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I - Apreciação e aprovação da Ata da:

44ªReunião Ordinária da Congregação Provisória da Faculdade de Tecnologia. pág. 06.

II - Expediente:

Afastamentos:

01)Prof. Dr. André Franceschi de Angelis, nos dias 14 e 15/05/13, para participar da avaliação de curso na FATEC de Carapicuíba – SP. pág. 09.

02)Profa. Dra. Simone Andréa Pozza no dia 23/05/13, para participar do evento “Seminário Sobre Inventários de Emissões de Gases de Efeito Estufa em Cidades” em São Paulo – SP. pág. 10.

03)Profa. Dra. Maria Aparecida Carvalho de Medeiros, nos dias 23 e 24/05/13, para participar da avaliação de curso na FATEC para reconhecimento – CEE em Praia Grande – SP. pág. 11.

04)Prof. Dr. Peterson Bueno de Moraes, no período de 25 à 28/05/13, para participar da 36ª Reunião Anual da Sociedade Brasileira de Química em Águas de Lindóia – SP. pág. 12.

05)Profa. Dra. Simone Andréa Pozza, no dia 27/05/13, para ministrar palestra na Semana de Meio Ambiente no ISCA Faculdades, Limeira – SP. pág. 13.

06)Prof. Dr. Francisco José Arnold no período de 03 à 05/06/13, para participar de Visita in Loco – Avaliação Institucional de Credenciamento do IES em Joatuba – MG. pág. 14.

07)Profa. Dra. Gisela de Aragão Umbuzeiro, no dia 12/06/13, para ministrar seminário na UNESP em Bauru – SP. pág. 15.

08)Profa. Dra. Regina Lúcia de Oliveira Moraes, no período de 24/06/13 à 05/07/13, para participar do The 43rd Annual IEEE/IFIP International Conference on Dependable Computing and Fault Tolerance – Encontro de Pesquisa projeto DEVASSES, em Budapeste – Hungria e Coimbra – Portugal. pág. 16.

09)Prof. Dr. Cristiano de Mello Gallep, no período de 20 à 26/07/13, para apresentar trabalho no Summer School on Biophotons no Pro-MAS / Hotel Raj Mahal, Castrop-Rauxel – Alemanha. pág. 17.

10)Profa. Dra. Simone Andréa Pozza, no período de 31/08 à 02/09/13, para participar do 1º Congresso Nacional do Projeto Rondon na USP de Ribeirão Preto. pág. 18.

11)Profa. Dra. Carmenlúcia Santos Giordado Penteado, no dia 23/05/13, para participar de Seminário no Instituto de Eletrotécnica e Energia da USP em São Paulo. pág. 19.

12)Profa. Dra. Rosa Cristina Cecche Lintz, no período de 29/08/13 à 04/11/13, para participar de atividades de pesquisa e ensino na Universidade Politécnica de Valência, Valência - Espanha. pág. 20.

13)Profa. Dra. Cassiana Maria Reganhan Coneglian, no dia 28/05/13, para participar de bancas de dissertação de mestrado na UNESP de Rio Claro. pág. 22.

14)Profa. Dra. Simone Andréa Pozza, no dia 21/06/13, para participar de bancas de mestrado na USP de São Carlos. pág. 23.

15)Prof. Dr. Mauro Menzori, no período de 22 à 24/05/13, para participar de lançamento de livro e encerramento de contrato com a Editora da Universidade Federal de Juiz de Fora, Juiz de Fora – MG. pág. 24.

16)Resultado do Prêmio PAEPE da FT – Edição 2013. pág. 25.

17)Parecer CG 41/2013 - Prorrogação de mandato de membros não natos da Comissão de Graduação. pág. 27.

18)Assuntos diversos.

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III - Ordem do Dia:

01)Apreciação sobre o parecer do projeto de iniciação científica da aluna Bruna Magalhães Prates Pereira, orientanda da Profa. Dra. Gisleiva Cristina dos Santos Ferreira. pág. 28.

02)Apreciação sobre o parecer do projeto de iniciação científica da aluna Sarah Raquel de Lima, orientanda da Profa. Dra. Ana Estela Antunes da Silva. pág. 37.

03)Apreciação sobre o parecer do projeto de iniciação científica do aluno Ciro Eduardo Pereira Bueno Junior, orientando da Profa. Dra. Eloisa Dezen-Kempter. pág. 44.

04)Apreciação sobre o parecer do projeto de iniciação científica da aluna Adrielle Reis de Oliveira, orientanda da Profa. Dra. Eloisa Dezen-Kempter. pág. 52.

05)Apreciação sobre o parecer do projeto de iniciação científica do aluno Robson Fernando de Jesus Pereira, orientando da Profa. Dra. Eloisa Dezen-Kempter. pág. 59.

06)Apreciação sobre o parecer do projeto de iniciação científica da aluna Rafaella do Amarante Carneiro, orientanda da Profa. Dra. Eloisa Dezen-Kempter. pág. 67.

07)Apreciação sobre o parecer do projeto de iniciação científica do aluno Ruan Carneir Cavalcante de Miranda, orientando da Profa. Dra. Eloisa Dezen-Kempter. pág. 75.

08)Apreciação sobre o parecer do projeto de iniciação científica da aluna Gabriela Fátima Barboza da Mata, orientanda da Profa. Dra. Marta Siviero Guilherne Pires. pág. 82.

09)Apreciação do Parecer Final do Concurso para Professor Titular na área de Saneamento Ambiental. pág. 91.

10)Parecer CG 40/2013 - Núcleo Docente Estruturante – NDE. pág. 93.

11)Parecer CPG 11/2013 - Credenciamento do Prof. Dr. Fábio Kummrow como professor visitante no Programa de Pós-Graduação da FT. pág. 95.

12)Parecer CPG 12/2013 - Credenciamento da Profa. Dra. Maria Beatriz Borer Morel como professora visitante no Programa de Pós-Graduação da FT. pág. 96.

13)Parecer CPG 13/2013 - Credenciamento do Prof. Dr. Ricardo da Silva Torres como professor visitante no Programa de Pós-Graduação da FT. pág. 97.

14)Parecer CPG 07/2013 - Credenciamento do Prof. Dr. Ivani Rodrigues da Silva como professor visitante no Programa de Pós-Graduação da FT. pág. 98.

15)Parecer CPG 14/2013 – Resultado das Eleições para Membros da CPG. pág. 99.

16)Parecer CE 08/2013 – Reestruturação do Curso de Especialização do Curso de Governança em tecnologia da Informação. pág. 102.

17)Prorrogação da bolsa de Professor Visitante do Prof. Dr. Stefano Mambretti na FT pelo período de mais um ano. pág. 123.

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���������������� ��

Professor Visitante do Exterior

Prof. Dr. Stefano Mambretti

Exterior: Via Ferruccio Parri 81

20097 San Donato Milanese (MI)

Italia

Brasil: Rua Independencia 40

Vila sao Joao

Limeira/SP.

Email: [email protected]

Campinas, 7 de maio de 2013.

Ao Diretor da Faculdade de Tecnologia

Universidade Estadual de Campinas

Prof. Dr. José Geraldo Pena de Andrade

RE: Pedido de prorrogação de bolsa professor visitante do exterior

Prezado Prof. Dr. José Geraldo Pena de Andrade,

Venho, por meio deste, apresentar minha solicitação para prorrogação de bolsa professor visitante

do exterior, conforme o termo de outorga de bolsa para professor visitante de instituição

estrangeira.

Estou à disposição para dirimir eventuais dúvidas à respeito desta solicitação.

Atenciosamente,

________________________________

Prof. Dr. Stefano Mambretti

� �

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ork

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desc

rib

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in t

he f

inal p

ap

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tance

will b

e b

ase

d o

n t

he f

ull-l

eng

th p

ap

er, w

hic

hif a

ccep

ted

fo

r p

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licati

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ust

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nu

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cliff

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p o

verl

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kin

g t

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ão

s b

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ho

tel

near

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only

5km

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mP

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Page 132: ˘ ˇ ˆ˙˝˛ ˚ ˜ ! #ˇ - ft.unicamp.br · 5 III - Ordem do Dia : 01) Apreciação sobre o parecer do projeto de iniciação científica da aluna Bruna Magalhães Prates Pereira,

Urb

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ms. T

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eeti

ng

has b

een r

eco

nvened

follo

win

g t

he s

uccess o

f th

e f

irst

co

nfe

rence h

eld

in t

he N

ew

Fo

rest,

ho

me t

o t

he W

essex I

nsti

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of

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log

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012

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Wate

r syste

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th

e u

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envir

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pp

ly n

etw

ork

s a

s w

ell a

ssew

ag

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nd

sto

rm d

rain

ag

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ms. T

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inte

ract

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ach

oth

er

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d w

ith

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bo

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s s

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as r

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, an

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cit

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xp

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urb

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ap

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d t

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pacts

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od

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a g

reat

deal o

fatt

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rom

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d p

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nal

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e w

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d t

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Tsunami F

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Email:[email protected]

www.witpress.com

WITPRESS

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Tsunami

From

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Tsunami From Fundamentals to Damage Mitigation S. MambrettiSAFETY & SECURITY ENGINEERING

9781845

647704

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TSUNAMIFrom Fundamentals to Damage Mitigation

Edited by

Stefano MambrettiUniversidade Estadual de Campinas, Brasil

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Published by

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Stefano MambrettiUniversidade Estadual de Campinas, Brasil

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Contents

Preface ....................................................................................................... ix

Chapter 1

Tsunami: from the open sea to the coastal zone and beyond .......................... 1

G. Mastronuzzi, H. Brückner, P.M. De Martini & H. Regnauld

1 Premise ............................................................................................................. 1

2 Genesis of a tsunami ........................................................................................ 5

3 Evidence of the impacts of tsunami ................................................................. 9

3.1 Offshore evidence .................................................................................. 10

3.2 Evidence on the coastal plain ................................................................. 14

3.3 Evidence on the beach dune system ....................................................... 17

3.4 Evidence on rocky coasts ....................................................................... 20

4 Field evidence and risk assessment ................................................................ 23

Acknowledgments .............................................................................................. 26

References .......................................................................................................... 27

Chapter 2

An inverse algorithm for reconstructing an initial Tsunami waveform ...... 37

Tatyana Voronina

1 Introduction .................................................................................................... 37

2 Statement of the problem ............................................................................... 42

3 Inverse method ............................................................................................... 44

4 r-solution ........................................................................................................ 44

5 Discretization of the problem ......................................................................... 45

6 Numerical experiments: description and discussion ...................................... 48

7 Conclusion ..................................................................................................... 54

Acknowledgements ............................................................................................ 56

References .......................................................................................................... 56

