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Research Papers: Piper and Riser Technology

Transient Flow in Natural Gas Pipelines Using Implicit Finite Difference Schemes

[+] Author and Article Information
Jan Fredrik Helgaker

Polytec Research Institute,
Haugesund, Norway;
Department of Energy and Process Engineering,
Norwegian University of Science and Technology,
Trondheim, Norway
e-mail: jan.fredrik.helgaker@polytec.no

Bernhard Müller, Tor Ytrehus

Department of Energy and Process Engineering,
Norwegian University of Science and Technology,
Trondheim, Norway

1Corresponding author.

Contributed by the Ocean, Offshore, and Arctic Engineering Division of ASME for publication in the JOURNAL OF OFFSHORE MECHANICS AND ARCTIC ENGINEERING Manuscript received June 17, 2013; final manuscript received January 25, 2014; published online April 1, 2014. Assoc. Editor: Rene Huijsmans.

J. Offshore Mech. Arct. Eng 136(3), 031701 (Apr 01, 2014) (11 pages) Paper No: OMAE-13-1066; doi: 10.1115/1.4026848 History: Received June 17, 2013; Revised January 25, 2014

Transmission of natural gas through high pressure pipelines has been modeled by numerically solving the governing equations for one-dimensional compressible flow using implicit finite difference methods. In the first case the backward Euler method is considered using both standard first-order upwind and second-order centered differences for the spatial derivatives. The first-order upwind approximation, which is a one-sided approximation, is found to be unstable for CFL numbers less than 1, while the centered difference approximation is stable for any CFL number. In the second case a cell centered method is considered where flow values are calculated at the midpoint between grid points. This method is also stable for any CFL number. However, for a discontinuous change in inlet temperature, the method is observed to introduce unphysical oscillations in the temperature profile along the pipeline. A solution strategy where the hydraulic and thermal models are solved separately using different discretization techniques is suggested. Such a solution strategy does not introduce unphysical oscillations for discontinuous changes in inlet boundary conditions and is found to be stable for any CFL number. The one-dimensional flow model is validated using operational data from a high pressure natural gas pipeline.

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References

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Poloni, M., Winterbone, D. E., and Nichols, J. R., 1987, “Comparison of Unsteady Flow Calculations in a Pipe by the Method of Characteristics and the Two-Step Differential Lax-Wendroff Method,” Int. J. Mech. Sci., 29(5), pp. 367–378. [CrossRef]
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Ytrehus, T., and Helgaker, J. F., 2013, “Energy Dissipation Effect in the One-Dimensional Limit of the Energy Equation in Turbulent Compressible Flow,” ASME J. Fluids Eng., 135(6), p. 061201. [CrossRef]
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Osiadacz, A. J., and Yedroudj, M., 1987, “A Comparison of a Finite Element Method and a Finite Difference Method for Transient Simulation of a Gas Pipeline,” Appl. Math. Model., 13(2), pp. 79–85. [CrossRef]
Osiadacz, A. J., and Chaczykowski, M., 2001, “Comparison of Isothermal and Non-Isothermal Pipeline Gas Flow Models,” Chem. Eng. J., 81(1-3), pp. 41–51. [CrossRef]
Helgaker, J. F., and Ytrehus, T., 2012, “Coupling Between Continuity/Momentum and Energy Equation in 1D Gas Flow,” Energy Proc., 26, pp. 82–89. [CrossRef]
Barley, J., 2012, “Thermal Decoupling: An Investigation,” Proceedings to 43rd PSIG Annual Meeting, Santa Fe, New Mexico, May 15–18.
Modisette, J., 2012, “Instability and Other Numerical Problems in Finite Difference Pipeline Models,” Proceedings to 43rd PSIG Annual Meeting, Santa Fe, New Mexico, May 15–18.
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Figures

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Fig. 1

Stencil used in the finite difference method

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Fig. 2

Local errors for p, m·, and T as a function of grid points N

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Fig. 3

Outlet boundary condition for mass flow for the hydraulic model

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Fig. 4

Results for hydraulic model. Left: Inlet mass flow, and right: outlet pressure.

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Fig. 5

Difference between implicit cell centered method and backward Euler upwind method for the hydraulic model. Left: Difference in inlet mass flow, and right: difference in outlet pressure.

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Fig. 6

Validation of hydraulic model using the implicit cell centered finite difference method. Left: Inlet mass flow, and right: outlet pressure. Modeled results show good agreement with measured values.

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Fig. 7

Effect of grid refinement for results in Fig. 6. Data presented above is the absolute difference between the selected grid and the high resolution solution (Δx = 0.25 km).

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Fig. 8

Boundary conditions used in full nonisothermal model. Left: Inlet mass flow f(t), and right: inlet temperature g(t).

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Fig. 9

Modeled flow results. Top: Inlet pressure, middle: outlet mass flow, and bottom: outlet temperature.

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Fig. 10

Outlet gas temperature computed using the implicit cell centered method for two different grid spacings, Δx = 0.5 km and Δx = 1 km. For a finer grid the oscillations in outlet temperature are reduced.

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Fig. 11

Temperature profile along 200 km pipeline computed using the implicit cell centered method with Δx = 1 km. (a) t = 1 h. Before the inlet temperature is reduced the temperature profile is nice and smooth. (b) t = 1 h 40 min. A discontinuous change in inlet temperature introduces nonphysical oscillations in the temperature profile. (c) t = 2 h 10 min. Oscillations are still present but have been slightly damped. (d) t = 4 h 40 min. Oscillations still present and have moved along the pipeline with the gas velocity u.

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Fig. 12

Left Unphysical oscillations which occur when a discontinuous change in inlet temperature is introduced using the implicit cell centered method and the backward Euler centered method. The oscillations are most dominant in the implicit cell centered method. Right: Temperature profile at the inlet of the pipe using the implicit cell centered method for different grid sizes.

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Fig. 13

Temperature profile along 200 km pipeline using the two solution strategies. The difference between the two solution strategies is how the energy equation is solved. In the first case the implicit cell centered method is used (CC) and all governing equations are solved simultaneously, while in the second case the hydraulic and thermal model are solved separately (Split). The hydraulic model is solved using the implicit cell centered method while the thermal model is solved using the backward Euler upwind method. (a) t = 1 h 40 min. A discontinuous change in inlet temperature introduces nonphysical oscillations in the case of the implicit cell centered method. (b) t = 1 h 40 min. A close up view of the temperature profile shows that the when the hydraulic and thermal model are solved separately using different discretizations no such oscillations are introduced. (c) t = 2 h 10 min. Oscillations have been damped, but are still present in the implicit cell centered method. (d) t = 4 h 10 min. A difference between the two solution strategies can still be seen.

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Fig. 14

Outlet temperature as a function of time using the two solution strategies. Solving all three governing equations simultaneously using the implicit cell centered method (CC) introduces small oscillations in the outlet temperature. When solving the hydraulic and thermal model separately using the backward Euler upwind method for the thermal model (Split) no such oscillations are present.

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Fig. 15

Validation of nonisothermal model using operational data from 650 km offshore pipeline. Top: Inlet pressure, middle: outlet mass flow, and bottom: outlet temperature.

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