nelcircumferenceinthez-planecanbemappedconformallyontotwocirclesofradiusandα,inthetransformedζ-planebytheanalyticfunction(Fig.)(VerruijtandBooker,))(iAwz()whereA=h(-α)/(+α),histhetunneldepthandαisaparameterdefinedashror)(rhhr()Then,Eq.()canberewrittenintermsofcoordinateξ-η()Byconsideringtheboundaryconditions,thesolutionforthetotalheadonacirclewithradiusρintheζ-planecanbeexrforaconstanttotalhead)alongthetunnelcircumference,usedintheexistingsolutions.Thesolutionsfortwodifferentboundaryconditionsarere-derivedwithinacommontheoreticalframeworkbyusingtheconformalmappingtechnique.Thedifferenceinthewaterinflowpredictionsamongtheapproximateandexactsolutionsisre-comparedtoshowtherangeofappli-cabilityofapproximatesolutions..DefinitionoftheproblemConsideracirculartunnelofradiusrinafullysaturated,homogeneous,isotropic,andsemi-infiniteporousaquiferwithahorizontalwatertable(Fig.).Thesurroundinggroundhastheisotropicpermeabilitykandasteady-stategroundwaterflowconditionisassumed.Fig..Circulartunnelinasemi-infiniteaquifer.AccordingtoDarcy’slawandmassconservation,thetwo-dimensionalsteady-stategroundwaterflowaroundthetunnelisdescribedbythefollowingLaplaceequation:yx()where=totalhead(orhydraulichead),beinggivenbythesumofthepressureandelevationheads,orZpW()p=pressure,W=unitweightofwater,Z=elevationhead,whichistheverticaldistanceofagivenpointaboveorbelowadatumplane.Here,thegroundsurfaceisusedastheelevationreferencedatumtoconsiderthecaseinwhichthewatertableisabovethegroundsurface.NotethatETani()usedthewaterlevelastheelevationreferencedatum,whereasKolymbasandWagner()usedthegroundsurface.InordertosolveEq.(),twoboundaryconditionsareneeded:oneatthegroundsurfaceandtheotheralongthetunnelcircumference.Theboundaryconditionatthegroundsurface(y=)isclearlyexpressedasHy)(()Inthecaseofadrainedtunnel,however,twodifferentboundaryconditionsalongthetunnelcircumferencecanbefoundintheliterature:(Fig.)()Case:zerowaterpressure,andsototalhead=elevationhead(ElTani,)yr)(()()Case:constanttotalhead,ha(Lei,;KolymbasandWagner,)arh)(()ItshouldbenotedthattheboundaryconditionofEq.()assumesaconstanttotalhead,whereasEq.()givesvaryingtotalheadalongthetunnelcircumference.Byconsideringthesetwodifferentboundaryconditionsalongthetunnelcircumference,twodifferentsolutionsforthesteady-stategroundwaterrinflowintoadrainedcirculartunnelarere-derivedinthenext..Analyticalsolutions..ConformalmappingThegroundsurfaceandthetunnelcircumferenceinthez-planecanbemappedconformallyontotwocirclesofradiusandα,inthetransformedζ-planebytheanalyticfunction(Fig.)(VerruijtandBooker,))(iAwz()whereA=h(-α)/(+α),histhetunneldepthandαisaparameterdefinedashror)(rhhr()Then,Eq.()canberewrittenintermsofcoordinateξ-η()Byconsideringtheboundaryconditions,thesolutionforthetotalheadonacirclewithradiusρintheζ-planecanbeex()与ElTani()公式(.)H=时表达式相同。()例:总水头恒定,ha。应用公式()给边界条件,lncos)(ln)(CHhChnCCHaannn()则,lnlnHHha().涌水量计算方法圆形排水隧道单位长度涌水量体积可有两种不同情况获得:)ln()()(lnQrhrhHAkHAkdk())ln()()(lnrhrhHhkHhkdkQaa()注意公式()是ElTani()H=表达式,而公式()与Kolymbas和Wagner()是相同解决方法。公式()和公式()之间一个明显区别就是:公式()中是A(=h(-α)/(+α)而公式()中是ha,这是由于隧道掌子面边界条件不同。还应指出,地下水位在地面之上这种情况也用到公式()与()。如果对地下水位低于地面,则将潜水面作为参考基准面。