Analytical Solutions for Unsteady Groundwater Flow in an Unconfined Aquifer under Complex Boundary Conditions

Abstract

The response laws of groundwater dynamics on the riverbank to river level variations are highly dependent on the river level fluctuation process. Analytical solutions are widely used to infer the groundwater flow behavior. In analytical calculations, the river level variation is usually generalized as instantaneous uplift or stepped, and then the analytical solution of the unsteady groundwater flow in the aquifer is derived. However, the river level generally presents a complex, non-linear, continuous change, which is different from the commonly used assumptions in groundwater theoretical calculations. In this article, we propose a piecewise-linear approximation to describe the river level fluctuation. Based on the conceptual model of the riverbank aquifer system, an analytical solution of unsteady groundwater flow in an unconfined aquifer under complex boundary conditions is derived. Taking the Xiluodu Hydropower Station as an example, firstly, the monitoring data of the river level during the period of non-impoundment in the study area are used to predict the groundwater dynamics with piecewise-linear and piecewise-constant step approximations, respectively, and the long-term observation data are used to verify the calculation accuracy for the different mathematical models mentioned above. During the reservoir impoundment period, the piecewise-linear approximation is applied to represent the reservoir water level variation, and to predict the groundwater dynamics of the reservoir bank.

Full text



Water2020,12,75;doi:10.3390/w12010075www.mdpi.com/journal/water
Article
AnalyticalSolutionsforUnsteadyGroundwaterFlow
inanUnconfinedAquiferunderComplexBoundary
Conditions
YawenXin,ZhifangZhou*,MingweiLiandChaoZhuang
SchoolofEarthScienceandEngineering,HohaiUniversity,Nanjing210098,China;
yawenxin@hhu.edu.cn(Y.X.);mingwei@hhu.edu.cn(M.L.);zchao1990@hhu.edu.cn(C.Z.)
*Correspondence:zhouzf@hhu.edu.cn
Received:23October2019;Accepted:21December2019;Published:24December2019
Abstract:Theresponselawsofgroundwaterdynamicsontheriverbanktoriverlevelvariationsare
highlydependentontheriverlevelfluctuationprocess.Analyticalsolutionsarewidelyusedtoinfer
thegroundwaterflowbehavior.Inanalyticalcalculations,theriverlevelvariationisusually
generalizedasinstantaneousupliftorstepped,andthentheanalyticalsolutionoftheunsteady
groundwaterflowintheaquiferisderived.However,theriverlevelgenerallypresentsacomplex,
non‐linear,continuouschange,whichisdifferentfromthecommonlyusedassumptionsin
groundwatertheoreticalcalculations.Inthisarticle,weproposeapiecewise‐linearapproximation
todescribetheriverlevelfluctuation.Basedontheconceptualmodeloftheriverbankaquifer
system,ananalyticalsolutionofunsteadygroundwaterflowinanunconfinedaquiferunder
complexboundaryconditionsisderived.TakingtheXiluoduHydropowerStationasanexample,
firstly,themonitoringdataoftheriverlevelduringtheperiodofnon‐impoundmentinthestudy
areaareusedtopredictthegroundwaterdynamicswithpiecewise‐linearandpiecewise‐constant
stepapproximations,respectively,andthelong‐termobservationdataareusedtoverifythe
calculationaccuracyforthedifferentmathematicalmodelsmentionedabove.Duringthereservoir
impoundmentperiod,thepiecewise‐linearapproximationisappliedtorepresentthereservoir
waterlevelvariation,andtopredictthegroundwaterdynamicsofthereservoirbank.
Keywords:complexboundaryconditions;piecewise‐linearapproximation;piecewise‐constantstep
approximation;analyticalsolutions;unsteadygroundwaterflow

1.Introduction
Theinteractionbetweengroundwaterandsurfacewaterisauniversalphenomenoninthe
naturalworldandanimportantpartoftheterrestrialhydrologicalcycle[1,2].Inthecaseofhydraulic
connectionsbetweensurfaceriversandunconfinedwater,theriverlevelvariationisthekeyfactor
affectingthegroundwaterdynamicsonbothsidesoftheriver[3–5].Whentheriverlevelrises,
surfacewaterinfiltratesintothegroundwater.Thiscanleadtoenvironmentalandgeological
problems[2,4,6–8]suchassolutetransport,groundwaterpollution,reservoirimmersion,andland
salinization.Therefore,understandingsurface–subsurfacewaterinteractionsandbeingableto
computetheeffectoftheseinteractionsareessentialtoaddressenvironmentalandgeological
problems.

