# A Front-Line and Cost-Effective Model for the Assessment of Service Life of Network Pipes

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

- Required data, which are only the pipe failure records, as well as the basic pipe characteristics (material, length and diameter). Therefore, the model attention is only focused on the failure times, not on potential causes, influential factors, costs or physical models.
- Complexity of analysis, which is reduced to statistical fittings to usual distributions, mainly Weibull, but normal, log-normal, etc., can be also used.

## 2. Materials and Methods

#### 2.1. Development and Application of the BPLA

- $\mathrm{LtF}$ (life till failure) is the time elapsed between the installation date (Id) and the fail date (Fd) in case the sub-pipe has one failure assigned, or between the installation date (Id) and the end date (Ed) of the availability window in case the sub-pipe has not a failure assigned. Later analysis of $\mathrm{LtF}$ will provide the estimation for the average service life of the pipe group.
- $\mathrm{TtF}$ (time till failure) is the time elapsed between the start date (Sd) of the availability window and the fail date (Fd) in case the sub-pipe has one failure assigned, or between the start date (Sd) and the end date (Ed) of the availability window in case the sub-pipe has not a failure assigned. Analysis of $\mathrm{TtF}$ will provide the location on the bath curve and the hazard rate for future failures forecasting.

#### 2.2. Quick Summary of the WPHM

## 3. Case Study and Results

#### 3.1. Basic Description of the System

#### 3.2. Application and Results of BPLA

- -
- Homogeneity within each pipe group will be based on the combination of same material and same diameter. The age is only being taken into account in case the results are not statistically acceptable.
- -
- All four main materials must be represented in the pipe groups.
- -
- For each material, the diameter group chosen will be the one with the highest number of recorded failures.

#### 3.3. Application and results of WPHM

- -
- Logarithm of pipe length (Ln.length): Following Martins [34], the logarithm of the length has been considered (instead of the length itself as Alvisi and Franchini do [16,37]) assuming that the hazard rate should be proportional to the length of the pipe—which is a suitable assumption for the particular network under study.
- -
- Diameter (Diam.): In all four pipe groups, pipes belong to the diameter group 2 (Table 6).
- -
- Total number of failures (Fails): This covariate is constant for each pipe, and, therefore, for all the interarrival times derived from it. As explained by Martins [34], this is a much practical way of keeping it in the analysis, as opposed to other options, such as neglecting it [16] or taking it as a dynamic covariate [15], which could cause convergence problems in the subsequent Monte Carlo simulations.
- -
- Age left (AL): As shown in Figure 7, this is a dynamic covariate that takes a different value for each interarrival time on the same pipe.

