The document summarizes the key points of a PhD dissertation on catchment transport and travel time distributions. It presents (1) the development of a unified theory of water age and life expectancy distributions, (2) how age mixing occurs in advection-dispersion systems, and (3) applications of these concepts to model conservative and non-conservative solute transport in three catchments. The dissertation advances understanding of catchment functioning through the use of time-variant age distributions to simulate reactive transport processes.

1. Catchment Transport and Travel Time Distributions:
Theoretical Developments and Applications
Paolo Benettin
PhD Days di Ingegneria delle Acque – Trento, 6-8 Luglio 2015

2. the Ph.D. dissertation
Cin
- Travel-time distributions
- Hydrologic and solute transport

3. University of Padua,
Padua, Italy
Gianluca Botter
EPFL,
Lausanne, CH
Andrea Rinaldo
PH.D. SUPERVISORS

4. Introduction

5. age = time since entrance
age T
𝒑 𝑸(𝑻, 𝒕)
Distribution of water parcels
time
𝐶 𝑡 =
0
∞
𝑐 𝑇 𝒑 𝑸 𝑻, 𝒕 𝑑𝑇
fundamental link
between water age
and water quality

6. source: ARPAV
Ecological
Status 2013
Sufficient
Poor
GoodVenice Lagoon drainage basin
Nitrate Loads
!
2001 2003 2005 2007 2009 2011
why studying water age

7. Sanford and Pope, Env. Sci. and Technol., 2013
why studying water age

8. 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)

9. Kirchner et al., 2000, Nature
WATER
CHLORIDE
why hydrologic transport isn’t so simple (1)

10. data from Plynlimon UHF catchment, UK
[mm/h][mg/l]
silica
chloride
[mg/l]
dry
period
wet
period
why hydrologic transport isn’t so simple (2)

11. why hydrologic transport isn’t so simple (3)
McDonnell et al., 2010, HP
realistic distributionsideal distributions

12. ages in storage
T
𝒑 𝑺(𝑻, 𝒕)
0
time
tt2t1
INJECTION TIMES
t3
S(t)
ages in the discharge
T
𝒑 𝑸(𝑻, 𝒕)
hydrologic transport processes

13. What relationship exists between
particles in storage
and particles in the fluxes?
‘StorAge-Selection’
functions
𝑝 𝑄(𝑇, 𝑡)
𝑝 𝑆(𝑇, 𝑡)
= 𝜔(𝑇, 𝑡)
flux

14. age-selection
ω 𝑇, 𝑡 =
𝑝 𝑄(𝑇, 𝑡)
𝑝 𝑆(𝑇, 𝑡)
preference for
younger ages
no preference
(random sampling)
preference for
older ages
age
𝜔 < 1
𝜔 > 1
𝜔[−]
1

15. My contribution

16. 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 order
Chronological order
3
2
4
1
5

17. Ype van der Velde Sjoerd van der Zee
NL
outlet
Conservative solutes from an agricultural area (Chapter 3)
Can we use
time-variant age
distributions
to model chloride
transport?

18. SOIL
STORAGE
GROUNDWATER
STORAGE
a simple transport model
soil water mean [d] ≈ 90
st.dev. [d] ≈ 20
groundwater
mean [d] ≈ 1100
st.dev. [d] ≈ 120

19. Model results
shorter (30-100 d)
travel times
Q[mm/h]
longer (2-3 y)
travel times

20. how do ages mix
in advection-
dispersion processes?
Age mixing in advection-dispersion models (Chapter 2)
flux
𝑝 𝑆 𝑻, 𝑡 STORAGE age
𝑝 𝑄 𝑻, 𝑡 DISCHARGE age
𝜌 𝒙, 𝑻, 𝑡 ‘age mass density’
Ginn, 1999, WRR
Benettin et al., WRR, 2013b

21. Age mixing in advection-dispersion models (Chapter 2)
STORAGE age
‘‘StorAge-Selection’’ function𝜔 𝑇, 𝑡 =
𝑝 𝑄(𝑇, 𝑡)
𝑝 𝑆(𝑇, 𝑡)
=
𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙
𝑣𝑜𝑙𝑢𝑚𝑒 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙
𝑝 𝑆 𝑇, 𝑡 =
1
𝑀(𝑡) 𝑉
𝜌 𝒙, 𝑇, 𝑡 𝑑𝒙
𝑝 𝑄 𝑇, 𝑡 =
1
𝜑 𝑜𝑢𝑡 𝑡 𝑆
𝑢 𝒙, 𝑡 𝜌 𝒙, 𝑇, 𝑡 − 𝑫 𝒙, 𝑡 𝛻𝜌 𝒙, 𝑇, 𝑡 𝒏 𝑑𝜎
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 age
VS
storage age
Ginn, 1999, WRR

22. Pe = 1
relative age [%]
ω[-]
𝜔 𝑇, 𝑡 =
𝑝 𝑄(𝑇, 𝑡)
𝑝 𝑆(𝑇, 𝑡)

23. Experiences abroad
Kevin J. McGuire
James W. Kirchner
period abroad at
Virginia Tech University
and
AGU fall meeting 2013

24. 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?

