condition assessment of water pipelines using a modified...

Condition Assessment of Water Pipelines using a Modified Layer 1

Peeling Method 2

3

Wei Zeng1, Jinzhe Gong2, Aaron C. Zecchin3, Martin F. Lambert4, Angus R. 4

Simpson5, Benjamin S. Cazzolato6 5

1PhD Candidate; School of Civil, Environmental and Mining Engineering, University of 6

Adelaide, SA 5005, Australia; Email: [email protected] 7

2Postdoctoral Research Fellow; School of Civil, Environmental and Mining Engineering, 8

University of Adelaide, SA 5005, Australia; Email: [email protected] 9

3Senior Lecturer; School of Civil, Environmental and Mining Engineering, University of 10


4Professor; M.ASCE; School of Civil, Environmental and Mining Engineering, University of 12


5Professor; M.ASCE; School of Civil, Environmental and Mining Engineering, University of 14


6Professor; School of Mechanical Engineering, University of Adelaide, SA 5005, Australia; 16

Email: [email protected] 17

18

ABSTRACT 19

Pipe wall condition assessment is critical for the targeted maintenance and for failure 20

prevention in water distribution systems. This paper proposes a novel approach for 21

condition assessment of water pipelines by adapting the layer peeling method. This 22

method was previously developed for, and applied to, tubular musical instruments. In 23

mailto:[email protected]





the proposed approach, the impulse response function (IRF) of a pipeline is obtained 24

using measured pressure traces resulting from transient events. The original layer 25

peeling method is further developed for application to water transmission pipelines by 26

1) modifying the end boundary from being an acoustic source tube to a closed valve 27

boundary condition; 2) incorporating the effects of unsteady friction and pipe wall 28

viscoelasticity into the layer peeling algorithm; and 3) incorporating frequency-29

dependent wave reflections and transmissions. Using the IRF and the modified layer 30

peeling method, the impedance of a pipeline can be estimated section by section from 31

downstream (the dead-end) to upstream of the pipeline. The distribution of wave speeds 32

and wall thickness can then be determined. In this study, numerical verifications were 33

conducted using the pipeline pressure responses simulated by the method of 34

characteristics (MOC). The deteriorated pipe sections (sections with changes in 35

impedance) were accurately detected using the new approach. Experimental 36

verification of the result was conducted on a laboratory copper pipeline. A short section 37

of pipe with a thinner wall thickness was successfully detected. 38

Keywords: hydraulic transient; water hammer; pipeline condition assessment; impulse 39

response function; layer peeling method; unsteady friction; viscoelastic effects. 40

1. Introduction 41

Water distribution systems (WDS) are important infrastructure assets. They normally 42

consist of buried pipeline networks that are massive in scale, and often old and 43

deteriorated. However, the condition of these pipes is extremely difficult and expensive 44

to determine. Over the past two decades, a number of non-invasive, hydraulic, transient-45

based methods have been developed for fault detection in water pipelines (Chaudhry 46

2014). Hydraulic transient-based fault detection in a pipeline system is conducted with 47

a transient disturbance, typically a pulse or a step pressure wave introduced by abruptly 48

operating a valve. The resulting transient waves propagate along the pipeline, and any 49

physical changes or anomalies such as leaks or blockages may result in specific wave 50

reflections (Lee et al. 2007). Anomalies in the pipes are located by analyzing time series 51

and wave speed (Brunone 1999). The size of a leak or a blockage can be determined by 52

examining the magnitude of the wave reflection. 53

Most of the transient-based methods focus on the detection of discrete elements, such 54

as leaks (Brunone 1999; Vítkovský et al. 2007; Ferrante et al. 2009; Shamloo and 55

Haghighi 2009; Duan et al. 2011; Gong et al. 2014a) and blockages (Wang et al. 2005; 56

Lee et al. 2008; Sattar et al. 2008; Meniconi et al. 2011). In addition to discrete faults, 57

extended blockages caused by tuberculation (Duan et al. 2012) and extended sections 58

of pipe degradation caused by spalling of the cement mortar lining and widespread 59

corrosion are common in ageing WDS; however, research on transient-based pipe wall 60

condition assessment is limited. Distributed pipe deterioration may reduce the water 61

transmission efficiency (Tran et al. 2010), create water quality problems (Vreeburg 62

and Boxall 2007) and may develop into bursts or severe blockages over time 63

(Zamanzadeh et al. 2007). Thus, developing cost-effective techniques for pipeline 64

condition assessment is essential in enabling strategically targeted pipe maintenance, 65

replacement and rehabilitation. 66

The inverse transient analysis (ITA) (Stephens et al. 2013) and the reconstructive 67

method of characteristics (MOC) analysis (Gong et al. 2014b) are two available, 68

transient-based techniques for continuous pipe-wall condition assessment. The inverse 69

transient method (ITA) was first proposed by Liggett and Chen (1994) for detecting 70

leaks in a pipe network. It was further developed by Vítkovský et al (2000; 2007) and 71

Covas et al.(2010) and was first applied to pipeline condition assessment by Stephens 72

et al. (2008; 2013). 73

In ITA, the transient responses of a numerical pipeline model with assumed parameters 74

(e.g. wave speeds) are simulated by the MOC. Then, an optimization process is 75

conducted to minimize error between numerical and experimental values. By 76

iteratively modifying the parameters in the model, the best possible match of the 77

numerical pressure traces with the traces measured from a real pipe can be obtained. 78

The numerical pipe model that provides the best match is used to interpret the condition 79

of the real pipeline. However, ITA is not computationally efficient, especially for 80

pipelines of substantial length which involves a large number of parameters to calibrate. 81

