# Analysis of MNF and FAVAD Model for Leakage Characterization by Exploiting Smart-Metered Data: The Case of the Gorino Ferrarese (FE-Italy) District

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

^{3}/s) from an orifice as [21,22]:

^{2}), g is the gravitational acceleration (m/s

^{2}), h is the pressure head (m), and C

_{q}is a flow coefficient ( ).

^{3-N1}/s), and N1 is the leakage exponent ( ). This relationship is one of the most commonly used equations, and it has also been adopted by the IWA [14,24,25]. However, making the exponent of the leakage equation a variable severs it from its fluid mechanics foundations and turns it into a purely empirical equation [26]. While the value of N1 should be 0.5 to be consistent with the hydraulic of orifices, field tests have shown that the coefficient N1 can take values even greater than 2 [2,24].

_{0}is the initial area at zero head differential (m

^{2}), m is the head-area slope (m

^{2}/m), and h is the pressure head.

_{q}is the flow coefficient—usually equal to 0.65—A

_{0}is the initial area at zero head differential, and m is the head-area slope. This model was first proposed by May in 1994 [17] and is commonly called the Fixed and Variable Area Discharge (FAVAD) equation. The first term of Equation (4) is the orifice equation and describes the flow through a fixed initial area of the leak. The second term in the equation describes the flow through the expanded area of the leak [26].

_{1}and h

_{1}are the leakage flow and pressure head before the pressure maneuver, and Q

_{2}and h

_{2}are leakage flow and pressure head after the pressure maneuver.

_{0}. First, an analytical exploration is performed by matching the two relationships (Equation (2)) and (Equation (4)):

## 3. Materials and Methods

^{2}. It corresponds to a natural DMA with only one water inflow point, and it supplies 294 users, of which 277 are residential and the remaining 17 represent commercial activities or non-domestic users.

_{0}of the FAVAD equation (Equation (4))) were estimated using different methods. As stated above, these coefficients are, in fact, generally estimated by using a couple of values of pressure and corresponding leakages observed before and after a marked pressure variation. Therefore, the pressure reduction maneuver performed every day at 23:00 was considered, and the pressure and corresponding leakages values observed before and after the reduction were extrapolated from the time series for each of the 45 days monitored, resulting in two pairs of values for each day (Figure 5).

_{0}(Equations (16) and (17)) for the specific day each time, and, regarding the value of pressure, the average of the two pressure values before and after the maneuver for the same day was considered. For MET2 and MET3, the corresponding coefficients and the averages of all pressure values of the pairs were considered. For MET4 and MET5, the corresponding coefficients and the averages of all the pressure values in the time series or only the averages of the data from 22:00 until 5:00 were considered.

## 4. Analysis of Results

_{0}, which do not have a physical meaning in network systems like the one considered here.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Relationship between the leakage exponent N1 of the power equation and the leakage number LN of the Fixed and Variable Area Discharge (FAVAD)equation.

**Figure 2.**(

**a**) A satellite view and (

**b**) the network layout and user connections of the district metered area of Gorino Ferrarese (FE-Italy).

**Figure 3.**Trends in the pressure with a 5 min time step over two days highlighting the operation of pressure reduction through two ellipses.

**Figure 4.**(

**a**) District metered area (DMA) inflow (

**red**) and the sum of all the users’ water consumption (

**blue**) with a 5 min time step over one day. (

**b**) Trends in the leakage with a 60 min time step over the period of 45 days.

**Figure 5.**Values of pressure before (

**red**) and after (

**orange**) the pressure reduction maneuver and values of leakage before (

**blue**) and after (

**light blue**) the same maneuver, which were extrapolated from the hourly time series for each of the 45 days of monitoring indicated on the x axis.

**Figure 6.**(

**a**) Night consumption from set 1 (empty black dots) in L/min and their mean value (full black dots) fitted by a straight line, as well as values from set 2 (empty yellow dots) and their mean value (full yellow dots), compared with values from the literature (green dots). (

**b**) Addition of minimum inflow (blue) and minimum total consumption (magenta) from set 3.

**Figure 7.**(

**a**–

**e**) Values of the coefficient m plotted as a function of the time step estimated with the different methods: MET1–MET5. (

**f**–

**j**) Values of the coefficient N1 plotted as a function of the time step estimated with the same methods.

**Figure 8.**FAVAD equations with coefficients evaluated with data with a 60 min time step using (

**a**) MET1 in blue, MET2 in magenta, and MET3 in green, as well as (

**b**) MET4 and MET5 in gray and black, respectively.

**Figure 9.**Theoretical equation for converting between the FAVAD and power equations (black line) compared with the values of N1 and LN obtained with data with a 60 min time step using MET1 in blue, MET2 in magenta, MET3 in green, MET4 in gray, and MET5 in black.

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**MDPI and ACS Style**

Marzola, I.; Alvisi, S.; Franchini, M.
Analysis of MNF and FAVAD Model for Leakage Characterization by Exploiting Smart-Metered Data: The Case of the Gorino Ferrarese (FE-Italy) District. *Water* **2021**, *13*, 643.
https://doi.org/10.3390/w13050643

**AMA Style**

Marzola I, Alvisi S, Franchini M.
Analysis of MNF and FAVAD Model for Leakage Characterization by Exploiting Smart-Metered Data: The Case of the Gorino Ferrarese (FE-Italy) District. *Water*. 2021; 13(5):643.
https://doi.org/10.3390/w13050643

**Chicago/Turabian Style**

Marzola, Irene, Stefano Alvisi, and Marco Franchini.
2021. "Analysis of MNF and FAVAD Model for Leakage Characterization by Exploiting Smart-Metered Data: The Case of the Gorino Ferrarese (FE-Italy) District" *Water* 13, no. 5: 643.
https://doi.org/10.3390/w13050643