## 7 Discussion  
During tripping operation, the axial movement of pipes generates fluid disturbance; hence, differential pressure develops. In this study, we developed a simple analytical hydraulics model that predicts the pressure gradient as pipe moves axially in the power-law and yield-power-law fluids. The model assumes that the fluid is incompressible and under steady-state conditions. The model performance accuracy analysis has been conducted by comparing the model prediction with the swab/surge experimental test data measured in 0.9 eccentric annuli filled with two power-law and two yield-power-law fluids by [PERSON] et.al. (2017) [12]. In Section 3, due to the non-linear nature of the newly derived model, we simplify the problem by momentarily neglecting the yield stress and consistency index. In this way, a linear model was obtained. To take into account the effect of the omitted stress/viscosity ratio (Equation (7)), two modifications were included in the linear model and the R-factor of [PERSON] et al. (1990) [25]. The reason for the modification in the R-factor is that [PERSON] et al. 's (1990) [24] R-factor is a function of power-based parameters only. In Section 4, we compared the model's effectiveness using [PERSON] et al. 's (2017) [12] experimental dataset. Moreover, the model prediction was also analyzed by comparing it to [PERSON] et al. 's (2017) [12] model. The first part of the model performance evaluation was conducted by using the \(R_{PL}\) (Equation (14)) and \(R_{YPL}\) (Equation (15)) modified R-factors (Equation (13)), which are used for the power-law fluids and the yield-power-law fluids, respectively. For these R factors, the optimized \(\beta^{\star}\) values have been obtained. The surprising result was that both power-law fluids showed \(\beta^{\star}\) values of 15.5, and the yield-power-law fluid showed a \(\beta^{\star}\) value of 10. The mean absolute sum between the model prediction and the measurement values computed results showed that the newly developed model's performance is relatively better than [PERSON] et al. 's (2017) [12] model. The beta factor (\(\beta^{\star}\)) for the power-law fluid is higher than that of the yield-power-law fluid when using the modified R-factors (Equations (14) and (15)). However, in Section 6, using [PERSON] et al. 's (1990) [25] R-factor, the magnitudes of the beta factor of the two fluids were analyzed. Results show that the best-fit model prediction for the power-law and the yield-power-law fluids were obtained when beta values were \(\beta^{\star}\) = 15.8 and \(\beta^{\star}\) = 9.5, respectively. The model performance analysis also shows that the new model prediction reduced the error between the model and the measurement results from Sections 4 and 6, indicating that both modeling results are fairly similar and show improved performances. Moreover, in Section 5, fictitious fluids have been formulated (Fluids 5-8) by varying the flow index and yield stress values while keeping the consistency index constant. The attempt was to investigate the impact of flow index in the assumed power and yield-power fluids. Results showed that the pressure loss gradient increased as the curvature exponent rose from 0.43 to 0.73. The lower \"\(n\)\" value showed higher R-factors. However, the \"\(n\)\" value in the pressure was dominant for the higher R-factor. One can also observe that the impact of \"\(n\)\" on the power-law fluid showed a higher R-factor than the yield-power-law fluid by 0.01 when the diameter ratio is about 0.9. The difference is insignificant for the lower diameter ratio. On the other hand, the power-law-based modified R-factor results are almost the same as the [PERSON] et al. (1990) [25] R-factor results. The model comparison and sensitivity study analysis performed well, and the trends are as expected. However, the model performance evaluation should be further assessed with other literature and field datasets. The model is limited to the impact of drill string axial speed and the fluid flow effect.