Using Travel Times in Reactive Transport

Transcription

1 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Usng Travel Tmes n Reatve Transport ENIGMA Summer Shool 2018 June 30, 2018, Cargèse Olaf A. Crpka

2 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Adveton-Dsperson-Reaton Equaton of Compound s q D aqueous-phase onentraton [mol/l] sold-phase onentraton [mol/kg] spef-dsharge vetor [m/s] dsperson tensor [m 2 /s] θ w volumetr water ontent [-] ρ b r (*) t s t ( ) ( w) ( s) θ D = θ r ρ r θ w + ρb + q w w + dry bulk mass densty of the solds [kg/l] reaton rate wthn phase * [on./s] b

3 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Adveton-Dsperson-Reaton Equaton of Compound t s t Addtonally needed: Law for mass-exhange between the aqueous and sold phases Rate laws for the reatons Intal and boundary ondtons ( ) ( w) ( s) θ D = θ r ρ r θ w + ρb + q w w + Parameters, parameters, parameters... b

4 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Dvde by the Volumetr Water Content t + ρb θ w s t + v ( ) ( w) b ( s) D = r + r wth the seepage-veloty vetor v = q/θ w Requres the 3-D vetor feld of velotes... whh s spatally varable n heterogeneous domans Large CPU tmes for spatally explt shemes Varous ssues wth Euleran numeral dsretzaton (numeral osllatons; numeral dsperson) Partle-based methods have ssues too (express rate laws as partle-partle nteratons; hardly any partles n low-veloty zones;...) ρ θ w

5 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Dlemma of Reatve Transport Smulatons n Heterogeneous Domans Nonlnear rate laws + Spatal varablty of hydraul and hemal parameters (wth unknown detals) + Need fne resoluton to overome numeral problems of spatally explt shemes Very hgh omputatonal effort Unertanty of parameters (and onepts) all out for ensemble-based stohast analyss But ensemble runs are hardly affordable

6 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Major Objetve Fnd a oneptually smplfed desrpton of reatve transport that 1. overs the major ontrols 2. s omputatonally effent and an thus be used n ensemble alulatons Try to get rd of a spatally varable 3-D desrpton n the alulaton of nonlnear reatve transport! Can only be aheved under spef ondtons.

7 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Gross Smplfatons Steady-state flow Neglet dsperson ( reatants are ether already mxed n the nflow or mx due to exhange wth the sold phase) Unform ntal ondtons Unform reaton parameters Spatally unform onentratons n the nflow ( dffuse nput)

8 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe A Favorte Case of Mne that Does Not Work Steady-state plume where the reaton partner mxes n from the sde

9 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Adveton-Reaton Equaton of Compound Introdue the advetve travel tme τ a : and apply the han-rule of dfferentaton: ), ( ), ( ) ( ) ( s s v s w b w w b r r t s t θ ρ θ ρ + = + + =1 a τ v a a a τ τ τ = = v v ), ( ), ( ) ( ) ( s s s w b w a w b r r t s t θ ρ τ θ ρ + = + +

10 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Adveton-Reaton Equaton of Compound n Travel-Tme Doman t + ρb θ w s t + τ a = r ( w) (, s) + ρb θ (, s) 3 spatal oordnates replaed by 1 travel-tme oordnate Varable veloty replaed by onstant fator of 1 Unform hemal parameters and ntal ondtons mply that the reaton rates depend on onentratons only (whh depend on reatve transport) Self-organzaton of onentraton fronts Compute everythng n the 1-D travel-tme doman and map results to the spatal doman (x,t) = (τ a (x),t) w r ( s)

11 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Example Calulatons of Knemat Age

12 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Effets of Loal Dsperson Stll steady-state flow, unform ntal & nflow onentratons, unform hemal parameters Longtudnal dspersve mxng: = a a a a D D x D x τ τ τ τ τ v v l l l l 1

13 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Effets of Loal Dsperson Transverse dspersve mxng Also mxes old and young water! Parameterzaton by effetve dsperson? τ τ ( ) D D eff, τ What s the rght oeffent D eff,τ (τ)?

