Pathogen intrusion may occur in water pipes when negative pressures allow external flows to enter through failures or leaks and then mix with safe water. Based on the Fixed and Variable Area Discharge (FAVAD) theory and the orifice equations, an analysis is proposed to estimate the intrusion flow across defects in pipes considering different types of failure. The equivalent diameter of different round hole failures was considered in order to obtain the dimensions of the split failures that presented the same pressure drop. In addition, experimental scenarios were made with external porous media to model the intrusion flow in buried pipes. An inverse method for the orifice equation is proposed to obtain the intrusion flows generated by the variations of two section failures produced by the pressure drop inside the pipe. The orifice equation properly represents the intrusion flow by adjusting the discharge coefficient. Furthermore, the considerable variations in the failures area with negative pressures should be taken into consideration in the expressions that estimate the intrusion flow.

## ABBREVIATIONS

*A*area

*C*_{d}discharge coefficient

*C*_{di}intrusion discharge coefficient

*d*_{eq}equivalent diameter

*g*gravitational acceleration

*h*head over the orifice

*N1*exponent on equivalent FAVAD equation

*P*_{w}wetted perimeter of the orifice

*q*_{L}leakage flow

*Q*flow

- Re
Reynolds number

*S*system expansion coefficient

*v*flow velocity

*v*_{intrusion}intrusion velocity

*Δh*pressure head for intrusion flow

## INTRODUCTION

*et al.*2002; Karim

*et al.*2003; LeChevallier

*et al.*2003; Friedman

*et al.*2004; Pedley & Bartram 2004; Van Lieverloo

*et al.*2006; Ramalingam

*et al.*2009; Besner

*et al.*2010, 2011; Yang

*et al.*2011; Ebacher

*et al.*2012; Lee

*et al.*2012).

The failures in pipes depend on the pipe material, its age, the soil surrounding the pipe, water quality, water pressure and the pipe workmanship. The consequence of a failure is either a leak or an intrusion flow. The majority of the literature focuses on leaks. Pipe failures occur when environmental and operating conditions jeopardise the pipe structural integrity through corrosion, degradation, poor installation or manufacturing defects (Almeida & Ramos 2010). Rajani & Klein (2001) classified the types of failure into three categories: (1) circumferential breaks caused by longitudinal tension; (2) longitudinal breaks caused by cross-sectional tension (radial tension); and (3) cracks in joints caused by a cross-sectional tension. This classification must be complemented with additional breaks, such as holes due to corrosion (Mora-Rodríguez *et al.* 2011). Several authors (Rajani *et al.* 1996; Makar 2000; Lambert 2001; Rajani & Kleiner 2001; Hu & Hubble 2007; Silva *et al.* 2007; Clair & Sinha 2012; Haghighi *et al.* 2012) have studied the types of failure and their mechanisms.

The Fixed and Variable Area Discharge (FAVAD) theory and its variations have recently become one of the most commonly used theories to model pipe leakage (May 1994; Lambert 2001; Cassa & Van Zyl 2013, 2014; Van Zyl & Cassa 2014). Since the 1990s, diverse values of leakage flow at corresponding pressure have been found for different types of failure using this theory with diverse pipe materials such as metal and plastic, both in the laboratory and data from water district systems (Lambert 2001; López *et al.* 2007; Greyvenstein & Van Zyl 2007; Ávila & Saldarriaga 2011).

Intrusion has not received as much attention in the literature as leakage (Kirmeyer & Martel 2001; LeChevallier *et al.* 2003; Boyd *et al.* 2004a, b; McInnis 2004; López *et al.* 2008; Mora-Rodríguez *et al.* 2012, 2013, 2014; Yang *et al.* 2014). Authors such as Walski *et al.* (2006) and Collins & Boxall (2012) assert that the orifice equation quantifies the leakage and intrusion flows related to pipe failures, due to the relationship between flow and pressure through the orifice. Low or negative pressures in a distribution network are mainly caused by uncontrolled pump shutdowns, inadequate valve operation, network maintenance and the use of hydrants (Friedman *et al.* 2004).