## 4. Discussion

#### 4.1. Comparison of Results

#### 4.2. Comparison of Procedures

- The sequential order of the calculations. This involves performing two stages of calculation to obtain the last result.
- The different nature of the calculations themselves. The first one is a multivariate statistical fitting, for which more than basic software is required—in this case, it has been the R package. The second calculation is a series of simulations (up to 1000 iterations per pipe) for which some sort of programming is needed. In this case, it has been performed through VBA for MSExcel
^{®}.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Shamir, U.; Howard, C.D.D. An Analytic Approach to Scheduling Pipe Replacement. J. AWWA
**1979**, 5, 248–258. Available online: http://132.68.226.240/urishamir/files/documents/1979%20-%20Shamir%20and%20Howard%20-%20Pipe%20Replacement%20-%20JAWWA.pdf (accessed on 20 January 2020). - Lauer, W.C. Water Distribution Operator Training Handbook; American Water Works Association: Denver, CO, USA, 2013; ISBN 9781583219546. [Google Scholar]
- Rostum, J. Statistical Modelling of Pipe Failures in Water Networks. Ph.D. Thesis, Norwegian University of Science and Technology, Trondheim, Norway, 2000. [Google Scholar]
- Kleiner, Y.; Nafi, A.; Rajani, B. Planning renewal of water mains while considering deterioration, economies of scale and adjacent infrastructure. Water Sci. Technol. Water Supply
**2010**, 10, 897–906. [Google Scholar] [CrossRef] - Fuchs-Hanusch, D.; Kornberger, B.; Friedl, F.; Scheucher, R. Whole of life cost calculations for water supply pipes. Water Asset Manag. Int.
**2012**, 8, 19–24. [Google Scholar] - Scholten, L.; Scheidegger, A.; Reichert, P.; Mauer, M.; Lienert, J. Strategic rehabilitation planning of piped water networks using multi-criteria decision analysis. Water Res.
**2014**, 49, 124–143. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Amaitik, N.M.; Amaitik, S.M. Prediction of pipe failures in water mains using artificial neural network models. In Proceedings of the International Arab Conference of information Technology (ACIT’2010), Benghazi, Libya, 14–15 December 2010. [Google Scholar]
- Christodoulou, S.; Deligianni, A. A neurofuzzy decision framework for the management of water distribution networks. Water Resour. Manag.
**2010**, 24, 139–156. [Google Scholar] [CrossRef] - Kutylowska, M. Neural network approach for failure rate prediction. Eng. Fail. Anal.
**2015**, 47, 41–48. [Google Scholar] [CrossRef] - Kabir, G. Planning Repair and Replacement Program for Water Mains: A Bayesian Framework. Ph.D. Thesis, University of British Columbia, Vancouver, BC, Canada, 2016. [Google Scholar]
- Martínez, P.G. Análisis de Variables Explicativas en Modelos de Predicción de Roturas en Redes de Tuberías. Ph.D. Thesis, Universidad Politécnica de Madrid, Madrid, Spain, 2017. Available online: http://oa.upm.es/id/eprint/47857 (accessed on 20 January 2020).
- Motiee, H.; Ghasemnejad, S. Prediction of pipe failure in Tehran water distribution networks by applying regression models. Water Supply
**2019**, 19, 695–702. [Google Scholar] [CrossRef] [Green Version] - Di Nardo, A.; Di Natale, M.; Giudicianni, C.; Greco, R. Complex network and fractal theory for the assessment of water distribution network resilience to pipe failures. Water Supply
**2018**, 18, 767–777. [Google Scholar] [CrossRef] - Kutylowska, M. Forecasting failure rate of water pipes. Water Supply
**2019**, 19, 264–273. [Google Scholar] [CrossRef] [Green Version] - Le Gat, Y.; Eisenbeis, P. Using maintenance records to forecast failures in water networks. Urban Water
**2000**, 2, 173–181. [Google Scholar] [CrossRef] - Alvisi, S.; Franchini, M. Comparative analysis of two probabilistic pipe breakage models applied to a real water distribution system. Civ. Eng. Environ. Syst.
**2010**, 1, 1–22. [Google Scholar] [CrossRef] - Kimutai, E.; Betrie, G.; Brander, R.; Sadiq, R.; Tesfamariam, S. Comparison of statistical models for predicting pipe failures: Illustrative example with the city of Calgary water main failure. J. Pipeline Syst. Eng. Pract.
**2015**, 1–2. [Google Scholar] [CrossRef] - Santos, P.; Amado, C.; Coelho, S.T.; Leitão, J.P. Stochastic data mining tools for pipe blockage failure prediction. Urban Water J.
**2017**, 14, 343–353. [Google Scholar] [CrossRef] - Debon, A.; Carrion, A.; Cabrera, E.; Solano, H. Comparing risk of failure models in water supply networks using ROC curves. Reliab. Eng. Syst. Saf.
**2010**, 95, 43–48. [Google Scholar] [CrossRef] - Davis, P.; Burn, S.; Moglia, M.; Gould, S. A physical probabilistic model to predict failure rates in buried PVC pipelines. Reliab. Eng. Syst. Saf.
