frequency response coding for the location of leaks in ...

FREQUENCY RESPONSE CODING FOR THE LOCATION OF LEAKS IN SINGLE PIPELINE SYSTEMS Pedro J. Lee l , John P, Vitkovskf Martin F, Lambert3 Angus R., Simpson4 and James A. Liggett5 1 Postgraduate Student, Department of Civil & Environmental Engineering, The University of Adelaide, Adelaide SA 5005, Australia; Email: [email protected] 2 Research Associate, Department of Civil & Environmental Engineering, The University of Adelaide, Adelaide SA 5005, Australia; Email: [email protected] 3 Senior Lecturer, Department of Civil & Environmental Engineering, The University of Adelaide, Adelaide SA 5005, Australia; Email: [email protected] 4 Associate Professor, Department of Civil & Environmental Engineering, The University of Adelaide, Adelaide SA 5005, Australia; Email: [email protected] 5 Professor Emeritus, School of Civil & Environmental Engineering, Cornell University, Ithaca, NY 14853-3501, USA. Email: [email protected] Abstract

The petroleum and water industries depend on pipelines for the mass transport of fluids and this has led to the increased development of pipeline integrity monitoring systems. The use of fluid transients for this purpose has gained popularity over the recent years as they travel at high speeds and are susceptible to changes in the internal conditions of the pipe. Analysis of the pressure responses from the pipeline in the frequency domain generates the frequency response diagram for the system and the shape of this diagram is indicative of the location and size of a leak within the pipeline. This paper outlines the use of the frequency response diagram as a means of leak location and size estimation in a reservoir-pipe-valve-reservoir system. An analytical solution describing the leak induced impact on this frequency response diagram can be used to fmd the exact location of any number of leaks within the pipe and their corresponding sizes. Keywords: Leakage, Water Pipelines, Frequency response, Random vibration, Resonance, Transfer functions, Transient flow, Linear systems

Introduction Fluid transients travel at high speeds within pipelines with a pattern of behaviour that is indicative of the flow condition within the system, making them an attractive means of determining system integrity. Numerous published methods utilize transient behaviour for the purpose of leak detection. These include the inverse transient method (Liggett and Chen 1994, Vitkovsky et al. 1999), the transient damping method (Wang, et al. 2002), and time-domain reflectometry techniques (Covas and Ramos 1999). The idea of custom designing pressure signals specifically for leak detection is a new development in this field. These designed transients, also known as "coded transients", have inherent properties that allow extraction of the pipeline integrity information. Lee et al. (2002a, 2002b) proposed the use of a generated time-invariant signal as a means of leak detection. Signal-processing techniques are used to analyze these signals in the frequency domain. Wang et al. (2002) determined that the presence of a leak within the pipeline imposes a frequency dependent impact on the damping of fluid transients. Lee et al. (2002a, 2002b) showed that a similar frequency dependent leak impact exists in the frequency response characteristics of a

pipeline system, and the shape of the frequency response diagram contains information concerning the leak. This paper presents an analytical solution that describes the impact of a leak on the frequency response of a pipeline system. This solution can be used to individually detect, quantify and locate mUltiple leaks in a single pipeline configuration.

Frequency Response of a Single Pipeline System Figure 1 - Single pipeline section used in coding derivation Position i

i+1 1+2

i+~

I

I

I

I I I I I

H~ "I \(

I

I I

>~(

L1

Ern

L2

~~P' Oscillatory Valve

Leak

Consider the single pipeline system of Figure 1. When an in-line valve downstream of the system is forced to flutter in a sinusoidal fashion, the magnitude of the pressure response measured just upstream of the valve is dependent on the frequency of the flutter. While a pipeline reinforces and transmits input signals of a particular frequency (e.g., the resonance frequencies), others are effectively absorbed within the system (e.g., the anti-resonance frequencies). It follows that pipeline systems can be considered as filters, the characteristics of which are determined by system properties such as boundary conditions, friction, and wave speed. Additional details can be found in Lee et al. (2002a, 2002b). Figure 2 - Impact of changing leak size and position on the frequency response diagram

8

I

Q)

CIl

c

a

~

7

--()--- Leak at 1400m. C A = 0.00014

---0-- Leak at 700m, CdAo

-+-- Leak at 700m, CdA =0.00014

- - No Leak

do

=0.00028

0

6

3

2

Q)

a::::

1

2

4

Frequency Ratio

(0

R

6 (=roI(O )

8

10

th

The degree that each frequency component is absorbed or transmitted within the pipeline can be described by a frequency response diagram (FRD), also known as the transfer function for the system (Lynn, 1973). This diagram relates both the magnitude and phase of the system output to the system input for different frequencies.