A physical model of intrusion flow has been developed through different types of failure to evaluate the relationship between the intrusion flow and the head loss through the orifice. Owing to the pressure variation, the main failure section also varies, affecting the coefficients in FAVAD and orifice equations. Furthermore, the pathogen intrusion in porous soil is a little-studied topic in terms of experiments (Collins & Boxall 2012; Yang *et al.* 2014). In this paper, the experiments of the intrusion flow with and without porous media around pipes are analysed in order to better understand this phenomenon.

## TYPES OF FAILURE

This section describes the representative types of failure in pipes used in the model: (1) round hole, (2) longitudinal split and (3) circumferential break (Rajani & Kleiner 2001).

### Round holes

*Q*is the flow,

*A*is the area of orifice,

*C*

_{d}is the discharge coefficient,

*g*is the gravitational acceleration and

*h*is the head over the orifice. The variables

*Q*,

*h*and

*C*

_{d}are obtained from diverse considerations related to the pipes and failures.

*N1*equation (Equation (3)) that is a generalisation of the orifice equation. The

*N1*equation is obtained by fitting to data points before and after pressure management and then dividing one equation by the other. This equation is the most commonly used equation that relates the leak flow with pressure in the leak zone: where

*N1*is an exponent that may vary between 0.50 and 2.50, depending on the type of leak. The leakage flow (

*Q*and

_{1}*Q*) and the pressure head inside the pipe (

_{0}*h*and

_{1}*h*) are the data points before and after pressure management.

_{0}Similarly, the *N1* equation applied for longitudinal splits in plastic pipes postulates that the orifice area varies with pressure (Lambert 2001). If the split opens in one dimension, the exponent of the pressure in (Equation (3)) changes to 1.5 and if the split opens up in two dimensions (longitudinally and radially), the exponent of the pressure changes to 2.5. Furthermore, recent studies (Cassa & Van Zyl 2013; Van Zyl & Cassa 2014) have stated that this conventional power equation using *N1* does not provide a perfect representation of elastic leaks. These authors proposed a dimensionless leakage number that provides a more consistent characterisation. In future research, this new formula will be considered for representing leak behaviour in elastic materials.

The flow ranges used to determine the round hole diameter of the orifices are described in Table 1, according to the leakage classification proposed by McKenzie (1999), in relationship with its detection.

Type of leak | q (m_{L}^{3}/s) |
---|---|

Background leak and undetectable | q < 2.8 × 10_{L}^{−6} |

Background leak and difficult detection | 2.8 × 10^{−6} < q < 5.5 × 10_{L}^{−5} |

Reported leak and easy detection | 5.5 × 10^{−5} < q < 1.4 × 10_{L}^{−4} |

Reported leak | 1.4 × 10^{−4} < q _{L} |

Type of leak | q (m_{L}^{3}/s) |
---|---|

Background leak and undetectable | q < 2.8 × 10_{L}^{−6} |

Background leak and difficult detection | 2.8 × 10^{−6} < q < 5.5 × 10_{L}^{−5} |

Reported leak and easy detection | 5.5 × 10^{−5} < q < 1.4 × 10_{L}^{−4} |

Reported leak | 1.4 × 10^{−4} < q _{L} |

The head over the orifice (*h*) is considered in a range from 0 to −98.1 kPa, which are the limits that generate an intrusion into a pipe, without cavitation.

The discharge coefficient (*C*_{d}) varies with the Reynolds number (Lambert 2001). As indicated in Van Zyl & Cassa (2014), some models have been proposed to characterise the behaviour of orifice discharge coefficients, as Idelchik (1994) indicated. In particular, it is known that the *C*_{d} directly depends on the flow rate and diverse formulas can be considered for different orifice shape. For laminar flows (Reynolds numbers below 10), the relationship between flow rate and pressure head is linear, although for Reynolds numbers smaller than 1,000 this dependence already exists (Idelchik 1994).