Water2020,12,752of14
Analyticalmethodsandnumericalsimulationsareoftenusedforstudyingtheunsteady
groundwaterflowcausedbyriverlevelvariations.Ingeneral,thenumericalsimulationmethod
requirescomplicatedalgorithmstosolvetheproblemofconvergenceandconservationofmassfor
specificmodels[9–11]andtoobtainanapproximatesolution.Theanalyticalsolutionneedstoobey
strictassumptions[12].However,ithasintuitiveandconciseexpression,anditisrelativelyeasyto
obtainunsteadygroundwaterflowlawsandgiveaneffectiveapproachtoevaluatethecalculation
accuracy.Forsimplifiedmathematicalmodels,thecommonlyusedanalyticalsolutionsareusually
basedontheconditionthattheriverlevelremainsconstantaftertheinstantaneouschangehasbeen
completed.Inaddition,theJ.G.Ferrismodel[13]isaclassicalmethodtosolvesuchproblems.
However,theactualchangeoftheriverlevelisdifferentfromtheassumptionsingroundwaterflow
theoreticalcalculations,whicharefrequentlyaffectedbymanyfactors,mainlydependingonthe
topographic,geomorphological,hydrological,andmeteorologicalconditionsofthebasin;the
hydrauliccharacteristicsofthebasinitself;andtheinfluenceofartificialregulation.Meanwhile,the
riverlevelvariationhasitsowncharacteristics:Firstly,thewaterlevelvariesnonlinearlyand
continuouslywithtime;secondly,thewaterlevelpresentsperiodicvariation;thirdly,thewaterlevel
variessharplyinthewetseasonandgentlyinthedryseason.Formathematicalsimplicity,the
piecewise‐constantstepapproximationisthemostcommonlyusedapproachtorepresenttheriver
levelvariation,andthensuchaproblemisanalyzedbythesuperpositionprinciplecombinedwith
thesolutionoftheclassicalmodel[14].However,whentheriverlevelvariationismorecomplex,the
piecewise‐constantstepapproachneedsmoresegmentstodescribethevariationprocess,andthe
calculationismorecumbersome.Moreover,thecommonlyusedapproachtorepresenttheriverlevel
variationmaynotalwaysbesufficienttocaptureimportantdetailsintheobservedhydraulic
transientandcontinuousvariationcharacteristics,whichwillalsoresultingreaterrorstothe
unsteadygroundwaterflowcalculation.Forcaseswheretheriverlevelvariabilityexhibitsincreasing
ordecreasinglineartrends,astep‐changeapproximationisgenerallynotsuitableunlessalarge
numberofcloselyspacedstepchangesareintroduced.Therefore,scientificandreasonable
approximationstoboundaryconditionswilldirectlyaffectthesolutionaccuracy.Forthat,wecanbe
aidedbytheuseofspecialfunctionsthatarenotcommonlyused,orwecanconstructspecial
functionstorepresentboundaryconditions[12,14–17].
Thepiecewise‐linearapproximationapproachiswidelyusedinsolvingnonlinearproblems[18–
22],anditisanidealapproachforrepresentingtime‐varyingfunctions.Songtag[23]considersthat
piecewise‐linearapproximationcansupervisenonlinearproblems;Zhuang[21]derivesananalytical
solutiontoestimatethehydraulicparametersinanaquitardbyapiecewise‐linearapproximationof
long‐termobservedwaterleveldatainadjacentaquifersandanalyzestheaquitardhysteresis
characteristics.Piecewise‐linearapproximationofthepumpingratevariablehasbeenillustratedin
thepetroleumengineeringliterature(Stewart[20]).Phoolendra[19]appliespiecewise‐linear
approximationtorepresentthesinusoidallyvaryingpumpingtest,andtheclosecorrespondence
betweenanalyticallycomputeddrawdownswithandwithoutpiecewise‐linearapproximations
validatestheproposedmethodology.
Inthisarticle,thepiecewise‐linearapproximationisextendedandappliedtothegeneralization
oftheriverlevelvariation.Basedontheconditionthattheriverboundaryisononesideofthemodel
andtheimperviousboundaryisontheothersideandtheflowisconsideredtobeinastablestateat
theinitialtime,theanalyticalsolutionofunsteadyflowintheunconfinedaquiferundercomplex
boundaryconditionsisderived.Itisnoteworthythattheanalyticalsolutionproposedinthisstudy
takesintoaccountthethicknessvariationintheunconfinedaquifer.Atthesametime,theriverlevel
isgeneralizedbythecommonlyusedmethodofpiecewise‐constantstepapproximation,andthe
correspondinganalyticalsolutionisalsoderived.TakingtheXiluoduHydropowerStationasan
example,twomethodologies(piecewise‐linearandpiecewise‐constantstepapproximations)areused
tocalculatewaterlevelvariationsinanunconfinedaquifercausedbytheriverlevelvariation.The
resultsarecomparedwiththoseforthelong‐termobserveddatatoillustratethecalculationaccuracy
ofthemethodologies,whichdemonstratedthatthepiecewise‐linearapproximationhaslessrelative
error.Aftervalidatingourmethodology,weanalyzethereservoirimpoundmentprocessbythe

Water2020,12,753of14
piecewise‐linearapproximationandpredicttheunsteadygroundwaterflowinanunconfinedaquifer
causedbyreservoirimpoundment.
2.TheoryandMethodology
2.1.LinearizationandSolutionoftheGroundwaterFlowModelNeartheRiverbank
Dupuit’sassumptionandBoussinesq’sequationplayimportantrolesinsolvinggroundwater
flowinunconfinedaquifersandarestillwidelyadoptedinhydrogeologicalcalculations.Inorderto
studytheunsteadygroundwaterflowunderriverboundaryconditions,aconceptualmodelof
groundwaterflowneartheriverbankisestablished,asshowninFigure1,withthefollowing
assumptions:(a)theunconfinedaquiferishomogeneousandisotropic,wheretheimpermeablebase
ishorizontalandtheupperinfiltrationcanbeneglected;(b)thegroundwaterflowdivertedbyrivers
canberegardedasone‐dimensional,andtheflowintheunconfinedaquiferfollowsDarcy’slaw;(c)
theinitialhydraulicheadintheunconfinedaquiferisuniformandisconsistentwiththeriverlevel;
(d)theboundaryconditionsoftheestablishedmodelfollowthattheleftsideistheriver’slevel
boundary,andtherightsideistheimperviousboundary,consideringthestudyareaasacertain
distancefromthevalleyratherthanonthewholestratigraphicscale.Thecorresponding
mathematicalmodelcanbeexpressedas:


























)0(
)0(
)0(
)0,0(
0),(
)()0,(),(
)0,(),(
)(
0
0
t
t
lx
tlx
txh
tgxhtxh
xhtxh
t
h
x
h
H
x
K
lxx
x
t

(1)
whereKisthehydraulicconductivity,μisthespecificyield,histhewaterlevel,Histheaquifer
thickness,H=h,xisthehorizontaldistance,tisthetime,andg(t)isthefunctionoftheboundary
waterlevelvariationwithtime.

