Distributed deterioration detection and location in ...

Distributed deterioration detection and location in single pipes using the impulse response function by Gong, J., Lambert, M. F., Simpson, A. R., and Zecchin, A. C.

14th Water Distribution Systems Analysis Conference (WDSA 2012)

Citation: Gong, J., Lambert, M. F., Simpson, A. R., and Zecchin, A. C. (2012). “Distributed deterioration detection and location in single pipes using the impulse response function”. 14th Water Distribution Systems Analysis Conference (WDSA 2012), American Society of Civil Engineers, Adelaide, South Australia. For further information about this paper please email Angus Simpson at [email protected]

DISTRIBUTED DETERIORATION DETECTION IN SINGLE PIPES USING THE IMPULSE RESPONSE FUNCTION Jinzhe Gong, Martin F. Lambert, Angus R. Simpson and Aaron C. Zecchin School of Civil, Environmental and Mining Engineering, University of Adelaide, Adelaide SA 5005, Australia

ABSTRACT This paper describes a novel distributed deterioration detection method in single pipelines using a time-domain pressure response trace and the impulse response function (IRF) of the pipeline. The hydraulic impedance of a deteriorated section is determined from the size of the first deteriorationinduced reflection. The wave speed, internal diameter, and wall thickness of the deteriorated section can be estimated if the anomaly impedance is due solely to a change in wall thickness. Sensitivity analyses are performed to study the relationship between the change in wall thickness and the size of the reflected and transmitted waves. The IRF of the pipeline system is used to determine the arrival time and duration of the first deterioration-induced reflection, which are then used to determine the location and length of the deterioration. Experimental verification has been performed on a single pipeline that includes a thinner-walled pipe section. The hydraulic impedance, wave speed, wall thickness, length and location of this section have been determined successfully. INTRODUCTION The high speed and wide operational range of fluid transients (water hammer waves) (Chaudhry 1987) make them attractive and promising for detecting anomalies in pipeline systems. In the last two decades, a number of transient-based fault detection methods have been developed for water transmission pipelines. Most of the existing methods only focus on the detection of discrete faults, such as leaks (Colombo et al. 2009; Puust et al. 2010) and discrete blockages (Vítkovský et al. 2003; Wang et al. 2005; Mohapatra et al. 2006; Lee et al. 2008; Sattar et al. 2008). However, the detection of areas of widespread deterioration has not attracted much attention, although non-discrete deterioration such as extended areas of corrosion, extended blockages and cement mortar lining spalling, commonly exist in real pipelines (Stephens et al. 2008). Distributed deterioration can reduce pipeline operation efficiency, introduce contamination into the transmitted fluid, and may also lead to the development of bursts or severe blockages over time. In addition, multiple reflections from a deteriorated section can distort transient response traces and make the detection of other faults even more difficult. As a result, understanding the behaviour of distributed deterioration in transient responses and developing effective distributed deterioration detection methods are critical. At present, only a few approaches have been developed for detecting extended anomalies in pipes and they all have limitations in real applications (Wu 1994; Scott and Satterwhite 1998; De Salis and Oldham 1999; Adewumi et al. 2000). Investigating distributed deterioration detection in water transmission pipelines, Arbon et al. (2007) conducted field studies on extended blockage detection in water distribution pipelines, and Stephen et al. (2008) performed field studies on the detection of changes in pipe wall thickness. The two studies both used fluid transients and inverse transient analysis (ITA) (Liggett and Chen 1994). The field tests undertaken by Stephens et al. (2008) indicated that the loss of cement mortar lining could lead to wall corrosion and significant changes in wave speed. However, the structural complexity and parametric uncertainties of real pipes are serious obstacles to an efficient and accurate ITA.

ISBN 978-1-922107-58-9 © Engineers Australia, 2012. All rights reserved.

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14th Water Distribution Systems Analysis Conference

Gong et al. (2012) proposed a technique for detecting single distributed deterioration by analyzing the time-domain transient pressure trace resulting from a step transient wave. A reservoir-pipelinevalve (RPV) system was used, in which an incident wave was generated by an abrupt closure of the valve at the end of the pipe. Analytical analysis demonstrated that if a deteriorated section is located within the pipeline, which can be modelled as a pipe section with anomalous hydraulic impedance, a square shape perturbation will theoretically appear in the first half period of the pressure response trace measured at the valve. The magnitude of the square perturbation is related to the hydraulic impedance of the deteriorated section, while the arrival time of this deterioration-induced reflection is indicative of the location. The technique was verified using experimental data, and the location and impedance of a pipe section with a thinner wall thickness were estimated successfully. One challenge of this technique is that, in the experimental pressure trace, the deterioration-induced reflection does not have a clear square shape as shown in the analytical analysis, but rather a perturbation with smooth boundaries. This is mainly due to the curved wave front of the incident wave, which in turn results from the limitation in the maneuverability of transient generation valves. The smooth boundaries bring difficulty in the estimation of the arrival time and duration of the deterioration-induced reflection. This research is an extension to the work presented in Gong et al. (2012). Sensitivity analyses are performed to study the relationship between the change in wall thickness and the size of the deterioration-induced reflection. The impulse response function (IRF) (Vítkovský et al. 2003; Lee et al. 2007b) of the pipeline system is estimated from the original pressure trace, and it is then used to determine arrival time and duration of the deterioration-induced reflection. In the IRF, the square deterioration-induced reflection is transferred to two spikes, yielding a better estimation of the critical times. The experimental data reported in Gong et al. (2012) is used to validate the proposed technique. BACKGROUND This section provides some background information on the application of the time-domain reflectometry (TDR) and the impulse response function (IRF) to pipeline fault detection. The definition of pipeline characteristic impedance and the wave speed formula are also reviewed. Time-domain reflectometry. In the transient-based pipeline fault detection research literature, the time-domain reflectometry (TDR) has mainly been used for locating leaks (Jönsson and Larson 1992; Silva et al. 1996; Brunone 1999; Lee et al. 2007a) along with minor loss elements (orifices) (Contractor 1965). When using TDR, a transient pressure wave is generated (usually by changing the opening of a valve) and transmitted along a pipeline. The speed of the wave, which can be determined theoretically or experimentally, is related to properties of the pipe and the fluid. Theoretically, any physical change in the pipeline can introduce reflections when a transient wave propagates through it. The reflected signals can be registered by pressure transducers located along the pipeline. TDR utilises the arrival time of the partially reflected signals together with the wave speed to determine the origin of the reflection. TDR-based fault location techniques are attractive due to their simplicity both in application and analysis. However, in real applications, measured pressure traces are usually distorted due to signal dissipation and dispersion (Vítkovský et al. 2007). Reflections are smoothed, and as a result the arrival time and duration of any reflected signal are hard to accurately read from the raw timedomain pressure trace. To improve the accuracy of the determination of the arrival time, the impulse response function (IRF) of a pipeline system can be employed (Vítkovský et al. 2003; Lee et al. 2007b). Impulse response function. The impulse response function (IRF) describes the response of a system when a unit impulse is applied as the input. The IRF for a specific pipeline system is unique