**2007**, 9, 1258–1266. [Google Scholar] [CrossRef] - Davis, P.; De Silva, D.; Marlow, D.; Moglia, M.; Gould, S.; Burn, S. Failure prediction and optimal scheduling of replacements in asbestos cement water pipes. J. Water Supply
**2008**, 4, 239–252. [Google Scholar] [CrossRef] - Punurai, W.; Davis, P. Prediction of asbestos cement water pipe aging and pipe priorization using monte carlo simulation. Eng. J.
**2017**, 2, 1–13. [Google Scholar] [CrossRef] - Yoo, D.; Kang, D.; Jun, H.; Kim, J. Rehabilitation priority determination of water pipes based on hydraulic importance. Water
**2014**, 6, 3864–3887. [Google Scholar] [CrossRef] [Green Version] - D’Ercole, M.; Righetti, M.; Raspati, G.S.; Bertola, P.; Ugarelli, R.M. Rehabilitation planning of water distribution network through a reliability-based risk assessment. Water
**2018**, 10, 277. [Google Scholar] [CrossRef] [Green Version] - Kleiner, Y.; Rajani, B. Comprehensive review of structural deterioration of water mains: Statistical models. Urban Water
**2001**, 3, 131–150. [Google Scholar] [CrossRef] [Green Version] - Rajani, B.; Kleiner, Y. Comprehensive review of structural deterioration of water mains: Physically based models. Urban Water
**2001**, 3, 151–164. [Google Scholar] [CrossRef] [Green Version] - Scheidegger, A.; Leitao, J.P.; Sholten, L. Statistical failure models for water distribution pipes–A review from a unified perspective. Water Res.
**2015**, 83, 237–247. [Google Scholar] [CrossRef] - van Vossen-van den Berg, J.; van Laarhoven, K.; Hillebrand, B.; Diemel, R. Predicting future failure rates of pipes as a ground for strategic asset management. In Proceedings of the IWA Specialized Conference: LESAM 2019, Vancouver, BC, Canada, 25–27 September 2019. [Google Scholar]
- Azeitona, M.; Vitorino, D.; Marques, J.; Pina, A.; Coelho, S.T. Network failure prediction: Applying statistics to open data and continuous data acquisition. In Proceedings of the Conference: 18° ENASB/18° SILUBESA, Porto, Portugal, 7–9 October 2018. [Google Scholar]
- Yin, T.; Becelaere, T. NETSCAN: The First Step of Dynamic Asset Management; Singapore International Water Week: Singapore, 2018. [Google Scholar]
- Kropp, I.; Baur, R. Integrated failure forecasting model for the strategic rehabilitation planning process. Water Supply
**2005**, 5, 1–8. [Google Scholar] [CrossRef] [Green Version] - Ebeling, C.E. An Introduction to Reliability and Maintainability Engineering; Waveland Press: Long Grove, IL, USA, 2019; ISBN-13: 978-1478637349. [Google Scholar]
- Mora, L.A. Mantenimiento, Planeación, Ejecución y Control; Alfaomega Grupo Editor: Ciudad de México, México, 2009; ISBN 13: 9789586827690. [Google Scholar]
- Martins, A.D.C. Stochastic Models for Prediction of Pipe Failures in Water Supply Systems. Master’s Thesis, Instituto Superior Tecnico, Lisboa, Portugal, 2011. [Google Scholar]
- García-Mora, B.; Debón, A.; Santamaría, C.; Carrión, A. Modelling the failure risk for water supply networks with interval-censored data. Reliab. Eng. Syst. Saf.
**2015**, 144, 311–318. [Google Scholar] [CrossRef] - Levin, R.I.; Rubin, D.S.; Rastogi, S.; Siddiqui, M.H. Statistics for Management; Pearson: London, UK, 2008; ISBN-13: 978-8177585841. [Google Scholar]
- Ramirez, R. Desarrollo de un Modelo Estadístico para la Estimación de la Vida útil y Predicción de Fallos en Tuberías de agua Potable y su Aplicación en la Gestión de Activos. Master’s Thesis, Universitat Politecnica de Valencia, Valencia, Spain, 2019. [Google Scholar]
- Nwobi, F.N.; Ugomma, C.A. A comparison of methods for the estimation of weibull distribution parameters. Metodološki Zvezki
**2014**, 11, 65–78. [Google Scholar] - Lei, Y. Evaluation of three methods for estimating the Weibull distribution parameters of Chinese pine (Pinus tabulaeformis). J. For. Sci.
**2008**, 12, 566–571. [Google Scholar] [CrossRef] [Green Version] - Rinne, H. The Weibull Distribution: A Handbook; Chapman & Hall/CRC Press-Taylor & Francis Group: Giessen, Germany, 2009; ISBN-13: 978-1420087437. [Google Scholar]
- Datsiou, K.C.; Overend, M. Weibull parameter estimation and goodness-of-fit for glass strength data. Struct. Saf.
**2018**, 73, 29–41. [Google Scholar] [CrossRef] - Therneau, T.M.; Lumley, T. Package survival. Available online: https://cran.r-project.org/web/packages/survival/survival.pdf (accessed on 29 February 2020).
- Christodoulou, S.E. Water network assessment and reliability analysis by use of survival analysis. Water Resour. Manag.
**2011**, 4, 1229–1238. [Google Scholar] [CrossRef]

