Determining the Leak Detection Range of Transient Wave-based Reflection method in Water Conveyance P

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Determining the Leak Detection Range of Transient Wave-based Reflection method in Water Conveyance Pipes.

Asiedu Sarpong1 1College of Water Conservancy & Hydropower Engineering, Hohai University, China.

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Abstract – The transient wave-based reflection method

the hydro-environment cannot be underestimated. To effectively conserve water resources and its usage, it is imperative that these pipe anomalies are well attended to, mitigated, or possibly, totally prevented from occurring.

The focus of this paper is on the leak anomaly, which is a very common threat to pipelines in all sections of engineering field. Leakage causes both environmental degradation, health issues (water contaminants through leakages from biological, chemical, and physical means), and economic loss. There are so many leak detection techniques which have been in practice over the past decades. To mention a few; Ground Penetrating Radar [1], Acoustic Correlator [2], Tethered method [3], and Transient wave-based methods [4,5]. The transient wavebased methods have been extensively been researched and applied to urban water pipes in the past few years.

They can further be grouped under two main categories; (i) full-waveform inversion (FWI) method, and (ii) local feature based method [6–8]. The full-wave method is grouped into either, (a) time domain method [7,9,10], or (b) frequency domain method [7,8,11–13]. The local feature-based utilizes partial leak-induced information, and it is divided into the following; (i) reflection-based method [7,14,15]; (ii) damping-based method [7,16–18]; and (iii) frequency response function (FRF) peak patternbased method [7,19,20]. This work focuses on the reflection-based method, which is also known as the transient wave-based reflection (TWR) method [21]. This method takes into consideration the partial reflection of the pressure wave that takes place at the leak, which then makes it possible for the location and discharge behaviour of the leak to be assessed [12]. The TWR utilizes the first period of the transient wave to detect and localize the leak along the pipeline. The main objective of this work is to determine at what range along the pipeline is the TWR capable of localizing a leak, particularly in long conveyance pipes. This work would be executed as follows; (i) to highlight the governing equations for the model; (ii) to numerically analyse the transient wave in leakage pipes; (iii) to validate the model with laboratory experiment; (iv) to determine the leak detection range based on the critical signal-to-noise-ratio (SNR), with the application of wavelet transform; (v) analyse and discuss the results, and; (vi) draw some vital conclusions from the work.

which utilizes the first period of the transient wave to localize a leak has extensively been applied in urban water pipes. To critically analyze its effectiveness in conveyance pipes, the objective of this work is to determine at what range the method is capable of detecting and localizing leakage in the target pipe. The 1-D water hammer equations are used which are converted into total differential equations by the method of characteristics with the finite difference method, of which the Brunone’s unsteady friction is incorporated. The model is assumed to be a reservoirpipe-valve system, in which the side discharge valve located downstream is suddenly closed to generate the transient event. To execute this work, first of all the transient waves in both intact and leaking pipes are analyzed, and then validated practically with laboratory experiment, which yielded accurate results. Thereafter, the leak detection range is determined based on the critical signal-to-noiseratio, with the use of wavelet transform as a signal processing technique for deterministic feature extraction in a white Gaussian noise environment. Based on the results, it is shown that the leak detection range of the transient wave-based reflection method in conveyance pipes largely depends on the size of the leak, the standard deviation in the reflected signal, and the location of the leak from the measurement section.

Key Words: Conveyance pipe, Leak, Localization, Transient wave, Wavelet transform, Signal-to-noiseratio, White noise.

1.INTRODUCTION The available fresh water on earth is about 0.5 %, which is generally inadequate for human beings’ survival if not well managed. Water plays a major role in every aspect of mankind, and its mismanagement poses a serious detriment to all living things. Pipelines are one of the major means of water transportation, typically, from the source to the point of use. However, pipelines are prone to various anomalies such as, leakage, blockage, and corrosion. Most of the constructed pipelines are aging, been laid under the sea and tall buildings, or been exposed to adverse weather conditions (i.e. rainfall and sunshine).

Conveyance pipes (transmission main) used in this paper, transports water from both short and long distances to the point of usage. The importance of conveyance pipelines in

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2. GOVERNING EQUATIONS The 1-D water hammer equations describe the characteristics of transient in water pipelines. These equations are the dynamic and continuity equations, which are given respectively as [22] Figure 1: Schematic of an RPV system.

