This study investigated the feasibility of characterising the pressure-leakage response of water distribution systems using the FAVAD (Fixed and Variable Area Discharges) equation, instead of the conventional N1 power equation. The study was based on 300 network models with randomly distributed leaks and 35 networks generated through a sensitivity analysis. It was found that the leakage rate and average zone pressure head (AZP) before and after pressure reduction may be used in conjunction with the FAVAD equation to estimate the initial leakage area (A0S) and head-area slope (mS) of any system. In addition, A0S and ms were shown to provide good estimates respectively of the sum of the initial areas and head-area slopes of all the individual leaks in the system. The study found that a dimensionless leakage number may be calculated for any system and used to characterise the pressure-leakage response. Finally, the study showed that it is possible to convert between N1 and the leakage number using a simple equation.

## LIST OF SYMBOLS

• A

Leakage area

•
• A0

Initial leakage area (leakage area at zero pressure)

•
• A0S

Initial leakage area of a system.

•
• AZP

•
• C

Leakage coefficient

•
• Cd

Discharge coefficient

•
• CdS

System discharge coefficient

•

Fixed And Variable Area Discharges

•
• g

Acceleration due to gravity

•
• h

•
• hAZP

•
• hAZP_1

Average zonal pressure head before reduction

•
• hAZP_2

Average zonal pressure head after reduction

•
• LN

Leakage number

•
• LNS

System leakage number

•
• m

•
• mS

•
• N1

Leakage exponent

•
• PMA

Pressure management area

•
• Q

Leakage rate

•
• Q1

Total system leakage rate before pressure reduction

•
• Q2

Total system leakage rate after pressure reduction

## INTRODUCTION

Pressure management is applied internationally as an effective method to reduce leakage from water distribution systems (Farley & Trow 2003; Lambert et al. 2013), and has also been shown to produce other benefits such as reduced pipe failure rates, extend pipe service life and lower consumption (Lambert et al. 2013).

Pressure management refers to the practice of reducing excess pressures in an isolated network zone called a pressure management area (PMA) using a pressure-reducing valve. Leakage in a PMA is estimated by first measuring the minimum night flow and then subtracting the estimated consumption from it.

Estimates of system leakage and average zone pressure head (AZP) before and after pressure management can be used to estimate the leakage coefficient C and leakage exponent N1 in the equation:
1
where Q is the leakage flow rate and hAZP the AZP. Equation (1) is also called the N1 power law or the N1 equation. The latter term will be used in this paper.
The N1 exponent is widely used to characterise the pressure-leakage response of water distribution systems. Applying Equation (1) to conditions before and after pressure management, and dividing one equation by the other, allows C to be eliminated and N1 obtained from the equation:
where hAZP_1and Q1 are the AZP and leakage flow rate before pressure management, and hAZP_2 and Q2 the AZP and leakage flow rate after pressure management.

The link between pressure and leakage is dependent on both C and N1, but since the exponent has a much greater influence on the relationship, only N1 is used in practice.

Despite its wide use in practice, the N1 equation has a number of disadvantages:

• It is an empirical equation not founded on fundamental fluid mechanics theory.

• The values of C and N1 for a given system are not constant, but are functions of the pressures at which they are estimated (Van Zyl & Cassa 2014).

• It is dimensionally awkward, since the units of C include the variable N1.

Hydraulic theory predicts N1 to be 0.5 for an orifice, but field studies on water distribution systems in the UK, Japan, Brazil, Cyprus and Malaysia have reported leakage exponents ranging from 0.36 to 2.79 (Lambert 1997). Greyvenstein & Van Zyl (2007) found N1 values between 0.4 and 2.3 for individual leaks in an experimental investigation. The following reasons have been proposed for the leakage exponent differing from the theoretical value of 0.5 (May 1994; Van Zyl & Clayton 2007; Schwaller & Van Zyl 2014b):

• Leakage areas are not constant, but vary with pressure.

• The leak flow regime may be laminar, transitional or turbulent.

• Darcy laminar flow may occur in the soil surrounding buried pipes.

• Impact of pressure variation on the water consumption component of the minimum night flow.

• The combined effect of many leaks with different properties and at different elevations in a PMA.