Figure1.Conceptualmodelofgroundwaterflowneartheriverbank.
Whenh(x,t)−h(x,0)≤0.1hm(dividingthewholecalculationtimetintoseveraltimeunits,hmis
theaquiferaveragethicknessinthetimeunit,whichcanbesatisfiedinmanypracticalproblemsin
anunconfinedaquifer),wecanapplythefirstlinearmethodoftheBoussinesq’sequationtothe
mathematicalmodel,whereuisthewaterlevelvariation,u(x,t)=h(x,t)−h(x,0),aisthehydraulic
diffusivityinunconfinedaquifer,a=Khm/μ,andEquation(1)canbetransformedinto:

Water2020,12,754of14
























)0(
)0(
)0(
)0,0(
0),(
)(),(
0),(
0
0
2
2
t
t
lx
tlx
txu
tgtxu
txu
t
u
x
u
a
lxx
x
t

(2)
thegeneralsolutionofthemathematicalmodelisgivenas:
))
2
12
(exp()0(
)12(
4
)(
))()
2
12
(exp(
)12(
4
)(
2
12
sin)(
2
12
sin)()(),(
2
0
2
11
t
l
n
ag
n
tH
dt
l
n
a
g
n
tG
x
l
n
tHx
l
n
tGtgtxu
n
t
n
n
n
n
n





















 



(3)
2.2.SimpleRepresentationofthePiecewise‐LinearApproximationforRiverLevelBoundary
Theriverlevelvariationhistoryisapproximatedaslinearsegmentscomposedofnstages,
consideringthattheriverlevelhistoryisrecordedasg0,g1,g2,...,gnatdiscretetimeintervalst0,t1,t2,
...,tn,andt0=0.Expressingtheriverlevelvariationasapiecewise‐linearfunctionallowswritingg(t)
as:
)()()()( 1
1
1
11 ii tt
i
j
jjjii HHtttttg 









 





(4)
wheretiisthetimeturningpointatthejunctionbetweentheithandthe(i+1)thlinearstageofthe
riverlevelvariationhistory,andβi=(gi−gi−1)/(ti−ti−1)istheslopeofithlinearriverlevelvariation
element.Ifβi=0,theithstageoftheriverlevelvariationiszero,andifβi=βi+1forallivalues,then
stagesofriverlevelvariationreducetoonestageunderaconstantvariationslope.Htiisaunitstep
function,whichequalsonewhent≥tiandremainszeroelsewhere.
Thepiecewise‐linearapproximationcannotonlyexpressthegeneralfunctions,whichchange
regularlywithtime,suchassinusoidalfunction,exponentialfunction,andsoon,buttheycanalso
approximatetherandomvariationoftheriverlevelwithtime,whichisdifficulttoexpressinthe
generalfunctionform,anditcanbegeneralizedtodifferentdegreesaccordingtotherequired
accuracy.ShownasinFigure2,theboundaryriverlevelvarieswithtimeandpresentsdifferent
sinusoidal,g1(t)=2.0+sin(30t/π),exponential,g2(t)=1.2exp(0.2+0.08t),andrandom,g3(t),
characteristics,approximatedbypiecewise‐linearapproximationto19,4,and20segments,
respectively,whichreflectthevariationcharacteristicsoftheoriginaldataandthekeypoints.The
numberoflinearsegmentsisautomaticallydividedbytheprogramaccordingtotheinputaccuracy
requirements,notmanuallypartitioned.Thehighertheaccuracyrequirement,themorelinear
segmentsaredivided.Thepiecewise‐linearapproximationcanalsobeappliedtothetidalvariation
andvariable‐ratepumpingtests.

Water2020,12,755of14

Figure2.Riverlevelvariationsforsinusoidal(g1),exponential(g2),andrandom(g3)characteristics
withpiecewise‐linearapproximation.
g(t)isacontinuousfunctionwithrespecttot,butitsderivativewithrespecttotisdiscontinuous
attheturningpointti.Inordertoavoidthephenomenaofsharpbreakpointsinthedefinitiondomain
causedbythisproblem,wehavetoderivethewaterlevelvariationdistributionintheunconfined
aquifersequentially,namelymakingthefinalwaterlevelvariationdistributioninthe(i−1)thstage
astheinitialconditionfortheithstage.Usingthevariableseparationapproachtheanalyticalsolution
fortheinitial‐boundaryvalueproblemofEquations(1)to(4)isderivedas:
x
l
n
tt
l
n
a
jitt
l
n
a
na
l
tttttxu
j
j
n
i
j
j
jj
i
j
jii






2
12
sin
)]()
2
12
(exp[
)]()()
2
12
(exp[
)12(
116
)()(),(
1
2
2
1
3
1
3
2
1
1
1
1

































(5)





 )(1
)(0
)( ij
ij
ji


(6)
Whentheriverlevelchanges,theinfiltrationofriverwaterintothebankslopeisregardedasa
processofsaturatedgroundwaterflow,andtheunconfinedaquiferthicknesschangesatanytime.In
ordertogiveconsiderationtothecalculationsimplicityandthevariationofactualunconfinedaquifer
thickness,theunconfinedaquiferthicknessisregardedasaconstantvalueinthecalculatedtimeunit,
andtheinitialunconfinedaquiferthicknessistakenasthedifferencebetweentheriverlevelandthe
impermeablebasebeforethecalculatedtimepoint.Following,theunconfinedaquiferthicknessin
thenexttimeunitisthesumoftheinitialthicknessandtheaveragewaterlevelriseintheprevious
timeunitbasedontheiterativemethods.Therefore,themathematicalmodelproposedinthispaper
takesintoaccounttheunconfinedaquiferthicknessvariationfordiscretetimestepswhencalculating
theunsteadygroundwaterflow.Theaveragewaterlevelrisevalueistheaveragehydraulicresponse
oftheunconfinedaquiferinthehorizontaldirection,whichisachievedbyintegratingu(x,t)in
Equation(5)inthehorizontaldirectionanddividingitbythehorizontallengthL.Theanalytical
solutionoftheaveragehydraulicresponseinthepiecewise‐linearapproximationisasfollows:

Water2020,12,756of14































)]()
2
12
(exp[
)]()()
2
12
(exp[
)12(
132
)()(
),(
)(
1
2
2
1
4
1
4
2
1
1
1
1
0
j
j
n
i
j
jjj
i
j
jii
l
tt
l
n
a
jitt
l
n
a
na
l
tttt
l
dxtxu
tu