Chapter 3

Tsunami maximum flooding assessment in GIS environment ...................... 61

G. Mastronuzzi, S. Ferilli, A. Marsico, M. Milella, C. Pignatelli, A. Piscitelli,

P. Sansò & D. Capolongo

1 Introduction .................................................................................................... 61

2 Coastal geomorphology ................................................................................. 64

3 Materials and methods ................................................................................... 68

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4 Tsunami height and manning number ............................................................ 70

5 The flooding area assessment ......................................................................... 74

6 Discussion and conclusions ............................................................................ 75

Acknowledgements ............................................................................................ 77

References .......................................................................................................... 78

Chapter 4

Tsunami early warning coordination centres ................................................. 81

J. Santos-Reyes & A.N. Beard

1 Introduction .................................................................................................... 81

2 A systemic disaster management system model ............................................ 83

2.1 The basic structural organization of the model ...................................... 83

2.2 System 2: early warning coordination centre ......................................... 88

3 Modelling EWCC for the case of an Indian Ocean Country .......................... 88

4 Conclusions and future work ......................................................................... 91

Acknowledgements ............................................................................................ 91

Annexure A......................................................................................................... 92

References .......................................................................................................... 92

Chapter 5

RC buildings performance under the 2011 great East Japan Tsunami ....... 95

C. Cuadra

1 Introduction .................................................................................................... 95

2 Characteristics of the earthquake and tsunami ............................................... 96

2.1 Tsunami source ...................................................................................... 96

3 Damages due to tsunami .............................................................................. 100

3.1 Selected area ........................................................................................ 100

3.2 Damages on buildings .......................................................................... 104

3.2.1 Damages on wooden houses ...................................................... 104

3.2.2 Damages on steel structures ...................................................... 105

3.2.3 Damages on RC structures ........................................................ 107

4 Conclusions .................................................................................................. 109

References ........................................................................................................ 109

Chapter 6

Infrastructure maintenance and disaster prevention measures on Islands:

example of the Izu Islands near Tokyo ......................................................... 111

H. Gotoh, M. Takezawa & T. Murata

1 Introduction .................................................................................................. 112

2 Outlines of Izu Islands ................................................................................. 112

3 Population, aging, and industrial structures of Izu Islands ........................... 121

4 Land uses, infrastructures, and tourism of Izu Islands ................................. 123

5 Living standards and environmental hygiene in Izu Islands ........................ 128

6 Disaster prevention measures ....................................................................... 129

7 Conclusions .................................................................................................. 137

References ........................................................................................................ 137

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Chapter 7

Health-related impacts of Tsunami disasters ............................................... 139

Mark E. Keim

1 Background nature of tsunamis .................................................................... 139

1.1 Definition ............................................................................................. 139

1.2 Causes of tsunamis ............................................................................... 140

1.3 The physics of tsunami phenomenon ................................................... 141

2 Scope and relative importance of tsunamis .................................................. 142

3 Factors that contribute to the tsunami problem ............................................ 144

4 Factors affecting tsunami occurrence and severity ...................................... 144

5 Public health impact: historical perspective ................................................. 145

6 Factors influencing mortality and morbidity ................................................ 145

6.1 Mortality trends .................................................................................... 145

6.2 Tsunami-associated illness and injury .................................................. 146

6.3 Infectious diseases ................................................................................ 146

6.4 Worsening of chronic diseases ............................................................. 148

6.5 Psychosocial consequences .................................................................. 149

7 Conclusion ................................................................................................... 149

Disclaimer ......................................................................................................... 150

References ........................................................................................................ 150

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WATER PARTICLE KINEMATICS QUANTUM APPROACH: A CHALLENGE FORSPRINKLER IRRIGATION SYSTEMS†

D. DE WRACHIEN1*, G. LORENZINI2 AND S. MAMBRETTI3

1Department of Agricultural and Environmental Sciences, University of Milan, Italy

2Department of Industrial Engineering, University of Parma, Italy3Faculdade de Tecnologia, University of Campinas, Brazil

ABSTRACT

Designing a sprinkler irrigation plant is always associated with a full understanding of the kinematics of droplets during theiraerial path. This requires a very complicated modelling of the problem, as many variables affect one another in contributingto the whole process. The literature offers different descriptive methods among which is the ballistic one, to which theauthors have recently given a novel contribution, and which is also reported here. In addition to this, the present paperintroduces two novel quantum approaches applied to describe droplet kinematics, based on the time-dependent Schrödingerequation and on the Scale Relativity Theory. Such an idea not only completes the classical description with a mean moretightly describing the microscopic phenomenon but also gives a broadly applicable tool to describe the actual kinematicsof water droplets in sprinkler irrigation. Copyright © 2013 John Wiley & Sons, Ltd.

key words: droplet kinematics; spray flow; theoretical modelling; quantum mechanics; Schrödinger equation; Scale Relativity Theory

Received 29 November 2011; Revised 12 November 2012; Accepted 13 November 2012

RÉSUMÉ

La conception d’une installation d’irrigation par aspersion est toujours associée à la pleine compréhension de la cinématiquedes gouttelettes d’eau au cours de leur parcours aérien. La modélisation du problème s’avère très compliquée, car autant devariables s’influencent mutuellement en contribuant à l’ensemble du processus. La littérature propose différentes méthodesdescriptives, dont la méthode balistique, à laquelle les auteurs ont récemment donné une nouvelle contribution, qui estégalement rapportée ici. En plus de cela, le présent document présente deux nouvelles approches de mécanique quantiqueappliquée pour décrire la cinématique de gouttelettes, sur la base de l’équation de Schrödinger dépendante du temps et dela théorie de la relativité d’échelle. Une telle idée non seulement complète la description classique avec une plus étroitedescription du phénomène microscopique, mais aussi donne un outil largement applicable pour décrire la cinématique réellede gouttelettes d’eau dans l’irrigation par aspersion. Copyright © 2013 John Wiley & Sons, Ltd.

mots clés: cinématique des gouttelettes; débit de pulvérisation; modélisation théorique; mécanique quantique; équation de Schrödinger; théorie de larelativité d’échelle

INTRODUCTION

In every scientific context where they appear, the descrip-tion of jets, even of every single jet component, is definitelyone of the more challenging issues because of the mutual

affections of each particle and variable involved. This istrue also in sprinkler irrigation jets, where the in-flightkinematics of every water droplet is closely related to alarge number of parameters and to the non-linear depen-dence of one on another. This is why, even in well-established literature contributions (Kinzer and Gunn,1951; Edling, 1985; Kincaid and Longley, 1989; Kellerand Bliesner, 1990), a fully satisfactory description of theprocess cannot be arrived at, the studies being consistentlyconditioned by the many empiricisms adopted, resulting in

* Correspondence to: Stefano Mambretti, Faculdade de Tecnologia –

University of Campinas (UNICAMP), R. Paschoal Marmo, 1888 - CEP:13484-332 - Jd. Nova Itália - Limeira, SP, Brazil. Email: [email protected]† La dynamique quantique appliquée aux particules d’eau: un défi pour lessystèmes d’irrigation par aspersion.

IRRIGATION AND DRAINAGE

Irrig. and Drain. 62: 156–160 (2013)

Published online 31 March 2013 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/ird.1724

Copyright © 2013 John Wiley & Sons, Ltd.

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strongly case-dependent descriptions and failing to givean accurate description of the actual phenomenon. This iswhy a ballistic approach based on classical mechanicswas recently proposed and validated, the main results ofwhich will be summarised in the next section, able toreach a closed-form solution of the governing equationsbased on a few simplifying (but not empirical) hypotheses(Lorenzini, 2002, 2004, 2006; De Wrachien and Lorenzini,2006). It was the whole literature picture and the specificmodelling experience that suggested to the authors ofthis paper that there was a great need for a closer examina-tion of the kinematics affecting in-flight water particles,imagining the challenge of a quantum approach to such aproblem, also justified by the general dimension of thesystem faced. In general, one could say that in classicalmechanics the condition of a particle may be described byspecifying position and momentum as accurately asdesired; in contrast, in quantum mechanics this is not possi-ble, as Heisenberg’s Uncertainty Principle states (Dirac,1931). What is here presented, consequently, is a novelchallenge in the droplet-related scientific literature pano-rama: a double quantum approach based on the Schrödingerequation, on the one hand, and on the Scale RelativityTheory (Nottale, 1992) in the form of a Riccati equation,on the other.

WATER PARTICLE KINEMATICS: THENEWTONIAN APPROACH

A complete picture of the main state-of the-art results inNewtonian droplet kinematics modelling is available inLorenzini (2004, 2006) and De Wrachien and Lorenzini(2006); for more detailed thematic information we sug-gest reference is made to those publications, as a fullreview of such a topic is not the specific aim of the pres-ent investigation. In addition it may be observed that, inrecent decades and mainly relying on a Lagrangiandescription approach, the spray (or small droplets) kine-matics modelling problem has been broadly investigated,both in sprinkler irrigation and in chemical spray contexts(Teske et al., 1998a, 1998b; Keller and Bliesner, 1990;Teske and Ice, 2002). Spray drift modelling insights inthe classical Lagrangian formulation have been reportedin Hewitt et al. (2002) and in Bird et al. (2002). In theabove-quoted publications by De Wrachien and Lorenzini,a simplified droplet kinematics analytical model was alsodefined and validated, based on a system of parametricequations to describe a sprinkler droplet aerial path,obtained integrating a system of differential equations.Synthetically the parametric equations for position andmotion arrived at are:

x tð Þ ¼ m

kln

v0xk

mt þ 1

!

y tð Þ ¼ h$ m

kln

cos arctan

ffiffiffiffi

k

m

r

v0yffiffiffiffiffiffiffi

n

mg

r

0

B

B

B

@

1

C

C

C

A

cos arctan

ffiffiffiffi

k

m

r

v0yffiffiffiffiffiffiffi

n

mg

r $ t

ffiffiffiffiffiffiffi

kngp

m

0

B

B

B

@

1

C

C

C

A

(1)

and

x&tð Þ ¼ mv0x

m þ kv0xt

y&tð Þ ¼ $

ffiffiffiffiffi

ng

k

r

tan $ffiffiffiffiffiffiffiffi

ngkp

mt þ arctan

ffiffiffiffiffi

k

ng

s

v0y

!" #

(2)

where: x [m] is the horizontal coordinate; y [m] is the verticalcoordinate; x

&[m s-1] is the horizontal velocity component;

y&[m s-1] is the vertical velocity component; v0x [m s-1] is

the horizontal velocity component at t = 0 s; v0y [m s-1]is the vertical velocity component at t = 0 s; t [s] is time;m [kg]is the particle mass; n [kg] is the particle mass reduced becauseof buoyancy; g [m s-2] is gravity acceleration; h [m] is the y

coordinate value at t = 0 s; k ¼ frA2 k [kg m-1] is the friction

coefficient, depending on the dimensionless friction factoraccording to the Fanning f [-] (Bird et al., 1960). Such a modelallows for satisfactory agreement with a number of the litera-ture data (Edling, 1985; Thompson et al., 1993), as broadlydemonstrated (Lorenzini, 2004, 2006; De Wrachien andLorenzini, 2006). What is extremely relevant in such anapproach is that, being fully analytical, its results can beapplied to any particular system and process configurationwhich may occur, provided that the conditions for which theequations have been written are respected.However, the bestresults were achieved for high Re [-] values.

