Lecture 3 Contaminant Transport Mechanisms and PrinciplesBASIC DEFINITIONSMICRO VIEW OF UNSATURATED ZONEPARTITIONING RELATIONSHIPSHENRY’S LAW CONSTANTNOTE ON SOIL GAS CONCENTRATIONVOLUME REPRESENTATIONVOLUME-RELATED PROPERTIESCONTAMINANT CONCENTRATIONIN SOILNOMENCLATURE FOR DARCY’S LAWADVECTIVE FLUXDIFFUSIVE FLUXDIFFUSIVE FLUXTORTUOSITYNOTES ON DIFFUSIONMECHANICAL DISPERSIONMECHANICAL DISPERSIONMECHANICAL DISPERSIONTRADITIONAL VIEW OF HYDRODYNAMIC DISPERSIONACTUAL OBSERVATIONS OF PLUMESUSGS Cape CodResearch SiteMONITORING WELL ARRAYUSGS MONITORING NETWORKOBSERVED BROMIDE PLUME – HORIZONTALVIEWOBSERVED BROMIDE PLUME – VERTICAL VIEWLONGITUDINALDISPERSION VS. LENGTH SCALELateral and vertical dispersivityTRANSPORT EQUATIONTRANSPORT EQUATIONTRANSPORT EQUATIONTRANSPORT EQUATIONTRANSPORT EQUATIONTRANSPORT EQUATIONSOLUTION OF TRANSPORT EQUATION1-D SOLUTION OF TRANSPORT EQUATIONMoving front of contaminant from constant sourceEffect of dispersion coefficientEffect of retardation1-D SOLUTIONS2-D SOLUTIONS3-D SOLUTIONSLecture 3Contaminant Transport Mechanisms and PrinciplesBASIC DEFINITIONSGround surfaceBelowgroundsurface(BGS)Vadose zone,unsaturated zoneCapillary fringeWater tableSaturated zoneConfining bedConfined aquifer orartesian aquiferWater-table, phreatic,or unconfined aquiferCapillary fringe may be >200 cm in fine siltIn capillary fringe water is nearly saturated, but held in tension in soil poresMICRO VIEW OF UNSATURATED ZONEairwatersolidContaminant concentrations:Cw, mg/L concentration in waterCg, mg/L or ppmvconcentration in gasCs, gm/kgconcentration in solidsPARTITIONING RELATIONSHIPSSolid ↔ waterKd= partition coefficientWater ↔ vaporH = Henry’s Law constant watermg/Lsolid mg/kgKCCdws== watermg/mair mol/mHCC33wg==HENRY’S LAW CONSTANTH has dimensions: atm m3/ molH’ is dimensionlessH’ = H/RTR = gas constant = 8.20575 x 10-5atm m3/mol °KT = temperature in °KNOTE ON SOIL GAS CONCENTRATIONSoil gas is usually reported as:ppmv = parts per million by volumeg/mole weight molecularmL/mole 24,000 (mg/L) C(ppmv) Cgg×=VOLUME REPRESENTATIONGas volume, VgWater volume, VWSolid volume, VSTotal volume, VTVoid volume, VVVOLUME-RELATED PROPERTIESBulk density = ρb= mass of solidstotal volumePorosity = n = θ = VV/VTVolumetric water content or water-filled porosity = θW= VW/VTSaturation = S = VW/VVGas-filled porosity = θg(or θa) = Vg/VTθW+ θg= nCONTAMINANT CONCENTRATIONIN SOILTotal mass in unit volume of soil:CT= ρbCs+ θwCW+ θgCgIf soil is saturated, θg= 0 and θW= nCT= ρbCs+ n CWNOMENCLATURE FOR DARCY’S LAWQ = K i AK = hydraulic conductivityi = hydraulic gradient = dh/dLA = cross-sectional areaVelocity of ground-water movementu = Q / n A = q / n = K i / n = average linear velocityn A = area through which ground water flowsq = Q / A = Darcy seepage velocity = Specific dischargeFor transport, n is ne, effective porosityADVECTIVE FLUXFlowing ground water carries any dissolved material with it → Advective FluxJA= n u C mass / area / time= mass flux through unit cross section due to ground-water advectionn is needed since no flow except in poresDIFFUSIVE