(7)
2.3.SimpleRepresentationofthePiecewise‐ConstantStepApproximationforRiverLevelBoundary
Inpreviousstudies,thecontinuousvariationoftheriverlevelisusuallyapproximatedtobea
piecewise‐constantstepvariation,thatis,theriverlevelineachperiodisregardedasafixedvalue,
andtheriverlevelvariationbetweenadjacentperiodsisconsideredtobeinstantaneousreturnwater.
ThevariablevalueoftheriverlevelisdefinedasH0,H1,H2,...,Hnattimeturningpointt0,t1,t2,...,tn,
andt0=0.Then,thepiecewise‐constantstepvariationofriverlevelwithtimecanbeexpressedas:
i
i
i
tttHtg 
1
)( (8)
thecorrespondinganalyticalsolutionofthepiecewise‐constantstepapproximationisgivenas:
)()
2
12
(
1
1
1
1
2
)(
2
12
sin
)12(
4
),( 






 


 j
tt
l
n
a
i
j
jj
n
ieHHx
l
n
n
Htxu




(9)
andtheanalyticalsolutionoftheaveragehydraulicresponseinthepiecewise‐constantstep
approximationisasfollows:
)()
2
12
(
1
1
1
22
01
2
)(
)12(
8
),(
)( 









 j
tt
l
n
a
i
j
jj
n
i
l
eHH
n
H
l
dxtxu
tu



(10)
Inthiscase,thehydraulicdiffusivityaintheu(x,t)isvariable,thatis,theunconfinedaquifer
thicknessisavariable,whichisdeterminedbytheinitialthicknessandtheaveragehydraulic
responsevalue.
Asshownfromtheu(x,t)expressions,theseriestermexistsbothinpiecewise‐linearor
piecewise‐constantstepapproximationsforriverlevelvariation.Theseriestermindicatesthatwhen
theboundarywaterlevelchanges,thevariationrangeofthewaterlevelateachpointinan
unconfinedaquiferisnotequaltotheboundarywaterlevelvariation,namelydamping.
Theoretically,thespecificyieldvalueofunconfinedaquiferisusually3–4ordersofmagnitudelarger
thanthespecificstoragevalueoftheconfinedaquifer,sothegroundwaterflowintheunconfined
aquiferisdominatedbygravityconduction.Differingfromthepressureconductioninconfined
aquifers,thewaterreleasecausedbytheriverlevelriseinanunconfinedaquiferisaprocessthatis
notinstantaneous,which,dependingontheprogressionofthesaturatedzoneandleadingtothe
hydraulicresponse,isrelativelyslow.Whentheboundarywaterlevelrises,thewaterlevelateach
pointintheunconfinedaquifermaybelowerthanthatinaperiodoftime.Thegroundwaterlevel
reactswithdelaytowardstheriverlevelvariation,thatistosay,thereisatimelag.Inaddition,the
phenomenonismoreobvious,especiallywhentheboundarywaterlevelchangessharply.
Consideringthattheimpermeableboundaryisfarfromtheriverbank,thelimitvalueoftherisein
groundwaterlevelintheunconfinedaquiferwillnotexceedthemaximumvariationoftheriverlevel,
whichiscompletelyconsistentwiththeresultsoftheanalyticalsolution.


Water2020,12,757of14
3.FieldApplication
3.1.Background
TaketheXiluoduHydropowerStationasanexample.TheXiluoduHydropowerStationis
locatedonthemainstreamofJinshaRiverinLeiboCountyofSichuanProvinceandYongshan
CountyofYunnanProvince.ThelocationofthestudyareaisshowninFigure3.Thereservoirhasa
normalwaterlevelof600m.TheXiluoduHydropowerStationcooperateswiththeThreeGorges
ProjecttoimprovethefloodcontrolcapacityofthemiddleandlowerreachesoftheYangtzeRiver,
whichgivefullplaytothecomprehensivebenefitsoftheThreeGorgesProject.Besides,itpromotes
thedevelopmentofthewesternregionandrealizesthesustainabledevelopmentofthenational
economy.Theriverbedrockandthevalleyslopesonbothsidesofthedamsitearecomposedof
Emeishanbasalt(P
2
β)oftheUpperPermian,andtheMaokoulimestone(P
1
m)oftheLowerPermian
isburied70mbelowthedamfoundation.Theresearchsuggeststhatthepatternsofvalleyand
foundationdeformationcausedbyreservoirimpoundmentprocessesarecloselyrelatedtothe
groundwaterflowintheaquifers.Therefore,itisofgreatsignificancetocalculatetheinfluenceof
reservoirimpoundmentongroundwaterlevel[24].

Figure3.Mapshowingthegeographiclocationofthestudyarea.
3.2.RepresentationofRiverLevel
Intheanalyticalsolution,itisnecessarytogeneralizetheriverlevelindifferentformsinorder
tocalculatetheunsteadyflowintheunconfinedaquifercausedbytheriverlevelvariationand
comparethepiecewise‐linearmodelwiththetraditionalpiecewise‐constantstepmodelproposedin
thisarticle.ConsiderthewaterlevelmonitoringdataofJinshaRiverin1995asoriginaldatatofit.As
showninFigure4,theriverlevelvariationinahydrologicalyearpresentsseasonalvariation.The
riseofriverlevelisconcentratedinJune–September,duringwhichthereisobviousfluctuationsof
waterlevellikelyrelatedtoseasonalrainfall.Onthewhole,thewaterlevelvariationiscomplex,and
itisdifficulttoexpressitschangewithsimplefunctions.Theriverlevelisdividedinto7and17
segmentsbypiecewise‐linearandpiecewise‐constantstepapproximations.Themoresegmentsare
divided,theclosertheapproximatevaluesaretotheoriginaldata.Thetimesegmentsare
automaticallydividedbythewrittenprogramaccordingtotheinputaccuracyrequirements,not
manuallypartitioned.Thepiecewise‐linearapproximationproposedinthispaperisusedto
generalizetheriverlevel,whichcannotonlyreflectthewaterlevelchangetrendsbutalsoshowthe
characteristicwaterlevelandcorrespondingtimepoints.However,thetraditionalpiecewise‐
constantstepapproximationcanonlyrepresenttheoverallchangecharacteristicsandignoresthe
stagechangecharacteristics.Thisdemonstratesthat,forthecaseswhereriverlevelvariationsare
betterrepresentedbypiecewise‐linearapproximation,asriverlevelvariationsmaynotbeadequately

Water2020,12,758of14
expressedbyarelativelysmallnumberofstepsegments,thepiecewise‐constantstepapproximation
forrepresentingriverleveltransientsisnotalwayssufficienttoaccuratelyrepresenttheobserved
data.