A constant value is initially used to characterise pipe failures. Based on Lambert's (2001) experiments on round holes of 1 mm diameter and Reynolds number above 2,000, a *C*_{d} value of 0.75 was obtained. Nevertheless, drawing from other experiments diverse values of *C*_{d} have been found. For laminar flow, the *C*_{d} values were from 0.15 to 0.77; in transitional flow range, the *C*_{d} oscillates between 0.42 and 0.85; and in turbulent flow, the values obtained were from 0.56 to 0.74 (Lambert 2001; Yang *et al.* 2014). Some other authors have used constant values such as 0.60 (Ebacher *et al.* 2012), 0.70 and 0.80 (Lee *et al.* 2002, 2005), or obtained values between 0.71 and 0.76 (Walski *et al.* 2006). In another research project, the *C*_{d} values were specified according to the shape of the failure, 0.80 for a hole and 0.60 for a crack (Tabesh *et al.* 2009). The majority of the researchers relate the *C*_{d} to the Reynolds number. In this paper, the *C*_{d} is a calibration parameter at the end of the experiments.

*Q, h, C*

_{d}), diverse diameters were estimated for the experiments on pipe defects. The diameters, chosen according to the types of leak in the range of pressure mentioned above, are shown in Figure 2.

Four round holes failures with varying diameters were considered to simulate the intrusion flow and its relationship with the classification of leaks. In the case of the intrusion flow, according to the pressure range, the classification is proposed for two types of failure:

A large intrusion flow for round holes of 4.0 and 5.0 mm diameter represents the flow of a reported leak that is easy to detect.

A small intrusion flow for round holes of 1.0 and 1.5 mm diameter represents a flow for a background leak flow that is difficult to detect.

### Longitudinal splits and circumferential break

*d*

_{eq}is the equivalent diameter,

*A*is the area of the orifice and

*P*is the wetted perimeter of the orifice.

_{w}Type of failure | Length (mm) | Width (mm) | A (mm^{2}) | d_{eq} (mm) | ID. |
---|---|---|---|---|---|

Large longitudinal split | 44.5 | 0.135 (average) | 6.30 | 1.6 | LS44 |

Small longitudinal split | 4.0 | 0.28 | 1.1 | 1.0 | LS4 |

Large circumferential break | 48.3 | 0.19 (average) | 10.2 | 2.1 | CB48 |

Small circumferential break | 3.1 | 0.39 (average) | 1.4 | 1.2 | CB3 |

Type of failure | Length (mm) | Width (mm) | A (mm^{2}) | d_{eq} (mm) | ID. |
---|---|---|---|---|---|

Large longitudinal split | 44.5 | 0.135 (average) | 6.30 | 1.6 | LS44 |

Small longitudinal split | 4.0 | 0.28 | 1.1 | 1.0 | LS4 |

Large circumferential break | 48.3 | 0.19 (average) | 10.2 | 2.1 | CB48 |

Small circumferential break | 3.1 | 0.39 (average) | 1.4 | 1.2 | CB3 |

The failure areas, which were obtained with equivalent diameters, are similar to the reported leaks (difficult detection) in the pressure range shown in Figure 2. Therefore, the failures were similar to small round holes, according to the Huebscher equation, and the equivalent diameters obtained from a similar flow through the orifices showing the same pressure drop.

## MODEL OF INTRUSION FLOW

The physical model consists of a pipe with a failure placed inside a tank that maintains a constant water level, which represents the source of external flow. With this model, the volume of water that can be introduced into the pipe was measured during a negative pressure event for the eight failures depicted in the section ‘Types of failure’.

### Set-up description

The objective of the physical model was to verify the relationship between the pressure drop and the intrusion flow, analogous to the *N1* and orifice equations. Two different experiments were performed: the first one considers the pipe exterior without soil and the second one with soil. Below are the data and results of both experiments.

### Experiments without soil around the pipe

The model PVC pipe is 32 mm nominal diameter and 2.4 mm thick and the inner diameter is 27.2 mm. The exterior tank had a constant water level of 0.37 m over the pipe. The negative pressure inside the pipe varied between −8.8 and −68.6 kPa. Depending on the type of failure, four to seven scenarios of negative pressure were simulated. The number of repetitions for each pressure scenario was the minimum required to establish the statistical significance. Seven to fifteen repetitions of each scenario were carried out to guarantee the results were consistent with a minimum error in the experimental data. Pressure scenarios and repetitions were randomly chosen. The process for these scenarios is described in Table 3.

