Lecture 3 Contaminant Transport Mechanisms and Principles BASIC DEFINITIONS Ground surface Vadose zone, unsaturated zone Capillary fringe Water table Saturated zone Confining bed Below ground surface (BGS) Water-table, phreatic, or unconfined aquifer Confined aquifer or artesian aquifer Capillary fringe may be >200 cm in fine silt In capillary fringe water is nearly saturated, but held in tension in soil pores MICRO VIEW OF UNSATURATED ZONE Contaminant concentrations: water Cw, mg/L concentration in water air Cg, mg/L or ppmv concentration in gas solid Cs, gm/kg concentration in solids PARTITIONING RELATIONSHIPS Solid ↔ water Cs mg/kg solid = Kd = Cw mg/L water Kd = partition coefficient Water ↔ vapor Cg mol/m3 air =H= 3 Cw mg/m water H = Henry’s Law constant HENRY’S LAW CONSTANT H has dimensions: atm m3 / mol H’ is dimensionless H’ = H/RT R = gas constant = 8.20575 x 10-5 atm m3/mol °K T = temperature in °K NOTE ON SOIL GAS CONCENTRATION Soil gas is usually reported as: ppmv = parts per million by volume Cg (ppmv) = Cg (mg/L) × 24,000 mL/mole molecular weight g/mole VOLUME REPRESENTATION Gas volume, Vg Void volume, VV Water volume, VW Solid volume, VS Total volume, VT VOLUME-RELATED PROPERTIES Bulk density = ρb = mass of solids total volume Porosity = n = θ = VV/VT Volumetric water content or water-filled porosity = θW = VW/VT Saturation = S = VW/VV Gas-filled porosity = θg (or θa) = Vg/VT θW + θg = n CONTAMINANT CONCENTRATION IN SOIL Total mass in unit volume of soil: CT = ρb Cs + θw CW + θg Cg If soil is saturated, θg = 0 and θW = n CT = ρb Cs + n CW NOMENCLATURE FOR DARCY’S LAW Q = KiA K = hydraulic conductivity i = hydraulic gradient = dh/dL A = cross-sectional area Velocity of ground-water movement u = Q / n A = q / n = K i / n = average linear velocity n A = area through which ground water flows q = Q / A = Darcy seepage velocity = Specific discharge For transport, n is ne, effective porosity ADVECTIVE FLUX Flowing ground water carries any dissolved material with it → Advective Flux JA = n u C mass / area / time = mass flux through unit cross section due to ground-water advection n is needed since no flow except in pores DIFFUSIVE FLUX Movement of mass by molecular diffusion (Brownian motion) – proportional to concentration gradient ∂C JD = −DO ∂x in surface water !!! DO is molecular diffusion coefficient [L2/T] DIFFUSIVE FLUX In porous medium, geometry imposes constraints: ∂C ∂C JD = − τ DO n = −D*n ∂x ∂x τ = tortuosity factor D* = effective diffusion coefficient Factor n must be included since diffusion is only in pores TORTUOSITY Solute must travel a tortuous path, winding through pores and around solid grains Common empirical expression: L = straight-line distance Le = actual (effective) path τ ≈ 0.7 for sand ⎛L ⎞ τ = ⎜⎜ ⎟⎟ ⎝ Le ⎠ 2 NOTES ON DIFFUSION Diffusion is not a big factor in saturated groundwater flow – dispersion dominates diffusion Diffusion can be important (even dominant) in vapor transport in unsaturated zone MECHANICAL DISPERSION C B C B A A A arrives first, then B, then C → mechanical dispersion MECHANICAL DISPERSION