EPANET is one of the most widely used software packages for water network hydraulic modelling, and is especially interesting for educational and research purposes because it is in the public domain. However, EPANET simulations are demand-driven, and the program does not include a specific functionality to model water leakage, which is pressure-driven. Consequently, users are required to deal with this drawback by themselves. As a general solution for this problem, this paper presents a methodology for including leakage in EPANET models by following a two-stage process. Firstly, leakage is spatially distributed among the nodes, according to the characteristics of the network. Secondly, leakage is modelled through an emitter at each node. The process is described in detail and two numerical examples illustrate the applicability and advantages of the method. In addition, free access through a URL is provided to the leakage modelling tool that has been developed.

## INTRODUCTION

*et al*. (2010). The information needed to build a network model is clear. It basically includes knowledge of all the features of the system's physical assets, as well as the operational conditions of the dynamic elements. Information about water flows and consumption provided by the customer meter management system enables estimates of the magnitude and location of demand. Flow meters located at the network inlet points register the input water, and so total network water losses can be estimated through a water balance (Lambert & Hirner 2000). One software tool for water network modelling, which is widely used today, is EPANET (EPA 2013). Apart from its calculation capabilities in daily network management, EPANET is of particular interest for university education and research purposes because it is public-domain software. It has already enabled results to be successively achieved and published (Walski & Draus 1995; Colombo & Karney 2002; Almandoz

*et al*. 2005; Hernández

*et al.*2010; Arunkumar & Mariappan 2011; García

*et al.*2012; Ameyaw

*et al*. 2013). However, one of the main characteristics of EPANET is that its hydraulic calculation engine is demand-driven. The implication is that the data on water output at each node, defined as base demand, are input information that is needed to run each simulation. Flows in pipes and more importantly, pressures at nodes, are the simulation results. Under such formulation, there is no functionality to represent leakage in an explicit, straightforward, and reliable manner. Users must rely on their own knowledge and experience to overcome this drawback. Classical works on hydraulic modelling (Germanopoulos 1985; Germanopoulos & Jowitt 1989) have already published methods to include the pressure-dependent nature of water leakage in a hydraulic model where

*q*is the leakage of pipe

_{i}*i*,

*β*is a leakage coefficient,

_{i}*L*is the length of the pipe, is the average pressure of the pipe, and

_{i}*α*is the leakage exponent. More recently, Giustolisi

_{i}*et al*. (2008) developed a new algorithm to represent leakage more accurately by integrating the algorithm used by EPANET (Todini & Pilati 1988) with a pressure-driven model. Other methods have been employed by professionals. The common characteristic among such methods is that leakage is modelled as an additional nodal demand. Total network leakage is determined based on the water balance and then distributed among the nodes. Hence, each node has at least two demands: consumption and leakage. A worthwhile improvement would be decoupling the leakage demand from the time-pattern ruling the consumption demand. Leakage demand could remain as a steady value, or better, it could be assigned a specific leakage time-pattern obtained from the network operation records. However, the main problem is still that leakage flow is modelled as a fixed and independent quantity, although it is a variable and pressure-dependent parameter.

## METHOD PROPOSED

In the authors’ opinion, the best way to represent leakage in a hydraulic network model is not by means of an additional demand, but rather by adding a leak valve to each node *j*, as illustrated in Figure 1, where *Q _{BDj}* is the node base demand (consumption),

*P*is the node pressure,

_{j}*K*is the leak valve coefficient and

_{j}*Q*is the leakage flow. Under this representation, the pressure at the node behaves as the leakage driving factor, and the flow leaked through the valve may be estimated accordingly (Al-Ghamdi 2011).