WATER PARTICLE KINEMATICS: THEQUANTUM APPROACH

Quantum mechanics descriptions are based on the doublenature, wave and particle, of the elements considered. Theanalytical mean through which their kinematic analysis canbe performed is the time-dependent Schrödinger equation,which utilises the definition of a wave function c relatedto the probability of finding a particle, among a group ofother ones, at a specific location at a specific instant in time.The challenge in the present investigation is to apply such an

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approach to a water spray, each element of which is consid-ered as a particle and simultaneously also as a wave: the mo-tion of the latter is given in accordance with the solution ofSchrödinger’s equation; for the former velocity is definedstarting from the particle initial location and the wave func-tion phase gradient. If a set of initial positions is considered,which defines a particle motions system (i.e. quantum orBohmian trajectories) ruled by a specific wave function,then the probability of finding an element in a certain spaceand time location will be given by the quantum mechanicsprobability density (Holland, 2011). In general it can bestated that a quantum approach to kinematics allows one toface and solve both steady and transient states in relationto wave–particle systems and to Lagrangian or Euleriandescriptions (Goldstein et al., 2011). F [N] being the force,m [kg] the particle mass and Q [m] its trajectory, the quan-tum form of Newton’s second law of dynamics for a singleparticle may be written as

md2Q tð Þdt2

¼ F tð Þ (3)

while for a multiple system of N particles (k = 1, . . ., N) onemay arrive at

mk

d2Qk tð Þdt2

¼ rk Vð jVctquÞQ tð Þ (4)

[would be mkd2Qk tð Þdt2

¼ $rkV )Q tð Þ in classical mechanics,where the trajectory could also be expressed in the form ofa Bohmian, provided that V is modified to account fortime dependence (Lopreore and Wyatt, 1999)] where r k

is the 3-D gradient operator in relation to the kth particle;

V is a potential function; and Vctqu is the quantum potential:

Vctqu ¼ $

X

N

j¼1

ℏ2

2mj

r2j cj jcj j 1≤j < k≤Nð Þ (5)

containing the Dirac constant ħ = 1.055* 10$ 34J s.

SINGLE PARTICLE QUANTUM TRAJECTORY

As may be deduced from the general theory, when thequantum potential tends to zero, quantum and Newtoniantrajectories become comparable. This allows one to build abridge between these two fluid-dynamic contexts. Consideringa multi-particle system (the special application to waterdroplets in this paper comes within this description), oneelement of which affects another (see the concept of vectorpotential), a single particle quantum motion can be formulatedby means of the time-dependent Schrödinger equation:

D2r2c!x; t $ 1

2)m)V!

x; t)c!x; t ¼ $i)D) @

@t)c!x; t (6)

where D ¼ ℏ

2:m [m2 s-1] is the diffusion coefficient; c x!; t ¼

R x!; t)expS x

!; t , R x

!; t and S x

!; t being the wave function

amplitude and phase, respectively. Wyatt (2005) and Ghosh(2011) started from the Schrödinger equation (Equation 6),splitting it into a system of two fluid dynamic equations (knownas ‘quantum fluid dynamics equations’), namely the continuityequation and the Euler-type equation of motion (respectively):

@

@tr x

!; t þ rr x; t) x! x

!; t ¼ 0 (7)

d

dtv!x!; t , @

@tþ v

!x!; t)r v

!x!; t

¼ $1m

)rV x!; t þ Q x

!; t (8)

wherer x!; t ¼ R2 x!; t [kg m-3] is the fluid density and v! x

!; t ¼

2)D)rS x!; t the velocity field. Equations (7) and (8) allow for

full comprehension and assessment of a fluid particle motion,in case the force field by which such a particle is affected is thatdriven by the Newtonian potential Vx

!; t and by the additional

quantum potential, which can be expressed by

Q x!; t ¼ $2)m)D2)r2Rx

!; t

R x!; t(9)

The whole picture provided shows how a particle trajectorydescription relying on a quantum approach may help completemodelling of the process because of the substantial and clearformal similarity to the Newtonian case of a particle in motionproduced by a system of forces associated with the gradient of apotential representing the mutual interactions, thus providing aunified method for the computation of particle kinematicstaking into account both Newtonian and quantum features.It is of course a challenging task, that of starting from Equations(7) and (8) for particle motion description, to arrive at a full so-lution in closed form, i.e. by analytical means. This is why al-most exclusively non-analytical solutions are reported in therelated scientific literature. As in Kendrick (2011), two main ap-proaches are generally followed: ‘numerical approximations’and ‘dynamic approximations’. Especially for what pertainsto the former, a big effort has recently been made, also in rela-tion to the increased computer and software performance butmore developments are expected.

SINGLE PARTICLE SCALE RELATIVITYTHEORY TRAJECTORY

As well defined by Nottale et al. (1992), the Scale RelativityTheory may be applied to quantum mechanics for a particleof any dimension in a problem involving a space-time

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frame. The first application of Scale Relativity Theory todescribe a 1-D path of a free-falling particle was achievedby Hermann (1997), who worked on the Schrödinger time-dependent equation using a probability density function. Inparticular, for a semi-infinite domain, the Newton equationwritten in the complex field is

$ru ¼ m)dtV (10)

u a scalar potential being and V a complex velocity. WritingEquation (10) again in form of its real and imaginary parts,one arrives at (Hermann, 1997)

$D)ΔU $ U)rð ÞU ¼ $rU@

@tU ¼ 0

(

(11)

In the last equations U is the imaginary part of complexvelocity and D is the diffusion coefficient, previouslydefined. As in Al-Rashid et al. (2011), the first equation ofsystem (11) for a 1-D domain assumes the form of a Riccatiequation:

d

dxUx ¼ $m

ℏ)U2xþ 2 )ux $ c)m (12)

where c is a case-dependent constant of integration. Defin-ing y(x) as an arbitrary function of x, the first equation insystem (11) may be also written as (Al-Rashid et al., 2011)

d2

dx2yx$ 2)m

ℏ2 )ux $ c)m)yx ¼ 0 (13)

The novelty represented by Equation (13) is that of inves-tigating a 1-D particle moving in a quantum contextavoiding the direct use of the Schrödinger equation, thusgiving a potentially easier tool to compute any sort of trajec-tory. Apart from this main outcome, in general (Hermann,1997) some numerical codes to analyse the motion of quan-tum particles in the light of the Scale Relativity Theory arecurrently at the researcher’s disposal, providing reasonablyaccurate results, when compared to the traditional quantumones, achieved solving the Schrödinger time-dependentequation.

CONCLUSIONS

The general (and generic) problem of describing an in-flightparticle trajectory is common in many scientific and techni-cal contexts, such as in fire suppression, sprinkler irrigationand chemical spray dispersion. Obviously, depending on theparticular field of application, different aspects may becomecrucial in the investigations performed (e.g. the heat transferimplications when considering fire suppression). Moreover,depending on the specific systems and processes investi-gated, one may decide on different approaches to be applied

and/or on different hypotheses to be superimposed. In thissense the scientific literature offers a broad panorama of ref-erences which one may refer to. The present paper, in anovel way, faced the challenge of putting together twoapparently different kind of approaches, the Newtonianand the quantum, in relation to the problem of a particle ki-nematics investigation aimed in perspective at the full char-acterisation of water droplet kinematics in sprinklerirrigation. Having such a purpose in mind, the classicalNewtonian approach of a ballistic analytical model was pro-posed to describe both the water particle kinematics and adouble quantum approach formalised by the ‘traditional’Schrödinger time-dependent equation written in the splitform of quantum fluid dynamics equations, on the one hand,and by a Scale Relativity Theory application, on the other.This allowed for the examination, even if in a simplifiedway, of the full kinematic picture related to a particle in mo-tion and opened the field to a new idea of quantum dropletdynamics in an agricultural context. Future studies willdeepen Newtonian–quantum kinematics when referring toin-flight particles and will explore specific numerical de-scriptions from both points of view.

REFERENCES

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ð

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160 D. DE WRACHIEN ET AL.

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A NEW GENERAL ANALYTICAL SOLUTION FOR INFILTRATION STRUCTURES ��

DESIGN ��

��

��

M. Riva1, S. Mambretti

2, S. Chaynikov

3, P. Ackerer

4, O. Fasunwon

5 and ��

A. Guadagnini6 ��

��

1 Politecnico di Milano, Dipartimento Ingegneria Idraulica, Ambientale, Infrastrutture Viarie, Rilevamento, Piazza �

L. Da Vinci, 32, I-20133 Milano, Italy. E-mail: [email protected]. ���

���

2 Universidade Estadual de Campinas (UNICAMP), Faculdade de Tecnologia, R. Paschoal Marmo, 1888 - ���

CEP:13484-332 - Jd. Nova Itália - Limeira, SP, Brazil. E-mail: [email protected]. ���

���

3Politecnico di Milano, Dipartimento Ingegneria Idraulica, Ambientale, Infrastrutture Viarie, Rilevamento, Piazza ���

L. Da Vinci, 32, I-20133 Milano, Italy. E-mail: [email protected].

���

���

4 Laboratoire Hydrologie et Géochimie de Strasbourg, Univ. Strasbourg/EOST, CNRS UMR 7517, 1 rue Blessig, ��

67000 Strasbourg, France. E-mail: [email protected]. ��

���

5 University of Regina, Physics Department, Regina, Saskatchewan, Canada. E-mail: [email protected]. ���

���

6 Politecnico di Milano, Dipartimento Ingegneria Idraulica, Ambientale, Infrastrutture Viarie, Rilevamento, Piazza ���

L. Da Vinci, 32, I-20133 Milano, Italy. E-mail: [email protected] ���

���

!"#$%&'()