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and independent of the shape of the input signal. A complicated input signal can be considered a sequence of weighted and time-lagged impulses, and the corresponding output is a sum of the scaled and time-lagged impulse responses (Suo and Wylie 1989), which can be described as t

y (t ) = I (t ) ∗ x(t ) = ∫ I (t − τ )x(τ ) dτ 0

(1)

where I (t ) represents the impulse response function; x(t ) and y (t ) are input and output signal, respectively; the symbol ‘ ∗ ’ denotes a convolution operation. Suo and Wylie (1989) employed IRF to predict the head response (output) through a convolution process between the IRF and the input signal. The head response prediction procedure was also adopted in Ferrante and Brunone (2003) and Kim (2005). Liou (1998) proposed a leak detection method by analysing the maximum magnitude of the impulse response extracted from various positions along a pipe. Vítkovský et al. (2003) utilised the IRF in place of the original transient trace to increase the accuracy of leak and discrete blockage location. In their numerical modelling, the reflected signals from discrete hydraulic elements appeared to be sharp impulses with a well-defined peak. Lee et al. (2007b) further developed the IRF-based leak detection method and performed experiments in the laboratory. In Lee et al. (2007b), the discharge perturbation at the valve during the valve movement is used as the input to the system. It is termed ‘induced discharge perturbation’ and can be determined from the pressure variation during the valve manoeuvre for a fast valve movement using Joukowsky equation. The frequency response function (FRF) and impulse response function (IRF) of the tested pipeline can be derived through correlation-based analysis (Lee et al. 2007b). The use of the IRF in place of the raw transient trace can improve the operation of the TDR-based distributed deterioration detection procedure. The IRF refines the output signal by converting each reflected signal into a sharp impulse, so that the critical times can be determined more easily and accurately. Hydraulic impedance and the wave speed formula. The characteristic hydraulic impedance of a uniform pipeline is defined as B = a / ( gA)

(2)

where a is the wave speed; A is the internal cross-sectional area of the pipeline; and g is the gravitational acceleration. The impedance is sensitive to changes in wall thickness, which not only alter the internal cross-sectional area but also alter the wave speed. The wave speed and the wall thickness are related by the wave speed formula (Wylie and Streeter 1993):

a2 =

K/ρ 1 + ( K / E )( D / e)c1

(3)

in which K is the bulk modulus of elasticity; ρ is the density of fluid; E is Young’s modulus of the material; D is the internal diameter of the pipeline; e is the wall thickness; and c1 is a factor depending on the restraint condition (Wylie and Streeter 1993).

TRANSIENT WAVE ANALYSIS In the following, the behaviour of a transient wave propagating through a deteriorated section of pipe with lower impedance is analysed by the method of characteristics (MOC). The process of

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wave reflection and transmission is illustrated in Fig. 1 with corresponding hydraulic grade line (HGL). Hi

