## ABSTRACT

A pipe condition assessment model is required to implement effective and economical planned maintenance of the water distribution system. The application of such a model requires sufficient accuracy, which, however, is limited by the complexity of the pipe deterioration process and data storage capacity of the water utility. The majority of previous studies have focused on the improvement of assessment algorithms for data mining. In this study, a mechanistic deterioration point assignment (MDPA) model is developed to make advancements in the modes of data input and result output to enhance the model's accuracy and application scope for cast iron and steel pipes. In this MDPA model, (1) indicators/sub-indicators on external corrosion, external load, internal corrosion, and internal load are constructed and can be obtained by data estimation or techniques and (2) assessment results include both pipe overall condition and detailed conditions on corrosion and load, offering evidence for primary maintenance measures. The weights of the indicators/sub-indicators are estimated using the Bayesian statistics theory. The modelling results of pipe samples demonstrate that this MDPA model is an effective tool for pipe condition assessment.

## HIGHLIGHTS

The condition of indicators/sub-indicators of this MDPA model for pipe condition assessment can be obtained either by data estimation or techniques.

The MDPA model is capable of exporting pipe detailed conditions for suggesting primary maintenance measures.

The MDPA model can be tested or verified by advanced techniques to guarantee model accuracy.

## INTRODUCTION

The water distribution system (WDS) typically accounts for 50–80% of the whole construction investment and operating cost of water supply; therefore, it is the most crucial and valuable part. Varied environmental and operational factors deteriorate water pipes, which lead to pipe failures that cause huge water losses, operating cost rises, risks of water quality contamination, and even ground collapses (Wang *et al.* 2022; Sakai 2024). The American Society of Civil Engineers (2021) report card showed that there was a water main break every 2 min, and 6 billion gallons of treated water were lost every day in the USA. The water loss from the distribution system was about 68.70 billion cubic meters in 2022 in China (Ministry of Housing and Urban-Rural Development 2022). The energy used for water pumping is wasted in translating water to soils or through severely corroded pipes with narrow effective diameters. Water quality contamination risks arise due to the severe corrosion of pipe inner walls or the low pressure owing to negative transients caused by breaks (Qi *et al.* 2018). Ground collapses may happen due to soil fluidization between high-pressure water and surrounding soils (Latifi *et al.* 2022a, 2022b). To prevent the above side-effects, it is necessary and economical to identify pipes that mostly need maintenance or replacement before the failure of those pipes (Forero-Ortiz *et al.* 2023). Therefore, pipe condition is required to help make water asset management decisions and improve the performance of WDSs under limited funding (Forero-Ortiz *et al.* 2023; Le Gat *et al.* 2023).

Pipe condition includes structural integrity and its ability to meet service requirements (Opila & Attoh-Okine 2011). Models play a significant role in the research of water pipe condition assessment due to their ability to comprehend the intricate interplay among water, the environment, and pipes (Kleiner & Rajani 2002; Clair & Sinha 2012; Fitchett *et al.* 2020; Forero-Ortiz *et al.* 2023). One typical type is the deterioration point assignment (DPA) model, which defines a series of related factors contributing to pipe failure and calculates the relative weight of each factor to obtain a total score to determine pipe condition (Grigg 2006). The DPA models directly establish numerical relationships between pipe condition and relative factors due to management interests, which do not represent the processes of pipe deterioration mechanisms. Different DPA models consist of different combinations of factors (Table 1). Those factors are independent of each other and show certain characteristics of pipes (e.g., age and material) but cannot indicate pipe detailed condition (e.g., corrosion and load). Factors are divided into three categories: physical, environmental, and operational. This classification illustrates the basic logical structure of factors qualitatively and produces no effect on assessment results quantitatively. Several statistical and machine learning algorithms have been adopted in previous literature for weight calibration (Table 1).

