DTIC ADA640177: Transmissivity Resolution Obtained from the Inversion of Transient and Pseudo-Steady Drawdown Measurements

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MODFLOWandMore2003:UnderstandingthroughModeling-ConferenceProceedings,Poeter,Zheng,Hill&Doherty-www.mines.edu/igwmc/
Transmissivity Resolution Obtained from the Inversion of Transient and
Pseudo-Steady Drawdown Measurements.
(cid:0) (cid:0) (cid:1)
(cid:2)
TomClemo ,PaulMichaels ,R.MichaelLehman
(cid:3)
BoiseStateUniversity,[email protected],[email protected],Boise,IDUSA
IdahoNationalEnvironmentalandEngineeringLaboratory,[email protected],IdahoFalls,IDUSA
ABSTRACT
Akeyaspectofestimatingtransmissivitydistributionsfrompumpingtestsisthespatialresolutionofthe
estimation. Someassumptionofspatialstructureisneededtomakeaninversionpossible. Boththenature
ofgroundwater(cid:3)owandassumedstructurecauseasmearedestimate. Thisresearchisfocusedonthe
howinformationcontentofthedatasmearstheestimate. Theinversionprocedurenearthesolutionis
approximatedasiterativelinearupdates. Linearresolutionanalysisofthedistributedsensitivityto
transmissivityindicatesthelimitofdetailpossiblefromtheinversion.
Toenablethisresearch,adjoint-sensitivitycalculationswereaddedtoMODFLOW-2000. The
adjoint-sensitivitycalculationssigni(cid:2)cantlyspeedthecalculationofthesensitivitymatrix.
Tohighlighttheconcepts,weuseahomogeneousaquiferwithknownstorativity. Theanalysisindicates
resolutionoftransmissivityis(cid:2)nernearthepumpingandmonitoringwellsthanintheregionsbetween
wells. Transientdataarehighlyredundantwithrespecttotransmissivityinformation. Veryearlytimedata
doprovidedifferentspatialinformationthanlatertimedatado.
INTRODUCTION
Animportantquestioninanyendeavoris(cid:147)Howwellcanwedoit?(cid:148) Thisisoftenaverycomplicated
question. Boundingtheanswerratherthanansweringitdirectlyoftenturnsadif(cid:2)cultquestionintoonewe
cananswer. Ourendeavoristoestimatesubsurfaceproperties. Resolutionanalysisprovidesinsightinto
howwellwecanestimatetheproperties. Resolutionanalysisisacommontechniqueinthegeophysical
community(Menke,1989;Tarantola,1987). WiththeexceptionofVasco et. al. (1997),resolutionanalysis
hasnotreceivedattentioninthegroundwaterliterature.
Resolutionanalysisusesalinearapproximationofthedependenceofmeasurementsonestimated
parameterstodeterminehowtheinversionestimatesrelatetotheactualsubsurfaceproperties. Thisis
trulyaboundinganalysisbecauseweassumethatourgroundwatermodelisaccurateandthatthe
estimatedparametersarecloseenoughtocorrectthatnon-linearitycanbeignored. Wealsopartially
neglectthein(cid:3)uenceoferrorsinthemeasurementsusedintheinversion.
Inthispaper,weconsidertheestimationofthedistributionoftransmissivityinahypotheticalaquiferfrom
drawdownmeasurementsinobservationwellsaboutapumpingwell. Theemphasisisontheresolution
analysisnotonaspeci(cid:2)cinversionprocedure. AsthisisaconferenceonMODFLOW,theadditionofthe
adjointstatemethod(Sun,1994;TownleyandWilson,1985)ofcalculatingsensitivestoMODFLOW-2000
(Harbaughetal.,2000;Hilletal.,2000)anditsuseinthisinvestigationishighlighted.
SENSITIVITYCALCULATIONS
Thesimulatedhypotheticalaquiferhasacentralregion12.9mby7.6mofconstantgridspacingof0.25
m. TwelvewellsarepositionedinroughlyconcentricringsaboutthecentralwellasshowninFigure1. The
centralregionissurroundedbyagridthatexpandswithagrowthratioof1.5to150m. Thegridextends
(cid:1)
beyondthecentralregionfarenoughthatnoappreciabledrawdownoccursattheboundaryduringthe
simulationperiod. Aconstantstorativityof0.3isused. Auniformtransmissivityof0.0032m /sisused. A
constantpumpingrateof5l/sismaintainedfor10,000s(170min). Noappreciabledrawdownoccursat
theboundaryduringthesimulationperiod. Atotalof122observationsweremadefromdrawdownatthe
wells. Fourdrawdownobservationsperlogbasetentimeintervalwererecorded. Earlyobservations,
beforeawellresponds,werenotincluded.