Wave

Wr2

H0 H0 Q0

H0 Q0 B0

Hi Qi

B1

Hj2 Qj2

H0 Q0

B0

B0

B1

(c) HGL at time t0+∆t2 Wr1

Hi Hj1 H0

Wt1

B0

Hi Qi

Hj1 Qj1

B0

(a) HGL at time t0

H0 Q0

Hi Hj1 H0

Wr1

Hj2

Wt2

H0 Q0

Hj1 Qj1

B1

B0

Hj3

Hj2

Wr3

Wt2

Hi Qi

H0 Q0

Hj2 Qj2 B0

Wt3

Hj3 Qj3 B1

Wr1 H i Hj1 H0

Hj1 Qj1

Hi Qi

B0

(d) HGL at time t0+∆t3

(b) HGL at time t0+∆t1

Figure 1: Evolution of hydraulic grade line (HGL) when an incident wave propagating through a section of pipe with an impedance change The hydraulic impedance of the original pipeline and the deteriorated section are B0 and B1 , respectively. The deteriorated section is assumed to have lower impedance than the original pipe, i.e. B1 < B0 . Two impedance discontinuities, or interfaces, exist at the boundaries of the degraded section (the two vertical solid lines in the pipeline). In Fig. 1(a), the steady-state head is H 0 and the steady-state flow rate is Q0 , directed from left to right. An incident wave W0 with a magnitude of H i is approaching the right boundary of the deteriorated section from the right hand side. In Fig. 1(b), the first reflection occurs at the right boundary. Part of the wave is reflected as Wr1 and the rest is transmitted as Wt1 . By considering three characteristic lines at the right boundary as described in Wylie (1983), the head ( H j1 ) and flow ( Q j1 ) of the wave after the first reflection and transmission are derived as: 2 B1 (Hi − H0 ) B0 + B1 2 B0 Q j1 = Q0 + (Qi − Q0 ) B0 + B1

H j1 = H 0 +

(4) (5)

The first transmitted wave Wt1 , which has a head of H j1 and a flow rate of Q j1 , will be reflected and transmitted again at the left boundary of the deteriorated section [see waves Wr 2 and Wt 2 in Fig. 1(c)]. Using a similar approach based on the characteristic lines, the head ( H j 2 ) and flow ( Q j 2 ) of the wave after the second reflection and transmission are derived as H j2 = H0 +

4 B0 B1 (Hi − H0 ) ( B0 + B1 ) 2

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(6)

Q j 2 = Q0 +

4 B0 B1 (Qi − Q0 ) ( B0 + B1 ) 2

(7)

When the second reflected wave, Wr 2 , comes across the right boundary, the third transmission and reflection occur [see waves Wr 3 and Wt 3 in Fig. 1(d)]. The head ( H j 3 ) and flow ( Q j 3 ) of the wave after the third reflection and transmission can be derived as H j3 = H0 +

6 B02 B1 + 2 B13 (Hi − H 0 ) ( B0 + B1 )3

(8)

Q j 3 = Q0 +

2 B03 + 6 B0 B12 (Qi − Q0 ) ( B0 + B1 )3

(9)

As B1 is assumed to be smaller than B0 , according to Eq. (4), the value of H j1 will be lower than the original incident wave head H i [Fig. 1 (b)]. A pressure drop will be observed when the first reflected wave ( Wr1 ) arrives at a downstream side transducer. At the second (left) boundary, when the wave propagates from the lower impedance section to the higher impedance section, the value of the head H j 2 after reflection and transmission will be higher than H j1 [Fig. 1(c)]. Similarly, when the second reflected wave, Wr 2 , comes across the first (right) boundary, the head is increased again [Fig. 1(d)]. A pressure rise will then be observed in the pressure transient trace. As a result, a head perturbation with a square shape will appear in the time-domain head response trace. The head of the perturbation is H j1 , which can be used to determine the impedance of the deterioration. The process of wave propagation is also demonstrated in the x-t plan as shown in Fig. 2. t

T0+2T1

Hj3 T0+T1

Hj1

T0 Hi

x

L0+L1

L0

0

Deteriorated Transient generator section and transducer Figure 2: Process of wave transmission and reflection along a pipeline with a deteriorated section, illustrated in the x-t plane with arrows to represent the direction of wave propagation

In Fig. 2, an incident wave starts from the transient generator from time 0. The first reflected wave ( Wr1 ) with a head of H j1 arrives at the transducer at time Ta , which is located at the same location of the generator. The head value H j1 lasts until the wave Wt 3 [shown in Fig. 1(d)] that has a head of

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H j 3 arrives at time Ta + Tb . Higher dimensional transmission and reflection continue, but they have

no impact on the first square-shaped head perturbation that has a head value of H j1 and a duration of Tb . SENSITIVITY ANALYSES A sensitivity analysis is now performed to investigate the effect of the change in wall thickness on the pipeline characteristic impedance. An additional analysis investigating the effect of the deterioration impedance on the magnitude of the pressure perturbation of the first reflected wave ( Wr1 ) and the second transmitted wave ( Wt 2 ) is also conducted. Finally, relationships between the degree of changes in wall thickness and the normalised head perturbations of the reflected and transmitted waves are obtained. Analysis of the impedance change due to changes in wall thickness. Wall thickness change is a major reason for an impedance change because the wave speed is closely related to the wall thickness. Numerical analysis is now performed to investigate how a change in the wall thickness of a pipeline affects on the value of impedance. Defining the impedance ratio between damaged and intact sections as Br = ( B1 / B0 ) , according to the definition of the characteristic impedance of a pipeline Eq. (2), the impedance ratio Br can be rewritten as Br =

a1 D02 a0 D12

(10)

where variables with a subscript of ‘1’ denote parameters of the deteriorated section, and those with a subscript of ‘0’ represent parameters of the original pipeline. Assuming that the same wave speed formula [Eq. (3)] is applicable to both the original pipeline and the deteriorated section, and the external diameter is constant, i.e. D0 + 2e0 = D1 + 2e1 , where e0 and e1 are wall thickness of the original pipe and the deteriorated pipe, respectively, the wave speed in the degraded section can be derived as a12 =

( K / ρ )(1 + erc ) ( K / ρ ) / a02 + erc (1 − ( K / E )2c1 )

(11)

where erc is the relative change in wall thickness and is defined as erc = (e1 − e0 ) / e0 . A conservative bounding estimate of the range of erc is from -0.5 to 0.5, which means the thickness varies from half the original wall thickness to one and half of the original wall thickness. Given that the value of c1 is usually around one, and for steel or iron transmission pipelines the value of E is much greater than the value of K (Wylie and Streeter 1993), the small coefficient ( K / E )2c1 ≪ 1 can be neglected, and finally, Br can be rewritten as