Factors . | Weight calibration methods and their advantages . | Reference . |
---|---|---|

AG, CF, DA, DI, GW, LK, MT, OP, PR, ST, TD | The analytic hierarchy process provides logical decisions based on analytical methodology and eliminates the chances of challenge. | Al-Barqawi & Zayed (2006) |

AG, BD, DI, ER, IP, IQ, MT, OP, ST, PR, TD | Artificial neural network is an efficient and cost-saving methodology. | Geem et al. (2007) |

AG, BR, CR, DA, DI, GW, LK, MT, IP/OP, PR, WQ, SA, ST, TD | The hierarchical fuzzy expert system deals with imprecision and qualitative aspects that are associated with pipe condition. | Fares & Zayed (2010) |

AG, BD, DI, ER, IP, IQ, MT, OP, PR, ST, TD | Bayesian statistic theory fuses previous studies to obtain a more reliable assessment result. | Wang et al. (2010) |

AG, BR, CF, DI, GW, IQ, MT, TD, ST, WQ | The Fuzzy analytic network process and Monte Carlo simulation address the interdependency and accumulated uncertainty among factors for accurate and realistic results. | El-Abbasy et al. (2019) |

AG, BD, DI, LE, MT, WT | Machine learning algorithms are more accurate among known neuro-fuzzy-based methodologies. | Elshaboury & Marzouk (2022) |

AG, DI, BR, MP, MT, PR, TJ, WT | Monte Carlo simulation delivers advanced optimization and accurate estimation. | Dawood et al. (2022) |

Factors . | Weight calibration methods and their advantages . | Reference . |
---|---|---|

AG, CF, DA, DI, GW, LK, MT, OP, PR, ST, TD | The analytic hierarchy process provides logical decisions based on analytical methodology and eliminates the chances of challenge. | Al-Barqawi & Zayed (2006) |

AG, BD, DI, ER, IP, IQ, MT, OP, ST, PR, TD | Artificial neural network is an efficient and cost-saving methodology. | Geem et al. (2007) |

AG, BR, CR, DA, DI, GW, LK, MT, IP/OP, PR, WQ, SA, ST, TD | The hierarchical fuzzy expert system deals with imprecision and qualitative aspects that are associated with pipe condition. | Fares & Zayed (2010) |

AG, BD, DI, ER, IP, IQ, MT, OP, PR, ST, TD | Bayesian statistic theory fuses previous studies to obtain a more reliable assessment result. | Wang et al. (2010) |

AG, BR, CF, DI, GW, IQ, MT, TD, ST, WQ | The Fuzzy analytic network process and Monte Carlo simulation address the interdependency and accumulated uncertainty among factors for accurate and realistic results. | El-Abbasy et al. (2019) |

AG, BD, DI, LE, MT, WT | Machine learning algorithms are more accurate among known neuro-fuzzy-based methodologies. | Elshaboury & Marzouk (2022) |

AG, DI, BR, MP, MT, PR, TJ, WT | Monte Carlo simulation delivers advanced optimization and accurate estimation. | Dawood et al. (2022) |

**Physical factors**: AG, age; CF, C factor/Hazen–Williams coefficient; DI, diameter; IP, inner protection; LE, length; MT, material; OP, outer protection; TJ, type of joints; WT, wall thickness.

**Environmental factors**: BD, buried depth; DA, damage to surroundings/serviced area; ER, electric recharge; GW, groundwater depth; IQ, installation quality; LD, load; ST, soil type.

**Operational factors**: BR, breakage rate/leakage; CR, cost of repair; MP, maintenance practice; PR, pressure/hydraulic factor; TD, traffic disruption/surface type; WQ, water quality.

The DPA models make good use of various statistical and machine learning algorithms to deal with the complexity and uncertainty of WDSs. While previous studies have been conducted to improve model accuracy by employing novel assessment algorithms to capture the relationship between pipe condition and factors (Al-Barqawi & Zayed 2006; Geem *et al.* 2007; Fares & Zayed 2010; Wang *et al.* 2010; El-Abbasy *et al.* 2019; Dawood *et al.* 2022; Elshaboury & Marzouk 2022), there are still limitations or challenges in implementing the DPA model in practice as follows: (1) Models demand factor data with reliability and comprehensiveness to ensure the outputs adequately represent pipe condition (Gómez-Martínez *et al.* 2017; Dawood *et al.* 2022; Elshaboury & Marzouk 2022). It is hard for some water companies to record whole life cycle data for all the pipes, considering that some factors may vary irregularly for intangible reasons or even difficult to monitor (e.g., traffic load). (2) There exists unnegligible assessment uncertainty due to the complexity and randomness of the pipe deterioration process (Barton *et al.* 2021; Forero-Ortiz *et al.* 2023). It is difficult to accurately predict pipe condition, depending on the deterioration patterns of other pipes. Overfitting may happen when the model appears to fit well with the current data but fails to validate with future datasets. (3) Models output total scores for individual pipes while generally ignoring detailed pipe condition (Al-Barqawi & Zayed 2006; Geem *et al.* 2007; Fares & Zayed 2010; Wang *et al.* 2010; El-Abbasy *et al.* 2019; Dawood *et al.* 2022; Elshaboury & Marzouk 2022), which is important to suggest primary and targeted maintenance measures. The above limitations or challenges restrict the application effect of the DPA models.