!"#$%&'('%)*%+*,-!*.+%/.-01#("2)3%/.-01#("2)42+5%

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ABSTRACT ���

We develop an analytical solution to assist in designing and sizing stormwater ���

infiltration structures. As original elements, our solution allows to estimate the hydraulic head ��

in the infiltration structure as a function of (i) storm temporal dynamics, making use of an ��

appropriate Intensity Duration Frequency (IDF) curve, (ii) the model adopted to describe the ���

response of the catchment to a given input rainfall, and (iii) the evolution of the wetted front ���

advancing in a homogeneous soil where the infiltration device is located. Our solution allows ���

highlighting the effects of the various simplifications associated with existing formulations ���

employed in the common engineering practice. Our results indicate that typically adopted ���

methodologies based on the assumptions of a uniform infiltration rate and/or negligible flow ���

rate along the lateral surface of the structure may lead to overestimating the key design ���

parameters. ���

��

INTRODUCTION ��

The increased degree of urbanization and resulting surface runoff arising from ���

impermeable areas is a key element to be considered for the assessment of proper functioning ���

of natural and man-made waterways, including sewer systems and rivers. Limitations to the ���

water discharged to a public sewer system typically rely on employing facilities for local ���

infiltration of rainfall in the subsoil. This practice is regulated by local authorities and is ���

included in the EU Water Framework Directive (WFD 2000/60/EC) and the GroundWater ���

Daughter Directive (GWDD 2006/118/EC). ���

In this context, an infiltration drainage system (or soakaway) is a structure which is ���

widely used to dispose of a target rainfall volume. These systems must be sized appropriately ��

to (a) provide effective storage for the water runoff and (b) facilitate infiltration of the water in ��

the subsoil so that the soakaway is able to cope with a sequence of storms. Adoption of ���

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infiltration drainage systems is widespread in several Countries. Notable examples are provided ���

by Japan (Nozi et al. 1999), Germany (Zimmer et al. 1999), Sweden (Bennerstedt 1999) and ���

the UK (albeit with an initially slow adoption trend, e.g., Abbott and Comino-Mateos 2001), ���

where routine implementation of this practice is provided. ���

Although there is no general consensus on design standards and procedures for ���

infiltration practices (Akan 2002a-b), the following three international best management ���

practice (BMP) guidelines for drainage design are typically adopted in the United Kingdom ���

and Germany: (1) United Kingdom Building Research Establishment (BRE 1991); (2) ��

Construction Industry Research and Information Association (CIRIA) (Bettess 1996; Martin et ��

al. 2000); and (3) German Association for Water, Wastewater and Waste design (ATV-���

DVWK-Arbeitsgruppe 2002). These design practices (a) rely on empirical or semi-empirical ���

methods which are based on assumptions that pose serious limitations to their effectiveness and ���

(b) are known to lead to system failure during the first few months of operation (e.g., Haf et al. ���

2004; Scholz 2004; Zheng et al. 2006). ���

Abbott and Comino-Mateos (2001) compare the maximum water depth, hmax, which is ���

monitored in a perforated concrete ring soakaway following rainfall events over a 20 months ���

period against predictions obtained through the CIRIA design guidelines. These authors ���

consider scenarios corresponding to (a) 100% runoff or (b) actual runoff values representing a ��

percentage of the total precipitation and recorded for each event during their analyzed field ��

tests (adopted runoff values ranged between 46-85% with an average value of 63% of the total ���

precipitation). The authors note that predicted values of hmax always overestimate significantly ���

(by 86-250% or 28-44%, respectively for scenario (a) and (b)) their experimental counterparts. ���

Akan (2002a) proposes a methodology to design infiltration structures which (a) relies ���

on the assumption of uniform rainfall intensity, (b) is based on the rational method (e.g., ���

Kuichling 1889; Crobeddu et al. 2007) to compute inflow discharges to the structure, and (c) ���

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disregards out-flow from the lateral surface of the structure. This widely used assumption ���

implies that vertical hydraulic gradients dominate the infiltration process when the infiltration ���

structures are properly maintained. ��

Indeed, infiltration devices require periodic maintenance to prevent clogging related to ��

the effects of total suspended solids (TSS) (Emerson et al. 2010) and/or formation of biomat ��

(Radcliffe and West 2009) in the inflowing water leading to reduction of the soil hydraulic ��

conductivity. The useful life and effective functionality of a soakaway is related to the ��

frequency of maintenance and the risk of sediment being introduced into the system. In this ��

context, the structure requires periodic maintenance and pre-treatments are recommended to ��

reduce the risk of blockage and facilitate maintenance operations. Regular maintenance ��

requires a periodic (e.g., annual) removal of sediments and debris from pre-treatment devices. ��

However, because of the possibility of bottom clogging, several authors (e.g., Emerson et al. ��

2010, and references therein) suggest a cautious design of the infiltration devices by neglecting �

the base outflow. �

The objective of this work is to develop and present an analytical solution for the design ��

and sizing of local infiltration devices upon removing several of the limiting and unrealistic ��

assumptions which are typically adopted in the literature. Our manuscript focuses on the ��

behavior of relative small infiltration structures, or soakaways, with planar characteristic length ��

of about 1 – 2 m (e.g., Woods-Ballard et al. 2007). The device is considered to be placed within ��

a homogeneous soil and infiltration occurs both through the bottom and the sides of the ��

structure. These infiltration structures are commonly employed in several Countries due to ��

space constrains. We start by reviewing the details of the most common procedures and ��

assumptions employed for the design of an infiltration system in Section 2. Section 3 is devoted �

to the development of our analytical solution. The robustness of the Green-Ampt assumption �

underlying our solution is assessed through a comparison against detailed numerical results ����

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obtained upon solving the partially saturated flow problem within the homogeneous soil ����

surrounding the structure by means of a mixed finite element formulation (Belfort et al. 2009). ����

Section 4 presents and discusses some examples highlighting the effect of the assumptions ����

presented in Section 2 on the key features of the processes observed and their consequences on ����

practical applications. ����

����

PROBLEM STATEMENT ����

The temporal dynamics of the hydraulic head, h(t), in an infiltration system are ���

governed by the following mass balance equation ���

(1) ����

Here, water and the porous medium are assumed to be incompressible, is the area of the ����

bottom surface of the infiltration structure, φ is the porosity of the material employed to fill the ����

device (φ = 1 if the structure is not filled with any material), and is the inlet volumetric ����

flow rate [L3T

-1]; = is the outlet flow rate, QOB and QOL respectively being the ����

contributions of the bottom and lateral surfaces of the infiltration structure. ����

The model proposed by CIRIA is based on the following assumption (Bettess 1996; ����

Martin et al. 2000): (a) the rainfall intensity, i [LT-1

], is constant over the extent of the basin ����

area, Ad; (b) the total amount (100%) of rainfall generates surface runoff; (iii) the catchment (or ���

basin) concentration (or corrivation) time, Tc, (i.e., the largest travel time associated with water ���

particles displacing to the drainage system) is negligible; (iv) the infiltration flux, q [LT-1

], is ����

constant along the bottom and lateral surfaces of the infiltration structure. These assumptions ����

lead to ����

(2) ����

( )b IN O IN OB OL

dhA Q Q Q Q Q

dtφ = − = − +

bA

INQ

OQ

OB OLQ Q+

IN dQ i A=

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(3) ����

p being the perimeter of the bottom area, Ab, of the infiltration structure. Substituting (2) and ����

(3) into (1) and integrating the latter with the initial condition h(t = 0) = 0 (i.e., assuming the ����

system is empty before the beginning of the storm) yields ����

(4) ���

Equation (4) clearly indicates that the hydraulic head (or the water level) in the infiltration ���

structure increases monotonically with time to the asymptotic value ����

. Note that CIRIA recommends to estimate the head in the ����

system as the largest value provided by (4) on the basis of a 10 years return period storm ����

associated with a range of likely durations (e.g., ranging from 10 mins to 24 hrs). ����

Neglecting the outflow component from the lateral side of the infiltration structure or ����

considering small observation times allows simplifying (4) as ����

� (5a) ����

resulting in a linear temporal increase of the hydraulic head in the system. On the other hand, ����

disregarding the contribution to the outflow from the bottom of the structure leads to ���

(5b) ���

Note that (4) and (5b) practically coincide when the total area of the system (given by the ����

bottom and lateral surfaces of the infiltration structure) is negligible with respect to the ����

catchment area, a condition which is often encountered in practical applications. ����

The model proposed by the United Kingdom Building Research Establishment (BRE, ����

1991) is based on the same assumptions leading to (5b) (i.e., Ab is assumed to be completely ����

( )( )O bQ q A ph t= +

( ) 1 1 b

qpt

Ab d

b

A Aih t e

p q A

φ−� �� �

= − −� �� �� �� �� �

( )( ) d b dh t A p i q A A→ ∞ = −

( ) 1d

b

Aq ih t t

q Aφ

� �= −� �

� �

( ) 1 b

qpt

AdA i

h t ep q

φ−� �

= −� �� �� �

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clogged by fine particles at large observation times) and conservatively reduces the ����

contribution of the lateral outflow from the infiltration structure by 50%. These assumptions ����

can result in a significant oversizing of the structure, as shown in the following. ����

Similarly to the CIRIA approach, the BRE procedure suggests adopting a rainfall ���

associated with a 10-year return period. The (constant) infiltration flow rate is estimated ���

experimentally as ����

(6) ����

where is the storage volume of water available in the infiltration device between 75% ����

and 25% of its excavation depth, is the time for the water level to decrease from the 75% ����

to 25% depth, is the internal surface area (including Ab) of the structure evaluated for a ����

depth equal to 50% of the total excavation depth of the infiltration device. ����

The German Association for Water, Wastewater and Waste (ATV-DVWK-����

Arbeitsgruppe, 2002) proposes the following formulation to determine the maximum hydraulic ����

head, hmax, for an infiltration basin ���

max max

max2

bi d be

h hA A i q A p f

� �� ���� � � �� ��� �� � �

(7) ���

Here, it is assumed that there is no filling material in the soakaway (i.e., φ = 1.0), imax is the ����

largest rainfall intensity associated with rainfall duration , and f is a safety factor (the ATV-����

DVWK-A 138 method considers f = 1.15). The drainage system considered by (7) is a ring ����

soakaway with outer diameter, De = 1.10 Di, Di being the inner diameter. In this context, (7) ����

includes the difference between the internal (Abi) and the external (Abe) horizontal area of the ����

infiltration structure, and the perimeter (pe) is evaluated via De. Comparison of (7) and (1) ����

reveals that (7) expresses mass balance by (a) approximating the evolution term through the ����

75 25

50 75 25

p

p p

Vq

a t

=

75 25pV −

75 25pt −

50pa

ϑ

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incremental ratio between the final and initial conditions of the system, (b) expressing by ����

