## 5 Parametric Sensitivity Analysis
### Flow Index Effect on the Pressure Loss in Power-Law and Yield-Power-Law Fluids  
The sensitivity analysis presented in Section 5.1 has revealed a pronounced impact of the pipe-well diameter ratio on the power law model, in contrast to its effect on yield-power-law (YPL) fluids. To further evaluate the model's reliability, we have devised a series of new experiments to investigate the model's sensitivity to changes in the fluid index \"\(n\)\" value of the fluids. In these experiments, the consistency index of both fluids remains constant, while the fluid index \"\(n\)\" values are systematically adjusted to 0.43 and 0.73. Notably, the yield strengths of the yield-power-law fluids remain constant throughout these experiments. The fluid properties are provided in Table 6.  
The simulation was conducted with an eccentricity of 0.9 and a tripping speed of 0.5 ft/s. Figure 8 presents the results of this simulation. It is evident that for n = 0.73, both power-law (PL) and yield-power-law (YPL) fluids exhibited a higher pressure loss, with the effect being more pronounced in the power law model than in the yield power law  
Figure 6: Effect of diameter ratio and eccentricity in power-law fluids. Figure 7: Effect of diameter ratio and eccentricity in the yield-power-law fluid. model. Notably, the R-value of the power law effect remains consistent with the R-factor established by [PERSON] et al. (1990) ([PERSON] et al. (1990)) ([PERSON] et al. (1990)). This outcome is primarily attributed to the fact that the coefficient of the first term was left unaltered, and the influence of the last term on the behavior of the considered fluids was not significant. In contrast, for the yield power law model, adjustments were made to the first terms, transitioning from 0.072 to 0.08, while the effect of the last included term on the behavior of the considered fluids remained inconsequential. As Figure 9 illustrates, the behavior of the yield power law deviates from that found by [PERSON] et al. (1990) and the power law R-value, particularly for higher diameter ratios. However, this discrepancy is not exceedingly pronounced. The elevated pressure observed in both the power law and yield power law models can be attributed to the \(\beta^{*}\) values derived from the experimental dataset. Additional comparative assessments are required to validate the model further. Figure 8: Effect of flow index (\(n\)) on the PL and YPL fluids’ pressure loss vs. diameter ratio. Figure 9: Effect of flow index (\(n\)) on the R-factors (\(R_{PL}\) (Equation (15)), \(R_{YPL}\) (Equation (14)), [PERSON] et al. (1990) ([PERSON] et al. (1990)) (Equation (13))) vs. diameter ratio for \(e\) = 0.9 and \(k\) = 2.75 lbf-\(\text{s}^{n}/100\)\(\text{f}^{2}\). Determination of Power and Yield Power Law \(\beta^{*}\) Constant for [PERSON] et al. 's R-Factor  
In the following section, we aim to couple the power-law and yield-power-law fluids parameters to the R-factor proposed by [PERSON] et al. (1990) ([PERSON] et al., 2017). The sensitivity study results shown in Figure 9 indicate that the R-value variation is 0.01 when the diameter ratio is higher. In this subsequent section, we aim to determine the beta factors when both fluids have the R-factor, as [PERSON] et al. (2017). The analysis results show the optimized beta factor for power-law fluids with \(\beta^{*}\) = 15.8 and yield-power-law fluids with \(\beta^{*}\) = 9.5. Figures 10-13 show the model predictions and the measured datasets. Table 7 shows the summary of the model's performance accuracy analysis. For the power-law Fluid 1, the new beta factors reduced the error but produced nearly equal performance to [PERSON] et al. 's (2017) model for Fluid 2. On the other hand, the new beta factors performance for fluids 3 and 4 are better than the [PERSON] et al. 's (2017) (2017) model's prediction. Figure 11: Pressure loss prediction with [PERSON] et al.’s (2017) R-factor and measured power law—Fluid 2 (2017).