# Experimental Analysis of Shale Cuttings Migration in Horizontal Wells
## 4 Theoretical Model and Validation
### Validation
#### 4.3.1 No Rotation of the Drill Tool  
In the case where the drill tool was not rotating, the experimental data for the critical return velocity of the shale particles showed a good agreement with the theoretical values. The average relative error between the theoretical predictions and the experimental data was approximately 9.04%. Specifically, the maximum error observed was 11.76%, while the minimum error was 7.32%. As the eccentricity and particle size increased, both the theoretical and experimental values of critical start-up velocity increased correspondingly. This trend confirms that the model is capable of accurately predicting the critical velocity for non-rotating conditions within an acceptable margin of error.  
Tables 2-4 present detailed comparisons between the experimental and calculated critical start-up velocities for different particle types (Types 1, 2, and 3) under varying eccentricities and flow rate conditions. For spherical particles, the model tends to overestimate the start-up velocity, with an average error ranging from 15.50% to 17.56%. In contrast, for non-spherical particles, the model predictions are more accurate, with errors falling within a range of 8.63% to 9.61%.  
The theoretical model calculates the critical start-up velocity for both spherical and non-spherical particles by considering forces such as drag, buoyancy, gravity, and lifting forces. However, it consistently overestimates the start-up velocity for spherical particles due to its idealized assumptions, which do not fully account for real-world factors such as particle shape irregularities and complex fluid dynamics. In actual conditions, spherical particles, with their smooth surfaces, face less resistance than the model predicts, allowing them to move more easily. This discrepancy leads to lower actual start-up velocities and an average error of around 15%. Conversely, the model performs better for non-spherical  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{**Eccentricity**} & \multirow{3}{*}{**Drilling Fluid Rate** **(L/s)**} & \multirow{3}{*}{**Experimental** **Data**} & \multicolumn{5}{c}{**Calculated Data**} \\ \cline{3-6}  & & & **Spherical Particle** **(m/s)** & **Error (\%)** & **Non-Spherical Particle (m/s)** & **Error (\%)** \\ \hline  
\hline \hline \end{tabular}
\begin{tabular}{c c c c c} \hline \hline \hline \multirow{3}{*}{**Expertricity**} & \multirow{3}{*}{**Drilling Fluid Rate** **(L/s)**} & \multirow{3}{*}{**Experimental** **Data} & \multicolumn{5}{c}{**Calculated Data**} \\ \cline{3-6}  & & & **Spherical Particle** **(m/s)** & **Error (\%)** & **Non-Spherical Particle (m/s)** & **Error (\%)** \\ \hline
0 & 7.24 & 0.37 & 0.32 & 15.63 & 0.40 & 8.  
particles, like flaky or irregular shapes, as it incorporates the drag coefficient and shape factors, leading to more accurate predictions with errors typically between 8% and 9%.  
Eccentricity further influences the model's accuracy, especially for spherical particles. As the annular eccentricity increases, errors between experimental and theoretical values grow. This suggests that the model struggles to account for the complex fluid flow patterns in eccentric annuli, where non-uniform velocity distributions and turbulence are more pronounced. Consequently, the model's predictions for spherical particles become less reliable in these conditions, contributing to the discrepancies observed as eccentricity increases.