(2) adopting i = imax, and (c) evaluating by (3) upon replacing by max

/2 (as suggested by ���

BRE, 1991). ���

In conclusion, all procedures described above provide estimates of head dynamics in an ����

infiltration structure on the basis of mass balance considerations according to the following key ����

assumptions: (i) the rainfall intensity is constant over the basin area, (ii) no infiltration over the ����

basin area occurs (i.e., runoff is 100% of rainfall), (iii) the time of concentration of the ����

catchment is negligible, and (iv) q is constant and is estimated by means of empirical ����

expressions. We derive in Section 3 original analytical solutions (with different degrees of ����

complexity) upon removing these limiting assumptions. We also describe the impact of the ����

various simplifications on the final solution to the problem. ����

ANALYTICAL SOLUTIONS AND DISCUSSION OF RESULTS ���

We start our analysis by removing the unrealistic assumption that the rainfall intensity ���

over the drainage basin is constant in time. Adoption of this assumption renders an inflow ���

hydrograph which is too simplified to represent the actual rainfall scenario (e.g., Hong 2010). ���

We introduce the Intensity Duration Frequency (IDF) curve that provides a mean ���

intensity, , for a given duration, �, and return period, TR, as ���

(8) ���

A [LTC-1

], B [T] and C[ ] being coefficients that depend on TR. Adoption of an IDF curves of ���

the kind expressed in (8) might be limited to scenarios where only limited information are ���

available. However, we note that historical series of rainfall are not accessible in most cases, ���

IDF curves being the only viable tool for rainfall characterization. Here, we employ the ��

INQ

OQ h h

( ), Ri Tϑ

( )( )

, R C

Ai T

ϑ=

+

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procedure developed by Keifer and Chu (1957) and introduce the synthetic hyetograph defined ��

as ���

� �� � � �

1, ,

p

R C C

p p

AC t tAi t T

B t t B t t�

� � (9) ���

where tp is the peak time, i.e., the time where the peak value of i is observed, which is ���

estimated as = r� (with 0< r <1), r usually ranging between 0.3 and 0.5 (Keifer and Chu ���

1957). Note that (9) is expressed in terms of � �, ,R

i t T� , while (8) is in terms of . The ���

synthetic hyetograph (9) provides the same mean intensity (around the peak) of the IDF (8) for ���

a given value of �. Losses from rainfall are neglected for the purpose of our application. Note ���

that the functional format of the hyetograph is not influenced by these losses. Therefore, the ���

formulation we propose can be adopted also in the presence of rainfall losses when these are ��

properly evaluated. The total flow rate entering the infiltration structure can then be estimated ��

as ����

( ) ( ) ( ) ( ) ( )0 0

, , , , , ,

t t

IN R d R d RQ t T A i T u t d A i t T u d= − = −� �ϑ τ ϑ τ τ τ ϑ τ τ (10) ����

where is the instantaneous unit hydrograph (IUH). Note that = , ����

being the Dirac delta function, in the limiting case of vanishing concentration time. Assuming ����

constant rainfall intensity and negligible concentration time allows rewriting (10) as (2). ����

In the following we analyze the effect of (a) assuming a constant infiltration flow rate ����

(as considered in the formulations presented in Section 2) or (b) embedding the temporal ����

dynamics of the infiltration rate in the model on the basis of the groundwater flow equations. ����

Fok et al (1982) provide experimental evidence supporting a depiction of the vertical and ���

horizontal infiltration flow rate as one-dimensional infiltration processes, as suggested by ���

pt

( ), Ri Tϑ

( )u t τ− ( )u t τ− ( )tδ τ− δ

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���

Green and Ampt (1911). Then, the total flow rate infiltrating from the bottom of the structure ����

can be written as ����

(11) ����

where Ks and pf are the hydraulic conductivity and the characteristic suction head of the ����

underlying soil, and z(t) is the depth of the advancing wetted zone below the bottom of the ����

infiltration structure. This quantity can be evaluated by mass conservation as ����

(12) ����

where n is the porosity of the underlying soil and total saturation is assumed. Bower (1969) ����

showed that the Green-Ampt equation (12), despite its simplicity, renders results that compare ���

relatively well and satisfactorily against those which can be obtained on the basis of relatively ���

complex models (e.g., the Philip equation, Philip 1969). Bower (1969) also shows that Ks in ����

(11) is well approximated as one-half the saturated vertical hydraulic conductivity because of ����

the effect of entrapped air in the pores within the wetted zone. ����

The contribution of the lateral surface of the infiltration structure to the total outgoing ����

flow rate can then be evaluated as ����

(13) ����

Here, the coordinate x(t) measures the lateral advance of the wetted zone and is estimated by ����

mass balance arguments as ����

(14) ���

In the following subsections we employ (9) – (14) and derive new solutions for the ���

evolution of the hydraulic head in the infiltration structure under conditions of negligible or ����

non-negligible concentration time. ����

( )( ) ( )

( )f

OB s b

z t h t pQ t K A

z t

+ +=

( ) ( )( )

( )

f

s

z t h t pdz tn K

dt z t

+ +=

( ) ( )( )

( )f

OL s

h t pQ t K ph t

x t

+=

( )( )

( )

f

s

h t pdx tn K

dt x t

+=

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���

Head dynamics in the infiltration device for negligible concentration time ����

The case analyzed here is representative of a scenario associated with a concentration ����

time that is much smaller than the time-scale of the infiltration process from the soakaway. In ����

this case, = and (10) simplifies as ����

(15) ����

Substituting (11), (13) and (15) into (1) leads to ����

(16) ���

Equation (16) must be solved jointly with (12) and (14), governing the evolution of the wetting ���

zone. The system (16), (12), (14) can be cast in dimensionless form as ����

(17a) ����

(17b)

����

(17c)

����

where , , , , , , ����

* /R R sT T K= λ , λ being a characteristic length describing the structure geometry, e.g., λ = Ab/p. ����

Recalling (9), is given by ����

(17d) ����

where and . In the following, we solve the system (17) with ���

the initial condition * * * 0h z x= = = (i.e., the infiltration structure is empty at the beginning of ���

the storm and the soil is dry). The system (17) is general (within the limits of the Green-Ampt ����

( )u t τ− ( )tδ τ−

( ) ( ) ( ) ( )0

, , , , , ,

t

IN R d R d RQ t T A i T t d A i t Tϑ τ ϑ δ τ τ ϑ= − =�

( )( ) ( )

( )( )

( )( )

, ,f f

b d R s b s

z t h t p h t pdhA A i t T K A K ph t

dt z t x tφ ϑ

+ + += − −

( )* * * * **

* * * * *

* * *, ,

f fdR

b

z h p h pAdhi t T h

dt A z xφ ϑ

+ + += − −

* * **

* *

( ) fz h pdz t

ndt z

+ +=

* **

* *

fh pdx

ndt x

+=

* /h h λ= * /z z λ=* /f fp p λ= * /x x λ= * /st t K λ=

* /sKϑ ϑ λ=

* / si i K=

( )

* *** * * *

* * ** * *

( , , ) 1R C

C t rAi t T

B t rB t r

ϑϑ

ϑϑ

� − �= −

+ − �+ − �

( )* 1/ c c

sA A Kλ −= * /sB B K λ=

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���

formulation) and includes some of the models presented in Section 2 as particular cases. For ����

example, the system (17) can be solved upon disregarding the last term in (17a) when the ����

lateral outflow rate is neglected. The third term appearing in (17a) should be disregarded if the ����

outflow rate from the bottom of the system is negligible. In case the infiltration flow rate is ����

assumed to be constant, as adopted in several applications (e.g., Sieker 1984), and ����

approximated as a fraction of the conductivity, with m < 1 (usually a value m = 0.5 is ����

used; Sieker 1984), the system (17) can be solved analytically (see Appendix A) to yield ����

( )* * * * *1exp , , , , 1 1d

R

b

Amh t t T m

A

� � = − + − � �

� � ϕ ϑ φ

φ φ (18) ���

where ���

(19) ����

If the lateral outflow discharge is negligible, (18) can be further simplified into ����

(20) ����

To assess the robustness of adopting the Green-Ampt hypothesis along the vertical and ����

horizontal directions during the derivation of (17), we solve numerically the Richards’ equation ����

describing unsaturated flow using the well documented and tested mixed finite elements ����

formulation of Belfort et al. (2009) for an infiltration structure with circular base. Pressure-����

water content and pressure-relative hydraulic conductivity relationships are described by the ����

Mualem-Van Genuchten model (VGM) as detailed in Van Genuchten (1980). The numerical ���

simulations are performed on a two-dimensional vertical domain, representing a homogeneous ���

porous medium. Only half of the domain is considered due to symmetry. The size of the ����

domain and the space-time grid discretization have been selected to avoid boundary effects and ����

to obtain accurate numerical solutions with an affordable CPU time. The final selected grid ����

sq mK=

*

* * * * * * * * *

0

( , , , , ) ( , , ) exp

t

R R

mt T m i T dϕ ϑ φ τ ϑ τ τ

φ

� = �

� �

*

* * * * * *

0

( , , ) *

t

dR

b

A mh i T d t

Aτ ϑ τ

φ φ= −�

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���

forms a square domain with side equal to 30 m. Discretization is performed by setting a fine ����

grid with uniform size of 0.5 cm around the infiltration structure and with grid size which ����

gradually increases with distance from the structure to attain a maximum size equal to 1 m at ����

the domain boundary. The simulation time step is set to 0.1 s. The initial conditions describe a ����

porous medium corresponding to dry sand. Impervious boundaries are applied everywhere ����

except at the infiltration structure where the total QIN is prescribed. The spatial distribution of ���

the inflow rate along the lateral and bottom surfaces of the structure is typically unknown. It is ���

then set in the model by (i) prescribing pressure along the edge of the elements of the ���

infiltration structure (bottom and lateral sides) at each time-step according to the mass-balance ���

equation and to the given QIN, (ii) solving Richards’ equation for the subsequent time step, and ���

(iii) evaluating QO and updating the pressure at the structure edges accordingly. Water inflow ���

into the structure due to seepage during decreasing water level in the infiltration structure is ���

neglected. This procedure is adopted throughout the simulation. ���

Figure 1 contrasts the time evolution of *h obtained by the mixed finite element ���

numerical solution based on the code of Belfort et al. (2009) (dashed curve) and solving (17). ���

The total CPU time required by the numerical simulation is 17520 seconds (close to 5 hours, on ��

a standard PC with an Intel Core I5 M460 processor) while the solution of (17) takes only a ��

few seconds. For illustration purposes, the results of Figure 1 are calculated on the basis of the ���

parameters of the IDF obtained for the region of the city of Milano with TR = 10 years (Becciu ���

et al. 1997), i.e., C = 0.84, A = 53 mm hC-1

and B = 0.155 h, KS = 10-4

ms-1

(as representative of ���

sandy soils) λ = 0.5 m, and = 7500 s (i.e., ). We further set / =100, φ = 1.0, n ���

= 0.3, r = 0.5. The parameters selected for the VGM correspond to a sand soil matrix (Huang et ���

al. 1996). The two solutions for *h are quite similar, the one based on (17) displaying a slight ���

temporal delay with respect to its numerical counterpart relying on Richards' equation. This ���

ϑ ϑ* = 1.5 dA bA

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���

observed behavior is likely due to the fact that the occurrence of an unsaturated zone is ���

neglected in (17). However, we note that for design purposes we are mainly interested in ��

estimating the maximum hydraulic head, , developing in the infiltration structure. The ��

peak values of h obtained with the two solutions are almost identical (Figure 1), the time at ����

which is attained being slightly overestimated by (17). ����

In the following we compare the solution obtained by solving (17) against those derived ����

by employing a constant q and/or neglecting the flow rate along the bottom or the lateral ����

surface of the infiltration structure for several durations of the precipitation, . The remaining ����

parameters are set to the values adopted in the simulation described above. In the case where q ����

is constant, is attained at the end of the precipitation period, i.e., at time = . When ����

q is variable, can be equal to or smaller than the duration of precipitation depending on . ����

This behavior is illustrated in Figure 2, which depicts the dependence of the ratio / on ���

. Figure 2 juxtaposes the complete solution based on (17) and the approximations obtained ���

upon neglecting the flow that occurs along the bottom or the lateral surface of the infiltration ����

structure. It is clear that the component of the outflow along the bottom of the infiltration ����

device does not influence significantly . On the other hand, neglecting the lateral ����

component of the outflow from the infiltration device results in a notable overestimation of ����

that appears to be always equal to . ����

Figure 3 reports the dependence of the (dimensionless) maximum value of the head ����

attained in the infiltration device, , on for the same scenarios reported in Figure 2 ����

(continuous curves). The solutions obtained with a constant infiltration rate (evaluated by ����

setting m = 0.5, as proposed in several applications) are also reported for reference (dashed ���

curves in Figure 3). Neglecting the outflow from the bottom surface has a marginal effect. The ���

*

maxh

*

maxh

ϑ

*

maxh*

maxt*ϑ

*

maxt*ϑ

*

maxt*ϑ

*

maxt

*

maxt

*

maxh*ϑ

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���

percentage error in the evaluation of varies from 3% to 10% for the ranges of ����

considered. Neglecting the outflow from the lateral surface of the infiltration device leads to ����

negligible percentage errors (< 5%) for < 1.5. Increasing , this simplification leads to ����

significantly overestimating regardless the model employed to describe the infiltration ����

rate. Figure 3 clearly demonstrates that the simplifying assumption of considering a constant ����

value for q along the lateral surface of the device (even as it is evaluated with the suggested ����

small value m = 0.5) can lead to a notable over-sizing of the structure for large storm durations. ����

Similar qualitative results have been obtained with different values of λ, / , φ, Ks and r ����

(not shown). ���

Estimation of the head in the infiltration device for non negligible concentration time ���

When the concentration time is non negligible with respect to the other time-scales ����

involved in the infiltration process, one can use an experimentally-based expression for the ����

IUH in (10) for the determination of QIN. For illustration purposes, we adopt the widely used ����

runoff model according to which the area of the basin contributing to the inflow scales linearly ����

with time (e.g., Hall 1989). This leads to the following expression for the IUH ����

(21) ����

where Tc is concentration time. Making use of (21), (10) becomes ����

(22) ����

Here, ���

*

maxh *

maxh

*

maxh*

maxh

*

maxh

dA bA

( )1/

0

c c

c

T t Tu t

t Tτ

≤�− = �

>�

( ) ( ), , , , , ,IN R c d R cQ t T T A I t T Tϑ ϑ=

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���

(23) ���

with given by (9). Substituting (11), (13) and (22) into (1) and writing the resulting ����

equation in dimensionless form yields ����

(24) ����

Here, , and . Equation (24) must be solved jointly with (17b, 17c), ����

driving the evolution of the wetting zone. Equation (24) should be solved upon removing the ����

last term if the lateral outflow rate can be neglected. If the outflow rate from the bottom of the ����

soakaway is negligible, then the second term on the right hand side of (24) can be neglected. In ����

the particular case of constant infiltration flow rate from the bottom and lateral surfaces of the ����

system, then (24) has the following analytical solution ���

(25) ���

where ����

(26) ����

When the lateral outflow discharge is negligible, (25) can be further simplified as ����

(27) ����

As already noted in Section 3.1, tmax and hmax are only marginally affected by the ����

simplification = 0, while neglecting the lateral outflow leads to a considerable ����

( )

( )

( )

0

1, , 0

1, , , , ,

0

C

t

R c

C

t

R c R c c

C t T

c

i T d t TT

I t T T i T d T t TT

t T

τ ϑ τ

ϑ τ ϑ τ ϑ

ϑ

�≤ ≤�

���

= ≤ ≤ +��� > +���

( , , )Ri t Tϑ

( )* * * * **

* * * * * *

* * *, , ,

f fdR c

b

z h p h pAdhI t T T h

dt A z xφ ϑ

+ + += − −

* / sI I K= * /c c sT T K λ=

* * * * * *1exp ( , , , , , ) 1 1d

R c

b

Amh t t T T m

Aϑ φ

φ φ

� � = − Θ + − � �

� �

*

* * * * * * *

0

( *, *, , , , ) ( , , , ) exp * *

t

R c R c

mt T T m I T T dϑ φ τ ϑ τ τ

φ

� Θ = �

� �

** * * * * *

0

( , , , )

t

dc

b

A mth I T T d

Aτ ϑ τ

φ φ= −�

OBQ

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���

overestimation of both tmax and hmax. For this reason, in the following we show only the results ����

obtained employing (25), i.e., including ≠ 0 and ≠ 0. This implies that construction of ����

these types of infiltration structures requires posing particular attention to allocate drainages ���

close to the lateral walls, which are the most performing parts of the system. Impervious walls ���

should then be avoided. On the other hand, the effect of bottom clogging due to the sediments ����

transported by the sewer system appears to be negligible when hmax >> �. ����

Figure 4 reports the dependence of / on for three selected values of the ����

concentration time. It is observed that can be significantly larger or smaller than the ����

precipitation period depending on and . Figure 5 illustrates the effect of on ����

predictions of , i.e., increasing values of are associated with a slight reduction of the ����

excavation depth required for the infiltration structure. ����

Figures 3 and 5 suggest that considering temporal dynamics of the outflow is relevant to ����

a proper depiction of the system. Methods disregarding this process tend to overestimate the ���

design of the structure. These results appear to indicate that the water depth in the infiltration ���

device increases continuously with the rainfall duration to approach an asymptotic value at . ����

This result casts some doubts on the practice of founding the structure design on the concept of ����

critical duration and is consistent with the adoption of typical DDF/IDF curves. ����

CONCLUSIONS. ����

Our work leads to the following major conclusions. ����

1. We present an analytical solution rendering the temporal dynamics of the water level ����

within stormwater infiltration infrastructures (or soakaways) which are typically used in ����

the context of urban catchments. We present solutions of our expressions for a test case ����

associated with characteristic length scale = 0.5 m, representing, e.g., a soakaway ���

with a circular base of diameter equal to 2.0 m. This characteristic length scale is ���

OBQ

OLQ

*

maxt*ϑ *ϑ

*

maxt

*ϑ *

CT*

CT

*

maxh*

CT

λ

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��

typical of relatively small infiltration structures that are commonly used in several ���

Countries due to space constrains. ���

2. Our solution removes some of the limitations associated with typically used empirical ���

formulations and explicitly includes the effects of (i) storm temporal dynamics, (ii) the ���

model adopted to describe the response of the catchment to a given input rainfall, and ���

(iii) the evolution of the wetted front advancing in the soil. ���

3. Our solution relies on a Green-Ampt infiltration model along the vertical and horizontal ���

directions. We analyze the impact of this hypothesis on the evaluation of the maximum ���

head, , at the infiltration structure through a detailed numerical solution of the ��

unsaturated flow problem within the porous medium surrounding the infiltration ��

structure relying on the well documented and tested mixed finite elements formulation ���

presented by Belfort et al. (2009). The values obtained for are mutually consistent, ���

the solution based on (17) displaying only a slight temporal delay with respect to its ���

numerical counterpart obtained by removing the Green-Ampt assumption. Note that the ���

total CPU time required by the numerical solution and our (17) is about 5 hours and ���

only a few seconds, respectively, for the example tested. ���

4. Our results indicate that a best practice for stormwater infiltration devices design should ���

include proper modeling of the dynamics of the infiltration rate, while methodologies ���

based on the assumption of a uniform infiltration rate lead to overestimating the key ��

design parameters. As a practical implication, it is suggested that special care should be ��

devoted to a proper design of the lateral walls of the structure with particular emphasis ����

on their drainage efficiency, while clogging of the bottom exerts a secondary influence ����

on the performance of the system in our example. ����

Appendix A ����

*

maxh

*

maxh

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��

In the case of constant flow rate ( ), (16) becomes ����

(A1) ����

Integration of (A1) leads to

����

(A2) ����

Deriving (A2) with respect to t* leads to ���

(A3) ���

Substituting (A2), (A3) into (A1) yields ����

(A4)

����

( )*

* * * * *

1 2

0

1, , exp

t

dR

b

A mC i T m d C

A

� � = − − + � �

� � � τ ϑ τ τ

φ φ

(A5)

����

Substituting (A5) into (A2) yields ����

( )*

* * * * * * *

0

* *

2

1exp , , exp

1 exp exp

t

d dR

b b

A Am mh t i T m d

A A

m mt C t

� � � = − − − � � �

� � �

� � − + − + − � �

� �

� τ ϑ τ τφ φ φ

φ φ

(A6) ����

The constant C2 is determined by the initial condition. By assuming the system is empty ����

prior to the storm (i.e. ( )* * 0 0h t = = ) renders , so that (A6) coincides with (18). ����

sq mK=

** *

*

d

b

Adhmh i m

dt Aφ + = −

* *

1 expm

h C tφ

� = − �

** *1

1* *exp exp

dCdh m m mt C t

dt dtφ φ φ

� � = − − − � �

� �

* * * * *1

*exp ( , , ) d

R

b

AdCmt i t T m

dt Aφ ϑ

φ

� − = − ��

2 0C =

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���

References ����

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drainage system and field testing of current design procedures. J. CIWEM., 15, ���

198-202. ����

Akan, A.O., (2002a). Sizing stormwater infiltration structures, Journal of Hydraulic ����

Engineering, ASCE, 128 (5): 534-537. ����

Akan, A.O., (2002b). Modified rational method for sizing infiltration structures, Can. J. Civ. ����

Eng. 29: 539-542. (June) ����

ATV-DVWK-Arbeitsgruppe, (2002). Planning , Bau und Betrieb von Anlagen zur ����

Versickerrung von Neiderschlagswasser (Design, construction and operation of ����

rainwater drainage systems). ISBN 3-935669-83-6, ATV-DTWK (German ����

Association for Water, Wasterwater and Waste), Gesellschaft zur Förderung der ���

Abwassertechnik e.V., Hennef, Germany, Regelwerk A-138, ES-41. ���

Becciu G., Mambretti S., Paoletti A. (1997): Risk Design of Urban Drainage Networks on the ����

Basis of Experimental Data, Excerpta Vol. 11. ����

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doi:10.1002/hyp.7539 ����

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���

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SUDS Manual Ciria C697, London, UK, 606 pp. ���

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sustainable Urban Infiltration ponds treating storm runoff. Journal of Urban ���

Planning and Development, 132, (1), 36-41. ���

Zimmer, U., Geiger, W.F. and Caesperlein, G. (1999) Safety factors for the design of ���

infiltration facilities. In: Proc. of 8th

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I.B. and Ball, J.E. (Eds).), Sydney, Australia, pp 256-264. ���

��

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Figure 1 Dependence of h* on t* as calculated by the mixed finite element numerical

solution obtained through the code of Belfort et al. (2009) (dashed curve) and

solving (17) (continuous curve). Corrivation time is neglected and *ϑ = 1.5.