Another type of pipe condition assessment method besides models is the techniques that are widely available and commonly used in practice. Acoustic, visual, electromagnetic, ultrasonic, radiographic, and thermographic techniques provide direct insights into pipe condition on integrity, corrosion, load, and indirect elements (e.g. soil condition) (Li *et al.* 2015; Gnatowski *et al.* 2021; Latif *et al.* 2022). Recently, techniques have been combined with intelligent approaches to obtain pipe condition for better accuracy and precision. Wu *et al.* (2024) used convolutional neural networks to detect and locate pipe leakage based on measured vibrations from a fibre-optic cable. The convolutional neural networks are utilized to handle information obtained by acoustic emission for leak detection and size identification (Ahmad *et al.* 2023; Siddique *et al.* 2023). Those studies open new avenues for innovation in condition assessment, demonstrate the application potential of combining techniques and data processing methodologies, and increase the possibility of practical applications for large-scale and complex WDSs.

In this study, a mechanistic DPA (MDPA) model is developed for pipe condition assessment with a similar basic structure to the DPA models. Factors from Table 1 are combined based on the pipe deterioration mechanism to construct indicators/sub-indicators on external corrosion, external load, internal corrosion, and internal load in the MDPA model. Weights of the indicators/sub-indicators are estimated using the Bayesian statistics theory, and then total scores for pipe condition can be obtained. The MDPA model can achieve satisfactory accuracy through (1) an appropriate combination of factors in mechanism constraint due to an in-depth understanding of the relationships between pipe factors and their effects on pipe condition and (2) thoroughfares to techniques to test or verify the precision of assessment results obtained by data estimation. The application scope is enhanced by (1) inputting indicator/sub-indicator conditions through either data estimation or techniques to rid of the requirement of a sizeable high-quality dataset and (2) outputting detailed pipe condition on corrosion and load for targeted maintenance measures. Therefore, the MDPA model can be an effective and convenient tool for implementing pipe management in practice.

## THE MDPA MODEL

### Model structure

*G*is the pipe condition rate;

*i*is the serial number of pipe indicator or sub-indicator;

*I*is the total number of indicators/sub-indicator;

*λ*is the weight of the indicator/sub-indicator; Ind is the pipe indicator/sub-indicator.

The assessment objects of this MDPA model are cast iron and steel pipes, which currently occupy 49.9% of the buried pipes in China (China Urban Water Association 2019). The main causes of cast iron and steel pipe deterioration are widely acknowledged to be corrosion and load although entirely explicit mechanisms are still not completely clear because WDSs are buried underground, making it difficult to observe pipe deterioration behaviour (Rajani & Kleiner 2001; Seica & Pacher 2006; Wang *et al.* 2022). External corrosion includes galvanic, electrolytic, and microbiological corrosion, which reduces pipe residual stress. Internal corrosion leads to hydraulic deterioration including the increase in surface roughness and the reduction in the cross-sectional area, which decreases the sufficient ability of WDSs for adequate pressure, and water quality is also polluted by the galvanic and bacterial action from internal corrosion products (Yamini & Lence 2010). The buried pipe resists load mainly from the external environment of soil and traffic and internal operation pressure.

### Indicators/sub-indicators

#### External corrosion

Rossum (1969) developed a model to calculate the pit depth of the external wall, which considered soil and pipe properties as influential parameters. It was coincident with the survey that 67% of the respondents pointed out corrosive soils as the primary cause of external corrosion in their systems, 12% believed it was the direct connection of dissimilar metals, and 10% presented coating damage or degradation as the main reason conducted in the USA and Canada (Romer & Bell 2001).

*K*and

_{n}*n*are coefficients depending on the soil condition,

*K*equals 170, 222, and 355;

_{n}*n*equals 1/6, 1/3, and 1/2 when the soil condition is good, fair, and poor, respectively;

*T*is pipe age, year;

*K*

_{ae}is a coefficient that depends on pipe material and equals 0.4, 0.9, and 0.6 when pipe material is ductile cast iron, grey cast iron, and steel, respectively;

*K*

_{pe}is a coefficient depending on protection measure and equals 0.5 and 0.9 with and without coating.

