## 3 Results and Discussions
### Stick Slip Behavior at 300 m (Upscale) and 10 m (Downscale) Depth  
This phase of the investigation focused on the behavior of the drill strings with an upscale string operating at 300 m and a downscale string operating at 10 m. Thus, the subsequent comparisons give detailed information regarding the influence of depth and system size on the transmission of torsional forces and the stability of the rotational system. Results from these simulations are illustrated in Figures 8-10.  
In the first scenario, a torque of 500 kgfm was applied to the upscale string, while 0.04 kgfm was applied to the downscale string (Figure 8). At this torque level, both strings started by mimicking the top RPM. However, it is evident from the results that the stick-slip effect was more pronounced in the downscale string compared to the upscale string. This  
Figure 8: Upscale and downscale string RPM at torque values of 500 kgfm and 0.01 kgfm at 300 m (upscale) and 10 m (downscale).  
Figure 9: Upscale and downscale string RPM at torque values of 1500 kgfm and 0.04 kgfm at 300 m (upscale) and 10 m (downscale).  
aligns with previous findings, where smaller, lighter strings are more sensitive to rotational instabilities due to their reduced inertia and stiffness. Despite the downscale string's higher stick-slip severity, both systems managed to overcome the torsional oscillations within a short period, stabilizing their rotations and synchronizing with the top RPM. This outcome reflects the capacity of both systems to self-correct and reach a steady rotational state after passing through an initial phase of torsional instability.  
In the second scenario, the torque was increased to 1500 kgfm for the upscale string while the torque for the downscale string remained at 0.04 kgfm (Figure 9). As expected, both strings initially mirrored the top RPM's rotation. However, the upscale string entered the stick-slip phase earlier than the downscale string. This observation suggests that at higher torque levels, even the upscale string, despite its larger size and greater structural stability, became prone to torsional disturbances. Although the downscale string took longer to enter the stick-slip phase, it once again exhibited greater severity of oscillations compared to the upscale string. This behavior is consistent with the trend observed in previous tests, where smaller-scale systems experience more pronounced instabilities. The higher stick-slip severity could indicate that the smaller system has more difficulty dissipating the torsional energy efficiently, causing more abrupt rotational jumps.  
In the final scenario, the torque was raised to 4500 kgfm for the upscale string and 0.09 kgfm for the downscale string (Figure 10). At this heightened torque level, both strings again started by rotating in sync with the top RPM, but they eventually entered the stick-slip phase. A key observation at this torque level is that the upscale string now entered the stick-slip phase earlier than the downscale string. This reversal in behavior suggests that the upscale string's stability is more easily compromised under extreme torque conditions. Additionally, the severity of stick-slip oscillations for both strings increased at this higher torque. The larger rotational disturbances observed indicate that the transmission of excessive torque introduces significant challenges for maintaining smooth rotational behavior, even for the more robust upscale string. Both strings underwent severe torsional vibrations at one point, but over time, they stabilized and rotated in sync with the top RPM. This shows the inherent resilience of the system, demonstrating that both models, despite initial instability, can recover and resume stable rotation.  
Figure 10: Upscale and downscale string RPM at torque values of 4500 kgfm and 0.09 kgfm at 300 m (upscale) and 10 m (downscale).