theory and application of time-variable transit times in hydrologic systems

CUAHSI virtual short course Ciaran Harman, July 8th 2015

Theory and applicaAon of Ame-variable transit Ames in hydrologic systems

Theory and applicaAon of Ame-variable transit Ames

in hydrologic systems

CUAHSI Virtual short course Lecture 1, July 8th 2015

Ciaran Harman Geography and Environmental

Engineering, Johns Hopkins University





Why a short course on this? Lots of recent progress in this area

Steep learning curve for newcomers Need to connect theory with data

To introduce fundamental concepts Cut through confusion caused by varying terminology

To put pracAcal tools (i.e. code) in the right hands Help people with datasets use the new theory

To suggest direcAons for future research There is much to do!

2





What this short course is not

An unbiased textbook or review Narrower scope

Methods reviewed in McGuire and McDonnell (2006) will not be covered

Focus on pedagogy, rather than covering all literature

SeZled science and methods AcAve debates are in progress

Terminology and nomenclature vary Best approach not agreed on

3



Two types of transport models Eulerian ConcentraAon at xed points

Lagrangian Parcels moving through space

LTRANS Larval transport model (North et al 2006) from www.usglobec.org

DO above the bed Courtesy of Jeremy Testa, UMCES



Lumped or integrated Lagrangian transport

Inow Oublow How do we represent the emergent eect of

- External forcing (e.g. climate) - Internal processes (e.g. ow pathways) on the composiAon of the oublow?



Why do we want to do this?

Insight into data Observables are oden spaAally integrated Q & C at catchment outlet

Classify behavior How are sites dierent?

Dominant processes control emergence

Forward modeling PredicAons at scale

E.g. subwatersheds in a larger catchment model

Upscale heterogeneity Fewer parameters

parameter idenAcaAon and uncertainty

6



What is the transport analogue to a storage-discharge relaAonship?

Emergent empirical relaAonship at scale of interest

Result of myriad unresolved processes, but may reect few dominant processes

Parameters related to measurable properAes (?)

7

Kirchner (2009 WRR)





Upscaled transport conservaAon + closure relaAons?

8

(Reggiani et al, 1999 AWR)

Representa3ve Elementary Watershed (REW) approach Conserva0on Laws + Closure Rela0ons

+





Can TTD be used for transport closure relaAons?

At steady-state* ow, yes!

9

Maloszewski and Zuber (1982, J. Hydr)



Can TTD be used for transport closure relaAons?

TradiAonal methods struggle

External variability Inows AND oublows

Internal variability Shiding ow pathways Eect of spaAal averaging

10

McGuire et al 2006

What about 4me-variable ow systems?





A simple example

Luke Pangle Minseok Kim

Peter Troch Yadi Wang Charlene Cardoso Marco Lora NSF Hydrology grant EAR-1344664

Mini-LEO IrrigaAon + tracers

Discharge @ seepage face

1 m3 loamy sand





28 days of periodic irrigaAon

28 days

IrrigaAon

Discharge at seepage face

Total storage (from load cells)





Tracer injecAons show complex breakthroughs

NaCl

Deuterium

Deuterium + NaCl NaCl





Observed Ame-varying transit Ame distribuAons

External - TTD reects Ame history of irrigaAon Internal - TTD diers depending on the state of the system

Can internal and external controls be separated? Can internal part be dis3lled into an underlying closure rela3on?

14

Wet condiAon Dry condiAon





A way forward

Recent work has developed/revived:

1. Rigorous theoreAcal foundaAon

2. ConservaAon laws for water age

3. A framework for developing closure relaAons

15





What well cover today

1. Basic terms, steady ow 2. Unsteady ow 3. The conservaAon law 4. Closure relaAons 5. Example 6. Q&A

16





Rest of the short course

Session 1 Basic SAS theory

Session 2 Use of the rSAS code

Session 3 Advanced theory and applicaAons

Session 4 ParAcipant presentaAons and feedback

17

1. BASIC TERMS, STEADY FLOW

Virtual short course: Theory and applicaAon of Ame-variable transit Ames in hydrologic systems

18

Some basic terminology Control volume, the system

Solute mass :

Storage :

Inow rate :

Mass ux in :

Oublow rate :

Mass ux out :


Theory and applicaAon of Ame-variable transit Ames in hydrologic systems 20

w

Inow

Oublow

ConcentraAon is dened by:





w Assump3on 1: OuBlow at 3me t is composed of parcels that arrived as inow at a distribu3on of earlier 3mes