It has been demonstrated in numerous individual leak studies that varying leakage areas may result in a wide range of N1 values, and this is widely accepted as the main reason for the observed variability in N1 (Lambert 2000).

Finite element studies under both linear elastic (Cassa & Van Zyl 2013) and viscoelastic (Ssozi 2015) conditions have shown that leakage areas vary linearly with pressure, irrespective of leak type, loading conditions or pipe material and section properties. Some experimental studies have also confirmed these results (Ferrante 2012; Ferrante et al. 2013). Thus the relationship between leakage area and pressure may be described with the equation:
2
where A is the leakage area at head h, A0 the initial leakage area (defined as the leakage area under zero pressure conditions) and m the head-area slope.
In a hydraulic sense, leaks are simply orifices and thus their hydraulics can be described by the orifice flow equation (see, for instance, Idelchik 1994):
3
where Cd is the discharge coefficient.
Replacing Equation (2) into Equation (3) results in the FAVAD (Fixed and Variable Area Discharges) equation, first introduced by May (1994):
4
The first term in the FAVAD equation is identical to the orifice flow equation and describes the discharge through the initial portion of the leak area (i.e. the area of the leak at zero pressure), while the second term describes the leakage flow through the expanded portion of the leak area.

It should be noted that practitioners often refer to the FAVAD concept when really using the N1 equation, sometimes describing N1 as the ‘equivalent FAVAD N1’ or ‘FAVAD N1’. In these cases, the FAVAD concept is invoked as motivation for the leakage exponent not being the theoretical value of 0.5, rather than a different equation. In this paper the N1 equation will strictly refer to Equation (1) and FAVAD equation to Equation (4).

Van Zyl & Cassa (2014) defined a dimensionless leakage number LN as the ratio of the flow through the expanded leak area to the initial leak area as:
5
For instance, a leakage number equal to one means that the initial and expanded portions of the leakage area contribute equally to the leakage flow, while a leakage number less than one means the initial portion has a greater contribution than the expanded portion.

While the pressure-leakage relationship of individual leaks has been shown to adhere to the FAVAD equation (Cassa & Van Zyl 2013; Van Zyl & Cassa 2014), the application of this equation to systems with many leaks has been limited. May and Lambert (Lambert 2014) considered systems with many leaks, with individual leaks either fixed (with N1 = 0.5) or flexible (with N1 = 1.5) and little variation in-between. Ferrante et al. (2014a, b) investigated the application of both the N1 and FAVAD equations to a system with several leaks. The two equations were perturbed and the results compared with simulations of 100 districts with 100 leaks each. When using the N1 equation they found that the system N1 is often higher than the mean local leak exponent. For the FAVAD equation they found that the effects of parameter perturbation were much less evident, and global and local leak laws show the same functional relationship between leakage and pressure.

The purpose of this study was to investigate the use of the FAVAD equation to characterise the pressure-leakage response of systems (or PMAs) with many leaks. The PMA leakage model is described in the next section, followed by an evaluation of the FAVAD equation to characterise its pressure-leakage behaviour, an investigation into the accuracy of the FAVAD parameters and the link between the FAVAD parameters and system N1. Finally, the implications for practical leakage management are discussed.

## NETWORK LEAKAGE MODEL

Schwaller & Van Zyl (2015) proposed a statistical model of the distribution and parameters of elastic leaks in a network based on current best knowledge. Their study showed that a large number of individual leaks, each adhering to the FAVAD equation, can explain the range of N1 values reported in most field studies. The same model was used as the basis for this study. Since a full description of the development and parameters of the model is available in Schwaller & Van Zyl (2015), only a brief overview is provided here.

A typical PMA consisting of 40 km of pipes and 2,500 service connections was used as basis for the model. A spread sheet leakage model was developed consisting of different numbers of leaks at random positions and with randomly distributed properties. The following statistical distributions were used for individual leak parameters in the PMA:

• The discharge coefficient Cd was modelled using a normal distribution.

• The initial leakage area Ao was modelled using a lognormal distribution bounded by zero for background leaks. Since it is more likely that large and potentially detectable leaks in the network were found and repaired quickly in the network, a normal distribution was chosen for potentially detectable leaks.

• The FAVAD head-area slope of individual leaks were modelled as a generalized power function of the initial leakage area A0, based on finite element study by Cassa & Van Zyl (2013).

