The use of numerical simulations to improve the management of water distribution networks (WDNs) has dramatically increased in recent decades. Nevertheless, the modeling of leakages is still a major issue to face when setting up a model of a WDN. Because water losses increase at increasing pressure, they are usually modeled by assuming a leakage node behaves like an emitter. Unfortunately, a definitive assessment of the leakage law exponent is as yet lacking in the literature. Consequently, a field analysis was carried out on an existing WDN: (a) to assess whether the power law can effectively represent the relationship between leakage and pressure; and (b) to calculate the value of the exponent. A 3-month measurement campaign was developed, aiming at assessing the dependence of the losses to the pressure. The analysis showed that the power law equation with an exponent around 0.7 can well represent the pressure–leakage relationship.

## INTRODUCTION

*Q*

_{L}is the node leakage flow,

*h*the node pressure head, and

*c*and

*n*are the coefficient and the exponent of the equation. The relationship is a more generalized form of the orifice equation (with

*n*= 0.5). Nevertheless, while the orifice equation is derived from the conservation of energy law, the power equation is an empirical relationship not founded on basic principles, used for mathematical convenience. Leakage exponents higher than that of the orifice equation depend on several factors, the most important of which is that the leak area increases with increasing pressure.

In principle, the leakage coefficient and exponent in Equation (1) vary at each node. In practice, the coefficient *c* can only be assessed by calibrating the network model, whereas the exponent *n* is assumed constant over the network. Due to the variation of the orifice area with pressure, *n* can deviate significantly from the theoretical value of 0.5 (orifice flow), particularly for plastic pipes. Values above 1 can be attained, depending on both the type of pipe and leak.

Lambert (2001) suggested that *n* could vary between 0.5 (for rigid pipes, which are insensitive to pressure variations) and 2.5 (for very flexible pipes, highly sensitive to pressure variations), depending on both the type of pipe and leak.

*Q*

_{L,N}can be calculated using the average zone point (AZP) approach: where

*c*

_{N}is the network leakage coefficient and

*h*

_{AZP}the pressure head at the AZP. Consequently, at time

*t*the network leakage can be calculated as (May 1994; Garcia

*et al.*2007): where

*t*

_{MNF}is the minimum night flow (MNF) time.

Cassa & van Zyl (2012) showed that the leakage exponent does not provide the best characterization of the pressure response of a leak. Van Zyl & Cassa (2014) proposed a more consistent characterization based on a dimensionless Leakage Number to provide a satisfactory modeling of elastic leaks.

*h*. By assuming a round hole in a plastic pipe, the equation can be written as (van Zyl & Clayton 2007): where

*C*

_{d}is the discharge coefficient,

*d*

_{0}the original hole diameter and

*A*a coefficient depending on the pipe characteristics (dimensions and material). Nevertheless, such a relationship is more difficult to use than the simple power law equation, because it requires hole diameter and pipe characteristics to be known. The small contribution to the leakage flow given by Equation (4) due to the terms with exponent 1.5 and 2.5 should also be noted.

Despite the limitations of using the power equation, such a model is very often used to characterize leakages in water systems. Experimental analysis is required for adequate characterization of the leakage exponent. Some authors have carried out laboratory tests. Greyvenstein & van Zyl (2007) performed laboratory experiments on the leakage from pipes both taken from the field and with artificial failures. Asbestos cement, steel and uPVC pipes were considered in the experiments and exponents up to 2.3 were measured during the tests. The highest leakage exponents occurred in corroded steel pipes, whereas values less than 0.5 were measured for uPVC pipe with circumferential cracks. Values of the exponent of less than 0.6 were measured by Walski *et al.* (2009) in laboratory experiments carried out on PVC pipes with diameters between 25 and 150 mm. Circular, longitudinal and circumferential breaks were considered and pressure varied between 0.69 and 4.83 bar. A 31 mm copper pipe was also tested to compare different materials.

De Paola & Giugni (2012) carried out laboratory experiments on orifices with different shapes and dimensions fitted on both steel and ductile iron pipes. The results showed that the orifice equation can reliably predict the outflow discharge. Laboratory tests performed by Ferrante *et al.* (2011) on an HDPE pipe with both regular and irregular leaks showed a hysteretic loop in the head-discharge plane, suggesting that different values in the pressure leakage relationship would be calculated on the loading and unloading curves.

Following a different approach, several authors have calibrated the leakage exponent on the basis of real network measurements. Ardakanian & Ghazali (2003) performed field experiments to obtain the relationship between pressure and leakage in Sanandaj (Iran) WDN. Very old cast-iron pipes were selected for tests and a 4 bar pressure variation was applied. The exponent of the power law equation was found to vary between 1.10 and 1.18 for the four investigated locations, showing high correlation coefficients (always greater than 0.90).