14 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Effets of Loal Dsperson Upon dspersve mxng, the groundwater age beomes a dstrbuton at eah pont Mean and varane of groundwater age meet: v µ τ v σ 2 τ ( D µ ) τ ( 2 D σ ) = 2 µ ( D µ ) τ = 1 τ τ

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16 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Parameterzaton of Dspersve Mxng n Travel- Tme Framework of Reatve Transport Stll 1-D model All mxng s between young and old water Defntely requres spatally unform nflow onentratons D eff,τ an vary wth (and only wth) µ τ Obtan D eff,τ from dependene of σ τ2 on µ τ ) ( ) (, s w b w eff w b r r D t s t θ ρ µ µ µ θ ρ τ τ τ τ + = + +

17 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Aerob Degradaton and Dentrfaton n Bank Fltraton (Rver-Water Infltraton) Oxygen and ntrate are the eletron aeptors Dssolved organ arbon s the eletron donor Elevated oxygen onentratons nhbt dentrfaton The reatants are mxed n the rver and jontly transported nto the rverbed Immoble aerob and dentrfyng mrobes grow upon ther reatons There s a maxmum bomass onentraton ( arryng apaty )

18 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Stohometry of Reatons Net reaton usng organ arbon as eletron donor: CH 2 O + (1-Y den ) 4/5 NO 31 (1-Y den ) 2/5 N 2 + Y den bo den + Aerob metabolsm: CH 2 O + (1-Y aer ) O 2 Y aer bo aer + wth spef yeld oeffents Y and bomass onentratons bo (measured n arbon) related to bomass type Double-Monod knets as typal rate law: r aer = DOC DOC + K DOC Ox Ox + K Ox µ aer,max Y aer aer bo

19 1-D Invason Traer DOC on. Oxygen Ntrate Aerobes Dentrfer

20 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe 2-D Heterogeneous Doman Referene ase: 2-D spatally explt, reatve transport model Several mappng approahes from travel-tme doman to spatal doman: Negletng dsperson altogether Consderng the loal longtudnal dsperson oeffent at the average veloty Consderng an effetve longtudnal dsperson oeffent that nreases lnearly wth mean travel tme

21 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Spatal Dstrbuton of Ntrate (worst) normalzed onentratons of spatally explt alulaton normalzed errors

22 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Spatal Dstrbuton of Ntrate (worst) normalzed errors

23 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Spatal Dstrbuton of Aerob Batera normalzed onentratons of spatally explt alulaton normalzed errors

24 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Spatal Dstrbuton of Aerob Batera normalzed errors

25 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe More on Relatonshp between Conentratons and Travel Tmes n Boreatve Transport Jont njeton of substrate, oxygen, and ntrate Durnal flutuatons of veloty magntude, but not dreton 2-D spatally explt advetve-dspersve-boreatve transport Lnes: Isohrones for tme-averaged veloty. Sanz-Prat et al. (2016)

26 2-D Results Mapped 1-D Results Dfferene Can explan most of spatal varablty n onentratons by that of travel tmes

27 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Lessons Learned Travel-tme based smulatons of reatve transport are good approxmatons f marosop transverse mxng s not desve (dffuse nput) hemal parameters are spatally unform dreton of flow does not vary n tme Effetve longtudnal mxng needs to be parameterzed at an nvadng front but s less mportant at late tmes

28 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Fate of Ntrate Controlled by e-donor Release from the Matrx Smlar alulatons as the prevous ones, but wth frst-order release of DOC rather than jont njeton. Release of DOC s restrted to zones of hgh Natural Organ Matter (NOM) n the matrx Bnary hemal heterogenety

29 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Let s Follow a Water Parel Through the Aqufer Zones of Reaton NOM NOM NOM O

30 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Exposure Tme Tme that a water parel has spent n the reatve zones: v τ ( D ) exp τ exp = 1 0 n reatve zones elsewhere Use smlar mappng approah as before, but wth exposure rather than travel tme

31 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Travel- and Exposure Tmes n a Heterogeneous Doman Two dstnt materals Sanz-Prat et al. (2016) Low-ondutvty materal releases DOC Hgh-ondutvty materal released no DOC