Viewed at micro-scale (i.e., pore scale) arrival times A, B, and C can be predicted Averaging travel paths A, B, and C leads to apparent spreading of contaminant about the mean Spatial averaging → dispersion MECHANICAL DISPERSION Dispersion can be effectively approximated by the same relationship as diffusion—i.e., that flux is proportional to concentration gradient: ∂C JM = −DM n ∂x Dispersion coefficient, DM = αL u αL = longitudinal dispersivity (units of length) TRADITIONAL VIEW OF HYDRODYNAMIC DISPERSION ACTUAL OBSERVATIONS OF PLUMES USGS Cape Cod Research Site Source: NOAA Coastal Services Center, http://www.csc.noaa.gov/crs/tcm/98fall_status.html Accessed 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 ARRAY USGS MONITORING NETWORK Source: http://ma.water.usgs.gov/CapeCodToxics/photo-gallery.html Photo by D.R. LeBlanc. OBSERVED BROMIDE PLUME – HORIZONTAL VIEW Significant longitudinal dispersion, but limited lateral dispersion OBSERVED BROMIDE PLUME – VERTICAL VIEW Limited vertical dispersion LONGITUDINAL DISPERSION VS. LENGTH SCALE Lateral and vertical dispersivity TRANSPORT EQUATION Combined transport from advection, diffusion, and dispersion (in one dimension): J =JA +JD +JM ∂C ∂C J = nuC − D * n − DM n ∂x ∂x ∂C J = nuC − D H ∂x DH = D* + DM = τ DO + αL u = hydrodynamic dispersion TRANSPORT EQUATION Consider conservation of mass over control volume (REV) of aquifer. REV = Representative Elementary Volume REV must contain enough pores to get a meaningful representation (statistical average or model) TRANSPORT EQUATION Change in contaminant mass with time ∂C T ∂t ∂C T ∂t Flux in less flux out of REV Sources and sinks due to reactions = −∇•J ± S/S (1) = ∂J − ∂x ± S/S (2) TRANSPORT EQUATION CT = = total mass (dissolved mass plus mass adsorbed to solid) per unit volume ρb CS + n CW = ρb CS + n C (3) Note: W subscript dropped for convenience and for Consistency with conventional notation Substitute Equation 3 into Equation 2: ∂ (ρbCS ) ∂ (nC) ∂ ⎛ ∂C ⎞ + =− ⎜ nuC − DHn ⎟ ± S/S ∂t ∂t ∂x ⎝ ∂x ⎠ ↑ no solid phase in flux term (4) TRANSPORT EQUATION CS = Kd C by definition of Kd Assume spatially uniform n, ρb, Kd, u, and DH and no S/S ∂C ∂C ∂ C (ρbK d + n) = −nu + nDH 2 ∂t ∂x ∂x 2 DH u ∂ C ∂C ∂C + =− 2 K n K n ρ + ρ + ∂t ⎞ ∂x ⎛ b d ⎞ ∂x ⎛ b d ⎜ ⎟ ⎜ ⎟ n n ⎝ ⎠ ⎝ ⎠ 2 (5) (6) TRANSPORT EQUATION “Retardation factor”, Rd ρbK d + n n = ρbK d 1+ n = Rd (7) Substituting Equation 7 into Equation 6: u ∂ C DH ∂ C ∂C =− + ∂t R d ∂x R d ∂x 2 2 (8) Effect of adsorption to solids is an apparent slowing of transport of dissolved contaminants Both u and DH are slowed SOLUTION OF TRANSPORT EQUATION Equation 8 can be solved with a variety of boundary conditions In general, equation predicts a spreading Gaussian cloud x - [ [ x-a x+a Relative Concentration C/C0 1.0 0.8 0.6 0.4 0.2 0.0 t1 t2 t0 Spreading of a solute slug with time due to diffusion. A slug of solute was injected into the aquifer at time t with a resulting initial concentration of C0 . 0 Adapted from: Fetter, C. W. Contaminant Hydrogeology. New York: Macmillan Publishing Company, 1992. 