 
Figure4.Riverheadvariationin1995isdividedinto7segments(a)and17segments(d),respectively
withpiecewise‐linearandpiecewise‐constantstepapproximations;iisthesegments;andcalculated
resultsofgroundwater‐levelvariationforlong‐termobservationwellsX35(b,e)andX50(c,f)with
piecewise‐linearandpiecewise‐constantstepapproximationsfordifferentsegments.
3.3.ModelCalculation
Inordertostudytheunsteadyflowoftheunconfinedaquifercausedbytheriverlevelvariation,
itisnecessarytoanalyzeanddeterminethehydrogeologicalparametersofthebasaltinthebank
slope.Reasonablevaluesofthehydrogeologicalparametersareveryimportantforthereliable
analysisofthecalculationresults.Thehydraulicconductivityofthebasaltareameasuredbyfield
testsoftheXiluoduHydropowerStationrangedfrom0.06m/dto2.70m/d,inwhichthehydraulic
conductivityandtheporosityofrockmassweresmall.Theaveragehydraulicconductivityand
porosity(K=0.90m/d,n=0.03)afterexcavationwereselectedtocalculateandspecificyieldwas
approximatedasporosity.TheinitialthicknessoftheunconfinedaquiferwasH=153m,andthe
horizontalextensiondistanceofthestudyareawas7.9km.
At85.7mand371.4mawayfromtherivervalley,thegroundwaterlevelattheselectedlocation
wascalculatedbypiecewise‐linearandpiecewise‐constantstepmodels,respectively.Atthesame
time,thedynamicobserveddataofthewaterlevelatthecorrespondinglong‐termobservationwells
X35andX50wereusedtoverifytheaccuracyofthetwomodels.Thecalculatedresultsareshownin
Figure4.Itisobviousfromthefittingdegreeofthecalculatedresultsandthelong‐termobservation
datathatthegroundwaterlevelcalculatedbythepiecewise‐linearmodelwasclosertothedynamic
observeddataofthelong‐termobservationwell.Inordertoquantifythecalculationaccuracyofthe
twomodels,theRMSE(rootmeansquareerror)andRE(relativeerror)wereusedtorepresentthe
deviationbetweenthecalculatedandtheobservedvalues,whichcouldbeexpressedwithEquations
(11)and(12)andthecalculatedresultsasshowninTable1.Theerroranalysisshowedthatthemore
segmentsweredivided,thehigherthecalculationaccuracywas,andthecalculationaccuracyofthe
piecewise‐linearmodelwashigherthanthatofthepiecewise‐constantstepmodel.

Water2020,12,759of14



n
i
imic HH
n
RMSE
1
2
,, )(
1(11)
%100
1
,
,,
1



im
imic
n
iH
HH
n
RE (12)
whereHc,idenotesthecalculationvalueofgroundwaterlevelattheithtimepoint,Hm,idenotesthe
correspondingmonitordata,andnisthetotalnumbermonitored.
Table1.Root‐Mean‐SquareError(RMSE)andRelative‐Error(RE)betweenobservedandcalculated
datawithPiecewise‐Linear(Columns3and4)andPiecewise‐ConstantStep(Columns5and6).
LinearApproximationStepApproximation
RMSERERMSERE
i=7X351.740.00332.330.0046
X502.350.00472.630.0053
i=17X351.500.00281.810.0036
X501.820.00362.520.0049
Ingeneral,thegroundwaterdynamicinthelong‐termobservationwellwasbasicallyconsistent
withtheriverleveldynamic,whichrepresentsgoodsynchronization.Themainreasonisthatthe
distancebetweentheselectedlong‐termobservationwellandtherivervalleywasrelativelyshort
comparedwiththewholestudyarea,andthesignaldampingphenomenonwasnotobvious.
However,thelong‐termobservationwellsX35andX50,whicharelocatedclosetoandfarfromthe
rivervalley,respectively,performeddifferentresponsedegreesofgroundwaterleveltoriverlevel
variation.Comparedwiththeriverlevelvaluesandthegroundwaterlevelobserveddata,withthe
distanceincreasing,theamplitudeofgroundwaterlevelvariationdecreased,andthegroundwater
levelvariationdelayphenomenonintheX50wasmorenotablethanthatintheX35,whichisrelated
totheflowresistanceoftherock.Fromthecalculationresults,theerrorsofX50weregreaterthan
thoseofX35,whichmaybeduetotheselectionofhydraulicparameters.Fortheconvenienceof
modelcalculation,wechosethesamehydraulicdiffusivityinanunconfinedaquifertocalculatethe
groundwaterdynamicatdifferentlocationsfromthevalley.However,thehydraulicdiffusivityinan
unconfinedaquiferneararivervalleyisgenerallylargerthanthatfarfromvalleys,whichisrelated
tothegoodpermeabilityofrockmassesandthefastdischargeofgroundwaterneartherivervalley.
Therefore,thesamehydraulicdiffusivityintheunconfinedaquiferselectedmayaffecttheaccuracy
ofgroundwaterlevelcalculationsatdifferentlocations.
3.4.PredictionofGroundwaterLevelFluctuationCausedbyReservoirWaterLevelVariation
Fromthemodelcalculationresultsandgroundwaterdynamicdatainthelong‐termobservation
well,thepiecewise‐linearmodelhasahighreliabilityincalculatinggroundwaterflowcausedbythe
boundarywaterlevelvariation.Thisexamplehasshownthattheanalyticalmodelcanbeusedto
representtheflowwithsufficientaccuracy.TheXiluoduHydropowerStationhasbeenimpounded
since30October2012.Theinitialriverlevelwas383.45m.InNovember2014,theimpoundmenthad
reachedthenormalwaterlevelof600m,andintheobserveddataofthereservoirwaterleveluntil
26February2018,thewaterlevelfluctuatedperiodically.Takingtheinitialtimeof30October2012
andtheendtimeof26February2018for1943days,thereservoirwaterlevelvariationwas
generalizedbyapiecewise‐linearapproximation,whichwasdividedinto43continuouslinear
segments,asshowninFigure5.