(1)

With further integrations, the equations can be written as [23]

(2) (7) where is the flow discharge; is the wavespeed; is the piezometric head; is the distance along the pipeline; is the cross-sectional area of pipe; is the inner diameter of pipe; is the vertical time in space; is the gravitational acceleration; and is the Darcy-Weisbach friction factor.

(8) Solving for H at point P,

The above equations are in partial differential form in which their solutions are complex and subsequently impractical. Hence, they are converted into ordinary differential equations for convenience. The method of characteristics (MOC) is used herein for this purpose, and then the finite difference method is applied to find possible solutions to the unknown variables. Equations (1) and (2) becomes

(9) (10) in which B is the pipeline characteristic impedance

(11) (3) and R is the pipeline resistance coefficient

(12) (4) The compatibility equations are solved for the unknowns and of , at any interior grid intersection point, point P at section i. That’s

and

(5)

(13) (14)

(6)

in which the , , , and are coefficients, and they are known constants when the equations are applied.

Where

Details of the water hammer equations can be found in [22], and [23].

(15)

Let’s assume point is a node at the pipe where the initial values of and Q are known. The ith section is horizontally situated along the pipe, and the distance between A and section i is . Point P is an interior node at where the characteristic lines of and from point A and B meet, at time , along which their respective compatibility equations are valid.

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(16) then

can be found as

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whereas, could be obtained from either Equation (13), (14), or from

leak somewhere along the pipe, it would be observed that the transient wave would exhibit a distinct singularity which wouldn’t be noticed in the case of no leak. This distinct singularity is related to the arrival of a leakreflected wave at the measurement station [4].

(18) In real-life applications, usually transient events in pipelines are coupled with complexities such as unsteady friction. To account for this, the Brunone’s model for unsteady friction is incorporated in to the governing equations to take care of this issue. With that, the and can be reproduced as

When there is no leak in the pipe, the duration of the first peak is half reflection time [26,27]. However, if there is a leak, duration of is the time in which the pressure wave propagates from the valve to the leak and returns to the valve. Mathematically

(19)

(25) where is the distance from the leak to the upstream reservoir; is the reflection time of the leak; and is the initial time of transient event.

(10) in which [24,25],

2.2 The wavelet transform The wavelet transform as signal processing tool, is a local time-frequency analyzer capable of extracting local frequency components of a signal. The signal H(t) of a continuous wavelet transform is defined as [6]

(11) in which ,

,

(26)

where Brunone’s friction coefficient; Vardy’s decay coefficient for turbulent flow; = the direction of flow.

where = mother wavelet; = translation parameter; and = scale parameter.

Since its application, the wavelet transform has proven to be a robust tool for identification of local singularities in measured signals caused by reflected waves from pipe anomalies [6,28].

2.1 Derivation of the leak model Considering the law of the conservation of mass, then

(12)

Many functions could be used for the mother wavelet, of which some of these have very simple analytical form, whiles others are of much more complex by definition and do not have explicit form [29]. The mother wavelet used in this work is the type of Daubechies [30] of an order of n, (dbn).

(23) in which discharge at upstream of the leak location; discharge at downstream of the leak location; discharge at the leak; pressure head at upstream of the leak location; pressure head at downstream of leak location, and; the head at the leak.

The leak is considered to be an orifice, and the leak flow is discharged into the atmosphere, hence

3. RESULTS ANALYSIS AND DISCUSSIONS 3.1 Numerical simulation for leak detection

A reservoir with a constant head which is located at upstream of the system, with an initial pressure head 50 m, and initial flow rate, 0.2553 m3/h. The rest of the pipe parameters are as follows; 0.05 m; pipe length, 1000 m. A side discharge valve located at downstream is closed suddenly to generate the transient event. When there is no leak, the transient wave behavior is plotted in Figure 2, which serves as a benchmark for the TWR method for the leak detection.

(24) in which coefficient of discharge and area of the leak. It is assumed that the pipe system consist of a reservoir, single pipeline, and a valve. In other words, a reservoir-pipe-valve (RPV) system is used for the analysis in this work (see Figure 1). The reservoir is located at the upstream whiles the valve is situated downstream. As soon as the valve is suddenly closed, it generates a transient signal which propagates upstream. If there is a

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Meanwhile, when there is a leak, with an actual location of, 800 m from upstream reservoir, and actual leak size, 8.5e-4 m2, a sudden closure of the side discharge valve shows the characteristics of the transient wave in the target pipe, as shown in Figure 2.