**Figure 1.**Classification of statistical models for pipe survival analysis (based on Kabir [10]).

**Figure 7.**Main time variables for failure data of one pipe in the WPHM (according to Alvisi and Franchini [16]).

Pipe | Instal. Date | Length | Diameter | Material |
---|---|---|---|---|

P_{1} | Id_{1} | L_{1} | D_{1} | M_{1} |

… | … | ... | ... | ... |

P_{i} | Id_{i} | L_{i} | D_{i} | M_{i} |

P_{j} | Id_{j} | L_{j} | D_{j} | M_{j} |

… | … | ... | ... | ... |

P_{n} | Id_{n} | L_{n} | D_{n} | M_{n} |

Failure | Fail Date | Pipe | Fail Order in Pipe |
---|---|---|---|

F_{1} | Fd_{1} | Pi | P_{i}_O_{1} |

F_{2} | Fd_{2} | Pj | P_{j}_O_{1} |

F_{3} | Fd_{3} | Pn | P_{n}_O_{1} |

F_{4} | Fd_{4} | Pj | P_{j}_O_{2} |

F_{5} | Fd_{5} | Pj | P_{j}_O_{3} |

… | ... | … | ... |

F_{m} | Fd_{m} | Pn | P_{n}_O_{2} |

Sub-Pipe | Length | Diam. | Mat. | Fail Date | Instal. Date | LtF | TtF |
---|---|---|---|---|---|---|---|

