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ScienceDirect Procedia Engineering 70 (2014) 1513 – 1517

12th International Conference on Computing and Control for the Water Industry, CCWI2013

Implications of the known pressure-response of individual leaks for whole distribution systems J. Schwallera, J.E. van Zylb* b

a BIT Consult GmbH – Miltner AG, Am Storrenacker 1b, 76139, Karlsruhe, Germany University of Cape Town, Private Bag X3, Rondebosch, Cape Town, 7711, South Africa

Abstract Previous experimental and numerical studies have shown that leaks in water distribution systems expand linearly with pressure under elastic conditions. The flow rate from such leaks as a function of the pressure head is described by FAVAD equation. The aim of this study was to investigate whether a combination of individual leaks displaying FAVAD behaviour can explain the typical range of leakage exponents found for distribution systems in field studies. For this purpose, a spreadsheet model was developed with typical distributions of leak quantities, areas, discharge coefficients and head-area slopes. Using a repeatability analysis, it was found that individual leaks displaying FAVAD behaviour can indeed explain the most commonly used leakage exponent of around 1 as well as the observed variation in leakage exponents between 0.5 and 1.5. © Ltd. ©2013 2013The TheAuthors. Authors.Published PublishedbybyElsevier Elsevier Ltd. Open access under CC BY-NC-ND license. Selectionand andpeer-review peer-reviewunder underresponsibility responsibility CCWI2013 Committee Selection ofof thethe CCWI2013 Committee. Keywords: Leakage; Pressure; FAVAD; Leakage Exponent; Modelling

1. Introduction Pressure management is frequently used in practice as a water loss control strategy in water distribution systems (Farley & Trow, 2003). Results from various field studies have shown that leakage from distribution systems is often considerably more sensitive to changes in pressure than the Torricelli orifice equation predicts. If leaks behaved as conventional orifices, the leakage flow rate will be proportional to the square root of the pressure, i.e.

* Corresponding author. Tel.: +27 21 650 2584; fax: +27 21 689 7471. E-mail address: [email protected].

1877-7058 © 2013 The Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of the CCWI2013 Committee doi:10.1016/j.proeng.2014.02.166

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J. Schwaller and J.E. van Zyl / Procedia Engineering 70 (2014) 1513 – 1517

have a leakage exponent of 0.5. However, field studies have reported leakage exponents ranging between 0.36 and 2.95 (Wu, et al., 2011), with the most commonly used leakage exponents of 1.0 (Lambert, 2000) to 1.15 (Ogura, 1979). The aim of this study was to develop a spreadsheet model of the leaks occurring in a typical distribution system and use this model to investigate the combined behavior of leaks in the system with variations in pressure. In particular, the study aimed to determine whether such a model could explain the range and typical values of leakage exponents found in field studies. Since leakage flow was assumed to be small compared to peak demands, system hydraulics were not considered. The study was limited to static pressures and to elastic leaks. Ferrante et al (2011) showed that permanent deformation and hysteresis can occur at leaks in plastic pipes, but these phenomena were not considered in this study. 2. Background A leak in a pipe can be considered as an orifice, for which the flow rate is described as a function of the orifice area A and pressure head h by the Torricelli equation:

Q = Cd A 2 gh

(1)

where Cd is the discharge coefficient and g is acceleration due to gravity. While the Torricelli equation predicts leakage to be proportional to the square root of pressure, field tests found this relationship to be unsuitable for describing system pressure-leakage response, leading to the adoption of a more general equation in the form:

Q = Ch N 1

(2)

where Q is the leakage flow rate, C the leakage coefficient and N1 the leakage exponent. Leakage exponents are mostly not published directly, but obtained from indirect sources, such as the ranges in Table 1 obtained from Farley et al. (2003).

Table 1. Values of N1 determined from sector tests in three different countries (adapted from Farley and Trow,2003) Country

Number of sectors tested

Mean value of N1

Range of value of N1

UK (1977)

17

1.13

0.70 to 1.68

Japan (1979)

8

1.15

0.63 to 2.12

Brazil (1998)

13

1.15

0.52 to 2.79

While the leakage exponents in Table 1 vary between 0.36 and 2.95, the vast majority of leakage exponents occur between 0.5 and 1.5. For instance, in 75 tests on district meter areas (DMA) in the UK, 90 % of the leakage exponents were between 0.5 and 1.5, and a Japanese study by Ogura (1979) found 7 out of 8 successful field tests had leakage exponents in this range. In 1994, May (1994), introduced the FAVAD (Fixed And Variable Discharge) concept by assuming that some leaks are rigid, while others will expand with increasing pressure. May’s assumption that leak area is a linear function of pressure was confirmed by Cassa et al (2010), in a finite element study on the behavior of holes and longitudinal, circumferential and spiral cracks in different pipe materials (uPVC, cast iron, steel and asbestos cement) under two loading states. They concluded that elastically deforming leaks areas vary linearly with pressure