In the case of the intact pipe (leak-free), the pressure surge can be calculated by the Joukowsky’s equation, . From Figure 2, the results from the leaking pipe gives the maximum pressure head as, 165.75 m, and the head at the leak location as, m, reflecting at time 0.47 s. The leak location can be estimated by using Equation (25). It is obvious that there is heavy damping and a distinct singularity in the damaged pipe, indicating the presence of a leak.

Figure 3: Comparison of numerical and experimental transient waves.

3.3 Leak detection range for TWR To determine the leak detection range, a critical signal-to – noise-ratio (SNR) is to be established. This so called critical SNR is the minimum value of SNR which will allow for the localization of a leak in the target pipe. To achieve this, a numerical pressure signal with a step ( ) is generated and different tests are conducted, in which a white Gaussian noise of different SNR values are superimposed on the same pressure signal.

With an actual pressure variation (step), measured 4000 m from upstream reservoir on a pipe length of 20,000 m (2km), two test cases are carried out. The first test, 1A, with no superimposed white noise on the signal. With the use of the wavelet transform as a signal processing technique, the estimated variation location (step), after the analysis is 4000 m, with no(zero) error.

Figure 2: Transient wave behavior in a leaking, and intact pipe.

3.2 Experimental validation for leak detection The experiment is conducted at the laboratory of the College of Water Conservancy & Hydropower Engineering, Hohai University, China. The setup comprise of an upstream reservoir, and a downstream reservoir. A side discharge valve is fixed at the downstream section, of which a pressure transducer is installed near the upstream of the valve as the measuring device. In addition, a desktop computer which is connected with all the outlets and the transducers. The parameters for the setup are; pipe diameter, D = 0.025 m; length of pipe, L = 241.52 m; wavespeed, = 1350 m/s; initial pressure head, = 45 m; and initial flow rate, = 0.0125 m3/h.

For the second set of test, 2B, the same pressure signal is superimposed with white noise of different SNR values, SNR(-10, 0, 5, 7, 10, 20)dB. Twenty (20) different tests are conducted for each SNR, where the averaged localization error for each specific SNR is plotted. In Figure 5, it can be observed that the averaged location error estimated increases as the SNR values becomes lower. In other words, the higher the SNR, the lower the error encountered. Whiles the averaged localization error incurred for SNR = 10 is 5 m, which is acceptable, that of SNR = 7 and 5 are 9.75 and 15 m respectively. It is found that the error margin begins to widen below SNR = 7, and increases further as the SNR value gets lower. The SNR value for 7 dB, therefore becomes the critical SNR which is established to allow for leak localization in the target pipeline for this work. Its standard deviation ( ) can be determined as

A fast closure of the valve generated the transient event at time, 0.07 s. After the analysis, the estimated leak location is, 174 m. At the same time, the same experimental parameters are used for numerical simulation, and then compared with that of the experimental data. This is shown in Figure 3. The estimated numerical leak location gives, 173.4 m.

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whiles the relationship between the signal power (S) and the noise power can be described as .

which returns from the upstream reservoir in a half cycle, equals half the reflection time 2L/ is the arrival time at the measurement section, that’s 40 s. The distance from the leak to the measuring device for all the cases can be determined as

(29)

= the arrival time at the measurement

in which

section of a wave reflected by a singularity from a given distance, L.

Figure 4: Typical additive white noise of SNR = 10 dB.

Simulations for the different leak locations are analyzed and displayed in Table 1. It is worthy to note, that at every leak location, SNR of different are determined, and thereafter compared to that of the critical SNR, in which is a constant dependent on the characteristic of the pressure transducer [29].

The calculated at all the leak locations in ascending order of (2000 to 19850)m are, (18.9; 19; 19.35; 19.65; 19.86; 20.1; 20.16; 20.2; and 20.35)dB. The critical SNR = 7 dB can also be mathematically defined as

(30)

Figure 5: Identification of Critical SNR.

in which = the ratio between the average of the leak reflected wave and the standard deviation.

The SNR which can be defined as the measure of determining the level of desired signal to that of the level of background noise, is expressed as a single numerical value in decibels (dB). Mathematically,

Table 1: Different XL with their respective outcomes.

