Residence time theories of pollutants transport

TRAVEL TIME DISTRIBUTIONS AND

CATCHMENT- SCALE TRANSPORT MODELS:

RECENT ADVANCES AND NEW CHALLENGES

GIANLUCA BOTTER([email protected])

UNIVERSITA’ DEGLI STUDI DI PADOVADepartment of Civil Architectural and Environmental Engineering (DICEA)

the the the the chemical response chemical response chemical response chemical response is much more “damped” compared to the is much more “damped” compared to the is much more “damped” compared to the is much more “damped” compared to the

hydrologic signal hydrologic signal hydrologic signal hydrologic signal –––– different processesdifferent processesdifferent processesdifferent processes

HYDROLOGIC vs CHEMICAL SIGNALSHYDROLOGIC vs CHEMICAL SIGNALSHYDROLOGIC vs CHEMICAL SIGNALSHYDROLOGIC vs CHEMICAL SIGNALS

[Kirchner et al.., Nature 2000] [Kirchner et al.., Nature 2000] [Kirchner et al.., Nature 2000] [Kirchner et al.., Nature 2000]

the hydrologic response to a rainfall event is chiefly made by water the hydrologic response to a rainfall event is chiefly made by water the hydrologic response to a rainfall event is chiefly made by water the hydrologic response to a rainfall event is chiefly made by water

particles already particles already particles already particles already in storage before the event (old water)in storage before the event (old water)in storage before the event (old water)in storage before the event (old water)

THE OLD WATER THE OLD WATER THE OLD WATER THE OLD WATER PARADOXPARADOXPARADOXPARADOX

TRACKS OF PAST RAINFALL EVENTS IN TRACKS OF PAST RAINFALL EVENTS IN TRACKS OF PAST RAINFALL EVENTS IN TRACKS OF PAST RAINFALL EVENTS IN STREAMS… STREAMS… STREAMS… STREAMS…

LASTING FOR MONTHS/YEARS (LONG MEMORY)LASTING FOR MONTHS/YEARS (LONG MEMORY)LASTING FOR MONTHS/YEARS (LONG MEMORY)LASTING FOR MONTHS/YEARS (LONG MEMORY)

EVENT EVENT EVENT EVENT WATERWATERWATERWATER

THE AGE OF WATER THE AGE OF WATER THE AGE OF WATER THE AGE OF WATER & WATER & WATER & WATER & WATER QUALITY ISSUESQUALITY ISSUESQUALITY ISSUESQUALITY ISSUES

NUTRIENTSNUTRIENTSNUTRIENTSNUTRIENTS

PESTICIDESPESTICIDESPESTICIDESPESTICIDES ECOSYSTEM IMPACTSECOSYSTEM IMPACTSECOSYSTEM IMPACTSECOSYSTEM IMPACTS

SLUDGE SPILLSSLUDGE SPILLSSLUDGE SPILLSSLUDGE SPILLS


NUTRIENTSNUTRIENTSNUTRIENTSNUTRIENTS

PESTICIDESPESTICIDESPESTICIDESPESTICIDES ECOSYSTEM IMPACTSECOSYSTEM IMPACTSECOSYSTEM IMPACTSECOSYSTEM IMPACTS

SLUDGE SPILLSLUDGE SPILLSLUDGE SPILLSLUDGE SPILL

[Rinaldo et al., WRR 1989] [Rinaldo et al., WRR 1989] [Rinaldo et al., WRR 1989] [Rinaldo et al., WRR 1989]


LAND MANAGEMENT AND CATCHMENT RESILIENCELAND MANAGEMENT AND CATCHMENT RESILIENCELAND MANAGEMENT AND CATCHMENT RESILIENCELAND MANAGEMENT AND CATCHMENT RESILIENCE

XXXX0

XXXXt(t;X0)

X1

X3

X2

INJECTION

AREA

CONTROL CONTROL CONTROL CONTROL

VOLUMEVOLUMEVOLUMEVOLUMELagrangianLagrangianLagrangianLagrangian transporttransporttransporttransport model: model: model: model:

water water water water particlesparticlesparticlesparticles travelingtravelingtravelingtraveling throughthroughthroughthrough

a control volume a control volume a control volume a control volume

(MAINLY HILLSLOPES PRCESSES)(MAINLY HILLSLOPES PRCESSES)(MAINLY HILLSLOPES PRCESSES)(MAINLY HILLSLOPES PRCESSES)

[e.g., Taylor, 1921; [e.g., Taylor, 1921; [e.g., Taylor, 1921; [e.g., Taylor, 1921; DaganDaganDaganDagan, 1989], 1989], 1989], 1989]

