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Isotope Hydrology Shortcourse
Residence Time
Approaches using
Isotope Tracers
Prof. Jeff McDonnell
Dept. of Forest Engineering
Oregon State University
1
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Outline

Day 1
 Morning: Introduction, Isotope Geochemistry Basics
 Afternoon: Isotope Geochemistry Basics ‘cont, Examples
 Day 2
 Morning: Groundwater Surface Water Interaction, Hydrograph
separation basics, time source separations, geographic source
separations, practical issues
 Afternoon: Processes explaining isotope evidence, groundwater
ridging, transmissivity feedback, subsurface stormflow, saturation
overland flow
 Day 3
 Morning: Mean residence time computation
 Afternoon: Stable isotopes in watershed models, mean residence
time and model strcutures, two-box models with isotope time
series, 3-box models and use of isotope tracers as soft data
 Day 4
 Field Trip to Hydrohill or nearby research site
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University
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© Oregon State
University
How these time and space scales relate
to what we have discussed so far
3
Bloschel et al.,McGuire,
1995
OSU
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© Oregon State
University
This section will examine how we
make use of isotopic variability
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Outline
 What is residence time?
 How is it determined?
modeling background
 Subsurface transport basics
 Stable isotope dating (18O and 2H)
 Models: transfer functions
 Tritium (3H)
 CFCs, 3H/3He, and
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University
85Kr
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Residence Time
 Mean Water Residence Time
(aka: turnover
time, age of water leaving a system, exit age, mean transit time, travel
time, hydraulic age, flushing time, or kinematic age)
 tw=Vm/Q
 For
1D flow pattern: tw=x/vpw
where vpw =q/f
 Mean Tracer Residence Time

 tC (t )dt
Residence time distribution
I
tt 
0

 C (t )dt
I
0
© Oregon State
University
g (t ) 

CI (t )
 C (t )dt
 CI Q / M
I
0
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Why is Residence Time of Interest?
 It tells us something fundamental about
the hydrology of a watershed
 Because chemical weathering,
denitrification, and many
biogeochemical processes are
kinetically controlled, residence time
can be a basis for comparisons of water
chemistry
Vitvar & Burns,
2001
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University
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Tracers and Age Ranges
 Environmental tracers:

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University
added (injected) by natural processes, typically
conservative (no losses, e.g., decay, sorption),
or ideal (behaves exactly like traced material)
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Modeling Approach
 Lumped-parameter models (black-box
models):
 System
is treated as a whole & flow pattern
is assumed constant over modeling period

Used to interpret tracer observations in system outflow (e.g.
GW well, stream, lysimeter)
 Inverse procedure; Mathematical tool:
 The convolution integral
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
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Convolution
y (t ) 
 h(t   ) x(t )d

 A convolution is an integral which expresses
the amount of overlap of one function h as it
is shifted over another function x. It therefore
"blends" one function with another
 It’s frequency filter, i.e., it attenuates specific
frequencies of the input to produce the result
 Calculation methods:
Fourier transformations, power spectra
 Numerical Integration

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The Convolution Theorem
 f (t ) * g (t )  F ( )G( )


{ f (t ) * g (t )}     f ( x) g (t – x) dx  exp( i t ) dt
  

Proof:



  f ( x)   g (t  x) exp(–i t ) dt  dx









Trebino, 2002



© Oregon State
University
f ( x){G (  exp(–i x)} dx
Y()=F()G()
and
|Y()|2=|F()| 2 |G()|
2
f ( x) exp(–i x) dxG (   F ( G ( 
We will not go through this!!
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Convolution: Illustration of how it
works
g() = e -a
x()
Step

e
-(-a
g(-)
Folding
1

e -a(t-
g(t-)

t
Displacement
2
x()g(t-)
Multiplication
3
Integration
4
t
y(t)
t
© Oregon State
University
Shaded
area
t
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Example: Delta Function
f (t )   t  a) 


