Optimizing Particle Transport and Drag Coefficient in Petroleum Pipelines Using a Sensitivity Based Pontryagin Approach

Abstract

Local sensitivity analysis identifies the drag coefficient, mixture density, and particle diameter as dominant parameters that reduce particle transport, with sensitivity indices of −0.82, −0.65, and −0.50, respectively. In contrast, mixture velocity and control input exhibit strong positive influence, with sensitivity indices of +0.85 and +0.95. An optimal control strategy is formulated using Pontryagin’s Maximum Principle and solved numerically using a forward–backward sweep algorithm. Simulation results show that, without control, particle velocity exhibits overshoot and oscillatory behavior. With optimal control applied, the particle velocity converges smoothly to a stable value of approximately 0.4 m/s, closely tracking the mixture velocity with reduced energy expenditure. Comparison with published CFD and multiphase flow studies confirms that the predicted velocity range and stabilization trends are consistent with reported particle transport behavior. The proposed approach provides an efficient tool for stabilizing particle motion and improving flow assurance in pipeline systems.

Keywords

• Particle transport
• Sensitivity analysis
• Drag coefficient
• pontryagin optimization
• mixture density

Optimizing Particle Transport and Drag Coefficient in Petroleum Pipelines Using a Sensitivity Based Pontryagin Approach

Abstract

This study presents a sensitivity based optimal control framework for enhancing solid particle transport in petroleum pipelines. A nonlinear dynamic model is developed to describe particle velocity evolution under the effects of drag force, mixture density, particle size, and bounded flow disturbances. A control variable is introduced to regulate the effective mixture velocity and actively stabilize particle motion. Local sensitivity analysis identifies the drag coefficient, mixture density, and particle diameter as dominant parameters that reduce particle transport, with sensitivity indices of 0.82, 0.65, and 0.50, respectively. In contrast, mixture velocity and control input exhibit strong positive influence, with sensitivity indices of +0.85 and +0.95. An optimal control strategy is formulated using Pontryagins Maximum Principle and solved numerically using a forward backward sweep algorithm. Simulation results show that, with optimal control applied, the particle velocity converges smoothly to a stabilized value of 0.458 m/s, corresponding to 91.6% of the mixture velocity (0.5 m/s), with a relative deviation of less than 0.5% from the analytical steady state prediction.. Comparison with published CFD and multiphase flow studies confirms that the predicted velocity range and stabilization trends are consistent with reported particle transport behavior. The proposed approach provides an efficient tool for stabilizing particle motion and improving flow assurance in pipeline systems.

Keywords

• Particle transport
• Sensitivity analysis
• Drag coefficient
• pontryagin optimization
• mixture density

References

1. Ali, A. A., Abdul Majeed, G. H., & Al-Sarkhi, A. (2025). Review of multiphase flow models in the petroleum engineering: Classifications, simulator types, and applications. Arabian Journal for Science and Engineering, 50, 4413–4456. https://doi. org/10.1007/s13369-024-09302-0
2. Allababidi, S., Yan, W., & Yeung, H. (2012). Sand transportations and deposition characteristics in multiphase flows in pipelines. Journal of Energy Resources Technology, 134(2), Article 022901. https://doi.org/10.1115/1.4006433
3. Chen, S., Zhang, J., Hong, B., Duan, X., Chen, S., & Gong, J.(2019, June 16–21). Modelling of liquid-solid flow in horizontal pipe applying computational fluid dynamics method [Paper presentation]. The 29th International Ocean and Polar Engineering Conference, Honolulu, HI, United States.
4. Hou, Z., Jiang, E., Chen, Y., & Wang, H. (2023). A novel prediction model of the drag coefficient of irregular particles in power-law fluids. Processes, 11(11), Article 3213.
5. Mahindar, S., Momin, M., & Sharma, M. (2025). Effect of oil viscosities on pressure drop and flow characteristics in sudden contraction horizontal tube during oil water flows. Arabian Journal for Science and Engineering, 50, 13851–13867. https://doi.org/10.1007/s13369-024-09547-9
6. Ogunsesan, O. A., Hossain, M., Iyi, D., & Dhroubi, M. G. (2019). CFD modelling of pipe erosion due to sand transport. In M. Abdel Wahab (Ed.), Proceedings of the 1st International Conference on Numerical Modelling in Engineering (Lecture Notes in Mechanical Engineering, pp. 240–252). Springer. https://doi. org/10.1007/978-981-13-2273-0_22
7. Guo, X., Stoesser, T., & Nian, T. (2022). Effect of pipeline surface roughness on peak impact forces caused by hydrodynamic submarine mudflow. Ocean Engineering, 243, Article 110184.
8. Calisti, V., Lucardesi, I., & Scheid, J. F. (2023). Shape sensitivity analysis of a 2D fluid-structure interaction problem. Journal of Optimization Theory and Applications, 199, 36–79. https://doi. org/10.1007/s10957-023-02213-4

