Method Background Introduction Contents

Teknisk- naturvetenskaplig fakultet UTH-enheten

Besöksadress:

Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress:

Box 536 751 21 Uppsala Telefon:

018 – 471 30 03 Telefax:

018 – 471 30 00 Hemsida:

http://www.teknat.uu.se/student

Contents

1

Introduction

1.1 Purpose . . . 1

1.2 Literature review . . . 3

1.2.1 Volume-balance (VB) analysis . . . 3

1.2.2 Change point detection (CPD) . . . 4

1.2.3 Steady state inverse analysis . . . 4

1.2.4 Flow residual vector method . . . 5

1.2.5 Transient test-based technologies (TTBTs) . . . 5

1.2.6 Acoustic monitoring (AM) . . . 6

1.2.7 Pressure-point analysis (PPA) . . . 7

1.2.8 Negative pressure wave (NPW) method . . . 7

1.2.9 Critique and summary . . . 7

1.3 Goals . . . 10

1.4 Outline of the thesis . . . 10

2

Background

2.2 Pipe characteristics and trends . . . 14

2.2.1 Pipe materials and age . . . 14

2.2.2 Pipe diameter . . . 15

2.2.3 Pipe length . . . 15

2.3 Pipe deterioration . . . 16

2.4 Pipe failure mechanisms . . . 17

3

Method

3.2 WDN as a graph . . . 20

3.3 Deriving equations of the network . . . 22

3.4 Probability Leak Model/ Estimating leak behavior in a pipe . . . 26

3.4.1 Gamma distribution as probability distribution function . . . 27

3.4.2 Effect of shape and scale parameters . . . 28

3.4.3 Model parameterization . . . 31

3.5 Algorithm I . . . 36

3.5.1 Connectivity data . . . 37

3.6 Algorithm II . . . 43

4

Simulation

4.2 Layout . . . 48

4.3 Sensor Information . . . 51

4.4 Leak data simulation . . . 53

4.4.1 Leak generation . . . 53

4.4.2 Method of leak assignment . . . 54

4.5 Generating observed flow data . . . 55

5

Results

5.2 Simplified Results . . . 59

6

Conclusions

A

Complete Results

A.2 Case 2 . . . 67

A.3 Case 3 . . . 67

List of Figures

1.1 Flow data of several days to observe patterns . . . 2

2.1 Layout of water supply system[17] . . . 13

2.2 Rate of occurrence of failures in pipes . . . 16

3.1 Leak detection model . . . 20

3.2 WDN representation as a directed graph . . . 21

3.3 Flowdata representation . . . 22

3.4 Affect of scale parameter in Gamma distribution . . . 29

3.5 Affect of shape parameter in Gamma distribution . . . 30

3.6 Flow data readings (Green) observed for different flow sensor layouts 39 4.1 Overview of the Simulation Model . . . 48

4.2 Layout for the three cases . . . 50

4.3 Sensor positions for all cases . . . 53

5.1 Distribution of extra/missed quantity of leak . . . 61

List of Tables

1.1 Summary of leak detection methods . . . 9

2.1 Lifetime estimates of Swedish drinking water pipes (Malm, 2013) . . . 15

3.1 Conversion of a directed list into connectivity data . . . 40

4.1 Length and Diameter data for Case 1 . . . 51

5.1 Example result . . . 58

5.2 Simplified test results of all cases . . . 59

5.3 Results of exhaustive testing of all cases . . . 62

A.1 Complete results of Case 1 . . . 66

A.2 Complete results of Case 2 . . . 67

A.3 Complete results of Case 3 . . . 68

Chapter 1

Introduction

This chapter will explain the purpose and goals of the work presented. It also contains a critical review of some of the popular detection methods and an outline of the report.

1.1 Purpose

Urban water networks are an ongoing effort of human civilization since hundreds of years ago. There are immense resources that go into developing and maintaining a water supply structure, not to mention the cost of collecting, storing and treating water through the intricate network thereby fostering serious interest in making efficient water systems.

Consider the water data of Uppsala where Uppsala Vatten has reported 17.9% of all the water produced for 2019 has leaked out from supply pipes [1]. It is equal to 3.3 Mm³ of water, the energy used in just producing this water is about 2.3 GWh [1]. To put the numbers into perspective, all that water could be used to fill 1,320 Olympic sized pools or provide electricity to 170 people in Sweden for a day [2].

Water leaks are ever present in a water distribution network (WDN) due to multiple factors that are both inherent and obscure for a water supply system. Every urban water systems tries to keep the water losses low and to further decrease it by early detection and localization. Finding and fixing leaks has tremendous payoffs in terms of energy efficiency. Therefore water companies have cause to support research for leak detection and localization techniques.

Current leak detection techniques

Reactive repairs form the bulk of water loss management. The need to reduce leakage, save energy and water is expected to increase in the future due to climate change,

increased demand for resources and aging water supply networks. Small leaks are hard to detect. Over time they grow into bigger leaks and thus become easier to detect. This process of a small leak growing into a big leak might occur over a long period of time during which a lot of water can be lost before it gets big enough to be detected. This is a common practise by urban water suppliers to detect leaks.

To give an example of such practice Uppsala Vatten has remotely readable flow sensors placed around strategic locations within the water network in Uppsala. The information collected from this sensors is flow data, this data when represented as a time series graph shows the water demand pattern of a city. A graph showing the flow data collected by Uppsala Vatten for a city over several days is shown in Figure 1.1. It is clear that water demand pattern follows a diurnal cycle of high and low demands. Each day has two distinct peaks occurring around 6:00 and 18:00.

This can be related to the time most people leave for work or return to homes. It is exceptionally low during early hours of the morning around 3:00. This level of consumption is called the minimum night flow (MNF).

Figure 1.1: Flow data of several days to observe patterns

One of the leak detection strategy employed by Uppsala Vatten is to manually monitor the flow data during MNF. Since the demand is supposed to be low, if there is a jump in the data that is large enough to be distinctive, this is observed as a leak and necessary action is taken. Another strategy depends on the public to report water leakage and then act on it. These strategies are clearly geared towards finding leaks that lose a high quantity of water. This implies that smaller leaks usually slip under such methods. With the above mentioned techniques, Uppsala Vatten is currently losing 17% [1] of all produced water. Future demands to be efficient push them to decrease the percentage of leaks to be atleast 11%. This indicates a need for a water detection technique that is able to better detect and localise small leaks

with the help of flow sensor readings. This is the main goal of this thesis. To do that, classification of different kinds of leaks is required. Those definitions will help in deciding if an existing method can or cannot work on small leaks and also guide the exploration for a new method.

Types of leaks

For the purpose of this project, there are two main types of leaks: visible and hidden.

When pipes fail and water appears on the streets/ ground surface, it is a visible leak.

