# Uncertainty Propagation through a Point Model for Steady-State Two-Phase Pipe Flow
## 1 Introduction  
Multiphase flow models are used in a range of applications, such as petroleum transport, nuclear energy and biomechanics. Accuracy in the model output is required to ensure the models to be useful decision support tools. Consequently, there is a rapid development in methods for quantifying the uncertainty in these models. [PERSON] and [PERSON] [1] compared several types of uncertainty propagation methods, including Monte Carlo methods (MC), polynomial chaos expansions (PC), full-factorial numerical integration (FFNI) and univariate dimension reduction (UDR). They explain the relative strengths of each method, and one conclusion is that PC is most viable in comparison to FFNI and UDR when input distributions are normal but output distributions are not. This is the situation in our analysis. Later, [PERSON] et al. [2] discussed applications of the methods discussed in [PERSON] and [PERSON] [1] to multiphase flows. Furthermore, they asked in the short-term that vendors of multiphase simulators implement tools for propagating uncertainty and yield sensitivities and prediction intervals. It was also requested that the simulators incorporate uncertainty in closure laws and that researchers accompany experimental data with uncertainty estimates. In the long term, [PERSON] et al. [2] recommended producing scale-up data to improve extrapolation in multiphase models. A case study is presentedin [PERSON] et al. (2018); [PERSON] et al. (2018), where they demonstrated how uncertainty propagation may be used for flow assurance on the Shtokman gas and condensate field. Their analysis included pressure drop and liquid holdup predictions using a one-dimensional model in the software OLGA. They determined probability distributions for a selection of input variables and closure laws and they propagated these uncertainties through the multiflow model using a Monte Carlo method. The result is the 10 th, 50 th, and 90 th output percentiles for pressure drop and liquid holdup predictions and measures of sensitivity to the input variables. [PERSON] et al. (2018) used Monte Carlo simulations with OLGA in order to identify influential variables and closure laws in several groups of data with different flow conditions. They are only able to construct satisfactory probability distributions for each closure law when using a group of data where the closure law is significant. [PERSON] et al. (2018) modeled liquid holdup and pressure drop in two-phase pipe flow using the TUFFP Unified Model for two-phase flow. They assumed an uncertainty range for each input variable and compared perturbation, Taylor series approximations and Monte Carlo methods for uncertainty propagation. They concluded that Taylor series approximations overestimate the output uncertainty while the other methods perform well. [PERSON] et al. (2018) also demonstrated the importance of selecting an appropriate framework when handling uncertainty in multiphase modeling and highlighted the value of quantitative information about the input uncertainty distributions for decision making. Just recently, in [PERSON] et al. (2018), a Gaussian process and principal component analysis were applied to a complex two-phase flow model in order to explore the uncertainty and reduce the complexity of the model. [PERSON] and [PERSON] (2018) considered a one-dimensional model for two-phase pipe flow. Known distributions for input variables are propagated through the model using Monte Carlo methods to obtain first-order and total sensitivity indices as well as output distributions. In the thesis of [PERSON] (2018), a similar method was applied to the point model Shell Flow Correlations and the one-dimensional model PIPESIM by [PERSON]. Polynomial chaos expansions were also applied, and both analyses were implemented in the open-source software UQLab ([PERSON], 2019). In this work, we compare uncertainty estimates obtained by Monte Carlo simulations with those of polynomial chaos for the two-phase point model presented in [PERSON] et al. (2018).