The water loss is a phenomenon frequently observed within water distribution systems. A considerable part of water losses occurs either because of the incorrect assembly of joints or because of the fatigue and ageing of the material used to ensure a watertight seal. Moreover, such a leakage is very difficult to detect and to assess. In this work, we present a novel formulation for modelling the pressure effect on the background leakage through the joints. The proposed approach is based on the preliminary evaluation of the enlargement Δ*ω*, due to pressure, of the existing space between the outer side of the spigot end of a pipe and the inner wall of the hub end of the adjacent pipe (which is characterised by the area *ω*_{atm} at atmospheric pressure). Furthermore, the whole procedure is based on the evaluation, by field data or calibration, of a parameter ξ representing the rate of enlarged area *ω* that, for several reasons, may be not covered by the gasket, *ω* being the value, at pressure *p*, of the area above defined.

## INTRODUCTION

The volumes of water introduced into a water distribution system (WDS) often exceed the volumes actually supplied to consumers and accounted for. The difference between the volumes of water introduced into the network and those actually accounted for consists of many contributions, commonly known as ‘illegitimate uses, under-metered uses, unmetered uses, losses at storage facilities, physical losses from pipes and physical losses from joints’ (van Zyl & Clayton 2005; Puust *et al.* 2010). Sometimes, physical losses from pipes and joints (*leakage*) constitute the greatest part of the total water loss, amounting to a significant percentage of the total volume of water introduced into the WDS.

When pipelines are connected by joints, besides the losses from holes and longitudinal or circumferential cracks, a considerable part of water losses occurs either because of the incorrect assembly of joints or because of the fatigue and ageing of the material used to ensure a watertight seal or because of insufficient mass of material forming the seal. The knowledge of the actual operating conditions of the pipe joints after their implementation is crucial in order to determine the total volume of water lost (Cassa *et al.* 2010).

To clarify the specific contribution that the authors intend to provide in the present work, a preliminary distinction must be made between *leakage bursts* and *background leakage*. Leakage bursts are localised water losses, detectable by direct monitoring systems, e.g., leakage detection systems (Geiger 2008; Puust *et al.* 2010) and/or by indirect analysis, e.g., computational pipeline monitoring (API RP 1130 2007; Puust *et al.* 2010). Leakage bursts can also be found because of their adverse effects (for example, ground level subsidence caused by earth erosion due to piping phenomena). As a consequence of the typically short time interval associated with burst activity before their detection and, at the same time, of the much more extended presence of the leakages from joints, holes and cracks, most of the physical water losses at yearly scale can be attributed to the background leakages, which are very difficult to individuate and evaluate.

In all cases, the leakage discharges depend on the local pressure and can be modelled by means of appropriate pressure-discharge equations. This observation suggests that the correct management of undetectable leaks (background leakages) could be achieved by minimising the nodal pressures throughout the WDS (Ulanicki *et al.* 2000; Thornton 2003; Lambert & Thornton 2011; Lenzi *et al.* 2013), almost to the necessary values to ensure proper service of the water network (Gargano & Pianese 2000; Pirozzi *et al.* 2002; Cozzolino *et al.* 2005). To this aim it is necessary to identify the relations between pressures at different points of the network and losses that there can be realised.

While numerous works related to the evaluation of water losses from holes and longitudinal cracks in the pipes can be found in literature (Ogura 1979; Osterwalder & Wirth 1985; Greyvenstein & van Zyl 2007; Cassa *et al.* 2010; Cassa 2011; Ferrante 2012), a reduced number of works has been devoted to losses from junctions (May 1994; Milano 2006).

The lack of knowledge of the amount of losses which occur at the joints, together with the uncertainty about the formula to be used to relate these losses to the pressure, represents a serious limitation for those studies aimed at the correct identification of the interventions to be undertaken in order to reduce losses from several water distribution networks. Indeed, in the past, most urban WDSs have been assembled using iron pipes or cast/ductile iron pipes with *hub and spigot joints* (Figure 1).

