# Determination of Pipeline Leaks Based on the Analysis the Hurst Exponent of Acoustic Signals

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

^{0.5}

^{H}

#### 2.1. R/S Analysis Algorithm

- 1.
- The smallest own divisor m of the sample n is determined. Sample n is divided into k = n/m groups.

_{i}.

- 2.
- For each group, the average is calculated$$\overline{{t}_{k}}=\frac{1}{m}{\displaystyle \sum}_{i=1}^{m}{t}_{i},\frac{1}{m}{\displaystyle \sum}_{i=m+1}^{2m}{t}_{i}\dots \frac{1}{m}{\displaystyle \sum}_{i=\left(k-1\right)m+1}^{n}{t}_{i}$$
_{i}$${X}_{1}={t}_{1}-\frac{1}{m}{\displaystyle \sum}_{i=1}^{m}{t}_{i},{X}_{2}={t}_{2}-\frac{1}{m}{\displaystyle \sum}_{i=1}^{m}{t}_{i}+{X}_{1}\dots {X}_{m}={t}_{m}-\frac{1}{m}{\displaystyle \sum}_{i=1}^{m}{t}_{i}+{X}_{m-1}$$ - 3.
- Calculate the normalized diaposon for each group$${R}_{k}=max\left({X}_{i}\right)-min\left({X}_{i}\right)$$
- 4.
- For each group, the standard deviation S
_{k}calculated according to the standard formula$${S}_{k}=\sqrt{\frac{1}{m}{\displaystyle \sum}_{i=1}^{m}{\left({t}_{i}-\overline{{t}_{k}}\right)}^{2}}$$ - 5.
- The R/S index for each group is defined as R
_{k}/S_{k}. Then, the average range of the variation is found$$\overline{R/{S}_{j}}=\frac{1}{k}{\displaystyle \sum}_{i=1}^{k}\frac{R}{{S}_{i}}$$

- 6.
- The procedure described above is repeated for all possible proper divisors as m. At the last step, m = n/2.Thus, a selection is obtained $\overline{R/{S}_{j}}$The number of elements in the sample match the number of proper divisors.
- 7.
- A graph of dependences of log R/S on log m is being built and using the method of least squares a regression equation of the form$$\mathrm{log}R/S=H\times \mathrm{log}m+\mathrm{log}c$$

#### 2.2. Experimental Stand

## 3. Results and Discussion

- The position estimate is calculated $\overline{H}$.
- The spread estimate S is calculated as a standard deviation.
- For a given significance level α confidence interval is constructed:$$\overline{H}+S\times t\left(1-\frac{\alpha}{2},m-2\right)$$

^{2}/s.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Experimental device: 1—valve; 2—pipeline (length 2 m, external diameter 0.159 m, wall thickness 6 mm); 3—vibration acceleration sensor AR2038R; 4—defect; 5—manometer; 6—capacity; 7—pump LEO XKJ-900 I; 8—matching device AG01-3; 9—analog-to-digital converter NI USB-6229; 10—computer.

**Figure 6.**Values of the Hurst exponent at a pump discharge pressure of 1.5 bar: (

**a**) for X-axis signals, (

**b**) for Y-axis signals.

**Figure 7.**Values of the Hurst exponent at a pump discharge pressure of 2 bar: (

**a**) for X-axis signals, (

**b**) for Y-axis signals.

**Figure 8.**Values of the Hurst exponent at a pump discharge pressure of 2.5 bar: (

**a**) for X-axis signals, (

**b**) for Y-axis signals.

**Figure 9.**Values of the Hurst exponent at a pump discharge pressure of 3 bar: (

**a**) for X-axis signals, (

**b**) for Y-axis signals.

**Figure 10.**Values of the Hurst exponent at a pump discharge pressure of 3.5 bar: (

**a**) for X-axis signals, (

**b**) for Y-axis signals.

**Figure 11.**Comparison of the spectra of acoustic signals of the pipeline at a pressure of 2 bar (the spectrum of a defect–free pipeline is indicated in black; the spectrum of a pipeline with a leak is indicated in red): (

**a**) for a defect diameter of 1 mm, (

**b**) for a defect diameter of 2 mm, (

**c**) for a defect diameter of 3 mm, (

**d**) for a defect diameter of 4 mm, (

**e**) for a defect diameter of 5 mm, (

**f**) for a defect diameter of 6 mm, (

**g**) for a defect diameter of 7 mm, (

**h**) for a defect diameter of 8 mm.

**Figure 13.**Spectra of acoustic signals of the pipeline at a pressure of 2 bar (the spectrum of a defect–free pipeline is indicated in black; the spectrum of a pipeline with a leak is indicated in red): (

**a**) for a defect diameter of 1 mm, (

**b**) for a defect diameter of 2 mm, (

**c**) for a defect diameter of 3 mm, (

**d**) for a defect diameter of 4 mm, (

**e**) for a defect diameter of 5 mm, (

**f**) for a defect diameter of 6 mm, (

**g**) for a defect diameter of 7 mm, (

**h**) for a defect diameter of 8 mm.

Pressure, Bar | Consumption, L/Min |
---|---|

1.5 | 45 |

2 | 38 |

2.5 | 31 |

3 | 24 |

3.5 | 16 |

Hole Diameter, Mm | Maximum Pressure, Bar |
---|---|

1 | 3.5 |

2 | 3.5 |

3 | 3 |

4 | 2.5 |

5 | 2.5 |

6 | 2 |

7 | 2 |

8 | 2 |

**Table 3.**The results of calculating the Reynolds number for different modes of water flow in the pipe.

Pressure, Bar | Reynolds Number |
---|---|

1.5 | 6461 |

2 | 5456 |

2.5 | 4451 |

3 | 3446 |

3.5 | 2441 |

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**MDPI and ACS Style**

Zagretdinov, A.; Ziganshin, S.; Vankov, Y.; Izmailova, E.; Kondratiev, A. Determination of Pipeline Leaks Based on the Analysis the Hurst Exponent of Acoustic Signals. *Water* **2022**, *14*, 3190.
https://doi.org/10.3390/w14193190

**AMA Style**

Zagretdinov A, Ziganshin S, Vankov Y, Izmailova E, Kondratiev A. Determination of Pipeline Leaks Based on the Analysis the Hurst Exponent of Acoustic Signals. *Water*. 2022; 14(19):3190.
https://doi.org/10.3390/w14193190

**Chicago/Turabian Style**

Zagretdinov, Ayrat, Shamil Ziganshin, Yuri Vankov, Eugenia Izmailova, and Alexander Kondratiev. 2022. "Determination of Pipeline Leaks Based on the Analysis the Hurst Exponent of Acoustic Signals" *Water* 14, no. 19: 3190.
https://doi.org/10.3390/w14193190