## 3 Results
### Optimization  
The optimization and validation of the experimental results were conducted through the utilization of regression equations derived from the response values, as outlined by [PERSON] et al. [14]. In this study, the focus was on investigating the influence of fly ash concentration and temperature of the base mud on the sag factor (SF) at various angles of inclination using a response surface design. The response surface methodology (RSM), facilitated by Minitab 18 software, was employed for this purpose, involving the design of experiments (DOE) and regression analysis. The experimental design encompassed the selection of appropriate ranges for fly ash concentrations and temperature values, as indicated in Table 3. Subsequently, the response surface design, illustrated in Table 4, was generated \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & \multicolumn{2}{c}{**This Study**} & \multicolumn{1}{c}{[6]} \\ \hline Anti-sagging additive & Fly ash & Fly ash & Vermiculite \\ Weighting agent & Barite & Ilmenite & Barite \\ The ratio of anti-sagging additive (lb/bbl) & 4 & 4 & 4 \\ Temperature (\({}^{\circ}\)F) & 250 & 250 & 250 \\ Pressure (psi) & 500 & 500 & 300 \\ Particle size distribution (D\({}_{50}\)) (\(\mu\)m) & 6.5 & 22 & 21.64 \\ Sag factor & 0\({}^{0}\) & 0.5021 & 0.5040 & 0.5100 \\
45\({}^{0}\) & 0.5151 & 0.5154 & 0.5200 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of comparisons between this study (fly ash) and [6]. Figure 15: Comparison between fly ash (This study) and vermiculite [6] under static conditions. based on the experimental data obtained using barite as the weighting agent, chosen for its stability in the sag factor compared to ilmenite. The optimization process sought to determine the optimal levels of fly ash concentration and temperature that would minimize sag factors across various wellbore inclinations (0\({}^{\circ}\), 30\({}^{\circ}\), 45\({}^{\circ}\), 60\({}^{\circ}\), 70\({}^{\circ}\), 80\({}^{\circ}\), and 90\({}^{\circ}\)). The reliability of the response surface tool was assessed through Analysis of Variance (ANOVA), as depicted in Table 5, revealing the significant effects of fly ash concentration, temperature, and their interaction on SF for fluids 3 and 4 (Barite densified). A significance level of \(p\)-value <0.05 was established as the threshold. Furthermore, Table 6 summarizes the model's reliability, with coefficients of determination (R\({}^{2}\), R\({}^{2}\)(adj), and R\({}^{2}\)(pred)) indicating the acceptability of the response surface models. Notably, higher R\({}^{2}\) values denote a better fit of the model to the observed data, with values approaching unity indicating a high degree of accuracy in prediction. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{ \begin{tabular}{c} **Run** \\ **Order** \\ \end{tabular}  } & \multicolumn{3}{c}{**Fly**} & \multicolumn{3}{c}{**Temperature**} & \multicolumn{3}{c}{**Wellbore Inclinations**} \\ \cline{2-9}  & **Ash Ratio (lb/bbl)** & \({}^{\circ}\)**F** & & & & & & \\ \hline
1 & 3 & 250 & 0.504000 & 0.512067 & 0.5155  
The optimization and validation process underscored the efficacy of the response surface methodology in identifying optimal conditions for minimizing the sag factor in barite-densified fluids, thereby enhancing the understanding and management of sagging phenomena in highly deviated geothermal drilling operations.