There are also many practical issues (e.g. wave dissipation and dispersion) that affect 82

the identifiability of the technique (Vítkovský et al. 2007). 83

The reconstructive MOC analysis assesses the condition of a pipeline section by section 84

by inverting the conventional MOC calculation (Gong et al. 2014b). The method is 85

computationally efficient, because the pipeline properties are calculated analytically 86

instead of through an iterative, optimizer-driven model calibration process. However, 87

when the pipeline configuration is either complex, or wave dissipation is significant or 88

dispersion needs to be considered, it would be impossible to handle these situations 89

using the reconstructive MOC algorithm. 90

A technique that shares a similar principle to the reconstructive MOC analysis is the 91

layer peeling method, which was developed in the acoustics research field. An early 92

technique named acoustic pulse reflectometry was proposed by Ware and Aki (1969) 93

as a seismological technique for observing stratifications in the earth’s crust. The layer 94

peeling method was built on this, and applied reconstructions of the geometry of short 95

air ducts with varying cross sections (Amir and Shimony 1995b, a). It was then applied 96

to bore reconstruction of musical wind instruments in order to inspect their qualities 97

(Sharp and Campbell 1997). In these applications, an acoustic source tube with 98

properties designed to prevent or control wave reflections is attached to one end of the 99

duct/instrument to extract the impulse response function (IRF) of the system. The IRF 100

is then analyzed to determine the acoustic impedance and reflection coefficients moving 101

section by section away from the acoustic source. Further research increased the length 102

of reconstruction (Sharp 1998), enhanced the robustness (Forbes et al. 2003), increased 103

the axial resolution (Li et al. 2005), and coupled the higher mode acoustic waves in the 104

method (Hendrie 2007). However, there is no application of the layer peeling concept 105

to pipeline condition assessment using hydraulic transients to date. 106

The research reported in this paper develops a novel pipeline condition assessment 107

approach that uses the IRF of a pipeline and a modified layer peeling algorithm. The 108

key innovations include: the use of a dead-end boundary condition instead of an 109

acoustic tube and the determination of the directional IRF at the dead-end; the 110

incorporation of wave dissipation and dispersion induced by unsteady friction and pipe 111

wall viscoelasticity; and the incorporation of the frequency-dependent wave reflection 112

and transmission at cross-sections with an impedance change. 113

The new technique enables the distribution of pipeline impedance to be determined, 114

from which wall thickness and wave speed along the pipe can be reconstructed. 115

Compared with the ITA method (Stephens et al. 2013), the proposed approach is more 116

efficient, because it does not need a time-consuming, iterative, forward-modelling and 117

optimization process. Compared with the reconstructive MOC analysis, the proposed 118

approach is more accurate, because it can conveniently incorporate frequency-119

dependent effects, such as wave dissipation and dispersion. 120

To validate the new layer-peeling-based approach, extensive numerical simulations 121

have been conducted for pipes with and without both friction or viscoelasticity; with 122

uniformly and non-uniformly distributed deteriorations; and with and without 123

measurement noise. For all the numerical cases, the distribution of the wave speed along 124

the pipe was accurately reconstructed. Experimental verification was also conducted on 125

a copper pipeline in the laboratory. A pipe section with a thinner wall thickness was 126

clearly identifiable in the reconstructed pipe impedance. Towards the end of this paper, 127

a discussion of the limitations and practical challenges of the new method is presented. 128

Finally conclusions are made. 129

2. The Layer Peeling Method for Pipeline Condition Assessment 130

The principle of the new approach is illustrated in Fig. 1, which is a block diagram 131

describing the wave propagation and reflection process in a pipeline. In the figure, the 132

superscripts (+ and –) illustrate forward and backward directions respectively, the 133

subscripts 1, 2, and i represent the number of the pipe section, and the subscripts l and 134

r represent the left side and right side of the section. 135

The method includes the four following main steps. First, the IRF of a pipeline system 136

(P1,l ), which is the input to the modified layer peeling algorithm, is determined through 137

transient analysis first, and pre-processed to yield the directional IRF (𝑃1,𝑙+ and 𝑃1,𝑙

− ). 138

Second, the dissipation, and the dispersion of pressure waves during the propagation in 139

each pipe section, is formulized as a transfer function (hi). Thirdly, the wave 140

transmission and reflection at the interface between pipe sections is represented using 141

reflection ratios (ri,i+1 and ri+1,i) and transmission ratios (si,i+1 and si+1,i). Finally, a 142

recursive procedure is applied to reconstruct the reflection and transmission ratios 143

section by section, using the IRF and the propagation functions. These four components 144

are discussed in detail in subsequent sections. 145

146

147

Fig. 1 Block diagram describing the wave transmission and reflection process in a 148

pipeline. 149

2.1 Impulse response function (IRF) 150

This section illustrates the two steps to get the directional IRF that is the input of the 151

layer peeling method. First, a singular value decomposition is used to obtain the IRF of 152

the pipeline system. The formulae to transfer the system IRF to the directional IRF are 153

then given. 154

2.1.1 Methods for obtaining the system IRF 155

The IRF 𝑧(𝑡) is defined as the response measured at the output when an ideal impulse 156

input is injected into a system. In a linear transmission system, a real signal input 𝑥(𝑡) 157

with N data samples, such as a pulse, can be treated as N scaled impulses with different 158

starting times. Each scaled impulse generates a scaled response that starts at the same 159

time as the input impulse. Thus, the overall response of the system 𝑦(𝑡) is the sum of 160

these scaled responses, which may be written as (Smith 1999) 161

𝑦(𝑛) = ∑ 𝑥(𝑛 − 𝑚)𝑧(𝑚)∞𝑚=−∞ (1) 162

The method used to obtain the IRF in this research is based on deconvolution in the 163

time domain using the singular value decomposition (Agulló et al. 1995). Equation (1) 164

can be rewritten in the matrix form 𝐲 = 𝐗𝐳, in which 𝐗 is a triangular matrix of 𝑥𝑖𝑗 =165

𝑥𝑖−𝑗+1 for 𝑗 ≤ 𝑖 and 𝑥𝑖𝑗 = 0 for 𝑗 > 𝑖 , and 𝐲 is the column vector of the wave 166

reflections. Applying the singular valve decomposition to 𝐗 gives 𝐗 = 𝐏𝚲𝐐𝑇, where 167