This can mean there is water flowing out on streets or a connection at an apartment is burst or leaking. These leaks are most likely to be reported quickly by onlookers.

The other type is called hidden leaks. Pipes under this situation leak the water into underground and remain hidden from sight. They can consistently be leaking small quantities or increase gradually the amount of leaking. These are most likely to be found only when the leak is causing substantial changes in demand patterns, during a maintenance or leak investigation.

1.2 Literature review

A thorough understanding of related previous works is required to get inspiration for a new model. A critical review of such works can help identify why they cannot be used for the current scenario and where they failed.

1.2.1 Volume-balance (VB) analysis

MacTaggart and Myers [3] implemented a Volume-balance (VB) based detection system on a 140 km long crude oil pipeline. The volume-balance approach’s main principle is expressed as:

V_in− V_out = ∆LP

The above equation says that the difference of volume entering and leaving the pipeline over a fixed time interval must be equal to change of linepack inside the pipe over the same time interval. Linepack is the total volume of fluid contained over the time interval within the concerned system. It is extra fluid stored in long distance pipelines to enhance operational flexibility during high demands or shortfall in supply source. In a leak free situation the change in linepack is zero otherwise it is a non zero value representing the amount of fluid leaking.

In the study [3] pressure and temperature were measured at all injection points of the pipeline. All flows in and out of the pipeline were also measured. The sampling interval of all these measurements was approximately 30 seconds.

The pipeline had varying temperatures due to the fact that some of the pipeline was underwater. Other factors inherent to transporting crude oil in a pipeline also caused various temperature fluctuations. This caused changes in the pressure and volume of the crude oil.

Considering all these intricacies the linepack was recalculated for each sampling interval and integrated over different time intervals (called windows) ranging from few minutes to few days to be analyzed. The linepack must cross an alarm threshold to be considered as a leak. This threshold value for the linepack depended on the length of these windows. For a short window, the thresholds were sized to detect “rupture sized leaks” while for a long window, thresholds were sized to detect smaller leaks.

This was claimed to avoid false alarms due to transients, instrument repeatability error and measurement bias. A 1 l/s leak was estimated to be detected in 4 hours and a complete failure was detected in less than 3 minutes.

1.2.2 Change point detection (CPD)

With the use of flow data from Uppsala Vatten, a purely data based Change point detection (CPD) algorithm was implemented by Enander [4]. This technique used dif- ferent methods to create varied algorithms. These algorithms checked for an abrupt change in a time series during MNF. The algorithms implemented change-point de- tection, which gave the time when a change occurred in the time series. This is easy to implement and large instantaneous leaks can be found quickly. However, leaks that gradually increase in size will slip past the CPD as well as small leaks.

1.2.3 Steady state inverse analysis

A study by Pudar and Liggett [5] introduces the problem of detecting and localizing leaks while measurements of the pipe network are known as an inverse problem.

The study uses pressure data supplemented by few flow data points. However, it is not necessary to have all the measurements at every pipe of the network to use their method. The study considers the flow/pressure reading at a node (end of a pipe) as a “demand”. They have divided their work into overdetermined where number of known demands are greater than number of unknown demands, even- determined where a pressure or flow rate is given for every unknown demand and, underdetermined where number of unknown demands are greater than number of known demands. The study assumes all leaks to occur at nodes and that a more sophisticated approach would be needed if a higher resolution of leak location is required. Because detecting the actual location of leak in a pipe will require a model that is able to simulate the true behavior of the network, with respect to water flow

(including the leaks). To put forth such a model, the data and methods by the study are insufficient.

The authors of [5] put forth two non-linear equations for each pipe. The equations expressed pressure drop and total flow pertaining to a pipe. The total flow included leak at the pipe node as well. This leak was expressed in terms of pressure (referred to as head) using the general orifice flow equation [6]. The orifice areas of possible leaks are considered the unknowns. These areas were calculated by minimizing the difference between the measured and calculated pressure readings.

A surprising result of this study revealed that an overdetermined system did not yield better results than a even-determined system. It was observed that with increasing number of measurements for an underdetermined case, the results were increasingly conclusive. Sensitivity matrices were used to express confidence (in the solution) which was found to be influenced by location of the measurements.

1.2.4 Flow residual vector method

So far all the discussed studies focused on finding which pipes are leaking without knowing the exact location of the leak on the pipe. Detecting such leak locations on a pipe requires far more sophisticated techniques and an extensive measurement network. A study by Javier et al. [7] uses only flow data to localize leak locations on a pipe . In this study demands are assumed to occur at nodes and leaks occur between nodes. A WDN was modeled as a set of nonlinear partial derivative equations in terms of the flow rate. Each equation describes a pipe of the WDN. These continuous equations were discretized into ordinary differential equations. This way a flow in a pipe is broken into a consecutive set of discrete flows. A set of residuals for each of these discrete flows were calculated by subtracting them from the mean nominal flow of that pipe. It is stated that residuals can be greater than/lesser than/equal to zero, which represented that the leak is downstream of/upstream of/in that section of pipe. The section is given by the residual notation. This was tested on a small network example with good results. It is interesting to note that even though they had excellent results, the study claims that their method is not intended to be a replacement to methods based on pressure readings because, pressure based detection offers cheaper and faster detection [8].

1.2.5 Transient test-based technologies (TTBTs)

In a study by Asada et al. [9] it is claimed that manometers are cheaper than flowme- ters and can be easily installed at air valves on pipelines. It is noted that a pressure change can be undetectable if the leaks are small or if the water pressure is low. They suggest that a pressure wave large enough to be detected can be generated with the

help of water hammer events. This occurs in incompressible fluids such as water when valves are shut down rapidly. The momentum of flowing water has no where to escape and thus causes a pressure wave which oscillates between ends of a pipeline.

From an arbitrary point in the pipeline, this pressure wave is observed to have a cyclical nature. But in presence of leaks, the pressure wave has an increased damp- ing rate and new leak-reflected signals are created. Transient test-based technologies (TTBTs) study these effects to detect leaks in water systems.

The effect of leak-reflected signals in pressure transients has been studied and the detection of leaks using this effect demonstrated in some studies. But these failed when the leak was small or when noise due to pipe structure was present. Using both leak-reflected signals and leak-induced damping were found to be computationally enormous. So the study [9] proposed that other methods can be used to narrow down the leak location and decrease the complexity of computations. A previous work [10] by the authors of this study [9] concluded that leak-induced damping is minimally affected by noise compared to the leak-reflected signals. The study [10]

also concluded that the damping rate of a pressure wave was faster due to energy dissipation from a leak. Therefore the study [9] modeled leak-induced damping to calculate energy dissipation and localize the leak location on the pipe.