#### External load

*C*

_{d}is the load coefficient;

*ρ*is the soil density, lb/ft

^{3};

*B*

_{d}is the trench width, ft;

*t*is the pipe wall thickness, ft;

*K*is the surface load coefficient of rigid pavement;

*D*is the pipe external diameter, ft;

*P*is the wheel load, 9,000 lb;

*F*is the impact factor, 1.5.

#### Internal corrosion

*a*

_{p}is the roughness growth rate, calculated with the expression of 0.0833 exp (1.9–0.5 pH), ft per year;

*d*is the pipe internal diameter.

#### Internal load

*p*is the water pressure, kg/cm

^{2}.

*et al.*2012; Wright

*et al.*2019; Latif

*et al.*2022), external load, soil load, traffic load (Gnatowski

*et al.*2021), internal corrosion, pipe internal property (Tohmyoh & Suzuki 2009; Rayhana

*et al.*2020), pH and internal load (Qi

*et al.*2018), and techniques that demonstrate pipe condition as leakage or burst (Li

*et al.*2015; Latif

*et al.*2022). The condition obtained by detection techniques is available for testing or validating assessment results reached by data analysis of the MDPA model or fed into the MDPA model directly as input data for the indicators/sub-indicators. This helps to conquer the limitations of data requirements in the application of pipe condition assessment models.

### Weight estimation method

*θ*is the estimation parameter;

*p*(

*θ*) is the prior probability of

*θ*;

*p*(

*θ*|

*y*) is the posterior probability of parameter

*θ*, which is the parameter conditional distribution after data estimation;

*p*(

*y*|

*θ*) is the likelihood function, which represents the occurrence probability of the condition

*y*under different realizations.

*et al.*2010). We use to express the uncertainty of model simulation error of the MDPA model by the Bayesian statistics theory, as shown in Equation (13):where

*ε*is the model simulation error, which is assumed to be independent and normally distributed with a mean of zero and variance

*σ*

^{2},

*ε*−

*N*(0,

*σ*

^{2}).

In this study, the Markov Chain Monte Carlo (MCMC) method is used to generate model posterior distribution. The stationary distribution of the Markov process is the model posterior distribution in Bayesian inference and then the simulation process is run long enough to generate an approximation of the distribution.

Three measures of fit are employed to test the simulation result of the MDPA model.

- (1) The deviance information criterion (DIC), a larger DIC value reveals a poorer fit between the observations and simulations. The DIC has been used in a variety of literature for models comparing and discussed for improvement (Spiegelhalter
*et al.*2014). With unknown parameters*θ*, data*y*, and the likelihood function*P*(*y*|*θ*), the deviance of a Bayesian model can be shown as follows:

- (2) The coefficient of determination
*R*^{2}provides a measure of how well the model predicts and is calculated as follows:where SS_{E}is the sum of squared errors; SS_{T}is the total sum of squares;*y*is the observation data; is the mean value of model simulation; is the mean of the observations_{i}*y*;_{i}*m*is the number of observations. (3) Model error

## DATA SOURCE

The data inputted to the MDPA model to generate indicators/sub-indicators weights and perform some statistical analysis used in sections 4.1–4.3 is the same data that was used in Geem *et al.* (2007) and Wang *et al.* (2010), which include factor data to construct indicators/sub-indicators and pipe condition rates. Factors include pipe age, diameter, trench depth, inner coating, outer coating, soil density, bedding, number of roads, soil type, pipe material, pH and pressure (Figure 1). Pipe condition rates, ranging from zero to one, are calculated using pipe data on outer corrosion, cracks, pinholes, inner corrosion, and the H–W coefficient, with their weights equalling 0.2. A value of zero means that a pipe is in excellent condition and no maintenance action is needed, while a value of one indicates that a pipe is in critical condition and immediate repair or replacement is required.

The data to make real-world validation is from distribution mains that were buried in communities in a city in northern China and under critical conditions. Detailed information on the data used in this study is provided in Supplementary Materials.

