# Rate of Penetration Estimation with Parameter Correction
## 1 Introduction  
### Depth Corrections  
Well depth measurement is a fundamental aspect of drilling operations in the Oil and Gas industry, serving as a crucial parameter for a variety of operational decisions, including the estimation of the Rate of Penetration (ROP), wellbore positioning, and drilling efficiency optimization. The accuracy of well depth measurements is critical because errors in these measurements can lead to significant operational inefficiencies, increased costs, and safety risks. However, achieving precise well depth measurement is challenging due to various factors, such as the physical properties of the drill string, dynamic drilling conditions, and inherent limitations of measurement techniques. To address these challenges, numerous studies have focused on measured depth correction methods, as discussed in [1, 2, 3, 4]. Traditionally, well depth is estimated by measuring the length of the drill string deployed into the wellbore, assuming that this length corresponds directly to the depth of the well. However, this approach does not account for several dynamic factors that can affect the actual depth, such as drill string elongation due to tension, compression, thermal expansion, and pressure effects. Moreover, the dynamic nature of drilling operations, including changes in weight on bit (WOB), rotational speed, and fluid dynamics, further complicates the accurate measurement of well depth. These factors can cause the drill string to stretch, contract, or bend, leading to deviations from the true well depth. One of the primary sources of error in well depth measurement is the elongation of the drill string due to tension. As the weight of the drill string increases with depth, it causes the string to stretch, leading to an overestimation of the well depth. In [5], the authorsemphasized the importance of accounting for tension elongation in well depth measurement and proposed a correction model that incorporates drill string compliance. Their work demonstrated that neglecting tension elongation could result in significant depth measurement errors, particularly in deep wells. Moreover, hydraulic effects, including pressure ballooning and flow-induced lift, can also cause errors in well depth measurement. In [6], these hydraulic effects are explored and correction models are developed to account for them, showing that these factors become increasingly significant in high-pressure environments. In addition, thermal expansion of the drill string, caused by temperature variations along the wellbore, is another factor that can lead to inaccuracies in well depth measurement. As the drill string is exposed to higher temperatures at greater depths, it expands, causing an overestimation of the well depth. Another critical factor influencing well depth measurement is the correction of WOB, which significantly affects the accuracy of depth estimation, particularly in deep wells. WOB directly impacts the force exerted at the bit, influencing both ROP and wellbore stability [7]. However, inconsistencies in WOB measurements can introduce errors in depth calculations due to drill string deflection, dynamic interactions with the wellbore, and unaccounted physical forces acting along the drill string. One approach to improving WOB accuracy involves compensating for the discrepancies between surface-derived WOB and true downhole conditions. For example, in [8], the authors analyze the limitations of deriving WOB and Torque on Bit (TOB) from surface measurements, emphasizing the impact of unmodeled physical processes occurring between the bit and surface equipment. To address these discrepancies, they propose real-time compensation methods based on torque, drag, and hydraulic calculations, which refine WOB estimations and ultimately enhance drilling efficiency and bit performance. Another crucial refinement to WOB correction focuses on the accuracy of reference hook load measurements.