Catchment Transport and Travel Time Distributions:Theoretical Developments and ApplicationsPaolo Benettin

PhD Days di Ingegneria delle Acque – Trento, 6-8 Luglio 2015

the Ph.D. dissertation

Cin

Cout

- Travel-time distributions

- Hydrologic and solute transport

University of Padua, Padua, Italy

Gianluca Botter

EPFL, Lausanne, CH

Andrea Rinaldo

PH.D. SUPERVISORS

Introduction

age = time since entrance

age T

𝒑𝑸(𝑻 ,𝒕)

Distribution of water parcels

time

𝐶 (𝑡 )=∫0

∞

𝑐 (𝑇 )𝒑𝑸 (𝑻 ,𝒕 )𝑑𝑇fundamental link

between water ageand water quality

source: ARPAV

EcologicalStatus 2013

Sufficient

Poor

GoodVenice Lagoon drainage basin

Nitrate Loads

!

2001 2003 2005 2007 2009 2011

why studying water age

Sanford and Pope, Env. Sci. and Technol., 2013

why studying water age

propagation of a pressure wave rather than actual water

‘new’ rainfall

discharge‘old’ stored water

CONCEPTUAL EXPLANATION:

why hydrologic transport isn’t so simple (1)

Kirchner et al., 2000, Nature

WATER

CHLORIDE

why hydrologic transport isn’t so simple (1)

data from Plynlimon UHF catchment, UK

[mm

/h]

[mg/

l]

silica

chloride

[mg/

l]

dryperiod

wetperiod

why hydrologic transport isn’t so simple (2)

why hydrologic transport isn’t so simple (3)

McDonnell et al., 2010, HP

realistic distributionsideal distributions

ages in storage

T

𝒑𝑺(𝑻 ,𝒕 )

0time

tt2t1

INJECTION TIMESt3

S(t)

ages in the discharge

T

𝒑𝑸(𝑻 ,𝒕)

hydrologic transport processes

What relationship exists between particles in storage

and particles in the fluxes?

‘StorAge-Selection’ functions

𝑝𝑄 (𝑇 , 𝑡)𝑝𝑆(𝑇 ,𝑡)

=𝜔(𝑇 , 𝑡)

flux

Storage

age-selection

ω (𝑇 , 𝑡 )=𝑝𝑄(𝑇 , 𝑡)𝑝𝑆(𝑇 , 𝑡)

preference for younger ages

no preference(random sampling)

preference for older ages

age

𝜔<1

𝜔>1𝜔[−

]

1

My contribution

dissertation overview

• 1. Unified theory of water age and life expectancy distributions

• 2. Kinematics of age mixing in advection-dispersion systems

• 3. Application to conservative fertilizer transport in a dutch catchment

• 4. Application to chloride transport in a highly monitored UK catchment

• 5. Application to non-conservative solutes in a forested US catchment

Logical orderChronological order

32415

Ype van der Velde Sjoerd van der Zee

NL

outlet

Conservative solutes from an agricultural area (Chapter 3)

Can we use time-variant age

distributionsto model chloride

transport?

SOILSTORAGE

GROUNDWATERSTORAGE

a simple transport model

soil water mean [d] 90 st.dev. [d] 20

groundwater

mean [d] 1100 st.dev. [d] 120

Model results

shorter (30-100 d)travel times

Q [

mm

/h]

longer (2-3 y) travel times

how do ages mixin advection-

dispersion processes?

Age mixing in advection-dispersion models (Chapter 2)

flux

Storage

𝑝𝑆 (𝑻 , 𝑡 ) STORAGE age

𝑝𝑄 (𝑻 , 𝑡 ) DISCHARGE age

𝜌 (𝒙 ,𝑻 ,𝑡 ) ‘age mass density’Ginn, 1999, WRR

Benettin et al., WRR, 2013b

Age mixing in advection-dispersion models (Chapter 2)

STORAGE age

‘‘StorAge-Selection’’ function𝜔 (𝑇 , 𝑡 )=𝑝𝑄(𝑇 , 𝑡)𝑝𝑆(𝑇 ,𝑡)

=𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙𝑣 𝑜𝑙𝑢𝑚𝑒𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙

𝑝𝑆 (𝑇 , 𝑡 )= 1𝑀 (𝑡)∫𝑉

❑

𝜌 (𝒙 ,𝑇 , 𝑡 )𝑑 𝒙

𝑝𝑄 (𝑇 , 𝑡 )= 1𝜑𝑜𝑢𝑡 (𝑡 )∫𝑆

❑

[𝑢 (𝒙 ,𝑡 )𝜌 (𝒙 ,𝑇 , 𝑡 )−𝑫 (𝒙 , 𝑡 )𝛻𝜌 (𝒙 ,𝑇 ,𝑡 ) ]𝒏𝑑𝜎 DISCHARGE age

𝜕𝜌 (𝒙 ,𝑇 , 𝑡 )𝜕𝑡

+𝜕 𝜌 (𝒙 ,𝑇 ,𝑡 )

