# Transient Wave Scattering and Its Influence on Transient Analysis and Leak Detection in Urban Water Supply Systems: Theoretical Analysis and Numerical Validation

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement and Study Framework

## 3. Models and Methods

#### 3.1. One-Dimentional (1D) Transient Model

_{w}is wall shear stress; x is spatial coordinate; and t is temporal coordinate. In the numerical simulations, the wall shear stress is modelled by the Darcy–Weisbach formula, where only the steady state friction is included. The method of characteristics (MOC) is used for the 1D numerical simulations in this study, and the details for implementing this numerical scheme into above transient model can refer to the classic textbooks and references in this field [9,10]. While in the analytical analysis, the friction effect (wall shear stress term in the equation) is excluded due to the mathematical complexity and difficulty of analytical derivation, and so as to highlight the effect of wave scattering during transient flow process.

#### 3.2. Multi-Scale Perturbation Method

_{1}= εx, x

_{2}= ε

^{2}x,

_{1}= εt, t

_{2}= ε

^{2}t,

_{1}, x

_{2}and t, t

_{1}, t

_{2}) correspond to wave oscillations, initial wave modulation, and the modulation by randomness when waves propagate along the pipeline, respectively; ε characterizes the ratio of different scales and ε << 1. The derivatives with respect to x and t are transformed in accordance with the chain rule as [11],

_{0}, P

_{1}and P

_{2}correspond to the above three scales of wave propagation and modification, respectively. It is important to note that high order terms (>2) with regard to ε from Equations (9) and (10) are neglected in the following analytical analysis under the assumption of a relatively small extent of non-uniformities of pipe diameters or cross-sectional areas. This assumption will be validated and discussed through numerical applications later in this study.

## 4. Analytical Results and Analysis

#### 4.1. Results of Regular Non-Uniformities

_{0}) following a periodic co-sinusoidal variation along the pipeline, as follows:

_{0}is the mean value of the pipe cross-sectional area, assuming A

_{0}= 1.0 m

^{2}in this study for simplification; and λ

_{b}is the periodic length of pipe diameter disorders.

- Subcritical detuning: $0<\mathsf{\Omega}<{\mathsf{\Omega}}_{0}$,$$\{\begin{array}{l}T\left({x}_{1}\right)=\frac{\mathsf{\Omega}\mathrm{sinh}K\left(L-{x}_{1}\right)+\mathrm{i}aK\mathrm{cosh}K\left(L-{x}_{1}\right)}{\mathsf{\Omega}\mathrm{sinh}\left(KL\right)+\mathrm{i}aK\mathrm{cosh}\left(KL\right)}\\ R\left({x}_{1}\right)=\frac{{\mathsf{\Omega}}_{0}\mathrm{sinh}K\left(L-{x}_{1}\right)}{\mathsf{\Omega}\mathrm{sinh}\left(KL\right)+\mathrm{i}aK\mathrm{cosh}\left(KL\right)}\end{array},$$
_{0}and Ω represent the incident wave frequency and pipe disorder variation frequency, respectively, and Ω_{0}= δAω/2, Ω = aλ_{b}; x_{1}is the distance along the disordered section in the pipeline; i is the imaginary unit, and i^{2}= −1; L is the total length of disordered section along the pipeline; and K is the detuning (group) wave number and, $K=\sqrt{\left|{\mathsf{\Omega}}_{0}^{2}-{\mathsf{\Omega}}^{2}\right|}/a$. - Supercritical detuning: $\mathsf{\Omega}>{\mathsf{\Omega}}_{0}$,$$\{\begin{array}{l}T\left({x}_{1}\right)=\frac{\mathsf{\Omega}\mathrm{sin}K\left(L-{x}_{1}\right)+\mathrm{i}aP\mathrm{cos}K\left(L-{x}_{1}\right)}{\mathsf{\Omega}\mathrm{sin}\left(KL\right)+\mathrm{i}aK\mathrm{cos}\left(KL\right)}\\ R\left({x}_{1}\right)=\frac{{\mathsf{\Omega}}_{0}\mathrm{sin}K\left(L-{x}_{1}\right)}{\mathsf{\Omega}\mathrm{sin}\left(KL\right)+\mathrm{i}aK\mathrm{cos}\left(KL\right)}\end{array}.$$
- Bragg resonance: $\mathsf{\Omega}={\mathsf{\Omega}}_{0}$,$$\{\begin{array}{l}T\left({x}_{1}\right)=\frac{\mathrm{cosh}{\mathsf{\Omega}}_{0}\left(L-{x}_{1}\right)/a}{\mathrm{cosh}{\mathsf{\Omega}}_{0}L/a}\\ R\left({x}_{1}\right)=-\mathrm{i}\frac{\mathrm{sinh}{\mathsf{\Omega}}_{0}\left(L-{x}_{1}\right)/a}{\mathrm{cosh}{\mathsf{\Omega}}_{0}L/a}\end{array}.$$

