Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.
STORAGE	
  SELECTION (SAS)	
  FUNCTIONS:	
  A	
  TOOL	
  
FOR CHARACTERIZING DISPERSION PROCESSES AND
CATCHMENT-SCALE SOLU...
RIVER HYDROCHEMISTRY and CATCHMENT SCALE TRANSPORT
…why RIVER HYDROCHEMISTRY ?
Water quality has well known implications f...
the chemical response is much more “damped” compared to the
hydrologic signal – different processes
HYDROLOGIC vs CHEMICAL...
THE OLD WATER PARADOX
‘new’ rainfall
discharge‘old’ stored water
the hydrologic response to a rainfall event is chiefly ma...
THE OLD WATER PARADOX
TRACKS OF PAST RAINFALL EVENTS IN STREAMS…
LASTING FOR MONTHS/YEARS (LONG MEMORY)
EVENT WATER
the hy...
NON-POINT SOURCES & CATCHMENT-SCALE SOLUTE TRANSPORT
NUTRIENTS
PESTICIDES ECOSYSTEM IMPACTS
SLUDGE SPILLS
WATER RESOURCES AND WATER QUALITY
...NOT ONLY BECAUSE OF REDUCED WATER AMOUNTS,
BUT ALSO BECAUSE OF INSUFFICIENT WATER QUA...
THE AGE OF WATER & WATER QUALITY ISSUES
LAND MANAGEMENT AND CATCHMENT RESILIENCE
A CHALLENGING PROBLEM...
SPATIAL and TEMPORAL
PATTERNS of SOLUTE INPUT
LANDSCAPE HETEROGENITY
TEMPORAL VARIABILITY
OF CLIM...
SPATIALLY DISTRIBUTED
MODELS
LUMPED
FORMULATIONS
TWO COMPLEMENTARY APPROACHES
VS.
SPATIALLY DISTRIBUTED
MODELS
LUMPED
FORMULATIONS
TRAVEL
TIMES
TWO COMPLEMENTARY APPROACHES
X0
Xt(t;X0,t0)
X1
X3
X2
INJECTION
AREA
CONTROL
VOLUME
Lagrangian transport model:
water parcels traveling through a
contro...
KERNEL of SPATIALLY INTEGRATED INPUT-OUTPUT CONVOLUTIONS
AGE DISTRIBUTION of the outflows
T
T	
  
OUT(t)
Storage
IN (t)
TR...
THE FOKKER PLANK EQUATION
( )=0|, ttg x
[Benettin, Rinaldo and Botter, WRR 2013]
displacement pdf (injection in t0)
ADVECT...
AGE MASS DENSITY
T
 T
 T
AGE MASS DENSITY [Ginn, WRR 1999]
...REPRESENTS THE AGE (T) DISTRIBUTION
AT A GIVEN POINT x AND A...
SPATIAL INTEGRATION OF THE FOKKER PLANCK EQUATION
AGE PDF IN THE OUTFLOW (TRAVEL TIME PDF):
)(
)(
),(
),(),(
tM
t
tTp
T
tT...
The particles leaving the system are sampled among those in storage,
and so their age:
ω(T, t)pout (T,t) = pS(T, t)
PREFER...
The particles leaving the system are sampled among those in storage,
and so their age:
1	
  
SAMPLING	
  through	
  SAS	
 ...
SAS as SPATIALLY INTEGRATED DESCRIPTORS of TRANSPORT
SAS seen from a full 3D KINEMATIC FORMULATION ...
SURFACE INTEGRAL: f...
1D ADVECTION DISPERSION WITH CONSTANT u AND D
VELOCITY FIELD and BC:
>> 1D FINITE DOMAIN
>> CONSTANT D, u
>> ABSORBING/REF...
1D CONVECTION DISPERSION WITH CONSTANT u AND D
normalized age
SASSASPDFPDF
STORAGE SELECTION FUNCTION	
  
