Many of the transient models used to simulate gas and liquid pipelines are based on the method of finite differences (FD). FD models represent continuous model properties like pressure, flow rate, and temperature on a discrete grid of points in space and time, thereby approximating the true continuous partial differential equations (PDEs) governing fluid flow as a finite set of algebraic equations. This approach can be efficient and accurate but is subject to numerical instability and a host of other numerical problems if the grids aren't chosen appropriately. This article explores the situations in which these problems arise, how to diagnose them, and and what can be done about them. Explicit, fully implicit, and partially implicit FD models are examined. Recommendations for model building and model use are developed based on the results of a variety of test scenarios.
All computer pipeline models embody some or all of a set of differential equations representing conservation of mass, momentum, energy, and possibly species and some other quantities. While a solution of the underlying PDEs is guaranteed to represent what would actually happen in the pipeline, the conversion of them into algebraic equations brings in the additional possibility of numerical errors: even if the PDEs were right, the solution might still be wrong. For the purposes of this paper we will consider only the conservation of mass, momentum, and energy PDEs. These are the required equations to make a pipeline model that solves for the pressure, temperature, and flow rate everywhere in the pipe system as a function of location and time. Just these three PDEs provide several distinct opportunities for the introduction of serious numerical errors into the solution. The process of converting a set of PDEs on paper to a FD solution scheme is known as "discretization".