## 4 Results and Discussions
### Sensitivity Analysis  
As ANNs were shown to be the most accurate model for predicting pressure gradients in this study, they were used to conduct a sensitivity analysis so that each input variable's relationship with the output variable can be identified. Figures 8-13 represent changes in the pressure gradient for changes in each variable as per the current ANN predictions. This analysis was performed by applying the ANN model developed in this study with varying values of one independent variable at a time, while others were kept constant at their average values. For example, if velocity was observed to affect the pressure gradient, then all other parameters were fixed at their average values (as shown in Table 1), while velocity was changed within a predetermined range. This approach for sensitivity analysis was employed in previous studies as well e.g., [17; 32].  
Figure 8: Effect of average velocity.  
Figure 9: Effect of oil viscosity.  
Figure 11: Effect of oil density.  
Figure 12: Effect of water viscosity.  
As shown in Figures 8-14, average velocity and oil viscosity have positive effects on the pressure gradient. Specifically, increasing the magnitudes of these variables tends to increase friction losses. The boosting effect of velocity is highly evident in fluid dynamics studies. It should be mentioned that \(\Delta P\)/\(L\) is proportional to \(V^{2}\) for a single-phase pipe flow. The impact of oil viscosity is a WAF-specific phenomenon. Higher oil viscosity most likely increases the degree of wall-fouling, thereby increasing \(\Delta P\)/\(L\)[4; 5; 7].  
Water density and oil density have opposite effects on pressure losses. Varying the fluid density tends to affect the core eccentricity. Although the effect of eccentricity in WAF is not a well-studied phenomenon [8], the current study sheds some light on the topic. Water density seems to have a linear effect, whereas oil density has an inverse influence on friction losses. Oil density increases the pressure gradient within the lower range, while the gradient decreases as the density exceed 900 kg/m\({}^{3}\). A more practical investigation is required in this field.  
Figure 14: Effect of pipe diameter.  
Figure 13: Effect of water fraction.  
Water viscosity did not have a significant impact on pressure losses, as its magnitude was essentially constant (~1 mPa.s). Similar to water viscosity, water fraction was found to have a negligible effect on friction losses. A detailed analysis of the experimental measurements also demonstrated comparable results [30]. Pipe diameter had an inverse influence on the WAF pressure gradient, which was expected since there is a nearly proportional correlation between \(\Delta P/L\) and \(D^{-1}\) for the single-phase flow in a pipeline.  
The outcome of this sensitivity analysis highlights another advantage of using ANN, as the same model can capture varying degrees of relationships between variables. Traditional parametric and analytical models lack this ability or require prior information about the problem that needs to be customized in a specific way to fit the model. On the other hand, the ANN model was developed in the present study without using any priory information.