𝐏, 𝐐 are orthonormal matrices composed of column vectors 𝐩𝒊 and 𝐪𝑖, respectively, 168

and 𝚲 is a diagonal matrix composed of the singular values λ𝑖 sorted in descending 169

order of size. Then 𝐗 can be written as 𝐗 = ∑ λ𝑖𝐪𝑖𝐩𝑖𝑇𝑁

𝑖=1 and the IRF 𝐳 = 𝐗−1𝐲 can be 170

written as (Agulló et al. 1995) 171

𝐳 = (∑ 𝜆𝑖−1𝐩𝑖𝐪𝑖

𝑇𝑁𝑖=1 )𝐲 (2) 172

However, a small value of λ𝑖 when i approaches N makes a significant contribution 173

to 𝐗−1 , which results in distorted deconvolution. A combination of a truncation 174

regularization and a Tikhonov’s regularization gives (Forbes et al. 2003) 175

𝐳 = (∑λ𝑖

𝜆𝑖2+𝛼𝑐

𝐩𝑖𝐪𝑖𝑇𝐽

𝑖=1 ) 𝐲 (3) 176

where J is the truncation point and J<N, 𝛼𝑐 is an regularization parameter. A smaller J 177

or a larger 𝛼𝑐 results in a smoother but less sharp IRF. The two parameters can be 178

determined by trial and error until satisfactory results are achieved (Forbes et al. 2003). 179

The algorithm in Eq. (3) is used in this paper. 180

181

2.1.2 From the system IRF to the directional IRF 182

In the conventional layer peeling method, which has previously been applied to musical 183

instruments (Sharp 1996), a long, uniform, source tube, which is two times longer than 184

the musical instrument, was attached to one end of the instrument, as shown in Fig. 2 185

(a). The source tube enables the identification and separation of the forward signal input 186

and the backward wave reflections. Thus, the IRF calculated from the system response 187

is directional. It represents the impulse reflections from the instrument only (i.e. no 188

impact from the source tube). 189

In a water transmission line systems, it is not feasible to connect a long source tube to 190

a water pipeline. Instead, a dead-end boundary condition is considered by closing an 191

inline valve, as shown in Fig. 2 (b). A pressure wave generator and a transducer are 192

installed close to the dead-end. 193

194

Fig. 2 Schematic diagram of the testing systems; (a) musical instrument, and (b) 195

water pipeline. 196

Due to the dead end, the directional impulse reflection 𝑃1,𝑙− would be further reflected 197

by the end boundary, and will again enter into the pipeline system, as shown in the first 198

dashed box in Fig. 1. Neglecting any loss in the reflection at the dead-end, i.e. the 199

reflection coefficient is unity, the magnitude of the system IRF at the dead end [𝑃1,𝑙, 200

may be directly determined from Eq. (3)] is effectively two times that of the directional 201

impulse reflections from the pipe discontinuities (backward propagating wave, 𝑃1,𝑙− ), i.e. 202

𝑃1,𝑙 = 2𝑃1,𝑙− (4) 203

As shown in Fig. 1(a), the forward propagating wave into the pipeline is described as 204

𝑃1,𝑙+ = 𝑃𝑖𝑚𝑝𝑢𝑙𝑠𝑒

+ + 𝑃1,𝑙− (5) 205

where 𝑃𝑖𝑚𝑝𝑢𝑙𝑠𝑒+ represents an impulse signal. 206

The two directional IRF signals, which represent the impulse reflections from the pipe 207

discontinuities and the forward-propagating wave into the pipeline respectively, can be 208

obtained through Eqs. (4) and (5), and written as 209

𝑃1,𝑙− = 0.5𝑃1,𝑙 (6) 210

𝑃1,𝑙+ = 𝑃𝑖𝑚𝑝𝑢𝑙𝑠𝑒

+ + 0.5𝑃1,𝑙 (7) 211

2.2 Wave dissipation and dispersion 212

For a fluid-filled pipe, the wave speed (𝑎) in the fluid depends on the properties of the 213

fluid and the pipe wall. For deteriorated sections of a metallic pipeline, the change in 214

wall thickness affects the wave speed governed by the following formula (Wylie and 215

Streeter 1993), 216

𝑎 = √𝐾 𝜌⁄

1+(𝐾 𝐸⁄ )(𝐷 𝑒⁄ )𝑐1 (8) 217

where 𝐾 represents the bulk modulus of the water; 𝜌 is the density of water; 𝐸 is the 218

Young’s modulus of elasticity of the pipe wall; 𝐷 is the pipe’s inner diameter; 𝑒 is the 219

wall thickness of the pipe; and 𝑐1 is the pipeline restraint factor. The value of 𝑐1 220

depends on whether the pipe is thin walled (D/e > 25) or thick walled (D/e ≤25) (Wylie 221

and Streeter 1993). For a structurally degraded, viscoelastic pipeline, the wave speed 222

may also change due to the degradation of the pipe elasticity. 223

Pressure waves in pipelines experience frequency-dependent dissipation and dispersion 224

due to friction and viscoelasticity from the pipe wall (Gong et al. 2015a). The wave 225

dissipation and dispersion in the i th pipe section (within which the properties are 226

assumed uniform) can be described by a transfer function ℎ𝑖 such that 227

𝑝𝑖,𝑟+ = 𝑝𝑖,𝑙

+ ∗ ℎ𝑖 (9) 228

𝑝𝑖,𝑟− = 𝑝𝑖,𝑙

− ∗ ℎ𝑖−1 (10) 229

where ∗ means the convolution in the time domain, 𝑝 is the Fourier transformation of 230

𝑃, and the inverse in (10) refers the inverse convolution, that is, for function a(t), 𝑎 ∗231

𝑎−1 = 𝛿(𝑡). 232

The following part of this section illustrates the processes used to obtain the analytical 233

expression of the transfer function. 234

The one–dimensional (1D) momentum equation for transient fluid flow (Wylie and 235

