Figure 4 shows a real-life case study, in Southern Italy, where two main reservoirs (Res1 and Res2) feed, through several transmission pipes, a water network composed of three consumption centres Net1, Net2 and Net3 corresponding to three small towns, and five water tanks, upstream to other consumption centres (other small towns) as reported in Fig. 5 with the label Ext.

Thus, the two sources of water feed more than three consumption centres, while the system under analysis is focused to three of them. In addition, Fig. 6 shows the upstream and downstream DMA in which the system is divided.

Figure 5 shows the spatial location of private connections (i.e., georeferenced customer meters), while indicating five water tanks upstream to other consumption centres (other small towns) fed by the system under analysis. Thus, the two sources of water feed more than three consumption centres, while the system under analysis is focused to three of them. Figure 6 shows the upstream and downstream DMA in which the system is divided.

To the demonstration, a second possible configuration is assumed. A third optional reservoir (Res3) feeding Net1 and Net2, while a closed valve disconnects Net3 from Res3, see Fig. 4, which remains fed by Res1and Res2. The DMAs are not altered by the new configuration.

The configuration adding Res3 is not fictitious. It occurs when a pumping station from a well is activated; the fictitious reservoir Res3, characterized by a specific level pattern, represents the pumping pressure and the source of water (well) feeding Net1 and Net2.

In both configurations, the main transmission pipes are fed by a pump next to the Res1, which feeds Res2, and the upstream DMA composed of Net2 and Net3. While the downstream DMA, corresponding to Net1, is fed by Res2 in the first configuration and by Res3 (well and pumping station) in the second one. Figure 4 shows transmission pipes delivering water volumes to five tanks upstream of other towns (Ext1, Ext2, Ext3, Ext4 and Ext5). Such feeds to towns are included in the hydraulic model as fictitious consumers located in the outer nodes (red dots in Fig. 4), because of the availability of inflow measurements to tanks.

The total length of the system layout is about of 110 km, including 20 km of transport pipelines. The main information of the system layout, including length of each DMA and number of its customers, is shown in Table 1:

Table 1

Length of pipelines and number of customers for each part of the WTS case study

| L [km] | Customers [-] |

Net1 | 23.46 | 4,125 |

Net2 | 33.33 | 4,219 |

Net3 | 34.92 | 4,902 |

Water Mains | 18.84 | 27 |

System | 110.5 | 13,273 |

The hydraulic analysis of the entire WDS has been performed using advanced hydraulic modelling. Among other advancements, it allows the pressure-driven leakage calculation at pipe level [23] depending on pressure and asset deterioration parameters, which is relevant for model calibration and internal water balance of DMAs.

The calibration aimed at separating at DMA level pressure-dependent water loss from daily variable pattern and level of customer water consumption, based on available flow and pressure measurements [21]. It was performed using data of five representative operating cycles (days): weekdays and summer holidays, weekdays and winter holiday and New Year's Day.

Table 2 reports in five rows labelled OC the five operative cycles and the last of mean values. The first three columns refer to the water volumes of: consumption, *D**c*; outflow, *D**out*; and loss, *W**Leak*. The fourth, fifth and sixth columns to the density of water: consumption, D1*a**c*; outflow, D1*a**c*; and their summation, D1*a*. In the seventh column the linear water loss indicator, M1*a*, and in the last two columns the two ways of calculating M1*b* assuming, as above discussed, two definitions of inflow at the denominator: the amended, *W**IN* = *W**Leak*+*D**c*, M1*b*(am.); and the original, *W**IN* = *W**Leak*+*D**c*+*D**out*, M1*b*(orig.).

Table 2

Daily and mean values of M1*a* and D1*a**W*, [m3/km/day], M1*b*, [%], of the entire system.