Br =

1 (1 − 2erc e0 / D0 ) 2

( K / ρ )(1 + erc ) K / ρ + erc a02

(12)

It can be seen from Eq. (12) that the value of Br not only depends on the relative change in wall thickness ( erc ), but it is also related to the original wave speed ( a0 ) and the ratio of wall thickness

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to the internal diameter of the original pipeline ( e0 / D0 ). For example, assuming that the ratio of e0 / D0 is 0.04 (a typical value of real pipelines), and the transmitted fluid is water, three curves corresponding to specific wave speeds can be plotted to illustrate the response of Br to changes in erc , as shown in Fig. 3. 1.4

Impedance ratio, Br

1.3 1.2

a0 = 800 m/s a0 = 1000 m/s a0 = 1200 m/s

1.1 1 0.9 0.8 0.7 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Relative change in wall thickness, erc

Figure 3: Variation of the impedance ratio ( Br ) according to the relative change in wall thickness ( erc ) Fig. 3 shows that the value of Br varies from 0.7 to 1.25 under the above assumption. In addition, a reduction in wall thickness can introduce more change in impedance than an increase in thickness of the same magnitude. Further more, a greater variation in impedance can be observed for the same relative change in wall thickness for a lower original wave speed. Analysis of the head of the reflected and transmitted waves. The previous transient wave analysis shows that head values of the reflected and transmitted wave are related to the impedance of the deteriorated section. An analysis is now performed to study the detail of this relationship. The normalised head perturbation of the first reflected wave can be defined as the head perturbation over the initial head increase caused by the incident wave, i.e.

H r* = ( H i − H j1 ) /( H i − H 0 )

(13)

Substituting Eq. (4) into Eq. (13), and using the impedance ratio Br , the normalised head of the first reflected wave can be rewritten as H r* = (1 − Br ) /(1 + Br )

(14)

The plot of Eq. (14) is shown in Fig. 4 as the dashed line. Assuming the value of Br ranges from 0.5 to 1.5 (which is a little wider than the range of Br shown in Fig. 3), the normalised head perturbation of the reflected wave decreases from positive to negative almost linearly. This result indicates that, compared with the impedance of the original pipeline, a degraded section with a lower impedance value ( Br < 1 ) introduces a perturbation with a head value lower than the head of the incident wave ( H r∗ > 0 , thus H j1 < H i ); while a deteriorated section with an impedance value higher than the original pipe ( Br > 1 ) produces a perturbation with a head value higher than the

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incident wave ( H r∗ < 0 , thus H j1 > H i ). The head perturbation is more sensitive to deterioration with lower impedance, with more than 30% head perturbation for Br = 0.5 and only around -20% for Br = 1.5 .

Normalised head perturbation

0.5 H*r

0.4

H*t

0.3 0.2 0.1 0 -0.1

-0.2 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Impedance ratio, Br

Figure 4: Variation of the normalised head perturbation for the reflected wave ( H r* ) (dashed line) and variation of the normalised head perturbation for the transmitted wave ( H t* ) (solid line) according to changes in the impedance ratio ( Br ) The normalised head perturbation of the second transmitted wave Wt 2 can be defined as

H t* = ( H i − H j 2 ) /( H i − H 0 ) and finally written as H t* = (1 − Br ) 2 / (1 + Br )2

(15)

The plot for H t* is shown in Fig. 4 as the solid line. It is observed that the value of H t* is always positive for any value of Br , which means that the head of the transmitted wave is always lower than the head of the original incident wave. This phenomenon is easy to understand, as part of the energy is reflected back at the boundaries of the deteriorated section in the form of reflected waves. Consistent with the finding derived from Eq. (14), the value of H t* is more sensitive to deterioration with lower impedance ( Br < 1 ), with more than 10% head perturbation for Br = 0.5 and only around 4% for Br = 1.5 . This indicates that a degraded pipe section with a lower impedance value can introduce more energy reflections than a deteriorated section with an impedance increase of the same magnitude. However, the normalised head perturbation of the transmitted wave is less than 2% when the impedance ratio ( Br ) is within -0.75 to 1.33. This finding indicates that the head drop for an incident wave propagating through a deteriorated section of pipe can be neglected if the impedance change is not significant (within ± 30%).

Relationship between changes in wall thickness and the normalised head perturbations. Based on the previous two analyses, relationships between the relative change in wall thickness ( erc ) and the normalised head perturbations of the reflected and transmitted waves ( H r* and H t* ) can be obtained by substituting Eq. (12) into Eqs (14) and (15). Plots for these results are depicted in Fig. 5 and Fig. 6.

