水文地质与工程专业外文文献翻译Word文档下载推荐.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.

tunnelinasemi-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

mapping

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.

surfaceboundarycondition

TheconstantC1canbeobtainedbyconsideringtheboundaryconditionatthegroundsurfacewithρ=1intheζ-plane,

（10）

Fig.2.Planeofconformalmapping.

tunnelboundarycondition

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）

solutionforgroundwaterinflow

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）.

inwaterinflowpredictions

Inordertoinvestigatethedifferenceinwaterinflowpredictionsamongtheexactandapproximatesolutionsandtherangeofapplicabilityofapproximatesolutions,therelativeerror,previouslyshowninFig.3ofElTani（2003）,areobtainedagainfrom

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.

ofHintheunderwatertunnel

TheeffectofHonthewaterinflowpredictionintheunderwatertunnelisinvestigatedbyusingtheapproximateandexactsolutions.Fig.4showstheresultsofwaterinflowwithrespecttor/hwithdifferentb（=H/h）.TheinflowisobtainedfromEq.（16）forQ1orEq.（19）forQA2consideringha=-handh﹥﹥r.

ThesolidlinerepresentstheresultsforQ1,whereasdottedlineindicatestheresultforQA2.ItcanbeseenfromFig.4thatthewaterinflowincreaseswithincreasingb.Forb=0.5and1,theinflowratebyQ1increasesgreatlyasr/h

1,asexpected.TheapproximatesolutionQA2slightlyoverestimat