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Are Pipeline Leaks Deterministic or Stochastic?

This guest blog post is part of a series written by Edward J. Farmer, PE, ISA Fellow and author of the new ISA book Detecting Leaks in Pipelines. To download a free excerpt from Detecting Leaks in Pipelines, click here. If you would like more information on how to purchase the book, click this link. To read all the posts in this series, scroll to the bottom of this post for the link archive.

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.

How to Optimize Pipeline Leak Detection: Focus on Design, Equipment and Insightful Operating Practices
What You Can Learn About Pipeline Leaks From Government Statistics
Is Theft the New Frontier for Process Control Equipment?
What Is the Impact of Theft, Accidents, and Natural Losses From Pipelines?
Can Risk Analysis Really Be Reduced to a Simple Procedure?
Do Government Pipeline Regulations Improve Safety?
What Are the Performance Measures for Pipeline Leak Detection?
What Observations Improve Specificity in Pipeline Leak Detection?
Three Decades of Life with Pipeline Leak Detection
How to Test and Validate a Pipeline Leak Detection System
Does Instrument Placement Matter in Dynamic Process Control?
Condition-Dependent Conundrum: How to Obtain Accurate Measurement in the Process Industries
Are Pipeline Leaks Deterministic or Stochastic?
How Differing Conditions Impact the Validity of Industrial Pipeline Monitoring and Leak Detection Assumptions
How Does Heat Transfer Affect Operation of Your Natural Gas or Crude Oil Pipeline?
Why You Must Factor Maintenance Into the Cost of Any Industrial System
Raw Beginnings: The Evolution of Offshore Oil Industry Pipeline Safety
How Long Does It Take to Detect a Leak on an Oil or Gas Pipeline?
Book Excerpt + Author Q&A: Detecting Leaks in Pipelines

About the Author
Edward Farmer has more than 40 years of experience in the “high tech” part of the oil industry. He originally graduated with a bachelor of science degree in electrical engineering from California State University, Chico, where he also completed the master’s program in physical science. Over the years, Edward has designed SCADA hardware and software, practiced and written extensively about process control technology, and has worked extensively in pipeline leak detection. He is the inventor of the Pressure Point Analysis® leak detection system as well as the Locator® high-accuracy, low-bandwidth leak location system. He is a Registered Professional Engineer in five states and has worked on a broad scope of projects worldwide. His work has produced three books, numerous articles, and four patents. Edward has also worked extensively in military communications where he has authored many papers for military publications and participated in the development and evaluation of two radio antennas currently in U.S. inventory. He is a graduate of the U.S. Marine Corps Command and Staff College. He is the owner and president of EFA Technologies, Inc., manufacturer of the LeakNet family of pipeline leak detection products.

Connect with Ed


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