21

Assump3on 2: Solute is conserva3ve, so ouBlow conc. is the average of the inow conc., weighted by

this distribu3on





w

22

This is the backward transit 3me pdf

Distribu3on is normalized

Assump3ons 3 & 4: Flow is steady, and this distribu3on is xed

is the age (more later)

FracAon of oublow with exit age between and

is the Amestep





w

23

This is the backward transit 3me distribu3on

Distribu3on is normalized

FracAon of discharge younger than T

Discrete:

ConAnuous:





w

24

Now, consider the distribu3on of ages T that parcels entering at 3me ti will aUain at the 3me they leave





Consider all the parcels of water that:

leave at Ame t

Arrive as inow at Ame ti

at age T

What is the mass of water in these parAcular parcels?

in terms of the backward TTD in terms of the forward TTD

At steady state ow





Thus at steady state:

forward and backward TTD are iden3cal!





4. Cancelling Q(t):

Discrete convolu3on

5. In the limit of small :

Con3nuous convolu3on

1. Mass of solute in oublow is:

2. Using concentraAon instead of mass:

6. Sewng

Equivalent form

3. But since





w

28

Assump3ons 5: We can assign a

concentra3on Cold to old water that entered before t0

Unknown conc.

Unobserved fracAon

Including unobserved fracAon:



2. UNSTEADY FLOW


29





A simple case to introduce some concepts

30

well-mixed or, uniform selecAon





1. ConservaAon of mass for a control volume and conservaAve solute:

3. The soluAon comes out as a convoluAon with a Ame-varying backwards transit Ame distribuAon!

2. The well-mixed or uniform selecAon assumpAon allows us to write:

see e.g. BoZer et al (2011 GRL)





Linear reservoir

Fixed storage

Flow weighted age

reference discharge

Turnover rate at Q=Q0

Steady-state





w

33

The backward TTD is now 3me-varying!

Assump3ons 3 & 4 Flow is NOT steady

This distribu3on is NOT xed

DistribuAon depends on (condiAonal on) the exit Ame





What about the forward transit Ame distribuAon?

Niemis theorem

(Niemi 1977)

leave at Ame t

arrive as inow at Ame ti

at age T

Recall the equivalence of water parcels that:





A claricaAon of terms

35

Transit Ame

Age Residence Ame

Life expectancy

Time Entry Ame Birth

Exit Ame Death

e.g. Benewn et al (2015)





A claricaAon of terms

36

Backward transit 0me distribu0on DistribuAon of age at death for parAcles with

Forward transit 0me distribu0on DistribuAon of life expectancy at birth for parAcles with

Residence 0me distribu0on DistribuAon of ages of parcels inside the system at

Life expectancy distribu0on DistribuAon of life expectancy of parcels inside the system at





The shorthand used here

37

Backward transit 0me distribu0on DistribuAon of age at death for parAcles with

Forward transit 0me distribu0on DistribuAon of life expectancy at birth for parAcles with

Residence 0me distribu0on DistribuAon of ages of parcels inside the system at

Life expectancy distribu0on DistribuAon of life expectancy of parcels inside the system at





Niemis theorem Oublow at Ame t

Inow at Ame ti

FracAon of oublow at Ame t with age at death

FracAon of inow at Ame ti with life-expectancy at birth





The forward TTD is also 3me-varying!

w

Assump3ons 3 & 4 Flow is NOT steady

This distribu3on is NOT xed

DistribuAon depends on (condiAonal on) the input Ame





NoAce how at T=0 the scaled distribuAons match





In short, under unsteady ow:

Forward and backward transit Ame distribuAons dont look like nice funcAons

Reect the Ame-variability of the external uxes and the variable internal processes

(in this case simple diluAon)





w

But what if my hydrologic system has mulAple oublows (e.g. discharge, ET)? is not well-mixed? has Ame variable internal processes?

i.e. it is a real system!





To construct a framework that can deal with these complexiAes we have to start thinking about storage: 1. How does the populaAon of parcels

stored in the system change as parcels arrive and depart?

2. How are parcels in the oublow

selected from the parcels in storage?

ConservaAon Law

Closure RelaAons



3. THE CONSERVATION LAW


46





w

w

Residence Ame PMF/PDF FracAon of storage at Ame with an age, or residence Ame, between and

Residence Ame CDF FracAon of storage at Ame with an age, or residence Ame, less than





To normalize, or not to normalize

48

There are advantages to leaving all the distribu3ons un-normalized, and dening:

Age-ranked storage

Age-ranked ux

Age-ranked storage density

Age-ranked ux density





DerivaAon of a conservaAon law What happens to the water that enters between Ame ti and t?

