# Sewer Pipes Condition Prediction Models: A State-of-the-Art Review

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## Abstract

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## 1. Introduction

## 2. Sewer Pipe Deterioration

## 3. Factors Affecting Deterioration of Sewer Pipes

## 4. Deterioration Models for Sewer Pipes

## 5. Statistical Models

#### 5.1. Linear Regression Models

#### 5.1.1. Model Description

_{i}and X

_{i}as shown in Equation (1) [21].

_{i}is dependent variable for facility i, β

_{0}, and β

_{1}are parameters to be estimated, X

_{i}is independent variable, and ϵ

_{i}is random error term. Multiple linear regression can be used to predict condition of sewer pipe with consideration of more than one independent variables. When deterioration of sewer pipes is modeled, condition state of the pipe is the dependent variable and independent variables contain pipe attributes such as pipe age, material, length, slope, and other environmental and operational factors. As the condition states of the sewer pipes are discrete values, the linear regression may have trouble predicting the categorical variables.

#### 5.1.2. Previous Studies

#### 5.1.3. Model Discussion

#### 5.2. Markov Chain Models

#### 5.2.1. Model Description

_{ij}is the transition probability that, given the system in state i at time t, will be at state j at time (t + 1). Generally, the transition probability matrix (m × m matrix) is used to calculate the transition probabilities. For example, consider a set of pipe state condition, C = {C1, C2, C3, C4, C5}. When a sewer pipe is in condition 1, a series of probabilities P11, P12, P13, P14, and P15 determine the condition state of pipe in the next period. The deterioration process starts in one of the states and moves from one to another. If the sewer pipe is currently in condition C3, it moves to condition C4 in the next step with a probability of P34. This probability is called transition probability and only takes into account the current condition of pipe, without considering the historical data and previous conditions. The transition probability matrix is given in Equation (3).

#### 5.2.2. Previous Studies

#### 5.2.3. Model Discussion

#### 5.3. Logistic Regression Model

#### 5.3.1. Model Description

_{1}, X

_{2}, …, X

_{p}are independent variables, α is the intercept parameter for category i, and β is the regression coefficients. Multinomial logistic regression is used when multiple levels of categorical response variables are in the model. Equation (6) shows the multinomial logistic regression formula.

_{1}, X

_{2}, …, X

_{p}are independent variables, α is the intercept parameter for category i, and β is the regression coefficients associated with dependent category i. The probability than Y = 1 can be measured using an exponential transformation as shown in Equation (7).

#### 5.3.2. Previous Studies

## 6. Discussion and Conclusions

## 7. Future Research Needs

## 8. Acronyms

AI | Artificial Intelligence |

ASCE | American Society of Civil Engineers |

AWWA | American Water Works Association |

C | Pipe State Condition |

CCTV | Closed-circuit Television |

CIPP | Cured-in-Place Pipe |

EPA | Environmental Protection Agency |

Equation | Equation |

i | Facility Index |

I/I | Infiltration/Inflow |

m | Matrix |

MCMC | Markov Chain Monte Carlo |

NetCoS | Network Condition Simulator |

P | Probability |

P_{ij} | Transition Probability |

PVC | Polyvinyl Chloride |

t | Time |

U.S. | United States |

X_{i} | Independent Variable |

Y_{i} | Dependent Variable for Facility |

α | Intercept Parameter |

ϵ_{i} | Random Error Term |

β | Regression Coefficient |

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Physical Factors | Environmental Factors | Operational Factors |
---|---|---|

Connections | Backfill type | |

End invert elevation | Bedding material | |

Installation method | Ground movement | Blockages |

Joint type | Groundwater level | Burst history |

Pipe length | pH | Debris |

Pipe shape | Road type | Flow velocity |

Pipe slope | Root interference | Hydraulic condition |

Sewer age | Soil corrosivity | Infiltration/exfiltration |

Sewer depth | Soil fracture potential | Previous maintenance |

Sewer pipe material | Soil moisture | Sediment level |

Sewer size | Soil type | Sewer function |

Start invert elevation | Sulfate soil | Surcharge |

Surface type |

Authors | Year | Model | Independent Variables |
---|---|---|---|

Davies et al. | 2001 | Logistic regression | Age, Material, Diameter, Depth, Length, Sewer Type, Location, Corrosivity, Road Type, Other Factors |