𝜕𝑇+𝛻 ∙ [𝑢 (𝒙 , 𝑡 ) 𝜌 (𝒙 ,𝑇 , 𝑡 ) ]=−𝛻 ∙ [𝑫 (𝒙 ,𝑡 )𝛻𝜌 (𝒙 ,𝑇 ,𝑡 ) ]

discharge ageVS

storage age

Ginn, 1999, WRR

Pe = 1

relative age [%]

ω [-

]

𝜔 (𝑇 , 𝑡 )=𝑝𝑄(𝑇 , 𝑡)𝑝𝑆(𝑇 ,𝑡)

Experiences abroad

Kevin J. McGuire

James W. Kirchner

period abroad atVirginia Tech University

andAGU fall meeting 2013

High-frequency chloride at Plynlimon (UK) (Chapter 4)

Benettin et al., 2015, WRR

Upper Hafren Catchment (UK)2 years of 7-hour measurements

chloride

How can we explain the observed

high-frequency solute dynamics?

CALIBRATED HYDROCHEMICAL MODEL

Parameters posterior distributions

age dynamics

Tracer response

input variability+general affinity

for younger ages

younger ages older ages

StorAge Selection functionsCumulative age distributions

MOBILEWATER

MINERAL

silica transport in a forested catchment (Chapter 5)

Benettin et al., in review

How can we use age distributions

to model age-dependent

transport?

Travel time distributions

NS= 0.62

Silicon (Si)

Nov-2006 Nov-2007 Nov-2008

Age-dependent transport

dry days:

many old particles

wet days:

many young particles

𝐶 (𝑡 )=∫0

∞

𝐶𝑒𝑞 (1−𝑒−𝑘𝑻 ) �́�𝑄 (𝑻 ,𝑡 )𝑑𝑻

𝐶𝑒𝑞 𝑐 (𝑇 ) 1° order chemical kinetics:

Silica and sodium at Hubbard Brook Watershed 3

data kindly provided by G. Likens and D. BusoNS= 0.42 - 0.76

Silicon (Si) Sodium (Na)

NS= 0.34 - 0.66

1/𝑘 10−13𝑑𝑎𝑦𝑠

OUTLET

ti te

TT = te- ti

time

TRAVEL TIME

t

AGE LIFE EXPECTANCY

back to basics

Tracer injection experiment

Water samples at a catchment outlet

Cin

Cout

time

Queloz et al., WRR, 2015a,b

PAST entrance times

FUTUREexit times

time

mg/

L

Kirchner and Neal, PNAS, 2013

What is the link between

‘forward’ and ‘backward’ age

tracking?

Backward and forward age tracking (Chapter 1)

Q (t)

J (t)

S (t)

Definitions: distributions of particles

Governing equations

𝑑𝑆 (𝑡)𝑑𝑡

=𝐼𝑁 (𝑡 )−𝑂𝑈𝑇 (𝑡 )• Hydrologic Balance:

• CONTINUITY for each age class T (either or )

𝜕𝜕𝑡

𝑁𝑺(𝑇 ,𝑡 )+𝑐𝜕𝜕𝑇

𝑁𝑺(𝑇 , 𝑡)=∑𝑖

𝐹 𝑖 (𝑡 )𝑝𝑭 𝒊(𝑇 , 𝑡)

• forward , , BC

• backward ,

Master Equation generator:

(define: volumetric quantity)

Benettin et al., 2015, Hydrol. Process.

Life-expectancytracking

Age tracking

modeling implications

0 10 20 30 40 50 60 70 80 900

1000000

2000000

3000000

4000000

5000000

age [years]

1940

source: CDC/NCHS, National Vital Statistics System, USA

US population by age class

1950196019701980199020002010

h𝑢𝑚𝑎𝑛𝑟𝑒𝑠𝑖𝑑𝑒𝑛𝑡 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛=�́�𝑆(𝑇 , 𝑡)

water particles as a dynamic population

source: CDC/NCHS, National Vital Statistics System, USA

US age at death, 1940 - 2010

0 10 20 30 40 50 60 70 80 900.00

50000.00

100000.00

19401950196019701980199020002010

age [years]

h𝑢𝑚𝑎𝑛𝑜𝑢𝑡𝑓𝑙𝑜𝑤 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛=�́�𝑄(𝑇 ,𝑡)

water particles as a dynamic population

0 10 20 30 40 50 60 70 80 900.00

0.01

0.02

pS(T,t)19401950196019701980199020002010

age [y]

pdf [

1/y]

0 10 20 30 40 50 60 70 80 900.00

0.02

0.04

pQ(T,t)19401950196019701980199020002010

age [y]

pdf [

1/y]

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

age selection𝜔(PS, )𝑡19401950196019701980199020002010

transformed age [-]

pdf [

-]

Progress

water particles as a dynamic population

Theory Applications

• development of the master equation generator (introduction of the forward formulation)

• definition of age concepts in general advection-dispersion systems

• generation of time-variant age dynamics through simple hydrochemical models

Summary of the results

• modeling of 3 diverse real-world catchments

• exploration of catchment functioning

• use of age distributions for reactive transport

acknowledgments

Plynlimon data:

Ype van der Velde

Hupsel Brook data:

Hubbard Brook data:

S. Bailey, JP Gannon, M. Green, J. Campbell, G. Likens, D. Buso