Maximum values of h* are also reported.

0

5

10

15

20

0.1 1 10*t

*h

Analytical solution

Numerical solution

15.77 15.75 *1.5ϑ =

!"#$%&'

!"#$%&'('%)*%+*,-!*.+%/"01('2%/"01('345+6%

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Figure 2 Dependence of *

maxt / *ϑ on *ϑ when corrivation time is neglected.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 0.5 1 1.5 2

* *

max/t ϑ

0; 0OL OBQ Q= ≠

0; 0OL OBQ Q≠ ≠

0; 0OL OBQ Q≠ =

!"#$%&'

!"#$%&'('%)*%+*,-!*.+%/"01('2%/"01('%345+6%

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Figure 3 Dependence of *

maxh on *ϑ when corrivation time is neglected. Dashed

curves are obtained considering a constant infiltration flow rate with m = 0.5

*

maxh

0

5

10

15

20

25

0 0.5 1 1.5 2

!" !" !"

!"!" !"

!"

!"!"

!"!"!"

!"!"

!"0; 0OL OBQ Q= ≠

0; 0OL OBQ Q≠ ≠

0; 0OL OBQ Q≠ =

!"

!"#$%&'

!"#$%&'('%)*%+*,-!*.+%/"01('2%/"01('%345+6%

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Figure 4 Dependence of *

maxt / *ϑ on *ϑ for three values of corrivation time. Dashed

curves are obtained considering a constant infiltration flow rate with m = 0.5.

0

1

10

100

0 0.5 1 1.5 2 *ϑ

* *

max/t ϑ

TC

*= 0.1

T

C

*= 1.0

T

C

*= 0.5

!"#$%&'

!"#$%&'('%)*%+*,-!*.+%/"01('2%/"01('%345+6%

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Figure 5 Dependence of *

maxh on *ϑ for three values of corrivation time. Dashed

curves are obtained considering a constant infiltration flow rate with m = 0.5!

*

maxh

0

4

8

12

16

0 0.4 0.8 1.2 1.6 2

TC

*= 0.1

T

C

*= 0.5

T

C

*= 1.0

!"!" !"!"!"

!"

!"

!"

!"

!"!" !"!"

!"!"

!"

!"

!"

!"!"

!"#$%&'

!"#$%&'('%)*%+*,-!*.+%/"01('2%/"01('%345+6%

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Figures Caption

Figure 1 Dependence of on as calculated by the mixed finite element

numerical solution obtained through the code of Belfort et al. (2009) (dashed

curve) and solving (17) (continuous curve). Corrivation time is neglected and

= 1.5. Maximum values of are also reported.

Figure 2 Dependence of / on when corrivation time is neglected.

Figure 3 Dependence of on when corrivation time is neglected. Dashed

curves are obtained considering a constant infiltration flow rate with m = 0.5

Figure 4 Dependence of / on for three values of corrivation time. Dashed

curves are obtained considering a constant infiltration flow rate with m = 0.5

Figure 5 Dependence of on for three values of corrivation time. Dashed

curves are obtained considering a constant infiltration flow rate with m = 0.5

!"#$%&'()*!+,&-!.*

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Evolutionary Computation Techniques toAssess Losses in Water Supply Networks

Stefano Mambretti, Paulo S. Martins & Regina L. MoraesSchool of TechnologyUniversity of Campinas, UNICAMP, Brazil

Abstract

In this work, a methodology that combines both simulation and measure-ments of pressures and discharges on the water supply network is applied toa case study. The demands of the model nodes are systematically changedby means of two evolutionary algorithms and the network is simulated inorder to match the readings of the instruments. By comparing the mea-surements with the simulated values it is possible to assess the losses andestimate their locations.Specifically, two methods have been tested and applied to a case study:

the former based on Simulated Annealing (SA) and the latter on GeneticAlgorithms (GAs). The simulations show that the methods based on GAsperform much better and are able to detect the different hypothesized sce-narios, while the single individual used by SA risks to be trapped in aunfeasible zone in its search. Moreover, the solution obtained by GAs canbe further improved by means of a simple Hill Climbing procedure (HC),thus achieving satisfactory results.Keywords: water supply networks, water losses, Genetic Algorithms, hydraulicmodeling.

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1 Introduction

Water loss in water distribution networks is gaining more attention recentlyin the research community due to their scale (up to 50% - 70% in somecountries) and economic impact on the society. Non-Revenue Water (NRW)or lost water is the difference between the volume entering a distributionsystem and the volume billed to customers. This volume is a serious eco-nomic damage for the companies, and the challenge is compounded by thefact that sources might become scarcer due to pollution and the increase indemand. To this end, methodologies that aim at detecting, predicting, pre-venting or avoiding water losses are welcome, in order to help managementto make well-informed decisions and ultimately mitigate (or eliminate) thisproblem.In particular, management systems need to know where and how to inter-

vene (e.g. repair or substitution of a pipe) [1], a challenge that is usuallyformulated as a multi-objective optimization problem. The objective func-tion (O.F) is represented by the performance of the network and the costs ofthe rehabilitation [2, 3]. Common objectives functions are the not-deliveredwater volumes or the number of customers affected by interruptions causedby pipe bursts [4].Such condition led to the development of models that are either able to

generate pipe breaks [1] or have available good databases about previousbreakages [5, 6]. Another objective to be pursued is the increase of thenetwork efficiency through the reduction of water losses. However, the lim-ited funds available constrain the invested annual budget and increase theimportance of scheduling interventions.A new methodology has already been presented that identifies the areas

where losses are mostly expected [7]. It is based on data collection (dis-charge and pressure) from instruments positioned on the water supply net-work, and successive comparison of the data collected with those simulatedby software. The results of the model should match the readings of theinstruments. Under the hypothesis that the model is a good representationof the real network, the differences between simulated and recorded dataare due to the different demands imposed at the nodes. The optimizationwas based on a classic Genetic Algorithm.This paper focus on different methods of Evolutionary Computation in

order to establish the best procedure to minimize the Objective Function.Moreover, we assessed whether or not the value of the Objective Functionis a reliable indicator of the goodness of the presented solution.The remainder of this paper is organized as follows: Section 2 discusses

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the optimization methods used in this work. Section 3 introduces the casestudy (city of Castegnato). Finally, in Section 4 we present our remarks andconclusions.

2 Evolutionary Computation

Complex and multi-objective optimization problems are often solved bymeans of Evolutionary Computation. The term Evolutionary Computation(EC) [8] represents a large spectrum of heuristic approaches to simulateevolution, including (for example) Genetic algorithms (GAs) [9, 10], Sim-ulated Annealing (SA) [11], Particle Swarm Optimization [12], and AntColony Optimization [13]. In this work, two approaches have been tested(Simulated Annealing and Genetic Algorithms), as described in the follow-ing sections.

2.1 Simulated Annealing (SA)

The first method used in this work is based on Simulated Annealing, wherewe admit the possibility, at the beginning of the optimization, of a solutionthat worsens the objective function (O.F.). This is in order to explore alarger space and to arrive to the best solution, while also avoiding beingtrapped in the local optima as would happen with the simple Hill-Climbingtechniques. For the application of SA, a random number F ǫ ]0,1[ is selectedand the new discharge Q is calculated by equation (1):

Q = λ · [−ln(1− F )]1k (1)

where λ and k are the parameters of the distribution. The average of thedistribution µ is:

µ = λ · Γ(

1 +1

k) (2)

The initial temperature T0 = - δf+

ln(p0)is computed setting the initial value

of the probability of acceptance of a positive scenario p0. During the opti-mization, the temperature decreases following the law:

Ti+1 = Ti ·W = T0 ·Wi (3)

where W=0.99995, having decided to perform one million simulation runs.The results of the optimization are reported in Table 4. Notice that T =0 means that the implemented procedure is a Hill Climbing; increasing the

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initial temperature and the value of k causes the procedure to have morevariability. One million simulations have been performed for each couple ofparameters (k, T ).

2.2 Genetic Algorithm

The second optimization method used in this work is a Genetic Algorithmwith mutation and crossover operators and roulette wheel selection [14].This algorithm has been tested using one and two points for crossing over

the chromosomes. As it is known, these algorithms are able to find pointsclose to the best solution, but not the best solution itself; therefore, at theend of the application of the GAs, a procedure that applies the so-calledHill Climbing is also used in order to find the best possible solution.As for the GAs, the parameters used in the algorithm are:• Number of individuals per population: 2500• Number of generations: 100• Elitism: 20%Notice that as there are 440 nodes in the network, there is also the same

number of parameters to be calibrated; the number or individuals of thepopulation is taken as more than 5 times the number of parameters.

3 Case Study

The case study considered is the water supply network of Castegnato, asmall town in the North of Italy with around 7900 inhabitants and witha network divided in two disconnected parts. The characteristics of thetown and its water supply networks have been presented by Mambretti andOrsi [7]. As over the years the board of water supply managers recordedmore than 50% of water losses, a number of transducers have been installedthroughout the network. Fig. 1 illustrates the transducers which are uniquelyidentified by an integer ID and represented by a circle. The dots representnodes and the edges the pipes. These elements of the graph are furtherdetailed in Table 1.In order to understand whether the number and position of devices are

appropriate or not to locate the leakages, five scenarios have been simulated.These scenarios impose losses in the different areas of the town. They allowus to verify whether or not they can be reconstructed by the algorithms

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Figure 1: ID and position of measurement devices (Castegnato)

mentioned in Section 2. The O.F to be minimized is given by equation 4:

O.F = min

(

#control nodes∑

i=1

|hmeas − hcomp|

|hmeas|·WH

+

#control links∑

i=1

|qmeas − qcomp|

|qmeas|·WQ +

|qgloballyExpected − qgloballyComputed|

|qgloballyExpected|·WGE

)

(4)

where W are weights, which are all set to 1 for the theoretical scenarios.Clearly, their actual value ultimately depends on the expected precision ofthe real devices.