Streeter 1993) is given as 236

1

𝐴

𝜕𝑄

𝜕𝑡+

𝜕𝑃

𝜌𝜕𝑥+ 𝑔ℎ𝑓𝑇 = 0 (11) 237

in which Q represents the volumetric flow rate, g is acceleration due to gravity, A is the 238

internal cross-section area of the section, and the head loss ℎ𝑓𝑇, which is the sum of a 239

steady-state component ℎ𝑓𝑠 and an unsteady-state component ℎ𝑓𝑢. 240

The continuity equation for the 1D transient flow incorporating the viscoelastic 241

behavior of the pipe wall is given as (Covas et al. 2005) 242

𝐴

𝑎2

𝜕𝑃

𝜕𝑡+ 𝜌

𝜕𝑄

𝜕𝑥+ 2𝜌𝐴

𝜕𝜀𝑟

𝜕𝑡= 0 (12) 243

where 휀𝑟 is the retarded circumferential strain in the pipe wall. 244

Using the concept of steady-oscillatory flow with h, q and 휀𝑟∗ representing the time 245

varying oscillatory components of flow, head and retarded strain, Eq. (11) and Eq. (12) 246

can be transformed into the frequency domain (Gong et al. 2016) as 247

1

𝐴𝑗𝜔𝑞 +

𝜕𝑝

𝜌𝜕𝑥+ 𝑅𝑔𝑞 = 0 (13) 248

𝐴

𝑎2 𝑗𝜔𝑝 + 𝜌𝜕𝑞

𝜕𝑥+ 2𝜌𝐴𝑖𝜔휀𝑟

∗(𝑗𝜔) = 0 (14) 249

Where j is the imaginary unit, 𝑅 is the linearized resistance per unit length, and can be 250

described by a summation of the steady friction part 𝑅𝑠 and unsteady friction part 𝑅𝑢, 251

i.e. 252

𝑅 = 𝑅𝑠 + 𝑅𝑢 (15) 253

where 𝑅𝑠 = 𝑓𝑄0/(𝑔𝐷𝐴2), with f representing the Darcy-Weisbach friction factor. The 254

expression of 𝑅𝑢 depends on the selection of the unsteady friction model in the time 255

domain. If the Vardy and Brown unsteady friction model for smooth-pipe turbulent 256

flow (Vardy and Brown 1995) is chosen, the expression of 𝑅𝑢 can be written based on 257

the derivation by Vítkovský et al. (2003) as 258

𝑅𝑢 =2𝑗𝜔

𝑔𝐴(

𝑗𝜔𝐷2

4𝜈+

1

𝐶∗)−1 2⁄

(16) 259

where 𝜈 is kinematic viscosity of fluid and 𝐶∗ is the shear decay coefficient. 𝐶∗ =260

0.00476 for laminar flows and 𝐶∗ = 7.41/𝑅𝑒𝑘 for smooth pipe turbulent flow with 261

𝑘 = log10(14.3/𝑅𝑒0.05) and Re representing the Reynolds number. 262

With the multi-element Kevin-Voigt model (Covas et al. 2005) for viscoelastic effects, 263

and the corresponding frequency domain transformation (Gong et al. 2016), 휀𝑟∗(𝑖𝜔) in 264

Eq.(14) can be written as 265

휀𝑟∗(𝑗𝜔) = 𝐶ℎ ∑

𝐽𝑘

𝑗𝜔𝜏𝑘+1

𝑁𝑘=1 (17) 266

where 𝐶 = 𝑐1𝐷𝜌𝑔/2𝑒, 𝐽𝑘 is the creep-compliance of the k th Kevin-Voigt element 267

and 𝜏𝑘 is the retardation time of the dashpot of the k th Kevin-Voigt element. 268

Combining Eqs. (13), (14) and (17) yields 269

𝜕2𝑝

𝜕𝑥2 + (𝜔2 − 𝑔𝐴𝑅𝑗𝜔) (2𝐶

𝑔∑

𝐽𝑘

𝑗𝜔𝜏𝑘+1

𝑁𝑘=1 +

1

𝑎𝑒2) 𝑝 = 0 (18) 270

The equation can be also written as 271

𝜕2𝑝

𝜕𝑥2+

𝜔2

𝑎𝑐2 𝑝 = 0 (19) 272

with the general solution for the oscillatory pressure wave 273

𝑝(𝑥, 𝑡) = 𝑒𝑗𝜔(𝑡−𝑥 𝑎𝑐)⁄ (20) 274

Thus, the pressure wave after propagating a distance of ∆𝑥 can be written as 275

𝑝(𝑥 + ∆𝑥) = 𝑝(𝑥)𝑒𝑗𝜔∆𝑥 𝑎𝑐⁄ (21) 276

in which 𝑎𝑐 is the complex wave speed described by 277

𝑎𝑐 = √𝜔2

(𝜔2−𝑔𝐴𝑅𝑗𝜔)(2𝐶

𝑔∑

𝐽𝑘𝑗𝜔𝜏𝑘+1

𝑁𝑘=1 +

1

𝑎2) (22) 278

If friction is neglected (Gong et al. 2016) and then 𝑅 = 0, then 279

𝑎𝑐 = √1

(2𝐶

𝑔∑

𝐽𝑘𝑗𝜔𝜏𝑘+1

𝑁𝑘=1 +

1

𝑎2) (23) 280

If viscoelastic behavior of the pipe wall is neglected, then the expression for the 281

complex wave speed simplifies to 282

𝑎𝑐 = 𝑎√1

(1−𝑔𝐴𝑅𝑗 𝜔⁄ ) (24) 283

If both the friction and the viscoelastic effects of the pipe wall are neglected, the 284

expression of the complex wave speed 𝑎𝑐 collapses to the elastic wave speed 𝑎. 285

The complex wave speed 𝑎𝑐 can be also treated in the following form 𝑎𝑐 = 𝑎𝑟 + 𝑗𝑎𝑖 286

(Suo and Wylie 1990) such that 𝑎𝑒 and 𝜇𝑒 in Eq.(25) can represent the frequency 287

dependent wave speed and attenuation respectively with the following expressions: 288

𝑎𝑒 =|𝑎𝑐|

𝑎𝑟 𝑎𝑛𝑑 𝜇𝑒 =

𝜔𝑎𝑖

|𝑎𝑐| (25) 289

For a pressure wave propagating from the left boundary to the right boundary of the i 290

th pipe section, the following equation can be obtained through Eqs. (21) and (25). 291