## RESULTS AND DISCUSSION

We started with values recommended by previous literature and generated the best values of the indicator/sub-indicator weights using the Bayesian statistic theory. In our study, the indicator/sub-indicator weights are supposed to be normal distributions. Since there were no similar indicators/sub-indicators constructed in previous literature, we added the factor weights relating to the indicators/sub-indicators of the MDPA model together to guide the prior distributions of the weights. The weights of indicator of external corrosion, external load, internal corrosion, and internal load were 0.36, 0.14, 0.24, and 0.06 obtained by Al-Barqawi & Zayed (2006), and 0.4, 0.2, 0.34, and 0.08 obtained by El-Abbasy *et al.* (2019). Then the prior normal distributions of these four indicators are (0.24, 0.02), (0.1, 0.02), (0.4, 0.02), and (0.15, 0.02) to the MDPA model. The weights of sub-indicators of soil effect, pipe external property, soil load, traffic load, pH effect, pipe internal property, and internal load were 0.13, 0.24, 0.09, 0.05, 0.10, 0.24, and 0.08 obtained by Al-Barqawi & Zayed (2006) and El-Abbasy *et al.* (2019). Then the prior normal distributions of these seven sub-indicators are (0.2, 0.02), (0.18, 0.02), (0.2, 0.02), (0.2, 0.02), (0.2, 0.02), (0.2, 0.01), and (0.2, 0.01) to the MDPA model. We use the software of WinBUGS (version 1.4.3) to fit data for complex statistic calculation using the Bayesian theory, the calculation process of which is similar to that in the article of Wang *et al.* (2010).

### Fitting of the MDPA model

*G*separately, as described in Equation (1). Most of the pipe condition rates calculated are contained within the 2.5 and 97.5% credible intervals (Crls) as shown in Figure 3. The posterior estimates for the normal distributions of indicator/sub-indicator weight are shown in Table 2, which are comparable to the weights obtained in the articles of Al-Barqawi & Zayed (2006) and El-Abbasy

*et al.*(2019) shown in the previous paragraph. The detailed condition of corrosion and load is demonstrated by condition rates of indicators and sub-indicators, and water utilities can take targeted measures. More attention should be paid to external corrosion monitoring of pipe 5 since external corrosion condition is worse than external load, internal corrosion, and internal load condition (Figure 4(a)). This situation is nearly equally due to soil effect and pipe external property since those two sub-indicators obtain similar condition rates (Figure 4(b)). No maintenance action is required due to its low condition rate, as shown in Figure 3.

Indicator/sub-indicator . | Weight . | R^{2}
. | DIC . | ||
---|---|---|---|---|---|

Mean . | 2.5% . | 97.5% . | |||

External corrosion | 0.46 | 0.29 | 0.61 | 0.72 | − 18.38 |

External load | 0.37 | 0.22 | 0.51 | ||

Internal corrosion | 0.33 | 0.16 | 0.33 | ||

Internal load | 0.13 | 0.01 | 0.35 | ||

Soil effect | 0.20 | 0.03 | 0.38 | 0.78 | − 22.56 |

Pipe external property | 0.23 | 0.06 | 0.40 | ||

Soil load | 0.13 | 0.01 | 0.31 | ||

Traffic load | 0.12 | 0.01 | 0.27 | ||

pH effect | 0.19 | 0.02 | 0.39 | ||

Pipe internal property | 0.12 | 0.01 | 0.29 | ||

Internal load | 0.23 | 0.050 | 0.41 |

Indicator/sub-indicator . | Weight . | R^{2}
. | DIC . | ||
---|---|---|---|---|---|

Mean . | 2.5% . | 97.5% . | |||

External corrosion | 0.46 | 0.29 | 0.61 | 0.72 | − 18.38 |

External load | 0.37 | 0.22 | 0.51 | ||

Internal corrosion | 0.33 | 0.16 | 0.33 | ||

Internal load | 0.13 | 0.01 | 0.35 | ||

Soil effect | 0.20 | 0.03 | 0.38 | 0.78 | − 22.56 |

Pipe external property | 0.23 | 0.06 | 0.40 | ||

Soil load | 0.13 | 0.01 | 0.31 | ||

Traffic load | 0.12 | 0.01 | 0.27 | ||

pH effect | 0.19 | 0.02 | 0.39 | ||

Pipe internal property | 0.12 | 0.01 | 0.29 | ||

Internal load | 0.23 | 0.050 | 0.41 |

We examined the prior rationality of the MDPA model by doubling (i.e., cutting the variance) and cutting (i.e., increasing the variance) in half the precision of the input priors. The former examinations lead to a slightly better coefficient of determinations of 0.80 and 0.75 when input data are 7 sub-indicators and 4 indicators, while the latter examinations lead to significant reductions in fitting effect. The above results indicate that the prior probabilities selected by the MDPA model meet the assessment requirements and can ensure simulation accuracy. The ranges of the priors are neither too wide to induce easy formation of multiple optimization values nor too narrow to induce difficulty in computing the optimal solution.