(cid:4)(cid:6)(cid:5)(cid:8)(cid:7)
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1. REPORT DATE 3. DATES COVERED
2003 2. REPORT TYPE 00-00-2003 to 00-00-2003
4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
Transmissivity Resolution Obtained From The Inversion Of Transient
5b. GRANT NUMBER
And Pseudo-Steady Drawdown Measurements
5c. PROGRAM ELEMENT NUMBER
6. AUTHOR(S) 5d. PROJECT NUMBER
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13. SUPPLEMENTARY NOTES
MODFLOW and More 2003: Understanding through Modeling - Conference Proceedings, Government or
Federal Purpose Rights License
14. ABSTRACT
A key aspect of estimating transmissivity distributions from pumping tests is the spatial resolution of the
estimation. Some assumption of spatial structure is needed to make an inversion possible. Both the nature
of groundwater flow and assumed structure cause a smeared estimate. This research is focused on the how
information content of the data smears the estimate. The inversion procedure near the solution is
approximated as iterative linear updates. Linear resolution analysis of the distributed sensitivity to
transmissivity indicates the limit of detail possible from the inversion.
15. SUBJECT TERMS
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MODFLOWandMore2003:UnderstandingthroughModeling-ConferenceProceedings,Poeter,Zheng,Hill&Doherty-www.mines.edu/igwmc/
10
Themodelhasatotalof9409cells. Estimatingadistributionof
transmissivityreferstoallowingeachcelloftheMODFLOWmodel
C6
C1 tobeassignedadifferentvalue. Asaconsequence,amatrixof
5
s parametersensitivities,thesensitivitiesofthedrawdown
Meter B6 B1 B2 observationstotransmissivity,mustbecalculatedforeachcellinthe
e in 0 C5 A1 C2 meqoudaetli.onCaplrcoucleastisngofthMeOsDeFnLsiOtivWit-y2m00a0trrixequusiirnegstahesespeanrsaitteivictyalculation
nc B5 B3
a B4 foreachcell. Thiscalculationrequired19hona1GHzPC.These
st
Di calculationsagreewellwiththeanalyticsolutionspresentedbyOliver
−5
C3 (1993). Anadjointstatesbasedcalculationofdrawdownsensitivityto
hydraulic conductivity was added to MODFLOW-2000 following the
C4
proceduredescribedinCarreraandMedina(1994).
−10
−10 −5 0 5 10 .
Distance in Meters The procedure of Carrera and Medina requires one adjoint states
calculationperobservationlocation. Thesecalculationstook2.3min-
X4Figure1WellField utes. Thecalculationswerenotsuf(cid:2)cientlyaccurateforourpurpose.
Theydisagreeby10%to20%withthesensitivityequationbased
calculations. Errorswerelargerforcellswithsmallsensitivitybuttheseerrorsarenotrelevant. The
inaccuracymayrepresentacodingerrororafundamentallimitation. Improvementstothecalculationsare
beinginvestigated. Whilenotsuf(cid:2)cientfortheresolutionanalysis,20%accuracymaybeacceptableforthe
earlystagesofaninversionwherenon-linearitymakesanycalculationinaccurate.
Theadjointsensitivitycalculationsthatwereusedwerebasedonaseparateadjointstatescalculationfor
eachofthe122observations. Thesecalculationstook94minutestocomplete,12timesfasterthanthe
sensitivityequationcalculation. Errorsforsigni(cid:2)cantsensitiveswerelessthan0.1%forobservationsthat
coincidedwiththeendofatimestepandbetween0.1%and2%forinterpolatedobservations.
Improvementstospeedareanticipated. Wehavenotedthatifonlythe(cid:2)rstthirdoftheadjointstates
calculationwereperformedanadditionfactorofthreespeed-upcouldbeobtainedwithoutappreciable
increaseinerror.
RESOLUTIONMATRIX
Theresolutionmatrixisbasedonthesensitivitycalculationofthelaststepofanon-linearinversionproce-
dure(Tarantola,1987). Atthelaststeptheinversioncanbelinearizedto
(cid:9)(cid:11)(cid:10)(cid:13)(cid:12)(cid:13)(cid:14) (cid:15)
1)
(cid:9) (cid:10) (cid:14)
where isthesensitivitymatrix, isanunknownerrorinthetransmissivityparameters,and isthe
(cid:9) (cid:10)(cid:17)(cid:12)(cid:18)(cid:9)(cid:20)(cid:19)(cid:22)(cid:21)(cid:23)(cid:14) (cid:9)
differencebetweentheobservationwiththecurrentparameterestimat(cid:16)esandthemeasuredobservations.
