**Prediction
of pipeline erosion uncertainty for scale-up**

Wei Dai, Ravi Nukala, Selen Cremaschi

^{a}Russell School of Chemical
Engineering, The University of Tulsa, 800 S Tucker Drive, Tulsa,OK, USA

Abstract

In this study, a dimensional analysis is used for quantification of erosion-rate prediction uncertainty 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. Erosion in pipelines is defined as the material removal from the solid surface due to solid particle impingement. The phenomena that leads to this type of erosion, especially in multiphase flow systems, is very complex and depends on many factors including fluid and solid characteristics, the pipeline 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, Oka et al. (2005) developed their erosion model using particle impingement in air with empirical constants based on particle properties and hardness of the target materials. Their model is one of the most commonly cited in the literature. Another semi-mechanistic model called 1-D SPPS (Zhang, 2007), which is widely used for predicting erosion rates by the oil and gas industry, was developed with several empirically estimated parameters, like the sharpness factor of particles, Brinell hardness and the empirical constants in the impact angle function. These empirical parameters are calculated using experimental observations. However, the experimental data used in these calculations and also for model validation and uncertainty quantification are, for the most part, collected in small pipe diameters (from 2 to 4 inches). These small pipe sizes do not coincide with the field conditions, where the pipe diameters generally exceed 8 inches. Hence, the predictions of erosion models are routinely extrapolated to conditions where experimental data or even operating experience is not available, and the estimation of erosion-rate prediction uncertainty becomes crucial especially for systems too-costly to fail. The goal of this study is to develop a systematic approach to estimate erosion-rate prediction uncertainty for extrapolations. To achieve this goal, the erosion-rate model discrepancy, which is defined as the difference between experimental erosion rates and the corresponding erosion rate predictions, is modeled using Gaussian Process Modeling (GPM, Rasmussen, 2006). We used functions of dimensionless numbers that are relevant to erosion phenomena as the inputs to the GPM. Use of dimensionless numbers as inputs enables the scale-up of uncertainty estimates.

The GPM models erosion-rate model discrepancy as a Gaussian random process, which is defined by mean and covariance functions assuming a multivariate normal distribution (Zhen, 2013). The most likely values of mean and covariance function parameters are determined by Maximum Likelihood Estimation (MLE) using experimental data. The GPM model not only provides prediction in locations where experimental data is not available but also constructs the prediction confidence using the covariance functions through conditional probability distribution.

The approach developed to determine dimensionless groups and the functions as inputs to GPM has four main steps: (1) Identify all possible dimensionless groups that are relevant to erosion phenomena using Buckingham ¦Ð theorem. Here, we considered pipe diameter, particle size, density of particle and flow, viscosity of flow, flow rates, gravitation constant and surface tension as the dimensional variables. Time, length and mass are the three basic units. Therefore, we obtain 67 sets of 8 dimensionless numbers. Because some dimensionless numbers are repeated in different sets, the Buckingham ¦Ð theorem yields 200 distinct dimensionless numbers. Including pipe geometry, particle hardness and sharpness, which are already dimensionless, results in 203 distinct dimensionless numbers. (2) Calculate the correlation between each dimensionless number and model discrepancy, and find the sum of the correlation coefficient values that are greater than 0.5 (these refer to strong correlations) for each set. (3) Select the dimensionless group sets with the five highest correlation sums, and flag their corresponding dimensionless numbers as candidate inputs to GPM. (4) Formulate and solve the optimization problem to select the best function of candidate dimensionless numbers. Here, the best is defined as the function that minimizes the selected performance metric, i.e., area metric.

The performance of different input sets and functions as inputs to GPM is assessed using a modified area metric (Ferson, 2008). Area metric is defined as the integral of disagreement area between the estimated erosion rate and experimental data. A smaller area metric represents a better prediction of the model discrepancy. It can also be used to locate under-prediction regions of the erosion model.