SP_{1_1} | 1 | D_{1} | M_{1} | - | Id_{1} | Ed − Id_{1} | Ed − Sd |

... | 1 | D_{1} | M_{1} | - | Id_{1} | Ed − Id_{1} | Ed − Sd |

SP_{1_L1} | 1 | D_{1} | M_{1} | - | Id_{1} | Ed − Id_{1} | Ed − Sd |

... | ... | ... | ... | ... | ... | ... | ... |

SP_{i_1} | 1 | D_{i} | M_{i} | Fd_{1} | Id_{i} | Fd_{1} − Id_{i} | Fd_{1} − Sd |

SP_{i_2} | 1 | D_{i} | M_{i} | - | Id_{i} | Ed − Id_{i} | Ed − Sd |

... | 1 | D_{i} | M_{i} | - | Id_{i} | Ed − Id_{i} | Ed − Sd |

SP_{i_Li} | 1 | D_{i} | M_{i} | - | Id_{i} | Ed − Id_{i} | Ed − Sd |

SP_{j_1} | 1 | D_{j} | M_{j} | Fd_{2} | Id_{j} | Fd_{2} − Id_{j} | Fd_{2} − Sd |

SP_{j_2} | 1 | D_{j} | M_{j} | Fd_{4} | Id_{j} | Fd_{4} − Id_{j} | Fd_{4} − Sd |

SP_{j_3} | 1 | D_{j} | M_{j} | Fd_{5} | Id_{j} | Fd_{5} − Id_{j} | Fd_{5} − Sd |

SP_{j_4} | 1 | D_{j} | M_{j} | - | Id_{j} | Ed − Id_{j} | Ed − Sd |

... | 1 | D_{j} | M_{j} | - | Id_{j} | Ed − Id_{j} | Ed − Sd |

SP_{j_Lj} | 1 | D_{j} | M_{j} | - | Id_{j} | Ed − Id_{j} | Ed − Sd |

... | ... | ... | ... | ... | ... | ... | ... |

SP_{n_1} | 1 | D_{n} | M_{n} | Fd_{3} | Id_{n} | Fd_{3} − Id_{n} | Fd_{3} − Sd |

SP_{n_2} | 1 | D_{n} | M_{n} | Fd_{6} | Id_{n} | Fd_{6} − Id_{n} | Fd_{6} − Sd |

SP_{n_3} | 1 | D_{n} | M_{n} | - | Id_{n} | Ed − Id_{n} | Ed − Sd |

... | 1 | D_{n} | M_{n} | - | Id_{n} | Ed − Id_{n} | Ed − Sd |

SP_{n_Ln} | 1 | D_{n} | M_{n} | - | Id_{n} | Ed − Id_{n} | Ed − Sd |

$\mathsf{\tau}$ | Fail Date | Pipe | Fail Order in Pipe | Interarr. Time | AL | Length | Diam. | Mat. |
---|---|---|---|---|---|---|---|---|

$\tau $_{1} | - | P_{1} | - | Ed − Sd | Sd − Id_{1} | L_{1} | D_{1} | M_{1} |

... | ... | ... | ... | ... | ... | ... | ... | ... |

$\tau $_{i} | Fd_{1} | P_{i} | P_{i}_O_{1} | Fd_{1} − Sd | Sd − Id_{i} | L_{i} | D_{i} | M_{i} |

$\tau $_{i + 1} | - | P_{i} | - | Ed − Fd_{1} | Fd_{1} − Id_{i} | L_{i} | D_{i} | M_{i} |

$\tau $_{i + 2} | Fd_{2} | P_{j} | P_{j}_O_{1} | Fd_{2} − Sd | Sd − Id_{j} | L_{j} | D_{j} | M_{j} |

$\tau $_{i + 3} | Fd_{4} | P_{j} | P_{j}_O_{2} | Fd_{4} − Fd_{2} | Fd_{2} − Id_{j} | L_{j} | D_{j} | M_{j} |

$\tau $_{i + 4} | Fd_{5} | P_{j} | P_{j}_O_{3} | Fd_{5} − Fd_{4} | Fd_{4} − Id_{j} | L_{j} | D_{j} | M_{j} |

$\tau $_{i + 5} | - | P_{j} | - | Ed − Fd_{5} | Fd_{5} − Id_{j} | L_{j} | D_{j} | M_{j} |

... | ... | ... | ... | ... | ... | ... | ... | ... |

$\tau $_{m + n − 2} | Fd_{3} | P_{n} | P_{n}_O_{1} | Fd_{3} − Sd | Sd − Id_{n} | L_{n} | D_{n} | M_{n} |

$\tau $_{m + n − 1} | Fd_{m} | P_{n} | P_{n}_O_{2} | Fd_{m} − Fd_{3} | Fd_{3} − Id_{n} | L_{n} | D_{n} | M_{n} |

$\tau $_{m + n} | - | P_{n} | - | Ed − Fd_{m} | Fd_{m} − Id_{n} | L_{n} | D_{n} | M_{n} |

Materials | Network Length (%) | Avg. Age (Years) |
---|---|---|

Asbestos cement (AC) | 30.2 | 41.4 |

Ductile iron (DI) | 26.0 | 45.4 |

Cast iron (CI) | 17.4 | 62.8 |

Polyethylene (PE) | 25.5 | 42.7 |

Other (steel, PVC, concrete…) | 0.9 | 55.7 |

Diameter Groups | Network Length (%) | |
---|---|---|

ID Gr. | Diameter (mm) | |

1 | <63 | 6.0 |

2 | 63–140 | 47.6 |

3 | 140–200 | 23.3 |

4 | 200–280 | 13.5 |

5 | 280–400 | 4.5 |

6 | >400 | 5.1 |

Year | Material Gr. | Diameter Gr. | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

AC | DI | CI | PE | 1 | 2 | 3 | 4 | 5 | 6 | |

2017 | 70 | 27 | 57 | 192 | 61 | 227 | 33 | 25 | 0 | 0 |

2018 | 46 | 52 | 61 | 240 | 131 | 219 | 38 | 9 | 2 | 0 |

Total | 116 | 79 | 118 | 432 | 193 | 446 | 71 | 34 | 2 | 0 |

Pipe Group | Mat. Gr./Diam. Gr. | Total Pipes | Total Length (km) | Recorded Failures | |
---|---|---|---|---|---|