1-D SOLUTION OF TRANSPORT EQUATION For instantaneous placement of a long-lasting source (for example, a spill that leaves a residual in the soil), solution is: ⎛ R d x − ut ⎞ Co ⎟ erfc⎜ C(x, t ) = ⎜ 4R D t ⎟ 2 d H ⎠ ⎝ Where Co = C(x=0, t) = constant concentration at source location x = 0 Solution is a front moving with velocity u/Rd 1.0 0.9 0.8 0.7 0.6 0.50 0.4 0.3 0.16 0.2 Mean Relative Concentration C/C0 0.84 0.1 x 0.0 +s s x = ut/Rd The profile of a diffusing front as predicted by the complementary error function. Adapted from Fetter, C. W. Contaminant Hydrogeology. New York: Macmillan Publishing Company, 1992. Moving front of contaminant from constant source 10 C0 = 10 u=1 DH = 0.1 Rd = 1 9 Concentration, C(x,t) 8 7 t=1 ut = 1 6 t=3 ut = 3 t=5 ut = 5 5 4 3 2 1 0 0 1 2 3 4 5 Distance, x Moving front of contaminant from constant source 6 7 8 9 Effect of dispersion coefficient Effect of Rd on moving front of contaminant Effect of retardation 10 9 Concentration, C(x,t) t=3 ut = 3 Rd = 1 t=3 ut = 3 Rd = 2 8 7 C0 = 10 u=1 DH = 0.1 6 5 4 3 2 1 0 0 1 2 3 4 5 Distance, x 6 7 8 9 1-D SOLUTIONS Transport of a Conservative Substance from Pulse and Continuous Sources . Continuous Input of Mass Per Unit Time M Starting at Time t = 0 Dimensions Pulse Input of Mass M 1-D M exp - (x-vt) C= 1/2 1/2 4Dxt 2np t D x . M, M are instantaneous or continuous plane sources M . [ [ 2 x=0 v . ( x-vt C = M erfc 2nv 2 Dx t x=0 v ( . Continuous Input of Mass Per Unit Time M in Steady State . C= M nv ( for x > 0 ( x=0 M L2 M M 2 L T v to ∞ t=0 t = t1 Mass Front at input here time t Mass input here Adapted from: Hemond, H. F. and E. J. Fechner-Levy. Chemical Fate and Transport in the Environment. 2nd ed. San Diego: Academic Press, 2000. 2-D SOLUTIONS Transport of a Conservative Substance from Pulse and Continuous Sources Dimensions 2-D . M, M are instantaneous or continuous line sources [[ . [ [ M M M L M L-T Continuous . Input of Mass Per Unit Time M Starting at Time t = 0 Pulse Input of Mass M [ (x-vt)2 y2 M exp + C= 4Dx t 4Dy t 4np t Dx Dy v t = t1 t=0 y x [ . M C= 4np 1/2 (vr)1/2 [ [ ( ( exp (x-r)v erfc r-vt 2D x 2 Dx t Dy v Continuous Input . of Mass Per Unit Time M in Steady State [ [ . exp (x-r)v C= 1/2 1/2 2D x 2np (vr) Dy M v Plume at time t y y x Adapted from: Hemond, H. F. and E. J. Fechner-Levy. Chemical Fate and Transport in the Environment. 2nd ed. San Diego: Academic Press, 2000. x to ∞ 3-D SOLUTIONS Transport of a Conservative Substance from Pulse and Continuous Sources Dimensions 3-D . M, M are instantaneous or continuous point sources [[ . [ [ M M M L M T Continuous . Input of Mass Per Unit Time M Starting at Time t = 0 Pulse Input of Mass M exp - . M C= [ 8np 3/2 3/2 t (x-vt)2 4Dx t + Dx Dy D z y2 4Dy t + v z y z2 4Dz t [ M C= 8np r DyD z z y [ [ ( ( exp (x-r)v erfc r-vt 2D x 2 Dx t v Continuous Input . of Mass Per Unit Time M in Steady State . M C= 4np r DyD z z y x t=0 [ [ exp (x-r)v 2D x v to ∞ t = t1 Plume at time t Adapted from: Hemond, H. F. and E. J. Fechner-Levy. Chemical Fate and Transport in the Environment. 2nd ed. San Diego: Academic Press, 2000.