Lee et al. (2002a, 2002b) have illustrated that the FRD of a non-leaking pipeline results in equal magnitude peaks and troughs when measured at the perturbation boundary. The presence of a leak within the system was found to cause a pattern damping of the resonance peaks in the FRD. Figure 2 shows the FRD for the pipeline in Figure 1 for four difference leaking situations. The location of the leak was found to have an impact on the shape of the leak induced damping pattern, while the size of the leak affects only the magnitude of this damping. The analytical solution for the leak-induced impact on the FRD for a single pipeline system is now derived.

Derivation of Frequency Response Leak Impact Equations The frequency response equations are derived for a single pipeline system with a constant head upstream boundary with a single leak as shown in Figure 1. The excitation of the system occurs at the downstream boundary, using an oscillatory inline valve. Considering Pipe 1 in Figure 1, the linearised equations relating the variation of head and discharge between the extremities of a section of uniform pipe is given by Chaudhry (1987) as

q 1+1

= cosh(~)l))q

1 -

sinh(~)l))h

;

(1)

I

)

h where,

~

1+1

= -z) sinh(~)l) )qi + cosh(~Jl )h

i

(2)

is the propagation function

11=

(3)

and Z is the characteristic impedance 2

Z= ~ joogA

(4)

1= the length of the pipe section, qi and hi are the complex discharge and head fluctuation at the ith position in the system, 00 = oscillation frequency in radians, a = celerity of the pipeline, g = gravitational acceleration, A = cross-section area of the pipe and R is the frictional resistance, defmed as, R=(fQo)/(gDA2) for turbulent flow and R=(64u)/(gAD2) for laminar flow withf= Darcy-Weisbach friction factor, U = kinematic viscosity, Qo = ratio of the steady state to mean flow through the pipe and D = diameter of the pipeline. The subscript "1" denotes the pipe number associated with each parameter, and j = r-i . At the upstream reservoir boundary position, the head fluctuation is set as zero, hi = 0, substitution ofthis boundary condition into Eqs. (1) and Eqs. (2) yields,

= cosh(~J))ql = -Z) sinh(~)ll )qi

ql+l h 1+1

The linear, oscillatory equations for the leak orifice is given by Lee et al. (2002) as

(5) (6)

q i+2 _-

qi+1

QL- hi+1 --

(7)

2HL h i+2

=hl+l

(8)

where, QL, HL are the steady state discharge out of the leak, and the steady state head at the leak respectively. Substituting Eqs. (7) and (8) into Eqs. (5) and (6), resubstituting Eqs. (1) and (2) for pipe 2 and solving for the oscillatory valve boundary condition (Chaudhry, 1987) gives the measured head response upstream of the valve at position i + 3 in Figure 1 as h i +3

Lh = - . . , = - - - - - - -- - -- - - - - - ------=

(9)

where, I:!.Ho is the steady state head loss across the valve orifice, 'to is the steady state dimensionless opening of the valve. I:!.'t is the magnitude of the dimensionless valve

i, c =.!!...- , a gA

opening fluctuation and Qo is the steady state discharge across the valve, b =

and the subscripts denote the pipe number. Note that Eq. (9) assumes that the leak size is small, (on the order of 1% of the pipe area), and steady state friction plays a minor role in the leak induced pattern damping on the FRD. For small leaks, it was also found that the term }QLC2 Sin(b J co) sin(b2 ffi)/(2HL) in Eq. (9) plays a minor role in the magnitude of the FRD at resonance frequencies. At these frequencies, which occur at, (10)

where n is a positive integer, representing the peak number. Eq. (9) now becomes (11)

where XL * = LJ/(LJ+L2J. Inversion and taking absolute values gives

1

(12)

For a non-leaking pipeline, the frequency response at the resonance peaks is a constant and is equal to

