Gas dynamic principles are applied to construct a leak detection system for long gas pipelines of arbitrary connectivity that contain point resistances to flow. Scaling analysis reveals that inertia is negligible in an isothermal core, leading to a nonlinear diffusion equation for density supplemented by thermal boundary layers at inlets. Smallamplitude linearization is applied to quantify simulation accuracy and to reveal dynamic leak signatures. On operating pipelines, the density equation, coupled with compressible point resistance formulae, is solved by a novel finite element method that yields flow directly. Using only data at boundaries, leaks are identified from differences between simulated and measured flows.
Leak detection on gas pipelines is complicated by fluctuations of supply and demand that mask evidence of leaks. For example, the drop in pressure caused by a small leak may pass unnoticed when the system pressure is rising due to excess supply. Alternatively, a pressure slump from a spike in demand may be mistaken for a leak. On long pipelines, pressure is not uniform and the discrimination of leak and system dynamics is especially difficult. Many leak detection systems are based on empirical or highly idealized models of pipeline dynamics that necessarily require copious data (temporal or spatial) to compensate for the assumptions upon which the model is based. Wholly statistical monitoring systems that flag deviations from normal operating patterns [1] are an example of this class. Acoustical methods [2] are limited to short sections and consequently require measurements along the pipeline. Linear dynamic models are also restricted in range [3] and depend upon data at intermediate points to ensure accuracy. Managing a stream of special data at several locations and times is inconvenient and a barrier to application of these methods for leak detection. By contrast, a rigorous and detailed pipeline model offers highly accurate simulation from standard operating data at relatively few points. Pressure or flow at pipeline boundaries are sufficient and readily available: production is controlled to meet projected demands, and customer conditions are monitored for billing. Such models also require specification of pipeline segments (length, diameter, roughness), auxiliary equipment (valves, compressors, regulators), connections and basic gas properties (molecular weight, viscosity). The pipeline simulator described in this paper is based on an assembly of generic elements, illustrated in Figure 1. Segments of constant diameter pipe join at points of concentrated resistance to form branches. The network is formed by connecting branches at hubs of zero resistance. Note that branches may be comprised of segments of different diameters and that the resistance at points may be zero. Only two segments can join at a point, two or more branches connect at a hub. In this paper we show how scaling (asymptotic) analysis may be applied on long segments to rigorously reduce the full set of nonlinear gas dynamic equations to a simpler, more useful form. Compact equipment and connections are represented by point resistances that track segment dynamics in quasi-steady fashion.
