Quantifying model uncertainty in scarce data regions

Wei Dai, Selen Cremaschia

aDepartment of Chemical Engineering,The University of Tulsa, 800 S Tucker Drive, Tulsa,OK, USA

selen-cremaschi@utulsa.edu

Abstract

The complexity of physics-based engineering models and simulations has increased with increasing computational power. The inevitable uncertainty in these modelâ??s predictions originates from model inputs, numerical approximations and model form. Of all the three sources, the reliability of the models largely depends on how much the model form captures the underlying reality. Furthermore, most models are validated using a limited set of experimental data that does not usually coincide with the regions where the models would be used for predictions. In this talk, we present a non-parametric approach - Gaussian Process Modeling (GPM) - to statistically correct the discrepancy of a model, to quantify model uncertainty, and to expand model predictions and its uncertainty at regions where the experimental data for model validation is not available or scarce.
The Gaussian Process Modeling is represented by Gaussian random process which is defined by mean and covariance functions assuming a multivariate normal distribution1. The Gaussian random process characterizes the model with different settings of hyper-parameters. The most likely values of hyper-parameters are determined by Maximum Likelihood Estimation (MLE) using the set of known data. In general model-uncertainty-quantification formulation, the experimental response expected of system is expressed as the addition of the postulated physics-based modelâ??s output, a bias function which estimates the model discrepancy, and a random error accounting for the experimental uncertainty2. Both the model response (which includes uncertainties associated with estimated model parameters, numerical errors and interpolation uncertainties) and bias function can be expressed as Gaussian random processes with two sets of Gaussian hyper-parameters. The hyper-parameters are estimated by comparing M number of physics-based model runs with N number of experimental measurements according to MLE. Once the hyper-parameters are estimated, Gaussian processes are used to estimate model response and model discrepancy (bias function) at any input point. As the experimental variability (random experimental error) is estimated using the available N number of experimental measurements, the overall model uncertainty is quantified using the model response, model discrepancy, and the experimental variability. The outlined approach not only provides the prediction of interested points in the untested domain but also constructs the prediction confidence through conditional probability distribution.
The proposed Gaussian Process Modeling approach is demonstrated for prediction of expected erosion rate within pipelines that transport particles in multiphase gas-liquid flows. The transport of solids in multiphase flows is common practice in energy industries because of the
unavoidable extraction of solids from oil and gas bearing reservoirs either onshore or offshore sites. The safe and efficient operation of these pipelines requires reliable estimates of erosion rates, and production rates are generally limited to keep the effects of erosion at acceptable
levels. This type of erosion is defined as the material removal from the solid surface due to solid particle impingement. This erosion process, especially in multiphase flow systems, is a very complex phenomenon and depends on many factors including fluid characteristic, solid characteristics, the construction material properties and the geometry of the flow lines. Given
this complexity, most of the modeling work in this area focuses on developing empirical or semi- mechanistic models. For example, a semi-mechanistic model3, which is widely used for predicting erosion rates by the oil and gas industry, was developed with several empirically specified parameters. The experimental data for model validation and uncertainty quantification for erosion predictions are collected in small pipe diameters (from 2 to 4 inches). However, the model is used to predict erosion rates in field, where larger pipe diameters (>8 inches) are required. The outlined GPM approach is applied to generate erosion prediction confidence intervals of the semi-mechanistic model in the full-range of application domain including
untested areas. This talk will introduce the outlined GPM approach and its application to erosion rate prediction model.

Reference

1 Zhen J., Reliability-Based Design Optimization with Model Bias and Data Uncertainty, SAE International,

2013.

2 Kennedy, M.C. and Oâ??Hagan, A., Bayesian Calibration of Computer models, Journal of the Royal

Statistical Society Series B-Statistical Methodology. 63:425-450, 2001.

3 Mazumder, Q. H., Development and Validation of a Mechanistic Model to Predict Erosion in Single-

Phase and Multiphase Flow. PhD. Dissertation, Department of Mechanical Engineering, The University of

Tulsa; 2004.