If couldbeinverted,wecouldupdatethetransmissivityestimateby ,but isa138by9409
matrixwhichdoesnothaveaninverse. Instead,wewritetheupdateas
(cid:10)(cid:13)(cid:12)(cid:18)(cid:9) (cid:19)(cid:25)(cid:24) (cid:14) (cid:15)
(cid:16)
2)
(cid:26)(cid:27)(cid:19)(cid:25)(cid:28) (cid:10) (cid:10)
(cid:16)
isreferredtoasthegeneralizedinverse. Theupdate, ,canberelatedtotheerror, ,by
combining1and2.
(cid:10)(cid:17)(cid:12)(cid:17)(cid:9) (cid:19)(cid:29)(cid:24) (cid:9)(cid:11)(cid:10) (cid:15)
(cid:16)
3)
(cid:9) (cid:19)(cid:25)(cid:24) (cid:9)
(cid:9)(cid:20)(cid:19)(cid:29)(cid:24) (cid:9)
R= istheresolutionmatrix(Menke,1989;Tarantola,1987). Theresolutionmatrixdescribeshowthe
transmissivityupdateisrelatedtothetransmissivityerror. Because isnottheinverseof ,theupdate
isnotequaltotheerror. Itcanbeshownthat,ifthetransmissivityestimateisnearlycorrectthennon-linear
effectsarenotimportantandequation3holdsforthetransmissivityestimateitself. Thatis,
(cid:30) (cid:31)"!$# (cid:12)&% (cid:30)’#)(+*,(cid:31) (cid:15)
4)
(cid:4).-./
MODFLOWandMore2003:UnderstandingthroughModeling-ConferenceProceedings,Poeter,Zheng,Hill&Doherty-www.mines.edu/igwmc/
(cid:30) (cid:31)"!$# (cid:30)(cid:11)#0(1*,(cid:31)
where isthetransmissivityestimateoftheinversionand istherealtransmissivity. Equation4
neglectserrorsinthemodel,measurementserrors,andotherplaguesofinversion. Thesecaveatsaside,
theresolutionmatrixprovidesuswithatoolthatwecanusetosetourexpectations,designexperiments,
andcomparedifferentinversionstrategies.
ANALYSISUSINGTRUNCATEDSINGULARVALUEDECOMPOSITION
(cid:9)2(cid:19)(cid:25)(cid:24)
In equation 2, incorporates the constraints used in the inversion. These
0
constraintsmayincludepriorinformation,regularizationequationstoimposea
structureonthesolution,orapriorcovariancestructurearisingfromaBayesian
formulation of the inversion problem. The constraints must be included in the
( )( )llgg1010−2 rbeysuosluintigontraunnacalytseidssoifnaguplaarrtvicaululaerdinevceormsiopnospitrioocn(cid:9)ea(cid:19)(cid:25)sss(cid:24) .thWeewsaiymopflifcyaltchuelaatninaglytshies
oo
ll generalizedinverse. Menke(1989)referstothis asthenaturalgeneralized
−4 inverse. It is natural because it constrains the estimated distribution to only
have variations that are distinguishable from the measurements. Any form of
regularization introduces a bias into the estimation. The bias of the truncated
50 100
singular decomposition approach is based on the information content in the
eeiiggeennvveeccttoorr iinnddeexx
data. Ifavariation intransmissivitycannotbeclearlysupportedbythedata,it
isleftoutofthesolution.
Figure2Normalized (cid:9)
.
eigenvaluesobtained
Thesingularvaluedecompositionof resultsin
fromsingular (cid:9)3(cid:12)5476(cid:22)8(cid:11)9 (cid:15)
decompositionofthe
5)
sensitivitymatrix.
6 4 8
:(cid:25);
4 8
where isadiagonalmatrixofeigenvalues, ,and and arematricescontainingtheassociated
eigenvectors. Wereferto asthedataeigenvectorsand asthetransmissivityeigenvectors.
4 8
Becauseoftheorthogonalityandnormalitypropertiesof and
(cid:9) (cid:19)(cid:25)(cid:24) (cid:12)<8 6 (cid:19)(cid:22)(cid:21) 4 9 (cid:15)
6)
(cid:19)?> (cid:21)
= @BA
where isadiagonalmatrixwithdiagonalelements . Verysmalleigenvaluesindicatearedundancy
intheinformationaboutthetransmissivitycontainedintheobservations. Twomeasurementswithexactly
thesameinformationcontentwouldproduceaneigenvaluewithavalueofzero.