_{j}*Q*is the leakage flow rate at node

_{j}*j, P*is the pressure,

_{j}*K*is the emitter coefficient, and

_{j}*N*is the pressure exponent. This equation is consistent with the fixed and variable area discharges theory (May 1994; Thornton & Lambert 2005; Cassa & van Zyl 2013), where the pressure exponent mainly depends on the predominant pipe material (0.5 for metallic pipes, 1.2 or greater for plastic pipes, and about 1.0 for different materials more or less equally combined). In summary, the new approach begins by estimating the total network water losses, and then a two-stage method is used to model leakage. In the first stage, the spatial distribution of leakage is determined. In the second stage, the leakage is distributed among the nodes by calibrating the emitter parameters. The results are verified to ensure that the total amount of water leaked in the model is equal to that of the real network. A detailed description of the process is presented below.

### Stage 1 – spatial distribution of leakage

*et al*. (2008), the authors propose to characterise such influences in a more flexible way: each factor affecting pipe leakage is represented by a leak variable (

*γ*), not a pressure coefficient; and each pipe

*i*in the model is characterised by a particular leak variable value (

*γ*). Assuming that leakage is uniformly distributed along each pipe (Germanopoulos 1985), then each

_{i}*γ*will be equally divided between both pipe nodes. All the half

_{i}*γ*values are then assigned to every node

_{i}*j*from the pipes it connects, that is, where

*m*is the number of pipes connected to node

_{j}*j*. Finally, each

*γ*is normalised by where

_{j}*γ*

_{Net}is the total sum of all

*γ*. The Г

_{i}*thus obtained represents the relative importance of each node in terms of leakage, as compared to the whole of the network. In simple cases, for example, a small network whose pipes are homogeneous,*

_{j}*γ*could be just the pipe length. In networks that are not so homogeneous, the number of repairs per pipe length could be used in combination with the pipe length. It is a simple indicator that may include both reported breaks and unreported leaks (located through active leakage control). In cases where the available network maintenance records are sufficiently large and accurate, a more complex alternative consists of fitting the number of repairs per length in recent years to an exponential expression (Shamir & Howard 1979; Kleiner & Rajani 2002). Such an expression would provide the number of failures in recent years for each pipe or group of pipes, and this number could be considered, in combination with pipe length, as the leak variable for the purpose of distributing leakage. In more complex cases, other additional factors may be considered – such as diameters (Walski & Pelliccia 1982); age (Berardi

*et al*. 2005); service connections (Lambert & Hirner 2000); and material, soil, and water pressure (Kleiner & Rajani 2002). Therefore, the modeller may consider one leak variable

*γ*per factor

_{k}*k*and assign each to the pipes (

*γ*) and nodes (

_{k,i}*γ*) accordingly. The normalised values may then be calculated for each node (Γ

_{k,j}*) and, finally, they may be combined to obtain the lumped nodal value Γ*

_{k,j}*. This whole process is considered a multi-criteria decision problem and may require one of the available techniques, for example weighted sum model (Triantaphyllou 2002), analytical hierarchical process (Cabrera*

_{j}*et al*. 2011), fuzzy logic (Islam

*et al*. 2012; Parra

*et al*. 2013; Cavallo

*et al*. 2013), etc.

### Stage 2 – calibration of emitter coefficients at network nodes

*Q*

_{Net,real}is the real total network leakage over a period of 24 h, is the average pressure of the nodes, and

*N*is the pressure exponent. The second step is to distribute the network leakage coefficient among the nodes using the expression where

*K*

_{j}^{(h)}is the leak valve coefficient at node

*j*in iteration

*h*,

*K*

_{Net}

^{(h)}is the network leakage coefficient in iteration

*h*, and Γ

*is the normalised leak variable value for node*

_{j}*j*. The third step consists of simulating the network with the estimated emitter coefficients over a 24-h period. The final step is to calculate the simulated total network leakage and compare it to the known real total network leakage . where 0.005 is suggested as a general value for ɛ. Depending on the difference , the network leakage coefficient for the next iteration (

*K*

_{Net}

^{(h+1)}) is modified accordingly, and the sequence is repeated, from the second step on, until the convergence criteria is satisfied. An illustration of the iterative process is presented in Figure 2.