• The distribution of the pressure head was modelled as a uniform statistical distribution to represent a PMA with a constant elevation.

For each model, a typical parameter value was estimated, ranging from very low, to very high values as defined in Schwaller & van Zyl (2015) in an attempt to represent realistic parameter ranges found in the field. Individual leak behaviour was modelled using the FAVAD equation. A summary of the parameters used in the model is provided in Table 1.

Table 1

Summary of the leakage parameters used in this study

VariableComponentVery lowLowTypicalHighVery high
Pressure head Mean (m) 20 30 45 60 75
Range (m) ±5 ±10 ±20 ±45
Discharge coefficient Mean (-) 0.5 0.575 0.65 0.725 0.8
Standard deviation (-) 0.026 0.030 0.035 0.039
Background leaks Number 550 550 550 550 550
Standard deviation (mm²) 3.7 3.4 3.2 3.1 2.9
Potentially detectable (PD) leaks Percentage PD leaks 0.1% 0.4% 1% 3% 12.5%
Number 0.5 5.6 17.2 69.8
Pressure variation Practice values (m) 15 25 35 50
Modelling values (m) 0.001 0.01 0.1 10
VariableComponentVery lowLowTypicalHighVery high
Pressure head Mean (m) 20 30 45 60 75
Range (m) ±5 ±10 ±20 ±45
Discharge coefficient Mean (-) 0.5 0.575 0.65 0.725 0.8
Standard deviation (-) 0.026 0.030 0.035 0.039
Background leaks Number 550 550 550 550 550
Standard deviation (mm²) 3.7 3.4 3.2 3.1 2.9
Potentially detectable (PD) leaks Percentage PD leaks 0.1% 0.4% 1% 3% 12.5%
Number 0.5 5.6 17.2 69.8
Pressure variation Practice values (m) 15 25 35 50
Modelling values (m) 0.001 0.01 0.1 10

The parameters of a typical system were used to generate 100 random networks each comprising 100, 1,000 and 10,000 leaks based on the ‘typical’ column in Table 1. In addition, a sensitivity analysis was conducted by varying each parameter in the typical network to its very low, low, high and very high values, respectively. This resulted in 335 network leakage models that were used both in Schwaller & Van Zyl (2015) and this study.

It should be noted that the range of leakage generated through the above process extended well beyond that observed in practice. However, the purpose of this study was to evaluate the proposed method over a large range of conditions, and thus networks with excessive leakage generated through the method described above were included in the study.

The study assumed minimum night flow conditions, which implies negligible head losses and static nodal pressures. The total leakage rate was calculated as the sum of all the individual leakage rates in the system.

To characterize the pressure-leakage behaviour of a system, the leakage and AZP were estimated before and after a uniform reduction of the pressure at all points. This resulted in two data points: (Q1; hAZP_1) and (Q2; hAZP_2), where Q is the total system leakage, hAZP the AZP, and subscripts 1 and 2 refer to the system before and after the change in pressure respectively.

In Schwaller & Van Zyl (2015) the conventional approach was followed by using the two data points in the N1 equation (Equation (1)) to determine C and N1 for the system. However, in this study, the same two data points were used to determine the coefficients of the FAVAD equation (Equation (4)). To do this, the FAVAD equation was first written in the form:
6
where A0S is the system initial leakage area, mS the system head-area slope and CdS the system discharge coefficient. In this study, CdS was assumed to be equal to the average leak Cd in the model (0.65), which then allowed the other two parameters to be estimated using the equations:
7

8
The FAVAD parameters were determined for each of the 335 generated networks with leakage and then analysed. Good correlations between the system and individual FAVAD leakage parameters were observed as follows:
• The system initial leakage area A0S was found to be approximately equal to the sum of the individual leak initial areas as shown in Figure 1.

• The system head-area slope mS was found to be approximately equal to the sum of the individual head-area slopes as shown in Figure 2.

Figure 1

Comparison between the system initial leakage area A0S and the sum of the individual leakage areas for 335 networks.

Figure 1

Comparison between the system initial leakage area A0S and the sum of the individual leakage areas for 335 networks.

Figure 2

Comparison between the system FAVAD head-area slope mS and the sum of the individual leak head-area slopes for 335 networks.

