## 3 Results
### Response Surface Methodology Application
#### 3.4.2 Analysis of Mechanical Specific Energy (MSE)  
As seen in Section 3.4.1, residual plots and coefficients of determination (R\({}^{2}\)) were initially used to evaluate the model in use. Figure 7 details the residual plots resulting from the performed regression.  
Visual analysis of the residual plots shown in Figure 7 does not indicate a clear violation of the regression model assumptions for the fitted data. The model shows a robust fit of the quadratic regression model, as evidenced by the R\({}^{2}\) and Radj\({}^{2}\) of 96.27 and 96.12%, respectively. In addition, the predicted and adjusted coefficients of determination are very close, suggesting that the model's predicted values are highly consistent [49].  
However, the Breusch-Pagan test indicates a violation of the homoscedasticity assumption, i.e., the \(p\)-value of the test is less than 0.05. Thus, the null hypothesis of homoscedasticity of the residuals can be rejected. To solve this problem, the model was re-estimated  
Figure 7: Residual plots: (**a**) normal probability plot; (**b**) versus fits; (**c**) versus order.  
using the method to obtain heteroscedasticity-consistent standard errors (variant HC2). Table 3 shows the estimates of the significant effects of these factors on the MSE using a 5% significance level.  
The analysis of Table 3 shows the effects of the factors on MSE, including the effect of three second-order interactions, namely, WOB vs. WOB, TOR vs. TOR, and FLOW vs. FLOW, and the effect of three first-order interactions: WOB vs. TOR, WOB vs. FLOW, and TOR vs. FLOW. For the factors shown in Table 3, the statistical model for predicting MSE is expressed by Equation (11).  
\[\begin{array}{l}\text{MSE}=-7.22\times 10^{4}\;\text{X}_{1}+1.18\times 10^{4}\; \text{X}_{2}+3.77\times 10^{5}\;\text{X}_{3}-1.19\times 10^{3}\;\text{X}_{4}+4.52 \times 10^{2}\;\text{X}_{1}^{2}-3.56\times\\ 10^{1}\;\text{X}_{2}^{2}-5.09\times 10^{2}\;\text{X}_{3}^{2}-4.55\times 10^{2}\; \text{X}_{1}\cdot\text{X}_{3}+8.25\times 10^{1}\;\text{X}_{1}\cdot\text{X}_{4}-4.2 \times 10^{2}\;\text{X}_{3}\cdot\text{X}_{4}\end{array} \tag{11}\]  
The surface plots, as a result of having different interactions between the input variables (drilling parameters: WOB, RPM, FLOW, and TOR) and MSE, are shown in Figure 8.  
From the analysis of Figure 8, it can be seen that the values of WOB must be adjusted to values from 10 up to 25 klb. This is due to two different behaviors shown in the WOB plots. Initially, increasing the WOB value leads to a decrease in the MSE value; however, applying WOB greater than 25 klb results in a large increase in MSE, which is not desired in drilling operations. This optimization analysis is important because it shows that the highest values are not always the best for drilling a given region, i.e., it is possible to drill satisfactorily using low WOB values.  
In addition, the RPM and TOR values should be adjusted to be as low as possible, while the FLOW should be maintained at its highest. Low RPM and high FLOW help the drill-bit stability and efficient hole cleaning, respectively. However, these values must be tested in the field, since high FLOW can cause abrasion in the wall of the borehole.  
Another important point is about the TOR values. From the surface plots, it can be seen that TOR should be as low as possible. However, it is known that TOR is a variable that depends on other mechanical parameters, for example, WOB and RPM. Thus, to obtain low TOR values, it is necessary to test and evaluate different combinations of WOB and RPM. From these combinations of parameters, shown in Figure 8, it is possible to obtain low MSE values, which is desired in the drilling process.