9. Al-Safran, E., Rasheed, A., & Ghasemi, M. (2026). Flow pattern prediction of gas/liquid two phase flow over wide range of liquid viscosity and inclination angle using machine learning. Chemical Engineering Research and Design, 228, 290–300.
10. Li, B., Guo, Z., Zheng, L., Shi, E., & Qi, B. (2024). A comprehensive review of wax deposition in crude oil systems: Mechanisms, influencing factors, prediction and inhibition techniques. Fuel, 357, Article 129676. https://doi.org/10.1016/j.fuel.2023.129676
11. Cai, X. (2026). CFD-driven optimization of multiphase microfluidic systems for enhanced analytical sensitivity in chemical and biochemical assays. Micro-chemical Journal, 226, Article 118382.
12. Goes, M. R. R. T., & Guedes, T. A. L. (2023). Multiphase flow simulation in offshore pipelines: An accurate and fast algorithm applied to real-field data. Chemical Engineering Science, 268, Article 118438.
13. Ali, A.A., Abdul-Majeed, G.H. & Al-Sarkhi, A. Review of Multi-phase Flow Models in the Petroleum Engineering: Classifications, Sim-ulator Types, and Applications. Arab J Sci Eng 50, 44134456 (2025). https://doi.org/10.1007/s13369-024-09302-0.
14. Mahindar, S., Momin, M. & Sharma, M. Effect of Oil Viscosities on Pressure Drop and Flow Characteristics in Sudden Contraction Horizontal Tube During Oil Water Flows. Arab J Sci Eng 50, 1385113867 (2025). https://doi.org/10.1007/s13369-024-09547-9.
15. Ogunsesan, O.A., Hossain, M., Iyi, D., Dhroubi, M.G. (2019). CFD Mod-elling of Pipe Erosion Due to Sand Transport. In: Abdel Wahab, M. (eds) Proceedings of the 1st International Conference on Numerical Modelling in Engineering . NME 2018. Lecture Notes in Mechanical Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-13-2273-0_22.
16. Yan Xia,Zhaowu Lin,Yu Guo,Zhaosheng Yu. Modelling of filtered drag force for clustered particle-laden flows based on interface-resolved simulation data. Journal of fliud Mechanics. 57(2024),10.1017/jfm.2024.857.
17. Xinyue Cai.CFD-driven optimization of multiphase microfluidic systems for enhanced analytical sensitivity in chemical and biochemical assays. Microchemical Journal.226(2026),118382.
18. ingfu Zhang, Hui Tang, Mingming Zhang.Drag Reduction Through Traditional or Machine Learning-Based Flow Control. Mdpi Fluid. 11(2026).
19. Liu, K., Xiong, X., Dai, X. et al. Optimization of drag reduction performance and energy efficiency enhancement of polyethylene oxide in oil-water two-phase flow using response surface methodology. Sci Rep (2026). https://doi.org/10.1038/s41598-026-72269-5. 20. Calisti, V., Lucardesi, I. & Scheid, JF. Shape Sensitivity Analysis of a 2D FluidStructure Interaction Problem. J Optim Theory Appl 199, 3679 (2023). https://doi.org/10.1007/s10957-023-02213-4