_{1}) can be obtained according to Equation (14) and shown in Figure 5. The results of Figure 5 show clearly that, under a fixed disorder magnitude, the reflection coefficient (R) is decreasing along the pipeline, which indicates that the wave perturbation energy is decayed gradually by the disorder section. As expected, the reflection coefficient (R) at a fixed location of the pipe disorder section increases with the disorder magnitude (δA) due to more wave energy reflecting back at the initial disorder section in the larger disorder magnitude (δA) case. For example, the reflection coefficient at the starting location of the disorder section (i.e., X/L

_{0}= 0) could attain 0.9 when the disorder magnitude is about 10% of the mean value (i.e., ε~0.1). Under this situation, there would be very little wave energy (perturbations) remaining at the end of pipe disorder section (since the total energy in the entire pipeline is conserved), resulting in a relatively high decrease-gradient of the reflection coefficient curve for larger δA case as shown in Figure 5. More numerical validations to corroborate the analytical result are conducted later in this study.

#### 4.2. Results of Random Non-Uniformities

_{0}= amplitude of the incident wave; $\lambda ={\lambda}_{r}-i{\lambda}_{i}$ is complex wave number, with λ

_{r}and λ

_{i}= wave damping factor and wave phase change (frequency shift) factor, respectively, and

_{b}with λ

_{b}= correlation length which describes the spatial variability of pipe diameter non-uniformities.

^{−1}as shown in Figure 6. This parameter (L

_{loc}) is used later in this study for the evaluation of the wave scattering effect due to different pipe diameter non-uniformities.

_{cor}~ 1/α) is usually determinate (but maybe known or unknown for the analyst), and therefore, a dimensionless parameter (termed as wave scattering factor) can be further defined for better characterizing the wave scattering effect in that system as,

## 5. Numerical Validation

#### 5.1. Settings of Numerical Tests

_{w}= 1/k) and the characteristic/correction length (distance) of pipe diameter non-uniformities (λ

_{b}= 1/α) were considered for the evaluation. It is assumed that both types of non-uniformities (represented by pipe cross-sectional area) have a zero mean relative to the original nominal value. Note that λ

_{b}represents the periodic length of disordered diameters for the regular disorder case, while it represents the correlation length of disordered diameters in the pipeline for the random disordered cases (i.e., 1/α).

_{0}= initial steady pressure head level; R

_{f}= amplitude factor of incident wave and R

_{f}= 0.2 in this study; and ω = incident wave frequency.

#### 5.2. Validation for Regular Case

_{0}) and there were a total of 20 uniformly spaced reaches with each 100 m (λ

_{b}= 200 m). A continuously sinusoidal incident wave defined by Equation (20) was used in this study and the incident wave frequency was adjusted to achieve the three cases: λ

_{w}/2λ

_{b}> 1, λ

_{w}/2λ

_{b}= 1 and λ

_{w}/2λ

_{b}< 1. The envelope of the maximum and minimum pressure head profiles along the disordered pipe section was extracted from the numerical results and plotted in Figure 9a–c.