This was tested on an experimental setup where the network is a single pipeline with a spiral structure. On one end was the upstream reservoir and on the other end the pipeline had a manometer placed upstream of an air valve. The air valve led to a downstream reservoir. They tested three different cases, where a single leak was present in different locations of the pipe. The results were favorable in narrowing down the leak location and it was observed that the effectiveness of the model increased as the location was closer to the upstream reservoir.

1.2.6 Acoustic monitoring (AM)

In acoustic monitoring the acoustic sensors are fitted on the outer side of the pipe.

These sensors are programmed to collect data during MNF. The sound being collected is the sound of water flowing through the pipe. The frequency of this sound depends on the material and size of the pipe and also varies according to the size of the leaks.

Therefore these sensors should be able to capture data over a large bandwidth.

The data is checked to distinguish between leak and non-leak signals. The acoustic sensors can be moved to different areas functioning as a temporary survey tool.

This was shown to be cost ineffective due to the high installation and removal costs of the sensors in a study by van der Kleij and Stephenson [11]. In a study [12]

carried out by Sanchez et al. an acoustic sensor was placed every 160m over a total of a 75km network. Over a duration of 6 months 49 leaks were detected.

The study performed cost-benefit analysis which showed that the high installation

cost of permanent acoustic sensors could be justified in systems that have a higher deterioration rate.

1.2.7 Pressure-point analysis (PPA)

PPA is a widely used technique. It works on the principle that pressure in a pipe decreases in presence of a leak. This decrease in pressure is detected using the statistical properties of the pressure measurements. The approach implemented by Md Akib et al. [13] compared the statistical mean of the new incoming data with that of the previous data or the threshold value. If the mean of the new incoming data is found be smaller than the mean of previous data or smaller than a threshold value, a leak is suspected. Small leaks which cannot be easily detected by other methods can be detected using PPA. The low cost of pressure sensors and ease of implementation are advantages of this method.

1.2.8 Negative pressure wave (NPW) method

Whenever a leak occurs in a pipeline, the leak causes changes in pressure that are instantaneous. As the pressure decreases, a NPW originates at the leak location and it travels both upstream and downstream of the leak. This NPWs contains leak information that can be gathered by visual inspection and signal analysis. This was implemented by Sang et al. [14]. NPW detection requires little hardware for an entire network and is thus cost effective. This method has a fast response time for large instantaneous leaks and is widely implemented for leak localization.

1.2.9 Critique and summary

Table 1.1 summarizes all the methods discussed so far along with their weaknesses.

While all of the methods have various advantages. The scenario that will be explored by this work is an underdetermined case that has only few flow data points. To that extent the only methods that use flow data are CPD and flow residual vector. The inverse analysis is unsuitable because it uses a combination of flow and pressure data.

The CPD method has a few shortcomings, a hidden leak that gradually becomes bigger, causes no sudden change in a time series. This type of leak will go unnoticed when using CPD. There exists CPD techniques that will detect a small, growing leak eventually but it will take some time before being detected. The method ignores the structure of the WDN. It is reasonable to assume that making use of the network structure will give useful conclusions that go beyond the initial flow readings. The flow residual vector method is simply unfeasible to implement for the current work because of the high number of flow sensors placed throughout the network.

Some general conclusions made from the literature review are:

1. Methods that have continuous measurement data are quick to respond to leak.

2. Network structure needs to be utilized to gain more information.

3. Installing sensors in every node of a network is financially absurd.

4. Limited in-field testing makes it difficult to evaluate the performance of the existing methods.

5. Methods that are applicable only to pipelines for detecting leaks seem to require a thoroughly measured pipeline.

6. Repair of water pipes is often financially ineffective for the water company than letting the water to leak, especially for small leaks.

7. In reality every pipe in a network is probably leaking a small amount and it is more beneficial for the company to repair on moderately sized leaks.

Table 1.1: Summary of leak detection methods

Methods Principle Strengths Weakness

VB Change in volume of fluid going in and volume of fuid going out

Ease of implementation, low cost and ”rupture”

sized leaks are quickly found.

Not a network approach and requires many measurements, small leaks take long time to

be detected.

CPD change in data of a time series

Ease of implementation, low cost, applicable to entire network and large

instantaneous leaks are found quickly.

Leaks cannot be located and it cannot detect small leaks or gradually

increasing leaks.

Inverse Analy-

sis

Difference between measured and calculated

values

Low cost, network wide detection and gives confidence values of the

results.

Computationally complex and requires accurate pipe roughness

coefficients.

Flow residual

vector method

Difference between current discrete flow and

the mean nominal flow of that pipe

Ease of implementation for a network level leak

detection.

Requires sensors at every node and is high-cost

method.

TTBTs Study the effect of leak-induced damping on

pressure waves

Low cost and not affected by noise.

Requires suppressing flow rate in the pipeline

before sampling the data. Not applicable to

networks.

AM Analyze acoustic data to differentiate between

leak and no-leak

Ease of implementation for a network wide leak

detection.

High cost of installation and prone to false alarms due to noise.

PPA Detect leaks by finding change in pressure values

Low cost and easy to implement.

False alarms due to other pressure events. Not applicable for network

level leak detection.

Leak location is difficult to achieve [16].

NPW Analyze reflected pressure waves for leak

data

Low cost and fast response for large instantaneous leaks.

Works for an entire network.

Only applicable to large instantaneous leaks and data is prone to noise

deterioration. High probability of false

alarms.

1.3 Goals

The overall conclusion from the literature review is that a flow data based method that works on underdetermined scenarios by using the network’s structure has to be put forth. This method should give priority to detecting the biggest possible leak instead of many small leaks. The method should be automated to be quick in response. The following directives help to systematically achieve this.

Available methods of leak detection that use few data points and/or only flow data must be analyzed. Different pipe characteristics are to be studied to determine their trends in relation to pipe failures. From this, useful characteristics should be determined and applied to hypothesize a model that will adequately describe a pipe’s leak behavior. The flow data of a network should be formulated in such a way that it is enmeshed with the network structure. Finally the algorithms to detect leaks should be automated in a systemic way and their performance observed.

With these goals there is a primary limitation that should be addressed. Initially data from Uppsala Vatten was proposed to be used to replicate network scenarios, but the collaboration proved unsuccessful despite a lot of effort. Therefore the de- veloped detection techniques will be tested in simulated conditions. Water networks are highly complex and intricate structures. It is of utmost importance to test the hypothesis put forward by this thesis on simple water networks, to gauge its per- formance. Even a small improvement that helps save 1% of leaks is a big decrease overall. This thesis aims to strike a balance between the complexity of real life water networks and the models used.

1.4 Outline of the thesis

After this introduction in Chapter 1, Chapter 2 explains the different elements of an urban water network. The pipe failure trends according to its characteristics are shown. The reason for these trends and failures in general are briefly discussed.

