Residence Time Modelling
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Transcript Residence Time Modelling
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Isotope Hydrology Shortcourse
Residence Time
Approaches using
Isotope Tracers
Prof. Jeff McDonnell
Dept. of Forest Engineering
Oregon State University
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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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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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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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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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Tracers and Age Ranges
Environmental tracers:
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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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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
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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
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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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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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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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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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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
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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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Seasonal Variation in 18O of Precipitation
air temperature (°C)
18O (per mil SMOW)
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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
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University
T=-1[(B/A)2 –1)1/2
1300
1400
1500
Time [days]
1600
1700
1800
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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)
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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
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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
4D 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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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-h1, 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
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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)
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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
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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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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 )
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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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Soil Water Residence Time
Stewart &
McDonnell, 2001
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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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Model 3…
Stable deep
signal
Uhlenbrook et
al., 2002
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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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RIF
-7
Low
-3.5
0
High
Scale
500 m
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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
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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
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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
© Oregon State
University
70
McGuire, OSU
Tracers: RT
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
© Oregon State
University
71
McGuire, OSU