Figure2plotsthenormalizedeigenvaluesfromsingulardecompositionofthesensitivitymatrixusinga
semi-logscale. Theeigenvectorsarenormalizedwithrespecttothelargesteigenvalue. The(cid:2)gureshows
arapiddeclineintheeigenvaluesimplyingalargeredundancyininformation. Twosetsofeigenvaluesare
plotted. Oneusingallthe138observationsandoneusingonlythe13observationsacquiredat10,000s,
thelasttimestepofthesimulation. Theeigenvaluesofthese13observationsnearlycoincidewiththe
thirteenlargesteigenvaluesofthefulldataset. Theoverlapofthetwosetssuggeststhatobservations
fromdifferentlocationsprovidemuchmorenewinformationthancanbeobtainedbyaddingalarge
numberofnewobservationsatcurrentlocations,aconclusionthatissupportedbytheresolutionanalysis
presentedbelow.
(cid:10)DC
(cid:16)
Aconsequenceofsmalleigenvaluesisthatthereareweightedcombinationsoftransmissivity, that
E 8 E
hav;eaverysmallin(cid:3)uenceonthecalculateddrawdownobservations. Thesecombinationsarede(cid:2)nedby
(cid:14) C 4 8 (cid:14)
; ,where iFsa; constant. Thereisacorresp;ondin; gweightedcombinationofdrawdownmeasurements,
(cid:14) (cid:21) (cid:10)
de(cid:2)nedby thatare; in(cid:3)uencedbythe .@BAIf isimperfect,asofcourse(cid:16)itw;illbeusingmeasured
data,thentheerrorsin willbemagni(cid:2)edby intheresultingp:(cid:29)r;edictionof . Toavoidthisproblem,
wetruncatethesingularvaluedecompositionatsomeminimum . Thisisthemethodwherebyonly
variationsintransmissivitythatcanbesupportedbythedataareretained. Theappropriatetruncationlevel
isrelatedtotheuncertaintyintheobservations. Vogel(2003)providesanalgorithmforselectingan
appropriatetruncationlevel.
(cid:4).-HG
MODFLOWandMore2003:UnderstandingthroughModeling-ConferenceProceedings,Poeter,Zheng,Hill&Doherty-www.mines.edu/igwmc/
RESOLUTIONOFTRANSMISSIVITYESTIMATES
Theresolutionmatrixislarge;numberofparametersbynumberofparameters. Thespreadfunction
providesonewayofreducingtheresolutionmatrixtoacomprehensiblelevel. Menke(1989,pg68)de(cid:2)nes
thespreadfunctionfortheresolutionmatrixas N(cid:29)OQP N
%(cid:22)J (cid:12) L K LN K % JSR (cid:15)
I I ; ;
spread ;M (cid:21) M (cid:21) 7)
T
where istheidentitymatrix. Wede(cid:2)nethespreOadofeacN(cid:22)hOQcPeN llas
U (cid:12)WV LN K % JSR (cid:15)
; I ; ;
M (cid:21) 8)
Avalueof1foracellimpliesthatthecellcanbeestimatedwithgoodaccuracy. Italsoimpliesthatthe
estimateforthecellisnotin(cid:3)uencedbythetransmissivityofothercells. Asmallvalueimpliesthatthe
estimatesoftransmissivityforacellarestronglyin(cid:3)uencedbyothercells. Avaluenearzeroindicatesthe
thetransmissivityestimateforthecellwillnotrelatedtotheactualtransmissivityofthecell.
Figure3plotsthespreadfunctionsforthreeseparatecases;a)usingalltheeigenvectorsofthe138
observations,b)usingthelargest29eigenvectors,andc)usingonlythe13observationsacquiredatthe
lasttimestepofthesimulation. Theplotsarecolorbasedandeachplotisrescaledsothatthelargest
valuesarepurpleandandsmallvaluesarered. Thecolorscaleisdrawntotherightofeachplot. Cells
withvaluesbelowthecutoffonthecolorscalearenotdrawn.
10 10
− 0.992 − 0.568
5 − 0.908 5 − 0.517
ers − 0.815 ers − 0.46
et − 0.722 et − 0.404
e in M 0 −− 00..563269 e in M 0 −− 00..32497
nc − 0.443 nc − 0.233
Dista−5 −− 00..235469 Dista−5 −− 00..11726
− 0.163 − 0.0628
− 0.0699 − 0.00605
−10 −10
−10 −5 0 5 10 −10 −5 0 5 10
Distance in Meters Distance in Meters
(a) (b) (c)
Figure3Spreadfunctionsoftheresolutionmatrix. a)Usingalltheeigenvectors. b)Using
eigenvectorsofthe29largesteigenvaluesc)Usingonly10,000sobservations.
