Physics of Aquatic Systems II

Transcription

1 Contents of Session 9 Physics of Aquatic Systems II 9. Model concepts Werner Aeschbach-Hertig Institute of Environmental Physics University of Heidelberg 1 Model concepts used to interpret tritium data Fully mixed system (exponential model) Multicompartment models (linked mixed boxes) (Black) box or lumped parameter models Theory, types of models Applications Excel workbook for box models calculations Numerical model of hydrological systems in space and time Groundwater flow and transport models Literature: Mook, Vol. 3, ch. 2; Vol. 6, ch. 2 and 4 Kinzelbach et al., 22 and more literature on web. ( 2 Motivation for modeling Tracer ages in Lake Issyk-Kul "Naive" tracer dating ( 3 H, 3 H- 3 He, CFCs, SF 6, 85 Kr, 14 C, ) assumes isolated water parcels (no mixing). Hydrological systems always show some degree of mixing. Tracer ages usually deviate from "true" mean water ages. Problem becomes evident in: Tracer time series (e.g. tritium) Multitracer studies Solution: Use simple models concepts to relate tracer concentrations to mean water residence times. Classical case: Interpretation of tritium time series from Hofer et al., 22, L&O 47: Why do the different tracer ages deviate from each other? 4 Influence of mixing on transient gas tracer ages Non-linearity of the 3 H- 3 He age under mixing 25 τ = 4 τ = 3 τ = 2 τ = 15 2 γ = 3 H young / 3 H old 2 τ = 12 3 He [TU] 15 1 τ = 1 τ = 7 age [yr] 15 1 γ = 1/2 γ = 1 γ = 2 γ = 4 5 τ = 4 5 τ = 2 2-comp. mixing: effect depends on curvature (d 2 c/dt 2 ) of input fct. Linear increase of input: linear response of age on mixing SF 6 : d 2 c/dt 2 > : SF 6 ages tend to underestimate the true age CFCs in last ~1 yr: d 2 c/dt 2 < : CFC ages overestimate true age H [TU] fraction of the old component Age of mixture is biased towards the age of the 3 H-rich component Issyk-Kul: 3 H homogenous (γ = 1): slightly overestimated ages 6

2 Tracer ages in Lake Issyk-Kul Qualitative explanation of age deviations in Lake Issyk-Kul: CFCs: d 2 c/dt 2 < : apparent tracer age much too high SF 6 : d 2 c/dt 2 > : tracer age slightly too low 3 H- 3 He: γ = 1, tracer age slightly too high Tracer ages in the Ocean Good agreement between CFCs and 3 H- 3 He for low ages Older mixtures: 3 H- 3 He biased towards bomb peak age Maximum CFC age: ~ 5 yr 7 8 Tritium time series: Smoothing by mixing Example: Mixing of fast and slow runoff components in a river Mixed reactor model (exponential model) Simple model for tritium in aquatic systems: Fully mixed reservoir with constant throughflow, variable input concentration, and decay Q,C in V,C Q,C Mass balance: from Mook, 21 9 V: System volume [L 3 ] Q: Flow rate [L 3 T -1 ] k = Q/V: Exchange rate [T -1 ] τ = V/Q: Exchange time [T] C: System concentration [ML -3 ] C in : Input concentration [ML -3 ] C out = C: Output conc. [ML -3 ] λ = decay constant [T -1 ] 1 st order linear inhomog. diff. equation. General solution for C() = C : System in operation since infinite time: 1 Tritium output from mixed reactor (exp. model) Tritium time series from Wellenberg spring Input Samples 1 1 input decay exp mod τ = 3 exp mod τ = 7 exp mod τ =

3 8 7 6 Tritium time series from Wellenberg spring Input Samples Tritium time series in springs: exponential model τ = 3 τ = 6 τ = 1 Input Samples decay decay component model: C = f p C p + f g C g Tritium time series in rivers Precipitation (p): prompt (within-year) runoff, input curve Groundwater (g): long-term reservoir, exponential model Tritium time series in rivers Results of 2-component model Fractions of components Model parameters for old (groundwater) component f p f g from Mook, from Mook, Tritium time series in springs: multi-box models Grafendorf spring, Austria Generalisation: Box or lumped parameter models Input Hydr. System = Black Box Output Several boxes (cells, compartments) in series Each box fully mixed Total mean res. time: 7 yr from Kendall and McDonnell,

