1-De Paola, F. and Giugni, M.

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LEAKAGES AND PRESSURE RELATIONS: AN EXPERIMENTAL 
RESEARCH 
 Francesco De Paola and Maurizio Giugni
 
Department of Hydraulic Geotechnical and Environmental Engineering, University of Naples Federico II, 
via Claudio 21, 80125 Napoli. 
depaola@unina.it; giugni@unina.it 
Abstract 
Leaks in water systems are presently a frequent and increasing event that involves cost increase and poor 
service, not compliant to quality standards and modern management criteria. The most recent data available in 
Italy, resumed into the report issued by Control Committee for Water Resources Use ([1]), shows leakages with 
an average value of 37%. It is important, for maintenance perspective, to investigate occurrence and evolution 
of water leaks and the analytical links between the leaks and the network pressure, for a reliable calibration of 
water system quali-quantitative simulation models. The present work reports the results an experimental 
campaign started at Laboratory of Hydraulic of Department of Hydraulic, Geotechnical and Environmental 
Engineering of University of Naples Federico II in order to analyze the features of Qp(P) relation, which are 
compared with principal results issued in literature. 
Keywords 
Water leakages, pressure, experimental relations. 
1 INTRODUCTION 
Leaks in water systems are presently a frequent and increasing event causing cost increase and poor service 
levels which are not compliant to quality standards and modern management criteria. The most recent data 
available in Italy are resumed into the report issued by Control Committee for Water Resources Use [1] which 
shows leakages between 21% and 61%, with an average value around 37%. The recovery of part of these lost 
volumes would allow on one side to acquire immediately “new” resources and also doubtless economical 
benefits due to adduction, treatment and distribution cost savings. Losses reduction can obviously be achieved 
by well planned maintenance campaigns, both ordinary and extraordinary: they are, however, expensive actions 
to be undertaken from financial perspective and are often not rewarded by the economical value of the recovered 
resource. 
A different approach is to control network operation pressure by installing PRV (Pressure Reducing Valves) or 
realizing District Meter Areas (DMA). For this purpose it is important a research work in order to analyze the 
forming and evolution mechanisms of water leaks and the analytical links between the leaks Qp and the network 
pressure P. An experimental investigation on these topics has been started at the Hydraulic Laboratory of 
Department of Hydraulic, Geotechnical and Environmental Engineering of University of Naples Federico II [2], 
[3], whose results have been compared to the most interesting ones present in literature [4], [5], [6], [7], [8], [9], 
[10]. The present work provides further elements related to long lasting tests. 
2 EXPERIMENTAL SET-UP 
Experimental tests were carried out on the Hydraulic Laboratory high pressure circulation plant. The water 
discharge, detected by an electromagnetic meter, is led to a system of two parallel air vessels which allow to 
keep a constant pressure up to 10 bars, and from this to a cylindrical steel adductor (800 mm diameter) 
connected to the test plumbing (Figure 1). This latter in the first experimental tests was a steel pipe 100 mm 
nominal diameter (DN 100), 6.70 m long and flat; in the second experimental tests was a last generation cast iron 
pipe (BLUTOP - PAM Saint Gobain) nominal diameter (DN 125). 
At the downstream end of the experimental duct was fitted a rubber wedge sluice valve (Figure 1.), connected to 
a PVC 110 which collects discharge to a recirculation tank. At 3.35 m downstream of the adductor was 
installed a junction with an interception spherical valve at the end of which there is a nozzle used to simulate the 
occurrence of a leak (detail of Figure 1). 
 
 
 
 
Figure 1. Experimental installation layout. 
During the tests the water discharge flowing out of the nozzle is collected into a tank suitably calibrated, 
measuring filling time or, as alternative, is measured by a small size electromagnetic meter (DN25) (Figure 1). 
Pressure measure is run by a WIKA transducer with a 0-10 bar range and sample frequency equals to 10 Hz. For 
each test signals were acquired for a minimum of 60 s, calculating pressure average value. 
In the various experiments the following parameters were varied: 
1. feeding pressure, between 2 (min) and 7 (max) bars; 
2. water discharge, in a range between 3 and 45 l/s; 
3. nozzle geometry. In particular the presence of leaks with different shape and size was simulated by 
using suitable nozzles : round, rectangular, square, irregular (Figure 2); 
aiming to achieve, by tests analysis, water discharge and feeding pressure ratio. 
20.0 mm15.0 mm
12.0 mm9.0 mm6.0 mm3.0 mm
 
12.0 x 12.0 mm3.0 x 20.0 mm3.0 x 10.0 mm2.0 x 20.0 mm2.0 x 10.0 mm
0.0 mm 10 mm 20 mm 
IR-1
0.0 mm 10 mm 20 mm
IR-2
 
Figure 2. Test nozzles with different shape. 
 
