The post Are Pipeline Leaks Deterministic or Stochastic? first appeared on the ISA Interchange blog site.

Is there a distinct pattern in an observation, or a set of observations that is unique to leaks? Before the leak, the pipeline and all the observations we can make about it indicate the pipeline is doing something normal or expected for which there is “coherence” between expected operating conditions and actual observations. Process monitoring systems, in fact, use such observations to ensure the pipeline (or any process) is operating within design parameters and expectations. A leak is usually a stochastic (random) process.

Aside from the fundamental definition involving fluid unintentionally escaping from the pipe the specific conditions of that happening are generally random.

- The location can be anywhere, depending on the precipitating cause. Some locations (crossings, e.g., see
*Detecting Leaks in Pipelines*) produce more leaks than highly protected regions (e.g., in process plants). - Size can vary from seeping through corroding decomposition to a backhoe-induced full-pipe separation.
- Initial conditions of flow in the pipeline depends on the present operational objectives and the conditions under which operation is occurring. This can involve various pressure differentials, flow rates, viscosities, and densities. The pipeline always knows what it’s doing, we only get to know what we can observe about it.
- Leakage flow can be constant or variable. Corrosion-induced leaks, for example often start very small and increase over time as corrosion increases.
- Physical damage may be small, such breakage of a root valve supplying a measurement system impulse line, or huge, like frost-heave breaking and displacing portions of the line pipe.
- Leakage may begin large and decrease in minutes or hours as environmental conditions limit flow. In other words, the path through which the escaping fluid travels on its way into the infinite environment can be restricted by freezing, more earth movement, the flow impacting an impermeable obstruction, or changes in the mode of operation.
- Depending on the fluid, a leak may involve only a vapor component of a more complex fluid. The bulk of the flow, mostly liquid, may continue on its way while the reduced pressure at the leak site flashes some of the higher vapor pressure components into vapor which escapes from the pipeline. The mass flow terms, the percentage of the line flow that escape becomes a tiny percentage of the line flow rate.
- Cold days generally flow differently than warm ones, especially through a leakage path in which earth or sun is involved.
- What you get to see (observe) depends on how and where you look. Good instruments always matter, as does the location of the leak relative to those observations.
- You get the idea!

Fundamentally, what happens depends on the time-dependent events associated with the onset and maintenance of the leak. This has been well-studied and in simple terms reduced to “conservation of momentum,” often referred to as “The principle of impulse and momentum” in fluid systems. This concept suggests that the force (e.g., pressure multiplied by pipe area-of-flow) applied over an interval of time produces a change in velocity that depends on the mass of the affected fluid. Essentially, P x A x dt = M x dv (P is pressure, A is area of flow, dt is an interval of time, M is the mass on which the force of the pressure is acting, and dv is the resulting change in velocity).

*If you would like more information on how to purchase Detecting Leaks in Pipelines, click this link. To download a free 37-page excerpt from the book, click here.*

In more useful terms this is often expressed as:

** dv/dt = F/M = P x A / M**

This can be expressed several ways. A common one is “The change in velocity per unit time during the application of a force F is inversely proportional to the effected mass.” In the most general sense, and with the greatest respect for stochastic considerations that’s what we can depend upon. Another way of looking t this is dv/dt is the acceleration of the fluid resulting from the application of the force F to the mass M.

If we integrate this equation, we get the velocity V as:

**V = F / M x t + Vo where Vo denotes the initial velocity (at time t = 0)**

The astute reader will observe we are converging on the equation for mass flow conservation – the so-called “continuity equation.” Multiply both sides by the area of flow and you see that the flow in must be equal to the initial flow out plus and inflow added to the pipe run because of the forces increasing velocity.

One could also wonder if the integral of this equation might tell us something about conservation of energy. Bernoulli started in a different place with his energy equation and the combination of points of view are interesting fodder for another blog.

Some other things may or may not happen, such as the emission of acoustic noise, but there is no assurance those events will occur in any specific case. Impulse and Momentum, however, always manifests in one way or another and consequently is the forest best (and often the only one) suited for hunting down dinner on any given day.

Leaks occur in an inherently stochastic environment and context so there is no reason to presume there is anything deterministic about them. The “fingerprint” is not deterministically unique – it is as random as the process from which it comes. On the other hand, what if we had sufficient experience with (data from) a particular situation to understand its characteristics and limitations? That opens some doors! With a little proprietary magic, the strangest things can pull on our shirtsleeves and scream, “ME! ME! ME! That will have to wait for another blog.

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Are Pipeline Leaks Deterministic or Stochastic?

Source: ISA News