Scientific Bulletin. Physical and Mathematical Research


A numerical method is proposed for solving the problem of pipeline transportation of natural gas through a non-horizontal gas pipeline, when the law of change in the hydrostatic pressure in time is specified at the inlet of the section, and the law of change in the mass flow of gas is set at the output.

The nonlinear, quasi-one-dimensional equations of pipeline gas transport in the isothermal approach are transformed using the natural logarithm of dimensionless gas density and the introduction of forward and backward traveling waves. In the newly constructed system of equations, convective and resistance forces are nonlinear, and the remaining components of the equations of traveling waves are linearized.

The equations of traveling waves involving scaled values of the speed of sound, length of a section, and gas density under normal conditions are presented in dimensionless variables and are approximated using a two-layer implicit scheme and an “upstream” scheme by A.A.Samarsky. The calculation of the forward wave, taking into account the input pressure value, was carried out according to the increasing discrete coordinate, and the backward wave, according to the decreasing discrete coordinate, where the flow velocity at the end of the section was first determined. According to the introduced substitutions, the condition at the exit from the site was transformed to a transcendental equation for the flow velocity, which was solved by multi-step using the Newton tangent method.

Due to the nonlinearity of the equations being solved in the computational domain, an iterative process was organized for each time step.

As an initial condition, we used the results of numerical integration of the dimensionless momentum conservation equation for a given constant gas mass flow rate, which significantly reduced the calculation time than in the case of the flow establishment method.

A computational experiment proved the applicability of the method for various (sinusoidal and discontinuous) options for setting the boundary functions for the inlet pressure and output mass gas flow rate, as well as a curved path.

It has been revealed that in long curved gas pipelines, intensive changes in gas-dynamic indicators occur near the border, where the indicators change in time. If the average boundary indicators tend to increase and decrease, then the ubiquitous transition process proceeds slowly, according to the average value of the hydrodynamic flow velocity.

At small lengths of the section, transients proceed intensively, where the role of the resistance force is negligible.

Calculations carried out for long, medium, and short distances showed that the proposed numerical method and calculation algorithm can be used in other boundary conditions too. The list of such conditions includes the tasks of taking into account the characteristics of the compressor and receiver, the outflow of gas into unlimited space, which were not solved due to their non-linearity.

First Page


Last Page



1. Seleznev, V.E., Aleshin, V.V., Pryalov, S.N. (2007). Sovremennye komp'yuternye trenazhery v truboprovodnom transporte. Matematicheskie metody modelirovaniya i prakticheskoe primenenie [Modern computer simulators in pipeline transport. Mathematical modeling methods and practical application]. Moscov: MAKS Press.

2. Trofimov, A.S., Kocharyan, E.V., Vasilenko, V.A. (2003). Kvazilinearizatsiya uravneniy dvizheniya gaza v truboprovode [Quasilinearization of the equations of motion of gas in a pipeline]. Elektronnyy nauchnyy zhurnal ‘Neftegazovoe delo’. No 1.

3. Sadullaev R. and et. el. (2003). Raschet magistralnogo gazoprovoda s uchetom relyefa mestnosti [Calculation of the main gas pipeline taking into account the terrain]. Gazovaya promyshlennost. 8. Pp. 58-59.

4. Ermolaeva N.N. (2017). Matematicheskoe modelirovanie nestatsionarnykh neizotermicheskikh protsessov v dvizhushchikhsya neizotermicheskikh mnogofaznykh sredakh [Mathematical modeling of non-stationary non-isothermal processes in moving non-isothermal multiphase media]. Doctoral (Phys. Math.) Dissertation. St. Petersburg. 323 p.

5. Xu H., Kong W., Yang F. (2019). Decomposition characteristics of natural gas hydrates in hydraulic lifting pipeline. Natural Gas Industry B. Vol. 6. Issue 2. Pp. 159-167.

6. Krivovichev G.V. (2017). A computational approach to the modeling of the glaciation of sea offshore gas pipeline. International Journal of Heat and Mass Transfer. Vol. 115. Part B. Pp. 1132-1148.

7. Zemenkova M.Yu., Babichev D.A., Zemenkov Yu.D. (2007). Metody sistemnogo analiza v reshenii zadach upravleniya slozhnymi tekhnicheskimi sistemami. Elektronnyy nauchnyy zhurnal ‘Neftegazovoe delo’. [Online]. URL: http://ogbus.ru/files/ogbus/authors/Zemenkova/Zemenkova_1.pdf

8. Mohamed Kh., Brahim B., Karim L., Hassan H., Pierri H., Amin B. (2015). Experimental and numerical study of an earth-to-air heat exchanger for buildings air refreshment in Marrakech. Proceedings of BS2015: 14th Conference of International Building Performance Simulation Association, Hyderabad, India, Dec. 7-9. Pp. 2230-2236.

9. Fazlikhani Faezeh, Goudarzi Hossein, Solgi Ebrahim. (2017). Numerical analysis of the efficiency of earth to air heat exchange systems in cold and hot-arid climates. Energy conversion and management. Vol. 148. Pp. 78-89.

10. Elsharkawy A.M. (2004). Efficient methods for calculations of compressibility, density and viscosity of natural gases. Fluid Phase Equilibria.Vol. 218. Issue 1. Pp. 1-13.

11. Ebrahimi-Moghadam, A., Farzaneh-Gord, M., Arabkoohsar, A., Moghadam, A.J. (2018). CFD analysis of natural gas emission from damaged pipelines: Correlation development for leakage estimation. Journal of Cleaner Production. Vol. 199. Pp. 257-271.

12. Deng Y, Hu H, Yu B, Sun D, Hou L, Liang Y. (2018). A method for simulating the release of natural gas from the rupture of high-pressure pipelines in any terrain. Journal of Hazardous Materials. Vol. 342. Pp. 418-428.

13. Yuan Q. (2018) Study on the restart algorithm for a buried hot oil pipeline based on wavelet collocation method. International Journal of Heat and Mass Transfer. Vol. 125. Pp. 891-907.

14. Jang, S.P., Cho, C.Y., Nam, J.H. et al. (2010). Numerical study on leakage detection and location in a simple gas pipeline branch using an array of pressure sensors. J. Mech. Sci. Technol. 24. Pp. 983-990.

15. Gyrya V., Zlotnik A. (2019). An explicit staggered-grid method for numerical simulation of large-scale natural gas pipeline networks. Applied Mathematical Modelling. Vol. 65. Pp. 34-51.

16. Dorao C.A., Fernandino M. (2011). Simulation of transients in natural gas pipelines. Journal of Natural Gas Science and Engineering. Vol. 3. Issue 1. Pp. 349-355.

17. Enikeev R.D., Nozdrin G.A., Chernousov A.A. (2017). The Model and the Methods for Numerical Simulation of Wave Action of Real Working Fluids in Pipelines. Procedia Engineering. Vol. 176. Pp. 461-470.

18. Lewandowski A. (1995) New Numerical Methods for Transient Modeling of Gas Pipeline Networks. New Mexico: Pipeline Simulation Interest Group.

19. Charnyy I.А. (1975). Neustanovivsheesya dvijeniye realnoy jidkosti v trubax [Unsteady movement of real fluid in pipes]. Moscow: Nedra.

20. Khujaev, I., Mamadaliev, Kh. (2020). An iterative method for solving nonlinear equations of real gas pipeline transport. Journal of Physics: Conference Series. 1441:012145.



To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.