25. CALIBRATED HYDROCHEMICAL MODEL
Parameters
posterior distributions

26. age dynamics
Tracer
response
input
variability+general affinity
for younger ages
younger ages older ages
StorAge Selection functionsCumulative age distributions

27. MOBILE
WATER
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?

28. Travel time distributions
NS= 0.62

29. 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 𝑝 𝑄 𝑻, 𝑡 𝑑𝑻
𝐶𝑒𝑞 𝑐(𝑇) 1° order chemical kinetics:

30. 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 − 13 𝑑𝑎𝑦𝑠

31. OUTLET
ti te
TT = te- ti
time
TRAVEL TIME
t
AGE LIFE EXPECTANCY
back to basics

32. Tracer injection experiment
Water samples
at a catchment outlet
Cin
Cout
time
Queloz et al., WRR, 2015a,b
PAST
entrance times
FUTURE
exit 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)

33. Q (t)
J (t)
S (t)
Definitions: distributions of particles

34. Governing equations
𝑑𝑆(𝑡)
𝑑𝑡
= 𝐼𝑁 𝑡 − 𝑂𝑈𝑇 𝑡• Hydrologic Balance:
• CONTINUITY for each age class T (either 𝑇𝑅 or 𝑇𝐸)
𝜕
𝜕𝑡
𝑁𝑺(𝑇, 𝑡) + 𝑐
𝜕
𝜕𝑇
𝑁𝑺(𝑇, 𝑡) =
𝑖
𝐹𝑖 𝑡 𝑝 𝑭𝒊
(𝑇, 𝑡)
• forward → 𝑇 = 𝑇𝐸, 𝑐 = −1, 𝐹 = 𝑜𝑢𝑡𝑓𝑙𝑜𝑤𝑠, BC = 𝑖𝑛𝑓𝑙𝑜𝑤𝑠
• backward → 𝑇 = 𝑇𝑅, 𝑐 = +1, 𝐹 = 𝑖𝑛𝑓𝑙𝑜𝑤𝑠, BC = 𝑜𝑢𝑡𝑓𝑙𝑜𝑤𝑠
Master Equation generator:
(define: 𝑁𝑆 𝑇, 𝑡 = 𝑆 𝑡 𝑝 𝑆(𝑇, 𝑡) volumetric quantity)
Benettin et al., 2015, Hydrol. Process.

35. Life-expectancy
tracking
Age
tracking
modeling implications

36. 0
1
2
3
4
5
0 10 20 30 40 50 60 70 80 90
Millions
age [years]
1940
source: CDC/NCHS, National Vital Statistics System, USA
US population by age class
1950196019701980199020002010
ℎ𝑢𝑚𝑎𝑛 𝑟𝑒𝑠𝑖𝑑𝑒𝑛𝑡 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 = 𝑝 𝑆(𝑇, 𝑡)
water particles as a dynamic population

37. source: CDC/NCHS, National Vital Statistics System, USA
US age at death, 1940 - 2010
0.00
0.05
0.10
0 10 20 30 40 50 60 70 80 90
Millions
age [years]
1940
1950
1960
1970
1980
1990
2000
2010
ℎ𝑢𝑚𝑎𝑛 𝑜𝑢𝑡𝑓𝑙𝑜𝑤 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 = 𝑝 𝑄(𝑇, 𝑡)
water particles as a dynamic population

38. 0.00
0.01
0.02
0 20 40 60 80
pdf[1/y]
age [y]
pS(T,t)
1940
1950
1960
1970
1980
1990
2000
2010
0.00
0.02
0.04
0 20 40 60 80
pdf[1/y]
age [y]
pQ(T,t)1940
1950
1960
1970
1980
1990
2000
2010
0
5
10
15
0.0 0.2 0.4 0.6 0.8 1.0
pdf[-]
transformed age [-]
age selection
𝜔(PS,𝑡)
1940
1950
1960
1970
1980
1990
2000
2010
Progress
water particles as a dynamic population

39. 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

40. 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