Figure 2

Comparison between the system FAVAD head-area slope mS and the sum of the individual leak head-area slopes for 335 networks.

The results show that pressure management data can be used in the FAVAD equation to provide information on the physical properties of leaks in a system. The total initial areas of all the leaks provide a meaningful measure of the physical integrity of the system. Also, since Cassa & Van Zyl (2013) showed that the head-area slope of a leak can be linked to the properties of the pipe (diameter, material, wall thickness) and leak (type and size), the system head-area slope provides insights into the type of leaks present.

The model used in this study allows information on the leaks to be known perfectly, and thus errors in the system FAVAD parameters can only result from two factors:

• Differences in proportional variations in leak pressures. While the absolute pressure change in the model was identical for all leaks, the proportional change in pressure varied due to the random variations in leak elevations (and thus static pressures).

• Variations in the individual leak discharge coefficients.

To illustrate this, the impact of pressure and discharge coefficients on the accuracy of the system initial area was investigated by doing a sensitivity analysis on a typical system (which happened to have an estimation error of 16%). The parameters used for this analysis are given in Table 1 and the results are shown in Figure 3.
Figure 3

Sensitivity of the system initial area estimate to variations in pressure and discharge coefficients.

Figure 3

Sensitivity of the system initial area estimate to variations in pressure and discharge coefficients.

The figure shows that variations in the discharge coefficient and mean system pressure have little impact on the system initial area estimate, but that the error is highly sensitive to the range of pressures (i.e. difference in elevations) of the system. For a horizontal system the estimation error reduces to 1%, but increasing the range of leak elevations to a very high value (90 m) increased the error to almost 100%.

Further work was done to investigate the errors in the system FAVAD parameter estimates. One way of improving the accuracy of the estimate would be to include the unknown discharge coefficient in the FAVAD parameter estimation rather than assuming an average value in advance. This can be done by modifying Equations (6) and (7) to estimate the effective area (CdSA0S) and effective head-area slope (CdSmS) respectively. This approach may be particularly useful in field applications where the mean and range of discharge coefficients are unknown.

To investigate the effect of elevation differences on the system initial leakage area estimate, the absolute errors for the 300 random networks were calculated for the typical, low and very low pressure ranges in Table 1. These correspond to variations of ±10 m, ±5 m and 0 m (i.e. a horizontal system) respectively. Cumulative distributions of the absolute values of the errors are shown in Figure 4.
Figure 4

Cumulative error in the FAVAD system area for variable elevation differences.

Figure 4

Cumulative error in the FAVAD system area for variable elevation differences.

Figure 5

Relationship between the system leakage number LNS and N1 for 335 systems.

Figure 5

Relationship between the system leakage number LNS and N1 for 335 systems.

The results confirm the sensitivity of the system initial area error to the pressure range: the median error for the typical, low and very low (horizontal) pressure ranges were found to be 8.7%, 4.6% and 0.8% respectively. Using the effective system area produced only marginal improvements for the typical and low pressure ranges. However, for no elevation variation the error reduced to zero for all systems when using the effective area.

While significant errors in the initial system area occurs throughout the range modelled (Figure 1), it can be seen from Figure 2 that the error in the system head-area slope is only significant for small head-area slopes. Further analysis showed that for a system with typical pressure range the errors are generally below 10% when the total head-area slope is above 10−6 m and below 5% when the total head-area slope is above 10−5 m. For systems with head-area slopes below 10−7 m, the estimation error can be substantial.

The system with a small slope shows only a minor improvement in accuracy. However, for horizontal systems, the errors were substantially smaller with a median error below 3% even for systems with the lowest head-area slopes.

As with the system initial area error, using the equivalent head-area slope only produced marginal improvements in accuracy for the typical and small slopes, but reduced the error to zero for all horizontal systems.

While the errors in the system FAVAD parameters can be significant as discussed above, it should be noted that these errors are still small when compared with the range of values that the parameters may adopt: in this study the system initial areas and head-area slopes varied by three and six orders of magnitude respectively. Thus even in the worst cases, the system FAVAD parameters still provide good order-size estimates of the actual system values.