## NUMERICAL EXAMPLE 1 – USING EMITTERS

The network illustrated in Figure 3 supplies water to 25,000 people through 8,000 service connections. Table 1 shows the network characteristics. All pipes have a roughness of 0.1 mm. The total network water inflow is 5,100 m^{3}/day (59.01 L/s). Meters at the consumer sites enable calculation of base demand and daily consumption patterns for each node (Table 1, Table 2 and Figure 4); the total water demand computed through aggregation is 3,500 m^{3}/day (40.5 L/s). The total leakage estimate is 1,600 m^{3}/day (18.5 L/s) and is calculated using a simplified water balance (network input = consumption + leakage); therefore, the average water efficiency is 68.5%. There is a fairly balanced combination of pipe materials in the network and so the pressure exponent is assumed to equal 1.1. The network pressure is about 40 m on average and about 15 m during peak demand time.

Pipe ID | Length (m) | Diameter (mm) | Node ID | Elevation (m) | Consumption (L/s) | Time Pattern ID |
---|---|---|---|---|---|---|

L1 | 6,250 | 300 | N2 | 55 | 1.1 | PatA |

L2 | 3,100 | 200 | N3 | 27 | 4.0 | PatB |

L3 | 1,800 | 200 | N4 | 25 | 3.4 | PatC |

L4 | 3,800 | 200 | N5 | 12 | 3.4 | PatA |

L5 | 3,700 | 80 | N6 | 3 | 4.5 | PatD |

L6 | 2,200 | 100 | N7 | 12 | 5.6 | PatA |

L7 | 1,600 | 150 | N8 | 23 | 3.4 | PatC |

L8 | 4,600 | 150 | N9 | 12 | 5.0 | PatD |

L9 | 1,400 | 150 | N10 | 32 | 3.4 | PatE |

L10 | 2,600 | 100 | ||||

L11 | 3,600 | 80 | Reservoir | 100 | ||

L12 | 1,500 | 200 | ||||

L13 | 3,800 | 250 |

Pipe ID | Length (m) | Diameter (mm) | Node ID | Elevation (m) | Consumption (L/s) | Time Pattern ID |
---|---|---|---|---|---|---|

L1 | 6,250 | 300 | N2 | 55 | 1.1 | PatA |

L2 | 3,100 | 200 | N3 | 27 | 4.0 | PatB |

L3 | 1,800 | 200 | N4 | 25 | 3.4 | PatC |

L4 | 3,800 | 200 | N5 | 12 | 3.4 | PatA |

L5 | 3,700 | 80 | N6 | 3 | 4.5 | PatD |

L6 | 2,200 | 100 | N7 | 12 | 5.6 | PatA |

L7 | 1,600 | 150 | N8 | 23 | 3.4 | PatC |

L8 | 4,600 | 150 | N9 | 12 | 5.0 | PatD |

L9 | 1,400 | 150 | N10 | 32 | 3.4 | PatE |

L10 | 2,600 | 100 | ||||

L11 | 3,600 | 80 | Reservoir | 100 | ||

L12 | 1,500 | 200 | ||||

L13 | 3,800 | 250 |

Hour | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

PatA | 0.14 | 0.11 | 0.06 | 0.06 | 0.06 | 0.07 | 0.30 | 0.50 | 0.90 | 1.50 | 1.80 | 2.20 | 2.30 | 2.40 | 1.80 | 1.40 | 1.10 | 0.70 | 0.70 | 0.50 | 0.90 | 1.70 | 1.80 | 1.00 |

PatB | 0.08 | 0.07 | 0.04 | 0.04 | 0.04 | 0.05 | 0.21 | 0.55 | 1.06 | 1.81 | 2.17 | 2.55 | 2.67 | 2.49 | 1.72 | 1.37 | 1.38 | 0.91 | 0.78 | 0.46 | 0.67 | 1.13 | 1.16 | 0.62 |