𝑝𝑖,𝑟+ = 𝑝𝑖,𝑙

+ ∗ 𝑒− 𝜇𝑒∆𝑥𝑖𝑒−𝑗𝜔∆𝑥𝑖 𝑎𝑒⁄ (26) 292

in which T/2 is the time needed for the wave to pass through the i th section, ∆xi is the 293

length of the section, 𝑒− 𝜇𝑒∆𝑥𝑖 represents the wave dissipation and 𝑒−𝑗𝜔∆𝑥𝑖 𝑎𝑒⁄ 294

represents the wave dispersion. The transfer function in Eqs. (9) and (10) is given by 295

ℎ𝑖 = 𝑒− 𝜇𝑒∆𝑥𝑖𝑒−𝑗𝜔∆𝑥𝑖 𝑎𝑒⁄ (27) 296

The transfer function ℎ𝑖 represents the wave dissipation and dispersion, and will be 297

used in the reconstruction process outlined in the following section 2.4. 298

2.3 Wave transmission and reflection 299

If an incident pressure wave p (in the frequency domain) meets a discontinuity in the 300

pipe (an interface with an impedance change), a reflected wave pr will be generated. 301

The amplitude of the incident wave will change to ps after passing the discontinuity. 302

The reflection coefficient 𝑟 (the ratio of the complex amplitude of the reflected wave 303

to that of the incident wave) and the transmission coefficient 𝑠 (the ratio of the complex 304

amplitude of the transmitted wave to that of the incident wave) are determined by the 305

pipeline characteristic impedance 𝐵 . They can be calculated using the formulae 306

(Chaudhry 2014) 307

𝑟𝑖,𝑖+1 =𝑝𝑟

𝑝=

𝐵𝑖+1−𝐵𝑖

𝐵𝑖+1+𝐵𝑖 (28) 308

𝑠𝑖,𝑖+1 =𝑝𝑠

𝑝=

2𝐵𝑖+1

𝐵𝑖+1+𝐵𝑖= 1 + 𝑟𝑖,𝑖+1 (29) 309

in which 310

𝐵 = 𝑎𝑐 𝑔𝐴⁄ (30) 311

is the complex characteristic impedance. According to Eq. (23), it can be seen that 𝑟𝑖,𝑖+1 312

and 𝑠𝑖,𝑖+1 are frequency-dependent for the viscoelastic model. For the unsteady friction 313

model, the complex wave speed 𝑎𝑐 is proportional to the elastic wave speed 𝑎 if A is 314

assumed to be constant in Eq. (24). Thus, 𝑟𝑖,𝑖+1 and 𝑠𝑖,𝑖+1 are frequency-independent 315

for the unsteady friction model when Eq. (24) is substituted into Eqs. (28) and (29). 316

If the incident wave propagates in the opposite direction, then the reflection coefficient 317

and the transmission coefficient can be calculated as: 318

𝑟𝑖+1,𝑖 = −𝑟𝑖,𝑖+1 (31) 319

𝑠𝑖+1,𝑖 = 1 − 𝑟𝑖,1+1 (32) 320

At the interface of two sections as shown in Fig. 3, the transmitted waves and reflected 321

waves are represented by the dot-dashed arrows and dashed arrows, respectively. 322

According to the directions of the arrows, it follows that the forward-propagating wave 323

𝑃𝑖+1,𝑙+ travelling into section i+1 is the sum of the transmitted wave of 𝑃𝑖,𝑟

+ and the 324

reflected wave of 𝑃𝑖+1,𝑙− . The backward-propagating wave 𝑃𝑖,𝑟

− travelling into section i 325

is the sum of the reflected wave of 𝑃𝑖,𝑟+ and the transmitted wave of 𝑃𝑖+1,𝑙

− . Combining 326

Eqs. (28) to (32), the following formulae can be written based on the analysis above: 327

𝑝𝑖+1,𝑙+ = 𝑝𝑖+1,𝑙

− (−𝑟𝑖,𝑖+1) + 𝑝𝑖,𝑙+ (1 + 𝑟𝑖,𝑖+1) (33) 328

𝑝𝑖,𝑟− = 𝑝𝑖,𝑟

+ (𝑟𝑖,𝑖+1) + 𝑝𝑖+1,𝑙− (1 − 𝑟𝑖,𝑖+1) (34) 329

The relationship of the wave transmission and reflection can also be represented by the 330

third dashed box in Fig. 1. By rearranging the two formulae, the relationship between 331

the travelling waves at either side of a section interface can be summarized as (Amir 332

and Shimony 1995b) 333

[𝑝𝑖+1,𝑙

+

𝑝𝑖+1,𝑙− ] =

1

1−𝑟𝑖,𝑖+1[

1 −𝑟𝑖,𝑖+1

−𝑟𝑖,𝑖+1 1] [

𝑝𝑖,𝑟+

𝑝𝑖,𝑟− ] (35) 334

335

Fig. 3 Wave transmission and reflection at a junction 336

In the time domain, the space-time diagram of the wave reflection and transmission are 337

shown in Fig. 4 for a pipeline discretized into sections. The forward-propagating waves 338

along the diagonal in the diagram are defined as the main transmitted waves 𝑃𝑖,𝑟+ (𝑖𝑇 2⁄ ) 339

(thick solid lines). The backward propagating waves, at the same time and position as 340

the main transmitted waves, are defined as the initial reflected waves 𝑃𝑖,𝑟− (𝑖𝑇 2⁄ ) 341

(dashed lines). If the frequency-dependent characteristics of the wave reflection and 342

transmission are neglected in the time domain, the initial reflected waves are only 343

caused by the reflection of the main transmitted waves, as shown in Fig. 4, which means 344

𝑟𝑖,𝑖+1𝑡 =

𝑃𝑖,𝑟− (𝑖𝑇 2⁄ )

𝑃𝑖,𝑟+ (𝑖𝑇 2⁄ )

(36) 345

in which 𝑟𝑖,𝑖+1𝑡 is the elastic wave reflection ratio in the time domain. If the elastic 346

characteristic impedance 𝐵𝑖𝑡 = 𝑎𝑖/𝑔𝐴𝑖 is known, the real characteristic impedance 347