Chapter 3 introduces the leak detection model. Then the representation of a network as a directed graph is presented. The different terminology used in this representation are explained. From this, general equations formulating the flows, leaks and known data of a network are derived. The consequences and possible alterations of these equations are presented. The leak behavior of a pipe is presented as probability function using the physical characteristics. The inner workings of the leak detection model as presented as two algorithms and explained in detail.

Chapter 4 describes the working of a simulation model that has been used to generate the required inputs to the leak detection model.

Chapter 5 of the thesis presents all results and describes notable observations

from these results.

Chapter 6 summarizes the work that has been done and provides critique. The conclusions made from results of the tests are also summarized here. Suggestions for future work are presented.

Chapter 2

Background

This chapter discusses the physical assets involved in a WDN. The layout of the net- work itself, pipe characteristics and failure mechanisms are explained. An overview of the type of failures are discussed.

2.1 General layout of water supply systems

Water supply systems can vary from complex to simple, but they are all made up of two divisions: a transmission and distribution network. Figure 2.1 from Misiunas [17]

is an excellent example to visualize the layout of a water supply system.

A transmission network is used to carry large quantities of water usually between two distant locations. For example, to transport water from a purification setup to a storage facility closer to human settlement. The water is pumped between such locations during off-peak hours to lower the energy cost. These pipes do not serve end customers directly, but instead branch off into distribution networks.

A distribution network includes distribution mains and service connections. A city’s WDN can be a mixture of looped and branched distribution mains pipes which ultimately end at a service connection. Each service connection can be an apartment or individual house or even a water hydrant. A looped structure offers redundancy to reach a location in case of damage to any of the other paths. A branched structure is desirable for its simplicity and low cost. It is also called a tree structure. Often, a part of the water distribution network is closed off by isolation valves to create District Metered Areas (DMAs). DMAs are created to ease water management for a large network by creating small self monitored sections. It is significantly easier to analyse water consumption of a DMA, which can help identify leaks. There are various types of fittings used to connect pipes throughout the network. Specialized mechanical equipment for the purpose of isolating parts of the network, managing

Figure 2.1: Layout of water supply system[17]

pressure between two connections, small reservoirs, various other control and main- tenance devices also make up the network.

It is easy to see from the description that there are many parts where a water leak is possible. This thesis however, focuses on dealing with pipe related leaks in a WDN. As we will see below water leaks at junctions are a possibility, however, this water must flow from a pipe leading to that junction. In this way leaks at junctions can also be detected.

2.2 Pipe characteristics and trends

In this section, a few important characteristics of pipes are presented. The reason for their importance with respect to pipe leaks is explained. The failure trends of these characteristics are also shown.

2.2.1 Pipe materials and age

Pipes have been made from different materials with various techniques over the past 100 years. Malm et al. [18] gives a clear picture of the material distribution by comparing their own findings with the Swedish Water & Wastewater Association’s data. The research showed that ductile iron along with cast iron made up almost 55% of the water network followed by PVC, PE and steel. It also notes that joints for early cast iron were not durable causing short lifetime estimate for the material.

Ductile iron before 1980’s were found to be far inferior to those made after 1980’s in the study. It can be seen clearly from the Table 2.1 that material of the pipe influences its service lifetime through physical degradation. More importantly the deduction that pipes that have recently been installed have little to no deterioration and that pipes that are old have deteriorated by a large percentage.

Type Lifetime (in years), % of pipes

100% Median 10%

Grey cast iron <1950 20-40 80-110 110-150 Grey cast iron ≥ 1950 30-50 90-120 120-160 Ductile iron <1980 20-40 40-60 60-100 Ductile iron ≥ 1980 40-60 110-140 140-180

PE 40-60 110-140 140-180

PVC <1970 20-40 40-60 60-80 PVC ≥ 1970 30-50 80-130 120-160 Other/Unknown 20-40 80-110 110-150

Table 2.1: Lifetime estimates of Swedish drinking water pipes (Malm, 2013)

2.2.2 Pipe diameter

From the works of Wengstr¨om [19] and Kettler and Goulter [20] a clear relationship between small diameter pipes and rate of breakage is observed. As the diameter gets bigger it is observed in [19] that the rate of breakage dropped drastically. This observation is also made by Stefano and Marco [21]. There are many explainations that try to reason this phenomenon. One proposal by Pelletier et al. [22] is that smaller diameter pipes have thinner walls and this makes them prone to breakage.

It was found in some cases by [19] that larger diameter pipes were found to have large amount of steel in the pipe material and this would have helped reduce the rate of failure. In the study [21] the smallest pipe diameters recorded were in the range of up to 100 mm and had the highest breakrates, the largest diameters being 500-1000mm with the lowest number of breaks.

2.2.3 Pipe length

Pipe length is often used to describe the total amount of pipe laid in an installation period. But pipe length as a parameter in defining rate of failure is of importance to this thesis.

For long pipes soil conditions which affect the rate of corrosion and pressure exerted by above-ground conditions can vary widely over the pipe. Andreou [23] ob- served that breakrates were proportional to the square root of its length. Wengstr¨om [19]

also observed a similar pattern.

2.3 Pipe deterioration

From the previous section it is clear that various materials of pipes are distributed throughout a water network. This gives rise to highly variable failure patterns. To find out the reliability of such a network, a good prediction of when its individual pipes will fail is required. The current age of a pipe is an important factor that directly affects its deterioration rate which leads to failure. But techniques that use physical examination of the extent of damaging existing pipes by excavation are classified destructive testing/evaluation. They change the structural or surface integrity of the component examined. After such testing the component analyzed becomes unfit for continued service. Without damaging parts of the WDN non destructive procedures give an accurate picture of the deterioration present in a pipe.

Such techniques sometimes use a camera to be sent along the pipe, or use complex radiographic, ultrasonic or magnetic imaging techniques. They are expensive and resource intensive as they require highly-specialized tools and certified personnel.

Hence there is a need to understand the mechanics of deterioration in pipes, but without physically digging up the entire pipe network.

Figure 2.2: Rate of occurrence of failures in pipes

A bathtub curve [Figure 2.2] is often used to represent a theoretical rate of failure for a product over its lifetime. It is applicable to pipes as well, to observe their temporal vulnerability. There are three stages of failure:

ˆ Burn-in/Infant Mortality

ˆ Normal

ˆ Wear-out

The reason for first stage failures is usually some kind of undetected produc- tion/installation fault. During the normal lifetime of the pipe, there are random events such as unusual loads which occasionally cause failures. During the wear out phase, deterioration of the pipe is due to various factors that increase with time such as : stress from ground above, collateral damage from constructions, leaching, corrosion due to the outside environment. The end life of a pipe is fraught with re- placements and necessary repairs of sections of pipes, which [24] noted; significantly affects the rate of failures in the wear-out phase.