STORAGE SELECTIO...
STORAGE SELECTION FUNCTIONS AND PECLET NUMBER
normalized age [%]
ω(T)
STORAGE SELECTION FUNCTIONS FOR DIFFERENT DEGREE OF ...
SPATIAL PATTERNS of CONCENTRATION and SAS FUNCTIONS
SPATIAL PATTERNS of
CONCENTRATION
..for low Pe:
C_out = mean C in (0,L...
SPATIALLY DISTRIBUTED INJECTIONS AND SAS FUNCTIONS
SPATIALLY DISTRIBUTED INJECTIONS... INCREASE SAS UNIFORMITY
storAGE sel...
WHY SHOULD WE CARE ABOUT SAS FUNCTIONS?
)(
)(
),(
),(),(
tM
t
tTp
T
tTp
t
tTp out
out
SS φ
−=
∂
∂
+
∂
∂
),(),(),( tTptTtTp...
WHY SHOULD WE CARE ABOUT SAS FUNCTIONS?
)(
)(
),(
),(),(
tM
t
tTp
T
tTp
t
tTp out
out
SS φ
−=
∂
∂
+
∂
∂
),(),(),( tTptTtTp...
ADVANTAGES of THE FORMULATION
DRY WET
INCORPORATES THE TIME VARIABILITY
of HYDROLOGIC FLUXES (dynamic TTDs)
10/2007 11/200...
DIRECT INTEGRATION OF HYDROLOGIC AND CHEMICAL DATA/MODELS
INCORPORATES THE TIME VARIABILITY
of HYDROLOGIC FLUXES (dynamic ...
«CREATION» OF UNAVAILABLE AGES IS NOT ALLOWED reducing
the risk of getting the right answer for the wrong reason
DIRECT IN...
DIRECT INTEGRATION OF HYDROLOGIC AND CHEMICAL DATA/MODELS
SPATIAL HETEROGENEITY
CAN BE REPRESENTED
«CREATION» OF UNAVAILAB...
INCLUDING SPATIAL HETEROGENEITY
Identify distinct INTERNAL UNITS (VERTICAL and/or HORIZONTAL
HETEROGENEITY) and then defin...
SAS-BASED LUMPED HYDROCHEMICAL MODEL @ PLYNLIMON (UK)
SERIES OF TWO
STORAGES WITH
UNIFORM SAS
+
LUMPED
HYDROLOGIC
MODEL
OB...
DYNAMICAL AGE SELECTION @ PLYNIMON (UK)
CATCHMENT-
SCALE
AGE SELECTION
is controlled by
the catchment
«STATE»
StorAGE SELE...
DYNAMICAL AGE SELECTION @ PLYNIMON (UK)
CATCHMENT-
SCALE
AGE SELECTION
is controlled by
the catchment
«STATE»
FAST flows
(...
DYNAMICAL AGE SELECTION @ PLYNIMON (UK)
StorAGE SELECTION FUNCTIONS	
  
YOUNG	
   OLD	
  normalized age
ω
[Benettin, Kirch...
OBSERVED AND MODELED Cl CONCENTRATIONS @ HUPSEL BROOK
SHORT TERM FLUCTUATIONS RELATED TO
THE ROOT ZONE (short travel times...
ATRAZINE CONCENTRATIONS @ MONCHALTORF (CH)
OBSERVED
MODEL
[Bertuzzo et al., AWR 2013]
LONG-TERM SILICA & SODIUM DYNAMICS @ HUBBURD BROOK (US)
RIVER HYDROCHEMISTRY is driven by the chemical
differentiation bet...
CONCLUDING REMARKS
High dispersion coefficients in 1D
advection – disersion models lead
to uniform SAS (random sampling)
U...
ACKNOWLEDGMENTS
K. McGuire, J. Kirchner
D. Tetzlaff, C. Soulsby
Andrea Rinaldo, Paolo Benettin, Enrico Bertuzzo
...more de...
Upcoming SlideShare
Loading in …5
×