1

=

Ih:'lh~elT* I

'to 211'tMlo

(13)

Eq. (12) shows that the response at the resonance peaks in a leaking pipeline is reduced from the non-leaking case by a periodic function of frequency XL *. The inversion of the response function was performed to allow a more accurate extraction of this periodic function. Note that the frequency is in terms of units lin, where n is the harmonic peak number. As XL * is defined as the dimensionless distance from the upstream system boundary to the leak, extraction of the dominant frequency in Eq. (12) will lead to the accurate location of the leak in the single pipeline system. Table 1 - Example pipeline parameters

PARAMETER Total Length Diameter HI H2 A Roughness height ~L

CdAOLeak / Apipe Location of Leak

VALUE 2000m 0.30m 50.0m 20.0m 1200 mls 0.000046 m 0.1 0.2% Varies

Figure 3 shows the Fourier decomposition of the peak magnitudes in the FRD for the pipeline in Figure 1 with leaks at four different locations. Each leak is the same size and the parameters of the pipeline is given in Table 1. From the sampling theorem, the highest frequency (the Nyquist frequency) that can be represented is lin = 0.5. In the situation where XL * < 0.5 the peaks of the FRD have a fluctuation pattern of frequency equal to x/ validating the result ofEq. (11). Note that for XL * > 0.5, the oscillation frequency exceeds the Nyquist frequency, and is aliased down to the lower frequency of (1- XL*). Leaks located beyond the center of the pipe are therefore indistinguishable from leaks located at the symmetric position at the other end of the pipeline based on the leak induced frequency. From this result, it can be concluded that for each observed peak pattern frequency, there are two possible symmetric leak locations. To determine the exact location of the leak, the phase offset of the pattern can be used. From Eq. (12), each perturbation function has an initial phase offset of (1t+7tXL), the additional1t arises from the negative in front of the cosine. For XL· > 0.5, aliasing of the function results in a phase reversal, and the phase of the oscillatjon becomes - (1t+nxL). It can be seen that for leaks located at XL * < 0.5, the phase of the leak-induced signal is located in the third quadrant of the unit circle, while leaks located at XL * > 0.5 have phase located in the first quadrant. The phases of the leak induced patterns in Figure 3 are shown in Table 2.

Figure 3 - Fourier decomposition of peak magnitudes in the FRD of two leaking pipes

E 0.035

--

..~ 0.03 CO E ~ 0.025

l' , , ,

1

,

\ ' = 0.138

-

-

\' =0.862

\' = 0.024

- - \'=0.384

CO

Q)

0.

c

0.02

a 0.015

:;::;

co

·c co >

0.01

~ 0.005 :::J

-g

0

~

~ -0.005 0

-'

0.1

0.2

0.3

0.4

0.5

Frequency in peak number =1/n

Table 2 - Phases of leak induced patterns

LEAK LOCATION XL· 0.138 0.024 0.862 0.384

PHASE (rads) 1.1381t

1.0241t -1.8621t 1.3841t

Once the frequency of the leak-induced pattern oscillation is determined, the starting phase of the oscillation can be used to detennine the exact location of the leak. Furthennore, the magnitude of the leak-induced pattern can be used to find the size of the leak. The magnitude of the leak-induced impact can be found from Eq. (12), and is given by (14)

The leak sizes for the four different leak location examples are all identical, and from Eq. (14) the magnitude of the oscillation is given by 0.02015 m- l , which corresponds to the Fourier decomposition results. Eq. (14) can be further rearranged to give the leak impedance parameter, QdHL . The lumped leak parameter, CttAo, where Cd = discharge coefficient of the leak and Ao = the area of the leak orifice, can be detennined from the orifice equation, (15)

Procedure for leak detection The results above can be used in a procedure for leak detection in a single pipeline system. The procedure is as follows; 1. Using a series of sinusoidal transients in frequency sweep, or a wide band transient as described in Lee et al. (2002a, 2002b), determine the frequency response diagram for the pipeline system. For an asymmetric system, this FRD should be extracted from the excitation boundary. 2. Perform a Fourier transform of the data containing only the inverted response at each resonant peak, identifyin the major frequency, f, within the response. The dimensionless leak location, XL , is either for 1-f. 3. Determine the phase of the oscillation to determine which half of the pipe the leak resides. For phase located in the third quadrant, the corresponding peak is located at XL * < 0.5 .. For phase located in the fIrst quadtrant, the corresponding peak is located atxL* > 0.5. 4. Determine the size of the leak. from the magnitude of the frequency peak.