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a = 800 m/s 0

0.15 of the reflected wave, H*r

Normalised head perturbation

0.2 a = 1000 m/s 0

a = 1200 m/s

0.1

0

0.05 0

-0.05 -0.1 -0.15 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Relative change in wall thickness, e rc

Figure 5: Variation of the normalised head perturbation of the reflected wave ( H r* ) according to relative changes in wall thickness ( erc )

a = 800 m/s of the transmitted wave, H*t

Normalised head perturbation

0.03 0

0.025

a0 = 1000 m/s a0 = 1200 m/s

0.02 0.015 0.01 0.005

0 -0.5-0.4-0.3 -0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 Relative change in wall thickness, e rc

Figure 6: Variation of the normalised head perturbation of the transmitted wave ( H t* ) according to relative changes in wall thickness ( erc ) Fig. 5 illustrates the changing pattern of H r* that results when the value of erc is varied. For the same degree of change in wall thickness, the perturbation is more evident in the pipeline that has a lower wave speed. Reduction in the thickness of the wall introduces a perturbation of greater magnitude than that caused by an increase in wall thickness with the same ratio. Fig. 5 also indicates that the magnitude of the perturbation is proportional to the degree of variation in wall thickness. Pipeline sections with a mild change in wall thickness can only introduce slight reflections. As seen in Fig. 5, when the variation of wall thickness is less than ± 10%, the reflected pressure perturbation has a relative magnitude of no more than ± 3%. This observation can be used to distinguish severe deterioration from natural variations in wall thickness. The variations of H t* shown in Fig. 6 indicate that a greater change in wall thickness, or a lower value of original wave speed, leads to a greater value of the normalised head perturbation for the transmitted wave, which means a greater decrease in the head of the transmitted wave thus more energy is reflected. Meanwhile, compared with an increase in wall thickness, a reduction in wall thickness of the same magnitude introduces a greater head decrease in the transmitted wave.

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However, the normalised head perturbation for the transmitted wave is no more than 1% for most erc values. This result indicates that, for an incident wave propagating through a deteriorated section of pipe with a change in wall thickness, the reflected energy is only a tiny part of the whole energy of the incident wave. DISTRIBUTED DETERIORATION DETECTION This section proposes a methodology on how to estimate the impedance, length and location of distributed deterioration in a single pipeline using the previously presented transient analysis. A reservoir-pipeline-valve system is adopted to illustrate the procedure. An error analysis is conducted to study the robustness of the proposed impedance estimation algorithm. Determination of the impedance of a deteriorated section. For a reservoir-pipeline-valve system, an incident wave can be generated by a fast valve closure. The point just upstream of the valve is the best position for measuring pressure because the signal will be reinforced at the closed pipe end (Lee et al. 2006). The head value after the full value closure ( H i ) can be calculated using either the Joukowsky head change equation (Chaudhry 1987) or the MOC analysis, which yields H i = H 0 + B0Q0 . The flow rate at the valve is zero when the valve is fully closed, i.e. Qi = 0 . Assuming effects of friction are negligible, the incident wave with a head of H i and a flow of zero will propagate upstream without attenuation, and the steady-state head at the deterioration is same as the steady-state head at the valve. Substituting the values H i and Qi into Eqs (4) and (5) yields the following expression for the head and flow after the first reflection and transmission 2 B0 B1 Q0 B0 + B1 B − B0 Q j1 = 1 Q0 B0 + B1

H j1 = H 0 +

(16) (17)

When the reflected wave Wr1 arrives at the end of the pipeline, where the transducer and the closed valve are located, the flow remains zero due to the dead end. Applying the Joukowsky head change equation again, the head increase due to the flow change is B0Q j1 , so that the head observed at the closed end is H1 = H j1 + B0Q j1 = H 0 +

3B0 B1 − B02 Q0 B0 + B1

(18)

The head stays at this level until the wave front of Wt 3 arrives at the transducer, which introduces a pressure rise. Provided the values of H0, Q0 and B0 are known, the value of B1 can be obtained by rearranging Eq. (18), which yields B1 =

B02Q0 + B0 ∆H 3B0Q0 − ∆H

(19)

where ∆H is the head difference between the head of the disturbance and the steady-state head, i.e. ∆H = H1 − H 0 .

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If the anomalous impedance is caused solely by a change in wall thickness, and the external diameter of the deteriorated section is known, the wall thickness and wave speed in this section can be determined using the impedance definition Eq. (2) and the wave speed formula Eq. (3). Error analysis of the estimated impedance. In real world applications, the head values registered in the transient trace usually contain errors due to background noise or intrinsic error of instruments. An error analysis is now performed to determine the effects of the measurement error in the head reading on the accuracy of the estimated impedance value. Assuming the theoretical head difference between the head value of the disturbance and the steadystate head is ∆H , and the value of the observed head difference (determined from the pressure trace) is ∆H (1 + β H ) , where β H can be described as the relative error of the head difference. As a result of the introduction of β H , the impedance of the deteriorated section determined by ∆H (1 + β H ) will have errors. The relative error of the derived impedance can be defined as β B = ( B1′ − B1 ) / B1 , where B1′ is the value of impedance derived from the observed head difference. Substituting ∆H (1 + β H ) into Eq. (19), the expression of B1′ can be derived. Then rearranging Eq. (18) to get the expression of ∆H and substituting it into the expression of B1′ , finally, β B can be derived as

βB =

βH 4 Br B − r βH (3Br − 1)(1 + Br ) 1 + Br

(20)

It can be seen from Eq. (20) that the error in the derived impedance value depends not only on the error in the head measurement, but also on the value of the theoretical impedance ratio Br . Fig. 7 shows the variation of β B according to changes in β H and with various Br values.

Relative error in impedance,βB

0.3 B = 1.2 r

0.2

B = 1.0 r

0.1

B = 0.8 r

0 -0.1 -0.2 -0.3 -0.2

-0.1 0 0.1 Relative error in head difference, β

0.2 H

Figure 7: Variation of the relative error in the derived impedance ( β B ) according to changes in the relative error in the head difference ( β H ) with various impedance ratio ( Br ) values The results shown in Fig. 7 indicate that for a deteriorated section of pipe with a higher impedance value than that of the original pipe ( Br > 1 ), the value of β B is similar to or slightly larger than β H . On the other hand, when Br < 1 , the value of β B is less than β H in most situations. These findings indicate the proposed impedance determination procedure for distributed deterioration is robust and error tolerant, as the error is not amplified during the calculation.