Step | Description |
---|---|

1 | To start the experiment, the pump equipment began and the valve upstream was partially closed to obtain a value of pressure between 0 and 98.1 kPa, according to the type of failure and the number of scenarios considered |

2 | The recirculation system of the intrusion tank was regulated based on the quantity of intrusion volume and on the type of failure in order to keep a constant water level and a low level of turbulence in the intrusion tank |

3 | After establishing a constant flow for the two recirculation systems, the pressure and the flow in the pipe were captured using LabVIEW© software and the intrusion flow was measured volumetrically |

Step | Description |
---|---|

1 | To start the experiment, the pump equipment began and the valve upstream was partially closed to obtain a value of pressure between 0 and 98.1 kPa, according to the type of failure and the number of scenarios considered |

2 | The recirculation system of the intrusion tank was regulated based on the quantity of intrusion volume and on the type of failure in order to keep a constant water level and a low level of turbulence in the intrusion tank |

3 | After establishing a constant flow for the two recirculation systems, the pressure and the flow in the pipe were captured using LabVIEW© software and the intrusion flow was measured volumetrically |

This process was repeated to define the relationship between the intrusion flow and the negative pressure for each scenario. Tables 4 and 5 show a summary of each experiment carried out for the round holes, and the longitudinal splits and the circumferential breaks, respectively.

Parameter | RH1 | RH1.5 | RH4 | RH5 |
---|---|---|---|---|

Number of scenarios | 5 | 7 | 5 | 4 |

Repetitions | 7 | 10 | 15 | 8 |

Max pressure (kPa) | −13.7 | −10.8 | −9.8 | −22.6 |

Min pressure (kPa) | −51.0 | −55.9 | −33.3 | −68.6 |

Min intrusion flow (m^{3}/s) | 3.7 × 10^{−6} | 6.8 × 10^{−6} | 4.5 × 10^{−5} | 4.8 × 10^{−5} |

Max intrusion flow (m^{3}/s) | 7.1 × 10^{−6} | 1.5 × 10^{−5} | 7.7 × 10^{−5} | 1.6 × 10^{−4} |

Parameter | RH1 | RH1.5 | RH4 | RH5 |
---|---|---|---|---|

Number of scenarios | 5 | 7 | 5 | 4 |

Repetitions | 7 | 10 | 15 | 8 |

Max pressure (kPa) | −13.7 | −10.8 | −9.8 | −22.6 |

Min pressure (kPa) | −51.0 | −55.9 | −33.3 | −68.6 |

Min intrusion flow (m^{3}/s) | 3.7 × 10^{−6} | 6.8 × 10^{−6} | 4.5 × 10^{−5} | 4.8 × 10^{−5} |

Max intrusion flow (m^{3}/s) | 7.1 × 10^{−6} | 1.5 × 10^{−5} | 7.7 × 10^{−5} | 1.6 × 10^{−4} |

Parameter | LS4 | LS44 | CB3 | CB48 |
---|---|---|---|---|

Number of scenarios | 4 | 3 | 4 | 5 |

Repetitions | 10 | 12 | 10 | 12 |

Max pressure (kPa) | −23.5 | −14.7 | −23.5 | −8.8 |

Min pressure (kPa) | −58.8 | −23.5 | −52.0 | −27.4 |

Min intrusion flow (m^{3}/s) | 7.7 × 10^{−6} | 9.6 × 10^{−7} | 5.6 × 10^{−6} | 6.7 × 10^{−5} |

Max intrusion flow (m^{3}/s) | 1.1 × 10^{−5} | 1.6 × 10^{−6} | 9.1 × 10^{−6} | 1.3 × 10^{−4} |

Parameter | LS4 | LS44 | CB3 | CB48 |
---|---|---|---|---|

Number of scenarios | 4 | 3 | 4 | 5 |

Repetitions | 10 | 12 | 10 | 12 |

Max pressure (kPa) | −23.5 | −14.7 | −23.5 | −8.8 |

Min pressure (kPa) | −58.8 | −23.5 | −52.0 | −27.4 |

Min intrusion flow (m^{3}/s) | 7.7 × 10^{−6} | 9.6 × 10^{−7} | 5.6 × 10^{−6} | 6.7 × 10^{−5} |

