1. Isotope Hydrology Shortcourse
Residence Time Approaches using Isotope Tracers
Prof. Jeff McDonnell
Dept. of Forest Engineering
Oregon State University

2. 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

3. How these time and space scales relate to what we have discussed so far
Bloschel et al., 1995

4. This section will examine how we make use of isotopic variability

5. 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 85Kr

6. Residence time distribution
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
Residence time distribution

7. 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

8. Tracers and Age Ranges Environmental tracers:
added (injected) by natural processes, typically conservative (no losses, e.g., decay, sorption), or ideal (behaves exactly like traced material)

9. 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

10. Convolution
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

11. The Convolution Theorem
Proof:
Y(w)=F(w)G(w) and
|Y(w)|2=|F(w)| 2 |G(w)| 2
Trebino, 2002
We will not go through this!!

12. Convolution: Illustration of how it works
x(t) t
g(t) = e -at
Step
g(-t) t
e -(-at)
Folding 1
g(t-t) t
e -a(t-t) 2
Displacement
x(t)g(t-t) t
Multiplication 3
y(t) t
Shaded area
Integration 4

13. Example: Delta Function
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

14. Matrix Set-up for Convolution
= [length(x)+length(h)]-1
= length(x) =S
y(t)
= x(t)*h
= 0

15. Similar to the Unit Hydrograph
Precipitation
Excess Precipitation
Infiltration Capacity
Excess Precipitation
Time
Tarboton

16. Instantaneous Response Function
Unit Response Function U(t)
Excess Precipitation P(t)
Event Response Q(t)
Tarboton

17. Subsurface Transport Basics

18. Subsurface Transport Processes
Advection
Dispersion
Sorption
Transformations
Modified from Neupauer
& Wilson, 2001

19. Modified from Neupauer
Advection
Solute movement with bulk water flow
t=t1
t2>t1
t3>t2
FLOW
Modified from Neupauer
& Wilson, 2001

20. Subsurface Transport Processes
Advection
Dispersion
Sorption
Transformations
Modified from Neupauer
& Wilson, 2001

21. Modified from Neupauer
Dispersion
Solute spreading due to flowpath heterogeneity
FLOW
Modified from Neupauer
& Wilson, 2001

22. Subsurface Transport Processes
Advection
Dispersion
Sorption
Transformations
Modified from Neupauer
& Wilson, 2001

23. Modified from Neupauer
Sorption
Solute interactions with rock matrix
FLOW
t=t1
t2>t1
Modified from Neupauer
& Wilson, 2001

24. Subsurface Transport Processes
Advection
Dispersion
Sorption
Transformations
Modified from Neupauer
& Wilson, 2001

25. Modified from Neupauer
Transformations
Solute decay due to chemical and biological reactions
MICROBE
CO2
t=t1
t2>t1
Modified from Neupauer
& Wilson, 2001

26. Stable Isotope Methods

27. Stable Isotope Methods
Seasonal variation of 18O and 2H in precipitation at temperate latitudes
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

28. Seasonal Variation in 18O of Precipitation
Vitvar, 2000

29. Seasonality in Stream Water
Deines et al. 1990

30. Example: Sine-wave Cin(t)=A sin(wt) Cout(t)=B sin(wt+j)
T=w-1[(B/A)2 –1)1/2

31. Convolution Movie

32. Transfer Functions Used for Residence Time Distributions

33. Common Residence Time Models

34. Piston Flow (PFM) Assumes all flow paths have transit time
All water moves with advection
Represented by a Dirac delta function:

35. 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

36. 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:

37. Exponential-piston Flow (EPM)
Combination of exponential and piston flow to allow for a delay of shortest flow paths
for t T (1-h-1), and
g(t)=0 for t< T (1-h-1)
Piston flow =

38. Heavy-tailed Models
Gamma
Exponentials in series

39. Exit-age distribution (system response function)
Confined aquifer
PFM: g(t’) = (t'-T)
Unconfined aquifer
EM: g(t’) = 1/T exp(-t‘/T) DM EM EM
EPM
PFM
PFM EM
Maloszewski and Zuber
Kendall, 2001