Water2020,12,7510of14

Figure5.Reservoirwaterleveldurationcurverepresentedwithpiecewise‐linearapproximation.
Weanalyzedtheunsteadyflowofunconfinedaquifercausedbythereservoirwaterlevel
variationaccordingtothepiecewise‐linearapproximationproposedinthispaper.Fourselectedtime
nodeswere9February2013,16March2014,29July2015,and14February2018correspondingto100,
500,1000,and1900d(basedontheinitialtimeofimpoundment)toanalyzethedistributionofthe
waterlevelateachpointinthestudyarea.Atthesametime,groundwaterlevelvariationswithtime
wereconsideredat0.2,0.5,1.0,2.0,3.0,5.0,and7.9km(thewaterbarrierboundary)awayfromthe
reservoirbank,asshowninFigure6.Arisingreservoirwaterlevelwillrechargetheunconfined
aquiferandcausealong‐termandwide‐rangingimpact.Theexplanationofthelong‐term
characteristicisthatbasalthasgoodintegrity,alargethickness,acompactstructure,low
permeability,andmostaredryundernaturalconditions.Inaddition,thehydraulicresponseof
groundwaterintheunconfinedaquifermainlydependsongravity,supplementedbypressure
transfer.Whenriverwaterrechargesunconfinedwaterinbasaltstrata,thedevelopedjoints,
interlayer,andinternalshearzonesaresaturatedfirstly.Theprocessofmountainsaturationis
extremelyslow,andthecorrespondingwaterlevelriseisslow,sotheriverwaterwillrechargethe
unconfinedaquiferforalongtime.Thewiderangewasduetothebroadhorizontalextensionof
basaltstratumandthelow,gentle,naturalgroundwaterlevelofunconfinedwater.Whatismore,
afterreservoirimpoundment,therewasalargedistributionarea,wheretheunconfinedsurface
elevationofthemountainbodywassmallerthanthereservoir’snormalwaterlevel.
Atthesametime,thewaterleveldecreasedgraduallyasitextendedfromthereservoirbankto
thesideawayfromthat,showingobviousdampingofgravitywaterrelease.Theunconfinedwater
surfaceadvancedcontinuouslyastimewenton.Whenthetimewas100d,thetransferdistanceof
groundwaterlevelfluctuationcausedbythereservoirwaterlevelvariationwas4200m,whichis
shownintheApointcoordinates(4200,383.45);whenthetimewas500d,ithadbeentransferredto
theimperviousboundary,wherethewaterlevelwas383.56m;whenthetimewas1000d,thewater
levelattheimperviousboundarywas390.33m;whenthetimewas1900d,thewaterlevelatthe
imperviousboundaryis434.25m.Theanalyzeddatashowthatthewaterlevelrisesslowlyandnever
equalsthereservoirwaterlevel,whichindicatesthattheunconfinedaquiferresponsetothereservoir
waterlevelvariationhasobviousdampingcharacteristics.Attime0d,thegroundwaterlevelisat
theinitialtime.Accordingtotheexistingsurveydata,attheinitialtime,thefluctuationrangeofthe
groundwaterlevelinthewholestudyareawasverysmall,only0.5m,sothegroundwaterlevelat
thistimewasapproximatelyaplane,withavalueof383.45m.
Comparingthetime‐varyingcurvesofgroundwaterlevelsat0.2,0.5,1.0,2.0,3.0,5.0,and7.9km
awayfromthereservoirbank,thegroundwaterlevelatthesamelocationshowedanoverallupward
trendwithtime.Zerokilometerindicatesthereservoirwaterlevelvariation.At0.2,0.5,and1.0km,
thegroundwaterlevelfluctuatedobviouslywithtime,showingagoodsynchronizationwiththe
reservoirwaterlevelvariations,althoughtheincreasewaslessthanthat.At2.0km,therewasslight
groundwaterlevelfluctuation.Withthedistanceincreasing,thefluctuationofthegroundwaterlevel
nearlydisappears,reducedgraduallyat3,5and7.9km,showingonlyanincreaseofthewaterlevel.

Water2020,12,7511of14
Inaddition,whenthereservoirwaterlevelchangedperiodically,thedampingofthehydraulic
responseintheunconfinedaquiferwasalsoreflectedinthephasedifferencebetweenthe
groundwaterlevelandthereservoirwaterlevel.Whenthereservoirwaterlevelchangedfrom
ascendingtodescending(descendingtoascending),thegroundwaterlevelatacertainpositionaway
fromthebankslopekeptascending(descending)foraperiodoftime,andthen,itbegantodescend
(ascend),whichindicatestheexistenceofacertainphasedifference.Thefartherawayfromthebank
slope,thebiggerthephasedifferenceis,anditcannotevenreflectthefluctuationsofthereservoir
waterlevel.Itcanbeseenthatthegroundwaterlevelchangedgreatlynearthebankslope,andthe
hydraulicdampingeffectwassmaller,whichcanreflectthechangecharacteristicsofthereservoir
waterlevel.However,thevariationrangeofgroundwaterlevelwassmallerfarfromthebankslope,
thehydraulicdampingeffecthadgreaterinfluence,andthechangecharacteristicsofthereservoir
waterlevelattenuatedaccordingly.Itisworthtopointoutthatthissystembehavior(phasedifference,
dampedfluctuation)wasinlinewithotherstudies[25].Sincethemodelisassumedtobean
imperviousboundary,thelimitvalueofthegroundwaterlevelvariationateachpointinthe
unconfinedaquiferwillnotexceedthemaximumrisevalueofthereservoirwaterlevel.Thesteady
stateflowofgroundwaterwasnotbeingachievedafter2000days.

