## 3 Results
### Machine-Learning-Based Modeling Result  
#### 3.2.1 Result 1--Simulated and Laboratory-Based Data Model  
The initial ML modeling was performed using a physics-based simulated dataset. The example presented here illustrates how ML models predicted software-generated dataset. For this, the drilling fluid shown in Figure 4 was circulated in the experimental well.  
Figure 18 shows the simulated synthetic dataset obtained from the power law (PL) and Robertson-Stiff (RS) models. Simulation stopped at the maximum allowable tripping speed that caused the well pressure to reach the weak point (i.e., fracture at the casing shoe, including safety factor). Figure 19 also shows the laboratory scale surge pressure gradient as a function of the tripping speed [7].  
Frictional pressure loss is proportional to the square of fluid flow. Similarly, the simulated (Figure 18) and the laboratory experimental (Figure 19) surge pressure variations with the pipe movement trend represented a second-order polynomial function. Therefore, the simulated and the experimental test data were modeled as a polynomial function that included both constant and middle terms. The maximum allowable well pressure was when the well pressure reached the casing pressure (\(P_{\mathrm{c}}\)). The variation in well pressure due to tripping speeds, obtained from Equation (11), is given as follows:  
\[P_{\mathrm{c}}=\beta_{0}+\beta_{1}V_{\mathrm{p}}+\beta_{2}V_{\mathrm{p}}^{2} \tag{15}\]  
where \(\beta_{0}\), \(\beta_{1}\), and \(\beta_{2}\) are the curve fitting parameters, determined from the green or blue datasets.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Swabbing**} & \multicolumn{2}{c}{**Surging**} \\ \cline{2-5}
**OBM/Well/Density** & **\% BP to PL Change** & **\% BP to RS Change** & **\% BP to PL Change** \\ \hline
90:10 Inclined 1.96 sg & \(-9.89\) & \(-2.42\) & \(-4.55\) & \(0.14\) \\
80:20 Inclined 1.96 sg & \(-3.21\) & \(-0.99\) & \(0.68\) & \(-0.51\) \\
90:10 Vertical 1.96 sg & \(-7.34\) & \(-2.73\) & \(-2.96\) & \(0.05\) \\
80:20 Vertical 1.96 sg & \(-6.05\) & \(-0.33\) & \(-1.24\) & \(-0.79\) \\ \hline \hline \end{tabular}
\end{table}
Table 8: Comparisons of 80:20 and 90:10 OBMs (from Figures 16 and 17, respectively) with 1.96 sg density in vertical and inclined wells.  
Figure 17: Surging comparisons of 2.0 sg 80:20 and 90:10 OBMs filled in deviated and vertical wells.  
Based on the knowledge of case shoe pressure, one can therefore write well pressure at the critical tripping speed using the inversion of Equation (15), given as:  
\[\mathrm{V_{p}}=\frac{-\beta_{1}+\sqrt{\beta_{1}^{2}-4\beta_{2}(\beta_{2}-P_{c})}}{ 2\beta_{2}} \tag{16}\]  
Using the synthetic and experimental data, polynomial-based modeling was performed to estimate the curve fitting parameters (\(\beta_{0}\), \(\beta_{1}\), and \(\beta_{2}\)); the results are presented in Table 9. Both models showed higher R\({}^{2}\) correlation values. However, it is essential to note that polynomial-based modeling was only applied for swab and surge field data or laboratory data if the pressure variations behaved as polynomials when the tripping speeds varied.  
Figure 19: Experimental surge pressure gradient vs. tripping speed data [7].  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Data** & \(\beta_{2}\) & \(\beta_{1}\) & \(\beta_{0}\) & \(\mathbf{R^{2}}\) \\ \hline Figure 19 [7] & \(-0.1675\) & \(0.4851\) & \(0.0452\) & \(0.9992\) \\ Figure 18 [PL] & \(-9\times 10^{-10}\) & \(1\times 10^{-5}\) & \(1.9954\) & \(0.9965\) \\ Figure 18 [RS] & \(-4\times 10^{-10}\) & \(1\times 10^{-5}\) & \(1.9941\) & \(0.9997\) \\ \hline \hline \end{tabular}
\end{table}
Table 9: Polynomial function correlation factors of simulated and experimental data.  
Figure 18: Drillbench-software-simulated surging pressure as a function of tripping speed up to the fracture point.