### Evaluations of the indicators/sub-indicators

We estimated the weights of pipe indicators/sub-indicators individually to evaluate their statistical impact on model performance through numerical experiments on the MDPA model. For these tests, the weight of one indicator/sub-indicator was estimated and the weights of the remaining four indicators/six sub-indicators were assigned fixed values. As Table 3 shows, there is no significant difference in goodness of fit among all tests with similar model error uncertainties. These findings can be interpreted as evidence that pipe condition rates are equally sensitive to the prior distribution specifications of indicators/sub-indicators, meaning that the range of predicted output is affected nearly equally by the distributions of input indicators/sub-indicators. The reasons for this ‘similar’ predictive model error can be that prior specification change of the indicators/sub-indicators may not lead to an exploration of suboptimal regions of the model posterior, and no relationship between the indicators/sub-indicators and pipe condition rates is particularly steeper or higher order than others.

Indicator/sub-indicator . | Model error . | R^{2}
. | ||
---|---|---|---|---|

Mean . | 2.5% . | 97.5% . | ||

External corrosion | 0.14 | 0.10 | 0.21 | 0.70 |

External load | 0.14 | 0.11 | 0.20 | 0.69 |

Internal corrosion | 0.14 | 0.11 | 0.21 | 0.71 |

Internal load | 0.14 | 0.11 | 0.21 | 0.65 |

Soil effect | 0.14 | 0.11 | 0.21 | 0.72 |

Pipe external property | 0.14 | 0.11 | 0.21 | 0.72 |

Soil load | 0.13 | 0.10 | 0.19 | 0.75 |

Traffic load | 0.13 | 0.10 | 0.19 | 0.70 |

pH effect | 0.14 | 0.11 | 0.20 | 0.76 |

Pipe internal property | 0.13 | 0.10 | 0.19 | 0.75 |

Internal load | 0.14 | 0.11 | 0.20 | 0.72 |

Indicator/sub-indicator . | Model error . | R^{2}
. | ||
---|---|---|---|---|

Mean . | 2.5% . | 97.5% . | ||

External corrosion | 0.14 | 0.10 | 0.21 | 0.70 |

External load | 0.14 | 0.11 | 0.20 | 0.69 |

Internal corrosion | 0.14 | 0.11 | 0.21 | 0.71 |

Internal load | 0.14 | 0.11 | 0.21 | 0.65 |

Soil effect | 0.14 | 0.11 | 0.21 | 0.72 |

Pipe external property | 0.14 | 0.11 | 0.21 | 0.72 |

Soil load | 0.13 | 0.10 | 0.19 | 0.75 |

Traffic load | 0.13 | 0.10 | 0.19 | 0.70 |

pH effect | 0.14 | 0.11 | 0.20 | 0.76 |

Pipe internal property | 0.13 | 0.10 | 0.19 | 0.75 |

Internal load | 0.14 | 0.11 | 0.20 | 0.72 |

*et al.*2007). From estimation results, the variation coefficients (standard deviation⁄ mean value) of external corrosion, external load, internal corrosion, and internal load are reduced from 0.30, 0.37, 0.47, and 0.79 to 0.18, 0.20, 0.27, and 0.69, respectively; the variation coefficients of soil effect, external pipe property, soil load, traffic load, pH effect, internal pipe property, and internal load are reduced from 0.70, 0.78, 0.70, 0.70, 0.70, 0.70, and 0.59 to 0.45, 0.38, 0.64, 0.52, 0.58, 0.60, and 0.41, respectively. The prior and posterior distributions of the indicators/sub-indicators are shown in Figure 5. All the variation coefficients are reduced from the prior distributions to the posterior distributions, and there is no substantial difference among the variation coefficients. Thus, the weight estimation of all the indicators/sub-indicators of the MDPA model needs statistical calculation besides expert experience.