FORMULATION of FORMULATION of FORMULATION of FORMULATION of TRANSPORT AT THE CATCHMENT SCALE TRANSPORT AT THE CATCHMENT SCALE TRANSPORT AT THE CATCHMENT SCALE TRANSPORT AT THE CATCHMENT SCALE

),();(

tXVdt

XtdXt

t =0

particle’s trajectory:

INPUT

OUTPUT

“Time interval elapsed from the entry to the leave”

TRAVEL (or TRANSIT) TIME tT : the AGE when you die

“ Time spent by a water/solute particle inside a catchment since entry”RESIDENCE TIME tR : the AGE

RESIDENCE TIMES vs TRAVEL TIMES

...RANDOM VARIABLES (space-time heterogeneityof the relevant processes and enviroments)

RESIDENCE TIME PDF evaluated at time t: pRT (tR , t )

( ) ( ) ( )∫∞−

−=t

iiRTiINs dttttptCtC ,average storage concentration

(STORAGE MEMORY of the INPUT)


AGE DISTRIBUTION of living people

RESIDENCE TIME PDF evaluated at time t: pRT (tR , t )

( ) ( ) ( )∫∞−

−=t

iiRTiINs dttttptCtC ,average storage concentration

(STORAGE MEMORY of the INPUT)


TRAVEL TIME PDF conditional to the exit time: p’OUT (tT , t )

( ) ( ) ( )∫∞−

−=t

iiOUTiINOUT dttttptCtC ,'output flux concentration

(OUTPUT MEMORY of the INPUT)

AGE DISTRIBUTION of living people AGE DISTRIBUTION of people dying

RESIDENCE TIME PDF: TEMPORAL EVOLUTION

STORAGE S(t)INPUT OUTPUT

low

high

N(t

R,t)

N(tR,t) = S(t) pRT(tR,t)

number of particles with age tR at time t

RESIDENCE TIME tR

IN(t

)

OU

T(t)

t

t

PURE AGEING: 0=∂

∂+

∂∂

R

RR

tttN

tttN ),(),(

PLAYPAUSE


N(t

R,t)

RESIDENCE TIME tR

BOUNDARY CONDITION: N(tR=0,t)=IN(t) + AGEING: 0=∂

∂+

∂∂

R

RR

tttN

tttN ),(),(

PURE AGEING: 0=∂

∂+

∂∂

R

RR

tttN

tttN ),(),(

INPUTIN

(t) t

STORAGE S(t) OUTPUT

low

high

OU

T(t)

t

PAUSEPLAY


N(t

R,t)

RESIDENCE TIME tR

BOUNDARY CONDITION: N(tR=0,t)=IN(t) + AGEING: 0=∂

∂+

∂∂

R

RR

tttN

tttN ),(),(

=∂

∂+

∂∂

R

RR

tttN

tttN ),(),(

INPUTIN

(t) t

STORAGE S(t) OUTPUT

low

high

OU

T(t)

t

AGEING + SAMPLING VIA OUTPUT )(),(' tOUTttp ROUT−

PAUSEPLAY


N(t

R,t)

RESIDENCE TIME tR

INPUTIN

(t) t

STORAGE S(t) OUTPUT

low

high

OU

T(t)

t

)(),(')],()([)],()([ tOUTttpt

ttptSt

ttptSROUT

R

RRTRRT −=∂

∂+

∂∂

The particles leaving the system are sampled among those in storage, The particles leaving the system are sampled among those in storage, The particles leaving the system are sampled among those in storage, The particles leaving the system are sampled among those in storage,

and so their age: and so their age: and so their age: and so their age:

w(tR, t) p’OUT(tR,t) = pRT(tR, t)

AGES SAMPLED AGES AVAILABLE PREFERENCE

AGE FUNCTIONAGE FUNCTIONAGE FUNCTIONAGE FUNCTION

(MIXING)(MIXING)(MIXING)(MIXING)

LOW AVAILABILITY or LOW LOW AVAILABILITY or LOW LOW AVAILABILITY or LOW LOW AVAILABILITY or LOW PREFERENCE PREFERENCE PREFERENCE PREFERENCE IMPLIES LOW IMPLIES LOW IMPLIES LOW IMPLIES LOW SAMPLINGSAMPLINGSAMPLINGSAMPLING

–––– AGES AGES AGES AGES NOT REPRESENTED IN THE OUTPUT NOT REPRESENTED IN THE OUTPUT NOT REPRESENTED IN THE OUTPUT NOT REPRESENTED IN THE OUTPUT