Max intrusion flow (m^{3}/s) | 1.1 × 10^{−5} | 1.6 × 10^{−6} | 9.1 × 10^{−6} | 1.3 × 10^{−4} |

### Experiments with soil around the pipe

In these experiments, 88% of the external granular material had a diameter between 5 and 1.25 mm. The experimental process for this scenario was the same as described in Table 3. The results of pressure and intrusion flow were compared with the pipe with a round hole of 1 mm without porous media in the range of −13.7 to −51.0 kPa.

## EXPERIMENTAL RESULTS

The results obtained for the diverse types of failure from the physical model are presented, first for experiments on intrusion flow without soil around the pipe and then for the experiments on intrusion flow with soil around the pipe.

### Intrusion flow without soil around the pipe

In the first group, the failures obtained exponent values under 0.5: the round holes of 4 mm (0.45), 1.5 mm (0.46) and 5 mm (0.47).

In the second group, the failures obtained exponent values near to 0.5: the round hole of 1 mm (0.50) and the longitudinal split of 4 mm (0.52).

In the third group, the failures obtained exponents with values higher than 0.50: the circumferential break of 48 mm (0.55), the longitudinal split of 44 mm (0.60) and the circumferential break of 3.1 mm (0.63).

The experimental results were adjusted to the potential equation and the coefficient of determination (*R*^{2}) was estimated in order to indicate how well experimental results fit to the potential equation. The round holes of 1, 1.5, 4.0 and 5.0 were the best fit (*R*^{2} around 0.9986). Then, the circumferential break of 48 mm and the longitudinal split of 44 mm had an acceptable fit (*R*^{2} around 0.9949). Finally, the circumferential break of 3.1 mm fitted with a coefficient *R*^{2} = 0.9857 and the longitudinal split of 4 mm fitted with a coefficient *R*^{2} = 0.9754.

### Intrusion flow with soil around the pipe

*N1*equation (Figure 9), the exponent reduces its value from 0.50 to 0.30 when compared with the previous section.

In the case of the intrusion flow with porous media, fine gravel has an influence on the exponent, which decreases to 0.30. The presence of the porous media induces a resistance in the flow near the defect, as there is some fraction of water retention in the external media (White 2009). The property relating the porosity with the water retention in soils is the specific retention: the existence of pores in the external porous media provide for the passage or retention of water within the soil profile (Fetter 2001). This affects the flow passing across the holes in the intrusion process and therefore the exponent in the equation.

## ANALYSIS OF RESULTS: DISCUSSION

The experimental results for intrusion flow, modelled according to the *N1* and orifice equations, showed that some failures fit well according to the intrusion flow related to the leak classification.

Failures with maximum intrusion flow (between 0.20 and 0.40 m

^{3}/h): circumferential break of 48 mm and round holes of 5 and 4 mm. These are similar to the flow of reported leak and easy detection.Failures with medium intrusion flow (between 0.02 and 0.05 m

^{3}/h): round holes of 1 and 1.5 mm, the longitudinal split of 4 mm and the circumferential break of 3 mm. According to the dimensions of the failures, the longitudinal split of 4 mm and the circumferential break of 3 mm have similar intrusion flows as the circular failures of 1 and 1.5 mm. The longitudinal and small circumferential failures obtained similar intrusion flows to that of the equivalent round holes. These intrusion flows are similar to background leak and difficult detection.

However, two failures did not obtain intrusion flows similar to the leak classification.

The intrusion flow of the circumferential break of 48 mm was higher than the intrusion flow obtained with the equivalent diameter of 2.1 mm. In fact, this failure obtained higher intrusion flow than the round hole of 5 mm.

The longitudinal split of 44.5 mm was the failure that produced the smallest intrusion flow and was only obtained when negative pressure was close to zero. The intrusion flow was between 0.003 and 0.005 m

^{3}/h, similar to the flow of an undetectable leakage. The flow obtained with the longitudinal split of 44 mm was 10 times smaller than the flow obtained with the equivalent diameter calculated of 1.6 mm.

