河海大学土力学英文教案Chapter 2 Permeability of Soil.docx

河海大学土力学英文教案Chapter 2 Permeability of Soil.docx

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河海大学土力学英文教案Chapter2PermeabilityofSoil

Chapter2PermeabilityofSoil

2.1Introduction

Soilsareassemblagesofsolidparticleswithinterconnectedvoidsthroughwhichwatercanflowfromapointofhighenergytoapointoflowenergy.Thestudyoftheflowofwaterthroughporoussoilmediaisimportantinsoilmechanics.Thisisnecessaryforestimatingthequantityofundergroundseepageundervarioushydraulicconditions;forinvestigatingproblemsinvolvingthepumpingofwaterforundergroundconstructions;andformakingstabilityanalysesofearthdamsandearth-retainingstructuresthataresubjectedtoseepageforces.

（i）HydrostaticGroundwaterCondition（静水条件）

Pore-waterpressure（u,孔隙水压力）atadepthzbelowthewatertableisdefinedas

（3.1）

wherewisunitweightofthewater（水的重度）andzisdepth（深度）ofthewaterbelowgroundwatertable.

（ii）TotalHead（总水头）

Thetotalhead（h）isrelatedtothepore-waterpressure（u）andelevationhead（z）bythefollowingequation:

（3.2）

zofequation（3.2）isdefinedwithrespecttoachosendatum.

（iii）GroundwaterFlow

Waterwillflowthroughthesoil（sayfrompointAtoB）ifhydraulicgradient（oradifferenceintotalhead）existsbetweenpointsAandB.

2.2Darcy’sLaw（达西定律）

In1856,Darcypublishedasimpleequationforthedischargevelocityofwaterthroughsaturatedsoils,whichmaybeexpressedas

（3.3）

where

q=flowrate（m3/s）,渗流量

i=hydraulicgradient水力梯度

h=totalheaddifference（m）水头差

L=lengthofflow（m）渗径长度

k=coefficientofpermeability（m/s）渗透系数

A=cross-sectionalareaofthespecimen（m2）横截面积

Equation（3.3）impliesthattheflowrate（q）bearsalinearrelationshiptothehydraulicgradient（i）.However,anon-linearrelationshipbetweenqandiisfoundforclay[叁考:

土力学p.69图3-3].Forpracticalreason,thefollowingrelationshipbetweenqandiisassumedforclay:

（3.3a）

whereibisstartinghydraulicgradient（起始水力梯度）.Forgravellysoil（砾土）,linearrelationshipbetweenqandiappearsatsmallhydraulicgradientonly.Ashydraulicgradientincreasesbeyondacriticalvalue,therelationshipbecomesnon-linearastheflowbecomesturbulent（紊流）[叁考:

土力学p.69图3-3].

Darcy’slawisvalidforlaminarflow（层流）conditionwhereReynoldsnumbersissmallerthanorequalto1.Reynoldsnumbers（雷诺数）isdefinedasfollows:

（3.4）

whereisdensityofwater（水的密度）,visvelocityofwater（流速）,isviscosityofwater（水的粘滞系数）anddisaveragediameterofsoilgrains（土粒子平均粒径）.As=1g/cm3,v=0.25cm/s,=0.0131g/scmattemperatureof10C,discalculatedbyequation（3.4）

ThusDarcy’slawcanbeappliedtosoilswhicharefinerthancoarsesand.

2.3DeterminationofCoefficientofPermeability（渗透系数的测定）

2.3.1Constantheadtest（常水头测试）

AtypicalarrangementoftheconstantheadpermeabilitytestisshowninFig.2.1.Inthesetup,thetotalwaterheadisalwaysmaintainedconstantduringtheperiodofthetest.Theconstantheadtestissuitableforcoarse-grainedsoils.Thecoefficientofpermeabilityiscalculatedbythefollowingequation:

（3.5）

whereVisvolumeofwatercollected,Lislengthofthespecimen,Aiscross-sectionalareaofthespecimen,histotalheaddifferenceandtisdurationofthetest.

2.3.2FallingHeadTest（变水头测试）

AtypicalarrangementofthefallingheadpermeabilitytestisshowninFig.2.2.Itissuitableforfine-grainedsoils.Therateofflowofwaterthroughthespecimenatanytimetisgivenby

（3.6a）

whereVisvolumeofwatercollected,Lislengthofthespecimen,A1iscross-sectionalareaofthespecimen,A2iscross-sectionalareaofthestandpipeandhistotalheaddifference.Re-arrangingequation（3.6a）into

（3.6b）

Integratingequation（3.6b）

（3.6c）

ThevaluesofkfordifferenttypesofsoilaretypicallywithintherangesshowninTable2.1.

2.4SeepageandFlowNets（渗流和流网）

2.4.1Laplace’sequation

Laplacedifferentialequationofcontinuityisusedtodescribethetwo-dimensionalsteadyflowconditionforagivenpointinthesoilmass.LetusconsiderasoilelementasshowninFig.2.3.Theelementhasdimensionofdxanddzinxandzdirection,respectively.Waterflowsthroughtheelementduetoahydraulicheaddifferencebetweenupstreamanddownstreamside.Letvxandvzbethedischargevelocityinthexandzdirection,respectively.Theratesofflowofwaterintotheelementinthexandzdirectionare

（3.7a）

（3.7b）

Theratesofflowofwateroutoftheelementinthexandzdirectionare

（3.7c）

（3.7d）

Assumingwaterisincompressibleandnovolumechangeinthesoilmassoccurs,thetotalrateofinflowshouldbeequaltothetotalrateofoutflow

（3.7e）

UsingDarcy’slaw,vxandvzcanbeexpressedas

（3.7f）

（3.7g）

Substitutingequations（3.7f）and（3.7g）intoequation（3.7e）

（3.7h）

Ifthesoilisisotropickx=kz,equation（3.7h）becomes

（3.7i）

2.4.2FlowNets

Thecontinuityequation[equation（3.7i）]inisotropicmediumrepresentstwoorthogonalfamiliesofcurves–thatis,theflowlines（流线）andequipotentiallines（等势线）.