The time intervals between the stages of failure vary considerably. Failure pattern is uniquely specific for a particular pipe as noted in [17]. Wengstr¨om [19] reviewed data collected from different studies and detailed on various parameters that seemed to significantly affect failure rate in pipes. It is important to understand that age, pipe diameter, material, soil environment, pressure, pipe length, temperature fluctu- ations and even previous repairs and failures affect the deterioration of a pipe. All these factors complicate the search for duration of each of the deterioration phase.

A conclusion from the study of pipe characteristics and their breakrate trends is that age is notably favored as one of the main reason for a pipe’s rate of deterioration, followed strongly by its diameter. Length and pipe material are also seen as good choices to predict a pipe’s rate of failure. All of these parameters are properties of a WDN that are often documented and thus easy to obtain. Pipe material is a tricky parameter, materials produced in different years often possess different structural strengths. This is due to improving laws governing manufacture processes. This shows that a pipe material is a useful parameter as long as the year of production is also known. This information is however difficult to obtain and/or simulate. Due to these reasons it is proposed that age, length and diameter of the pipes in a WDN be used to model leak behavior of a pipe.

2.4 Pipe failure mechanisms

After understanding when a pipe is most likely to fail, it is essential to know how it can fail. Pipe failures can be caused due to low strength of pipe or high stress loads being carried in the pipe. These factors are caused by root causes such as corrosion, stress, manufacturing and installation flaws.

Corrosion: Data from the study [25] suggest that corrosion pits formed on the exterior of buried pipes are the most common failure mechanisms. The environment in which pipes are buried in give rise to external factors affecting the corrosion rate of pipe. Some of the external factors are stray electric currents, soil moisture content

and soil chemical content. Internal corrosion is also possible by the chemical prop- erties of the contents. Properties of treated/untreated water such as temperature, microbiological activity, alkalinity affect internal corrosion. This type of failure is predominantly observed in cast and ductile iron pipes. Long term deterioration of plastic pipes (PVC/PE) are not well documented because they have only been in use for the last 40—60 years.

High stress: Rajani & Kleiner [25] have shown that the structural capacity of a buried pipe is dependent on the environmental and operational loads. These loads are formed due to:

1. soil pressure : changes with temperature and soil movement

2. traffic loading: stress caused from mass of water being transported daily in WDNs

3. frost loads: caused by soil pressure and due to water-temperature interaction 4. third party interference: excavation for rehabilitation, maintenance installation

or repair works

5. operational pressure: stress from the pressure changes a WDN is subjected to, during its lifetime by regular operation

Manufacturing defects Faulty manufacturing techniques give raise to many failure mechanisms. If the pipe thickness varies over the length of the pipe, it may not be sufficient to handle the stress put on it. Another example would be if the material composition was faulty, the end result will be a material that is perhaps weaker than intended.

Different pipe failure mechanisms cause different kinds of pipe breakage. O’day et al. [26] has classified three different types of pipe breakages:

ˆ Circumferential cracking is more common in smaller diameter pipes.

ˆ Longitudinal cracking occurs frequently in large diameter pipes.

ˆ Bell splitting in small diameter cast iron pipes is a normal occurrence.

There are further classifications but for this project the above pipe characteristics are sufficient to understand the probability of failures they may cause.

Chapter 3

Method

In this chapter the leak detection model that achieves the goal of detecting leaks in an underdetermined scenario is presented. The chapter begins with a simple overview of the leak detection model. Then the representation of a WDN as a directed graph is described. The various terms used to explain this representation are explained.

Using this graph a set of general equations that give the relationship between flows, leaks and known flow data are formulated. The consequences of these equations are considered. Using the physical characteristics of the pipes their leak behavior is presented. The different algorithms used by the model that utilize the general equations and the leak behavior are presented.

3.1 Overview of the leak detection model

The leak detection model put forth in this thesis requires the layout of the network, the physical characteristics of the pipes, supply flows to the network and observed flows of the network. The model works on these input data to first determine the total size of the leak present in the network which is the total supply going into the WDN. Then, Algorithm I first determines presence of zero leak pipes. These are a set of pipes which have been detected as having no leak present in them, then it localizes the entire leak down to only certain areas of the network called leak areas.

A leak area is a set of connected pipes which have been determined of having a leak.

Algorithm II inspects these leak areas and localizes leaks down to a specific pipe.

This process is a simplified overview of the leak detection model and is shown in Figure 3.1.

Network layout Pipes' physical characteristics

Observed flows

Leak Detection Model

Supply flows

Total leak size

Leak areas

Detected leaks Algorithm

I

Algorithm II Zero leak

pipes

Sensor tolerance

Figure 3.1: Leak detection model

3.2 WDN as a graph

A WDN is a complicated structure. Using graph theory, such a network of pipes can be modeled as a simple set of nodes and edges. The edges describe the pipe connections, and nodes are junctions where pipes are connected.

Water always flows according to the pressure gradient of a network, such that the edges of a network will be directed. It is impossible for water to flow in circles unless it is pumped around on purpose. But since the pipes are part of a bigger pressure regulated network; a circular flow does not occur. It is very likely that huge leaks change the local pressure of a pipe which may cause abnormal flows, but such a large leak will be a visible leak or so large that it can be easily detected with change point algorithm as in [4] and hence not studied in this thesis. The direction of water flow is thus an important factor when representing a WDN using graph theory. It is assumed that pipes have a directional attribute that governs the direction of water flow. This attribute can be described by using directed edges to represent the pipes.

So a natural conclusion is to use a directed graph to visualize and analyze flows in a WDN.

Consider Figure 3.2 as an example. The number of edges going into a node is defined as in-degree and the out-degree is the number of outgoing edges. It is clear that all nodes of this network have an in-degree of only one, so it is classified as a directed tree structure. Each pipe originates from a source node and terminates at a target node. Pipe 1 is the supply pipe, which connects the reservoir to the distribution network. The reservoir storage is represented by supply node 1 here. The order of flow or node numbering has no explicit meaning, i.e, even though a supply node is represented by node 1, it should not be expected that in all cases only node 1 is the

Figure 3.2: WDN representation as a directed graph

supply node. It is assumed that, the exact amount of water entering the WDN is always known. This is due to the fact that every water company meticulously counts the amount of water being pumped in and out of a reservoir. If there was a leak at a storage facility itself, it would be regarded as highly noticeable and usually fixed on priority. This leads to a conclusion that the supply pipe’s leak (if any) is always known. For the sake of simplicity, we assume in this work that supply pipes never leak.

Nodes that do not have outgoing physical connections (out-degree of zero) are called demand nodes and the pipes going into them are called demand pipes. They represent service connections. During usual hours of usage, most demand nodes have demand flows that are non-zero values. But during MNF demand flows are very low and thus will be assumed to be equal to zero.