In these cases, the large longitudinal and circumferential failures (LS44 and CB48) obtained different ranges of intrusion flows than the equivalent round holes, although the *N1* equation did not obtain values of the exponent superior to 0.60. In these two cases, the failure section variation must be considered in the equations to obtain the flows based on the orifice equation (Table 6).

Type of failure | D_{equiv} (mm) | Scenario | V_{intrusion} (m/s) | Re | Regime |
---|---|---|---|---|---|

Large longitudinal split (LS 44) | 1.6 | 1 | 0.15 | 244 | Transition |

2 | 0.18 | 284 | |||

3 | 0.21 | 331 | |||

Large circumferential break (CB 48) | 2.1 | 1 | 6.6 | 13,937 | Turbulent |

2 | 8.2 | 17,278 | |||

3 | 10.1 | 21,170 | |||

4 | 11.2 | 23,484 | |||

5 | 12.5 | 26,355 |

Type of failure | D_{equiv} (mm) | Scenario | V_{intrusion} (m/s) | Re | Regime |
---|---|---|---|---|---|

Large longitudinal split (LS 44) | 1.6 | 1 | 0.15 | 244 | Transition |

2 | 0.18 | 284 | |||

3 | 0.21 | 331 | |||

Large circumferential break (CB 48) | 2.1 | 1 | 6.6 | 13,937 | Turbulent |

2 | 8.2 | 17,278 | |||

3 | 10.1 | 21,170 | |||

4 | 11.2 | 23,484 | |||

5 | 12.5 | 26,355 |

The method used to verify the failure section variation was estimating a correction of the discharge coefficient for the longitudinal and circumferential failures. The correction is based on obtaining an equivalent discharge coefficient similar to the round hole sections that yield the same Reynolds number.

*C*) of the round holes is calculated by Equation (6): where

_{di}*Q*is the intrusion flow,

*A*is the area of the orifice and

*Δh*is the pressure head over the failure. Figure 11 shows the relationship between the

*C*

_{di}and the Reynolds number. Reynolds number is defined as the ratio between the product of mean velocity times diameter in the orifice, over the kinematic viscosity of water. As the flow across the failure changes, Reynolds number ranges from transitional up to turbulent flows. In this case, the range of the

*C*

_{di}is from 0.9 to 0.7. The value of the

*C*

_{di}diminishes with an increase in Reynolds number and the round hole diameter.

The Reynolds number for the flows through the large dimensions failures were estimated in order to obtain the values for validating the *C*_{di}. The intrusion velocity and the Reynolds number are calculated for the equivalent *C*_{di} for the longitudinal and circumferential failures in every scenario.

To obtain an equivalent *C _{di}*, the width of the longitudinal and circumferential failures is calibrated; and, consequently, the Reynolds number calculated is modified in order to verify the failure section variation with the correction of the discharge coefficient. Table 7 shows the equivalent

*C*

_{di}with the width proposed for both failures. The values of the

*C*

_{di}were proposed from the estimations obtained with the similar round hole that produced a similar intrusion flow. In the case of the LS44 failure, the intrusion flow is transitional. Therefore, based on Lambert's experiments, the

*C*

_{di}value obtained is close to 0.38 (Lambert 2001).

Type of failure | C_{di} objective | Thickness proposed (mm) | Scenario | New V_{intrusion} (m/s) | New Re | C_{di} adjust |
---|---|---|---|---|---|---|

LS44 | 0.38 | 0.01 | 1 | 2.0 | 643 | 0.38 |

2 | 2.4 | 754 | 0.38 | |||

3 | 2.8 | 900 | 0.38 | |||

CB48 | 0.78 | 0.45 | 1 | 3.1 | 10,503 | 0.75 |

2 | 3.9 | 13,020 | 0.75 | |||

3 | 4.7 | 15,954 | 0.79 | |||

4 | 5.3 | 17,697 | 0.77 | |||

5 | 5.9 | 19,861 | 0.79 |

Type of failure | C_{di} objective | Thickness proposed (mm) | Scenario | New V_{intrusion} (m/s) | New Re | C_{di} adjust |
---|---|---|---|---|---|---|