49

Amount in the control volume Amount of water younger than T = t - ti at Ame t

Rate of increase The inows are always of age less than T

Rate of decrease Rate of oublow of water younger than T = t - ti at Ame t





DerivaAon of a conservaAon law

Therefore by conservaAon of mass:

50

SubsAtuAng ti = t - T and using the chain rule gives:





ConservaAon law #1

51

Rate of change of storage younger than a given age T

Rate of inow Rate of oublow younger than a given age T

Rate water in storage is gewng older than a given T

Harman (2015, WRR)

Boundary condiAon:





ConservaAon law #2

52

van der Velde et. al. (2012, WRR)

Taking the derivaAve of the previous equaAon with respect to T gives:





ConservaAon law #2

53

van der Velde et. al. (2012, WRR)

As pointed out by Porporato and Calabrese (2015, WRR) this is actually the McKendrick-von Foerster equaAon for age-structured populaAon dynamics:

Births [#/Ame]

Deaths [#/Ame]

There is a rich history of research on this equaAon.





ConservaAon law #3

54

BoZer et. al. (2011, GRL)

SubsAtuAng the normalized PDFs back in and expanding the derivaAves gives the master equaAon:





Extension to mulAple oublows

Easy to account for more than one path out:

55

PrecipitaAon

Discharge EvapotranspiraAon Groundwater

Note: Niemis Theorem and the forward TTD become more complicated see BoZer et al (2010 WRR), Harman (2015 WRR)





But to solve any of these we need more equaAons!

56

In parAcular: how do we determine the ux out?

So we have the conservaAon law

CumulaAve form:

Density form:

5 MIN BREAK Lets take a

57

But to solve any of these we need more equaAons!

58

In parAcular: how do we determine the ux out?

So we have the conservaAon law

CumulaAve form:

Density form:



4. CLOSURE RELATIONS


59





Several ways to express the problem

60

Loss funcAon, or death rate

Porporato and Calabrese (2015 WRR) McKendrick (1925), von Foerster (1959)

Discharge of age T

Storage of age T

BoZer et al (2011 GRL) Age funcAon

absolute StorAge SelecAon funcAon (aSAS)

% Discharge of age T

% Storage of age T

van der Velde et al (2011 WRR) STOP funcAon

frac3onal StorAge SelecAon funcAon (fSAS)

% Discharge of age percenAle PS

% Storage of age percenAle PS

Harman (2015 WRR) rank StorAge SelecAon funcAon (rSAS)

% Discharge of age-rank ST Storage of age-rank ST





Some contrived examples

61



absolute StorAge SelecAon funcAon





fSAS uses a change of variables

64

w

Residence Ame CDF FracAon of storage at Ame with an age, or residence Ame, less than

For every T, there is a PS E.g. when T = 10 we have PS = 0.4

So let us dene a new CDF where:

when

Taking the derivaAve w.r.t. T gives:

is a PDF on the interval [0,1] The fSAS func0on



frac3onal StorAge SelecAon funcAon

rSAS uses a dierent change of variables

66

Age-ranked storage Amount of storage at Ame with an age, or residence Ame, less than

For every T, there is an ST E.g. when T = 10 we have ST = 3.8 m3

So let us dene a new CDF where:

when

Taking the derivaAve w.r.t. T gives:

is a PDF on the interval [0,S(t)] The fSAS func0on



rank StorAge SelecAon funcAon



The transport column analogy

What do these look like for a well-mixed system?

69


Discharge of age T

Storage of age T

Age funcAon absolute StorAge SelecAon funcAon (aSAS)

% Discharge of age T

% Storage of age T

STOP funcAon frac3onal StorAge SelecAon funcAon (fSAS)

% Discharge of age percenAle PS

% Storage of age percenAle PS

rank StorAge SelecAon funcAon (rSAS)

% Discharge of age-rank ST Storage of age-rank ST





Which one to use? aSAS has normalizaAon problem

This implies cannot be chosen unless is known!

mu predicts Q amount and composiAon

Increases the stakes in choosing mu to match data





Which one to use?

Recall: we are looking for a way to express the emergent principle underlying transport

Both aSAS and mu assume age controls the death rate Reasonable in populaAon models, but what about hydrologic/hydraulic systems?

76





Consider a simple plug ow system What do the dierent storage selecAon funcAons look like?

Hint: All discharge is drawn from the oldest water in the system

- funcAon





Closure relaAons for plug ow

78


Age funcAon absolute StorAge SelecAon funcAon (aSAS)

STOP funcAon frac3onal StorAge SelecAon funcAon (fSAS)

rank StorAge SelecAon funcAon (rSAS)

Tmax will vary due to variaAons in the inow

Both these expressions are invariant in Ame

* The version of this slide used in the presentaAon contained an error that has been corrected





Which one to use?