PatC | 0.2 | 0.2 | 0.2 | 0.2 | 0.6 | 0.7 | 0.8 | 1.2 | 1.5 | 1.5 | 1.5 | 1.6 | 1.7 | 1.7 | 1.5 | 1.4 | 1.3 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 0.2 |

PatD | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.1 | 0.3 | 1.1 | 2.1 | 2.5 | 2.6 | 2.6 | 2.5 | 2.1 | 2.0 | 1.7 | 1.3 | 0.8 | 0.7 | 0.6 | 0.5 | 0.4 | 0.1 |

PatE | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 1.5 | 2.5 | 3.0 | 3.7 | 4.0 | 2.8 | 1.0 | 0.5 | 1.9 | 1.1 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Hour | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

PatA | 0.14 | 0.11 | 0.06 | 0.06 | 0.06 | 0.07 | 0.30 | 0.50 | 0.90 | 1.50 | 1.80 | 2.20 | 2.30 | 2.40 | 1.80 | 1.40 | 1.10 | 0.70 | 0.70 | 0.50 | 0.90 | 1.70 | 1.80 | 1.00 |

PatB | 0.08 | 0.07 | 0.04 | 0.04 | 0.04 | 0.05 | 0.21 | 0.55 | 1.06 | 1.81 | 2.17 | 2.55 | 2.67 | 2.49 | 1.72 | 1.37 | 1.38 | 0.91 | 0.78 | 0.46 | 0.67 | 1.13 | 1.16 | 0.62 |

PatC | 0.2 | 0.2 | 0.2 | 0.2 | 0.6 | 0.7 | 0.8 | 1.2 | 1.5 | 1.5 | 1.5 | 1.6 | 1.7 | 1.7 | 1.5 | 1.4 | 1.3 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 0.2 |

PatD | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.1 | 0.3 | 1.1 | 2.1 | 2.5 | 2.6 | 2.6 | 2.5 | 2.1 | 2.0 | 1.7 | 1.3 | 0.8 | 0.7 | 0.6 | 0.5 | 0.4 | 0.1 |

PatE | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 1.5 | 2.5 | 3.0 | 3.7 | 4.0 | 2.8 | 1.0 | 0.5 | 1.9 | 1.1 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

To illustrate the application of the method, leakage was modelled in two different ways, which were then compared. Model M1 is obtained by assuming the total network leakage (18.5 L/s) as an additional steady demand at each node (2.055 L/s) with no time pattern. Model M2 introduces leakage using the proposed method. Two numerical examples are provided below. In this first example, attention is only focused on the procedure for emitter calibration, and the only criterion considered for the spatial distribution of leakage is pipe length. Consequently, the value for *γ _{i}* for each pipe is simply the pipe length in metres. Table 3 shows the

*γ*for each node, as well as the resulting Г

_{j}_{j}.

Node | Length | K_{Net}^{(1)} | K_{j}^{(1)} | K_{Net}^{(9)} | K_{j}^{(9)} | |
---|---|---|---|---|---|---|

ID | γ (m) _{j} | Г_{j} | (L/s)/(m)^{1.1} | (L/s)/(m)^{1.1} | (L/s)/(m)^{1.1} | (L/s)/(m)^{1.1} |

N2 | 10,400 | 0.260 | 0.319 | 0.239 | 0.249 | 0.065 |

N3 | 2,450 | 0.061 | 0.319 | 0.056 | 0.249 | 0.015 |

N4 | 4,100 | 0.103 | 0.319 | 0.094 | 0.249 | 0.026 |

N5 | 3,750 | 0.094 | 0.319 | 0.086 | 0.249 | 0.023 |

N6 | 4,750 | 0.119 | 0.319 | 0.109 | 0.249 | 0.030 |

N7 | 1,900 | 0.048 | 0.319 | 0.044 | 0.249 | 0.012 |

N8 | 3,850 | 0.096 | 0.319 | 0.089 | 0.249 | 0.024 |

N9 | 3,000 | 0.075 | 0.319 | 0.069 | 0.249 | 0.019 |

N10 | 5,750 | 0.144 | 0.319 | 0.132 | 0.249 | 0.036 |

Node | Length | K_{Net}^{(1)} | K_{j}^{(1)} | K_{Net}^{(9)} | K_{j}^{(9)} | |
---|---|---|---|---|---|---|