LS44 | 0.38 | 0.01 | 1 | 2.0 | 643 | 0.38 |

2 | 2.4 | 754 | 0.38 | |||

3 | 2.8 | 900 | 0.38 | |||

CB48 | 0.78 | 0.45 | 1 | 3.1 | 10,503 | 0.75 |

2 | 3.9 | 13,020 | 0.75 | |||

3 | 4.7 | 15,954 | 0.79 | |||

4 | 5.3 | 17,697 | 0.77 | |||

5 | 5.9 | 19,861 | 0.79 |

This method was implemented to quantify the variation of the failure area based on the numerical calibration of the *C*_{di} for large failures using the orifice equation. The area of intrusion could vary with the negative pressure. The flow variation of the last two failures (the circumferential break of 48 mm and the longitudinal split of 44 mm) could be affected by a variation in the failure's section during the experiment. According to the methodology described and considering the *C*_{di} measured in the experiments, the section variation was modified to obtain the equivalent discharge coefficient (Table 8).

Type of failure | Variation of the area of the failure (%) | Description of the variation |
---|---|---|

LS44 | −93 | Considering the equivalent C_{di}, the area could be almost closed. This is probably due to the failure length and its small width |

CB48 | +112 | With the equivalent C_{di}, it shows that the calculated area is almost twice that of the area measured. The operating conditions of the negative pressure make this type of failure more vulnerable than others |

Type of failure | Variation of the area of the failure (%) | Description of the variation |
---|---|---|

LS44 | −93 | Considering the equivalent C_{di}, the area could be almost closed. This is probably due to the failure length and its small width |

CB48 | +112 | With the equivalent C_{di}, it shows that the calculated area is almost twice that of the area measured. The operating conditions of the negative pressure make this type of failure more vulnerable than others |

In the case of the longitudinal split, the negative pressure generated inside the pipe causes the section failure to practically close and the intrusion flow reaches values near to zero. On the other hand, the negative pressure inside the pipe caused the circumferential break to almost double its measured area.

## CONCLUSIONS

This research dealt with modelling the intrusion flow in buried and unburied pipes. In particular, this work presented a method to perform experiments on a physical model in a steady-state according to the analogy of *N1* and orifice equations. The final objective was to obtain a contrasted set of equations to represent the relationship between intrusion flow across defects and negative pressure head in different sort of leaks.

Eight different failures were experimented on to represent equivalent flows for easy and difficult detection. One experiment was carried out with external porous media in order to simulate the conditions of the intrusion phenomenon on buried water distribution pipes. It was observed that the intrusion had been reduced to almost half of the flow when the scenarios were carried out with porous media around the hole in the range of pressure simulated.

The three types of failure obtained exponent values around 0.5 for the relationship of the pressure and intrusion flow. Although the large size longitudinal failure obtained a high intrusion flow in relation to its equivalent diameter, the exponent was still near to 0.5, similar to the orifice equation.

This work also proposed a method to quantify the variation of the width failure on longitudinal splits and on circumferential breaks. The method is based on determining the failure section variation by estimating a correction of the discharge coefficient. The correction proposed an equivalent discharge coefficient for each failure, similar to the round hole sections that yield the same Reynolds number. The results showed that the longitudinal failure practically closed during the intrusion, and the circumferential failures were opened to almost twice its area.

The topic of intrusion flow with soil around the pipe is quite new, and just recently documented. In this sense, this paper contributes to the relationships obtained, to implement in computational models of pipe networks. The characteristics of the porous media and its specific retention diminish the exponent of the *N1* equation. This information will allow modellers to quantify the potential volume of external polluted water flow into pipes. With this knowledge, the model users and managers will have a very promising tool to estimate the potential risk and health implications of this phenomenon in real potable water systems.

## ACKNOWLEDGEMENTS

DAIP-UG project 2013: 1101.31A02.42.205000 is acknowledged. The use of English in this paper was revised by the DAIP translation services (*Servicios de traducción del Departamento de Apoyo a la Investigación y al Posgrado*) of the University of Guanajuato.