Aflowlineisalinealongwhichawaterparticlewilltravelfromupstreamtothedownstreamsideinthepermeablesoilmedium.Anequipotentiallineisalinealongwhichthetotalhead（总水头）atallpointsisthesame.Ifpiezometers（测压计）areplacedatdifferentpointsalonganequipotentialline,theheightofwaterwillrisetothesameelevationinallofthem.Acombinationofanumberofflowlinesandequipotentiallinesiscalledaflownet.

Tocompletethegraphicalconstructionofaflownet,theflowandequipotentiallinesaredrawninsuchawaythat（i）theequipotentiallinesintersecttheflowlinesatrightangle,（ii）theflowelementsareapproximatesquares,（iii）theupstreamanddownstreamsurfacesofthepermeablelayerareequipotentiallinesand（iv）theboundaryoftheimpervouslayerisaflowline.

Basedontheflownet,thetotalflowrate（q）canbeestimatedby

（3.8）

whereHisthetotalheaddifferencebetweentheupstreamanddownstreamsides,NfisthenumberofflowchannelsandNdisthenumberofequipotentialdrops.

Inaddition,thepore-waterpressure（u）atagivenpointAdeterminedfromtheflownetis

（3.9a）

（3.9b）

whereHistotalheaddifference,Ndistotalnumberofequipotentialdrops,ndisnumberofequipotentialdropsatpointA,htistotalheadatpointAandheiselevationheadatpointA.

2.5Effectivestress（有效应力）

2.5.1Effectivestressprinciple

Effectivestress（’）isdefinedas

（3.10a）

whereNsisthesumofinter-contactforcesandAisthecross-sectionalareaofthesoilmassunderconsideration.

Totalstress（）isdefinedas

（3.10b）

whereAsiscross-sectionalareaofthesoilmassoccupiedbysolid-to-solidcontacts.AstheratioAs/Aissmallandcanbeneglectedforpressurerangesencounteredinpracticalproblems.Thus,equation（3.10b）becomes

（3.11）

2.5.2Pore-waterpressureandeffectivestressunderhydrostaticcondition

Thetotalstressatdepthzis

（3.12a）

Thepore-waterpressureatdepthzis

（3.12b）

Theeffectivestressatdepthzis

（3.12c）

wheresatissaturatedunitweightofsoil,wisunitweightofwaterand’issubmergedunitweightofsoil.

2.5.3Pore-waterpressureandeffectivestressunderseepage

（i）Downwardseepage

Thepore-waterpressureatdepthzisreducedbecauseofthedownwardflowofwater

（3.13a）

wherehisthehydraulicheaddifferencebetweengroundsurfaceanddepthz.

Theeffectivestressatdepthzbecomes

（3.13b）

（ii）Upwardseepage

Thepore-waterpressureatdepthzisincreasedbecauseoftheupwardflowofwater

（3.14a）

wherehisthehydraulicheaddifferencebetweengroundsurfaceanddepthz.

Theeffectivestressatdepthzbecomes

（3.14b）

2.5.4Criticalhydraulicgradient（临界水力梯度）

Atboilingorquickconditionunderacriticalhydraulicgradient（ic）,theweightofthesoilwillbebalancedbytheupwardseepageforcesuchthattheeffectivestressbecomeszero.Fromequation（3.14b）

（3.15）

whereeisvoidratioandnisporosity.

Table2.1Typicalvaluesofcoefficientsofpermeability（k）

Soiltype

k（cm/s）

Gravel（砾石）

>10-1

Mixtureofgravelandsand（砾石与砂混合物）

10-3–10-1

Finesand（细砂）

10-5–10-3

Mixtureofsand,siltandclay（砂,粉土与粘土混合物）

10-7–10-5

Clay（粘土）

<10-7

Keywords

Clay粘土

Constantheadtest常水头测试

Cross-sectionalarea横截面积

Darcy’sLaw达西定律

Density密度

Diameterofsoilgrains土粒子粒径

Dischargevelocity渗透速度

EffectiveStress有效应力

Equipotentialline等势线

Fallingheadtest变水头测试

Flowline流线

Flownet流网

Flowrate渗流量

Gravel砾石

HydraulicHead水头

Hydraulicheaddifference水头差

Hydraulicgradient水力梯度

Breakinghydraulicgradient起始水力梯度

Criticalhydraulicgradient临界水力梯度

Hydrostaticcondition静水条件

Laminarflow层流

LaplaceEquation拉普拉斯方程

Mixture混合物

Permeability渗透性

Coefficientofpermeability渗透系数

Pore-waterpressure孔隙水压力

Porosity孔隙率

Reynoldsnumbers雷诺数

Sand砂土

Seepage渗流

SeepageForce渗流力

Silt粉土

TotalStress总应力

Turbulentflow紊流

Viscosity粘滞系数

Voidratio孔隙比

Fig.2.1Setupofconstantheadpermeabilitytest

Fig.2.2Setupoffallingheadpermeabilitytest

Fig.2.3Seepagethroughasoilelement

Chapter3Stressesinsoil（土体应力）

3.1Introduction

Soilmasscanbearloadingsfroms