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When a pipeline is tested in the laboratory, the absolute error in the head difference reading can be controlled within ± 0.2 m if all the measurement devices are used properly. Provided the theoretical value of ∆H is 10.0 m, a ± 0.2 m absolute error in the pressure difference reading only equates to a relative error ( β H ) of ± 2% only. Therefore, according to Eq. (20), the relative error in the estimated impedance value, β B , can be controlled within ± 2%. If multiple experiments are undertaken, the error can be further reduced by averaging the results from each experiment. Determination of the location and length of a deteriorated section. Time-domain reflectometry (TDR) (Lee et al. 2007a) can be used to determine the location and length of a deteriorated section. As illustrated in Fig. 2, when an incident wave starts from the transducer at time zero, the arrival time of the first deterioration-induced wave reflection is T0 = 2 L0 / a0

(21)

where L0 is the length between the transducer and the closest boundary of the deteriorated section. The duration of the square shaped disturbance is the time for the wave to travel twice the length of the degraded section of pipe, so that T1 = 2 L1 / a1

(22)

where L1 is the length of the deteriorated pipeline section. As a result, the location of the deterioration ( L0 ) and its length ( L1 ) can be determined, provided the wave speed and critical time points are known. The accuracy of the location and length determined from TDR is dependent on the accuracy of the time readings and the wave speed. In real world applications, the wave speed in the original pipeline is usually determined by experiments. The wave speed in a deteriorated section can be estimated using the wave speed formula [Eq. (3)] once the impedance of the section has been determined. Errors in the wave speed estimates usually can be controlled within an acceptable range. Difficulties exist, however, in accurately determining critical times from experimental pressure traces. Due to the rising front slope of the experimental incident waves (Gong et al. 2012) and signal dissipation and dispersion in real pipelines (Vítkovský et al. 2007), measured pressure perturbations resulting from a deteriorated section do not have clear or sharp boundaries. As a result, it is challenging to read the critical time points accurately from a raw transient trace. One method to improve the accuracy of reading the critical times is to use the impulse response function (IRF) of the tested pipeline system. Use of the IRF in analysing a pipeline system is discussed in the experimental verification section. Effects of friction. The proposed distributed deterioration detection technique is based on an assumption that the effect of friction is negligible, which could be a limitation in real world applications. However, this assumption is valid in practice if the steady-state flow velocity in the tested pipeline is small enough, as the head loss and wave attenuation due to friction are proportional to the square of the steady-state flow velocity. This is the case for the proposed distributed deterioration detection technique that uses the transient wave generated by closing the inline valve at the pipe end. According to the Joukowsky head change equation, the head rise caused by a fast and full closure of an inline valve at the end of a pipeline is aV0 / g , where V0 is the steady-state flow velocity. When the wave speed ( a ) is 1000 m/s, a steady-state flow velocity of 0.1 m/s can introduce a head rise of 10 m, which is high enough for implementing the detection

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technique. If the pipeline under test has a Darcy-Weisbach friction factor of 0.02 and an internal diameter of 0.04 m, the steady-state head loss due to friction is around 0.25 m per 1000 m in length. For the incident wave propagating upstream, according to the extended Joukowsky equation in Leslie and Tijsseling (2000), which takes the influence of quasi-steady Darcy-Weisbach friction into account, the head attenuation is around 1.25% after travelling 1000 m along the example pipeline discussed above. Unsteady friction in real pipes may introduce extra attenuation to the incident wave, but the influence is usually negligible for the wave travelling in the first few seconds. For pipelines with a larger diameter, the steady-state head loss and transient wave attenuation due to friction are even smaller. Therefore, the effects of friction can be neglected in real world applications, as long as the initial steady-state flow velocity is small and detection range is within a couple of kilometres. EXPERIMENTAL VERIFICATION Laboratory experiments have been conducted to validate the proposed distributed deterioration detection method. The pipeline system layout is shown in Fig. 8. The test pipeline is a 37.46 m straight copper pipe with an original internal diameter of 22.14 mm and a wall thickness of 1.63 mm (external diameter is 25.4 mm). The pipeline is rigidly fixed to a foundation plate at an interval of approximately 0.5 m in the axial direction to prevent fluid-structure interaction (FSI) during transient events. One boundary of the pipeline is connected to an electronically controlled pressurized tank and another boundary is a closed in-line valve. The wave speed in the original pipeline was measured to be 1328 m/s, which is consistent with the theoretical result determined by the wave speed formula ( E =121.4 GPa, K =2.19 GPa, ρ =998.2 kg/m3, c1 =1.006) (Kim 2008). Deteriorated section 1.649 m