Operating Cycle | *D**c* [m3] | *D**out* [m3] | *W**Leak* [m3] | D1*a**c* | D1*a**out* | D1*a* | M1*a* | M1*b* (am.) | M1*b* (orig.) |

OC 1 | 2,910 | 81,465 | 3,628 | 26.32 | 736.77 | 763.09 | 32.82 | 55.49 | 4.12 |

OC 2 | 2,615 | 74,393 | 3,788 | 23.65 | 672.81 | 696.46 | 34.26 | 59.16 | 4.69 |

OC 3 | 1,673 | 64,938 | 4,219 | 15.13 | 587.30 | 602.43 | 38.16 | 71.61 | 5.96 |

OC 4 | 1,604 | 67,170 | 4,266 | 14.50 | 607.48 | 621.98 | 38.58 | 72.68 | 5.84 |

OC 5 | 2,061 | 65,843 | 4,147 | 18.64 | 595.48 | 614.12 | 37.50 | 66.80 | 5.76 |

Mean OC | 2,173 | 70,762 | 4,010 | 19.65 | 639.97 | 659.62 | 36.26 | 64.86 | 5.21 |

It is worth noting that the advanced modelling and calibration allows calculating variation of the leakages and water consumption. Leakages varies over days because of the change of pressure regime due to different daily water consumption as pattern and level.

*Inconsistency of the percentage water loss indicator*

The last column of the Table 2 shows that M1*b* calculated at the daily scale, considering the inflow volume *W**IN* = *W**Leak*+*D**c*+*D**out*, is strongly dependent on the variability of *D**out*, which dominates *W**Leak*+*D**c*, and it is very low because the outflow to the five downstream tanks is very high with respect to water loss. The indicator, then, indicate a very low level of leakages because of the hydraulic system transfer a large amount of water to others, but the inconsistency is related to the fact that *D**p**out* contains the component of water loss of the five consumption centres downstream to the tanks. In fact, *D**p**out* is the summation of the downstream water loss *W**Leak**out* and consumption *D**p**W−out*. Therefore, to correct the indicator it is necessary to write:

$${\text{M1}}_{b}^{{}}=100 \cdot \frac{{{W_{Leak}}+W_{{Leak}}^{{out}}}}{{{W_{Leak}}+D_{{}}^{c}+D_{{}}^{{out}}}}=100 \cdot \frac{{{W_{Leak}}+W_{{Leak}}^{{out}}}}{{{W_{Leak}}+W_{{Leak}}^{{out}}+D_{{}}^{c}+D_{{}}^{{c - out}}}}$$

13

However, the calculation requires an unknown water loss *W**Leak**out* or water consumption *D**c−out* of the other consumption centres.

Furthermore, the original formulation of Eq. (13) causes the dependence of the values of M1*b* in the last column of Table 2 even depending on the downstream water loss *W**Leak**out*. Consequently, a reduction of *W**Leak**out* due to a policy of leakage management in the other consumption centres, increases the indicator.

Similarly, a reduction of *D**c−out*, due to a policy of containing water consumption in other consumption centres, increases the indicator, but this is the same problem caused by a policy reducing *D**c* of the hydraulic system under analysis.

*Amending the percentage water loss indicator*

To correct the indicator, it is possible to write:

$${\text{M1}}_{b}^{{}}=100 \cdot \frac{{{W_{Leak}}}}{{{W_{Leak}}+D_{{}}^{c}}}$$

14

Then, *D**out* is not considered assuming *W**IN* = *W**Leak*+*D**c*, in other words the *W**IN* is now defined as the difference between inflow and outflow of the system under analysis.

M1*b* calculated using Eq. (14) at the daily scale, reported in the ninth column of Table 2, depends on the daily variability of *D**c* only. The dependence is strong because of the low daily values of D1*a**c*, which is lower during the winter days causing the increase of M1*b* well beyond the actual increase of the water loss volume.

In fact, the daily variation of M1*a* is much lower and depends on D1*a* for hydraulic reasons, i.e., because of the lower pressure when the water consumption increases, but not because of the formulation of the linear water loss indicator.

In more details, considering the different days M1*b* spams from 72.68–55.49% with a mean value of 64.86%, while M1*a* spams from 32.52 m3/km/d to 38.58 m3/km/d with a mean value of 36.26 m3/km/d. This fact shows that M1*a* indicates a consistent average level of water loss, in contrast with M1*b* which varies improperly.