 
Air vessels 
Measurement tank 
Cylindrical pipe (  800) 
Experimental pipe (  100 steel or  125 cast-iron) 
Experimental section 
Pump 
Collection tank 
Central tank 
3.0 mm 6.0 9.0 12.0 
2.0 x 10.0 2.0 x 20.0 3.0 x 10.0 3.0 x 20.0 12.0 x 12.0 mm 
 15.0 20.0mm 
3 EXPERIMENTAL RESULTS 
Experimental tests were run with an approach both static and dynamic, in order to simulate various operating 
conditions of a water system. The static tests simulate the night operating conditions (suitable to test more 
precisely leaks occurrence), while dynamic tests simulate day running. 
During each test, as mentioned before, pressure was detected at least for 60 s, in order to evaluate pressure 
variation features due to the leak. 
3.1 Steel pipe 
3.1.1 Static tests 
From theoretical perspective, the relation between water leaks (Qp) and pressure (P) is as follows : 
b
p PaQ 
 
(1) 
The experimental tests allowed to calculate for the various nozzle shapes, with a very high correlation rate 
(always above 0.99), the value of coefficient (a) and exponent (b) in equation (1), as reported in Figure. 3 and 
summarized in the tables. Coefficient (a) increases, in both cases, as obvious, according to nozzle size; (b) 
exponent, instead, is always little far from the theoretical value of 0.50. 
 
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020
A/Atubo
a (
b)
A= area della fessura
Atubo = area tubazione di studio.
b
p PaQ 
a
b
Valore medio: 0.511
B x H [mm] a b 
1 12.0 x 12.0 1.25 0.4929 
2 3.0 x 20.0 0.48 0.5171 
3 2.0 x 20.0 0.37 0.5184 
4 3.0 x 10.0 0.26 0.5127 
5 2.0 x 10.0 0.19 0.5053 
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.00 0.05 0.10 0.15 0.20 0.25
d/D
a (
b)
d = diametro della fessura;
D = diametro della condotta.
a
b
b
p PaQ 
Valore medio: 0.501
d [mm] a b 
1 20.0 2.83 0.4959 
2 15.0 1.52 0.4976 
3 12.0 0.99 0.4888 
4 9.0 0.58 0.4820 
5 6.0 0.25 0.4849 
6 3.0 0.06 0.5199 
 
 (a) (b) 
Figure 3. Outflow law coefficients according to: (a) surface ratio a/A (Anozzle/Apipe); (b) diameter ratio (d/D) 
(round nozzles). Static test. 
3.1.2 Dynamic tests 
Dynamic tests simulate, as already mentioned, the daily demand variation, by generating different dischargevalues (varying from about 3 to 45 l/s) detected by an electromagnetic meter (fig.1). 
In these tests, leaks tend to increase, as expected, as pressure and nozzle size increase. It is evident the 
tendency of b exponent to a constant value, close about to the theoretical value 0.50, also at dynamic test 
conditions. (Fig. 4). 
It is also interesting to see that coefficient (a) has an almost linear trend as surface ratio Anozzle/Apipe varies for 
nozzles having either square or rectangular section, while for round section it shows a parabolic trend 
according to d/D ratio (d=nozzle diameter; D=pipe diameter) (Figure. 4). 
0.511 
0.501 
a/A 
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
0.0000 0.0020 0.0040 0.0060 0.0080 0.0100 0.0120 0.0140 0.0160 0.0180 0.0200
Aorifizio/Atubo
a (
b)
b
p PaQ 
a
b
Valore medio: 0.479
Aorifizio= area della fessura
Atubo = area della tubazione 
 
Figure 4. Outflow law coefficients according to : (a) surface ratio a/A(Anozzle/Apipe) (rectangular nozzles) ; (b) 
diameters ratio (d/D) (round nozzles). Dynamic tests. 
 
 
3.2 Cast-iron pipe 
Experimental tests for cast-iron pipes, still in progress, were run with the same described procedures, but with 
longer duration (almost three hours), in order to analyze leaks variations as discharge and pressure vary in the 
feeding pipe. 
Figure 5 shows the close correlation existing between outflow discharge and pipe pressure. It is also shown in 
Figure 6, the time progress of flow and pressure for the 9 mm circular nozzle. On the same figure, the points 
given by (1) were also reported for different values of (a) and (b) extracted either through the "static" tests and 
by calibration with a specific genetic algorithm (using GanetXL software). Both models are effective in the 
calculation of losses, nevertheless, the model that best replicates experimental results is due to genetic algorithm. 
Moreover, we note that the calibration leads to a value of the exponent (b) 0.462, slightly lower than the 
theoretical value 0.50. 
 
