The Wertenbach Equation in QRA

Introduction

Taking a trip down memory lane with the Wertenbach Equation in QRA and some might argue it's a bit “old-fashioned”, but I felt it was worth sharing for nostalgic purposes.

The Wertenbach Equation is a widely recognized semi-empirical model used in Quantitative Risk Assessment (QRA) to estimate the probability of pipeline failures. This blog post provides a mathematical breakdown of the equation, its components, real-world applications, limitations, and a worked example.

General Form of the Wertenbach Equation

The general form of the Wertenbach Equation is as follows:
p = C D^a P^b e^(c M + d * E)

Where:

• p: Probability of failure (per unit length per year).

• C: Empirical constant derived from historical failure data.

• D: Diameter of the pipeline (in meters or inches).

• P: Operating pressure (in bar or psi).

• M: Material factor (considering tensile strength, corrosion resistance, etc.).

• E: Environmental factor (external conditions like soil type, corrosion rate, and third-party interference).

• a, b, c, d: Exponents calibrated based on empirical datasets.

Parameter Definitions

1. Pipeline Diameter (D): Larger pipelines often carry higher loads, increasing structural risk. D^a accounts for scaling effects related to size.
2. Operating Pressure (P): High-pressure pipelines have a greater risk of rupture due to stress. P^b models this exponential relationship.
3. Material Factor (M): A material-specific coefficient considers tensile strength, corrosion resistance, and age.
4. Environmental Factor (E): Includes soil properties, cathodic protection effectiveness, and third-party interference.

Example Calculation

Estimate the failure probability for a gas pipeline with the following characteristics:

• Diameter (D) = 0.5 m

• Operating Pressure (P) = 100 bar

• Material Factor (M) = 0.8 (corrosion-resistant steel)

• Environmental Factor (E) = 0.5 (moderate corrosion risk)

• Empirical Constants: C = 1 x 10^-6, a = 2, b = 1.5, c = -0.3, d = -0.5

Substituting into the equation:
p= 1 x 10^-6 (0.5)^2 (100)^1.5 e^(-0.3 0.8 - 0.5 0.5)

Step-by-step Calculation:

1. D^a = (0.5)^2 = 0.25

2. P^b = (100)^1.5 = 10,000

3. c*M = -0.3 0.8 = -0.24

4. d*E = -0.5 0.5 = -0.25

5. e^(-0.24 - 0.25) = e^-0.49 ≈ 0.612

Final result:
p = 1 x 10^-6 0.25 10,000 0.612 = 1.53 x 10^-3 or….

p= 0.00153 failures per unit length per year).

Applications in QRA

1. Failure Estimation: Predicting the likelihood of incidents, such as ruptures or leaks.

2. Maintenance Prioritization: Identifying high-risk sections requiring immediate attention.

3. Compliance: Ensures adherence to safety regulations by quantifying and managing risks.

4. Geospatial Risk Mapping: Integrates with GIS tools to visually represent risk hotspots along pipeline routes.

Limitations and Challenges

1. Data Dependency: Requires accurate and comprehensive data to yield precise results.

2. Simplistic Assumptions: May not account for complex scenarios like multi-phase flows or unconventional materials.

3. Empirical Constants: Values may become outdated with advancements in materials and technology.

4. Modern Alternatives: Lags behind computational models leveraging AI and real-time IoT data.

Modern-Day Relevance

The Wertenbach Equation remains relevant for high-level risk assessments but often needs to be supplemented by modern technologies.

These include AI, IoT-enabled monitoring, and predictive analytics, which provide greater precision and adaptability.

Conclusion

• The Wertenbach Equation remains a foundational tool in pipeline risk assessment, offering valuable insights for high-level estimations.

• Has limitations in that it has dependency on precise empirical constants.

• While classic and empirically grounded, it is most effective when supplemented with modern methods like AI-driven analytics and IoT monitoring.

• Its value lies in bridging the gap between historical data and practical application, serving as a starting point for deeper risk analysis discussions.

References

• Industry guidelines (e.g., API 581).

• Semi-empirical models used in pipeline risk assessments. For instance, the U.S. Department of Transportation's Pipeline and Hazardous Materials Safety Administration (PHMSA) discusses various pipeline risk models in their technical documentation.

• Additionally, the paper "Assessment of Failure Frequency Methodology Applied to Dense Phase CO₂ Pipelines" examines semi-empirical fracture mechanics models for predicting pipeline failure probabilities.