f (t  u )  (u – a) du

 f (t  a)
Convolution with a delta function simply
centers the function on the delta-function.
This convolution does not smear out f(t).
Thus, it can physically represent piston-flow
processes.
Modified from
Trebino, 2002
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Matrix Set-up for Convolution
= length(x)
= [length(x)+length(h)]-1
=S
= x(t)*h
y(t)
=0
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Similar to the Unit Hydrograph
Precipitation
Excess Precipitation
Infiltration Capacity
Excess
Precipitation
Hydrographs for Event
2500
Time
Flow
2000
1500
1000
500
Tarboton
0
0
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University
1 Time(hrs) 2
3
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Instantaneous Response Function
Excess Precipitation P(t)
Unit Response Function U(t)
2500
2000
1500
1000
500
0
0
1
2
3
4
Event Response Q(t)
2500
2000
1500
Q ( t )   P ( ) U ( t  ) d 
1000
500
0
0
1
2
3
Tarboton
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University
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Subsurface Transport Basics
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Subsurface Transport Processes
 Advection
 Dispersion
 Sorption
 Transformations
Modified from
Neupauer
& Wilson, 2001
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University
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Advection
Solute movement with bulk water flow
t=t1
t2>t
1
t3>t
2
FLOW
Modified from
Neupauer
& Wilson, 2001
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University
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Subsurface Transport Processes
 Advection
 Dispersion
 Sorption
 Transformations
Modified from
Neupauer
& Wilson, 2001
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University
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Dispersion
Solute spreading due to flowpath heterogeneity
Modified from
Neupauer
& Wilson, 2001
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University
FLOW
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Subsurface Transport Processes
 Advection
 Dispersion
 Sorption
 Transformations
Modified from
Neupauer
& Wilson, 2001
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University
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Sorption
Solute interactions with rock matrix
FLOW
Modified from
Neupauer
& Wilson, 2001
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University
t=t1
t2>t1
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Subsurface Transport Processes
 Advection
 Dispersion
 Sorption
 Transformations
Modified from
Neupauer
& Wilson, 2001
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University
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Transformations
Solute decay due to chemical and biological reactions
MICROBE
CO2
Modified from
Neupauer
& Wilson, 2001
© Oregon State
University
t=t1
t2>t1
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Stable Isotope Methods
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Stable Isotope Methods
and 2H in
precipitation at temperate latitudes
 Seasonal variation of
18O
 Variation becomes progressively more
muted as residence time increases
 These variations generally fit a model
that incorporates assumptions about
subsurface water flow
Vitvar & Burns,
2001
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University
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Seasonal Variation in 18O of Precipitation
air temperature (°C)

18O (per mil SMOW)
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© Oregon State
University
0
Neversink watershed, 1993 - 1996
-5
-10
-15
-20
20
10
0
-10
Jan-93
Vitvar, 2000
Jan-94
Jan-95
Jan-96
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Seasonality in Stream Water
Deines et al. 1990
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Example: Sine-wave
Oxygen-18 (per mil)
-2
Cin(t)=A sin(t)
Cout(t)=B sin(t+j)
6
g(t)
0
-3
4
Mean = 235 d
2
Precipitation or recharge signal
-4
x 10
0
0
1000
2000
Time [days]
-6
-8
-10
Streamflow signal
-12
-14
1200
© Oregon State
University
T=-1[(B/A)2 –1)1/2
1300
1400
1500
Time [days]
1600
1700
1800
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© Oregon State
University
Convolution Movie
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Transfer Functions Used for
Residence Time Distributions
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Common Residence Time Models
-3
0.012
6
D/vx=5
D/vx=3.5
D/vx=2
D/vx=1
D/vx=0.1
D/vx=0.2
D/vx=0.5
D/vx=0.01
D/vx=0.05
D/vx=0.005
0.01
g(t)
0.008
0.006
3
0.002
1
1
1.5
Normalized time (t/T)
© Oregon State
University
4
2
0.5
2
eta=1
eta=1.25
eta=1.5
eta=1.75
eta=2
eta=2.25
eta=2.5
eta=2.75
eta=3
5
0.004
0
0
x 10
0
0
0.5
1
1.5
2
Normalized time (t/T)
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Piston Flow (PFM)
 Assumes all flow paths have transit time
 All
water moves with advection
 Represented by a Dirac delta function:
1
0.8
g (t )   (t  T )
g(t)
0.6
0.4
0.2
0
0
1
2
3
4
t/T
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Exponential (EM)
 Assumes contribution from all flow paths
lengths and heavy weighting of young
portion.
 Similar to the concept of a “well-mixed”
system in a linear reservoir model
0.16
0.14
0.12
g (t )  T 1 exp(t / T )
g(t)
0.1
0.08
0.06
0.04
0.02
0
0
© Oregon State
University
2
4
6
t/T
8
10
12
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Dispersion (DM)
 Assumes that flow paths are effected by
hydrodynamic dispersion or geomorphological
dispersion
 Arises from a solution of the 1-D advection-dispersion
equation:
C
 2C
C
t
D
x
2
v
x
0.01
0.008
 4D p t 