Ariaratnam et al. | 2001 | Logistic regression | Age, Material, Diameter, Depth, Sewer Type |

Wirahadikusumah et al. | 2001 | Markov chain | Material, Depth, Soil Type, Groundwater |

Lubini and Fuamba | 2001 | Logistic regression | Age, Material, Diameter, Length, Slope |

Micevski et al. | 2002 | Markov chain | Material, Diameter, Soil Type |

Koo and Ariaratnam | 2006 | Logistic regression | Age, Flow, Other Factors |

Chughtai and Zayed | 2008 | Linear regression | Age, Material, Diameter, Depth, Length, Slope, Bedding Type, Road Type |

Gat | 2008 | Markov chain | Age, Diameter, Sewer Type |

Ana et al. | 2009 | Logistic regression | Age, Material, Diameter, Depth, Length, Slope, Sewer Type, Location |

Salman and Salem | 2012 | Ordinal regression Logistic regression Binary regression | Age, Material, Diameter, Depth, Length, Slope, Sewer Type |

Sousa et al. | 2014 | Logistic regression | Age, Material, Diameter, Depth, Length, Slope |

Bakry et al. | 2016 | Multiple regression | Age, Material, Diameter, Depth, Length, Sewer Type, Soil Type, Groundwater, Surface Type, Traffic |

Gedam et al. | 2016 | Linear regression | Age, Material, Diameter, Depth |

Kabir et al. | 2018 | Bayesian logistic regression | Age, Material, Diameter, Depth, Length, Slope, Up Invert, Down Invert, Other Factors |

Malek Mohammadi et al. | 2019 | Logistic Regression | Age, Material, Diameter, Depth, Length, Slope, Groundwater, Soil Type |

Balekelayi and Tesfamariam | 2019 | Bayesian Regression | Age, Material, Diameter, Depth, Length, Slope, Groundwater, Residential and Commercial Connections, Repairs, Flushes, Cleaning, Degrease, Backups, and Root Cuts |

Applicability | Logistic Regression | Markov Chain | Linear Regression |
---|---|---|---|

Predicting condition of pipe groups | Moderate | Good | Poor |

Predicting condition of individual pipes | Good | Moderate | Good |

Predicting categorical dependent variables | Good | Moderate | Poor |

Conceptual and computational simplicity | Good | Poor | Good |

Identifying relationship between dependent and independent factors | Good | Poor | Moderate |

Calculation of condition probabilities | Good | Good | Poor |

Flexible to deficiency of data | Moderate | Poor | Good |

Prediction Models | Advantages | Disadvantages |
---|---|---|

Logistic Regression | Does not require too many computational resources, incredibly easy to implement, highly interpretable, not required input features to be scaled, capable of predicting probabilities, does not require normal distribution of independent variables, capable of predicting influence variables | not very useful for non-linear and complex problems, can only predict a categorical outcome |

Markov Chain | can predict categorical and continuous variables, strong statistical foundation, can be combined with other models, works for complicated distributions in high-dimensional spaces | difficult to implement and validate, requires large number of data, Markov assumptions may not be applicable for all datasets |

Linear Regression | very easy to implement, capable of predicting influence variables, highly interpretable | can only identify linear relationships between variables, only looks at the mean of the dependent variable, sensitive to outliers |

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## Share and Cite

**MDPI and ACS Style**

Malek Mohammadi, M.; Najafi, M.; Kaushal, V.; Serajiantehrani, R.; Salehabadi, N.; Ashoori, T. Sewer Pipes Condition Prediction Models: A State-of-the-Art Review. *Infrastructures* **2019**, *4*, 64.
https://doi.org/10.3390/infrastructures4040064

**AMA Style**

Malek Mohammadi M, Najafi M, Kaushal V, Serajiantehrani R, Salehabadi N, Ashoori T. Sewer Pipes Condition Prediction Models: A State-of-the-Art Review. *Infrastructures*. 2019; 4(4):64.
https://doi.org/10.3390/infrastructures4040064

**Chicago/Turabian Style**

Malek Mohammadi, Mohammadreza, Mohammad Najafi, Vinayak Kaushal, Ramtin Serajiantehrani, Nazanin Salehabadi, and Taha Ashoori. 2019. "Sewer Pipes Condition Prediction Models: A State-of-the-Art Review" *Infrastructures* 4, no. 4: 64.
https://doi.org/10.3390/infrastructures4040064