Debate conAnues but seem to be good reasons to choose fSAS or rSAS over aSAS or mu In hydrologic/hydraulic systems, exit probability:

more related to locaAon in storage (PS, ST) than to actual age (T)

84





Which one to use?

So what about rSAS vs fSAS? I prefer rSAS

Result is a probability distribuAon over storage More physically intuiAve More likely to elegantly capture storage-dependent transport dynamics DistribuAon insensiAve to (possibly arbitrary) decisions about the total size of the system E.g. where is the lower boundary of a watershed?

85



4. EXAMPLE


86





Back to the simple example

Luke Pangle Minseok Kim

Peter Troch Yadi Wang Charlene Cardoso Marco Lora NSF Hydrology grant EAR-1344664

Mini-LEO IrrigaAon + tracers

Discharge @ seepage face

1 m3 loamy sand





Empirically observed Ame-varying storage selecAon funcAons!

89





More informaAon on how this was done will be given in Week 3

90

Or you can read our paper: Harman and Kim (2014 GRL)





Smin iniAal delay zone

Sd dynamic zone

Sf fast zone

Ss slow zone





FuncAonal storage zones change as the system hydraulics change

92

Smin iniAal delay zone

Sd dynamic zone

Sf fast zone

Ss slow zone





The volume of these zones is directly related to the saturated pore volume

93



Empirical observaAons, and model-free validaAon of observed rSAS

FuncAonal zones inferred from Zed distribuAon

HYDRUS-2D model





So how can I use this in my system?

Next week: choosing an rSAS funcAonal form wng its parameters using tracer data

In the meanAme Look out for a problem sheet coming later today Keep gewng up to speed with Python (2.7+) Review this lecture and post quesAons on the rsas_users google group

95



6. QUESTIONS?


96





References

97

BoZer, G., E. Bertuzzo, and A. Rinaldo (2010), Transport in the hydrologic response: Travel Ame distribuAons, soil moisture dynamics, and the old water paradox, Water Resour. Res., 46(3), doi:10.1029/2009WR008371.

MKendrick, A. (1925), ApplicaAons of mathemaAcs to medical problems, Proc. Edinburgh Math. Soc., 44, 98130.

von Foerster, H. (1959), Some remarks on changing populaAons, in The KineAcs of Cellular ProliferaAon, edited by J. F. Stohlman, pp. 382407, Grune and StraZon, N. Y.

Harman, C. J., and M. Kim (2014), An ecient tracer test for Ame-variable transit Ame distribuAons in periodic hydrodynamic systems, Geophysical Research LeUers, 41(5), 15671575, doi:10.1002/2013GL058980.





References

98

Niemi, A. J. (1977), Residence Ame distribuAons of variable ow processes, The Interna3onal Journal of Applied Radia3on and Isotopes, 28(10-11), 855860, doi:10.1016/0020-708X(77)90026-6.

Benewn, P., A. Rinaldo, and G. BoZer (2015), Tracking residence Ames in hydrological systems: forward and backward formulaAons, Hydrological Processes, doi:10.1002/hyp.10513.

van der Velde, Y., P. J. J. F. Torfs, S. E. A. T. M. van der Zee, and R. Uijlenhoet (2012), QuanAfying catchment-scale mixing and its eect on Ame-varying travel Ame distribuAons, Water Resour. Res., 48, doi:10.1029/2011WR011310.

Porporato, A., and S. Calabrese (2015), On the probabilisAc structure of water age, Water Resour. Res., 51, 35883600, doi:10.1002/2015WR017027.

Kirchner, J. W. (2009), Catchments as simple dynamical systems: Catchment characterizaAon, rainfall-runo modeling, and doing hydrology backward, Water Resour. Res., 45(2), doi:10.1029/2008WR006912.

Reggiani, P., M. Sivapalan, S. M. Hassanizadeh, and W. G. Gray (1999), A unifying framework of watershed thermodynamics: consAtuAve relaAonships, Advances in Water Resources, 23, 1539.

Maoszewski, P., and A. Zuber (1982), Determining the turnover Ame of groundwater systems with the aid of environmental tracers, Journal of Hydrology, 57(3-4), 207231, doi:10.1016/0022-1694(82)90147-0.

McGuire, K., and J. J. McDonnell (2006), A review and evaluaAon of catchment transit Ame modeling, Journal of Hydrology, 330(3-4), 543563.

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