Transcription of Darcy’s Law and Hydraulic Head - University of Alabama
1 Darcy s Law and Hydraulic Head1. Hydraulic HeadQKhhLA= 12h1and h2are Hydraulic heads associated with points 1 and Hydraulic head, or total head, is a measure of the potentialof the water fluid at the measurement point. Potential of a fluid at a specific point is the work required totransform a unit of mass of fluid from an arbitrarily chosen stateto the state under consideration. Qdatumh1h2hp1hp2z1z2 Three Types of PotentialsA. Pressure potentialwork required to raise the water pressureReference stateCurrent statez= 0P= 0v= 0z= zP= Pv= vV: volume: density of water assumed to be independent of pressurewPwPPdPmmdPVmW === 00111 wB. Elevation potentialwork required to raise the elevationC. Kinetic potentialwork required to raise the velocityWmmgdzgzZ201== WmmadzmmdvdtdzvdvvZZ v300 02112== == Total [ Hydraulic ] head:hgPgzvgw== ++ 22 Unit [L](dz= vdt)Total potential: = + +Pgzvw 22 Unit [L2T-1]hPgzvgw=++ 22pressurehead [L]elevation [L]Kinetic termPiezometerdatumh1h2z1z2Pg1 gP 2A fluid moves from where the total head is higher to where it is lower.
2 For anidealfluid (frictionless and incompressible), the total head would stay constant. Total head or Hydraulic head:(Fetter, p141)Low elevation to high elevationLow pressure head to high pressure head Flow between points of same elevationFlow between points of same pressure headgvzgPhw22++= Kinetic term negligibleh= Hydraulic head [L]= pressure head [L]z= elevation head [L]Important: his relative to datum (reference state)For Groundwater FlowdatumABh2piezometersh1flowdirection? gPw The pressure exerted at each hole and the force of the jet depend on the height of water above that particular holeWater pressure varies with the water heightWays of water, p125A strong jet indicates that the recharge area may be far from the point!Example:PA= ?absolute pressurePFAPAo=+Relative (gage)PFAmgAHAgAgHAww== ==() Given: H= 10 mg = m/s2 w= g/cm3= 1000 kg/m3aAPmNmskgmmsmmkgP = = head at A?hPggHgHmpAAwww== == 10Po: atmospheric pressure = 1 atm= x 105 PaPAHIn the previous example, what is the Hydraulic head at A?
3 HAHdatumhA= ?how about PB?hB= ?BzPiezometerADatumz(elevationhead)(Pres surehead)h( Hydraulic head)Groundwater ()gPw Depth toGroundwaterPiezometer(measure water level at a point)Observation well or standpipe (measure water level along a section)Water enters thepipe through a pointWater enters thepipe through a piezometers to measure vertical Hydraulic gradientsMore on Hydraulic headgPzhzhp +=+=Surface ElevationDepth to piezometerDepth to waterABC22522522515010075807760 NestedPiezometersHead: hA=hB=hC=Pressure (hp)A= Head: (hp)B=(hp)C=Gradient between:AB = BC =AB C15010075elevation=225datum807766 Coastal aquifersDensity EffectWhen comparing heads of fluids with different densities, pressure heads must be converted to the same reference === = = By averagePoint-water heads for a system of three aquifers, each containing water with a different density??? m1100 m1040 m999 kg/m3 APoint-Water HeadElevation HeadWater DensityAquiferfs mAFresh-WaterHeadFresh-Water Pressure HeadPoint-Water Pressure HeadAquiferApplicability of Darcy s Law vdR=R: Reynolds number [-] : fluid density [ML-3]v: seepage velocity [LT-1]d: diameter of passageway for fluid [L] : viscosity [MT-1L-1]laminar flowturbulent flowDarcy s Law is only applicable to laminar flow with small Reynolds numbers (<1~10).
4 Groundwater behaves as laminar flow under most circumstances.
