# Rate of Penetration Estimation with Parameter Correction
## 5 Case Study  
### Case 1: MD Correction  
This simulation demonstrates how to correct MD by modeling both static and dynamic elongations using given parameters. The simulation is implemented using MATLAB (Version R2023a, The MathWorks, Inc., Natick, MA, USA) code, which calculates the total elongation over time and adjusts the MD accordingly. Table 1 summarizes the key parameters used in the analysis of the drill string dynamics. The parameters used in the drill string dynamic model are listed in Table 2. To ensure that the simulation conditions reflect realistic downhole drilling scenarios, the physical parameters used in the dynamic model were selected based on typical field values reported in the literature [5,18] and industry data sources (see Tables 1 and 2). The dynamic force and \begin{table}
\begin{tabular}{l c c} \hline
**Parameter** & **Value** & **Unit** \\ \hline Length of the drill string (\(L\)) & 5000 & m \\ \hline Young’s Modulus for steel (\(E\)) & \(2.05\times 10^{11}\) & Pa \\ \hline Cross-sectional area (\(A\)) & \(3.51\times 10^{-3}\) & m\({}^{2}\) \\ \hline Axial friction coefficient (\(\mu_{a}\)) & 0.25 & - \\ \hline Internal pressure (\(P_{\text{inner}}\)) & 35 & MPa \\ \hline External pressure (\(P_{\text{outer}}\)) & 15 & MPa \\ \hline Inner cross-sectional area (\(A_{\text{inner}}\)) & \(9.16\times 10^{-3}\) & m\({}^{2}\) \\ \hline Outer cross-sectional area (\(A_{\text{outer}}\)) & \(3.51\times 10^{-3}\) & m\({}^{2}\) \\ \hline Thermal expansion coefficient (\(\beta\)) & \(12\times 10^{-6}\) & 1/K \\ \hline Temperature difference (\(\Delta T\)) & 200 & \({}^{\circ}\)C \\ \hline \end{tabular}
\end{table} 
Table 1: Drill string parameters used in the analysis. torque components were generated using a bit-rock interaction model, with key parameters varied to simulate changes in formation properties. Specifically, the specific energy of rock (\(\epsilon\)) can be defined as a time-varying function with a wide range to mimic drilling through formations of varying hardness. Additionally, the cutter face inclination (\(\zeta\)) and the wear-flat length of the drill bit (\(\ell\)) can be chosen to reflect typical bit designs encountered in practical operations. Together, these modeling choices ensure that the simulation captures a broad and representative spectrum of real-world drilling conditions, thereby improving the applicability and credibility of the validation results. We simulate the system using the Euler method with a time step (0.01 s) over a duration of 10 s. The drill string's axial and angular velocities, dynamic force, and torque are computed iteratively at each time step. The simulation computes both static and dynamic elongations, allowing us to track the total elongation over time and compare it to the nominal drill string length. The initial conditions for position and velocity are set to zero. The simulation loop iteratively updates the state variables (position, velocity, angular velocity, etc.) using the dynamic force and torque to calculate changes in axial and angular velocity over time. The first part of the simulation compares the nominal drill string length (5000 m) with the corrected length, which accounts for static and dynamic elongation, see Figure 2. The corrected length shows a clear deviation from the nominal length, indicating the cumulative effect of elongation. Static elongation is the dominant factor, with significant contributions from thermal expansion and tension due to the drill string's weight. Dynamic elongation, though smaller in magnitude compared to static elongation, fluctuates over time due to transient forces like vibration and bit-rock interaction. This fluctuation can be seen in the plots of dynamic elongation over time, where periodic changes in elongation are evident.