MIXING: LINKING RESIDENCE and TRAVEL TIME PDF ’sMIXING: LINKING RESIDENCE and TRAVEL TIME PDF ’sMIXING: LINKING RESIDENCE and TRAVEL TIME PDF ’sMIXING: LINKING RESIDENCE and TRAVEL TIME PDF ’s

1

SAMPLING through AGE functions

11

uniform preference

over all agesω ω ω ω decreases for

older ages

� ��, �� , �� , � =const

ωωωω increases for

older ages

random random random random samplingsamplingsamplingsampling preference for old waterpreference for old waterpreference for old waterpreference for old water preference for new waterpreference for new waterpreference for new waterpreference for new water

��

� � �

RAINFALL

DISCHARGE

EVAPO-TRANSPIRATION

THE MASTER EQUATION FOR THE RESIDENCE TIME PDFTHE MASTER EQUATION FOR THE RESIDENCE TIME PDFTHE MASTER EQUATION FOR THE RESIDENCE TIME PDFTHE MASTER EQUATION FOR THE RESIDENCE TIME PDF

)()()()(

tQtETtJdt

tdS −−=

),(),(),(),(

ttFttpt

ttpt

ttpRRRT

R

RRTRRT =∂

∂+∂

∂

MASTER EQUATION FOR pRT:

−−+−=)()(

)],([)()(

)],([)()(

),(tStJ

tttStQ

tttStQ

ttFRETRQR

ωωωωωωωω 11

(WATER FLUXES, TYPE OF MIXING)

where:

RAINFALL

DISCHARGE

EVAPO-TRANSPIRATION

THE MASTER EQUATION FOR THE RESIDENCE TIME PDFTHE MASTER EQUATION FOR THE RESIDENCE TIME PDFTHE MASTER EQUATION FOR THE RESIDENCE TIME PDFTHE MASTER EQUATION FOR THE RESIDENCE TIME PDF

)()()()(

tQtETtJdt

tdS −−=

),(),(),(),(

ttFttpt

ttpt

ttpRRRT

R

RRTRRT =∂

∂+∂

∂

MASTER EQUATION FOR pRT:

−−+−=)()(

)],([)()(

)],([)()(

),(tStJ

tttStQ

tttStQ

ttFRETRQR

ωωωωωωωω 11

where:

.... .... .... .... ANALYTICAL SOLUTIONS for ANALYTICAL SOLUTIONS for ANALYTICAL SOLUTIONS for ANALYTICAL SOLUTIONS for ωωωω=1=1=1=1

TRAVEL & RESIDENCE TIME PDF’s MUST BE TRAVEL & RESIDENCE TIME PDF’s MUST BE TRAVEL & RESIDENCE TIME PDF’s MUST BE TRAVEL & RESIDENCE TIME PDF’s MUST BE CONSISTENT CONSISTENT CONSISTENT CONSISTENT WITH THE WITH THE WITH THE WITH THE

SET OF AGES AVAILABLE SET OF AGES AVAILABLE SET OF AGES AVAILABLE SET OF AGES AVAILABLE AND AND AND AND THE INPUT/OUTPUT THE INPUT/OUTPUT THE INPUT/OUTPUT THE INPUT/OUTPUT FLUXESFLUXESFLUXESFLUXES

(WATER FLUXES, TYPE OF MIXING)

THE CASE OF THE CASE OF THE CASE OF THE CASE OF RANDOM SAMPLING (RANDOM SAMPLING (RANDOM SAMPLING (RANDOM SAMPLING (ωωωω=1=1=1=1))))

−

−−= ∫

−

t

ttT

T

TOUTT

dxxSxJ

ttSttJ

ttp)()(

exp)()(

),('

TRAVEL TIME PDFTRAVEL TIME PDFTRAVEL TIME PDFTRAVEL TIME PDF

[Botter et al., WRR 2010]


−

−−= ∫

−

t

ttT

T

TOUTT

dxxSxJ

ttSttJ

ttp)()(

exp)()(

),('


[Botter et al., WRR 2010]

��

��

��

��

� �

�

Relationship with

advection/dispersion models:


ANALYTICAL (WELL MIXED)

NUMERICAL 3D SIMULATION

C=Cmax

C=0

−

−−= ∫

−

t

ttT

T

TOUTT

dxxSxJ

ttSttJ

ttp)()(

exp)()(

),('

EMERGING FEATURES:

1. TIME VARIANCE

2. DEPENDENCE ON ECO-

HYDROLOGICAL PROCESSES

� SERIES/PARALLEL OF MIXED SERIES/PARALLEL OF MIXED SERIES/PARALLEL OF MIXED SERIES/PARALLEL OF MIXED STORAGES, STORAGES, STORAGES, STORAGES, ifififif necessarynecessarynecessarynecessary

