Scientific Bulletin. Physical and Mathematical Research


Numerous problems based on equations of hyperbolic type are solved by the method of characteristics when boundary conditions of the first, second and third kinds are set on the boundary. These include, in particular, the problems of pipeline transportation of various media. The quasi-one-dimensional equations of conservation of mass and momentum are linearized and separate equations are drawn up for a specific indicator (flow rate, gas pressure) of the process. This approach is associated with the tradition of forming boundary conditions and solving problems only with respect to a specific indicator, in particu-lar, with respect to pressure, velocity, and flow rate of the medium.

Within the framework of the study, problems were formulated for the first and third stages of tes-ting the operation suitability of an elementary sec-tion of a gas pipeline: gas injection into the pipeline and gas outflow from the pipeline into the atmosp-here. When modeling these processes, the N.E.Joukowski formula was used on the limited gas outflow rate and the approach of a short pipe-line when the directional pressure change is caused only by the local component of the gas inertia force. The problem statement can also be used to study the state of the gas when starting an elementary section of a gas pipeline.

The method of characteristics was used for the first time for the case when a linear relationship between the sought functions holds for one of the boundaries. Formulas were obtained for the sought functions for the first conditional time period, which corresponds to the wave travel time from one end of the section to its other end. It is propo-sed to use these formulas as solutions in subsequent conditional periods, with new initial conditions ob-tained at the end of the previous conditional period.

The modified method of characteristics was implemented when solving the problem of gas out-flow from a pipeline with constant values of func-tions at initial and boundary conditions. Despite ensuring the continuity of initial conditions, dis-continuous solutions were obtained due to a pres-sure jump in the inlet section. The results with constant values were obtained at time points mul-tiple of the conditional period.

Calculations were performed for various valu-es of the initial pressure and the ratio of the flow choke area to the cross-sectional area of the pipe-line.

Numerical calculations showed that the pres-sure jumps over time at the points of the computa-tional domain have a stepwise character. With time, the jumps decrease in size, and, according to the re-sults obtained, the average pressure value in the section over time decreases exponentially.

It was revealed that the gas mass flow rate first decreases by jumps to a certain value, and then in-creases over time by an exponential law and tends to a state of rest.

The results obtained showed that with time, the gas velocity in the section also tends to zero. The gas velocity, as well as the gas mass flow rate, at the points of the computational domain (except for the closed end of the section), changes over ti-me by jumps, which become shorter with time.

The analytical solution to the problem obtai-ned allows us to consider the cases of variable functions at initial and boundary conditions, and the case when a temporary change in pressure is given at the inlet, and the gas is injected or outflo-wed into the infinite space of the atmosphere through the finite section.

First Page


Last Page





[1] Будак Б.М., Самарский А.А., Тихонов А.Н. Сборник задач по математической физике // М.: Наука, 1972. – 678 с.

[2] Тихонов А.М., Самарский А.А. Уравнения математической физики. – М.: Наука, 1977. – 736 с.

[3] Трофимов, А.С., Кочарян, Э.В., Василенко, В.А. (2003) Квазилинеаризация уравнений движения газа в трубопроводе // Электронй научнйй журнал «Нефтегазовое дело»

[4] Чарнй И.А. (1975) Неустановившеся движение реалной жидкости в трубах.– Москов: Недра.

[5] Грачев В.В., Уербаков С.Г., Яковлев Е.И. Динамика трубопроводных систем. – М.: Наука, 1987. – 438 с.

[6] Цой П.В. Системные методы расчета краевых задач тепломассопереноса: Прямые и обратные задачи не- стационарной теплопроводности и термоупругих напряжений. Гидродинамика и теплообмен в каналах сложного профиля / 3-е изд., перераб. и доп. – М.: Из- дателғство МЭИ, 2005. – 568 с.

[7] Krivovichev G.V. A computational approach to the modeling of the glaciation of sea offshore gas pipeline // International Journal of Heat and Mass Transfer Volume 115, Part B, December 2017, Pages 1132-1148. https://doi.org/10.1016/j.ijheatmasstransfer.2017.08.117

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

[9] Fazlikhani Faezeh, Goudarzi Hossein, Solgi Ebrahim. Nu-merical analysis of the efficiency of earth to air heat ex-change systems in cold and hot-arid climates // Energy conversion and management, 2017, №5, T: 148. – P. 78-89.