𝐵𝑖+1𝑡 can be estimated according to Eq. (28) as 348

𝐵𝑖+1𝑡 =

1+𝑟𝑖,𝑖+1𝑡

1−𝑟𝑖,𝑖+1𝑡 𝐵𝑖

𝑡 (37) 349

350

351

352

Fig. 4 Space-time diagram of the wave propagation 353

2.4 Procedures of the Modified Layer Peeling Method 354

The steps for reconstructing a pipeline with N sections using the modified layer peeling 355

method are shown in Fig. 5 (Note 𝐵1 is obtained as in Gong et al. (2014b)) and 356

described as follows. 357

358

359

Fig. 5 Main steps of the modified layer peeling method for pipeline condition 360

assessment. 361

Step 1: Use experimental data to calculate the system IRF through Eq. (3) 362

Step 2: Use the system IRF to calculate the directional IRF 𝑃1,𝑙− and forward-363

propagating wave 𝑃1,𝑙+ through Eqs. (6) and (7). 364

Step 3: Use the waves at the left side of the ith (i=1 for the first step) section 𝑝𝑖,𝑙+ and 𝑝𝑖,𝑙

− 365

to calculate the waves at the right side of the ith section 𝑝𝑖,𝑟+ and 𝑝𝑖,𝑟

− through Eqs. (9) 366

and (10). 367

Step 4: Use the waves at the right side of the ith section 𝑃𝑖,𝑟+ and 𝑃𝑖,𝑟

− (transferred to the 368

time domain) to calculate the elastic reflection ratio 𝑟𝑖,𝑖+1𝑡 through Eq. (36), and then 369

obtain the elastic characteristic impedance of the next section 𝐵𝑖+1𝑡 through Eq. (37), as 370

well as the complex characteristic impedance 𝐵𝑖+1 through Eqs.(22) and (30). 371

Step 5: Use the complex 𝐵𝑖 and 𝐵𝑖+1 to calculate the frequency-dependent wave 372

reflection ratio 𝑟𝑖,𝑖+1. Then, use the waves at the right side of the i th section 𝑝𝑖,𝑟+ , 𝑝𝑖,𝑟

− 373

and 𝑟𝑖,𝑖+1 to calculate the waves at the left side of the (i+1)th section 𝑝𝑖+1,𝑙+ and 𝑝𝑖+1,𝑙

− 374

through Eq. (35). 375

Step 6: Repeat steps 3 to 5 for i = 2, …, N-1 to calculate the characteristic impedances, 376

wave speeds and wall thicknesses for the remaining sections. 377

3. Numerical Verification 378

Numerical simulations were conducted on several reservoir-pipeline-valve systems to 379

verify the proposed approach for pipeline condition assessment. The MOC (Wylie and 380

Streeter 1993) has been used to obtain the transient pressure responses induced by a 381

flow fluctuation. For large pipes with a constant outer dimeter, a small change in wall 382

thickness slightly alters the inner diameter, but can cause large change in the wave 383

speed according to Eq.(8).Thus, only the wave speed varies along the pipeline to 384

simulate extended deterioration, and the inner diameter stays constant in the numerical 385

cases. 386

3.1 Case 1: Frictionless pipe 387

The first case study was conducted for a frictionless metallic pipeline with a uniformly 388

deteriorated section and a non-uniformly deteriorated section. The inner diameter of the 389

frictionless pipeline is assumed to be 600 mm throughout, and the wave speed in the 390

normal sections of the pipeline is 1000 m/s. The pipeline configuration and the 391

properties of the deteriorated sections are given in Fig. 6. The wave speed in the 392

uniformly deteriorated section is 800 m/s. The wave speed in the non-uniformly 393

deteriorated section has a gradual change from 1000 m/s to 800 m/s and back to 1000 394

m/s following one period of a cosine pattern (as shown in Fig. 8) of degradation over a 395

length of 108.3 m. A pressure pulse wave, shown in Fig. 7(a), induced by a flow pulse 396

was injected into the pipeline at the upstream face of the closed valve, and the wave 397

reflections were simulated using a frictionless MOC model (time step = 0.001 s for all 398

the numerical cases) that was given in Fig. 7 (b). The primary reflections marked in the 399

figure were directly induced by the deteriorated sections, and the other much smaller 400

reflections were caused by high-order reflections. Note that reflections from the closed 401

valve (dead-end) contributed to the signals seen in Fig. 7(b). 402

The proposed layer-peeling-based technique applied to the signals in Fig. 7 accurately 403

yields reconstructed wave speed distributions that are almost identical to the theoretical 404

values (as shown in Fig. 8). To determine how robust the proposed method is against 405

measurement noise, the reflection signal was contaminated with white Gaussian noise, 406

with a signal-to-noise ratio of 10. The mixed signal shown in Fig. 9(a) was then used 407

to calculate the system IRF [Fig. 9(b)] of the pipeline using Eq. (3). The modified layer 408

peeling method, without considering any wave dissipation or dispersion, was then 409

applied to reconstruct the pipeline. The outcome shown in Fig. 10 demonstrates that the 410

proposed technique can reconstruct the pipeline condition (wave speed in this case) for 411

both uniformly and non-uniformly deteriorated sections, even when measurement noise 412

is present. 413

414

Fig. 6 Pipeline configuration 1 415

416

Fig. 7 Input and reflected signals 417

418

Fig. 8 Wave speed reconstructed from the modified layer peeling method for the 419

frictionless pipe without measurement noise. (The two plots are virtually coincident) 420

421

422

Fig. 9 Reflection with noise and IRF 423

424

425


frictionless pipe with measurement noise. 427

3.2 Case 2: Effects of unsteady friction 428

A reservoir-pipeline, closed-valve system with one uniformly deteriorated section 429

(shown in Fig. 11) was considered to analyze the effect of the unsteady friction. The 430

inner diameter of the pipeline is 50.6 mm and the wave speed of a normal pipeline is 431

1000 m/s. The steady-state flow rate 𝑄0 is 0.1 L/s (V = 0.05m/s) and the Darcy-432

Weisbach factor 𝑓 is 0.02. A smaller-sized pipe (compared to the pipe in Fig. 6) was 433

chosen to highlight the effect of unsteady friction [unsteady friction effects are 434

insignificant and typically negligible for the primary reflections in large pipes (Stephens 435

et al. 2013)]. A pulse wave with the same duration and waveform as case 1 was injected 436

into the pipeline just upstream of the closed valve, and the pressure wave reflections at 437

the same point were simulated using an unsteady friction MOC model (Vardy and 438