These pipe categories exhibit durability and the ability to withstand traffic and trench loads; moreover, hub and spigot joints ensure quickness of installation and an easy replacement. Watertight connections are obtained by inserting elastomeric gaskets in the ring space existing between the hub end and the spigot end. Of course, ageing causes undesired polymerisation of the elastomeric gaskets, which can crack as a result of fatigue under the daily cycles of pressure loads or may be no longer suitable to follow the enlargements of the internal walls of the junctions and to cover the annular space between the outer wall of the spigot end and the inner wall of the hub end. Based on these considerations, in this work a new conceptual discharge formula aimed at evaluating the discharges lost through the above described joints is proposed. The analytical structure of the proposed formula is suitable to correctly interpret the background leakage, in relation to which the field observations show a specific link between the local pressure heads and the discharges flowing out from the pipes (Thornton & Lambert 2005). Furthermore, the adoption of such a formula is strongly suitable in the context of analysis of the type ‘pressure driven’.

## A CONCEPTUAL DISCHARGE FORMULA FOR THE EVALUATION OF LEAKAGES FROM HUB AND SPIGOT JOINTS

Consider a hub and spigot joint, where *D _{i}* and

*D*are the diameters of the outer wall of the spigot end and of the inner wall of the hub end, respectively, at atmospheric pressure

_{e}*p*

_{atm}(see Figure 1). The size of the empty space existing between the outer wall of the spigot end and the inner wall of the hub end is . Let

*ω*

_{atm}be the area of the annular section whose thickness is

*λ*

_{atm}. When the absolute pressure

*p*

_{abs}increases with respect to the atmospheric value

*p*

_{atm}, both

*D*and

_{i}*D*increase, becoming and , respectively.

_{e}*D*and the variation of

_{i}*D*with respect to the atmospheric pressure condition can be expressed as where

_{e}*p =*(

*p*

_{abs}−

*p*

_{atm}) is the relative pressure and

*s*and

*E*are the thickness of the pipe wall and Young's elasticity modulus of the pipe wall material, respectively. As a consequence of the internal pressure increases occurring in the ducts with respect to the phase of the laying of the pipes themselves, the size

*λ*

_{atm}of the gap between the outer wall of the spigot end and the inner wall of the hub end increases, becoming . Then, starting from Equation (1), the variation Δ

*λ*= (

*λ*′ −

*λ*

_{atm}) of

*λ*

_{atm}can be calculated by means of the equation The gap increment Δ

*λ*is usually small, but in some cases it may cause leakage from the junctions. In particular, this is what would happen if the hydraulic seals of the junctions were not guaranteed by perfectly elastic gaskets, able: (i) to expand/contract without any kind of limitation; (ii) to follow the variations in the gap between the outer wall of the spigot end and the inner wall of the hub end; (iii) to completely cover the increases Δ

*λ*.

Indeed, considering types of pipes commonly available on the European market, the construction of which is regulated by specific European Community rules, if the pressure head ranges in the interval [20, 100] m (typical values within a WDS), then:

for ductile iron pipes (

*E*≅1.2 × 10^{5}N/mm^{2}) with nominal pipe size (NPS) in the range [60, 1,600] mm, regulated by EN 545, Δ*λ*is varying in the range [0.00036, 0.05577] mm;for PVC pipes (

*E*≅ 3.5 × 10^{3}N/mm^{2}) with NPS in the range [40, 1,000] mm and pressure nominal (PN) in the range [0.6, 2.5] N/mm^{2}, regulated by EN 1452, Δ*λ*is varying in the range [0.00257, 0.06241] mm (having already considered that, for PVC pipes characterised by a nominal pressure equal to 0.6/0.8 N/mm^{2}, the pressure head cannot overcome 60/80 m, respectively).

As a consequence of the widening due to the pressure increase from *p*_{atm} to *p*_{abs}, a variation of the initial annular cross section *ω*_{atm}, which becomes *ω* at pressure *p*_{abs}*,* takes place.

In particular, when the values of the pressure head *h* = *p*/*γ* (with γ specific weight of water) fall within the range [20, 100] m, the Δ*ω* values vary as follows:

For ductile iron pipes having NPS in the range specified above, Δ

*ω*ranges within [0.11, 454.06] mm^{2};For PVC pipes having NPS and PN values specified above, Δ

*ω*ranges within [0.50, 1,185.67] mm^{2}.

