## 3 Results and Discussion
### Impact of Drilling Fluid
#### 3.2.4 Impact of Density  
Figure 31 illustrates the distribution patterns of annular velocity under varying drilling fluid densities. The annular velocity remains largely unchanged with increasing density, indicating that density exerts no significant influence on annular velocity distribution. Conversely, Figure 32 reveals that increased density elevates overall annular pressure, generating localized high-pressure zones (up to 2500 Pa) in the right annular gap, primarily induced by drill string rotation. Furthermore, the radial pressure gradient in the right annular gap intensifies with density, exhibiting lower pressures near the drill string and wellbore walls but peak values in the intermediate zone. This phenomenon is attributed to enhanced centrifugal forces on the drilling fluid under higher densities, which drive excessive fluid accumulation in narrow channels, thereby elevating pressure. While an elevated density prevents cuttings' deposition and wellbore collapse, the localized high-pressure zones heighten the risk of wellbore leakage. Consequently, intensified annular pressure monitoring is recommended when employing high-density drilling fluids. Figure 31: Distribution patterns of annular velocity under varying density of drilling fluid: (a) 1000 kg/m\({}^{3}\); (b) 1200 kg/m\({}^{3}\); (c) 1400 kg/m\({}^{3}\). Figure 30: SFED and TREE versus kinematic viscosity of drilling fluid. Figure 33 details the variation in frictional torque with drilling fluid density and its reduction rate relative to the condition without drilling fluid (approximately 15 N\(\cdot\)m, indicated by the dashed gray line). It can be observed that as the drilling fluid density increases from 1000 kg/m\({}^{3}\) to 1400 kg/m\({}^{3}\), the actual frictional torque significantly decreases from approximately 10.64 N\(\cdot\)m to about 8.14 N\(\cdot\)m. Although the torque continuously decreases, the marginal benefit of this reduction (i.e., the amount of torque decrease per unit increase in density) slightly decelerates. Correspondingly, the reduction rate of frictional torque, compared to the no-drilling-fluid scenario, substantially increases from about 28% to approximately 45.58%. This significant friction reduction effect is primarily attributed to the superior lubrication and support provided by higher-density drilling fluids: on the one hand, they help form a more stable lubricating film with a higher load-bearing capacity between the drill string and the wellbore wall, effectively isolating metallic contact; on the other hand, an increase in drilling fluid density enhances the buoyant force on the drill string, thereby reducing the normal force exerted by the drill string against the wellbore wall, which directly lowers frictional torque. Additionally, higher-density drilling fluids typically suspend and remove fine abrasive particles more effectively, improving lubrication conditions. The reduction in frictional torque is directly reflected in the SFED shown in Figure 34, which decreases from approximately 35.56 J/m at 1000 kg/m\({}^{3}\) to about 27.0 J/m at  
Figure 33: The effect of density of drilling fluid on friction torque. Figure 32: Distribution patterns of annular pressure under varying density of drilling fluid: (a) 1000 kg/m\({}^{3}\); (b) 1200 kg/m\({}^{3}\); (c) 1400 kg/m\({}^{3}\). 1400 kg/m\({}^{3}\), a reduction of about 24.07%. The TREE also decreases with an increasing drilling fluid density, from about 373.57 J/m (at 1000 kg/m\({}^{3}\)) to approximately 358.9 J/m (at 1400 kg/m\({}^{3}\)), a reduction of about 3.93%. The reason for TREE's smaller reduction magnitude compared to that of SFED and frictional torque may be that TREE not only includes the SFED component directly related to wellbore friction but also encompasses energy consumption from internal shearing of the drilling fluid, at the drill bit, or other components, with these additional energy expenditures potentially being less sensitive to changes in drilling fluid density.