_{w}/2λ

_{b}= 1 in Figure 9b, than other two cases, (λ

_{w}/2λ

_{b}> 1 in Figure 9a and λ

_{w}/2λ

_{b}< 1 in Figure 9c, which is consistent with the analytical results of Equations (12)–(14) and Figure 7. Meanwhile, for the cases of λ

_{w}/2λ

_{b}> 1 and λ

_{w}/2λ

_{b}< 1, the pressure wave envelopes were larger than that of the intact case because of the superposition of the scattered waves. To further validate the analytical solution, the reflection coefficients (R) of perfect resonance for case no. two were calculated and plotted in Figure 10. As shown in Figure 10, obvious discrepancies were observed between the analytical and numerical results, which were actually increasing with the disorder magnitude (δA). This result implies that the linearized analytical solution can provide good estimations for the wave scattering effect of a relatively small pipe disorder situation, which is due to the linearization assumption imposed by the analytical analysis.

#### 5.3. Validation for the Random Case

_{w}/2λ

_{b}> 1, λ

_{w}/2λ

_{b}< 1 and λ

_{w}/2λ

_{b}= 1, respectively. It is clear from these results that the pressure wave amplitude decays exponentially with distance along the pipe with random diameter non-uniformities. Moreover, the results for both uniform and triangular distributions of random non-uniformities indicate that the wave scattering effect behaves most significantly when λ

_{w}/2λ

_{b}= 1 (see Figure 12), which is similar to the results of the regular disorder cases analyzed above. The results also imply that the different probability distributions (uniform or triangular) for random non-uniformities along the pipeline have little impact on the wave-scattering effect, as long as the other parameters remain the same, e.g., mean, standard deviation and correlation.

## 6. Results Discussion and Implications

#### 6.1. Energy Analysis of Transient Wave Scattering

_{A}) because of more serious reflections by these non-uniformities in the pipeline. To further explain and understand the wave scattering effect, an energy analysis is performed based on the energy formulations in previous studies [14,15]. The results of case no. four in Table 2 are retrieved from the model and plotted in Figure 14. It is clearly shown in Figure 14 that the total energy in the pipeline system with random variation in diameters is always conserved, although each form of the energy (kinetic or internal) changes significantly with time. In other words, as a result of the wave scattering effect, the total energy has been re-distributed in the system due to the pipe diameter non-uniformities such that pressure wave envelopment is scattered significantly along the pipeline as indicated in the analytical solutions.

#### 6.2. Impacts on Transient Modelling and Analysis

#### 6.3. Impacts on Transient-Based Leak Detection

_{L}

^{*}& A

_{L}

^{*}) were normalized by the total pipe length and average pipe cross-sectional area, respectively. The relative errors of predicted dimensionless leak location by using these four methods are also listed in the table. The results of case T1 show that the additional pseudo leak is detected by all four methods, while actually there is no leak along the pipeline. In case T2, the maximum predicted error using the four methods can reach 39% and 40% for the leak location and size, respectively. This also indicates that the four leakage detection methods are invalid or inaccurate when wave scattering induced reflections and “damping” exist.

## 7. Conclusions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Bottom bar profile in Chesapeake Bay (adapted from [1]).

**Figure 2.**Cross-sectional views of aged water pipelines (adapted from [4]).

**Figure 3.**Different factors attributed to non-uniformities of pipe diameter (pictures adapted from online public sources on google websites).

**Figure 4.**Sketch of different types of pipe diameter non-uniformities (side-sectional profile): (

**a**) realistic and random situation; (

**b**) regular approximation by sinusoidal variation; (

**c**) regular approximation by step variation.

**Figure 7.**Dependence relationship of wave scattering factor on the incident waves and pipe diameter non-uniformities.

**Figure 8.**Schematic of numerical pipeline system: (

**a**) three-section pipeline; (

**b**) middle disorder section for testing.

**Figure 9.**Results of the pressure wave envelope for regular disorder cases: (

**a**) λ

_{w}/2λ

_{b}= 1 and δA = 0.05; (

**b**) λ

_{w}/2λ

_{b}= 1 and δA = 0.10; and (

**c**) λ

_{w}/2λ

_{b}= 1 and δA = 0.20.