The values of M1*b* suggest a massive pipes replacement with pressure control, whereas those of M1*a* suggest that investing in pressure control alongside with replacement of few selected pipes would turn into favourable returns. In this context, M1*a* guides investments more accurately, conversely to M1*b*, thereby influencing the overall quality of the expenditure.

Furthermore, a policy reducing *D**c* in not suggested by the amended indicator.

*Non-scalability to DMAs*

M1*a* and M1*b* have been computed also for each DMA identified by existing flow meters. Figure 5 shows DMAs of the system: DMA#1 comprises Net2 and Net3 and the surrounding transmission mains; DMA#2 includes Net1.

Table 3 reports the values of the same variables of Table 2, the mean on the five days for sake of synthesis without impairing the generality of the discussion, at the DMA level (DMA #1 and DMA #2) and for the entire system (System). In addition, calibration and hydraulic modelling allows calculating the average pressure (over the five days) of DMAs and the entire hydraulic system.

Table 3 refers to the normal operating condition, i.e., without pumping from well or Res3.

Table 3

Mean values over five days of M1*a* and D1*a**W*, [m3/km/day], M1*b*, [%] at the two DMAs

| DMA #1 | DMA #2 | System |

Length [km] | 84.7 | 25.8 | 110.5 |

Average Pressure [m] | 49.4 | 65.4 | 53.1 |

*W**Leak* [m3] | 2,880 | 1,130 | 4,010 |

*D**c* [m3] | 1,424 | 749 | 2,173 |

*D*out [m3] | 72,641 | 1,300 | 70,762 |

M1*a* | 34.0 | 43.8 | 36.3 |

D1*a**c* | 16.8 | 29.0 | 19.7 |

D1*a**out* | 857.6 | 50.4 | 640.4 |

D1*a* | 874.4 | 79.4 | 660.0 |

M1*b* (amended) | 66.9 | 60.1 | 64.9 |

M1*b* (original) | 3.7 | 35.5 | 5.2 |

Table 3 shows that the original definition of M1*b* is not scalable because the values strongly depend on DMAs outflow volumes. In fact, DMA #1 has M1*b* = 3.7%, which is very low with respect to M1*b* = 35.5% of DMA #2. This result is caused by the circumstance that the DMA #1 transfer most of the water to tanks (Ext1, Ext2, Ext3 and Ext 4 of Fig. 2), while DMA #2 only to Ext5. Therefore, the original definition of M1*b* indicates that the water loss is much higher in DMA #2 than DMA #1, incorrectly driving leakage management action in the first one. Instead, M1*a* indicate a greater value for DMA #2 (43.8) than for DMA #1 (34.0), whose difference can be related to pressure difference, as will be better discussed in the next section.

Table 3 shows that the amended definition of M1*b* is not scalable too, although the values are almost like that (64.86) of the entire hydraulic system under analysis. In fact, DMA #1 has M1*b* = 66.9%, and DMA #2 M1*b* = 60.1%. Nevertheless, those values are driven by the density of water consumption D1*a* with respect to M1*a*. In fact, DMA #1 has a lower D1*a* (16.80), which drives the M1*b* to a higher value (66.9%) due to a lower M1*a* (34.0). On the contrary, the higher value of D1*a* (29.0) of DMA #2 drives M1*b* to a lower value (60.1%) despite a higher M1*a* (43.8).

Therefore, the original definition of M1*b* indicates that the water loss is much higher in DMA #2 than DMA #1, incorrectly driving leakage management action in the first one. Instead, M1*a* indicate a greater value for DMA #2 (43.8) than for DMA #1 (34.0), whose difference can be related to pressure difference, as will be better discussed in the next section.

The amended definition of M1*b* is also indicating DMA #1 instead of DMA #2 because of a lower density of water consumption D1*a*.

Note that *D*out = 72,641[m3/km/day] for DMA #1 and *D*out = 1,300 [m3/km/day] DMA #2, i.e., the daily volume of Ext5, while *D*out = 70,762 [m3/km/day] for the entire system, i.e., the daily volume of all the feeding to Ext. Thus, the last volume is not the sum of the first two, because the outflow volume for DMA #1 is *D*out = 70,762 + 1,130 + 749 [m3/km/day], i.e., the daily volume of all the feeding to Ext plus the volumes of water loss and consumption of DMA #2. This is one of the reasons of non-scalability of M1*b*; the outflow of a DMA is an inflow for other DMA(s), i.e., portion of the system, thus a volume plays two time in the indicator, and it changes and might increase or decrease based on the DMAs configuration. This fact does not occur for the amended M1*b*, although it is weak and not scalable too.