Plot3awouldbeappropriateiftheeigenvalueswereallapproximatelythesamesizeindicatingthatallof
theeigenvectorscouldberesolvedbytheinversionprocess. Theplotshowsvaluesnear1forthecells
immediatelyadjacenttothepumpingwellandrelativelylargevaluesinthevicinityofthewell,indicating
(cid:4).-(cid:6)(cid:5)
MODFLOWandMore2003:UnderstandingthroughModeling-ConferenceProceedings,Poeter,Zheng,Hill&Doherty-www.mines.edu/igwmc/
thatthetransmissivityofadisturbedzonecouldberesolvedatthisgridsize. Awayfromthewellthe
spreadfunctiondiminishesrapidlywithonlya3mdiametercircleshowingastronganticipatedcorrelation
betweentheestimateandtheactuallocaltransmissivity. Thecellsimmediatelyadjacenttotheobservation
wellswouldalsobecorrelatedwiththeirestimates. Ifobservationswereacquiredmoreoften,thenaplot
similartoplot3awouldindicatehigherresolution,anunrealisticconclusion.
XZY (cid:19)(cid:29)[
Plot3bismorerealistic. Onlythe29eigenvectorsassociatedwithnormalizedeigenvalueslargerthan
wereusedtocreatethespreadfunction. Theplotindicatesthatnowherecantransmissivitybe
resolvedatthe25cmscaleofthegrid. Withalowercutoffof0.01,thelargeredareadoesnotindicatea
regionofevenmoderateresolution. Aswiththeeigenvaluecomparisonofthefullobservationsetwithjust
oneobservationateachwell,acomparisonofplots3cto3bindicatesthatthefullsetofobservationsonly
providesasmallimprovementasweconcludedearlierfromFigure2.
Onemightconcludethattransientdataarenotofvalueinestimatingatransmissivitydistribution. Wedo
notbelievethatsuchaconclusionisjusti(cid:2)ed. Investigationsofthesensitivitymatrixindicatethatsensitivity
isdifferentduringtheearlyportionofthedrawdowncomparedtothelateportionandthattheshapeofthe
sensitivity(cid:2)eldchangesrapidlyinthisregion. Thissuggeststhatmoremeasurementsatearlytimesmay
bevaluable. Thecruxofthematterliesinthesignaltonoiseratioofthemeasurements. Alargersignalto
noiseratiocorrelateswithalowereigenvaluecutoff,moreretainedeigenvectorsandmore(cid:2)nely
determinedtransmissivity. Unfortunately,themostimportantdataisrelatedtosmallvaluesofdrawdown.
Theresolution matrixcan beused to de(cid:2)nezoneswhere transmissivity
canbeestimatedwithcon(cid:2)dence. Thesizeofthezonescanbetailored
to optimize resolution uncertainty trade-off. Figure 4 demonstrates the
trade-off in a crude fashion. Zones of 16 grid cells each were de(cid:2)ned
(cid:1)
andthe resolution determined forthiszonation. Inthe centerthe zones
are1m . Attheedgeofthe(cid:2)gure,thezonesarelargerbecauseofthe
grid expansion. Potential resolution near the monitoring wells is larger
than revealed in the (cid:2)gure because the zonation pattern is combining
cells with positive observation sensitivity to transmissivity with cells of
negativesensitivity.
CONCLUSIONS
Wehavepresentedresolutionanalysisasatoolforinvestigatingtheabilityofinversiontoestimatea
distributionoftransmissivity. Theanalysisrequirescalculationofthesensitivityofeachobservationtothe
transmissivityofeachcellinthemodel. Incorporationofanadjointstatesbasedsensitivitycalculationinto
theMODFLOW-2000programsigni(cid:2)cantlyspeedsthedetermination ofthesensitivitymatrix. Byusing
truncatedsingularvaluedecompositionwehaveshownforourhypotheticalexamplethatmultiple
measurementsofdrawdownatasinglelocationhavesimilarinformationcontentswithrespectto
transmissivity. Resolutionanalysiscanprovideavaluabletoolforbothassessinganddesigningan
inversionprocedure.
Figure4Resolutionfor1m
zones ACKNOWLEGMENTS
ThisworkhasbeensupportedthroughInlandNorthwestResearchAlliancegrantBSU003andArmy
ResearchOf(cid:2)cegrantDAAD1900-1-0454. WethankMaryHillforencouragingtheadjointsensitivity
calulations. CGISScontribution124.
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