Gianluca Botter

446 views

Published on

StorAge Selection Functions: a tool for characterizing dispersion processes and catchment-scale solute tramsport

Published in: Science
  • Be the first to comment

Gianluca Botter

  1. 1. STORAGE  SELECTION (SAS)  FUNCTIONS:  A  TOOL   FOR CHARACTERIZING DISPERSION PROCESSES AND CATCHMENT-SCALE SOLUTE TRANSPORT G. Botter Dept. Civil & Environmental Engineering, University of Padova (ITALY) Workshop  on  coupled  hydrological  modling                                                                                  Padova  |  23  –  24    April  2015  
  2. 2. RIVER HYDROCHEMISTRY and CATCHMENT SCALE TRANSPORT …why RIVER HYDROCHEMISTRY ? Water quality has well known implications for human well being and ecosystem services In spite of the huge number of available models and datasets focused on water fluxes, catchment -scale transport models/datasets are less widespread River hydrochemistry provides important clues for process identification and hydrologic functioning
  3. 3. the chemical response is much more “damped” compared to the hydrologic signal – different processes HYDROLOGIC vs CHEMICAL SIGNALS [Kirchner et al.., Nature 2000]
  4. 4. THE OLD WATER PARADOX ‘new’ rainfall discharge‘old’ stored water the hydrologic response to a rainfall event is chiefly made by water particles already in storage before the event (old water)
  5. 5. THE OLD WATER PARADOX TRACKS OF PAST RAINFALL EVENTS IN STREAMS… LASTING FOR MONTHS/YEARS (LONG MEMORY) EVENT WATER the hydrologic response to a rainfall event is chiefly made by water particles already in storage before the event (old water)
  6. 6. NON-POINT SOURCES & CATCHMENT-SCALE SOLUTE TRANSPORT NUTRIENTS PESTICIDES ECOSYSTEM IMPACTS SLUDGE SPILLS
  7. 7. WATER RESOURCES AND WATER QUALITY ...NOT ONLY BECAUSE OF REDUCED WATER AMOUNTS, BUT ALSO BECAUSE OF INSUFFICIENT WATER QUALITY IN MANY REGIONS OF THE WORLD WATER RESOURCES ARE SHRINKING...
  8. 8. THE AGE OF WATER & WATER QUALITY ISSUES LAND MANAGEMENT AND CATCHMENT RESILIENCE
  9. 9. A CHALLENGING PROBLEM... SPATIAL and TEMPORAL PATTERNS of SOLUTE INPUT LANDSCAPE HETEROGENITY TEMPORAL VARIABILITY OF CLIMATE FORCING HYDROLOGIC PROCESSES
  10. 10. SPATIALLY DISTRIBUTED MODELS LUMPED FORMULATIONS TWO COMPLEMENTARY APPROACHES VS.
  11. 11. SPATIALLY DISTRIBUTED MODELS LUMPED FORMULATIONS TRAVEL TIMES TWO COMPLEMENTARY APPROACHES
  12. 12. X0 Xt(t;X0,t0) X1 X3 X2 INJECTION AREA CONTROL VOLUME Lagrangian transport model: water parcels traveling through a control volume [e.g. Dagan, 1989; Cvetkovic and Dagan, 1994; Rinaldo et al., 1989] TRAVEL TIME FORMULATION of TRANSPORT ),( ),;( 00 t dt ttd t t XV XX = particle’s trajectory: INPUT   OUTPUT   CONTROL PLANE CP KINEMATIC DEFINITION of TRAVEL TIME : CPtTt ∈);( 0XX