ID | γ (m) _{j} | Г_{j} | (L/s)/(m)^{1.1} | (L/s)/(m)^{1.1} | (L/s)/(m)^{1.1} | (L/s)/(m)^{1.1} |

N2 | 10,400 | 0.260 | 0.319 | 0.239 | 0.249 | 0.065 |

N3 | 2,450 | 0.061 | 0.319 | 0.056 | 0.249 | 0.015 |

N4 | 4,100 | 0.103 | 0.319 | 0.094 | 0.249 | 0.026 |

N5 | 3,750 | 0.094 | 0.319 | 0.086 | 0.249 | 0.023 |

N6 | 4,750 | 0.119 | 0.319 | 0.109 | 0.249 | 0.030 |

N7 | 1,900 | 0.048 | 0.319 | 0.044 | 0.249 | 0.012 |

N8 | 3,850 | 0.096 | 0.319 | 0.089 | 0.249 | 0.024 |

N9 | 3,000 | 0.075 | 0.319 | 0.069 | 0.249 | 0.019 |

N10 | 5,750 | 0.144 | 0.319 | 0.132 | 0.249 | 0.036 |

Considering the average network pressure of 40 m, *K*_{Net} has an initial value of = 18.5/(40)^{1.1} = 0.319. The first value for each emitter coefficient () is shown in Table 3. After the iterative process (nine iterations in this case), the final values were obtained for the emitter coefficients () as presented in Table 3.

### Model results comparison under usual operating conditions

One of the key variables to assess the validity of a hydraulic model is the network pressure after a simulation. Table 4 shows nodal pressures for low, mean, and peak consumption hours. However, as consumption increased from 9:00 to 13:00 the differences became wider, and eventually, unacceptable negative values were obtained at 13:00 with M1. Pressure results provided by M2 stay positive at that time and sufficiently close to the average 15 m pressure level.

Node | M1 9:00 | M2 9:00 | M1 11:00 | M2 11:00 | M1 13:00 | M2 13:00 |
---|---|---|---|---|---|---|

N2 | 29.9 | 30.0 | 8.0 | 14.3 | − 1.1 | 6.3 |

N3 | 50.0 | 50.5 | 17.2 | 26.8 | 2.7 | 13.8 |

N4 | 49.6 | 50.0 | 13.7 | 24.1 | − 2.3 | 9.7 |

N5 | 59.5 | 59.7 | 16.8 | 28.3 | − 2.7 | 10.4 |

N6 | 65.4 | 64.3 | 13.0 | 26.3 | − 4.6 | 12.1 |

N7 | 62.5 | 63.0 | 23.9 | 34.8 | 8.1 | 20.7 |

N8 | 55.9 | 56.0 | 25.7 | 34.3 | 13.3 | 23.3 |

N9 | 68.7 | 68.8 | 39.5 | 47.4 | 28.5 | 37.7 |

N10 | 48.5 | 48.5 | 20.2 | 28.2 | 8.4 | 17.6 |

Node | M1 9:00 | M2 9:00 | M1 11:00 | M2 11:00 | M1 13:00 | M2 13:00 |
---|---|---|---|---|---|---|