**Figure 10.**Comparisons of the numerical and analytical results of the reflection coefficients for the case of λ

_{w}/2λ

_{b}= 1: (

**a**) δA = 0.05; (

**b**) δA = 0.10; and (

**c**) δA = 0.20.

**Figure 11.**Results of the random disorder case with λ

_{w}/2λ

_{b}> 1: (

**a**) uniform distribution and α = 0.6 m

^{−1}; (

**b**) triangular distribution and α = 0.6 m

^{−1}.

**Figure 12.**Results of the random disorder case with λ

_{w}/2λ

_{b}=1: (

**a**) uniform distribution and α = 0.2 m

^{−1}; (

**b**) triangular distribution and α = 0.2 m

^{−1}.

**Figure 13.**Results of the random disorder case with λ

_{w}/2λ

_{b}< 1: (

**a**) uniform distribution and α = 0.05 m

^{−1}; (

**b**) triangular distribution and α = 0.05 m

^{−1}.

Type | Case No. | λ_{w}/2λ_{b} | A_{0} (m^{2}) | δA | Distribution Function | Correlation Function |
---|---|---|---|---|---|---|

Regular | 1 | >1 | 1.0 | σA/A_{0} = 0.20 | Degenerate (deterministic) | 0 for ζ ≠ 0 1 for ζ = 0 |

2 | =1 | |||||

3 | <1 | |||||

Random | 4 | >1 | 1.0 | σ_{A}/A_{0} = 0.23 | Uniform | e^{−α|ζ|} |

5 | =1 | |||||

6 | <1 | |||||

7 | >1 | 1.0 | σ_{A}/A_{0} = 0.23 | Upper triangular | e^{−α|ζ|} | |

8 | =1 | |||||

9 | <1 |

Case No. | Uniform Distribution | Upper Triangular Distribution | |||||
---|---|---|---|---|---|---|---|

4 | 5 | 6 | 7 | 8 | 9 | ||

Wave scattering factor (φ) | Analytical | 125.0 | 75.0 | 159.4 | 125.0 | 75.0 | 159.4 |

Numerical | 127.5 | 74.5 | 157.0 | 125.8 | 76.3 | 152.5 | |

Relative error (%) | 2.0 | 0.7 | 1.5 | 0.7 | 1.7 | 4.5 |

Case | Real Leak Information, x_{L}* & A_{L}* | Predicted Leak Information, x_{p}* & A_{p}* | Max. Error, |x_{L}* − x_{p}*| & |A_{L}* − A_{p}*| | |||
---|---|---|---|---|---|---|

TRM | TDM | SRFM | ITM | |||

T1 | No Leak | 0.50 & 0.01 | 0.50 & 0.038 | 0.25 & 0.024 | 0.46 & 0.031 | --- & --- |

T2 | 0.1 & 0.002 | 0.49 & 0.012 | 0.44 & 0.042 | 0.17 & 0.019 | 0.34 & 0.035 | 39% & 40% |

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**MDPI and ACS Style**

Duan, H.-F.
Transient Wave Scattering and Its Influence on Transient Analysis and Leak Detection in Urban Water Supply Systems: Theoretical Analysis and Numerical Validation. *Water* **2017**, *9*, 789.
https://doi.org/10.3390/w9100789

**AMA Style**

Duan H-F.
Transient Wave Scattering and Its Influence on Transient Analysis and Leak Detection in Urban Water Supply Systems: Theoretical Analysis and Numerical Validation. *Water*. 2017; 9(10):789.
https://doi.org/10.3390/w9100789

**Chicago/Turabian Style**

Duan, Huan-Feng.
2017. "Transient Wave Scattering and Its Influence on Transient Analysis and Leak Detection in Urban Water Supply Systems: Theoretical Analysis and Numerical Validation" *Water* 9, no. 10: 789.
https://doi.org/10.3390/w9100789