  13. 13. KERNEL of SPATIALLY INTEGRATED INPUT-OUTPUT CONVOLUTIONS AGE DISTRIBUTION of the outflows T T   OUT(t) Storage IN (t) TRAVEL TIME PDF conditional to the exit time t pout (T , t ) ( ) ( ) ( )∫∞− −= t iioutiINout dttttptCtC , output flux concentration (OUTPUT MEMORY of the INPUT) PDF UNSTEADY FLOW CONDITIONS, TYPICAL OF MOST HYDROLOGIC SYSTEMS
  14. 14. THE FOKKER PLANK EQUATION ( )=0|, ttg x [Benettin, Rinaldo and Botter, WRR 2013] displacement pdf (injection in t0) ADVECTION DISPERSION EULERIAN CONCENTRATION
  15. 15. AGE MASS DENSITY T T T AGE MASS DENSITY [Ginn, WRR 1999] ...REPRESENTS THE AGE (T) DISTRIBUTION AT A GIVEN POINT x AND AT A GIVEN TIME t mass input in t-T (age T) displacement pdf TIME SPENT INSIDE THE SYSTEM SINCE ENTRY (ages increase during the parcels’ journey within the control volume) AGE OF WATER/SOLUTE PARCEL T
  16. 16. SPATIAL INTEGRATION OF THE FOKKER PLANCK EQUATION AGE PDF IN THE OUTFLOW (TRAVEL TIME PDF): )( )( ),( ),(),( tM t tTp T tTp t tTp out out SS φ −= ∂ ∂ + ∂ ∂ xx dtT tM tTp V S ∫= ),,( )( 1 ),( ρ σρρ dtTttTt tΦ tTp outV out out nxxDxxu •∇−= ∫∂ )],,(),(),,(),([ )( 1 ),( SPATIALLY AVERAGED MASS AGE CONSERVATION AGE PDF IN THE STORAGE: ...as a function of spatially integrated fluxes and storage [Botter et al., GRL 2011]
  17. 17. The particles leaving the system are sampled among those in storage, and so their age: ω(T, t)pout (T,t) = pS(T, t) PREFERENCE StorAGE SELECTION FUNCTION LOW AVAILABILITY or LOW PREFERENCE IMPLIES LOW SAMPLING – AGES POORLY REPRESENTED IN THE OUTPUT [Botter et al., GRL 2011] OUT(t) pout(T,t) pS(T, t) AGES SAMPLED AGES AVAILABLE StorAGE selection: LINKING AGE DISTRIBUTIONS
  18. 18. The particles leaving the system are sampled among those in storage, and so their age: 1   SAMPLING  through  SAS  func?ons   1  1   uniform  preference   over  all  ages   ω decreases  for   older  ages   𝝎(𝑻, 𝒕)  𝝎(𝑻, 𝒕)  𝝎(𝑻, 𝒕)=const   ω  increases  for   older  ages   random sampling   preference for old water   preference for new water   T   T   𝑻   𝝎   𝝎   𝝎   ω(T, t)pout (T,t) = pS(T, t) AGES AVAILABLE PREFERENCE StorAGE SELECTION FUNCTION StorAGE selection: LINKING AGE DISTRIBUTIONS AGES SAMPLED
  19. 19. SAS as SPATIALLY INTEGRATED DESCRIPTORS of TRANSPORT SAS seen from a full 3D KINEMATIC FORMULATION ... SURFACE INTEGRAL: flux of ages across the boundaries VOLUME INTEGRAL: ages stored T [Benettin, Rinaldo and Botter, WRR 2013]
  20. 20. 1D ADVECTION DISPERSION WITH CONSTANT u AND D VELOCITY FIELD and BC: >> 1D FINITE DOMAIN >> CONSTANT D, u >> ABSORBING/REFLECTING BARRIERS SOLUTE INPUT: >> IMPULSIVE/CONTINUOUS >> POINT/DISTRIBUTED ​ 𝜕 𝐶/𝜕𝑡 + 𝑢​ 𝜕 𝐶/𝜕𝑥 = 𝐷​​ 𝜕↑2 𝐶/𝜕​ 𝑥↑2    