g (t )  
 T 
g(t)
0.006
0.004
1/ 2
2

t

  T 
1
t exp 1  



  T   4 D p t 
0.002
0
0
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University
2
4
6
t/T
8
10
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Exponential-piston Flow (EPM)
 Combination of exponential and piston
flow to allow for a delay of shortest flow
paths
h
 ht

g (t )  exp   h  1
T
 T

0.2
g(t)
0.15
for t T (1-h1, and
g(t)=0 for t< T (1-h-1)
0.1
0.05
Piston flow = 1
0
0
2
4
6
8
10
1
h
12
t/T
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Heavy-tailed Models
 Gamma
t  1
t / 
g (t )  
exp
 ( )
 Exponentials in series
 t 
 t 
1
g (t ) 
exp   
exp  
T1
T2
 T1 
 T2 

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Exit-age distribution (system
response function)
Unconfined aquifer
EM: g(t’) = 1/T exp(-t‘/T)
Confined aquifer
PFM: g(t’) = (t'-T)
DM
DM
EM EM EPM
EM
PFM
PFM
Maloszewski and Zuber
Kendall, 2001
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Exit-age distribution (system
response function) cont…
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 Partly Confined Aquifer:
EPM:
g(t’) = h/T exp(-ht'/T + h-1)
g(t’) = 0
for t‘≥T (1 - 1/h)
for t'< T (1-1/ h)
DM
Kendall, 2001
© Oregon State
University
Maloszewski and Zuber
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0
g(t)
-10
0.15
0.1
-15
0.05
-20
0
0
MRT = 12 months
D/vx = 0.3
0.2
g(t)
-5
-10
0.15
0.1
-15
0.05
-20
0
0
-10
-15
-20
0
MRT = 6 months
D/vx = 0.05
0.2
-5
0.15
0.1
0.05
20
40
60
Time (months)
© Oregon State
University
MRT = 6 months
D/vx = 0.3
0.2
-5
g(t)
O-18 (per mil)
O-18 (per mil)
O-18 (per mil)
Dispersion Model Examples
80
0
0
20
40
60
80
Time (months)
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Residence Time Distributions can be
Similar
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0.08
EPM: 21% Piston
MRT = 10.5 mon.
g(t)
0.06
0.04
DM: Dp = 0.36
MRT = 8.5 mon.
EPM 12% Piston
MRT = 9.5 mon.
DM: Dp = 0.27
MRT = 10.5 mon.
0.02
0.00
0
© Oregon State
University
500
1000
Time (d)
1500 0
500
1000
1500
Time (d)
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Uncertainty
Freq
30
0.15
10
0
10
FittedTransfer Functions
12
16
30
0.1
0.05
20
10
0
140
0
0
14
Piston Flow %
Freq
Function weighting [g(t)]
0.2
20
160
180
MRT
0.5
1
1.5
2
2.5
Normalized time [t/T]
3
3.5
4
-8
tracer content
Convolution
Output Obs
-9
-10
Simulation Results with Optimized Parameters
-11
40
45
50
55
60
65
70
75
time
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© Oregon State
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Identifiable Parameters?
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Review: Calculation of Residence Time

Simulation of the isotope input –
output relation:
t
C (t )   g (t   ) Cin (t ) d
0

Calibrate the function g(t) by assuming
various distributions of the residence
time:
1.
2.
3.
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Exponential Model
Piston Flow Model
Dispersion Model
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Input Functions
t
C (t )   g (t   ) Cin (t ) d
0
 Must represent tracer flux in recharge

Weighting functions are used to “amount-weight” the tracer
values according recharge: mass balance!!
 Methods:

Winter/summer weighting:
 in (t ) 
N i Pi
N
 P
i 1

Lysimeter outflow

General equation:
C (t ) 
© Oregon State
University

 g ( )w(t   )
0

in
(t   )d
 g ( )w(t   )d
0

i

 Cin  Cin
i i
where w(t) = recharge
weighting function
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Models of Hydrologic Systems
Model 1
Cin
Cout
Model 3
Cin
Model 2
Upper
reservoir
1- 
Cout

g
Direct runoff
Cin 1- g
1- g
g
Cout
Lower
reservoir
Maloszewski et
al., 1983
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University
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Soil Water Residence Time
Stewart &
McDonnell, 2001
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University
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Rietholzbach watershed, Switzerland
Mean baseflow residence time = 12.5 mo
-10
-11
18O
measured values
-12
dispersion model
Rietholzbach watershed, Switzerland
Mean groundwater residence time = 28 mo
-9
-10
-11
-12
measured values

-13
-14
-8
(per mil SMOW)
-9

18O
(per mil SMOW)
-8
Example from Rietholzbach
-13
exp/piston-flow model
1994
1995
1996
1997
-14
dispersion model
exponential model
1994
1995
1996
1997
Vitvar, 1998
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University
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Model 3…
Stable deep
signal
Uhlenbrook et
al., 2002
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University
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How residence time scales with basin area
172 degrees E
New Zealand
42 degrees S
Figure 1
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Contour interval
10 meters
Digital elevation model
and stream network
Figure 2
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1.0
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M15 catchment (2.6 ha)
0.5
Median sub-catchment size = 1.2 ha
0.0
Frequency
1.0
1
10
K catchment (17 ha)
0.5
100
Median sub-catchment size = 8.2 ha
0.0
1.0
1
10
PL14 catchment (80 ha)
0.5
100
Median sub-catchment size = 3.9 ha
0.0
1.0
1
10
100
Bedload catchment (280 ha)
0.5
Median sub-catchment size = 3.2 ha
0.0
Figure 3
1
10
100
Sub-catchment size ha
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Mean tritium age years
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K (17 ha)
2
Bedload
(280 ha)
PL14
(17 ha)
M15
(2.6 ha)
1
0
2
4
6
8
Median sub-catchment size ha
Figure 4
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University
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RIF
-7
Low
-3.5
0
High
Scale
500 m
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University
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Determining Residence Time of
Old(er) Waters
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What’s Old?
 No seasonal variation of stable isotope
concentrations: >4 to 50 years
 Methods:
 Tritium (3H)
 3H/3He
 CFCs
 85Kr
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Tritium
 Historical tracer: 1963 bomb peak of 3H in
atmosphere
1 TU: 1 3H per 1018 hydrogen atoms
 Slug-like input
 36Cl is a similar tracer

 Similar methods to stable isotope models
 Half-life (l) = 12.43
Tritium Input
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Tritium (con’t)
 Piston flow (decay only):
tt=-17.93[ln(C(t)/C0)]
 Other flow conditions:
t
C (t )   Cin (t ) exp(lt ' ) g (t  t ' )dt'
0
© Oregon State
University
Manga, 1999
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Deep Groundwater Residence Time
Spring: Stollen
t0 = 8.6 a, PD = 0.22
3H-Input
3H-Input-Bruggagebiet
1000
sim. 3H-Konzentrationen
3
H-Messungen mit analyt. Fehler
H-Input im Bruggagebiet
H [T.U.]
3
20
3
100
15
3
H-Konzentrationen [T.U.]
25
10
1950
1960
1970
1980
1990
2000
10
1992
1993
1994
Zeit [Jahre]
Time [yr.]
Uhlenbrook et
al., 2002
© Oregon State
University
1995
1996
1997
1998
1999
Time [yr.]
lumped parameter models
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3He/3H
 As 3H enters groundwater and
radioactively decays, the noble gas 3He
is produced
 Once in GW, concentrations of 3He
increase as GW gets older
 If 3H and 3He are determined together,
an apparent age can be determined:
3
*