[Rinaldo et al., WRR 2011]


GENERAL MODEL GENERAL MODEL GENERAL MODEL GENERAL MODEL STRUCTURE: TWO STRUCTURE: TWO STRUCTURE: TWO STRUCTURE: TWO RS COMPARTMENTSRS COMPARTMENTSRS COMPARTMENTSRS COMPARTMENTS

(Selective) ET(Selective) ET(Selective) ET(Selective) ETmay increase c(t)

)()()()(

tLtETtJdt

tdsnZ

R−−=

)()()()(

tMtMtFdt

tdMLET

s −−=

ROOT ZONE

DEEPER STORAGELEACHINGLEACHINGLEACHINGLEACHING

APPLICATION 1. CHLORIDE TRANSPORT IN THE HUPSEL BROOK (NL)APPLICATION 1. CHLORIDE TRANSPORT IN THE HUPSEL BROOK (NL)APPLICATION 1. CHLORIDE TRANSPORT IN THE HUPSEL BROOK (NL)APPLICATION 1. CHLORIDE TRANSPORT IN THE HUPSEL BROOK (NL)

CHLORIDES introduced

through FERTILIZERS...

...inducing high Cl

concentrations in

STREAMS


NEGATIVE

CORRELATION

between Q and C


NO FERTILIZATIONS

NEGATIVE

CORRELATION

between Q and C

PERSISTENCY of

the chemical

signal, despite the

input seasonality

(long memory)

OBSERVED OBSERVED OBSERVED OBSERVED AND MODELED AND MODELED AND MODELED AND MODELED DISCHARGES @ HUPSEL BROOKDISCHARGES @ HUPSEL BROOKDISCHARGES @ HUPSEL BROOKDISCHARGES @ HUPSEL BROOK

OBSERVED OBSERVED OBSERVED OBSERVED AND MODELED AND MODELED AND MODELED AND MODELED Cl CONCENTRATIONS @ HUPSEL BROOKCl CONCENTRATIONS @ HUPSEL BROOKCl CONCENTRATIONS @ HUPSEL BROOKCl CONCENTRATIONS @ HUPSEL BROOK

SHORT TERM FLUCTUATIONS RELATED TO

THE ROOT ZONE (short travel times)

in WINTER the Cl in WINTER the Cl in WINTER the Cl in WINTER the Cl concentrationconcentrationconcentrationconcentration isisisis sustainedsustainedsustainedsustained by GW (long by GW (long by GW (long by GW (long traveltraveltraveltravel timestimestimestimes))))

TIME VARIANCE OF THE RESIDENCE TIME PDFTIME VARIANCE OF THE RESIDENCE TIME PDFTIME VARIANCE OF THE RESIDENCE TIME PDFTIME VARIANCE OF THE RESIDENCE TIME PDF

ttttRRRR [[[[�]]]]ttttRRRR [[[[�]]]]

pp ppR

TR

TR

TR

T(( (( tt tt

RR RR,t,t ,t,t

)) ))[[ [[��]] ]]

pp ppR

TR

TR

TR

T(( (( tt tt

RR RR,t,t ,t,t

)) ))[[ [[��]] ]]

ttttRRRR [[[[�]]]]

pp ppR

TR

TR

TR

T(( (( tt tt

RR RR,t,t ,t,t

)) ))[[ [[��]] ]]

WINTER WINTER WINTER WINTER

(WET)(WET)(WET)(WET)

SUMMER SUMMER SUMMER SUMMER

(DRY)(DRY)(DRY)(DRY)

APPLICATION APPLICATION APPLICATION APPLICATION 2. 2. 2. 2. CHLORIDE TRANSPORT CHLORIDE TRANSPORT CHLORIDE TRANSPORT CHLORIDE TRANSPORT @ PLYNIMON (UK)@ PLYNIMON (UK)@ PLYNIMON (UK)@ PLYNIMON (UK)

INCREASE or DECREASE of

chloride concentrations during

floods (depending on the season)

APPLICATION APPLICATION APPLICATION APPLICATION 3. 3. 3. 3. PESTICIDE TRANSPORT IN THE MONCHALTORF (CH)PESTICIDE TRANSPORT IN THE MONCHALTORF (CH)PESTICIDE TRANSPORT IN THE MONCHALTORF (CH)PESTICIDE TRANSPORT IN THE MONCHALTORF (CH)

Huge PESTICIDE LOADS PESTICIDE LOADS PESTICIDE LOADS PESTICIDE LOADS introduced through farming...