FLUXMovement of mass by molecular diffusion (Brownian motion) – proportional to concentration gradientin surface water !!!DOis molecular diffusion coefficient [L2/T]xCDJOD∂∂−=DIFFUSIVE FLUXIn porous medium, geometry imposes constraints:τ = tortuosity factorD* = effective diffusion coefficientFactor n must be included since diffusion is only in poresxCn*DxCnDJOD∂∂−=∂∂−=τTORTUOSITYSolute must travel a tortuous path, winding through pores and around solid grainsCommon empirical expression:L = straight-line distanceLe= actual (effective) pathτ ≈ 0.7 for sand2eLL⎟⎟⎠⎞⎜⎜⎝⎛=τNOTES ON DIFFUSIONDiffusion is not a big factor in saturated ground-water flow – dispersion dominates diffusionDiffusion can be important (even dominant) in vapor transport in unsaturated zoneMECHANICAL DISPERSIONABCABCA arrives first, then B, then C → mechanical dispersionMECHANICAL DISPERSIONViewed at micro-scale (i.e., pore scale) arrival times A, B, and C can be predictedAveraging travel paths A, B, and C leads to apparent spreading of contaminant about the meanSpatial averaging → dispersionMECHANICAL DISPERSIONDispersion can be effectively approximated by the same relationship as diffusion—i.e., that flux is proportional to concentration gradient:Dispersion coefficient, DM= αLuαL= longitudinal dispersivity (units of length)xCnDJMM∂∂−=TRADITIONAL VIEW OF HYDRODYNAMIC DISPERSIONACTUAL OBSERVATIONS OF PLUMESUSGS Cape CodResearch SiteSource: NOAA Coastal Services Center, http://www.csc.noaa.gov/crs/tcm/98fall_status.htmlAccessed May 14, 2004. Source: U.S. Geological Survey, Cape Cod Toxic Substances Hydrology Research Site, http://ma.water.usgs.gov/CapeCodToxics/location.html. Accessed May 14, 2004.MONITORING WELL ARRAYUSGS MONITORING NETWORKSource: http://ma.water.usgs.gov/CapeCodToxics/photo-gallery.html Photo by D.R. LeBlanc.OBSERVED BROMIDE PLUME –HORIZONTALVIEWSignificant longitudinal dispersion, but limitedlateral dispersionOBSERVED BROMIDE PLUME –VERTICAL VIEWLimited vertical dispersionLONGITUDINALDISPERSION VS. LENGTH SCALELateral and vertical dispersivityTRANSPORT EQUATIONCombined transport from advection, diffusion, and dispersion (in one dimension):DH= D* + DM= τ DO+ αLu = hydrodynamic dispersionxCDnuCJxCnDxCn*DnuCJJJJJHMMDA∂∂−=∂∂−∂∂−=++=TRANSPORT EQUATIONConsider conservation of mass over control volume (REV) of aquifer.REV = Representative Elementary VolumeREV must contain enough pores to get a meaningful representation (statistical average or model)TRANSPORT EQUATIONS/SxJtCS/SJtCTT±∂∂−=∂∂±•∇−=∂∂Change incontaminantmass withtimeFlux in lessflux out ofREVSources and sinks due toreactions(1)(2)TRANSPORT EQUATIONCT= total mass (dissolved mass plus mass adsorbed to solid) per unit volume= ρbCS+ n CW= ρbCS+ n C(3)Substitute Equation 3 into Equation 2:()()S/SxCnDnuCxtnCtCHSb±⎟⎠⎞⎜⎝⎛∂∂−∂∂−=∂∂+∂ρ∂(4)↑ no solid phase in flux termNote: W subscript dropped for convenience and for Consistency with conventional notationTRANSPORT EQUATION()22dbHdb22HdbxCnnKDxCnnKutCxCnDxCnutCnK∂∂⎟⎠⎞⎜⎝⎛+ρ+∂∂⎟⎠⎞⎜⎝⎛+ρ−=∂∂∂∂+∂∂−=∂∂+ρ(5)CS= KdC by definition of KdAssume spatially uniform