As a consequence, the enlargements Δ*ω* caused by the pressure heads usually existing in WDSs are not negligible.

The values of Δ*ω*, evaluated for several commercial pipes regulated by European legislations (EN 1452, for PVC pipes; EN 545, for ductile iron pipes, respectively), are reported in Figure 2(a) and 2(b), where Δ*ω* is drawn for different materials as a function of *D _{i}* (Figure 2(a)) and, with reference to PVC pipes PN 10, for different values of

*D*and

_{i}*h*(Figure 2(b)), respectively.

For these pipes, very useful relationships between leakage and pressure or leakage and pressure head could be obtained to evaluate the leakage from hub and spigot joints.

*ω*

_{appr.}is an approximation of Δ

*ω*, we have where represents the mean diameter of the slice having thickness Δ

*λ*= (

*λ′*–

*λ*

_{atm}), due to enlargements caused by internal pressure. Because of different effects that the internal pressure has on the outer wall of the spigot and the inner wall of the hub, the value of

*K*must be assumed higher than 1. After several trials the authors have seen that its value can be assumed to be 1.5, regardless of the diameter of the pipe and its constituent material.

It is possible to observe that Δω and Δω_{appr.} (Equation (5)) are in good agreement, regardless of the diameter of the pipe and its constituent material.

Thus, in this work Equation (5) has been adopted to evaluate Δ*ω* in all subsequent relationships between leakage from a hub and spigot joint and pressure or between leakage from a hub and spigot joint and pressure head.

Let us consider that at atmospheric pressure the annulus of area *ω* is completely occupied by a water stop ring, made of a material that can be considered originally elastic. In consequence of this behaviour, the annulus is initially occupied by the sealing material (Figure 4(a)–4(c)), even though the pressure increases from *p*_{atm} to *p*_{abs}. Therefore, at this time there is no loss from the junction.

After a few years of operation, the material used for the gasket may alternatively: (i) continue to be perfectly elastic; (ii) behave as completely inelastic (such as when the seal becomes fully cured); or (iii) preserve only part of its elastic behaviour (as often occurs in reality). It can be assumed that, at atmospheric pressure, the polymerised gasket, though aged, is able to completely close the annular space of area *ω*_{atm} existing between the jointed pipe ends (see Figure 4(a)–4(c)). Similarly, it can be assumed that, in the case of a non-zero relative pressure, a completely (or partially) polymerised gasket cannot (perfectly) fit the enlarged ring-space Δ*ω* existing between the jointed pipe ends, allowing leakage to occur.

*ω*(see Figure 4(a)) for any value of pressure: in this condition, the leakages are null. Conversely (case ii), when the behaviour of the gasket becomes perfectly plastic (a completely polymerised gasket), the enlargement of the gasket is not possible and the seal is absolutely incapable of compensating for the increased ring-space Δ

*ω*(see Figure 4(b)). Accordingly, significant losses of water may occur from the joint, and the water tends to flow out from the entire increased annular space Δ

*ω*. Under typical operating conditions (case iii), and usually within a few years after the sealing of the joint, portions of the gasket may not be able to compensate for the increased Δ

*λ*of the gap (see Figure 4(c)), because of the polymerisation processes and fatigue phenomena resulting from the cyclic fluctuations of the pressures existing in the network during the day. Thus, leakage occurs, although such leakage is minor compared with the condition represented in Figure 4(b). Let

*ξ*be the fraction of the increased ring-space of thickness Δ

*λ*and area Δ

*ω*that cannot be covered by the gasket. Thus, an area from which the leakages occur will exist, and it is given by with 0 ≤

*ξ*≤ 1.

Generally speaking, also the coefficient *ξ* should be a function of the internal pressure. In fact: (i) for constructional reasons, because of the difficulty of assembly of the pipes that would result from the use of very thick gaskets, usually the material constituting the elastomeric gasket is not sufficient to cover the area Δ*ω*; (ii) in consequence of the aging of the material that constitutes the gasket, because of internal pressure this last tends to widen less than Δ*λ*. In both cases, the percentage of the annular space Δ*ω* not covered by the elastomeric gasket should increase with the internal pressure.