He
1
tt  l ln 3
 1
 H

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Determination of Tritiogenic He
 Other sources of 3He:
30
 Atmospheric
age (years)
solubility (temp dependent)
 Trapped air during recharge
20.5 years
 Radiogenic
production ( decay of U/Th20
series elements)
3He/3H
4He and
 Determined
by
measuring
10
other noble gases
0
1
5
10
50
Tage (years)
Modified from Manga, 1999
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Chlorofluorocarbons (CFCs)
 CFC-11 (CFCL3), CFC-12 (CF2Cl2), &
CFC-13 (C2F3Cl3) long atm residence
time (44, 180, 85 yrs)
 Concentrations are uniform over large
areas and atm concentration are
steadily increasing
 Apparent age = CFC conc in GW to
equivalent atm conc at recharge time
using solubility relationships
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85Kr
 Radioactive inert gas, present is atm
from fission reaction (reactors)
 Concentrations are increasing worldwide
 Half-life = 10.76; useful for young dating
too
 Groundwater ages are obtained by
correcting the measured 85Kr activity in
GW for radioactive decay until a point
on the atm input curve is reached
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85Kr
(con’t)
 Independent of recharge temp and
trapped air
 Little source/sink in subsurface
 Requires large volumes of water
sampled by vacuum extraction (~100 L)
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Model 3…
Uhlenbrook et
al., 2002
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Large-scale Basins
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Notes on Residence Time Estimation
• 18O and 2H variations show mean
residence times up to ~4 years only;
older waters dated through other tracers
(CFC, 85Kr, 4He/3H, etc.)
• Need at least 1 year sampling record of
isotopes in the input (precip) and output
(stream, borehole, lysimeter, etc.)
• Isotope record in precipitation must be
adjusted to groundwater recharge if
groundwater age is estimated
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Class exercise
ftp://ftp.fsl.orst.edu/pub/mcguirek/rt_lecture
 Hydrograph separation
 Convolution
 FLOWPC

Show your results graphically (one or several
models) and provide a short write-up that
includes:
– Parameter identifiability/uncertainty
– Interpretation of your residence time distribution in
terms of the flow system
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References
Cook, P.G. and Solomon, D.K., 1997. Recent advances in dating young
groundwater: chlorofluorocarbons, 3H/3He and 85Kr. Journal of Hydrology,
191:245-265.
Duffy, C.J. and Gelhar, L.W., 1985. Frequency Domain Approach to Water Quality
Modeling in Groundwater: Theory. Water Resources Research, 21(8): 11751184.
Kirchner, J.W., Feng, X. and Neal, C., 2000. Fractal stream chemistry and its
implications for contaminant transport in catchments. Nature, 403(6769): 524527.
Maloszewski, P. and Zuber, A., 1982. Determining the turnover time of groundwater
systems with the aid of environmental tracers. 1. models and their applicability.
Journal of Hydrology, 57: 207-231.
Maloszewski, P. and Zuber, A., 1993. Principles and practice of calibration and
validation of mathematical models for the interpretation of environmental tracer
data. Advances in Water Resources, 16: 173-190.
Turner, J.V. and Barnes, C.J., 1998. Modeling of isotopes and hydrochemical
responses in catchment hydrology. In: C. Kendall and J.J. McDonnell (Editors),
Isotope tracers in catchment hydrology. Elsevier, Amsterdam, pp. 723-760.
Zuber, A. and Maloszewski, P., 2000. Lumped parameter models. In: W.G. Mook
(Editor), Environmental Isotopes in the Hydrological Cycle Principles and
Applications. IAEA and UNESCO, Vienna, pp. 5-35. Available:
http://www.iaea.or.at/programmes/ripc/ih/volumes/vol_six/chvi_02.pdf
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Outline

Day 1
 Morning: Introduction, Isotope Geochemistry Basics
 Afternoon: Isotope Geochemistry Basics ‘cont, Examples
 Day 2
 Morning: Groundwater Surface Water Interaction, Hydrograph
separation basics, time source separations, geographic source
separations, practical issues
 Afternoon: Processes explaining isotope evidence, groundwater
ridging, transmissivity feedback, subsurface stormflow, saturation
overland flow
 Day 3
 Morning: Mean residence time computation
 Afternoon: Stable isotopes in watershed models, mean residence
time and model strcutures, two-box models with isotope time
series, 3-box models and use of isotope tracers as soft data
 Day 4
 Field Trip to Hydrohill or nearby research site
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