high ATRAZINE ATRAZINE ATRAZINE ATRAZINE CONCENTRATIONS CONCENTRATIONS CONCENTRATIONS CONCENTRATIONS in STREAMS during the SUMMER

(ATRAZINE is CANCEROUS)

ANNUAL

DYNAMICS of

DISCHARGE DISCHARGE DISCHARGE DISCHARGE and

ATHRAZINE ATHRAZINE ATHRAZINE ATHRAZINE

CONCENTRATIONCONCENTRATIONCONCENTRATIONCONCENTRATION

APPLICATION APPLICATION APPLICATION APPLICATION 3. 3. 3. 3. PESTICIDE TRANSPORT IN THE MONCHALTORF (CH)PESTICIDE TRANSPORT IN THE MONCHALTORF (CH)PESTICIDE TRANSPORT IN THE MONCHALTORF (CH)PESTICIDE TRANSPORT IN THE MONCHALTORF (CH)

OBSERVED OBSERVED OBSERVED OBSERVED AND MODELED At CONCENTRATIONS @ MONCHALTORF AND MODELED At CONCENTRATIONS @ MONCHALTORF AND MODELED At CONCENTRATIONS @ MONCHALTORF AND MODELED At CONCENTRATIONS @ MONCHALTORF

OBSERVED

MODEL

CHALLENGESCHALLENGESCHALLENGESCHALLENGES

Derive analytical or numerical solutions for Derive analytical or numerical solutions for Derive analytical or numerical solutions for Derive analytical or numerical solutions for arbitrary mixing functionsarbitrary mixing functionsarbitrary mixing functionsarbitrary mixing functions, , , ,

suitably parametrized suitably parametrized suitably parametrized suitably parametrized –––– relationship with relationship with relationship with relationship with spatially distributed spatially distributed spatially distributed spatially distributed modelsmodelsmodelsmodels

BetterBetterBetterBetter understandunderstandunderstandunderstand thethethethe rolerolerolerole ofofofof

evapotranspirationevapotranspirationevapotranspirationevapotranspiration andandandand plantplantplantplant

physiologyphysiologyphysiologyphysiology forforforfor thethethethe hydrohydrohydrohydro----

chemistrychemistrychemistrychemistry ofofofof riversriversriversrivers

Let the age functions be Let the age functions be Let the age functions be Let the age functions be dependent on the catchment’ dependent on the catchment’ dependent on the catchment’ dependent on the catchment’ state state state state (e.g. wet (e.g. wet (e.g. wet (e.g. wet

VS dry)VS dry)VS dry)VS dry)

FLOODSDROUGHTS

� AGE functions

CONCLUSIONSCONCLUSIONSCONCLUSIONSCONCLUSIONS

time time time time variance of travel variance of travel variance of travel variance of travel & & & &

residence residence residence residence time pdf’s time pdf’s time pdf’s time pdf’s @ @ @ @

multiple timescales multiple timescales multiple timescales multiple timescales

(variability (variability (variability (variability of of of of fluxes fluxes fluxes fluxes

and storages)and storages)and storages)and storages)

Robustness of the approach, Robustness of the approach, Robustness of the approach, Robustness of the approach,

suggested by the comparison suggested by the comparison suggested by the comparison suggested by the comparison

with field datawith field datawith field datawith field data

travel and residence time pdf’s are travel and residence time pdf’s are travel and residence time pdf’s are travel and residence time pdf’s are

related objects related objects related objects related objects dependent on mixing dependent on mixing dependent on mixing dependent on mixing

processes and hydrologic fluxesprocesses and hydrologic fluxesprocesses and hydrologic fluxesprocesses and hydrologic fluxes

THE RESEARCH TEAM (PADOVATHE RESEARCH TEAM (PADOVATHE RESEARCH TEAM (PADOVATHE RESEARCH TEAM (PADOVA----LAUSANNE)LAUSANNE)LAUSANNE)LAUSANNE)

University of Padova, Italy

Gianluca Botter

Paolo Benettin

EPFL, Lausanne, CH

Andrea Rinaldo

Enrico Bertuzzo

TRAVEL TIME DISTRIBUTIONS AND

CATCHMENT- SCALE TRANSPORT MODELS:

RECENT ADVANCES AND NEW CHALLENGES

GIANLUCA BOTTER([email protected])

UNIVERSITA’ DEGLI STUDI DI PADOVADepartment of Civil Architectural and Environmental Engineering (DICEA)

Residence time theories of pollutants transport

Education

Transcript of Residence time theories of pollutants transport