Figure6.Groundwater‐levelfluctuationcurvesatdifferentpointsinstudyarea(a)andgroundwater‐
leveldurationcurves(b)intheunconfinedaquifer.
4.ResultsandDiscussions
Theanalyticalsolutionsofgroundwaterlevelversustimeanddistancearedevelopedto
investigatetheimpactsoftheboundarywaterlevelapproximation,modelsaccuracy,anddamping
effectsinanunconfinedaquifer.Themathematicalmodelassumesthatrainfall,aquiferrecharge,
extractions,andbaseflowdischargearezero,whichmaybedifferentfromtheactualsituation.This
papermainlydiscussestheinfluenceofboundarywaterlevelongroundwaterflow.Inorderto
simplifytheanalyticalsolution,otherrechargeanddischargevaluesareconsideredaszero.
4.1.EffectofBoundaryWaterLevelApproximation
Analyticalsolutionsarederivedtostudythegroundwaterflowinthisarticle.Theanalytical
solutions,differentfromnumericalsimulation,andtheboundaryconditionscouldnotbe
representedasdiscretedatapoints.However,theoriginaltimeresolutionoftheriverlevelversus
timeseriescouldnotberepresentedwithanormalfunction.Therefore,weappliedthepiecewise‐
linearapproximationtogeneralizetheriverlevelvariation,soitcouldbeshownwithanexpression.
Linearsegments(7and17)weredividedintheriverlevel,asshowninFigure4a,b,andtheresults
indicatedthatthehigherthetimeresolution,thesmallerthedeviationbetweenthecalculatedvalues
andtheoriginaldata.


Water2020,12,7512of14
4.2.EffectofModelsAccuracy
Figure4displaysthecurvesofgroundwaterlevelversustimeforlong‐termobservationwells
X35andX50withpiecewise‐linearandpiecewise‐constantstepapproximations.Table1showsthe
erroranalysiswithdifferentapproximations.Bothdemonstratethattheaccuracyofthepiecewise‐
linearapproximationwashigherthanthatofthepiecewise‐constantstepwiththesametime
resolution.
4.3.EffectofDampingintheUnconfinedAquifer
Figure6presentsthegroundwaterresponselagsbehindthereservoirwaterlevelvariation.The
furtherawayfromthereservoirbank,themoreobviousthetimedelay.Itisobviousthatthefilling
ofthereservoiroverridesthissurfacesubsurfacewaterinteractioneffectinthefirst700days.In
addition,thegroundwaterlevelhasnotachievedasteadystate,evenafter2000days,andthe
maximumwaterlevelat7.9kmwasfarbelowthereservoirlevelinthesimulationperiod.Ifthe
reservoirwaterlevelstillchangesperiodicallyandthesimulationtimeisextended,thewaterlevelat
theimperviousboundarycouldreachtothereservoir’simpoundmentlevelatabout6000daysfrom
theinitialtime,whichisabouttheyear2030.
5.Conclusions
Theriverlevelusuallypresentsanon‐linearcontinuouschangeovertime,whichisdifferent
fromtheinstantaneousriseorpiecewise‐constantstepvariationofthewaterlevel.Accordingtothe
riverlevelvariationcharacteristics,apiecewise‐linearapproximationisproposedtogeneralizethe
riverlevelvariation.Basedontheactualsituation,theothersideofthestudyareaissetasan
imperviousboundary,andtheanalyticalsolutionofgroundwaterunsteadyflowcausedbytheriver
levelvariationisderived.
Resultswerecomparedwiththetraditionalpiecewise‐constantstepmodeltoillustratethe
calculationaccuracy.TakingtheXiluoduHydropowerStationasanexampletovalidatethemethod
proposedinthispaper,thepiecewise‐linearapproximationvaluestotheboundarywaterlevelare
closertotheactualwaterlevelvariationprocess,whichcanreflectthevariationcharacteristics.The
calculationaccuracyofthepiecewise‐linearmodelishigherthanthatofthepiecewise‐constantstep
modelthroughtheverificationofgroundwaterdynamicmonitordatainthelong‐termobservation
well.Theresponseofthegroundwaterleveldynamicinanunconfinedaquifertothereservoirwater
levelvariationhasobviousdamping,whichdemonstratesthatthefartherawayfromthebankslope,
themoreobviousthetimedelay.Thepiecewise‐linearapproximationofthetime‐varyingboundary
waterlevelcanalsobeappliedtopredictthehydraulicresponseintheconfinedaquifersand
aquitards.Itshouldbemadeclearthatthisexerciseisonlyrelevantfortheanalyticalsolutions.
Numericalmodelscanonlyusetheboundaryconditiontimeseriesontheresolutionofthemodel.
Basedonthegroundwaterlevelvariationobtained,wewillestimatethesurface‐subsurface
waterexchangeandstudytheinfluenceofrainfall,aquiferrecharge,extractions,andbaseflow
dischargeonthegroundwaterlevelandquantityinfurtherwork.
AuthorContributions:Conceptualization,Y.X.andZ.Z.;methodology,Y.X.;software,Y.X.,M.L.andC.Z.;
validation,Z.Z.,M.L.andC.Z.;formalanalysis,Y.X.;investigation,Y.X.;resources,Z.Z.;datacuration,Y.X.;
writing—originaldraftpreparation,Y.X.;writing—reviewandediting,Y.X.andZ.Z.;visualization,Y.X.;
supervision,Z.Z.Allauthorshavereadandagreedtothepublishedversionofthemanuscript.
Funding:ThisresearchwasfundedbytheNationalNaturalScienceFoundationofChina,grantnumberNo.
41572209,andChinaThreeGorgesCorporation,grantnumberNo.20188019316.
ConflictsofInterest:Theauthorsdeclarenoconflictsofinterest.