Cheung *et al.* (2009) carried out experiments on a district of the Capinzal (Brazil) WDN. The monitored network is made of PVC/PBA pipes, with a total length of around 6.7 km and diameters ranging between 20 and 100 mm. The resulting exponent was between 1.2 and 1.7 for the three monitored points.

Al-Ghamdi (2011) carried out field investigation to identify the pressure–leakage relationship in various districts of the Mekka (Saudi Arabia) WDN. Analysis showed that a power law equation correctly explains the relationship, and the leakage exponent was found to be 0.50 for the network of asbestos-cement pipes and 1.16 for the network of mixed pipes. The author also showed that the leakage rate increases with the pipe age, and a relationship depending on pressure and pipe age was inferred from the data of the two areas.

Alkasseh *et al.* (2013) modeled MNF to estimate water losses in Kinta Valley WDN (Malaysia), by monitoring flow discharge in 30 districts randomly chosen. A total of 20 factors accounting for physical, hydraulic, and operational variables were selected and correlated with MNF. Multiple linear regression was used to determine factors influencing the MNF, showing the major role of pipe length and pipe age, whereas no significant relationship was established between pressure and MNF due to the use of pressure reducing valves (PRVs).

Walski *et al.* (2006) pointed out that most representations of leaks are based on the assumption that the rate of the leakage is controlled by the orifice. However, in smaller leaks, additional head loss can occur between the leak and the soil surface as the water moves through the soil. The authors developed a simple relationship between orifice and soil head loss. An energy equation relating pressure and flow for both orifice and soil head losses was also suggested. Finally, an ‘orifice/soil number’ was proposed, to better understand the conditions during which orifice or soil head loss dominate. The same topic was further discussed by van Zyl & Clayton (2007), who stressed that the formation of hydraulic fractures takes place in the soil close to the leak. The flow occurs preferentially along these cracks and the flow rates rise by several orders of magnitude. In these conditions, conventional seepage analysis is no longer applicable and the fracturing is expected to produce flow increases that contribute to leakage exponents greater than unity.

## THE PROPOSED METHODOLOGY

*et al.*2002). Usually values smaller than 250–180 l/h are considered. At the node

*i*, BL can be schematized using Equation (1): in which the leakage coefficient

*c*

_{BL,i}was assumed to depend on both the connected properties and the material of the pipes connected to the node. For the sake of simplicity, for each node the served population instead of the number of properties was considered, thus resulting in the leakage coefficient: where

*α*is a coefficient of proportionality, to be further determined. The leakage coefficient was assumed proportional to both the population

*Ab*served by the node

_{i}*i*and the material of the pipe connected to the node by means of a pipe material coefficient

*c*

_{m,i}, calculated as: where

*c*is the material coefficient for the

_{j,i}*j*-th pipe connected to node

*i*and

*N*is the number of pipes connected to the

_{i}*i*-th node. The material coefficient accounts for the different susceptibility to leakage of pipe materials (e.g. due to corrosion). Consequently, the network BL was calculated as: where

*NN*is the number of nodes of the network. Coefficient

*α*in Equation (6) and the leakage exponent

*n*depend on the WDN characteristics and can be calculated by minimizing differences between measured and calculated water losses, by using Equation (8). Direct measurements or hydraulic simulation models should be used to assess the node pressure

*h*in Equation (8). Nevertheless, when head losses during night hours are negligible, node pressure can be calculated as the difference between the inlet pressure downstream of the PRV and the node elevation (i.e. hydrostatic pressure over the WDN during night hours can be assumed).

_{i}A simplified analysis, based on the AZP approach, can be used as well. In this case, the network leakage can be calculated by means of Equation (2). Renaud *et al.* (2012) discussed several approaches to estimate the pressure at the AZP (the ‘Topographic’ method, ‘Hydraulic model’ method and ‘Measurement’ method). The authors showed that application to a number of districts resulted in very similar results regardless of the method, when the district meter area (DMA) is served by a reservoir or a PRV set at a constant value.

## THE CASE STUDY: THE SANTA COLOMBA DISTRICT (BENEVENTO MUNICIPALITY)

The experimental assessment of the leakage exponent in Equation (1) was developed through extensive field measurements on a district of the Benevento WDN (Italy). The WDN serving the municipality of Benevento is a rather complex system, because of the extension (around 130 km^{2}) and the topography of the supplied area (with elevations ranging between 496 and 115 m asl). The average inflow is 291 l/s (year 2011) and the served population around 60,000, corresponding to a daily consumption of 410 litres per capita. At the end of the 1990s, the network was divided into 30 DMAs, to ensure adequate pressure over the network. A system for pressure regulation and flow measurement is available upstream of each DMA, for flow monitoring and leakage assessment.