0
1
2
3
4
5
6
7
0 1000 2000 3000 4000 5000 6000 7000 8000
t (s)
P 
(b
ar
)
Pressione Qpcal=a P[bar] 
b
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 1000 2000 3000 4000 5000 6000 7000 8000
t (s)
 Q
p 
(l/
s)
Qpmis Qpcal=a P[bar] 
b
 
 (a) (b) 
Figure 5. Trend of detected pressures (a) and outflow discharge (b) for a round nozzle (d= 9mm) for a 2 hours 
long test. 
0 
0.2 
0.4 
0.6 
0.8 
1 
1.2 
1.4 
1.6 
1.8 
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 
d/D 
a (b) 
Valore medio: 0.4953 
a 
b 
0.495 
0.479 
 
a/A 
0
1
2
3
4
5
6
7
0 1000 2000 3000 4000 5000 6000 7000 8000
t (s)
P 
(b
ar
), 
Q
p 
(l/
s)
Pressione Qpmis Qpcal (GanetXL - a=0.574, b=0.462) Qpcal(a=0.502, b=0.519)
Qpcal=a P[bar] 
b
 
Figure 6. Measured pressures and effluent flow rates for a circular nozzle (d = 9 mm) and their rate values 
calculated with relation (1) either with “static” tests (a = 0.502 and b = 0.519) or calibration of a genetic 
algorithm performed with experimental values derived from the long-term dynamic test (a = 0.574 and b = 
0.462). 
Such correlation Qp(P)can be useful to calculate probability distribution of x = Qp/P ratio and therefore to 
analyze leaks also from a probability perspective. In particular, the diagram of x ratio vs time (fig.7 (a)) shows 
variations between 0.20 and 2.5. In order to calculate the not overtaking probability starting from available data, 
it was calculated, by inferential analysis, the cumulative distribution function F(x) which resulted the two-
Parameter Gamma distribution [α=2.3041; β=0.3073 l/(s bar)]. Figure 7(b) shows that x values with the highest 
probability of not overtaking probability are the ones between 1 and 2, which means values referred to time span 
in which significant pressure gradients occur. As an example, for a value of 0.90 of the cumulative distribution 
function (90%), we get a value of x=1.3325. 
0
1
2
3
4
5
6
7
0 1000 2000 3000 4000 5000 6000 7000 8000
t [s]
P
Qp
x= Qp/P
Funzione di ripartizione
Campione Gamma
x
2.521.510.5
F(
x)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
 
 (a) (b) 
Figure 7. x=Qp/P (a) ratio and associated cumulative distribution function F(x) (b) ratio for a circular nozzle 
(d=9mm) in a 2 hours long test. 
4 CONCLUSIONS 
Starting from the first experiences run in UK [11], the relation between water leaks and pressure ratio was 
analyzed through the monomial equation (1). Some research works afterwords developed in Saudi Arabia [4], 
UK [12] and Iran [13] by the analysis of field data on different kind of pipes and different discharge range, 
confirmed such equation: available data are summarized in Table 1, which shows that exponent (b) resulted 
systematically higher than 0.50. 
)(
)(
)(
/





x
xF
 
Table 1 - Exponent (b) values as reported in literature 
Source b 
Thames Water ([12]) 1.00 
Inghilterra ([11]) 1.18 (average) 
Iran ([13]) 1.10 – 1.18 
Saudi Arabia([4]) 0.54 – 1.61 
The experiences run so far in Naples for metal pipes, with pressure varying within a wide range not investigated 
in other experimental works, have shown as follows : 
 the relation (1) allowed a reliable forecast of leaks vs pressure, whereas (a) and (b) coefficients were 
suitably defined; 
 (b) exponent was little far from the theoretical value of 0.50 both for round and rectangular nozzles; 
 (b) values resulted slightly dependent from test characteristics (static or long-term dynamic). 
Therefore, experimental data collected in Naples so far confirm the results achieved by other experiences for 
steel pipes with round nozzles [9], with reference to a range of pressures slightly smaller. 
The (b) values significantly higher than 0.50 reported in literature are, instead, related to steel pipes affected by 
corrosion and pipes made of asbestos concrete or plastic with longitudinal cracks. Probably they are due to a 
reduced consistence of material around the holes (due to corrosion) [7] or to the remarkable flexibility of plastic 
pipes [14], [15]. 
Further systematic experiences are in progress at the Hydraulic Laboratory of Naples on pipes of different 
material (PEad, PRFV), in order to acquire new elements aimed, in particular, to clarify the influence of the 
characteristics of the pipes and the features of the hole simulating the leak on the outflow mechanisms and, 
therefore, on the “leakage exponent” (b). 
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