Pipes with Failures | Total Failures | ||||

1 | AC/2 | 6149 | 165.7 | 62 | 75 |

2 | DI/2 | 8278 | 121.4 | 56 | 58 |

3 | CI/2 | 5242 | 120.1 | 70 | 87 |

4 | PE/2 | 5382 | 138.1 | 166 | 232 |

Pipe Group | LtF Analysis | TtF Analysis | ||
---|---|---|---|---|

${\mathsf{\beta}}_{\mathbf{L}\mathbf{t}\mathbf{F}}$ | ${\mathsf{\eta}}_{\mathbf{L}\mathbf{t}\mathbf{F}}$ | ${\mathsf{\beta}}_{\mathbf{T}\mathbf{t}\mathbf{F}}$ | ${\mathsf{\eta}}_{\mathbf{T}\mathbf{t}\mathbf{F}}$ | |

1 | 8.03 | 114.2 | 0.98 | 8700 |

2 | 9.96 | 108.8 | 1.33 | 635 |

3 | 4.76 | 197.6 | 1.27 | 1123 |

4 | 7.08 | 115.0 | 1.18 | 436 |

Pipe Group | LtF Analysis | TtF Analysis | ||||
---|---|---|---|---|---|---|

${\mathbf{D}}_{\mathbf{M}\mathbf{a}\mathbf{x}}$ | ${\mathbf{D}}_{\mathbf{C}\mathbf{r}\mathbf{i}\mathbf{t}}$ | $\left({\mathbf{D}}_{\mathbf{M}\mathbf{a}\mathbf{x}}<{\mathbf{D}}_{\mathbf{C}\mathbf{r}\mathbf{i}\mathbf{t}}\right)?$ | ${\mathbf{D}}_{\mathbf{M}\mathbf{a}\mathbf{x}}$ | ${\mathbf{D}}_{\mathbf{C}\mathbf{r}\mathbf{i}\mathbf{t}}$ | $\left({\mathbf{D}}_{\mathbf{M}\mathbf{a}\mathbf{x}}<{\mathbf{D}}_{\mathbf{C}\mathbf{r}\mathbf{i}\mathbf{t}}\right)?$ | |

1 | 0.0092 | 0.0034 | No | 0.0001 | 0.0034 | Yes |

2 | 0.0002 | 0.0038 | Yes | 0.0001 | 0.0039 | Yes |

3 | 0.0554 | 0.0043 | No | 0.0003 | 0.0043 | Yes |

4 | 0.0005 | 0.0036 | Yes | 0.0001 | 0.0036 | Yes |

Pipe Group | Mat. Gr./Diam. Gr./Age Gr. | Total Pipes | Total Length (km) | Recorded Failures | |
---|---|---|---|---|---|

Pipes with Failures | Total Failures | ||||

1* | AC/2/<50 | 156,397 | 156.4 | 52 | 64 |

2 | DI/2/all | 121,443 | 121.4 | 56 | 58 |

3* | CI/2/<60 | 84,694 | 84.7 | 36 | 51 |

4 | PE/2/all | 138,141 | 138.1 | 166 | 232 |

Pipe Group | LtF Analysis | TtF Analysis | ||
---|---|---|---|---|

${\mathsf{\beta}}_{\mathbf{L}\mathbf{t}\mathbf{F}}$ | ${\mathsf{\eta}}_{\mathbf{L}\mathbf{t}\mathbf{F}}$ | ${\mathsf{\beta}}_{\mathbf{T}\mathbf{t}\mathbf{F}}$ | ${\mathsf{\eta}}_{\mathbf{T}\mathbf{t}\mathbf{F}}$ | |