Taking inspiration from [5], leaks are assumed to occur only at source nodes of pipes. Therefore, leaks are represented by “dangling” edges at source nodes shown by red arrows in Figure 3.2. The study [5] states that such assumption allows leaks to become part of the water flows. This makes the leaks easy to quantify. The study [5] claims better sophisticated methods and measurements would be required if leaks are not assumed at nodes. With this assumption, the flow and leak notations of Figure 3.2 can be explained with the help of Figure 3.3. A pipe x connecting from node i to j will have l_x leak being lost at the source node (i) of pipe x because of the assumption, after which the remaining s_x flow reaches the target node. It is important to understand that l_x is not included in the reading of s_x.

Figure 3.3: Flowdata representation

3.3 Deriving equations of the network

During MNF we assume that, the total water supplied into the WDN must be equal to the total water leaking out. This yields the below simple equation that is applicable to the WDN as a whole.

Σwater supplied in = Σwater lost due to leaks (3.1) Nodal mass balance implies that the amount of water entering a node is equal to the amount of water leaving that node. The following general equation for every node shows the involved quantities:

Σ In flow = Σ leak flow + Σ out flow (3.2)

To better understand the above two general equations consider the WDN in Fig- ure 3.2 as an example. From the assumptions that there is no leak at the supply pipe and no demands during MNF, the following equations hold true:

l1 = 0

d₁ = d₂ = d₃ = 0 (3.3)

Applying nodal mass balance equations (3.2) to each node (except the supply node) gives the following equations for the demand nodes 3, 5 and 6:

s₂ = d₁ s₅ = d₂ s₄ = d₃

Substituting (3.3) in the above gives

s2 = s5 = s4 = 0 (3.4)

For node 4 the nodal balance equation is

s₃ = s₄+ s₅+ l₄+ l₅ Substituting (3.4) in the above gives

s₃ = l₄+ l₅ (3.5)

For node 2 the nodal balance equation is

s₁ = s₂+ l₂+ s₃+ l₃ Substituting (3.4) and (3.5) in the above gives

s₁ = l₂+ l₄+ l₅+ l₃ (3.6)

We see that (3.6) validates (3.1). If only supply flows into the network are known it is impossible to determine which pipe is leaking. With this information we only know the amount of leak in the network. From the above equations an important conclusion is that for any tree structure WDN, every single flow can be represented as a combination of leaks that are “downwards” of that flow. In the sense that, any pipe where water flowing through pipe x might flow through after x is considered downwards of pipe/flow x. Let column vectors s and l represent the flows and leaks of

the WDN respectively. The i^thelement represents the i^thflow/leak in these respective column vectors. The above set of linear equations can now be compactly described by

s = Dl (3.7)

The i^th row of D matrix represents the breakdown of the i^th flow of s vector as a sum of its downward leaks. The D matrix is always a square matrix of a size equal to number of pipes in a WDN. This gives a system of linear equations that completely describe a tree structure WDN and gives the numerical relationship between its flows and leaks. The D matrix can be obtained for non-tree networks as well. But the breakdown of multiple flows entering a same node may not be numerically accurate.

Because such flows will have the same set of downward leaks and it is difficult to calculate the percentage of contribution of each flow to its downward leaks. If any of such flows are used for leak detection, the nodal analysis for that node given by (3.2) is used and the out flows are replaced by their corresponding downward leaks using (3.7) to give a complete picture. It is possible to modify (3.7) so that the i^th row of that equation gives the nodal balance of the i^th node. But this implementation is better suited for complicated networks which have multiple nodes with an in-degree greater than 1 otherwise, it is simpler to use (3.7). The D matrix is not always invertible due to presence of linearly dependent columns and rows. This also rules out the possibility of using a pseudoinverse. So, even if the matrix s is known, the leaks would be non-trivial to determine. (3.7) can be used to sum up the linear equations derived for each flow for the example given by Figure 3.2 as:











 s₁ s₂ s₃ s₄ s₅













=













0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0













| {z }

D











 l₁ l₂ l₃ l₄ l₅













(3.8)

Since the main goal is to consider underdetermined scenarios, the number of sensors observing the flows should be less than the number of flows in the network. Let col- umn vector Y represent flow data observed from numerically ascending pipe numbers (according to s). This observed data (Y ) also includes measurement noise caused due to error inherent to sensors. It is given in the sensor’s data-sheet as its tolerance.

Let this measurement tolerance be tol. The noise caused by this is assumed to be Gaussian noise with zero mean and standard deviation σ where tol = 3σ. Then the observed data can be given by the following equation:

Y = y + H where y = Cs (3.9)

C is the observation matrix and is given as:

C_i,j =

(1 if i^th observed data is measuring s_j 0 otherwise

H is a column vector whose elements are the random Gaussian noise and column vector y represents the true flow data. (3.9) shows that the observed data are random variables due to the noise. This set of equations cannot be algebraically solved be- cause of the underdetermined nature causing more unknowns than known values and also due to the fact that the measurements are not deterministic but the leaks them- selves are. But it is possible to use y to move forward in leak detection. Substituting (3.7) in the true flow data equation (3.9) gives:

y = CDl (3.10)

The i^th true flow (y_i) is not known but the interval in which it must exist can be estimated with the help of the sensor tolerance:

y_i ∈ Y_i (3.11)

Y_i = [Y_i− tol, Y_i+ tol] (3.12) Each observed datum gives information about the size of leaks present downwards of the observed flow. For each observed datum, the estimated interval of the total downward leak is given by (3.12) and with (3.10) this total leak can be localized to a set of pipes. These set of pipes are called the leak area. Thus, from a set of observed data, leaks are now localized to only certain areas of the network. This is an improvement considering previously only the size of leaks in a network could be estimated.

A detected leak area may be a subset of another detected leak area. In such situations, it is possible to subtract the subset from the superset. This improves the leak area of the superset by making it smaller. A perfect leak area in this thesis would include just one pipe because the leak is localized to the required granularity.

However it is only possible when all the flows of the network or known. Thus, whenever possible the leak areas should be transformed into smaller leak areas. Of course their respective detected leaks should also be appropriately subtracted. These steps are performed by Algorithm I presented in Section 3.5. The algorithm uses the general equations presented here to detect any pipes that have zero leaks and then estimate leak areas from the observed data and the corresponding leak size. None of these leak areas is a subset to any of the others.