*Q*flowing out through the generic hub and spigot joint. In particular, it can be calculated as or where where

*μ 0.6*is the usual discharge coefficient,

*h*=

*p*/

*γ*is the pressure head,

*g*is the gravity acceleration,

*ρ*is the density of water.

## CALIBRATION OF THE PARAMETER ξ

The coefficient *ξ* introduced in Equations (6)–(8) accounts for the actual degree of joint opening due to the internal pressure. This coefficient is 0 for elastic non-polymerised gaskets with perfect joint implementation, and is 1 for a completely inelastic seal. Although the approach proposed in this paper does not seem to take into account the effects of viscoelasticity that can take place in elastomeric gaskets and pipes, as shown in Ferrante (2012), the approach may easily extended to incorporate these effects. To consider these effects and to verify the suitability of Equations (7) and (8), field data given by Milano (2006) for one ductile iron pipe and one PVC pipe were used, both just laid. The ductile iron pipe had the following characteristics: NPS = 300 mm; *D*_{int.} = 311.6 mm; *D _{i}* = 326.0 mm;

*s*= 7.2 mm;

*D*= 366 mm;

_{e}*E*= 1.07 × 10

^{5}N/mm

^{2}; while the PVC pipe had the following characteristics: NPS = 355 mm;

*D*

_{int.}= 341.0 mm;

*D*= 355.0 mm;

_{i}*s*= 7.0 mm;

*D*= 370.2 mm;

_{e}*E*= 3.0 × 10

^{3}N/mm

^{2}.

For the ductile iron pipe, 13 pairs of values (*h*, *Q*) are available, while 14 pairs of values (*h*, *Q*) are available for the PVC pipe.

The *ξ* values have been deduced by means of Equations (8) and (10), using the pairs of values (*h*, *Q*), and the geometrical and mechanical characteristics of the pipes, respectively, presented in the work of Milano (2006) have been shown to be actually variable with the internal pressure. The changes of the *ξ* values with the pressure head *h* are shown in Figure 5(a) (for the ductile iron pipe) and Figure 5(b) (for the PVC pipe), respectively.

Then, by using Equations (8), (10) and (11), it has been possible to estimate the values of the leakages from joints and to compare these values with those actually observed by Milano (2006) during field experiments.

A comparison between the observed and estimated leakages is shown in Figure 6.

It is possible to observe that, regardless of the diameter of the pipe and its constituent material, the agreement between the leakages observed and estimated is sufficiently good, allowing the possibility to use the proposed approach in the technical field.

Generally speaking, *ξ* must be considered a *model parameter*, whose value must be calibrated on a case-by-case basis, as explained in the following.

*Q*is the discharge flowing out from the

_{k,m}*kth*joint (

*k =*1, 2,

*…, N*) during the

_{j}*mth*time interval in which the day could be subdivided to carry out an extended period simulation (

*m*= 1, 2, …., NTI), the overall volume of water lost during the day by the link

*l*of the WDS (with

*l*= 1, 2, …, NL) is given by where

*NJ*is the number of joints that exist within the

_{l}*lth*link (approximately evaluated by the ratio

*L*

_{l}/L_{pipe,l}, where

*L*is the length of the link

_{l}*l*and

*L*

_{pipe}

*,*is the length of each pipe constituting the link);

_{l}*h*is the relative pressure head evaluated at the

_{k,l,m}*kth*joint of the

*lth*link during the

*mth*time interval;

*h*is the difference between the piezometric head and the elevation above a reference horizontal plane of the pipe axis, evaluated at the

_{k,l,m}*kth*joint of the

*lth*link during the

*mth*time interval; and

*ξ*is the value of the parameter

_{l}*ξ*related to link

*l*.

