半无限含水层中圆形排水隧道稳定Word文档格式.docx
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半无限含水层中圆形排水隧道稳定Word文档格式.docx
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Thisstudydealswiththecomparisonofexistinganalyticalsolutionsforthesteady-stategroundwaterinflowintoadrainedcirculartunnelinasemi-infiniteaquifer.Twodifferentboundaryconditions(oneforzerowaterpressureandtheotherforaconstanttotalhead)alongthetunnelcircumference,usedintheexistingsolutions,arementioned.Simpleclosed-formanalyticalsolutionsarere-derivedwithinacommontheoreticalframeworkfortwodifferentboundaryconditionsbyusingtheconformalmappingtechnique.Thewaterinflowpredictionsarecomparedtoinvestigatethedifferenceamongthesolutions.Thecorrectuseoftheboundaryconditionalongthetunnelcircumferenceinashallowdrainedcirculartunnelisemphasized.Ó
2007ElsevierLtd.Allrightsreserved.
Keywords:
Analyticalsolution;
Tunnels;
Groundwaterflow;
Semi-infiniteaquifer
1.Introduction
Predictionofthegroundwaterinflowintoatunnelisneededforthedesignofthetunneldrainagesystemandtheestimationoftheenvironmentalimpactofdrainage.Recently,ElTani(2003)presentedtheanalyticalsolutionofthegroundwaterinflowbasedonMobiustransformationandFourierseries.Bycompilingtheexactandapproximatesolutionsbymanyresearchers(Muscat,Goodmanetal.,Karlsrud,Rat,Schleiss,Lei,andLombardi),ElTani(2003)showedthebigdifferenceinthepredictionofgroundwaterinflowbythesolutions.KolymbasandWagner(2007)alsopresentedtheanalyticalsolutionforthegroundwaterinflow,whichisequallyvalidfordeepandshallowtunnelsandallowsvariabletotalheadatthetunnelcircumferenceandatthegroundsurface.
Whileseveralanalyticalsolutionsforthegroundwaterinflowintoacirculartunnelcanbefoundintheliterature,theycannotbeeasilycomparedwitheachotherbecauseoftheuseofdifferentnotations,assumptionsofboundaryconditions,elevationreferencedatum,andsolutionmethods.
Inthisstudy,weshallrevisittheclosed-formanalyticalsolutionforthesteady-stategroundwaterinflowintoadrainedcirculartunnelinasemi-infiniteaquiferwithfocusontwodifferentboundaryconditions(oneforzerowaterpressureandtheotherforaconstanttotalhead)alongthetunnelcircumference,usedintheexistingsolutions.Thesolutionsfortwodifferentboundaryconditionsarere-derivedwithinacommontheoreticalframeworkbyusingtheconformalmappingtechnique.Thedifferenceinthewaterinflowpredictionsamongtheapproximateandexactsolutionsisre-comparedtoshowtherangeofappli-cabilityofapproximatesolutions.
2.Definitionoftheproblem
Consideracirculartunnelofradiusrinafullysaturated,homogeneous,isotropic,andsemi-infiniteporousaquiferwithahorizontalwatertable(Fig.1).Thesurroundinggroundhastheisotropicpermeabilitykandasteady-stategroundwaterflowconditionisassumed.
Fig.1.Circulartunnelinasemi-infiniteaquifer.
AccordingtoDarcy’slawandmassconservation,thetwo-dimensionalsteady-stategroundwaterflowaroundthetunnelisdescribedbythefollowingLaplaceequation:
(1)
where
=totalhead(orhydraulichead),beinggivenbythesumofthepressureandelevationheads,or
(2)
p=pressure,
=unitweightofwater,Z=elevationhead,whichistheverticaldistanceofagivenpointaboveorbelowadatumplane.Here,thegroundsurfaceisusedastheelevationreferencedatumtoconsiderthecaseinwhichthewatertableisabovethegroundsurface.NotethatE1Tani(2003)usedthewaterlevelastheelevationreferencedatum,whereasKolymbasandWagner(2007)usedthegroundsurface.
InordertosolveEq.
(1),twoboundaryconditionsareneeded:
oneatthegroundsurfaceandtheotheralongthetunnelcircumference.Theboundaryconditionatthegroundsurface(y=0)isclearlyexpressedas
(3)
Inthecaseofadrainedtunnel,however,twodifferentboundaryconditionsalongthetunnelcircumferencecanbefoundintheliterature:
(Fig.1)
(1)Case1:
zerowaterpressure,andsototalhead=elevationhead(ElTani,2003)
(4)
(2)Case2:
constanttotalhead,ha(Lei,1999;
KolymbasandWagner,2007)
(5)
ItshouldbenotedthattheboundaryconditionofEq.(5)assumesaconstanttotalhead,whereasEq.(4)givesvaryingtotalheadalongthetunnelcircumference.Byconsideringthesetwodifferentboundaryconditionsalongthetunnelcircumference,twodifferentsolutionsforthesteady-stategroundwaterrinflowintoadrainedcirculartunnelarere-derivedinthenext.
3.Analyticalsolutions
3.1.Conformalmapping
Thegroundsurfaceandthetunnelcircumferenceinthez-planecanbemappedconformallyontotwocirclesofradius1andα,inthetransformedζ-planebytheanalyticfunction(Fig.2)(VerruijtandBooker,2000)
(6)
whereA=h(1-α2)/(1+α2),histhetunneldepthandαisaparameterdefinedas
or
(7)
Then,Eq.