Test

(27) in cases of pressure step (

(m)

)

(28) where = estimated SNR of a signal; y = reduction of pressure due to the leak; and = generic pressure step.

Having established the minimum SNR (critical SNR = 7 dB) allowable for localization of leak in the pipeline, the range within which the TWR method is capable for detecting the leak anomaly can now be determined. For that matter, numerical simulations are conducted on a damaged long pipeline of different leak locations. The following parameters are used; = 50 m; D = 0.05 m; L = 20, 000 m 2 (2km); = 9e-4 m ; = 1000 m/s; = 0.0668 m3/h; actual leak locations, (2000, 4000, 8000, 12000, 15000, 18000, 19500, and 19850)m. A sudden valve closure at downstream generated the transient event. The wave

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y (m)

(m)

(s)

(m)

1

2000

69.62

67.16

2.46

36.01

18005

2

4000

69.622

66.74

2.48

32.01

16005

3

8000

68.42

65.88

2.54

24.01

12005

4

12000

67.62

65.02

2.6

16.01

8005

5

15000

67.02

64.38

2.64

10.01

5005

6

18000

66.42

63.73

2.69

2.01

2005

7

19000

66.22

63.52

2.7

1.01

1005

8

19500

66.12

63.42

2.7

1.31

505

9

19850

66.05

63.34

2.71

1.31

155

The amplitude of the leak reflected wave (y) is plotted against the distance between the leak and the measurement section ( ) which is shown in Figure 8, in which the leak detection range is determined based on the critical SNR.

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when the distance between the leak and the measurement section increases. It is then found that the localization of a leak highly depends on the magnitude of the step (pressure variation) and the standard deviation of the signal.

4. CONCLUSIONS The main objective of this work is to determine the leak detection range of the TWR method in conveyance pipes, which has been successfully carried out. Based on the results, the following conclusions are made.

Figure 6: Relationship between leak wave and the standard deviation.

The characteristics of transient waves in pressurized water pipes of both intact and damaged leaking pipes were investigated. The TWR demonstrated effectively for leak localization both numerically and experimentally. The results in Figure 2 and 3 clearly shows the TWR method can accurately localize leakage in the target pipe.

As it is observed in Figure 8 that, the estimated averaged localization error increases as the SNR values get lower, this could be as a result of the increase in the noise power over the signal when the SNR approaches lowest values.

The signal-to-noise-ratio can either be of zero, positive, or negative value. However, when the SNR ratio is greater than zero, it indicates that the signal strength is higher than the noise level. On the other hand, when the SNR produces zero value, it indicates the signal has the same strength as that of the noise. Furthermore, when the SNR yields a negative value, then it indicates the signal power is weaker than the noise power. It is worth mentioning that, by increasing the signal strength and using higher SNR will significantly produce a satisfactory estimation of leak localization in presence of white Gaussian noise.

Figure 7: Relationship between the SNR and standard deviation.

It is obvious to note in Figure 6 that, the amplitude of the leak wave decreases as the standard deviation increases with respect to the increment of the distance. Similarly, in Figure 7, it can be found that, as the distance from the leak and measurement section gets further, the standard deviation gains increasing power whiles the signal SNR gets lower.

The wavelet transform with a mother wavelet of db1, is capable of extracting and identifying leak reflected wave even in the noisy environment. Below the critical SNR, the localization error increases exponentially due to the increase in the noise power as the SNR further approaches negative values.

The TWR method is therefore capable of localizing leakage in conveyance pipes, nonetheless, its detection range is affected by the size of the leak, the standard deviation in the reflected signal, and the location of the leak.

It is important to state here that, the detection range was numerically determined with a single pipe in this paper, and future works would involve validating it with complex experimental/real-life long conveyance pipe systems.

Figure 8: The leak detection range in target pipeline.

It can be deduced from Figure 8 that, as the strength of the leak reflected signal decreases, the noise power increases

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ACKNOWLEDGEMENT (Optional)

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based leak detection in the frequency domain considering fluid–structure interaction and viscoelasticity.

Mech Syst Signal Process 2021;153:107500.https://doi.org/10.1016/j.ymssp.2 020.107500.

The author wishes to acknowledge the College of Water Conservancy & Hydropower Engineering, Hohai University, China.

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