*ξ*can be taken as a constant within the whole WDS, the overall volume of water lost daily by the joints of the WDS is given by Equations (13) and (14) may be useful also to calibrate the value of the parameter

*ξ*to be used for subsequent analyses. For example: (a) the

*ξ*value for the

*lth*pipe may be estimated by using Equation (11), starting from the knowledge of the entire amount of water lost by that pipe in a given number NTI of time intervals in the day (for example, setting NTI = 1) and the pressure head distribution along the pipe for those time intervals; (b) the

*ξ*value for each group of NL pipes may be estimated by using Equation (14), starting from the knowledge of the water lost in a given number NTI of time intervals in the day (for example, setting NTI = 1) by that group of pipes, and the pressure head distributions along the pipes for those time intervals; (c) the

*ξ*value for the whole WDS, using Equation (14) again, starting from the knowledge of water lost in a given number NTI of time intervals in the day (for instance, setting NTI = 1) in the whole WDS and the pressure head distributions along the pipes for those time intervals.

To understand the importance of the conceptual model described above, it should be noted that: (i) a number of analyses, carried out over 100 field tests on sections of Japanese WDSs and Australian, Brazilian, Canadian, Malaysian, New Zealand, UK and US district metered areas, have shown that, to simulate the presence of holes and cracks in the pipe walls, formulas in which the leakage discharge is proportional to *p ^{b}* (or

*h*) must be used, where

^{b}*b*has to be chosen between 0.5 and 2.5 (Thornton & Lambert 2005); (ii) other laboratory experiments have led to values of

*b*ranging between 0.42 and 2.4 (van Zyl & Clayton 2005); (iii) according to Bargiela (1984), based on the experiments carried out by Godwin (1980) and the National Water Council (1980), the value

*b*= 1.18 could be considered (Germanopoulos & Jowitt 1989); (iv) a few tests carried out on WDSs in which all detectable (burst) leaks have been repaired or temporarily shut off, leaving only background (undetectable) leakage, tend to produce (Thornton & Lambert 2005) values of the exponent

*b*that are precisely equal to the value 1.5 in Equations (7) and (8).

Consequently, the simple model proposed in this paragraph, although unable to account for the soil head losses due to the movement of water through the soil after leaving holes or deteriorated joints, is capable of taking into account the background leakage by means of Equations (7) and (8). Therefore, it can be easily used within pressure-driven numerical models.

## CONCLUSIONS

To reduce losses within WDSs, several approaches may be adopted, such as reducing the pressures existing in the network to those strictly necessary to ensure the delivery of the discharges required by users. To identify actions to be implemented for this purpose, it is necessary to adequately simulate the operation of the network and, in particular, to use appropriate relationships between the pressure (or pressure heads) and leakages, able to allow the evaluation of losses in the network, including the background losses. Concerning this topic, several relationships have been proposed in the last 40 years. Most of them aim at the evaluation of water losses that occur from the holes and the longitudinal lesions present in the pipes. Very few studies are devoted instead to the evaluation of losses at joints. Starting from these considerations, in this work a new conceptual discharge formula has been proposed, aimed at evaluating the discharges and water volumes lost through the ‘hub and spigot’ joints, which are frequently used within the WDSs.

The proposed relationships between the leakage and pressure (or the leakage and pressure head, respectively) are physically based, and contain a single parameter calibration, which, in turn, is of clear physical meaning. For its calibration, different procedures have been illustrated, suitable to be used in realistic application. Among them, a calibration procedure based on experimental or field observations and the use of numerical analysis by a pressure driven approach has been presented.

However, the proposed relationships are far from the overall solution of the joints losses assessment. As a matter of fact, more theoretical and experimental work needs to be done to achieve a clear interpretation of the phenomena and a more precise evaluation of joints leakages.

Indeed, the field data used in this work are taken from just-laid pipes with very small leakages. Therefore, more data are needed to perform further analysis. In this perspective, the relationships proposed in this work seem suitable to provide sufficiently reliable estimates, at least as an order of magnitude, of the leakages at joints, only for just-laid pipes.

## ACKNOWLEDGEMENTS

The authors would like to thank the Editor, the Associate Editor and the three anonymous reviewers for their valuable observations and suggestions that greatly contributed to improving the paper. The present work was developed with financial contributions from the Campania Region, L.R. n.5/2002 – year 2008 – within the Project ‘Methods for the evaluation of security of pressurised water supply and distribution systems towards water contamination, also intentional, to guarantee to the users, and the optimisation design of water systems', prot. 2014.0293987 dated 29.04.2014 – CUP: E66D08000060002.