(1)canberewrittenintermsofcoordinateξ-η
(8)
Byconsideringtheboundaryconditions,thesolutionforthetotalheadonacirclewithradiusρintheζ-planecanbeexpressedas
(9)
whereC1,C2,C3andC4areconstantstobedeterminedfromtheboundaryconditionsatthegroundsurfaceandalongthetunnelcircumference.
3.2.Thesurfaceboundarycondition
TheconstantC1canbeobtainedbyconsideringtheboundaryconditionatthegroundsurfacewithρ=1intheζ-plane,
(10)
Fig.2.Planeofconformalmapping.
3.3.Thetunnelboundarycondition
Theotherconstantscanbeobtainedbyconsideringtwodifferenttunnelboundaryconditions.
zerowaterpressure.
Byconsideringζ=aexp(ίθ)intheζ-plane,theelevationheadaroundthetunnelcircumferencecanbeexpressedas
(11a)
Orintheseriesform(Verruijt,1996)
(11b)
AndthenapplyingtheboundaryconditionofEq.(4)gives
(12)
So,
(13)
NotethatEq.(13)isthesameformasEq.(4.1)inElTani(2003)forthecaseofH=0.
(2)Case2:
constanttotalhead,ha.
ApplyingtheboundaryconditionofEq.(5)gives
(14)
(15)
3.4.Thesolutionforgroundwaterinflow
Thesolutionforthegroundwaterinflow,whichisthevolumeofwaterperunittunnellength,intoadrainedcirculartunnelcanbeobtainedfortwodifferentcasesas
(16)
(17)
NotethatEq.(16)isthesamesolutionasElTani(2003)withH=0,whereasEq.(17)isthesamesolutionasKolymbasandWagner(2007).ThereisacleardifferencebetweenEqs.(16)and(17):
A(=h(1-α2)/(1+α2))inEq.(16)andhainEq.(17)duetothedifferentboundaryconditionsalongthetunnelcircumference.
Itisalsonotedthatthesolutions(16)and(17)areusedforthecaseinwhichthewatertableisabovethegroundsurface.Ifthegroundwatertableisbelowthegroundsurface,thegroundwaterlevelisusedastheelevationreferencedatum.Thesolutions(16)and(17)shouldbeusedwithH=0andh=thegroundwaterdepth(nottunneldepth).
4.Comparisonwithapproximatesolutions
FromtheexactsolutionEq.(17),thepreviousapproximatesolutionscanbeobtainedwiththeassumptionthatthetotalheadeverywhereatthetunnelcircumferenceisequaltothetotalheadat(x=±
r,y=-h),i.e.ha=-h(Lei,1999;
ElTani,2003).
(1)Approximatesolutionbyassumingha=-h.
Bysimplyassumingha=-handH=0,Eq.(17)canbesimplifiedas
(18)
WheresubscriptAmeansapproximatesolution.Eq.(18)wasindicatedasthesolutionbyRat,Schleiss,LeiinTable1ofElTani(2003).
(2)Approximatesolutioninthecaseofh﹥﹥r(deeptunnel)
Forh﹥﹥r,wehaveh+
andhenceEq.(18)canbefurthersimplifiedas
(19)
Eq.(19)wasindicatedasthesolutionbyMuskat,Goodmanetal.inTable1ofElTani(2003).
4.1.Differenceinwaterinflowpredictions
Inordertoinvestigatethedifferenceinwaterinflowpredictionsamongtheexactandapproximatesolutionsandtherangeofapplicabilityofapproximatesolutions,therelativeerror,previouslyshowninFig.3ofElTani(2003),areobtainedagainfrom
or
δ1andδ2showthedifferencesbetweenQ1(Case1)andQA1,QA2(approximatesolutionsofCase2)respectively.Here,H=0isused,andsothiscaseisthatthegroundwaterlevelisat/belowthegroundsurface.
Fig.3.Diffierenceamongsolutions(E1Tani,2003)
FromFig.3,δ1andδ2indicatethattheapproximatesolutions,QA1andQA2,overestimatetheinflowratebyabout10–15%whenr/h=0.5.InterestinglytheoverestimationbytheapproximatesolutionQA1increasesdrasticallyasr/h
1.Thismaybecausetheterm
and
asr/h
1.Thus,theapproximatesolutionQA2seemstogivebetterpredictionofgroundwaterinflowthanQA1.Since,theterm
asr/h
1,Q1givesstableresults.IfH≠0,howevertheterm
couldcauseinstabilityofQ1asr/h
1.Thiseffectisinvestigatedinthenext.
4.2.EffectofHintheunderwatertunnel
TheeffectofHonthewaterinflowpredictionintheunderwatertunnelisinvestigatedbyusingtheapproximateandexactsolutions.Fig.4showstheresultsofwaterinflowwithrespecttor/hwithdifferentb(=H/h).TheinflowisobtainedfromEq.(16)forQ1orEq.(19)forQA2consideringha=-handh﹥﹥r.
ThesolidlinerepresentstheresultsforQ1,whereasdottedlineindicatestheresultforQA2.ItcanbeseenfromFig.4thatthewaterinflowincreaseswithincreasingb.Forb=0.5and1,theinflowratebyQ1increase
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- 无限 含水层 圆形 排水 隧道 稳定