Brown 1995). 439

The pipeline was initially reconstructed using the modified layer peeling method which 440

is used in the frictionless case, without considering any signal dissipation or dispersion 441

(thus ignoring unsteady friction), and the result is plotted as the dashed line in Fig. 12. 442

Obvious errors are shown between 200 m to 350 m. 443

Another reconstruction was conducted using the modified layer peeling method 444

incorporating the transfer function that describes the unsteady friction using Eq. (24), 445

(25) and (27). It needs to be emphasized that the transfer function was renewed in every 446

step using the wave speed (and wall thickness in the experimental case), calculated in 447

the previous time step. The result shown as the dot-dashed line in Fig. 12 illustrates that 448

the error was eliminated along the pipeline except at points where the impedance 449

changed sharply (200 m and 300 m). The error at the impedance change interfaces was 450

induced by the ripples of the IRF when transformed from the wave reflections. 451

452


454

455

Fig. 12 Wave speed reconstructed from the modified layer peeling method for the unsteady 456

friction case. 457

3.3 Case 3: Effect of viscoelasticity 458

The pipeline system shown in Fig. 13 was used to analyze the viscoelastic effects of the 459

pipe wall. Friction was not considered, so that the viscoelastic effects would be 460

highlighted. The inner diameter of the pipeline was 50.6 mm and the wave speed of a 461

normal pipeline is 393 m/s. The physical details of the pipeline are adapted from the 462

experimental pipeline at Imperial College as reported in Covas et al. (2005), and the 463

viscoelastic parameters used are listed in Table 1. A pulse wave with the same duration 464

and waveform as case 1 was injected into the pipeline, and the pressure wave reflections 465

were simulated using a four-element Kevin-Voigt viscoelastic MOC model (Covas et 466

al. 2005). 467

468

Table 1 Viscoelastic parameters used in the numerical case 469

Retardation time

𝜏𝑘 (s)

Creep coefficients 𝐽𝑘

(Pa-1)

0.005 1.048E-10

0.5 1.029E-10

1.5 1.134E-10

5 8.083E-12

The pipeline was initially reconstructed using the modified layer peeling method, 470

without considering any signal dissipation or dispersion (Model 1) (thus ignoring 471

viscoelasticity). The reconstructed results plotted as the solid line in Fig. 14 include 472

significant errors at the deteriorated section. Another reconstruction was then 473

conducted using the model incorporating the viscoelastic effects using Eqs. (23), (25) 474

and (27), but excluding the frequency-dependent wave reflection and transmission 475

(Model 2). This means that the complex characteristic impedances in Eq. (28) were 476

replaced by the real characteristic impedances in this model. The results shown as the 477

dashed line in Fig. 14 illustrate that the error was reduced but still distinct. The third 478

reconstruction was conducted by further considering the frequency-dependent wave 479

reflection and transmission by using complex characteristic impedances in Eq. (28) 480

(Model 3). A high correlation was achieved between the reconstructed result (the dot-481

dashed line) and the theoretical values in Fig. 14. A small error remains in the part of 482

the pipeline close to the reservoir. This error was caused by neglecting the frequency-483

dependent characteristics of the wave reflection and transmission in Eqs. (36) and (37) 484

in the time domain. 485

486

487


489

490


viscoelastic case (Model 2 - with viscoelasticity and frequency-independent wave reflections 492

and transmissions; Model 3- with viscoelasticity and frequency-dependent wave reflections 493

and transmissions). 494

4. Experimental Verification 495

Laboratory experiments have been conducted on a single copper pipeline system in the 496

Robin Hydraulics Laboratory at the University of Adelaide to verify the proposed 497

technique for pipeline condition assessment. The experimental data was used to verify 498

the use of a direct wave, reflection-based technique (Gong et al. 2013), and the 499

reconstructive MOC analysis method (Gong et al. 2014b) for detecting a distributed 500

deterioration in a pressurized pipeline. 501

4.1 Experimental pipeline layout 502

The layout of the experimental pipeline system is given in Fig. 15. The pipeline was 503

connected to a pressurized tank at the upstream side, and a dead-end was created by 504

closure of the in-line valve at the downstream side. The basic geometry parameters are: 505

length L = 37.46 m, internal diameter D0 = 22.14 mm and wall thickness e0 = 1.63 mm 506

(D/e =13.6). A pipe section with a thinner pipe wall with L1 = 1.649 m, D1 = 22.96 mm, 507

e1 = 1.22 mm (D/e=18.8) and same material with the original pipeline was placed 508

17.805 m upstream from the in-line valve. It represents a pipe section with a uniform 509

wall thickness reduction due to internal corrosion. A side-discharge solenoid valve was 510

located 144 mm upstream from the closed in-line valve, for the generation of transient 511

waves. 512

The wave speed of the pipeline can be calculated using Eq. (8) with the following 513

parameters: E = 124.1GPa, K = 2.149 GPa, 𝜌 = 999.1 kg/m3 and 𝑐1 = 1.006 which is 514

assumed to be uniform for a thick-walled pipe. The theoretical wave speed calculated 515

using Eq. (8) for the intact pipeline was a0 = 1319 m/s and a1 = 1273 m/s for the thinner-516

walled section. 517

518

Fig. 15 System layout of the experimental pipeline system 519

4.2 Experimental data 520

A transient wave was generated by sharply closing the side-discharge solenoid valve, 521

which was adjacent to the closed in-line valve. The pressure traces were collected by a 522

Druck PDCR 810 pressure transducer with a 2 kHz sampling rate. Three experiments 523

were conducted using the same configuration, with the pressure traces shown in Fig. 524