1* | 7.48 | 126.3 | 0.90 | 11,600 |

2 | 9.96 | 108.8 | 1.33 | 635 |

3* | 16.36 | 88.9 | 1.07 | 2,070 |

4 | 7.08 | 115.0 | 1.18 | 436 |

Pipe Group | LtF Analysis | TtF Analysis | ||||
---|---|---|---|---|---|---|

${\mathbf{D}}_{\mathbf{M}\mathbf{a}\mathbf{x}}$ | ${\mathbf{D}}_{\mathbf{C}\mathbf{r}\mathbf{i}\mathbf{t}}$ | $\left({\mathbf{D}}_{\mathbf{M}\mathbf{a}\mathbf{x}}-{\mathbf{D}}_{\mathbf{C}\mathbf{r}\mathbf{i}\mathbf{t}}\right)?$ | ${\mathbf{D}}_{\mathbf{M}\mathbf{a}\mathbf{x}}$ | ${\mathbf{D}}_{\mathbf{C}\mathbf{r}\mathbf{i}\mathbf{t}}$ | $\left({\mathbf{D}}_{\mathbf{M}\mathbf{a}\mathbf{x}}-{\mathbf{D}}_{\mathbf{C}\mathbf{r}\mathbf{i}\mathbf{t}}\right)?$ | |

1* | 0.0004 | 0.0034 | Yes | 5 × 10^{−5} | 0.0034 | Yes |

2 | 0.0002 | 0.0038 | Yes | 0.0001 | 0.0039 | Yes |

3* | 0.0005 | 0.0046 | Yes | 0.0001 | 0.0046 | Yes |

4 | 0.0005 | 0.0036 | Yes | 0.0001 | 0.0036 | Yes |

Pipe Group | Characteristic Life (Years) | ${\mathbf{\beta}}_{\mathit{T}\mathit{t}\mathit{F}}$ | Location on the Bath Curve | Recommened Action | |
---|---|---|---|---|---|

Stage | Phase | ||||

1* | 126.3 | 0.90 | 1 | - | Corrective |

2 | 108.8 | 1.33 | 3 | 1 | Preventive |

3* | 88.9 | 1.07 | 3 | 1 | Preventive |

4 | 115.0 | 1.18 | 3 | 1 | Preventive |

Pipe. Group | Year 1 | Year 2 | Year 3 | Year 4 | Year 5 | |||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathsf{\lambda}}_{\mathbf{T}\mathbf{t}\mathbf{F}}$ | #Fails | ${\mathsf{\lambda}}_{\mathbf{T}\mathbf{t}\mathbf{F}}$ | Fails | ${\mathsf{\lambda}}_{\mathbf{T}\mathbf{t}\mathbf{F}}$ | Fails | ${\mathsf{\lambda}}_{\mathbf{T}\mathbf{t}\mathbf{F}}$ | Fails | ${\mathsf{\lambda}}_{\mathbf{T}\mathbf{t}\mathbf{F}}$ | Fails | |

1* | 2.0 × 10^{−4} | 31 | 1.8 × 10^{−4} | 29 | 1.8 × 10^{−4} | 28 | 1.7 × 10^{−4} | 27 | 1.7 × 10^{−4} | 26 |

2 | 2.5 × 10^{−4} | 30 | 3.1 × 10^{−4} | 38 | 3.6 × 10^{−4} | 43 | 3.8 × 10^{−4} | 48 | 4.2 × 10^{−4} | 51 |

3* | 3.0 × 10^{−4} | 26 | 3.2 × 10^{−4} | 27 | 3.3 × 10^{−4} | 28 | 3.3 × 10^{−4} | 28 | 3.4 × 10^{−4} | 29 |

4 | 9.1 × 10^{−4} | 125 | 1.0 × 10^{−3} | 142 | 1.1 × 10^{−3} | 153 | 1.2 × 10^{−3} | 161 | 1.2 × 10^{−3} | 167 |

Coefficient | Pipe Group 1* | Pipe Group 2 | Pipe Group 3* | Pipe Group 4 | ||||
---|---|---|---|---|---|---|---|---|

$\mathsf{\beta}$ | ${\mathbf{e}}^{\mathsf{\beta}}$ | $\mathsf{\beta}$ | ${\mathbf{e}}^{\mathsf{\beta}}$ | $\mathsf{\beta}$ | ${\mathbf{e}}^{\mathsf{\beta}}$ | $\mathsf{\beta}$ | ${\mathbf{e}}^{\mathsf{\beta}}$ | |

Intercept (${\mathsf{\beta}}_{0}$) | 5.38 | - | 8.15 | - | 20.07 | - | 3.81 | - |

Scale ($\mathsf{\sigma}$) | 1.18 | - | 0.58 | - | 1.30 | - | 0.88 | - |

Ln.length $({\mathsf{\beta}}_{1})$ | −0.206 | 0.813 | −0.287 | 0.750 | −0.174 | 0.839 | −0.236 | 0.789 |