Continuing with the WDN example, let flow data for pipes 1,3 and 5 be observed to be Y₁, Y₂ and Y₃ l/s respectively. Using (3.10) gives:

y =



 y₁ y₂ y₃



=





0 1 1 1 1 0 0 0 1 1 0 0 0 0 0















 l₁ l₂ l₃ l₄ l₅











 y₁ ∈ [Y₁ − tol, Y₁+ tol]

y₂ ∈ [Y₂ − tol, Y₂+ tol]

y₃ ∈ [Y₃ − tol, Y₃+ tol]

Using (3.12) for each observation, the required intervals for each y_i as shown above are obtained. The leak areas are obtained from the above equation. The leak area for observed data Y₁ includes pipes 2, 3, 4 and 5, for Y₂ observed data pipes 4 and 5 make up the leak area. The last observed data is taken over a demand pipe and it should be close to zero since there are no demands. The leak area of Y₂ is a subset of the leak area of Y₁ and using Algorithm I for this example will help avoid detecting a leak area that is subset of other leak areas.

The next step is to use these leak areas and localize each leak down to a specific pipe. To do so it is assumed that there is only one pipe that is leaking per leak area.

This assumption allows the detection of the largest possible leak of that leak area instead of estimating several small leaks. Such an assumption gives a result that is in line with the goals of this thesis. There is a possibility that the leak areas detected by Algorithm I overlap. Algorithm II examines the detected leak areas of Algorithm I in a way that ensures that each leak area used to localize a leak down to a specific pipe is disjoint to all the other leak areas used. To help localize a leak to a pipe, the physical characteristics of pipes are used to estimate how likely it is to have such a leak. This estimation is done by creating a probability distribution function for each pipe. The sample space for these distributions is in l/s. This is presented in the next section. The working of Algorithm II is presented in Section 3.6.

3.4 Probability Leak Model/ Estimating leak be- havior in a pipe

This section explains the designing of the probabilistic leak model that adequately describes a pipe’s leak behavior. This probability model is used to estimate how probable a detected leak is in pipes of a network.

3.4.1 Gamma distribution as probability distribution func- tion

It has been noted that pipe characteristics influence the number of breaks in a pipe, this quite logically affects the total amount of water leaking out of that pipe. With this understanding it is contemplated that the pipe characteristics directly influence the pipe’s leak behavior. Therefore pipe characteristics are appropriate to guide the designing of the probability leak model.

To adequately model the probability distribution, it is important to understand how the leak behavior changes. This is done by observing the different breakrate trends with respect to the pipe characteristics. As specified in Chapter 2, the charac- teristics which are assumed to be known are age, length and diameter. The breakrate trends for these characteristics were also discussed. To recall, with increasing age the number of breaks in a pipe increased. Pipes with small diameters had more breaks than pipes with large diameters. Long pipes had more breaks compared to short pipes.

These observations are taken to propose the following:

1. As a pipe ages, its structural integrity deteriorates and the possibility that a small break could cause the entire pipe to burst increases as well. This combined with the observation that older pipes have more breaks shows that they are highly probable to large leaks, while less likely to cause small leaks.

Younger pipes can also cause big bursts, but it is not very likely to occur because its structural integrity is still robust. Hence, young pipes have a low probability to cause large leaks while they are highly probable to leak small amounts of water.

2. The flow in a pipe is affected by its diameter. Large diameter pipes transfer a higher amount of water compared to small diameter pipes. So, a large diameter pipe is very likely to leak more water that a small diameter pipe.

3. As the length of a pipe increases, the number of breaks it has also increases.

This implies that the aggregate leak of a long pipe is larger than the aggregate leak of a short pipe.

Based on the above theories there are at least 2 factors that control a pipe’s leak behavior. Age is seen to control the possible range of leak values while diameter and length control the most likely leak value. Therefore to fully represent the proposed probability leak model a two parameter probability distribution should be used.

A Gamma distribution has two parameters, its probability distribution function

is represented as:

f (x; α, β) = ( _βα

Γ(α)x^α−1e^−βx x ≥ 0, α > 0, β > 0 0 otherwise

Gamma function Γ(α) is (α − 1)! for α >0 , α is the shape parameter and β is the scale parameter. A Gamma distribution is strictly bound on the lower end of x-axis by zero, and has no sharp upper bound. This supports the idea that a pipe’s leak can never be less than zero and that the maximum possible leak can be bound by calculations. The properties of Gamma distribution have been studied extensively and can be implemented without complex computation. Therefore the Gamma distribution is a good fit to express the probability of leak in a pipe.

3.4.2 Effect of shape and scale parameters

The parameters of the Gamma distribution should now be chosen carefully to reflect the effects of age, length and diameter. To accomplish this, the changes brought forth by each parameter of the Gamma distribution should be observed.

Let the mean and standard deviation be m and σ² for the Gamma distribution.

Their equations in terms of shape and scale parameters are given as:

m = αβ σ² = αβ²

Solving the above equations for α and β gives the below relationships between the shape parameter, mean and standard deviation as :

β = σ²

m (3.13)

The effect of scale parameter (β) is easier to observe out of the two. By definition scale parameter has the effect of stretching or compressing a distribution. If β is large then the distribution is more spread out. This is supported by (3.13) which shows β is proportional to variance and the variance controls the “spread” of the distribution.

The plot in Figure 3.4 shows the spread of the distribution being affected by varying scale parameters for a constant shape value. This spread effectively gives the range of possible values that can occur for a distribution. Therefore scale parameter (β) can be used to control the range of leak values that are possible for a particular pipe.

Figure 3.4: Affect of scale parameter in Gamma distribution

The Gamma distribution is a family of differently shaped distributions. The shape parameter (α) controls the shape of the distribution being presented. There are three fundamental shapes characterized by the following cases:

1. α < 1 : Gamma distribution is asymptotic to both the vertical and horizontal axes.

2. α = 1 : This special case represents an exponential distribution whose rate is

1 β.

3. α > 1 : The mass of distribution is concentrated to the left of the distribution and is thus skewed. The skewness reduces and the mode increases as the value of α increases.

The skewness and mode are given by the following equations:

mode = (α − 1)β α ≥ 1 (3.14)

skewness = 2

√α

The above equations defend the observations that for constant scale, increasing the shape parameter increases the mode value while decreasing the skewness. Therefore the shape parameter (α) can be used to control the most likely leak value of a pipe.

The cases mentioned above can all be observed in the plot shown in Figure 3.5.

Figure 3.5: Affect of shape parameter in Gamma distribution

To summarize, the scale parameter (β) controls the range of leak values that are probable for a pipe and the shape parameter (α) influences the highest probable leak value that can occur for a pipe. Thus the scale parameter (β) is apt to represent the age of pipes as it governs the range of possible leaks for a pipe. The shape parameter

(α) influences the most likely leak and is most suitable to represent the effect of length and diameter.

3.4.3 Model parameterization

The parameters of the Gamma distribution should be translated into numeric val- ues that sufficiently represent the effect of the physical characteristics on the leak behavior of a pipe.