N2 | 29.9 | 30.0 | 8.0 | 14.3 | − 1.1 | 6.3 |

N3 | 50.0 | 50.5 | 17.2 | 26.8 | 2.7 | 13.8 |

N4 | 49.6 | 50.0 | 13.7 | 24.1 | − 2.3 | 9.7 |

N5 | 59.5 | 59.7 | 16.8 | 28.3 | − 2.7 | 10.4 |

N6 | 65.4 | 64.3 | 13.0 | 26.3 | − 4.6 | 12.1 |

N7 | 62.5 | 63.0 | 23.9 | 34.8 | 8.1 | 20.7 |

N8 | 55.9 | 56.0 | 25.7 | 34.3 | 13.3 | 23.3 |

N9 | 68.7 | 68.8 | 39.5 | 47.4 | 28.5 | 37.7 |

N10 | 48.5 | 48.5 | 20.2 | 28.2 | 8.4 | 17.6 |

As water consumption (average and time pattern) and water losses (average) are the same in both models, the only reason that explains the worse performance of M1 is the time evolution of water leakage. Figure 5 shows the time evolution of water flows in both models. The dynamic behaviour of leakage is clearly appreciated in the case of M2. In contrast, the constant leakage flow rate of M1 is the cause of higher values of flow rates in the network at peak time, and the resulting negative pressure values were obtained.

At this stage, M1 would require further improvements to overcome the negative pressure problem. Such improvements would possibly consist of using an artificial time pattern that represents the leakage variation. Conversely, the calibrated emitters in M2 have already solved the problem in a consistent way, either in terms of hydraulic behaviour or well-adjusted water balance. If further calibration was required for M2, it would be quicker and more straightforward to implement than the calibration required for M1.

### Model results comparison under an unusual operating condition

Unusual operating conditions could be difficult to foresee, and when simulated, they may make models produce results that are not always valid. Single pipe closing was tested for M1 and M2. A 24-hour simulation was performed after closing each single pipe of the network. Assuming that water consumption is unaffected by the consequences of such closings, node pressures after the simulations were examined in search of a negative value at any hour. The results are summarised in Table 5 as the number of nodes with negative pressure values after each pipe shut-off. If that number is greater than zero, the model does not pass the pipe-closing test. The conclusion is that M2 is sufficiently robust for up to four pipes in the pipe closing test (L9, L10, L8 and L5), whereas M1 fails in all cases.

Pipe | M1 | M2 |
---|---|---|

L4 | 17 | 17 |

L2 | 17 | 13 |

L7 | 14 | 12 |

L3 | 13 | 11 |

L13 | 8 | 6 |

L12 | 7 | 4 |

L6 | 7 | 4 |

L11 | 3 | 1 |

L9 | 4 | 0 |

L10 | 3 | 0 |

L8 | 3 | 0 |

L5 | 3 | 0 |

Pipe | M1 | M2 |
---|---|---|

L4 | 17 | 17 |

L2 | 17 | 13 |

L7 | 14 | 12 |

L3 | 13 | 11 |

L13 | 8 | 6 |

L12 | 7 | 4 |

L6 | 7 | 4 |

L11 | 3 | 1 |

L9 | 4 | 0 |

L10 | 3 | 0 |

L8 | 3 | 0 |

L5 | 3 | 0 |

## NUMERICAL EXAMPLE 2 – SPATIAL DISTRIBUTION OF LEAKAGE

To illustrate the procedure for the spatial distribution of leakage, the network information was enhanced. In addition to pipe length, the number of service connections and failures were considered as leakage factors in the analysis. Table 6 shows the values of the coefficients *γ _{k},_{i}* which were assigned to each leakage factor and pipe. While the number of service connections has simple direct coefficient values, the number of failures per year was considered in a more comprehensive way. The reason is that an oversimplification, such as taking only the number of failures in the past year, may lead to biased results due to particular circumstances that may have occurred during such a year. In this example, we use an exponential model representing the relationship between number of failures per km-year and age in years (Shamir & Howard 1979; Kleiner & Rajani 2002). We assume the availability of historical records to construct three equations corresponding to pipes grouped by age. The values obtained from the equations are multiplied by pipe length to obtain the leakage coefficient for failures.