公式()与()在H=和h为地下水埋深(不是隋道深度)时适用。.与近似解比较由公式(),由隧道掌子面总水头处处等于(x=r,y=-h)总水头也即ha=-h假定,先前近似解可确定(Lei,;ElTani,).()通过假定ha=-h近似解。通过简单假设ha=-h且H=。公式()可以简化为)ln(QArhrhhk()其中下标A表示近似解。公式()是Rat,Schleiss指出解决方法,ElTani()表一()h﹥﹥r情况下近似解(深隧道)。由于h﹥﹥r,可得h﹥﹥h+hrhh,则公式()可进一步简化为)ln(QArhrhhk()公式()是由ElTani()表一中Muskat,Goodman等人提出来。.涌水量预测差别为了研究涌水量预报精确和近似解不同以及近似解适用范围,相对误差,如先前ElTani()图,由以下)(QAQQ或()QAQQδ与δ表明Q(例)与QA,QA(例近似解)之间差异,。其中,H=,所以这个例子是地下水水位在地面以下。表解差异由图,δ和δ表明近似解,QA和QA,当r/h=.时涌水量估计过量大约-。有趣是此过量估计是由于当r/h时近似解QA迅速增加。这也许是因为当r/h时)//ln(rhrh且AQ。因此,近似解QA似乎比QA能更好地预测地下水涌水量。由于当r/h时)//ln(rhrh,Q为稳定值。如H≠,当r/h时)//ln(rhrh可能导致Q值不稳定。下面研究其影响。..水下隧道H影响。通过采用近似与精确解来研究H对水下隧道涌水量预测影响。图显示涌水量和r/h和不同b(=H/h)相关结果。考虑到ha=-h且h﹥﹥r,涌水量由公式()中Q或者公式()中QA表示)()(khQrhrhb或)ln()(rhbkhQA表不同b值得涌水量预测实线表示Q结果,而虚线表示QA结果。可以从图中发现涌水量随b增加而增加。如预期那样,b=.和,当r/h,Q涌水率大大增加。近似解QA由于r/h≤.略高过估计值,但结果稳定。一般来说,不需考虑沿隧道边界条件r/h≤.(h≥.r)隧道现有精确和近似方法都可使用。隧道r/h>.时,精确解决方法应在适当沿隧道掌子面边界条件下使用。近似解QA似乎给出了水位在地面之上情况下稳定结果。.结论半无限含水层中圆形排水隧道稳定地下水流涌水量简单封闭分析解决方法在隧道掌子面两种不同边界条件(无水头以及恒定水头)共同理论框架内重新推求。近似解可用于r/h≤.(h≥.r)隧道。正确地估计隧道掌子面边界条件对于浅圆形排水隧道是重要。如果水位在地面以上,近似解QA似乎更适合实际应用。致谢第一作者谢谢两位学者对公式()正确性宝贵评论。参考文献ElTani,M.,.Circulartunnelinasemi-infiniteaquifer.Tunn.Undergr.SpaceTechnol.(),–.Kolymbas,D.,Wagner,P.,.Groundwateringresstotunnels–theexactanalyticalsolution.Tunn.Undergr.SpaceTechnol.(),–.Lei,S.,.Ananalyticalsolutionforsteadyflowintoatunnel.GroundWater,–.Verruijt,A.,.ComplexVariableSolutionsofElasticTunnelingProblems.GeotechnicalLaboratory,DelftUniversityofTechnology.Verruijt,A.,Booker,J.R.,.ComplexVariableAnalysisofMindlin’sTunnelProblem.DevelopmentofTheoreticalGeomechanicsrforaconstanttotalhead)alongthetunnelcircumference,usedintheexistingsolutions.Thesolutionsfortwodifferentboundaryconditionsarere-derivedwithinacommontheoreticalframeworkbyusingtheconformalmappingtechnique.Thedifferenceinthewaterinflowpredictionsamongtheapproximateandexactsolutionsisre-comparedtoshowtherangeofappli-cabilityofapproximatesolutions..DefinitionoftheproblemConsideracirculartunnelofradiusrinAnalyticalsolutionforsteady-stategroundwaterinflowintoadrainedcirculartunnelinasemi-infiniteaquifer:ArevisitKyung-HoParka,*,AdisornOwatsiriwonga,Joo-GongLeebaSchoolofEngineeringandTechnology,AsianInstituteofTechnology,P.O.Box,KlongLuang,Pathumthani,ThailandbDODAME&CCo.,Ltd.,F.,Anyang-Megavalley,Gwanyang-Dong,Dongan-Gu,Anyang,Gyeonggi-Do,RepublicofKoreaReceivedNovember;receivedinrevisedformFebruary;acceptedFebruaryAvailableonlineAprilAbstractThisstudydealswiththecomparisonofexistinganalyticalsolutionsforthesteady-stategroundwaterinflowintoadrainedcirculartunnelinasemi-infiniteaquifer.Twodifferentboundaryconditions(oneforzerowaterpressureandtheotherforaconstanttotalhead)alongthetunnelcircumference,usedintheexistingsolutions,arementioned.Simpleclosed-formanalyticalsolutionsarere-derivedwithinacommontheoreticalframeworkfortwodifferentboundaryconditionsbyusingtheconformalmappingtechnique.Thewaterinflowpredictionsarecomparedtoinvestigatethedifferenceamongthesolutions.Thecorrectuseoftheboundaryconditionalongthetunnelcircumferenceinashallowdrainedcirculartunnelisemphasized.ElsevierLtd.Allrightsreserved.Keywords:Analyticalsolution;Tunnels;Groundwaterflow;Semi-infiniteaquifer.IntroductionPredictionofthegroundwaterinflowintoatunnelisneededforthedesignofthetunneldrainagesystemandtheestimationoftheenvironmentalimpactofdrainage.Recently,ElTani()presentedtheanalyticalsolutionofthegroundwaterinflowbasedonMobiustransformationandFourierseries.Bycompilingtheexactandapproximatesolutionsbymanyresearchers(Muscat,Goodmanetal.,Karlsrud,Rat,Schleiss,Lei,andLombardi),ElTani()showedthebigdifferenceinthepredictionofgroundwaterinflowbythesolutions.KolymbasandWagner()alsopresentedtheanalyticalsolutionforthegroundwaterinflow,whichisequallyvalidfordeepandshallowtunnelsandallowsvariabletotalheadatthetunnelcircumferenceandatthegroundsurface.Whileseveralanalyticalsolutionsforthegroundwaterinflowintoacirculartunnelcanbefoundintheliterature,theycannotbeeasilycomparedwitheachotherbecauseoftheuseofdifferentnotations,assumptionsofboundaryconditions,elevationreferencedatum,andsolutionmethods.Inthisstudy,weshallrevisittheclosed-formanalyticalsolutionforthesteady-stategroundwaterinflowintoadrainedcirculartunnelinasemi-infiniteaquiferwithfocusontwodifferentboundaryconditions(oneforzerowaterpressureandtheotherforaconstanttotalhead)alongthetunnelcircumference,usedintheexistingsolutions.Thesolutionsfortwodifferentboundaryconditionsarere-derivedwithinacommontheoreticalframeworkbyusingtheconformalmappingtechnique.Thedifferenceinthewaterinflowpredictionsamongtheapproximateandexactsolutionsisre-comparedtoshowtherangeofappli-cabilityofapproximatesolutions..DefinitionoftheproblemConsideracirculartunnelofradiusrinafullysaturated,homogeneous,isotropic,andsemi-infiniteporousaquiferwithahorizontalwatertable(Fig.).Thesurroundinggroundhastheisotropicpermeabilitykandasteady-stategroundwaterflowconditionisassumed.Fig..Circulartunnelinasemi-infiniteaquifer.AccordingtoDarcy’slawandmassconservation,thetwo-dimensionalsteady-stategroundwaterflowaroundthetunnelisdescribedbythefollowingLaplaceequation:yx()where=totalhead(orhydraulichead),beinggivenbythesumofthepressureandelevationheads,orZpW()p=pressure,W=unitweightofwater,Z=elevationhead,whichistheverticaldistanceofagivenpointaboveorbelowadatumplane.Here,thegroundsurfaceisusedastheelevationreferencedatumtoconsiderthecaseinwhichthewatertableisabovethegroundsurface.NotethatETani()usedthewaterlevelastheelevationreferencedatum,whereasKolymbasandWagner()usedthegroundsurface.InordertosolveEq.(),twoboundaryconditionsareneeded:oneatthegroundsurfaceandtheotheralongthetunnelcircumference.Theboundaryconditionatthegroundsurface(y=)isclearlyexpressedasHy)(()Inthecaseofadrainedtunnel,however,twodifferentboundaryconditionsalongthetunnelcircumferencecanbefoundintheliterature:(Fig.)()Case:zerowaterpressure,andsototalhead=elevationhead(ElTani,)yr)(()()Case:constanttotalhead,ha(Lei,;KolymbasandWag Analyticalsolutionforsteady-stategroundwaterinflowintoadrainedcirculartunnelinasemi-infiniteaquifer:ArevisitKyung-HoParka,*,AdisornOwatsiriwonga,Joo-GongLeebaSchoolofEngineeringandTechnology,AsianInstituteofTechnology,P.O.Box4,KlongLuang,Pathumthani12120,ThailandbDODAME&CCo.,Ltd.,3F.799,Anyang-Megavalley,Gwanyang-Dong,Dongan-Gu,Anyang,Gyeonggi-Do,R