  21. 21. 1D CONVECTION DISPERSION WITH CONSTANT u AND D normalized age SASSASPDFPDF STORAGE SELECTION FUNCTION   STORAGE SELECTION FUNCTION   STORAGE   OUTFLOW   AGE DISTRIBUTIONS and SAS FUNCTIONS for POISSON INPUTS (...for selected times, but SAS are almost stationary) normalized age
  22. 22. STORAGE SELECTION FUNCTIONS AND PECLET NUMBER normalized age [%] ω(T) STORAGE SELECTION FUNCTIONS FOR DIFFERENT DEGREE OF DISPERSION   HIGH DISPERSION COEFFICIENTS (low Pe) INCREASES UNIFORMITY OF SAS (- RANDOM SAMPLING)
  23. 23. SPATIAL PATTERNS of CONCENTRATION and SAS FUNCTIONS SPATIAL PATTERNS of CONCENTRATION ..for low Pe: C_out = mean C in (0,L) BUT NOT A WELL MIXED SYSTEM storAGE selection [Benettin, Rinaldo and Botter, WRR 2013] RANDOM SAMPLING normalized age CONCENTRATION PROFILE
  24. 24. SPATIALLY DISTRIBUTED INJECTIONS AND SAS FUNCTIONS SPATIALLY DISTRIBUTED INJECTIONS... INCREASE SAS UNIFORMITY storAGE selection function (SAS) RANDOM SAMPLING normalized age
  25. 25. WHY SHOULD WE CARE ABOUT SAS FUNCTIONS? )( )( ),( ),(),( tM t tTp T tTp t tTp out out SS φ −= ∂ ∂ + ∂ ∂ ),(),(),( tTptTtTp Sout ω= >> derive ps(T,t) and pout(T,t) for water based on SAS and integral fluxes/storage ( ) ( ) ( )∫∞− −= t iioutiINout dttttptCtC , >> water age distributions can be used to compute concentrations of conservative (or reactive) solutes: SPATIALLY AVERAGED MASS AGE CONSERVATION { [Botter et al., GRL 2011; Botter WRR 2012; Rinaldo et al., WRR 2011]
  26. 26. WHY SHOULD WE CARE ABOUT SAS FUNCTIONS? )( )( ),( ),(),( tM t tTp T tTp t tTp out out SS φ −= ∂ ∂ + ∂ ∂ ),(),(),( tTptTtTp Sout ω= >> derive ps(T,t) and pout(T,t) for water based on SAS and integral fluxes/storage ( ) ( ) ( )∫∞− −= t iioutiINout dttttptCtC , >> water age distributions can be used to compute concentrations of conservative (or reactive) solutes: SPATIALLY AVERAGED MASS AGE CONSERVATION { [Botter et al., GRL 2011; Botter WRR 2012; Rinaldo et al., WRR 2011] RANDOM SAMPLING: ANALYTICAL SOLUTIONS
  27. 27. ADVANTAGES of THE FORMULATION DRY WET INCORPORATES THE TIME VARIABILITY of HYDROLOGIC FLUXES (dynamic TTDs) 10/2007 11/2007 DISCHARGE[mm/h]CONCENTRATION[mg/l] SILICA CHLORIDE (data from UHF @ Plynlimon, UK) Late OCT 2007 INPUT   Mid NOV 2007
  28. 28. DIRECT INTEGRATION OF HYDROLOGIC AND CHEMICAL DATA/MODELS INCORPORATES THE TIME VARIABILITY of HYDROLOGIC FLUXES (dynamic TTDs) ADVANTAGES of THE FORMULATION
  29. 29. «CREATION» OF UNAVAILABLE AGES IS NOT ALLOWED reducing the risk of getting the right answer for the wrong reason DIRECT INTEGRATION OF HYDROLOGIC AND CHEMICAL DATA/MODELS INCORPORATES THE TIME VARIABILITY of HYDROLOGIC FLUXES (dynamic TTDs) ADVANTAGES of THE FORMULATION
  30. 30. DIRECT INTEGRATION OF HYDROLOGIC AND CHEMICAL DATA/MODELS SPATIAL HETEROGENEITY CAN BE REPRESENTED «CREATION» OF UNAVAILABLE AGES IS NOT ALLOWED reducing the risk of getting the right answer for the wrong reason INCORPORATES THE TIME VARIABILITY of HYDROLOGIC FLUXES (dynamic TTDs) ADVANTAGES of THE FORMULATION