Length (m) | Serv. Con. | Age | Fail. Equation | Fail./ | Fail./year | |
---|---|---|---|---|---|---|

Pipe | γ_{1,i} | γ_{2,i} | (years) | (Fail./(km·year)) | (km·year) | γ_{3,i} |

L1 | 6,250 | 460 | 40 | 0.15 e^{0.081 t} | 3.83 | 23.9 |

L2 | 3,100 | 1,200 | 40 | 3.83 | 11.9 | |

L3 | 1,800 | 900 | 35 | 2.55 | 4.6 | |

L10 | 2,600 | 950 | 35 | 2.55 | 6.6 | |

L13 | 3,800 | 900 | 35 | 2.55 | 9.7 | |

L4 | 3,800 | 540 | 20 | 0.29 e^{0.063 t} | 1.02 | 3.9 |

L5 | 3,700 | 500 | 20 | 1.02 | 3.8 | |

L9 | 1,400 | 500 | 20 | 1.02 | 1.4 | |

L11 | 3,600 | 500 | 20 | 1.02 | 3.7 | |

L8 | 4,600 | 550 | 10 | 0.21 e^{0.048 t} | 0.34 | 1.6 |

L12 | 1,500 | 550 | 10 | 0.34 | 0.5 | |

L6 | 2,200 | 330 | 5 | 0.27 | 0.6 | |

L7 | 1,600 | 120 | 5 | 0.27 | 0.4 |

Length (m) | Serv. Con. | Age | Fail. Equation | Fail./ | Fail./year | |
---|---|---|---|---|---|---|

Pipe | γ_{1,i} | γ_{2,i} | (years) | (Fail./(km·year)) | (km·year) | γ_{3,i} |

L1 | 6,250 | 460 | 40 | 0.15 e^{0.081 t} | 3.83 | 23.9 |

L2 | 3,100 | 1,200 | 40 | 3.83 | 11.9 | |

L3 | 1,800 | 900 | 35 | 2.55 | 4.6 | |

L10 | 2,600 | 950 | 35 | 2.55 | 6.6 | |

L13 | 3,800 | 900 | 35 | 2.55 | 9.7 | |

L4 | 3,800 | 540 | 20 | 0.29 e^{0.063 t} | 1.02 | 3.9 |

L5 | 3,700 | 500 | 20 | 1.02 | 3.8 | |

L9 | 1,400 | 500 | 20 | 1.02 | 1.4 | |

L11 | 3,600 | 500 | 20 | 1.02 | 3.7 | |

L8 | 4,600 | 550 | 10 | 0.21 e^{0.048 t} | 0.34 | 1.6 |

L12 | 1,500 | 550 | 10 | 0.34 | 0.5 | |

L6 | 2,200 | 330 | 5 | 0.27 | 0.6 | |

L7 | 1,600 | 120 | 5 | 0.27 | 0.4 |

Table 7 shows the results of the spatial distribution of leakage to the network nodes. The three leakage factors were considered of equal importance, hence the final combined coefficient Г* _{j}* was obtained by a simple average. The coefficient represents each node's spatial incidence in terms of leakage. Finally, emitter coefficients

*K*were obtained, after an iterative process (six iterations) as described above. Differences can be easily noticed between those

_{j}*K*which were based on three criteria and previous examples (Table 3) that were based on only one criterion. However, as the network is simplified, the aim of the second example is not to decide whether to consider more criteria or not, but to illustrate the application of the method when more infrastructure data are available.

_{j}Node | Length (m) γ_{1,j} | Г_{1,j} | Serv. Con. γ_{2,j} | Г_{2,j} | Fail./year γ_{3,j} | Г_{3,j} | Г_{j} | K_{Net}^{(6)} (L/s)/(m)^{1.1} | K (L/s)/(m)_{j}^{(6)}^{1.1} |
---|---|---|---|---|---|---|---|---|---|

N2 | 10,400 | 0.261 | 1,370 | 0.171 | 31.9 | 0.439 | 0.290 | 0.238 | 0.069 |

N3 | 2,450 | 0.061 | 1,050 | 0.131 | 8.2 | 0.113 | 0.102 | 0.238 | 0.024 |

N4 | 4,100 | 0.103 | 1,200 | 0.150 | 6.4 | 0.088 | 0.114 | 0.238 | 0.027 |

N5 | 3,750 | 0.094 | 925 | 0.116 | 8.2 | 0.113 | 0.107 | 0.238 | 0.025 |

N6 | 4,750 | 0.119 | 995 | 0.124 | 7.1 | 0.097 | 0.113 | 0.238 | 0.027 |

N7 | 1,900 | 0.048 | 520 | 0.065 | 3.8 | 0.053 | 0.055 | 0.238 | 0.013 |

N8 | 3,850 | 0.096 | 665 | 0.083 | 2.9 | 0.040 | 0.073 | 0.238 | 0.017 |

N9 | 3,000 | 0.075 | 500 | 0.063 | 2.6 | 0.035 | 0.058 | 0.238 | 0.014 |

N10 | 5,750 | 0.144 | 775 | 0.097 | 1.5 | 0.021 | 0.087 | 0.238 | 0.021 |

Node | Length (m) γ_{1,j} | Г_{1,j} | Serv. Con. γ_{2,j} | Г_{2,j} | Fail./year γ_{3,j} | Г_{3,j} | Г_{j} | K_{Net}^{(6)} (L/s)/(m)^{1.1} | K (L/s)/(m)_{j}^{(6)}^{1.1} |
---|---|---|---|---|---|---|---|---|---|

N2 | 10,400 | 0.261 | 1,370 | 0.171 | 31.9 | 0.439 | 0.290 | 0.238 | 0.069 |

N3 | 2,450 | 0.061 | 1,050 | 0.131 | 8.2 | 0.113 | 0.102 | 0.238 | 0.024 |

N4 | 4,100 | 0.103 | 1,200 | 0.150 | 6.4 | 0.088 | 0.114 | 0.238 | 0.027 |

N5 | 3,750 | 0.094 | 925 | 0.116 | 8.2 | 0.113 | 0.107 | 0.238 | 0.025 |

N6 | 4,750 | 0.119 | 995 | 0.124 | 7.1 | 0.097 | 0.113 | 0.238 | 0.027 |

N7 | 1,900 | 0.048 | 520 | 0.065 | 3.8 | 0.053 | 0.055 | 0.238 | 0.013 |

N8 | 3,850 | 0.096 | 665 | 0.083 | 2.9 | 0.040 | 0.073 | 0.238 | 0.017 |

N9 | 3,000 | 0.075 | 500 | 0.063 | 2.6 | 0.035 | 0.058 | 0.238 | 0.014 |

N10 | 5,750 | 0.144 | 775 | 0.097 | 1.5 | 0.021 | 0.087 | 0.238 | 0.021 |

## AN AVAILABLE SOFTWARE TOOL

Undertaking the iterative leakage calibration process in practical cases with only the help of EPANET and several other software tools (databases and spreadsheets) may involve a significant amount of cumbersome and tedious work for engineers or utility staff. The authors have often applied the methodology presented above by means of case-specific programming combined with the EPANET Toolkit. The authors eventually decided to prepare one general application, and like EPANET, make it freely available on the web. At the moment, a beta version of the application has been released and can be downloaded from http://efficient.ita.upv.es/downloads/Instal_ITAfugas.zip.

## CONCLUSION

This paper presents a method to assist modellers in the task of simulating leakage in water distribution networks. The advantages of this method are that it enables a spatial distribution of leakage throughout the network and the adaptation of EPANET's use of emitters (mainly intended for nozzles or sprinklers) for modelling leakage in a hydraulically consistent manner. Furthermore, the spatial distribution is flexible enough to take into account and combine different leakage factors that the modeller may find relevant in the network and in the manner considered most appropriate. The calibration of the emitter coefficients is solved using a straightforward iterative method and a software tool is provided as an additional aid.

## ACKNOWLEDGMENT

The authors wish to thank the reviewers for their accurate comments and suggestions, which have definitely contributed to improving the paper.