# Uncertainty Propagation through a Point Model for Steady-State Two-Phase Pipe Flow
## 4 Discussion  
The sensitivity estimates are similar using Monte Carlo or polynomial chaos. The averages \(AS_{i}\) gives the clearest picture as they measure how much uncertainty each input brings to the output estimates across all cases. From the first panels of Figures 4 and 7, we see that the pipe diameter and liquid viscosity are important for both pressure drop and holdup, while the gas mass rate, gas density and pipe inclination only matters for the holdup estimates. We can utilize the sensitivity indices for efficiently reducing the uncertainty in the output estimates. The focus should be on reducing the measurement error of the most sensitive variables; in this case, the pipe diameter and the liquid viscosity. Reducing uncertainty in measurements of these variables will efficiently improve predictions. Keep in mind that sensitivity indices are not general but depend on the flow conditions. See [PERSON] et al. (2012) for a description of the experiments. The results can not directly be extrapolated to different experiments. A new analysis is required, but the methods described in Section 2 may be applied. Figure 7: Pressure drop sensitivity estimates averaged over 233 experiments. (**Left**) Weighted by the variance in each experiment. (**Right**) Plain averages with standard error bars. Figure 8: Frequency histograms of the pressure drop sensitivity estimates across 233 experiments. The sensitivity distribution for one input variable is given in each panel. The experiments are categorized as slug (blue) or stratified (red), and we compare estimates from Monte Carlo (solid lines) and PC (dotted). Figure 9: **(Left) Predicted versus measured pressure drop for 194 experiments. The diagonal line is where the coordinates are equal. The first axis gives measured value with two standard errors either way. The second axis is predicted mean, lower quantile and upper quantile. Slug experiments are blue, and stratified ones are red. (Right) The relative difference in PC predictions compared to MC predictions with respect to measured pressure drop. Upper quantiles (dashed), means (solid) and lower quantiles (dash-dotted). **  
Also note that because first-order and total indices are similar, there are no decisive uncertainty interactions. Furthermore, we cannot conclude that the flow regime is important for sensitivity estimates. The sensitivities are similar across each regime. However, the moments of the polynomial chaos expansions do not converge with order for some cases on the boundary between regimes. Two cases  
Figure 10: Frequency histogram of computational cost across 240 experiments. Each color is one type of simulation criterion, namely only convergence in sensitivities (blue) or also 90% prediction intervals (orange) or 95% (green). The number of cases that did not converge is given as an entry on the right. **(Left)** Computation time. **(Right)** Number of model evaluations. are on the boundary bubbly/slug and neither converge with PC. Among 24 cases on the boundary slug/stratified, five cases do not converge with PC. Thus, the current implementation of PC expansions is unreliable on the regime boundaries. The expansions do not capture the behavior of the flow model well on the boundaries because the model is not smooth there. Adding higher-order terms to the expansion would make the expansion better resemble non-smooth behavior, but this is not immediately possible due to computational expense. It is possible to construct high-order approximations with low complexity by applying variable selection, but this approach is less applicable. We have also explored the technical details in the uncertainty computations, specifically the performance of Monte Carlo simulations compared to that of polynomial chaos methods. For the fluid model in question, we clearly prefer MC because this method provides uncertainty estimates for all cases, and it does so in the least amount of time. In contrast, PC fails in many cases and has a larger computation cost.