Transient Flow in Gas Networks: Traveling waves

Martin Gugat 1  and David Wintergerst 1
  • 1 Department of Mathematics Friedrich-Alexander University Erlangen–N¨urnberg (FAU), Cauerstr. 11, 91058, Erlangen, Germany


In the context of gas transportation, analytical solutions are helpful for the understanding of the underlying dynamics governed by a system of partial differential equations. We derive traveling wave solutions for the one-dimensional isothermal Euler equations, where an affine linear compressibility factor is used to describe the correlation between density and pressure. We show that, for this compressibility factor model, traveling wave solutions blow up in finite time. We then extend our analysis to networks under appropriate coupling conditions and derive compatibility conditions for the network nodes such that the traveling waves can travel through the nodes. Our result allows us to obtain an explicit solution for a certain optimal boundary control problem for the pipeline flow.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • Banda, M.K., Herty, M. and Klar, A. (2006). Coupling conditions for gas networks governed by the isothermal Euler equations, Networks and Heterogeneous Media 1(2): 295-314.

  • Bounit, H. (2003). The stability of an irrigation canal system, International Journal of Applied Mathematics and Computer Science 13(4): 453-468.

  • Caputo, J.-G. and Dutykh, D. (2014). Nonlinear waves in networks: Model reduction for the sine-Gordon equation, Physical Review E 90(2): 022912.

  • Corless, R.M., Gonnet, G.H., Hare, D.E., Jeffrey, D.J. and Knuth, D.E. (1996). On the Lambert W function, Advances in Computational Mathematics 5(1): 329-359.

  • de Almeida, J.C., Vel´asquez, J. and Barbieri, R. (2014). A methodology for calculating the natural gas compressibility factor for a distribution network, Petroleum Science and Technology 32(21): 2616-2624.

  • Dos Santos Martins, V., Rodrigues, M. and Diagne, M. (2012). A multi-model approach to Saint-Venant equations: A stability study by LMIs, International Journal of Applied Mathematics and Computer Science 22(3): 539-550, DOI: 10.2478/v10006-012-0041-6.

  • Gugat, M., Hante, F., Hirsch-Dick, M. and Leugering, G. (2015). Stationary states in gas networks, Networks and Heterogeneous Media 10(2): 295-320.

  • Gugat, M., Schultz, R. and Wintergerst, D. (2018). Networks of pipelines for gas with nonconstant compressibility factor: Stationary states, Computational and Applied Mathematics 37(2): 1066-1097.

  • Gugat, M. and Ulbrich, S. (2017). The isothermal Euler equations for ideal gas with source term: Product solutions, flow reversal and no blow up, Journal of Mathematical Analysis and Applications 454(1): 439-452.

  • Kirchhoff, G. (1847). Ueber die Aufl¨osung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Str¨ome gef¨uhrt wird, Annalen der Physik 148(12): 497-508.

  • Lambert, J.H. (1758). Observationes variae in mathesin puram, Acta Helvetica 3(1): 128-168.

  • Li, D. (2010). Controllability and Observability for Quasilinear Hyperbolic Systems, American Institute of Mathematical Sciences Springfield, IL, USA.

  • Liu, M., Zang, S. and Zhou, D. (2005). Fast leak detection and location of gas pipelines based on an adaptive particle filter, International Journal of Applied Mathematics and Computer Science 15(4): 541-550.

  • Mugnolo, D. and Rault, J.-F. (2014). Construction of exact travelling waves for the Benjamin-Bona-Mahony equation on networks, Bulletin of the Belgian Mathematical Society Simon Stevin 21(3): 415-436.

  • Osiadacz, A. (1984). Simulation of transient gas flows in networks, International Journal for Numerical Methods in Fluids 4(1): 13-24.

  • Osiadacz, A. (1987). Simulation and Analysis of Gas Networks, Gulf Publishing Company, Houston, TX. Starling, K.E. and Savidge, J.L. (1992). Compressibility Factors of Natural Gas and Other Related Hydrocarbon Gases, American Gas Association, Washington, DC.

  • Veberic, D. (2010). Having fun with Lambert W(x) function, arXiv preprint 1003.1628.

  • Whitham, G.B. (1974). Linear and Nonlinear Waves, Wiley, New York, NY.


Journal + Issues