Higher-Resolution Numerical Solution for 2-D Transient Natural Gas Pipeline Flows
Authors: Ibraheem S.O. , Adewumi M.A. 
Published: December 2007

Abstract
Reliable modeling and prediction of transients features in transmission pipelines are desirable for optimal control of gas deliverability, design and implementation of active controls, and modeling of operational behavior of network peripheral equipment (e.g., chokes, valves, compressors, etc.). A numerical procedure is developed to simulate transient phenomena in a 2-D natural gas flow. Natural gas flow in a pipe is governed by hydrodynamic equations that are highly non-linear and hyperbolic. In this paper, a special Runge-Kutta method is employed to model accurate evolution of flow characteristics. The upwind method of van Leer, in which the numerical flux vectors are split to match the directions of wave propagation, is chosen as the base solution algorithm. Furthermore, the Total Variation Diminishing (TVD) technique is employed to extend to higher-order accuracy in order to resolve sharp discontinuous fronts. Several test cases were simulated to validate the developed code. The numerical algorithm is very stable, and although the problems solved were stiff and possess multiple and disparate time scales, excellent prediction for the unsteady features, especially sharp wave front and proper front tracking, were obtained.

Introduction
Natural gas is transported over thousands of miles of pipelines that crisscross several terrains and weather conditions. It is important to be able to predict the pressure surges that may be caused by either accidental and/or incidental occurrences for optimal design and safety purposes. Severe damage to pipelines and network peripheral facilities has been attributed to the effects of pressure surges which are often analyzed by inadequate models. In fact, oscillatory transient pressures are known to build up to unusually large magnitude with many undesirable consequences. Several works that have attempted to solve this problem have been based either on graphical methods or on the Method of Characteristics (MOC). Both methods have tremendously improved our knowledge of transient phenomena but are now becoming inadequate in certain difficult situations, especially multidimensions. A large body of literature is devoted to MOC, including the work of Taylor et al., Wylie et al., and more recently, the work of Rao and Eswaran. In most of these cases, several assumptions were made, such as neglecting the inertial term or the non-linear convective term, to alleviate computational problems. Most recently, however, the advanced techniques that are developed in the area of Computational Fluid Dynamics are now being investigated for solving the full conservation laws that govern the transport processes. Recent works in this area include those by Tan et al., Zhou and Adewumi, and Ibraheem and Adewumi. Tan et al. employed a center implicit method (CIM) to solve for pressure transients in pipelines. Although their CIM method was found to be faster than the MOC method, it yielded unsatisfactory results for sudden and sharp transients, even with the addition of artificial viscosity. It is further observed that a severe restriction is placed on their time steps. In the work of Zhou and Adewumi, a special type of hybrid TVD scheme with appropriate boundary conditions handling capability is developed, with Roe scheme as the underlying algorithm. The formulation employed a fixed point iteration rather than a time-stepping scheme for solution method. Ibraheem and Adewumi have also developed a different kind of TVD scheme with Runge-Kutta as the solution method that proved to be more efficient in tracking the frontal discontinuities.
In all the preceding work, one-dimensional model where the velocity and pressure are assumed to be uniform across the pipe sections were used. The results are in agreement with the experimental data only in high Reynold number flow.
P. 473