requiring monitoring of the low-order modes to prevent resonance.
2. Dominant frictional energy dissipation occurs from persistent drill string-wellbore contact (lower right). Eccentric annular fluid pressure significantly reduces this friction; under benchmark conditions, SFED and TREE decreased by 24.28% and 27.41%, respectively. While most parameter increases reduce SFED, TREE slightly rises if viscosity exceeds 2.5 \(\times\) 10\({}^{-5}\) m\({}^{2}\)/s, leading to laminar flow detrimental to wellbore cleaning.
3. Horizontal drill string displacement favors the right side, but a kinematic viscosity \(>\) 2.5 \(\times\) 10\({}^{-5}\) m\({}^{2}\)/s causes anomalous leftward shifts and introduces radial velocity gradients. Inherent circumferential flow asymmetry aids wellbore cleaning.
4. A critical friction coefficient of 0.15 stabilizes the drill string centrally. Excessive lateral displacement causes asymmetric cuttings' accumulation, impacting fracturing. Managing viscosity and drag reducers based on subcritical or supercritical friction is key to controlling deflection and cuttings buildup.
5. High-Young's-modulus formations increase drill string vibrations, the lateral coupled resonance risk, impact forces, and failure likelihood.  
Despite its computational efficiency, the recursive algorithm's robustness and convergence under challenging conditions (e.g., extreme deformations, complex contacts) require further evaluation. The current FSI framework, while foundational for friction-vibration-energy coupling, neglects shale anisotropy, gas influx, and cuttings interactions. Future work will enhance predictive capabilities by (a) integrating shale heterogeneity effects using well log data and (b) coupling advanced viscosity and cuttings' transport models. Nevertheless, the model's validated thresholds (e.g., viscosity 1-2.5 \(\times\) 10\({}^{-5}\) m\({}^{2}\)/s, critical friction 0.15) remain directly applicable to 76 mm slim-hole designs.  
Conceptualization, D.Z.; Data Curation, X.G. and Z.G.; Formal Analysis, X.G. and Y.Z.; Funding Acquisition, D.Z.; Investigation, X.G. and S.W.; Methodology, X.G. and D.Z.; Project Administration, D.Z.; Resources, D.Z.; Software, X.G. and Z.G.; Validation, Y.C. and Z.G.; Writing--Original Draft, X.G.; Writing--Review and Editing, Y.Z. and X.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Key Special Project of the National Key Research and Development Program of China (Grant No. SQ2024 YFC3200124).  
Not applicable.  
The original contributions presented in the study are included in the article. Further inquiries can be directed to the authors.  
[PERSON] was employed by the CRRC Dalian Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.