r

The above technique is powerful and can be further expanded to cover for a multiple leak situation. Figure 4 shows the Fourier decomposition of the response upstream of the valve when three leaks (XL * = 0.244, 0.427, 0.641) of the same size (CaAoLeatlApipe =0.02%) are located simultaneously in the pipeline of Figure 1. Note that the Fourier decomposition shows spikes at all the correct frequencies. A summary of the multi-leak results is shown in Table 3. The combination of the spike frequencies and their associated phases are able to accurately determine the exact location of each leak in the pipeline. Table 3 - Summary of multi-leak example

LEAK

LOCATION MAGNITUDE (11m)

PHASE (rads)

XL *

0.244 0.427 0.641

0.002 0.002 0.002

-2.37456 -1.79931 1.128967

PREDICTED LOCATION XL * 0.244 0.427 0.641

Conclusions Pipeline systems display frequency dependent behaviour and can therefore be considered as physical fIlters. Frequency response diagrams are a way of summarising this frequency dependency and are used in many system analysis applications. This paper introduces a method of detecting leaks in a single pipeline using the frequency response diagram. The frequency response diagram can be extracted from a pipeline system through the injection of continuous low amplitude fluid transients and is well suited for online monitoring of system integrity. The proposed method, derived from linearised equations of motion and mass balance, is able to accurately locate any number of leaks in a single pipeline confIguration without inverse calibration, making it one of the more powerful and effIcient fluid transient leak detection methods available. The use of resonance peaks for the analysis has the added advantage of increasing the signal to noise ratio while utilising the fact that leaks have the greatest impact at resonance frequencies. This technique can also be further adapted for the location and sizing of discrete blockages in a pipeline system.

Experimental work is currently underway at the University of Adelaide to validate the proposed technique. Figure 4 - Multi-leak situation

...........

0.0025

E

--

T"""

en ro

E ~

ro

0.002 0.0015

~

'.

1- '

(])

c.. c c

-

0.001

0

:;::::;

ro ·c ro

0.0005

>

"0 (])

U ::::J "0 C

~ ro

0 -0.0005

(])

....J

i

0

I

0.1

I

I

!!

0.2

,~--,,-I,-",-,---,

0.3

I

I

0.4

L-L-

0.5

Frequency in peak number =1/n

References Chaudhry, M. H. (1987). Applied Hydraulic Transients. Van Nostrand Reinhold Company Inc, New York, USA Covas, D. and Ramos, H. (1999). "Leakage detection in single pipelines using pressure wave behaviour," Water Industry System: Modelling and Optimisation Application, 1, pp 287-299. Lee, P.J., VitkovsJeY, J.P., Lambert, M.F., Simpson, A.R., and Liggett, J.A. (2002a). "Leak detection in pipelines using an inverse resonance method." 2002 Conference on Water Resources Planning & Management, ASCE, 19-22 May, Virginia, USA [CDROM] Lee, P.J., Vitkovsky, J.P., Lambert, M.F., Simpson, AR., and Liggett, J.A. (2002b). "Discussion of - Leak Detection in Pipes by Frequency Response Method Using a Step Excitation", submitted to Journal of Hydraulic Research, IAHR.. Liggett, J.A and Chen, L.C. (1994). "Inverse transient analysis in pipe network," Journal of Hydraulic Engineering, ASeE, 120(8), pp. 934-955. VitkovsJeY, J.P., and Simpson, AR. and Lambert, M.F. (2000). "Leak detection and calibration using transients and genetic algorithms," Journal of Water Resources Planning and Management, 126(4), pp. 262265. Wang, X.J., Lambert, M.F., Simpson, AR., Liggett, J.A, and VitkovsJeY, J.P. (2002). "Leak Detection in Pipeline Systems Using the Damping of Fluid Transients." Journal of Hydraulic Engineering, ASCE, 128(7), July, 697-711.