Now, Table 3 refers to the configuration with Res3, i.e., pumping from well. It reports the only changed variables with respect to Table 2.

Table 3. Mean values over five days of M1*a* and D1*a**W*, [m3/km/day], M1*b*, [%] at the two DMAs

| DMA #1 |

*D**p*out [m3] | 69,462 |

D1*a**out* | 820.1 |

D1*a* | 836.9 |

M1*b* (original) | 3.9 |

It is worth noting that, even a change of the technical scheme, i.e., a different source of water, changes *D**p*out term of DMA #1 and consequently the percentage indicator, while the linear one does not change.

Finally, it is then possible to state that the percentage of water loss indicator is not scalable at DMA level of interdependent portions of the system because it depends on outflow and water consumption volumes in the original definition and on water consumption in the amended one, being outflow and water consumption volumes dependent on spatial variability, i.e., on DMAs configuration and even on source change. Instead, M1*a* allows to explain the water loss at the DMA level also considering the pressure status.

Perspective of using Linear water loss indicator: the Asset Management Support Indicator

The previous sections demonstrate that the linear indicator, M1*a*, is an effective water loss indicator compared to percentage one. Previous literature recommended the Infrastructure Leakage Index (ILI) [24] [25], to assess the network conditions by considering the pressure level in the WDN in a heuristic way while M1*a* is scalable hydraulically based as clarified later. Nonetheless, several shortcomings are associated to ILI, including the methodology applied for identifying the pressure itself [26] [27], the uncertainty of the empirical estimate, which is independent of pipeline age, materials and diameters, the uncertainties in the confidence levels under different pressure scenarios [9] and the major inconsistencies with respect to actual leakage reduction achievements [28].

Therefore, the novel AMSI (Asset Management Support Indicator) is here proposed. It is based on M1*a* and inherits its features such scalability and linearity being a hydraulically based indicator better supporting the allocation of investments at any system scale.

Reminding that the unit of M1*a* is m3/km/day, the corresponding leakage outflow, *Q**Leak*, in m3/sec is scaled considering that 1 km = 1,000 m and in 1 day = 86,400 sec as reported in the following Eq. (15). In addition, Germanoupulos’ model [17] allows to write the leakage outflow, *Q**Leak*, as follows,

$${\text{M1}}_{a}^{{}}=86,400,000 \cdot {Q_{Leak}}=8.6 \cdot {10^7} \cdot \beta \cdot {P^{\alpha \approx 1}}$$

15

Eq. (15) makes explicit the dependence of M1*a* on a deterioration factor, *β*, and on the mean pressure, *P*, of the portion of the hydraulic system under analysis. *α* is an exponent that can be approximated to unit value as reported in recent studies [18] [19] [20] about the Fixed And Variable Area Discharge (FAVAD) formulation of leakage model. Note that M1*a* divided by 8.6·107 is the mean leakage outflow for unit length (one meter) pipe of the system under analysis.

AMSI is then defined as follows,

$${\text{AMSI}}\left( {asset} \right)=\frac{{{\text{M1}}_{a}^{{}}}}{{{P^{\alpha \approx 1}}}}=8.6 \cdot {10^7} \cdot \beta$$

16

Therefore, AMSI is a scaled deterioration factor of the portion of the system under analysis. Consequently, it depends on the number and size of leakage outflows in the mains and connections to the properties and depends on *asset* variables, like age, pipes material and diameter, number of connections to properties, etc. Its increase of rate over time depends on pressure, effects of fatigue (e.g., pressure variation due to unsteady flow or pressure control; traffic; etc.), environmental factors, etc. as reported in [8].