40. Exit-age distribution (system response function) cont…
Partly Confined Aquifer:
EPM: g(t’) = /T exp(-t'/T + -1) for t‘≥T (1 - 1/)
g(t’) = for t'< T (1-1/ ) DM
Kendall, 2001
Maloszewski and Zuber

41. Dispersion Model Examples

42. Residence Time Distributions can be Similar

43. Uncertainty

44. Identifiable Parameters?

45. Review: Calculation of Residence Time
Simulation of the isotope input – output relation:
Calibrate the function g(t) by assuming various distributions of the residence time:
Exponential Model
Piston Flow Model
Dispersion Model

46. Input Functions Must represent tracer flux in recharge Methods:
Weighting functions are used to “amount-weight” the tracer values according recharge: mass balance!!
Methods:
Winter/summer weighting:
Lysimeter outflow
General equation:
where w(t) = recharge weighting function

47. Models of Hydrologic Systems
Cin
Cout
Model 1
1- g g
Cout
Cin
1- b b
Model 3
Upper reservoir
Lower reservoir
Cout
Cin
1- g g
Model 2
Direct runoff
Maloszewski et al., 1983

48. Soil Water Residence Time
Stewart & McDonnell, 2001

49. Example from Rietholzbach
Vitvar, 1998

50. Model 3…
Stable deep signal
Uhlenbrook et al., 2002

51. How residence time scales with basin area
Figure 1

52. Digital elevation model
Contour interval
10 meters
Digital elevation model
and stream network
Figure 2

53. Figure 3

54. K (17 ha)
Bedload
(280 ha)
PL14
(17 ha)
M15
(2.6 ha)
Figure 4

55. 500 m
Scale -7
-3.5
Low
High
RIF

56. Determining Residence Time of Old(er) Waters

57. What’s Old?
No seasonal variation of stable isotope concentrations: >4 to 50 years
Methods:
Tritium (3H)
3H/3He
CFCs
85Kr

58. 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

59. Tritium (con’t) Piston flow (decay only): Other flow conditions:
tt=-17.93[ln(C(t)/C0)]
Other flow conditions:
Manga, 1999

60. Deep Groundwater Residence Time
Spring: Stollen
t0 = 8.6 a, PD = 0.22
3H-Input
3H-Input-Bruggagebiet
3H [TU]
3H [TU]
Time [yr.]
Time [yr.]
Uhlenbrook et al., 2002
lumped parameter models

61. 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:

62. Determination of Tritiogenic He
Other sources of 3He:
Atmospheric solubility (temp dependent)
Trapped air during recharge
Radiogenic production (a decay of U/Th-series elements)
Determined by measuring 4He and other noble gases
3He/3H age (years) 20 30 10
Tage (years)
20.5 years
Modified from Manga, 1999

63. 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

64. 85Kr
Radioactive inert gas, present is atm from fission reaction (reactors)
Concentrations are increasing world-wide
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

65. 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)

66. Model 3…
Uhlenbrook et al., 2002

67. Large-scale Basins

68. 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

69. 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

70. References
Cook, P.G. and Solomon, D.K., Recent advances in dating young groundwater: chlorofluorocarbons, 3H/3He and 85Kr. Journal of Hydrology, 191:
Duffy, C.J. and Gelhar, L.W., Frequency Domain Approach to Water Quality Modeling in Groundwater: Theory. Water Resources Research, 21(8):
Kirchner, J.W., Feng, X. and Neal, C., Fractal stream chemistry and its implications for contaminant transport in catchments. Nature, 403(6769):
Maloszewski, P. and Zuber, A., Determining the turnover time of groundwater systems with the aid of environmental tracers. 1. models and their applicability. Journal of Hydrology, 57:
Maloszewski, P. and Zuber, A., Principles and practice of calibration and validation of mathematical models for the interpretation of environmental tracer data. Advances in Water Resources, 16:
Turner, J.V. and Barnes, C.J., 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
Zuber, A. and Maloszewski, P., Lumped parameter models. In: W.G. Mook (Editor), Environmental Isotopes in the Hydrological Cycle Principles and Applications. IAEA and UNESCO, Vienna, pp Available:

71. 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