Material . | Ductile iron . | Steel . | HDPE . |
---|---|---|---|

% | 85.0 | 12.8 | 2.2 |

Material . | Ductile iron . | Steel . | HDPE . |
---|---|---|---|

% | 85.0 | 12.8 | 2.2 |

Diameter . | ≤ 80
. | 80 – 100 . | 150 . | 250 . |
---|---|---|---|---|

% | 35.1 | 13.6 | 45.2 | 6.1 |

Diameter . | ≤ 80
. | 80 – 100 . | 150 . | 250 . |
---|---|---|---|---|

% | 35.1 | 13.6 | 45.2 | 6.1 |

Inflow discharge and upstream node pressure were recorded at the DMA inlet at 15 min. time intervals. Pressure head was also measured at four significant locations within the network, so as to characterize time patterns over time (points P1 to P4 in Figure 1). Piezoresistive transducers were used for pressure measurement, whereas a Woltmann flow meter was used for flow discharge at the inlet node. Both pressure transducers and flow meter were coupled to electronic data-loggers for data storage.

Differences in the MNF recorded in corresponding periods of 2011 and 2012 and surveys carried out during July–August 2012 allowed the detection of two major breaks, which were repaired. As a consequence, only BL was assumed for water losses. UB could be present, but that would not alter the results significantly.

The water losses were calculated as the difference between the MNF and the night consumption (NC). The NC was calculated as the sum of domestic and non-domestic consumptions (McKenzie 1999). The domestic NC was estimated by assuming the total active population to be around 6% with an average consumption of 10 litres/head/hour (McKenzie 1999), resulting in an average flow discharge of around 1.5 l/s. Non-domestic NC (i.e. for public gardens and stadium irrigation, public-building consumptions) was estimated to be around 0.5 l/s, resulting in a total NC of 2.0 l/s. A small non-domestic use was considered in NC, because the WDN serves a mainly residential area.

Values of *α* and *n* were calculated by minimizing differences between measured and calculated water losses. Because of the low flow discharge during night hours, hydrostatic pressure over the district was assumed in calculating the node pressure head *h _{i}* (i.e. as the difference between inlet pressure downstream of the PRV and the node elevation). In any case, the EPANET model of the network was also developed, which confirmed the negligible head losses during night hours.

In addition, a material coefficient ten times larger than ductile iron was considered for *c _{j,i}*, because of the greatest susceptibility to leakage observed in the district being for steel pipes (mainly for pipe corrosion) and HDPE pipes (mainly for poor sealing at junctions). Nevertheless, the material coefficient was recognized to have low influence on the leakage model, due to the small percentage of pipe material other than ductile iron (see Table 1). As result,

*α*= 3.2·10

^{−5}and

*n*= 0.72 were calculated. Values of

*C*

_{BL,i}ranged between 0 (in nodes where no loss occurs) and 0.14.

*h*

_{AZP}was calculated as:

in which *I*_{NH} is the head at the network inlet, which is regulated by the PRV, and *GL _{i}* is the elevation of the

*i*-th node. The analysis returned

*c*

_{N}= 0.53 and

*n*= 0.75, with a negligible difference from a technical point of view. It was also confirmed that the value falls within the range suggested in the literature for non-plastic pipes, with expected values not far from the orifice flow exponent.

## CONCLUSIONS

In the paper the results of a field measurement campaign were given, aimed at assessing the exponent of the leakage–pressure relationship when the power equation is used. Data were collected by monitoring a district of the Benevento (Italy) WDN. The district serves around 2,800 properties with a total population of about 8,600. Pipelines are mainly in ductile iron, with a small percentage of steel and HDPE. A PRV and a flow meter were installed at the DMA inlet to regulate pressure and measure inflow. Pressure was measured downstream of the PRV and at four significant locations within the network. In order to highlight the relationship between pressure and leakage, DMA inlet pressure was varied between 40 m and 60 m during night hours and the corresponding MNF was measured. The field campaign was carried out over a period of about 3 months. Collected data made it possible to assess that the power equation can effectively model leakage within a WDN. The leakage equation at each node and the simplified model based on AZP returned very similar values. By minimizing differences between field measurements and models, an exponent around 0.7 was calculated.

As pointed out by the few experimental studies available in the literature, results confirm that values slightly greater than that of the orifice equation (*n* = 0.5) should be used for non-plastic pipes. Such values are independent of the pressure level and the flow discharge. Consequently, results can be confidently extended to WDNs with a small percentage of plastic pipes.

A more detailed analysis should be developed to assess the influence of the age of materials instead, because of the greater values of the exponent found for very old cast-iron pipes.