Water2020,12,7513of14
References
1. Lischeid,G.Driversofwaterlevelfluctuationsandhydrologicalexchangebetweengroundwaterand
surfacewateratthelowlandRiverSpree(Germany):Fieldstudyandstatistical.Hydrol.Processes2009,23,
2117–2128.
2. Mclachlan,P.J.;Chambers,J.E.;Uhlemann,S.S.;Binley,A.Geophysicalcharacterisationofthe
groundwater‐surfacewaterinterface.Adv.WaterResour.2017,109,302–319.
3. Chen,J.W.;Hsieh,H.H.;Yeh,H.F.;Lee,C.H.Theeffectofthevariationofriverwaterlevelsonthe
estimationofgroundwaterrechargeintheHsinhuweiRiver,Taiwan.Environ.EarthSci.2010,59,1297–
1307.
4. Zhou,Z.;Wang,J.GroundwaterDynamics;SciencePress:Beijing,China,2013.
5. Rushton,K.Representationinregionalmodelsofsaturatedriver‐aquiferinteractionforgaining/losing
rivers.J.Hydrol.2007,334,262–281.
6. Goerlitz,D.F.;Troutman,D.E.;Godsy,E.M.;Franks,B.J.Migrationofwood‐preservingchemicalsin
contaminatedgroundwaterinasandaquiferatPensacoia,Florida.Environ.Sci.Technol.1985,19,955–961.
7. Zhao,M.L.Finiteelementnumericalsimulationoftwo‐dimensionalgroundwatersolutemigration
question.Adv.Mater.Res.2011,301–303,352–356.
8. Luo,Z.;Yang,L.;Li,Z.;Wang,J.Three‐dimensionalnumericalsimulationforimmersionpredictionofleft
bankofSongyuanreservoirproject.Trans.Chin.Soc.Agric.Eng.2012,28,129–134.
9. Celia,M.A.;Bouloutas,E.T.;Zarba,R.L.Ageneralmass‐conservativenumericalsolutionforthe
unsaturatedflowequation.WaterResour.Res.1990,26,1483–1496.
10. Li,M.;Zhou,Z.;Yang,Y.;Xin,Y.;Liu,Y.;Guo,S.;Zhu,Y.;Jin,D.Immersionassessmentandcontrolofthe
rightbankofXingannavigationandpowerjunction.J.HohaiUniv.Sci.2018,46,203–210.
11. Pinder,G.F.;Sauer,S.P.Numericalsimulationoffloodwavemodificationduetobankstorageeffects.Water
Resour.Res.1971,7,63–70.
12. Yuan,F.;Lu,Z.Analyticalsolutionsforverticalflowinunsaturated,rootedsoilswithvariablesurface
fluxes.VadoseZoneJ.2005,4,1210–1218.
13. Ferris,J.G.;Knowles,D.B.;Brown,R.H.;Stallman,R.W.TheoryofAquiferTests;USGeologicalSurvey:
Denver,CO,USA,1962.
14. Mahdavi,A.Transient‐stateanalyticalsolutionforgroundwaterrechargeinanisotropicslopingaquifer.
WaterResour.Manag.2015,29,3735–3748.
15. Su,N.ThefractionalBoussinesqequationofgroundwaterflowanditsapplications.J.Hydrol.2017,547,
403–412.
16. Wu,D.;Tao,Y.;Lin,F.Applicationofunsteadyphreaticflowmodelanditssolutionundertheboundary
controlofcomplicatedfunction.J.Hydraul.Eng.2018,6,725–731.
17. Zhang,W.CalculationofUnsteadyGroundwaterFlowandEvaluationofGroundwaterResources;SciencePress:
Beijing,China,1983.
18. Bansal,R.K.Approximationofsurface–groundwaterinteractionmediatedbyverticalstreambankin
slopingterrains.J.OceanEng.Sci.2017,2,18–27.
19. Mishra,P.K.;Vessilinov,V.;Gupta,H.Onsimulationandanalysisofvariable‐ratepumpingtests.
Groundwater2013,51,469–473.
20. Stewart,G.;Watt,H.U.;Wittmann,M.J.;Meunier,D.Afterflowmeasurementanddeconvolutioninwell
testanalysis.InProceedingsoftheSPEAnnualTechnicalConferenceandExhibition,SanFrancisco,CA,
USA,5–8October,1983.
21. Zhuang,C.;Zhou,Z.;Zhan,H.;Wang,G.Anewtypecurvemethodforestimatingaquitardhydraulic
parametersinamulti‐layeredaquifersystem.J.Hydrol.2015,527,212–220.
22. Lee,J.;Wilson,D.Polyhedralmethodsforpiecewise‐linearfunctionsI:Thelambdamethod.Discret.Appl.
Math.2001,108,269–285.
23. Sontag,E.D.NonlinearRegulation:ThePiecewiseLinearApproach.IEEETrans.Automat.Contr.1981,26,
346–358.


Water2020,12,7514of14
24. Zhou,Z.;Li,M.;Zhuang,C.;Guo,Q.Impactfactorsandformingconditionsofvalleydeformationof
XiluoduHydropowerStation.J.HohaiUniv.Sci.2018,46,31–39.
25. Becker,B.;Jansen,M.;Sinaba,B.;Schüttrumpf,H.Onthemodelingofbankstorageinagroundwater
model:TheApril,1983,floodeventintheNeuwiederBecken(MiddleRhine).J.Water2015,7,1173–1201.

©2019bytheauthors.LicenseeMDPI,Basel,Switzerland.Thisarticleisanopenaccess
articledistributedunderthetermsandconditionsoftheCreativeCommonsAttribution
(CCBY)license(http://creativecommons.org/licenses/by/4.0/).
