## 3 Results
### Optimization
Below is a list of the linear regression (quadratic) equations in uncoded units that were derived for each wellbore inclination from the response surface tool:  
\[0^{\circ}=0.4806-2.1\times 10^{-4}\text{C}+1.43\times 10^{-4}\text{T}\ \text{-}\ 9 \times 10^{-4}\text{CT} \tag{5}\]  
\[30^{\circ}=0.49118+1.04\times 10^{-3}\text{C}+1.01\times 10^{-4}\text{T}\ \text{-}\ 7 \times 10^{-6}\text{CT} \tag{6}\]  
\[45^{\circ}=0.49474+5.9\times 10^{-4}\text{C}+1.01\times 10^{-4}\text{T}\ \text{-}\ 4 \times 10^{-6}\text{CT} \tag{7}\]  
\[60^{\circ}=0.49748-8.3\times 10^{-4}\text{C}+1.16\times 10^{-4}\text{T}\ \text{-}\ 1 \times 10^{-6}\text{CT} \tag{8}\]  
\[70^{\circ}=0.4970-5.0\times 10^{-4}\text{C}+1.05\times 10^{-4}\text{T}\ \text{-}\ 2 \times 10^{-6}\text{CT} \tag{9}\]  
\[80^{\circ}=0.5028-3.95\times 10^{-3}\text{C}+9.3\times 10^{-5}\text{T}+7 \times 10^{-6}\text{CT} \tag{10}\]  
\[90^{\circ}=0.5248-1.018\times 10^{-2}\text{C}+3\times 10^{-5}\text{T}+1.7 \times 10^{-5}\text{CT} \tag{11}\]  
where C is the concentration of the fly ash in lb/bbl, and T is the temperature in \({}^{\circ}\)F.  
As observed from quadratic Equations (5)-(11), it is evident that there is an empirical relationship between the concentration of fly ash (C) and temperature (T) that influences the SF differently at the wellbore inclinations investigated. Each of the quadratic equations above (5)-(11) was used to develop the contour and response surface plots below. The  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Static Sag Inclination** & **R-sq (\%)** & **R-sq (adj) (\%)** & **R-sq (pred) (\%)** \\ \hline \(0^{\circ}\) & 99.01 & 97.69 & 86.26 \\ \(30^{\circ}\) & 99.74 & 99.38 & 97.17 \\ \(45^{\circ}\) & 99.25 & 98.24 & 95.46 \\ \(60^{\circ}\) & 99.66 & 99.20 & 96.49 \\ \(70^{\circ}\) & 99.25 & 98.24 & 93.86 \\ \(80^{\circ}\) & 98.97 & 97.59 & 93.81 \\ \(90^{\circ}\) & 99.31 & 98.4 contour plots in Figure 16 depict the empirical relationship between the temperature (\({}^{\circ}\)F) and concentration of fly ash (lb/bbl) at different SF for the individual wellbore inclinations. With this, the optimal variables within the experimental limits can be easily identified. From physical observations, it can be inferred that there is a high interaction between temperature and fly ash concentrations as the slope of the SF was high for 0\({}^{\circ}\), 80\({}^{\circ}\), and 90\({}^{\circ}\) wellbore inclinations, while less interaction was noticed for 30\({}^{\circ}\)and 45\({}^{\circ}\). The average slope of SF may be said to be 60\({}^{\circ}\) and 70\({}^{\circ}\). This response may be due to the chemical reactions among the fly ash ratio, temperature, and wellbore inclination on the components of the BF. The 0\({}^{\circ}\) inclination (I-0) contour plot showed that a high sag factor (SF) of (>0.514 ppg) was achieved by blending 3.0-3.1(lb/bbl) of fly ash with Base Fluid (BF) at 390-400 \({}^{\circ}\)F, while a low SF (<0.502 ppg) was obtained from a blend of 3.9-4.0 lb/bbl of fly ash with BF at 250-255 \({}^{\circ}\)F. The contour plot is observed to bend outward (Convex shape). The highest SF usually occurs around 60 deg inclination (I-60), and at the highest temperature of exposure [7]. Consequently, an extremely high SF of (>0.524 ppg) was yielded by a combination of 3.0-4.0 lb/bbl of fly ash with BF at 360-400 \({}^{\circ}\)F. However, the concentration of fly ash was observed to decrease from 3.0 to 4.0 lb/bbl with increasing temperature. This further confirms that the sagging of the BF is temperature-dependent. A low SF of (<0.518 ppg) was achieved by formulating 3.0 to 4.0 lb/bbl of fly ash with BF at 250-275F. At 90\({}^{\circ}\) inclinations (I-90), the contour plot seems to have a high slope and is highly responsive to the variables (temperature and fly ash concentration).