16. 525

The pressure trace in the first 6 ms, as shown in Fig. 15, covered the full wave front and 526

was defined as the input signal to the system. The reflection signal could then be 527

obtained by subtracting the input signal from the original pressure trace. With the input 528

and reflection signals of each experiment, the system IRF of each experiment was 529

obtained using Eq. (3), represented as the dash-dotted lines in Fig. 17. To reduce the 530

background noise, an averaged IRF (represented as the solid line in Fig. 17) was 531

obtained by averaging these three sets of IRF traces. 532

533

Fig. 16 Experimental pressure traces 534

535

Fig. 17 Individual IRFs and averaged IRF 536

4.3 Reconstruction of the pipeline 537

The pipeline was reconstructed using the proposed approach, with the individual and 538

averaged IRFs. The reconstructed pipelines using the individual IRFs, with the model 539

incorporating the unsteady friction, are plotted in Fig. 18. A reconstruction using the 540

averaged IRF, using both frictionless and unsteady friction models, is shown in Fig. 19. 541

A clear dip in the reconstructed wall thickness, and a clear dip in the reconstructed wave 542

speed, which matches well with the theoretical values, can be observed. 543

Perturbations in the estimated wall thickness and wave speed of the reconstructed 544

pipelines are also illustrated in Fig. 18 and Fig. 19. They are caused by the joints in the 545

pipeline, natural variations in the pipeline parameter, fluid-structure interactions, and 546

other uncertainties associated with the experiments. 547

Slight differences can be seen in Fig. 20, in which the reconstructed results using the 548

frictionless model and the unsteady friction model are compared. The figure illustrates 549

that the frictionless model gives a conservative result. As the base flow in the pipeline 550

is zero (after the closure of the side-discharge valve), and the pipeline is short in length, 551

the effect of the unsteady friction is marginal in this experimental case. Overall, the 552

experimental results have validated the effectiveness of the proposed layer-peeling-553

based pipeline condition assessment technique. 554

555

Fig. 18 Reconstruction using individual IRFs: (a) reconstructed wall thickness 556

distribution compared with the theoretical values; and (b) reconstructed wave speed 557

distribution compared with the theoretical values. 558

559

Fig. 19 Reconstruction using the averaged IRF: (a) reconstructed wall thickness 560

distribution compared with the theoretical values; and (b) reconstructed wave speed 561

distribution compared with the theoretical values. 562

563

564

Fig. 20 An enlarged view of Fig. 19 565

5. Discussion 566

In this section, some practical issues when applying this new approach to field pipelines 567

are discussed. Strategies to refine this approach to enhance its practicality are also 568

included. 569

5.1 Resolution 570

The proposed technique enables continuous reconstruction of pipeline conditions, 571

which is an advantage over conventional time-domain reflectometry-based techniques 572

that only focus on major reflections and major deteriorations (Gong et al. 2013; Gong 573

et al. 2015b). The spatial resolution of the proposed method is limited by the effective 574

bandwidth of the incident waves, which is determined by the sharpness of the wave 575

front. Theoretically, one can accurately diagnose a deteriorated section with a length 576

longer than 𝑇𝑝𝑐/2, where 𝑇𝑝 is the duration of the pulse. The duration of the pulse, 577

generated by a side-discharge valve, is typically several milliseconds in the laboratory 578

and tens of milliseconds in the field due to limitations in the maneuverability of the 579

valve. Pressure generators that can generate high-frequency pressure waves, such as the 580

prototype spark-transient generator (Gong et al. 2018), will be helpful in increasing the 581

resolution. 582

5.2 Location of the generator 583

For the approach proposed in this paper, and for some other existing transient-based 584

methods for pipeline condition assessment, the pressure generator and transducer are 585

required to be installed close to the dead-end of the pipe. The dead-end can be achieved 586

by closing an in-line valve, but the installation of a generator and a transducer is not 587

always convenient. Further research will be conducted to extend the new approach to 588

other testing configurations that do not require a dead end. 589

590

591

5.3 Wave dissipation and dispersion 592

This research has demonstrated that errors are likely to occur if the wave dissipation 593

and dispersion are not properly considered in the algorithm. In field pipelines, there are 594

uncertainties and variations in the wave dissipation and dispersion, and they are difficult 595

to predict using theoretical models. Further research is needed to enable the in-situ 596

calibration of the wave dissipation and dispersion. 597

6. Conclusions 598

This paper has proposed a novel approach for pipeline condition assessment. The layer 599

peeling method previously applied to tubular musical instruments has been modified to 600

accommodate the differences between musical instruments and water pipelines. The 601

long source tube, which was used in the original method, has been eliminated. Unsteady 602

friction of the transient flow, and the viscoelastic effects of the pipe wall, have been 603

incorporated into the new method. Frequency dependent wave reflections and 604

transmissions were also incorporated into the method. This research has demonstrated 605

that the wall condition of water pipelines can be assessed using the reconstructed results, 606

using the modified layer peeling method. 607

Numerical simulations conducted in this research have demonstrated that 1) the 608

proposed approach can deal with multiple, deteriorated sections including non-609

uniformly distributed deteriorations; 2) reasonably accurate results can be achieved 610

even when the signal is contaminated by moderate background noise; 3) wave 611

dissipation and dispersion can be accounted for using a transfer function that is 612

constructed to consider the unsteady friction of the transient flow and the viscoelastic 613

effects of the pipe wall; and 4) frequency-dependent wave reflection and transmission 614

can be incorporated in the frequency domain, and that this is important for viscoelastic 615

pipes. 616

The experimental results further validated the new pipeline condition assessment 617

approach. Three sets of experimental data were used to obtain the averaged IRF, which 618

is the input for the modified layer peeling algorithm. The pipeline with a thinner-walled 619

section was successfully reconstructed both with and without the unsteady friction of 620

transient flow. 621

The proposed layer-peeling pipeline condition assessment technique is a promising 622

alternative to other existing methods because of both its computational efficiency and 623

its ability to assess a long pipe section continuously. The method is capable of 624

conveniently incorporating wave dissipation and dispersion, and frequency-dependent 625

reflection and transmission. Thus, it has the potential to allow engineers to conduct 626

reliable condition assessments for real pipelines, including viscoelastic pipes. 627

Acknowledgements 628

The research presented in this paper has been supported by the Australia Research 629

Council through the Discovery Project Grant DP170103715. 630

The author would like to acknowledge Leticia Mooney for her editorial assistance as 631

this paper was finalized. 632

633

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