## LEAKAGE NUMBER

Working with individual leaks, Van Zyl & Cassa (2014) showed that there is a one-to-one relationship between the leakage number and N1 in the form:
9
or
It can be shown from these equations that N1 is equal to one when the leakage number is 1. Van Zyl & Cassa (2014) demonstrated that in practical terms N1 is 1.5 when the leakage number is greater than 100, and 0.5 when the leakage number is less than 0.01 (but greater than 0).
The leakage number concept was applied to the systems in this study by rewriting Equation (5) in the following form:
10
where LNS is the system leakage number. The physical meaning of the system leakage number is that it describes the ratio of the leakage flow through the expanded to the initial portions of all the leak openings in the system.

The relationship between the leakage number and N1 was found to be also valid for systems with many leaks as shown for the 335 systems used in this study in Figure 5. The figure shows that the data points plot on or close to the theoretical relationship (Equation (9)).

The two points on the left hand side of the figure both had very large leaks at elevations very different to the AZP. It suggests that the FAVAD equation may result in N1 values greater than 1.5 under certain conditions. However, this will require further investigations and falls outside the scope of this study.

## APPLICATION IN PRACTICE

The findings of this study have several implications for characterising the pressure-leakage relationship of systems in practice. The main assumption underlying the FAVAD concept is that individual leak areas, and thus also system leakage areas, expand linearly with pressure. Thus it is necessary to evaluate the validity of this assumption for real systems.

Changing the system pressure, and thus the stresses in the pipe wall, can only result in the leakage area deforming in one of the following ways:

• Negligible deformation (fixed areas)

• Elastic deformation

• Viscoelastic deformation

• Plastic deformation

• Fracture

The leakage areas of the first three categories have been shown to expand linearly with pressure (Cassa & Van Zyl 2013; Ssozi 2015) and will thus satisfy the requirement for FAVAD behaviour.

The last two categories are likely to result in non-linear area variations with pressure. However, both plastic deformation and fracture are unlikely to occur when the pressure is reduced, as in the case of pressure management. Consequently the assumption of linear pressure-area behaviour seems reasonable for real systems.

Applying the FAVAD concept to systems in practice is feasible since no additional data are required beyond that currently used to estimate N1. The benefit of the FAVAD approach is that the system initial area and head-area slope provide physically meaningful properties that are independent of pressure.

Knowing the initial leakage area and head-area slope for a system allows its leakage number to be calculated, which provides the ratio of leakage through the expanded to the initial portions of the leak areas. Variations in the leakage number with pressure can be directly calculated due to the inclusion of pressure in Equation (10). In addition, Equation (9) may be used to convert between the system leakage number and N1. This, in turn, may be used to estimate how N1 will vary with system pressure.

It is recommended that the effective initial leakage area and head-area slope are used in field applications, since this avoids errors introduced by assuming a leak discharge coefficient. The leakage number is not affected by this, and the initial leakage area can always be estimated at a later point by assuming an average leakage coefficient.

Finally, the results show that the accuracy of system FAVAD parameters are sensitive to the slope of the system and that horizontal systems will provide the most accurate values. However, even for a system with a significant slope the error is likely to be small compared to the range of possible values, and thus the system FAVAD parameters will still provide a good estimate of the state of the system.

## CONCLUSIONS

This study used a stochastic model of leaks in a typical pressure management area to investigate the application of the FAVAD equation for characterising the pressure-leakage response of systems with many leaks.

Applying the FAVAD equation to a system provides estimates of the total initial areas of all the leaks as well as the sum of all the head-area slopes in the system. These are physical properties of the network that are independent of pressure and can be used to evaluate and monitor the extent and type of leaks present.

FAVAD parameters are sensitive to the slope of the system, and horizontal systems will have the lowest errors. However, even for large slopes, the estimation errors will be small compared to the several orders of magnitude the FAVAD parameter values may adopt in practice. Thus the FAVAD parameters of the system will still provide a good estimate of the physical state of the system.

The FAVAD parameters can be used to calculate the system leakage number, which is the ratio of the leakage flow through the expanded to the initial portions of all the leak openings in the system. A simple equation (Equation (9)) may be used to convert between the system leakage number and N1.

## ACKNOWLEDGEMENTS

The authors would like to express their appreciation to Mr Allan Lambert for his invaluable assistance in defining realistic network leakage parameters and providing information on international applications and testing of the N1 power law and FAVAD equation since 1994.

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