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and that the leak-pressure response of a leak can thus be characterized by its initial area A0 and head-area slope m in the form:

A = A0 + mh

(3)

Replacing this function into Equation 1 results in the FAVAD equation:

(

Q = Cd 2 g A0 h 0.5 + mh1.5

)

(4)

The first term of the equation describes leakage through the original area of the leak, while the second part describes the leakage through the expanding part of the leak. 3. Leakage Model The study was based on a spreadsheet model consisting of different numbers of leaks at random positions and with random parameters. This model implemented a small variation in system pressure at source, and then used the total leakage before and after the change in pressure to estimate the leakage exponent in Equation 2. This procedure reproduces the night tests used in the practice: input flow rate at night minus the known consumption is considered to be the leakage flow rate. Then, the pressure head is dropped in the system, a new leakage flow rate is calculated and the leakage exponent can be determined. An effort was made to set up a model with individual leak parameters variations resembling those of real systems. Since limited published information is available on individual leak parameters in distribution systems, specialist input and assumptions were used in this process. To model a system with many leaks, it was necessary to estimate the distributions and ranges of individual leak parameters in a typical system. The following forms were adopted for the different parameters: • The discharge coefficient Cd was modeled using a normal distribution. • The initial leak area A0 was modeled using separate normal distributions for background and bursts. • The head-area slope m was modeled as a general power function of the leak area based on (Cassa & van Zyl, Predicting the Head-Area Slopes and Leakage Exponents of Cracks in Pipes, 2011). • The pressure head h was modeled with a uniform distribution with a mean and range. 4. Results To determine the range of leakage exponents the model of individual leaks will produce in water distribution systems, three networks with 100, 1 000, and 10 000 leaks respectively were used as basis. One hundred randomized networks were then generated for each and the resulting leakage exponents analyzed. The results of the analyses are shown in Figure 1 and summarized in Table 2.

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('' $"!#%

'&*'

(''' $"!#% ('''' $"!#%

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Fig. 1. Leakage exponents found for 100 random instances of a distribution system with 100, 1000 and 10000 leaks.

Table 2. Leakage exponents for 100 random networks with 100, 1 000 and 10 000 leaks respectively. Number of system leaks

Mean N1

Range of N1

100 leaks

0,66

0,46 to 1,67

1000 leaks

0,92

0,46 to 1,59

10 000 leaks

1,08

0,81 to 1,26

The resulting networks displayed leakage exponents between 0.46 and 1.67, with the vast majority of values lying between 0.5 and 1.5. This compares reasonably well with the ranges of field studies as summarized in Table 1, thus showing that the combined effect of individual elastically deforming leaks can indeed produce the typical range of leakage exponents found in field studies. The results also show trends of reducing range and increasing mean leakage exponent with an increased number of system leaks. The average leakage exponent for a system with 10 000 leaks is close to the commonly used mean leakage exponent of 1.15. The system with 100 leaks shows a large fraction of low leakage exponents caused by small leaks in the absence of large leaks. 5. Conclusion This study modeled a hypothetical water distribution system with randomly distributed leaks. Individual leaks were modeled with the FAVAD equation that assumed a linear head-area relationship. The elevations, initial areas and discharge coefficients of individual leaks were determined from random distributions based on assumptions and specialist input. The study found that the combined behavior of individual leaks expanding elastically can explain the findings of pressure management field studies.


References Cassa, A. & van Zyl, J., 2011. Predicting the Head-Area Slopes and Leakage Exponents of Cracks in Pipes. University of Exeter, UK, s.n. Cassa, A., van Zyl, J. & Laubscher, R., 2010. A Numerical Investigation into the Effect of Pressure on Holes and Cracks in Water Supply Pipes. Urban Water Journal, 12 Febraury. Farley, M. & Trow, S., 2003. Losses in Water Distribution Network. London: IWA Publishing. Ferrante, M., Massari, C., Brunone, B., & Meniconi, S. (2011, November). Experimental Evidence of Hysteresis. Journal of Hydraulic Engineering(137-7), pp. 775-780. Greyvenstein, B. & van Zyl, J. E., 2007. An Experimental Investigation into the Pressure-Leakage Relationship of some Failed Water Pipes. Journal of Water Supply: Research and Technology—AQUA, 56(2). Hiki, S., 1981. Relationship between Leakage and Pressure. Japan Waterworks Association Journal, May, pp. 50-54. Lambert, A., 2000. What Do We Know About Pressure: Leakage Relationship in Distribution System?. Brno, Czech Republic, IWA. May, J. H., 1994. Leakage, Pressure and Control. The SAS Portman Hotel, s.n. Ogura, 1979. Experiment on the Relationship between Leakage and Pressure. Japan Water Works Association, June, pp. 38-45. Parry, J., 1881. Water: Its Composition, Collection ad Distribution. London: Frederick Warne And Co.. Wu, Z. Y. et al., 2011. Water Loss Reduction. Exton, Pennsylvania: Bentley Institute Press.

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