[10] Elsharkawy A.M. Efficient methods for calculations of compressibility, density and viscosity of natural gases // Fluid Phase equilibriaVolume 218, Issue 1, 1 April 2004, Pages 1-13. https://doi.org/10.1016/j.fluid.2003.02.003

[11] Deng Y. et al. A method for simulating the release of nat-ural gas from the rupture of high-pressure pipelines in any terrain // Journal of Hazardous Materials Volume 342, 15 January 2018, Pages 418-428. https://doi.org/10.1016/j.jhazmat.2017.08.053

[12] Yuan Q. Study on the restart algorithm for a buried hot oil pipeline based on wavelet collocation method // Interna-tional Journal of Heat and Mass Transfer Volume 125, Oc-tober 2018, Pages 891-907. https://doi.org/10.1016/j.ijheatmasstransfer.2018.04.127

[13] Dorao C.A., Fernandino M. Simulation of transients in natural gas pipelines // Journal of Natural Gas Science and engineering Volume 3, Issue 1, March 2011, Pages 349-355. https://doi.org/10.1016/j.jngse.2011.01.004

[14] Enikeev R.D., Nozdrin G.A., Chernousov A.A. The Mod-el and the Methods for Numerical Simulation of Wave Ac-tion of Real Working Fluids in Pipelines // Procedia engi-neering Volume 176, 2017, Pages 461-470. https://doi.org/10.1016/j.proeng.2017.02.345

[15] Lewandowski A. (1995) New Numerical Methods For Transient Modeling of Gas Pipeline Networks. – New Mexico: Pipeline Simulation Interest Group.

[16] Khujaev,I, Mamadaliev, Kh. An iterative method for solv-ing nonlinear equations of real gas pipeline transport // AMSD-2019 Journal of Physics: Conference Series 1441 (2020) 012145 IOP Publishing doi:10.1088/1742-6596/1441/1/012145

[17] Махкамов М.К., Бозоров О.Ш., Хужаев И.К. Конеч-норазностный метод расчета газопровода, проло-женного по пересеченной местности // Scientific Bulle-tin of the Andijan State University named after Zahiriddin Mukhammad Babur: Physical and Mathematical research. 2020, №1. – С. 77-85.

[18] Селезнев, В.Э., Алешин, В.В., Прялов, С.Н. (2007) Со-временне компъютерне тренажерй в трубопроводном транспорте. Математические методй моделирования и практическое применение / Под ред. В.Э. Селезнева. – Мосcов: МАКС Пресс

[19] Эрмолаева Н.Н. (2017) Математическое моделирова-ние нестационарнйых неизотермическиых процессов в движучиыхся неизотермическиых многофазнйх сре-даы: Дисс… докт. физ.-мат. наук. - СПб, 2017. – 323 с.

[20] Xu H., Kong W., Yang F. Decomposition characteristics of natural gas hydrates in hydraulic lifting pipeline // Natu-ral Gas Industry B Volume 6, Issue 2, April 2019, Pages 159-167. https://doi.org/10.1016/j.ngib.2018.07.005

[21] Zemenkova M.Yu., Babichev D.A., Zemenkov Yu.D. Metody sistemnogo analiza v reshenii zadach upravleniya slozhnymi tekhnicheskimi sistemami // Neftegazovoe delo. – 2007. – http://www.ogbus.ru/authors/ Zemenko-va/Zemenkova1.pdf.

[22] Jang S.P. et al. Numerical study on leakage detection and location in a simple gas pipeline branch using an array of pressure sensors // J Mech Sci Technol (2010) 24: 983. https://doi.org/10.1007/s12206-010-0216-8

[23] Ebrahimi-Moghadam A. et al. CFD analysis of natural gas emission from damaged pipelines: Correlation develop-ment for leakage estimation // Journal of Cleaner Produc-tion Volume 199, 20 October 2018, Pages 257-271. https://doi.org/10.1016/j.jclepro.2018.07.127

[24] Jang S.P. et al. Numerical study on leakage detection and location in a simple gas pipeline branch using an array of pressure sensors // J Mech Sci Technol (2010) 24: 983. https://doi.org/10.1007/s12206-010-0216-8



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.