Diam (${\mathsf{\beta}}_{2}$) | 0.006 | 1.006 | −0.022 | 0.977 | −0.065 | 0.937 | 0.016 | 1.016 |

Fails (${\mathsf{\beta}}_{3}$) | −3.319 | 0.036 | −2.353 | 0.095 | −2.497 | 0.082 | −0.728 | 0.482 |

AL (${\mathsf{\beta}}_{4}$) | 0.041 | 1.042 | −0.025 | 0.974 | −0.109 | 0.896 | −0.024 | 0.975 |

Pipe Group | Scale | ${\mathsf{\beta}}_{\mathbf{T}\mathbf{t}\mathbf{F}}^{\mathbf{e}\mathbf{q}\mathbf{u}\mathbf{i}\mathbf{v}}$ |
---|---|---|

1* | 1.18 | 0.847 |

2 | 0.58 | 1.724 |

3* | 1.30 | 0.769 |

4 | 0.88 | 1.136 |

Pipe Group | Year 1 | Year 2 | Year 3 | Year 4 | Year 5 |
---|---|---|---|---|---|

1* | 35 | 30 | 29 | 28 | 26 |

2 | 18 | 34 | 44 | 52 | 59 |

3* | 35 | 30 | 30 | 31 | 34 |

4 | 113 | 130 | 138 | 144 | 147 |

Pipe Group | $\mathsf{\beta}$ Obtained by Each Model | Relative Error (%) to WPHM | |||
---|---|---|---|---|---|

WPHM | BPLA | Pre-BPLA | BPLA | Pre-BPLA | |

1* | 0.85 | 0.90 | 0.87 | 6 | 2 |

2 | 1.72 | 1.33 | 1.33 | 23 | 23 |

3* | 0.77 | 1.07 | 0.93 | 39 | 21 |

4 | 1.14 | 1.18 | 1.12 | 4 | 2 |

Pipe Gr. | WPHM | BPLA | pre-BPLA | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 1 | 2 | 3 | 4 | 5 | 1 | 2 | 3 | 4 | 5 | |

1* | 35 | 30 | 29 | 28 | 26 | 31 | 29 | 28 | 27 | 26 | 25 | 23 | 21 | 21 | 20 |

2 | 18 | 34 | 44 | 52 | 59 | 30 | 38 | 43 | 48 | 51 | 31 | 38 | 44 | 48 | 52 |

3* | 35 | 30 | 30 | 31 | 34 | 26 | 27 | 28 | 28 | 29 | 18 | 17 | 16 | 16 | 16 |

4 | 113 | 130 | 138 | 144 | 147 | 125 | 142 | 153 | 161 | 167 | 86 | 94 | 98 | 102 | 104 |

Pipe Gr. | BPLA | pre-BPLA | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 1 | 2 | 3 | 4 | 5 | |

1* | 11 | 3 | 3 | 4 | 0 | 32 | 24 | 29 | 26 | 23 |

2 | 67 | 12 | 2 | 8 | 14 | 43 | 11 | 0 | 8 | 14 |

3* | 26 | 10 | 7 | 10 | 15 | 65 | 48 | 50 | 54 | 62 |

4 | 11 | 9 | 11 | 12 | 14 | 22 | 25 | 26 | 26 | 26 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ramirez, R.; Torres, D.; López-Jimenez, P.A.; Cobacho, R.
A Front-Line and Cost-Effective Model for the Assessment of Service Life of Network Pipes. *Water* **2020**, *12*, 667.
https://doi.org/10.3390/w12030667

**AMA Style**

Ramirez R, Torres D, López-Jimenez PA, Cobacho R.
A Front-Line and Cost-Effective Model for the Assessment of Service Life of Network Pipes. *Water*. 2020; 12(3):667.
https://doi.org/10.3390/w12030667

**Chicago/Turabian Style**

Ramirez, Roberto, David Torres, P. Amparo López-Jimenez, and Ricardo Cobacho.
2020. "A Front-Line and Cost-Effective Model for the Assessment of Service Life of Network Pipes" *Water* 12, no. 3: 667.
https://doi.org/10.3390/w12030667