Pipe characteristics vary widely in a network, especially urban areas where parts of the original network are expanded to suit future demands. These expansions can take place many times in different years and may possibly require some parts to be replaced with pipes which are newer than its surrounding pipes. An urban network thus has a heterogeneous collection of pipes.

To present continuous parameter ranges corresponding to each of the character- istics that can exist in a network requires a large amount of accurate information. It needs the amount of water a pipe of certain characteristic has been observed to leak.

Such data is expensive to obtain owing to the preciseness and additional measure- ment infrastructure it would require. It does have the potential to be more powerful in predictions. But in absence of such extensive data and to maintain a relatively simple mathematical framework, each pipe characteristic is classified into several categories. The following categorization criteria is formulated to help divide each characteristic into groups that are not too exclusive or inclusive in the presence of insufficient data:

1. Members of each category should be similar enough to each other in their breakrate trends.

2. The different categories of a pipe characteristic need to be distinct from each other in their breakrate trends.

The unimodality property helps in estimating the most likely leak of a pipe and thus it needs to be retained. Therefore the minimum shape parameter to represent the leak model is :

αmin = 1; (3.15)

Calculation of scale (β) parameter

To calculate the scale parameter of a pipe three age categories are proposed that classify a pipe as young, middle-aged and old. This is partly based on the classifica- tion presented in [18], where pipes were grouped into three categories ranging from youngest to oldest with their respective breakrates. These categories satisfy all of

the categorization criteria. In this thesis, the actual age of a pipe is not available. In- stead during simulation, a pipe is chosen to be in one of these age groups. Therefore the numeric age intervals of each of these age groups are not discussed.

An equation that relates the range of possible leaks and scale parameter needs to be theorized. To do this a Gamma distribution that has the minimum shape parameter (3.15) is considered. Its mode is calculated using (3.14):

mode = (α_min− 1)β

= (1 − 1)β

= 0

From the above it can be seen that the highest possible leak is zero for a Gamma distribution with the minimum shape parameter and that the scale parameter does not influence this value. Therefore this is a case of a pipe that is most likely to not leak at all. Even in this case, the effect of age should be present. Since the minimum leak value will always be zero (property of Gamma distribution), the maximum leak value should be controlled to effectively control the range of possible leak values.

In statistics the upper limit of data for a distribution is given by Tukey’s fences as:

upper limit = Q₃+ 1.5(IQR) (3.16)

where

Q₃ = third quartile, and

IQR = Q₃− Q₁ = Inter Quartile Range

A gamma distribution with α = α_min is an exponential distribution with a rate of

1

β. The quantile function is used to obtain the necessary quartiles. This function for an exponential distribution with a rate λ is:

Q(p) = −ln(1 − p)

λ for 0 ≤ p < 1

Using the above equation for a rate of _β¹, the first and third quartiles are given as Q(p = 1

4) = Q₁ = log(4 3)β Q(p = 3

4) = Q₃ = log(4)β

Substituting the above equations in (3.16) gives the the below equation:

upper limit = log(4)β + 1.5(log(4)β − log(4

3)β) (3.17)

This equation clearly shows the scale parameter controlling the upper limit of a dis- tribution, which effectively controls the range of possible leaks. The above equation can be thus solved for β by substituting different upper limit values.

The upper limit is taken to be 90% and 15% of the maximum theoretical flow that can reach the pipes for the old and young pipes respectively. This flow value is usually the supply flow going into that area/network. The limit of 90% is because, any leaks bigger than that are considered outlier cases where they become visible leaks and are detected through other means. The limit of 15% is to emphasize the difference in amount of leak that a young and old pipe can cause. The upper limit is not a hard stop on the possible leak values. It shows that leaks smaller than the upper limit are very likely and leaks greater that the upper limit can occur (outlier case) but they are not very likely. The scale parameters for young and old are represented as β_y and β_o, then the scale for middle aged pipes is given as:

β_m = β_y+ β_o 2

The scale parameter for each of the age groups have been chosen according to hy- pothesis (1). That is with increasing age, the range of possible leaks increases. The scale parameters in increasing order of value are:

β_y < β_m < β_o (3.18)

Calculation of shape (α) parameter

Pipe length and diameter need to be categorized and assigned their respective shape parameters according to the categorization criteria. The diameter categories chosen by [20] satisfies the categorization criteria. Each category has a difference of 50mm between its lower and upper limit. However all pipes with diameter ≤ 100mm are in one category. This is the smallest diameter category and according to hypothesis (2) causes small leaks and therefore has the smallest shape parameter out of all other diameter categories. Pipes that have a diameter > 300mm are not considered because there is no strong influence of such pipes on breakrates [20]. Therefore regardless of the characteristics present in a network each pipe can only belong to any of the five diameter categories. Let the shape parameter for each of these categories be represented as below:

αd 100, αd 150, αd 200, αd 250, αd 300

The subscript shows the upper limit (in mm) of the diameter category which the shape parameter represents.

These shape parameters must be calculated according to hypothesis 2. This implies that the shape parameter of these categories must increase as the size of the diameter increases. Therefore they are in the order:

α_{d 300} ≥ α_{d 250} ≥ α_{d 200} ≥ α_{d 150} ≥ α_{d 100} (3.19) Pipe length is categorized according to the data trends shown in the study by Wengstr¨om [19] to satisfy the categorization criteria. The upper and lower limits of each length category differs by 200mm. Pipes with length ≤ 200mm are in the shortest length category and are considered to cause no leaks compared to other length categories. The number of categories based on length depends on the char- acteristics of a network. Let the shape parameter for the first and last of the length categories be represented as below:

α_{l 200}, α_{l last}

The subscript shows the upper limit (in mm) of the length category which the shape parameter represents. Since it is not possible to know the last category’s upper limit, it is simply represented in the subscript as last.

The shape parameters must be calculated according to the hypothesis 3. This implies that the shape parameter of length categories must increase as its upper limit increases. Therefore they are in the order:

α_{l 200} ≤ α_{l 400}.... ≤ α_{l last} (3.20)

The shape parameter of a pipe is proposed to be a sum of its length shape parameter and diameter shape parameter. An exception to this exists and is presented further below.

Pipes in the smallest diameter category and the shortest length category are considered to cause no leaks and thus have the minimum shape parameter as shown in (3.15). Their shape parameters are thus:

α_{l 200}= α_{d 100} = α_min (3.21)

A pipe belonging to the shortest length category and the smallest diameter category is a special case. The shape parameter of such a pipe is 1 and not 2 to maintain the assumption that such a pipe would not cause any leaks. This is the exception mentioned above.

If the shape parameter of a distribution cannot be less than 1, then the highest possible leak is given by the mode of that distribution. This fact will be used to find the largest shape parameter of length and diameter categories.