4 Box models: Mathematics Time axis: t = (present) time, t' = time span before t, age Input: c in ( t t ) Transfer/response function, age/transit time distribution (TTD): ( ) ( ) g t with g t dt = 1 Mean age or transit time: () τ = t g t d t Output (convolution of input and transfer function plus decay): c out ()= t c in ( t t ) e λ t g () t d t 19 Parameter τ: Age Piston-flow Model (PM) Picks out the input of one specific age Describes closed system, no mixing Piston-flow age τ = apparent tracer age 2 Exponential Model (EM) Parameter τ: Mean age Weight of input events decreases exponentially with age recent input (t' << τ): highest contribution to output very old input (t' >> τ): very small contribution to output Describes completely mixed system (mixed reactor) Model age τ = mean residence time in mixed system Dispersion Model (DM) Parameter τ: Mean age Parameter δ: Dispersion Weight of input events: ~ Gauss-type distribution highest contribution to output for certain input age t' decreasing contribution to output for younger/older input Describes throughflow system with limited mixing (dispersion) Model age τ = mean transit time 23 24

5 .4.35 Exponential Model (EM) Transfer functions transfer function g(t') τ = 3 τ = 1 τ = 3.5 transfer function g(t') t' [yr] Dispersion Model (DM), δ = 1 τ = 3 τ = 1 τ = 3 transfer function g(t') Dispersion Model (DM), δ =.1 τ = 3 τ = 1 τ = t' [yr] t' [yr] 26 Dispersion parameter in the DM Exponential-Piston-Flow Model (EPM) Dispersivity α [L] and dispersion coefficient D [L 2 /T]: D = α v 1-D, constant throughflow, flow path x, flow time t: v = x D = α x τ τ Peclet-Zahl Pe [-] (Advektion/Dispersion): Pe v x D = x α Dispersion parameters δ [-] und d [T]: δ D v x = α x = 1 d 4D Pe v 2 = 4α v = 4τ α x = 4τ 1 Pe = 4τδ Gelhar et al. (empirical): α.1x Pe 1 δ.1 d.4τ Box models: Dispersion parameters describe apparent dispersion hydrodynamic dispersion, size of recharge area, screen length 27 Parameter τ: Mean age Parameter η: η -1 = mixed fraction of system volume Serial combination of mixed system and piston flow 28 Other possible model types Mixing with "old" Water Quite frequent situation: All tracer concentrations too low Explanation: Mixing with "old", tracer free water "Old" = pre ~195 (no bomb- 3 H/ 14 C, FCKWs, SF 6, 85 Kr) Add-on to any box-modell (PM, EM, DM, EPM, ) Parameter β: Fraction of old water (1 - β: young water) young c(t,τ,δ, ) (1-β)Q (1-β). c(t,τ,δ, ) From: Ozyurt & Bayari, 23, Computers & Geosciences 29: old c = βq 3

6 Example: Tritium in limestone aquifer, Poland Applications of box models to time series One tracer in one well (spring) at several times Schneealpe Austria Zuber et al., 24, J. Hydrol. 286: Rank et al., 1992, IAEA proceedings, Maloszewski et al., 22, J. Hydrol. 256: Applications of box models to multi-tracer studies Several tracers from several wells at one time Output of exponential model at time of sampling, for both tracers and variable values of τ Explaining differences of apparent tracer ages Results of a sampling in 2: Tritium: 18.8 ±.9 TU; 3 He tri : equivalent to 56.6 ±.7 TU age: 25 yr F-12: equivalent to 39 ± 1 pptv age: 15 yr 85 Kr: 25.7 ± 1.1 dpm/cc Kr age: 12 yr How can this be explained? 33 Tool: Excel workbook "Boxmodel", developed by K. Zoellmann and W. Aeschbach-Hertig (downloadable from 34 Box models versus numerical system models Box models Simple calculations, few parameters No spatial resolution No information on internal, physical structure of system Numerical Modelling Flow and transport in time and space Complex model definition, high computational needs More unknown parameters needs more data 35 Numerical groundwater models Equations of groundwater flow and transport (& boundary cond.): h ( K h) = S w t c σ w = v c + ( D c) + + ( cin c) t n n Discretisation and numerical solution on 2-D or 3-D grid, using finite difference method: h h h2 h1 = x x x2 x1 c c c2 c1 = t t t t e e

7 Numerical groundwater models Spatial discretisation of model area Example: 2-D model of Locust Grove, Delmarva Stepwise procedure using tracer ages to improve model calibration 37 from Reilly et al., 1994, Water Resour. Res. 3: Steps 1&2: Flowmodel and pathline analysis Step 3: Transport model for tritium 2-D vertical grid Flow model calibrated using hydraulic heads Calibration of dispersivity A) α L = α T = Comparison of path travel times with CFC-ages B) α L = α T =.15 m Recalibration of flow model from Reilly et al., 1994, Water Resour. Res. 3: from Reilly et al., 1994, Water Resour. Res. 3: Summary Need for models Simple "piston-flow" ages can be misleading Correct interpretation in case of mixing requires modeling Box models Simple, appropriate for quick analysis, few data Useful to interpret multitracer data sets Numerical models Powerful tools, but difficult to constrain and calibrate Tracer data can lead to substantial improvements of models 41