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

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

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

This section will examine how we make use of isotopic variability

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

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

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

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)

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

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

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!!

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

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

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

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

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

Subsurface Transport Basics

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

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

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

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

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

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

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

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

Stable Isotope Methods

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

Seasonal Variation in 18O of Precipitation Vitvar, 2000

Seasonality in Stream Water Deines et al. 1990

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

Convolution Movie

Transfer Functions Used for Residence Time Distributions

Common Residence Time Models

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

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

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:

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 =

Heavy-tailed Models Gamma Exponentials in series

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

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’) = 0 for t'< T (1-1/ ) DM Kendall, 2001 Maloszewski and Zuber

Dispersion Model Examples

Residence Time Distributions can be Similar

Uncertainty

Identifiable Parameters?

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

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

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

Soil Water Residence Time Stewart & McDonnell, 2001

Example from Rietholzbach Vitvar, 1998

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

How residence time scales with basin area Figure 1

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

Figure 3

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

500 m Scale -7 -3.5 Low High RIF

Determining Residence Time of Old(er) Waters

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

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

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

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

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:

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 1 5 10 50 Tage (years) 20.5 years Modified from Manga, 1999

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

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

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)

Model 3… Uhlenbrook et al., 2002

Large-scale Basins

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

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

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): 1175-1184. Kirchner, J.W., Feng, X. and Neal, C., 2000. Fractal stream chemistry and its implications for contaminant transport in catchments. Nature, 403(6769): 524-527. 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

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