For sake of completeness, AMSI can be also formulated using FAVAD formulation of the leakage model as follows:

$${\text{AMSI}}\left( {asset} \right)=\frac{{{\text{M1}}_{a}^{{}}}}{{\left( {1+LN} \right)\sqrt P }}$$

17

where *LN* is the leakage number of Van Zyl and Cassa [18], which better accounts for the system material and average pressure being FAVAD more physically based model than the Germanoupulos’ one [17].

AMSI is the daily volume of leakages for km of the system under analysis and for its unit pressure, and correctly characterizes its deterioration allowing to support the decision about the activities of asset management in any portion of the entire hydraulic system as reported in the following example.

AMSI is a relevant indicator because is the ratio of the density of water loss, M1*a*, and mean pressure of the system under analysis. It benefits of the fact that, being the pressure usually given in meters of water column, the numerator and denominator in Eqs. (16) and (17) has the same order of magnitude.

For example, if we have three consumption centres having a rather high M1*a* = 50 m3/km/day and we assume that *P* = 25, 100 and 50 m, AMSI = 2, 0.5 and 1.

Therefore, Eq. (16) provides insights though the interpretable M1*a*, offering guidelines for asset management in the following manner: (1) in the first town, water loss depends on system deterioration. Consequently, it needs prioritized plans for pipes replacement and implementing active leakage detection actions; (2) in the second town, water loss depends on high system pressure level. As a result, it requires pressure control actions, and (3) in the third town, water loss depend on both system pressure and system deterioration. This requests a combination of the approaches previously exposed to address the issue effectively.

Therefore, the structure and scalability of M1*a* facilitates a robust approach for steering investments for asset management of drinking water infrastructure, as today required worldwide. Achieved through the simple Eqs. (16) and (17), this method inherits the effectiveness of M1*a* and the scalability to focus on different portions of the entire DWI, including DMAs or consumption centres. Furthermore, the novelty of AMSI, with respect to their indicators, resides in the capability of supporting decision about the efficiency of the type of investment. In fact, it allows ex post and ex ante evaluations of the efficiency of investments about pipes replacement and active leakage detection versus pressure control, while M1*a* alone is much influenced by the efficiency of pressure control. AMSI also allows comparing different DWSs and understanding the type of investments for each water utility and where to allocate them inside each one as useful for a water agency such ARERA in Italy.

To complete the analysis of AMSI, Table 4 reports the data of referring to M1*a* like Table 2 but calculated with (the same of Table 2) and without the five outflows (Ext1, Ext2, Ext3, Ext4, and Ext5), e.g., assuming that the external tanks are fed by pumping from well, adding the row of AMSI.

Table 4

Mean values over five days of M1*a* and D1*a**W*, [m3/km/day], M1*b*, [%] at the two DMAs

| DMA #1 | DMA #2 | System |

**With External Fed** | | | |

Length [km] | 84.7 | 25.8 | 110.5 |

Average Pressure [m] | 49.4 | 65.4 | 53.1 |

*W**Leak* [m3] | 2,880 | 1,130 | 4,010 |

M1*a* | 34.0 | 43.8 | 36.3 |

AMSI | 0.69 | 0.67 | 0.68 |

**Without External Fed** | | | |

Average Pressure [m] | 51.9 | 70.3 | 56.2 |

*W**Leak* [m3] | 3,041 | 1,213 | 4,254 |

M1*a* | 35.9 | 47.0 | 38.5 |

AMSI | 0.69 | 0.67 | 0.68 |

The case study further substantiates the efficacy of both M1*a* and AMSI in guiding the quality of the investments. AMSI = 0.69 and 0.67 shows that both the DMAs exhibit minor deterioration, asking for pressure control actions rather than pipes replacement plans. In fact, such DMAs are both characterized by high average pressure, particularly the one downstream. Once feeding to external tanks end, the pressure at system level, as well as at DMA level, increase because of a lower water flow within the hydraulic system. Consequently, pressure dependent leakages escalated. Nonetheless, AMSI is unchanged since the deterioration factor remains unchanged. This observation highlights that AMSI inherits not only the M1*a* characteristics, but it also remains unaffected by pressure fluctuations, as required to be effectively representative of the trade-off between system deterioration and pressure level causing the density of water loss in any portion of the hydraulic system.