# Rate of Penetration Estimation with Parameter Correction
## 2 Well Depth Correction  
### Downhole Forces  
WOB is a crucial parameter in drilling operations, directly influencing the ROP and the mechanical energy applied to the formation. However, the measured WOB at the surface does not always accurately represent the true downhole WOB due to factors such as frictional drag, axial elongation of the drill string, and hydraulic forces acting on the system [11,16]. One of the primary sources of error in surface WOB measurement arises from frictional drag between the drill string and the wellbore. As the drill string moves along the well trajectory, contact with the wellbore wall generates axial friction forces that reduce the effective force transmitted to the bit. This effect becomes particularly pronounced in deviated and horizontal wells, where the lateral contact between the drill string and wellbore significantly increases frictional resistance. The frictional loss can be modeled using the following equation:  
\[F_{friction}=\mu_{a}\cdot F_{c}, \tag{1}\]  
where \(\mu_{a}\) is the axial friction coefficient, and \(F_{c}\) is the normal contact force along the wellbore, which is a key parameter affecting frictional losses along the wellbore. In [17], they present the Torque and Drag model that gives the detailed modeling of downhole forces, where \(F_{c}\) is given as  
\[F_{c}=\sqrt{(w\sin\theta-\mu_{a}F\theta^{\prime})^{2}+(\mu_{a}F\sin\theta \varphi^{\prime})^{2}}. \tag{2}\]  
Here \(w\) is the buoyant weight per unit length, \(\theta\) is the inclination, \(F\) is the total tension in the string, \(\varphi\) is the azimuth angle, and \({}^{\prime}\) denotes differentiation with respect to the measured depth. The axial force along the drill string, \(F\), includes contributions from the axial tension \(F_{t}\) and hydraulic forces acting on the string. It is calculated as  
\[F=-WOB+F_{t}+F_{\text{hydraulic}}, \tag{3}\]  
where the axial tension \(F_{t}\) is given by  
\[F_{t}=F_{r}+F_{friction}, \tag{4}\]  
and \(F_{r}\) is the tension at the top as  
\[F_{r}=\int_{0}^{L}w\cos\theta ds, \tag{5}\]  
and the hydraulic force component is given by  
\[F_{\text{hydraulic}}=\int_{0}^{L}(-A_{o}p_{q}^{\prime}-\pi d_{0}\tau_{0})ds, \tag{6}\]  
where \(p_{q}^{\prime}\) is the flow and motion-induced pressure gradient, \(\tau_{0}\) is the flow-induced axial shear stress at the outer string surface, and \(A_{o}\) is the respective cross-sectional areas. In the field, the WOB can be estimated from surface measurements using load cells installed on the top drive or hook. The measured WOB is derived from the hook load (\(F_{h}\)), accounting for the effects of the submerged weight of the drill string and frictional resistance in the wellbore. To accurately estimate the WOB and subsequently determine the axial tension force (\(F_{t}\)) and hydraulic force (\(F_{\text{hydraulic}}\)), an iterative correction algorithm based on measured hook load (\(F_{h}\)) is employed in this study. This approach ensures consistency between the measured and calculated forces, improving the reliability of the drilling force estimations. The algorithm begins by setting an initial estimate for WOB, denoted as \(WOB_{0}\). This initial guess is used in Equation (3) to compute the axial force \(F\) under the given wellbore and operational conditions. Once \(F\) is determined, the computed hook load \(F_{h,\text{calc}}\) is obtained using the following relationship:  
\[F_{h,\text{calc}}=F. \tag{7}\]  
The calculated hook load is then compared with the measured hook load (\(F_{h}\)). The difference between these values is defined as the following error:  
\[\Delta F_{h}=F_{h,\text{calc}}-F_{h}. \tag{8}\]  
The magnitude and sign of \(\Delta F_{h}\) determine whether a correction to WOB is necessary:  
* If \(\Delta F_{h}>0\), meaning the calculated hook load is greater than the measured value, the algorithm increases WOB by a predetermined step size to reduce the error.