# Uncertainty Propagation through a Point Model for Steady-State Two-Phase Pipe Flow
## 3 Results
### Holdup  
Figure 4 shows the estimated average sensitivity for the holdup predictions. The sensitivities quantify how sensitive the holdup predictions are to each input variable. In other words, it is the contribution of the uncertainty in each input to the uncertainty in holdup predictions. The estimated total sensitivity index and the estimated first-order index never differ more than 0.02. Thus, we use first-order indices in plots and refer to them simply as sensitivities. In the left panel, we see the averages weighted by the variance in each experiment. The right panel gives the plain averages with standard errors. The combined effect of liquid viscosity, pipe diameter and gas density account for ninety percent of the uncertainty in the holdup predictions. We have removed 7 out of the 240 cases from the results because the polynomial chaos expansions for pressure drop do not converge with polynomial order. The criteria for convergence is a change in estimates for the sensitivities, the output mean and the output variance from one order to the next less than 0.01. For the output mean and variance, we use the relative change. The criteria must be reached the latest at order 5. Table 3 contains information about the pressure drop statistics for the seven cases that do not converge. The holdup statistics actually change less than the threshold of 0.01, but we still exclude these results because we treat the pressure drop and the holdup as a joint variable in the simulations. For each case, we show the variable with the largest change from order 4 to 5 and the values of that variable for order 3, 4 and 5. All seven cases are on the border between two regimes, meaning that the model changes regime based on the sampled measurement error. The regimes assigned by the flow model are listed in the last column of the table. Figure 5 provides a more refined view of the holdup sensitivities. Each panel gives the histogram of sensitivity to one input based on the 233 cases. We plot separate histograms for slug flow (blue) and stratified flow (red). Furthermore, we compare MC (solid lines) to PC (dotted). There are only small deviations between the two methods. Half of the input variables have sensitivities consistently under 2%. The gas mass rate and density are moderately sensitive, while the diameter and liquid viscosity are in some cases highly sensitive, but not for all cases. Figure 4: Holdup sensitivity estimates averaged over 233 experiments. (**Left**) Weighted by the variance in each experiment. (**Right**) Plain averages with standard error bars. We compare predicted holdup to measured holdup in Figure 6. Only cases with converging prediction intervals from both MC and PC are included. In the left panel, each experiment is drawn as a cross. The horizontal part represents measurement error, and the vertical part (much smaller) represents prediction uncertainty. To be more precise, the crosses are the measured holdup with two standard deviations either way and the predicted mean and interval from MC simulations. The standard deviation in holdup measurements is set to 0.03. Furthermore, the diagonal line is where measurements and predictions are equal, and cases where the uncertainty box does not cover this line are highlighted. All cases with over-predicted holdup are observed slug flow. Conversely, under-predicted cases are stratified. The 90% intervals do not cover the observations in 20 (10.3%) out of 194 cases, while the 95% intervals are off in 12 (11.4%) out of 105 cases. The right panels of Figure 6 show the relative difference in the PC predictions and MC predictions. The mean holdup (solid line) is very similar, the upper quantile (dashed) is slightly larger, and the lower quantile (dash-dotted) is slightly smaller with PC. Thus, the PC predictions are overall similar to those of MC, but the intervals are wider.