### Comparison between the MDPA and DPA models

*et al.*(2010). It means this DPA model has the same input factor data as the MDPA model. The fitting effects of the MDPA and DPA models are shown in Figure 6. The MDPA model (seven sub-indicators) shows a slightly higher assessing accuracy with a better

*R*

^{2}value of 0.78 than 0.75 of the DPA model. One reason for this result should be the appropriate combination of pipe factors in the MDPA model. Taking the factor of age as an example, the MDPA model constructs a positive exponential relationship between age and pipe external corrosion, which is more conforming to the pipe deterioration process as the bath curve described. In contrast, the relationship between the age and model output using the DPA method is purely a linear function, which is not the actual pipe deterioration pattern. Therefore, the MDPA model achieves better accuracy with fewer model variables since the DPA model has 11 input factors and the MDPA model has 7 input sub-indicators. The detailed condition of pipe corrosion and load is outputted by the MDPA model, as shown in Figure 4(b) in section 4.1, while the DPA model can only output a total score for the whole pipe.

Data requirements are one of the main limitations in the application of pipe condition assessment models. This MDPA model can obtain input data either from pipe factor data or techniques. If critical data of a pipe are missing in the database, e.g. no pipe age data, pipe condition can be obtained in a novel way that the detailed condition of pipe corrosion is detected by techniques and the detailed condition of pipe load is calculated by Equations (6) and (7) to conquer the obstacle of insufficient pipe factor data. Besides serving as a means of obtaining pipe data, techniques can be used to test or verify the accuracy of assessment results and the structure of indicators/sub-indicators.

### A case study of the MDPA model

Seventy-three per cent of pipes obtained assessment rates higher than 0.7 (Figure 7), which aligns with the fact that all the pipes were in critical condition. There are assessment errors for pipes 6, 8, 14, and 15 with the output results of good condition. The detailed condition output from this MDPA model provides a basis for water utilities to decide what measures should be taken to address what problems. The modelling results of those pipe samples suggest that this MDPA model is an effective pipe condition assessment tool.

The structure of the indicators/sub-indicators of the MDPA model in this study is from classic mechanism models, which can be further improved and revised through the advancements in the pipe deterioration process promoted by emerging approaches. Finite element analysis has been utilized to elaborate local flow properties and grab complex interactions among mechanical stress, material properties, and environmental factors. The transient wave behaviours and wall shear stress can be investigated, and stress corrosion cracking can be predicted with better accuracy and efficiency (Zhang *et al.* 2022; Sarwar *et al.* 2024).

The selection of techniques depends on pipe material, diameter, pressure, velocity, types of defects, environmental factors, and the size of WDS. To improve the MDPA model, the application conditions of techniques need to be further summarized. How to use techniques to verify assessment results based on data estimation also requires organization. We obtain wrong assessment results for pipe samples in the case in this section, which may be avoided by proper integration with techniques.

More importantly, additional validation analyses on various WDSs should be conducted to accumulate and summarize practical experience. We believe water utilities can apply the MDPA model in a reasonable framework to integrate statistical analysis and techniques, depending on the practical situation of data storage, cost of techniques, and accuracy requirements.

## CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS

Pipe condition assessments are required for effective and economical planned maintenance to help water utilities rehabilitate hydraulic capacity, reduce operating costs, and guarantee water quality in WDSs. For the better employment of both pipe data and techniques, a MDPA model is developed in our study and provides advantages in two main ways: (1) the accuracy of indicator/sub-indicator inputted in the MDPA model can be tested or verified by advanced geophysical or sensing techniques, which integrates pipe detection techniques to guarantee model accuracy; (2) the indicators/sub-indicators are constructed from factors based on the pipe failure mechanism, while it can prevent illogical errors arising from purely data-driven methodologies and can also demonstrate pipe detailed conditions on corrosion and load to enrich assessment outputs.

The modelling results of pipes samples show that this MDPA model can output accurate pipe overall and detailed condition. The MDPA model can be flexibly and broadly applied by water utilities with the advantages of indicating detailed information on pipe condition, linking detection techniques, and requiring a small amount of input data.

Future research directions to enhance the MDPA model include the improvement of indicator/sub-indicator structures and the application principles of techniques. It is crucial to ensure that the indicators/sub-indicators represent detailed pipe conditions precisely in suitable structures in various network size, burying technologies, and surrounding environment. There is still much work to be done to clarify how techniques can be effectively utilized in the MDPA model for excellent accuracy and economical investment. Additional validations on various WDSs are anticipated to support the aforementioned research work for expanding the application scope of the MDPA model.

## ACKNOWLEDGEMENTS

This work was supported by the project funded by the National Natural Science Foundation of China (Grant No. 52200116).

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.