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Table 1: ID of the node or link where the device is positioned and type(pressure transducers P are positioned on nodes (N); flowmetersQ on links (L)

N N/L Type ID N N/L Type ID

01 N P 45 13 N P 363

02 N P 150 14 N P 300

03 N P 144 15 L Q 250

04 N P 191002 16 L Q 69

05 N P 41 17 N P 71

06 N P 224 18 L Q 66

07 N P 105 19 N P 381004

08 N P 4121002 20 L Q 185

09 N P 8 21 L Q 167

10 N P 690 22 N P 177100

11 N P 2431004 23 N P 680100

12 N P 237 24 - - -

4 Results

Results for SA are reported in Table 2. As it can be seen, they are quitedisappointing, as the final value of the O.F. is often larger than the initial oneafter one million simulations. The reason of the problematic performance ofthe SA is probably given by the presence of non-physical potential solutionswhich have been tested: these are given by a distribution of discharges thatwould induce situations in the network where pumps either cannot deliverenough flow or head, or the system has negative pressures. As can be seenin Fig. 3, once the potential solution travels in a field where the solutionis not acceptable, it remains trapped for long time before being able toescape; only then the O.F. start again to diminish (in gray we report theunacceptable solutions and in black those that are deemed acceptable). Ascan be seen, the initial O.F is much lower. The simulations reported in Fig.3 were performed with k = 10 and T = 5.The results for GAs are reported in Table 3 (one point crossover) and

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Table 2: O.F. values after using Simulated Annealing with different param-eters. The O.F. with the initial configuration is 5.3926.

N T = 0 T = 0.5 T = 1 T = 5 T = 10

1 2.6027 4.6943 5.5604 6.9473 13.1846

10 0.3397 6.9916 7.1754 11.1221 8.2630

100 0.4848 15.1249 5.0526 14.2636 14.7489

1000 0.9979 17.2065 17.4476 17.5842 17.9049

Figure 2: Losses in Castegnato according to theoretical scenarios

4 (two-points crossover). As GAs have random components, simulationshave been run 10 times for each scenario and for each method. It is tobe noted that the results can be further improved by repeatedly using theHill Climbing procedure. In the mentioned tables, the HC has been carried

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Figure 3: Path of the potential solution using SA method: results of 1 millionsimulations

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out performing one million simulations; however, the results can be furtherimproved. For example, the value obtained with the 2-point-crossover GAimproved with HC (Table 4, scenario 5 and simulation 1) with O.F = 0.1548.Hill Climbing was further reapplied several times, each time for one millionsimulation runs, obtaining subsequently O.F = 0.1309; O.F = 0.0988; andO.F = 0.0986.For the purpose of this research, the GAs seem to have a better perfor-

mance as they work in a group of individuals, and therefore if some of themfall in a field where the solution is not allowed, it is simply discarded in thefollowing population without affecting too much the final results.

Table 3: O.F. reached by GA1 point crossover and Hill Climbing

· scenario 1 scenario 2 scenario 3 scenario 4 scenario 5

N GA HC GA HC GA HC GA HC GA HC

01 0.3965 0.1053 0.3856 0.1095 0.3848 0.1110 0.3245 0.2920 0.4462 0.2170

02 0.1619 0.0916 0.4252 0.1151 0.3750 0.1155 0.4020 0.2723 0.4465 0.2081

03 0.1864 0.1045 0.4139 0.1153 0.1801 0.1006 0.4322 0.3001 0.3561 0.1619

04 0.1633 0.0962 0.3558 0.1088 0.2073 0.1074 0.3304 0.2970 0.4331 0.2278

05 0.2853 0.1078 0.4839 0.1190 0.4144 0.1026 0.4319 0.2984 0.4221 0.2105

06 0.1956 0.0949 0.5118 0.1122 0.3314 0.1072 0.4135 0.2827 0.3366 0.2098

07 0.1806 0.0926 0.5251 0.1170 0.4181 0.0947 0.4319 0.2935 0.4461 0.2046

08 0.1876 0.0932 0.4293 0.1084 0.3341 0.1031 0.3211 0.2882 0.3269 0.1918

09 0.1542 0.0945 0.4056 0.1088 0.3258 0.1144 0.3049 0.2482 0.2811 0.1523

10 0.1421 0.1022 0.4440 0.1121 0.5384 0.1044 0.4309 0.2913 0.4466 0.2187

5 Remarks and Conclusion

Water distribution networks are an essential component of water supplysystems and represent a critical infrastructure asset to the society. As such,they require effective and efficient energy saving management. In some coun-tries Unnaccounted for Water or Non-Revenue Water (NRW) is up to 70%of the water volume supplied, which is a serious economic damage.In this paper, a methodology to identify the areas where losses are mostly

expected was verified; the procedure requires data collection (discharge and

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Table 4: O.F. reached by GA2 point crossover and Hill Climbing

· scenario 1 scenario 2 scenario 3 scenario 4 scenario 5

N GA HC GA HC GA HC GA HC GA HC

1 0.1690 0.1052 0.2749 0.1052 0.1387 0.0978 0.2840 0.2501 0.1923 0.1548

2 0.1578 0.1020 0.2958 0.1045 0.1433 0.1047 0.2953 0.2632 0.2897 0.2012

3 0.1574 0.1024 0.3532 0.1052 0.1791 0.1031 0.3767 0.2463 0.2713 0.2069

4 0.1469 0.0953 0.1964 0.1029 0.1250 0.1107 0.3889 0.2738 0.4133 0.2286

5 0.1520 0.0913 0.3825 0.1103 0.1261 0.1037 0.3160 0.2868 0.2724 0.1933

6 0.1637 0.0994 0.3283 0.1056 0.1179 0.0970 0.3082 0.2771 0.2780 0.1743

7 0.2616 0.0912 0.2912 0.1005 0.1363 0.1107 0.2991 0.2708 0.4386 0.2160

8 0.1581 0.0965 0.3331 0.1089 0.1658 0.1218 0.3200 0.2910 0.1923 0.1654

9 0.1701 0.1066 0.2345 0.1086 0.1532 0.1063 0.3079 0.2738 0.4140 0.2223

10 0.1473 0.0963 0.2774 0.1030 0.1708 0.1015 0.3191 0.2842 0.3029 0.1904

pressure) from instruments positioned on the water supply network, andsuccessive comparison of the data collected with those simulated.Different methods to minimize the O.F. have been tested: those based

on Genetic Algorithms and those based on Simulated Annealing. In orderto test the methodologies, a case study has been identified, where devicesare to be installed and data collected. The position of the devices and thevalidity of the developed methodology were checked by means of a numberof scenarios before their installation.It is found that the areas where the largest losses are imposed are properly

identified; this allowed us to confirm that the positions of the devices placedwithin the network are correct.So far it has been found that the procedure based on GAs, preferably with

2 points crossover followed by a fine tuning with a Hill Climbing procedure,is able to minimize the O.F. It is now to be shown whether the minimizationof the O.F. allows the correct reconstruction of the initial scenario. In Fig. 3,scenarios 1 and 5 are reproduced, together with their best reconstruction.As it can be seen, the reduction of the value of O.F. actually brings thesolution towards the correct scenario reconstruction.The results achieved show that the scenarios are properly reconstructed,

even in case errors are obviously present; the goal to identify the areas wherelosses are concentrated seems to be reached. Among the selected methods,the GAs have shown a better performance, probably because they work

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with a number of individuals, while SA employs only one individual thatcan become trapped in a field where solutions are not allowed.In the next phase of this work the instruments will actually be positioned.

The procedure has the potential to be improved when real (actual) databecomes available. Undergoing research is related to the development ofalgorithms able to further minimize the O.F, as it is shown that this is a goodindicator of the goodness of the solution. Moreover, a needed improvementis the development of a method able to identify whether or not more thanone optimum is present. This would help in the decision about the minimumnumber of devices required to be installed.Future developments will comprise the analysis of real data collected on

the network, and the improvement of the computer program (i.e. simulator).Moreover, data will also be collected by means of a portable flowmeter. Thefinal goal of this research is the development of a new methodology that isable not only to locate areas where losses are mostly expected, but also theimprovement of the existing indicators of water supply management.

References

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[2] Halhal, D., Walters, G.A., Savic, D.A. & Ouazar, D., Scheduling ofwater distribution system rehabilitation using structured messy geneticalgorithms. Evolutionary Computation, 7(3), 1999.

[3] Giustolisi, O., Laucelli, D. & Savic, D., Development of rehabilitationplans for water mains replacement considering risk and cost-benefitassessment. J of Civil Engineering and Environmental Systems, 23(3),pp. 175–190, 2006.

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[5] Male, J.W., Walski, T.M. & Slutsky, A.H., Analyzing water mainreplacement policies. J Water Resour Plan Manage, 116(3), pp. 362–374, 1990.

[6] Sundahl, A., Using break data on water pipe systems for renewal plan-ning. COST Action C3 workshop, Brussels, 1996.

[7] Mambretti, S. & Orsi, E., Genetic algorithms for leak detection in water

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supply networks. 1st International Conference on Urban Water, NewForest, UK, pp. 25–27, 2012.

[8] Back, T., Fogel, D. & Michalewicz, Z., Handbook of evolutionary com-putation. IOP Publishing Ltd. and Oxford University Press, New Yorkand Oxford, 1997.

[9] Holland, J., Outline for a logical theory of adaptive systems. Journalof the ACM, 9(3), 1962.

[10] Holland, J., Adaptation in natural and artificial systems. University ofMichigan Press, 1975.

[11] van Laarhoven, P. & Aarts, E., Simulated Annealing: Theory and Appli-cations. Springer, 1987.

[12] Zhang, L., Zhou, C., Liu, X., Z, M., Ma, M. & Liang, Y., Solvingmulti objective optimization problems using particle swarm optimiza-tion. Proceedings of IEEE Congress on Evolutionary Computation 2003(CEC 2003), Canbella, Australia, pp. 2400–2405, 2003.

[13] Dorigo, M. & Sttzle, T., Ant Colony Optimization. The MIT Press, p.319, 2004.

[14] Goldberg, D., Genetic algorithms in search, optimization and machinelearning. Massachusetts: Addison-Wesley, Reading, 1989.