For estimating erosion-rate-prediction uncertainty, we compiled an experimental database of erosion rate measurements from literature. It contains 544 data points in single or multiphase carrier flows. Eighty percent of the data in the database are collected for gas dominated flows (i.e., gas only, annular, mist and churn flow). The experimental database covers a wide range of input conditions resulting in significantly different erosion rates.

We selected the 1-D SPPS model as our case study, quantified
its erosion-rate discrepancy using our developed approach. The 1-D SPPS model calculates
the maximum erosion by defining how a hypothetical representative particle will
impinge the target material. The abrasion caused by this particle is defined by
length loss in the target material, and is calculated using the momentum of
impingement. The maximum erosion rate model in 1-D SPPS calculates the target
material length loss per unit time, and uses a power law correlation of the
characteristic impact velocity. The 1-D SPPS model accounts for pipe geometry,
size and material, fluid properties (density and viscosity) and rate, and particle
sharpness, density and rate. The 1-D SPPS model discrepancy is calculated for
544 experimental data points. The model discrepancies and the corresponding
values of the candidate dimensionless numbers are used as inputs to our uncertainty
estimation framework. Then, the overall analysis is performed for each flow
regime and the corresponding average area metric is calculated. For mist flow, gas
velocity, density and pipe diameter are the repeating variables that yielded
the dimensionless groups with the smallest averaged area metric value, which
was equal to 0.047. For churn flow, particle size, gas density and surface
tension are the repeating variables that yielded the dimensionless groups with
the smallest averaged area metric value, which was equal to 9.2x10^{-4}.
For slug flow, pipe diameter, gas viscosity and surface tension are the
repeating variables that yielded the dimensionless groups with the smallest
averaged area metric value, which was equal to 8.1x10^{-4}. While for
annular flow, gas velocity, gas density and surface tension are the repeating
variables that yielded the dimensionless groups with smallest averaged area
metric value, which was equal to 0.0074. Those dimensionless groups identified
in each flow regime provided most influential variables in the quantification
of erosion-rate model discrepancy and also suggested possible modeling and
experimental improvements involving those variables. Besides, extrapolation of erosion
discrepancy prediction is possible based on the identified dimensionless groups
in each flow regime.

Our analysis indicate that the use functions of dimensionless numbers as inputs to GPM improves the erosion rate prediction discrepancy and reduces the confidence intervals of uncertainties compared to using dimensional inputs to GPM as evidenced by smaller area metrics. More specifically, compared to GPM results with dimensional inputs, our approach yields a 60% decrease of area metric value in mist flow, 23% decrease in churn, 78% decrease in slug and 12% increase in annular flow with respective best dimensionless groups.

**Acknowledgement**

This work is supported by the Chevron Energy Technology Company. Discussions and comments from the Haijing Gao, Gene Kouba and Janakiram Hariprasad of Chevron and Brenton McLaury, Siamack Shirazi of E/CRC at the University of Tulsa were highly acknowledged. <>Reference

Ferson, S., Oberkampf, W. L., and Ginzburg, L., 2008, Model Validation and Predictive Capability for the Thermal Challenge Problem, Computer Methods in Applied Mechanics and Engineering, Vol. 197, No. 29-32, pp 2408-2430.

Jiang, Z., Chen, W., Fu, Y., and Yang, R., 2013, Reliability-Based Design Optimization with Model Bias and Data Uncertainty, SAE International.

Oka, Y. I., Okamura, K., and Yoshida, T., 2005, Practical estimation of erosion damage caused by solid particle impact: Part 1: Effects of impact parameters on a predictive equation, Wear, 259(1-6), page 95-101.

Rasmussen, C.E. and Williams, C.K. I., 2006, Gaussian Processes for Machine Learning, The MIT Press.

Zhang, Y., Reuterfors, E.P., McLaury, B.S., Shirazi, S.A., and Rybicki, E.F., 2007, Comparison of Computed and Measured Particle Velocities and Erosion in Water and Air Flows, Wear, 263.

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