Tank

18.008 m D0

D1

Side-discharge valve Closed in-line 17.805 m valve D0

37.462 m

Figure 8: The experimental pipeline system layout

A 1.649 m long pipe section with a thinner pipe wall thickness (1.22 mm) located from 17.805 m to 19.454 m upstream from the in-line valve was placed in the original pipeline (as shown in Fig. 8). This pipe section was used to represent a deteriorated section of internal corrosion. While the material and external diameter of this section remained the same as those of the original pipeline, the internal diameter of this section was 22.96 mm (compared with 22.14 m for the main pipe). A transient wave was generated by sharply closing a side-discharge solenoid valve located 144 mm upstream from the in-line valve (around 4 ms closing time). Pressure responses were monitored at the side-discharge valve with a sampling rate of 2 kHz. Impedance estimation. The theoretical impedance of the original pipeline and the thinner-walled section are estimated as B0 = 3.516 ×105 s/m2 and B1 = 3.151× 105 from the pipeline properties and Eq. (2). The impedance of the thinner-walled section is then estimated from the measured pressure data and the algorithms described in Eq. (19). The details are described below. Fig. 9 shows part of the measured pressure transient data [which is the data of Test 1 in Gong et al. (2012)]. The transient response trace has a periodic pattern due to multiple reflections at two pipeline boundaries. Signal attenuation, dissipation and dispersion can also be observed. A pressure

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perturbation appears in the middle of the first half period in the transient trace. Complex perturbations due to multiple reflections can be observed in the following periods of the trace. 40

Head (m)

35 30 25 20 15 10 6.9

7

7.1

7.2

7.3

7.4

7.5

7.6

Time (s)

Figure 9: Experimental water hammer response trace Fig. 10 offers an enlarged view of the first half period of the transient response trace. A pressure drop followed by a pressure rise indicates that the reflection comes from a section of pipe with lower impedance. The value of the steady-state head H 0 = 25.55 m, which was determined by averaging a section of the pressure trace (6.92 s to 6.94 s) before the transient event. The head value of the incident wave caused by the valve closure is estimated as H i = 39.06 m using a similar data averaging strategy (6.945 s to 6.947 s). The steady-state flow rate Q0 = 0.0384 L/s, which was determined by the Joukowsky head change formula. The minimum head value of the deteriorationinduced head perturbation is H1 = 37.86 m at 6.9715 s, so that ∆H = 12.31 m.

Head (m)

39 34 29 24 6.92

6.94

6.96 6.98 Time (s)

7

Figure 10: An enlarged view of the first half period of the transient trace

Using Eq. (19) together with the theoretical impedance of the original pipe B0 , the impedance of the deteriorated section can be determined. Then, using the impedance definition Eq. (2) and the wave speed formula Eq. (3), the wall thickness and wave speed of the deteriorated section can be determined. These results are shown in Table 1 and they are close to the theoretical values. Table 1: Experimentally estimated properties of the deteriorated section Item Impedance (s/m2) Wave speed (m/s) Wall thickness (mm) Theoretical values 1282 1.22 3.151× 105 5 Estimated values 1292 1.29 3.217 × 10 Relative error* 2.1% 0.8% 5.7% *Relative error = |(Estimated value – Theoretical value)/ Theoretical value| ×100%

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Location and length estimation. To estimate the location and length of the deteriorated section using time-domain reflectometry (TDR), the arrival time and duration of the reflected signal are required. However, they are difficult to read accurately from the raw pressure transient trace (Fig. 10). The overall effects of the rising front slope of the incident wave, friction and unsteady flow make the shape of the first deterioration-induced head perturbation much smoother than the theoretical sharp square. To determine the critical times more accurately, the impulse response function (IRF) of the tested pipeline was derived. The input signal for the IRF determination is the induced discharge perturbation (Lee et al. 2006) calculated from the pressure variation during the valve manoeuvre and through the Joukowsky head change formula. The output signal is the head perturbation around the mean, which was obtained by subtracting the steady-state head from the measured pressure trace. Using the Matlab software, the plot of the IRF is obtained and presented in Fig. 11. 5

3

x 10

IRF magnitude (m -2s)

2 1 0 -1 -2 -3 -4

0

0.02 0.04 Time (s)

0.06

Figure 11: IRF extracted from the experimental water hammer response trace The reflected signal observed during the first half period of the experimental transient trace becomes two small impulses with clear positive and negative peaks in the IRF plot. The two reflected impulses are highlighted in a circle in Fig. 11 and enlarged in Fig. 12. 4

x 10

IRF magnitude (m -2s)

3 2 1 0 -1 -2 -3 0.025

0.027

0.029 Time (s)

0.031

Figure 12: An enlarged view of the two small reflected impulses in the IRF The arrival time of the first small impulse (0.027 s) and the time interval between the two impulses (0.003 s) are read from Fig. 12, which represent the arrival time and the duration of the original deterioration-induced perturbation, respectively. Using the wave speed in the original pipe and that 716