  31. 31. INCLUDING SPATIAL HETEROGENEITY Identify distinct INTERNAL UNITS (VERTICAL and/or HORIZONTAL HETEROGENEITY) and then define UNIT-SCALE SAS FUNCTIONS 𝝎1(T)  (unit  1) 1(T)  (unit  1) [see e.g. Birkel et al., WRR 2014; HP 2015] 𝝎2(T)  (unit  2) 2(T)  (unit  2) 𝝎3(T)  (unit  3) 3(T)  (unit  3) Bruntland Burn(UK): ongoing work in collaboration with C. Soulsby and D. Tetzlaff
  32. 32. SAS-BASED LUMPED HYDROCHEMICAL MODEL @ PLYNLIMON (UK) SERIES OF TWO STORAGES WITH UNIFORM SAS + LUMPED HYDROLOGIC MODEL OBSERVED   ROOT ZONE GROUNDWATER   OBSERVED MODEL   CHLORIDECONCENTRATIONDISCHARGE [Benettin et al., WRR 2015]
  33. 33. DYNAMICAL AGE SELECTION @ PLYNIMON (UK) CATCHMENT- SCALE AGE SELECTION is controlled by the catchment «STATE» StorAGE SELECTION FUNCTIONS   YOUNG   OLD  normalized age ω [Benettin, Kirchner, Rinaldo and Botter, WRR 2015]
  34. 34. DYNAMICAL AGE SELECTION @ PLYNIMON (UK) CATCHMENT- SCALE AGE SELECTION is controlled by the catchment «STATE» FAST flows (young) StorAGE SELECTION FUNCTIONS   YOUNG   OLD  normalized age ω [Benettin, Kirchner, Rinaldo and Botter, WRR 2015] INPUT   Mid NOV 2007
  35. 35. DYNAMICAL AGE SELECTION @ PLYNIMON (UK) StorAGE SELECTION FUNCTIONS   YOUNG   OLD  normalized age ω [Benettin, Kirchner, Rinaldo and Botter, WRR 2015] Late OCT 2007 CATCHMENT- SCALE AGE SELECTION is controlled by the catchment «STATE» FAST flows (young) vs GW flows (older)
  36. 36. OBSERVED AND MODELED Cl CONCENTRATIONS @ HUPSEL BROOK SHORT TERM FLUCTUATIONS RELATED TO THE ROOT ZONE (short travel times) in WINTER the Cl concentration is sustained by GW (long travel times) [Benettin et al., WRR 2013]
  37. 37. ATRAZINE CONCENTRATIONS @ MONCHALTORF (CH) OBSERVED MODEL [Bertuzzo et al., AWR 2013]
  38. 38. LONG-TERM SILICA & SODIUM DYNAMICS @ HUBBURD BROOK (US) RIVER HYDROCHEMISTRY is driven by the chemical differentiation between fast flows (short memory) and slow flows (long-memory) SILICON (Si) SODIUM (Na)
  39. 39. CONCLUDING REMARKS High dispersion coefficients in 1D advection – disersion models lead to uniform SAS (random sampling) Use of spatially distributed models to analyze SAS dynamics .. implications for lumped catchment-scale hydrochemical models Storage selection functions (SAS) are effective spatially integrated descriptors of mixing/dispersion processes in heterogeneous media The method provides consistent results in diverse settings (climate, solutes)
  40. 40. ACKNOWLEDGMENTS K. McGuire, J. Kirchner D. Tetzlaff, C. Soulsby Andrea Rinaldo, Paolo Benettin, Enrico Bertuzzo ...more details will be provided by Paolo Benettin tomorrow ...

×