32 Comparson between full 2-D and Exposure-Tme Based Results at Late Tmes Model annot Predt Bomass n Non-Reatve Zones

33 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Smplfatons and Extensons Buld-up of bomass takes days (aerob) to weeks (dentrfyng batera) Mrobes grow untl a loal steady state of DOC release and onsumpton s reahed, often lose to arryng apaty Mrobes an ope wth transent ondtons by dormany and bomass transport Explt onsderaton of mrobes not needed Wth exepton of nvadng fronts, dsperson s not needed Reatvty of the aqufer matrx not bnary

34 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Relatve Reatvty f(x) Release of the eletron donor from the matrx addressed as reatvty, relatve to a reatvty under referene ondtons: t = + v Ox Nt ; = r 0 fr 0 ( ) ( ) = r max Nt r max Ox Nt + Nt K Ox Nt + Ox K Ox Ox K + nh Ox K nh Ox

35 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Mathematal Trks n a Nutshell Adveton-reaton equaton wth varable relatve reatvty: + = () Total dervatve along a trajetory: = Introdue umulatve relatve reatvty: = (( )) along a trajetory Then, reatve transport along trajetory beomes: = Sngle ODE-system for one set of ntal onentratons!

36 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe (Cumulatve) Relatve Reatvty Relatve reatvty: How strong s the eletron-donor soure n omparson to referene ondtons? Depends on the type and amount of eletron-donor soure n the matrx Cumulatve relatve reatvty How muh (relatve) reatvty has a water parel seen on ts way to an observaton pont? Unts of tme: The lok tks more qukly n zones of hgh reatvty than n lowly reatve zones Can easly be evaluated by partle trakng

37 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Ageng of a Water Parel Causes Derease n Oxygen and Ntrate Conentraton Closed-form soluton for smplfed Mhaels-Menten knets

38 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Proof-of-Conept Applaton 1.5 Mo. ells Groundwater reharge ontans ntrate and oxygen usng MATLAB, MODFLOW-NWT, CUDA GPU Partle Trakng Computng tme: 12mn (mostly MODFLOW) Loshko et al. (2016)

39 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Hydraul Heads

40 Relatve Zentrum für Angewandte Reatvty Geowssenshaften (ZAG) n Hydrogeologe the Inluson

41 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Cumulatve Relatve Reatvty

42 Conentratons as Funton of Cumulatve Relatve Reatvty Spatal Dstrbuton of Cumulatve Relatve Reatvty Oxygen [mg/l] Ntrate [mg/l] F [s] Spatal Dstrbuton of Ntrate mappng Spatal Dstrbuton of Oxygen

43 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Aountng for Dereasng Dentrfaton Potental of the Aqufer Matrx Stohometres: 1O 2 :1C org & 1NO 31 :1.25C org Need a funtonal dependene of relatve reatvty on NOM-ontent of the matrx Consumpton of NOM n the matrx redues umulatve relatve reatvty along trajetores Regular update needed Spatally explt problem, solved along pathlnes that are dsretzed n travel-tme nrements

44 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe 3-D Monte-Carlo Smulatons Cells, 200 Realzatons Loshko et al. (2018)

45 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Breakthrough Curves Dssolved Oxygen Ntrate Boavalable NOM Loshko et al. (2018)

46 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Carry-Home Messages Travel-tme based approahes oneptually smplfy reatve transport smulatons Not the golden bullet for all setups... but sutable under spef ondtons A model does not need to nlude all proesses and propertes that you know of but those that ontrol the system. If hosen wsely, you an redue the omputatonal effort dramatally

47 Zentrum für Angewandte Geowssenshaften (ZAG) Hydrogeologe Pratal Challenges Inferene of travel tmes from age traers s less straghtforward than many beleve Chemal heterogenety (e.g., spatal dstrbuton of NOM) exsts but s less frequently assessed than physal heterogenety Catalogue of geologal strutures wth typal szes, hydraul and reatve parameters? Relatonshp between NOM ontent and aqufer reatvty?