in the deteriorated section, the location of the downstream boundary of the deteriorated section is derived to be 17.93 m upstream from the in-line valve using Eq. (21), and the length of this section is calculated to be 1.94 m using Eq. (22). The results are summarised in Table 2. Table 2: Experimentally estimated length and location of the deteriorated section Item Length (m) Location from valve (m) Theoretical value 1.649 17.805 Estimated value 1.94 17.93 Absolute error* 0.291 0.125 *Absolute error = |Estimated value – Theoretical value| Sources of error. The experimentally estimated impedance, wave speed, wall thickness, length and location of the thinner-walled pipe section are close to the theoretical values. However, the error shown in Tables 1 and 2 is just for this case study and cannot represent the average error level. As discussed in the error analysis on the derived impedance section, it is possible to keep the relative error in the estimated impedance within ± 2% in highly controlled laboratory, but specific value can vary from case to case. There are a variety of reasons for the error shown in Tables 1 and 2. Firstly, the wave front is a slope with a duration of 4 ms, rather than a vertical jump as shown in Fig 1. This is believed to be the major reason that makes the deterioration-induced head perturbation deviate from the theoretical squared-shape. Secondly, the pressure trace measured from real pipelines always possesses mild pressure fluctuations (even in the steady-state), so that the head and flow values determined from the pressure trace contain error. Thirdly, the equations used to interpret the pressure trace are derived from uniform and frictionless pipelines. In reality, friction and pipeline parameter natural variations can introduce distortions into the pressure trace. However, the influence of friction and pipeline parameter natural variations is not significant for the proposed method, because the initial flow velocity is small, and only the first half period of the pressure trace is used to identify the degraded section of pipeline, where the distortion is relatively slight compared to the following periods. Another source of errors is the limitation in the resolution of data acquisition. For a sampling rate of 2 kHz and a wave speed of 1328 m/s, the transient wave travels 0.664 m within one sampling interval. Other sources of errors include the error introduced in the IRF calculation procedure and the intrinsic error of the instruments. CONCLUSIONS A novel distributed deterioration detection method for single pipelines has been proposed based on time-domain reflectometry (TDR) and impulse response function (IRF) techniques. The impedance of a deteriorated section of pipeline is determined by the peak head value of the first deteriorationinduced perturbation shown in the first half period of a transient trace. The wave speed and wall thickness of the deteriorated section are then determined from the estimated impedance. Sensitivity analyses have been performed to study how the change in impedance or wall thickness in the deteriorated section affects the size of the deterioration-induced head perturbation. The IRF of the tested pipeline can be extracted from the measured transient trace, and it refines the reflected signals to impulses with clear peaks. The arrival time and duration of the deteriorationinduced perturbation can thus be determined easily and accurately. The location and length of the distributed deterioration can then be determined using the technique of TDR. Laboratory experiments on a single pipeline have been performed to validate the proposed method. A thinner-walled section of pipe is located in the experimental pipeline to represent the

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deterioration. The location, length, impedance, wave speed and wall thickness of the deteriorated section have been calibrated successfully. This research provided an understanding of the transient behaviour of a pipe section with anomalous impedance, and proposed a systematic method to detect the deteriorated section. The successful interpretation of the experimental data has proved the validity of the proposed technique. REFERENCES Adewumi, M. A., Eltohami, E. S., and Ahmed, W. H. (2000). "Pressure transients across constrictions." Journal of Energy Resources Technology, 122(1), 34-41. Arbon, N. S., Lambert, M. F., Simpson, A. R., and Stephens, M. L. (2007). "Field test investigations into distributed fault modeling in water distribution systems using transient testing." World Environmental and Water Resources Congress 2007, Tampa, Florida, United States. Brunone, B. (1999). "Transient test-based technique for leak detection in outfall pipes." Journal of Water Resources Planning and Management, ASCE, 125(5), 302-306. Chaudhry, M. H. (1987). Applied Hydraulic Transients, Van Nostrand Reinhold Company Inc, New York. Colombo, A. F., Lee, P., and Karney, B. W. (2009). "A selective literature review of transient-based leak detection methods." Journal of Hydro-environment Research, 2(4), 212-227. Contractor, D. N. (1965). "The reflection of waterhammer pressure waves from minor losses." Journal of Basic Engineering, ASME, 87, 445-452. De Salis, M. H. F., and Oldham, D. J. (1999). "Determination of the blockage area function of a finite duct from a single pressure response measurement." Journal of Sound and Vibration, 221(1), 180-186. Ferrante, M., and Brunone, B. (2003). "Pipe system diagnosis and leak detection by unsteady-state tests. 1. harmonic analysis." Advances in Water Resources, 26(1), 95-105. Gong, J., Simpson, A. R., Lambert, M. F., Zecchin, A. C., Kim, Y., and Tijsseling, A. S. (2012). "Detection of distributed deterioration in single pipes using transient reflections." Journal of pipeline systems engineering and practice, ASCE, Accepted (doi http://ascelibrary.org/doi/abs/10.1061/(ASCE)PS.1949-1204.0000111). Jönsson, L., and Larson, M. (1992). "Leak detection through hydraulic transient analysis." Pipeline systems, B. Coulbeck and E. P. Evans, eds., Kluwer Academic Publishers, 273-286. Kim, S. H. (2005). "Extensive development of leak detection algorithm by impulse response method." Journal of Hydraulic Engineering, ASCE, 131(3), 201-208. Kim, Y. (2008). "Advanced Numerical and Experimental Transient Modelling of Water and Gas Pipeline Flows Incorporating Distributed and Local Effects," PhD thesis, University of Adelaide, Adelaide. Lee, P. J., Lambert, M. F., Simpson, A. R., Vítkovský, J. P., and Liggett, J. A. (2006). "Experimental verification of the frequency response method for pipeline leak detection." Journal of Hydraulic Research, IAHR, 44(5), 693–707. Lee, P. J., Lambert, M. F., Simpson, A. R., Vítkovský, J. P., and Misiunas, D. (2007a). "Leak location in single pipelines using transient reflections." Australian Journal of Water Resources, 11(1), 53-65. Lee, P. J., Vítkovský, J. P., Lambert, M. F., Simpson, A. R., and Liggett, J. A. (2007b). "Leak location in pipelines using the impulse response function." Journal of Hydraulic Research, IAHR, 45(5), 643-652. Lee, P. J., Vítkovský, J. P., Lambert, M. F., Simpson, A. R., and Liggett, J. A. (2008). "Discrete blockage detection in pipelines using the frequency response diagram: numerical study." Journal of Hydraulic Engineering, ASCE, 134(5), 658-663.

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