## 3 Results and Discussion
### Impact of Drilling Fluid
#### 3.2.2 Impact of Flow Rate  
Rotating drill strings are subjected to frictional torque from the wellbore wall, while the annular fluid influences the contact between the drill string and the wellbore. Figure 23 illustrates the effect of drilling fluid flow rate on frictional torque over a 20 m length. As the flow rate increases, the frictional torque significantly decreases. Compared to scenarios without drilling fluid, the frictional torque between the drill string and the wellbore exhibits a maximum reduction of 35.55%. This indicates that the drilling fluid provides lubrication between the drill string and the wellbore wall, and increasing the flow rate effectively reduces interfacial friction. However, excessive flow rates may elevate pump pressure and intensify wellbore wall erosion. Therefore, maintaining wellbore stability should be prioritized when enhancing flow rates.  
Figure 21: Drill string modal frequencies under drilling fluid conditions.  
Figure 22: Effect of pump pressure on drill string vibration frequencies.  
In this study, the specific frictional energy dissipation per meter (SFED, J/m) is defined as the frictional energy dissipated per unit length of the drill string, calculated by Equation (23). The total rotational energy expenditure (TREE, kJ) is defined as the cumulative mechanical energy required to driving the drill string rotation, calculated by Equation (24). Combining Figure 23 and Equation (23), the SFED without drilling fluid is calculated as 46.965 J/m, and the TREE is 514.64 J. Figure 24 illustrates the variation trends of SFED and TREE with flow rate over a 1 s period. At a drilling fluid flow rate of 30 L/min, SFED and TREE decrease to 39.25 J/m and 403.66 J, respectively, representing reductions of approximately 16.43% and 21.56% compared to the no-fluid scenario. With an increasing flow rate, SFED and TREE exhibit similar declining trends. At 70 L/min, SFED decreases by 32.79% and TREE by 34.6%, demonstrating that elevated flow rates directly reduce frictional torque between the drill string and wellbore wall while enhancing drill string stability.  
\[\dot{E}_{f}=\frac{1}{L}\int_{I_{1}}^{I_{2}}T_{\text{fric}}\omega dt \tag{23}\]  
\[\text{TREE}=\frac{1}{L}\int_{I_{1}}^{I_{2}}T_{\text{drive}}\omega dt \tag{24}\]  
where \(T_{\text{fric}}\) and \(T_{\text{drive}}\) represent the frictional torque and the total driving torque, respectively, \(L\) is the length of the drill string, and \(\omega\) is the angular velocity of the drill string.  
Figure 25 shows the distribution patterns of annular velocity under varying drilling fluid flow rates. It can be observed that increasing the flow rate causes negligible spatial  
Figure 23: The effect of flow rate of drilling fluid on friction torque.  
Figure 24: SFED and TREE versus drilling fluid flow rate.  
displacement of the drill string. The circumferential distribution of annular velocity exhibits asymmetry: While the overall annular velocity increases with higher flow rates and radial velocity distribution remains relatively uniform, the variation amplitude of circumferential velocity remains limited. A higher flow rate implies increased axial velocity, which, under constant rotational speed and drilling fluid kinematic viscosity, weakens circumferential momentum transfer due to enhanced axial flow dominance. This directly reduces the centrifugal effect of drill string rotation on cuttings, hindering their radial migration. The diminished circumferential transport of cuttings elevates the probability of cuttings' bed formation, thereby compromising drill string stability. To mitigate this, adjustments to rotational speed and drilling fluid density can be implemented to maintain stability.