lecture 3 contaminant transport mechanisms and … · lecture 3 contaminant transport mechanisms...
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BASIC DEFINITIONS
Ground surface
Belowgroundsurface(BGS)
Vadose zone,unsaturated zone
Capillary fringe
Water tableSaturated zone
Confining bedConfined aquifer orartesian aquifer
Water-table, phreatic,or unconfined aquifer
Capillary fringe may be >200 cm in fine siltIn capillary fringe water is nearly saturated, but held in tension in soil pores
MICRO VIEW OF UNSATURATED ZONE
air
water
solid
Contaminant concentrations:
Cw, mg/L concentration in water
Cg, mg/L or ppmvconcentration in gas
Cs, gm/kgconcentration in solids
PARTITIONING RELATIONSHIPS
Solid ↔ water
Kd = partition coefficient
Water ↔ vapor
H = Henry’s Law constant
watermg/Lsolid mg/kgK
CC
dw
s ==
watermg/mair mol/mH
CC
3
3
w
g ==
HENRY’S LAW CONSTANT
H has dimensions: atm m3 / mol
H’ is dimensionless
H’ = H/RTR = gas constant = 8.20575 x 10-5 atm m3/mol °KT = temperature in °K
NOTE ON SOIL GAS CONCENTRATION
Soil gas is usually reported as:ppmv = parts per million by volume
g/mole weight molecularmL/mole 24,000 (mg/L) C
(ppmv) C gg
×=
VOLUME REPRESENTATION
Gas volume, Vg
Water volume, VW
Solid volume, VS
Total volume, VT
Void volume, VV
VOLUME-RELATED PROPERTIESBulk density = ρb = mass of solids
total volumePorosity = 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 CONCENTRATIONIN SOIL
Total mass in unit volume of soil:CT = ρb Cs + θw CW + θg Cg
If soil is saturated, θg = 0 and θW = nCT = ρb Cs + n CW
NOMENCLATURE FOR DARCY’S LAW
Q = K i AK = hydraulic conductivityi = hydraulic gradient = dh/dLA = cross-sectional area
Velocity 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 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
in surface water !!!
DO is molecular diffusion coefficient [L2/T]
xCDJ OD ∂
∂−=
DIFFUSIVE FLUX
In porous medium, geometry imposes constraints:
τ = tortuosity factorD* = effective diffusion coefficient
Factor n must be included since diffusion is only in pores
xCn*D
xCnDJ OD ∂
∂−=
∂∂
−= τ
TORTUOSITY
Solute must travel a tortuous path, winding through pores and around solid grains
Common empirical expression:
L = straight-line distanceLe = actual (effective) path
τ ≈ 0.7 for sand
2
eLL
⎟⎟⎠
⎞⎜⎜⎝
⎛=τ
NOTES ON DIFFUSION
Diffusion is not a big factor in saturated ground-water flow – dispersion dominates diffusion
Diffusion can be important (even dominant) in vapor transport in unsaturated zone
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:
Dispersion coefficient, DM = αL u
αL = longitudinal dispersivity (units of length)
xCnDJ MM ∂
∂−=
USGS Cape CodResearch Site
Source: 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.
USGS MONITORING NETWORK
Source: http://ma.water.usgs.gov/CapeCodToxics/photo-gallery.html Photo by D.R. LeBlanc.
OBSERVED BROMIDE PLUME –
HORIZONTALVIEW
Significant longitudinal dispersion, but limitedlateral dispersion
TRANSPORT EQUATION
Combined transport from advection, diffusion, and dispersion (in one dimension):
DH = D* + DM = τ DO + αL u = hydrodynamic dispersion
xCDnuCJ
xCnD
xCn*DnuCJ
JJJJ
H
M
MDA
∂∂
−=
∂∂
−∂∂
−=
++=
TRANSPORT EQUATION
Consider 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 EQUATION
S/SxJ
tC
S/SJt
C
T
T
±∂∂
−=∂
∂
±•∇−=∂
∂
Change incontaminantmass withtime
Flux in lessflux out ofREV
Sources and sinks due toreactions
(1)
(2)
TRANSPORT EQUATIONCT = total mass (dissolved mass plus mass adsorbed to
solid) per unit volume= ρb CS + n CW = ρb CS + n C (3)
Substitute Equation 3 into Equation 2:
( ) ( ) S/SxCnDnuC
xtnC
tC
HSb ±⎟
⎠⎞
⎜⎝⎛
∂∂
−∂∂
−=∂
∂+
∂ρ∂ (4)
↑ no solid phase in flux term
Note: W subscript dropped for convenience and for Consistency with conventional notation
TRANSPORT EQUATION
( )
2
2
db
H
db
2
2
Hdb
xC
nnK
DxC
nnK
utC
xCnD
xCnu
tCnK
∂∂
⎟⎠⎞
⎜⎝⎛ +ρ
+∂∂
⎟⎠⎞
⎜⎝⎛ +ρ
−=∂∂
∂∂
+∂∂
−=∂∂
+ρ (5)
CS = Kd C by definition of Kd
Assume spatially uniform n, ρb, Kd, u, and DH and no S/S
(6)
TRANSPORT EQUATION
ddbdb R
nK1
nnK
=ρ
+=+ρ
“Retardation factor”, Rd
2
2
d
H
d xC
RD
xC
Ru
tC
∂∂
+∂∂
−=∂∂
Substituting Equation 7 into Equation 6:
(8)
(7)
Effect of adsorption to solids is an apparent slowing of transportof dissolved contaminantsBoth 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
[ [
0.0
t1 t2 t0
0.2
0.4
0.6
0.8
1.0
x - a
x-
x + a
Spreading of a solute slug with time due to diffusion. A slug of solute wasinjected into the aquifer at time t with a resulting initial concentration of C .00
Adapted from: Fetter, C. W. Contaminant Hydrogeology. New York: Macmillan Publishing Company, 1992.
C/C
0R
elat
ive
Con
cent
ratio
n
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:
Where Co = C(x=0, t) = constant concentration at source location x = 0Solution is a front moving with velocity u/Rd
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
tDR4utxRerfc
2Ct,xC
Hd
do
+
0.0
0.2
0.3
0.10.16
0.50
0.84
0.4
0.6
0.7
0.8
0.9
1.0
Mea
n
σ σ
x =
x
ut/Rd
Adapted from Fetter, C. W. Contaminant Hydrogeology. New York: Macmillan Publishing Company, 1992.
C/C
0R
elat
ive
Con
cent
ratio
n
The profile of a diffusing front as predicted by the complementary error function.
Moving front of contaminant from constant source
Moving front of contaminant from constant source
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9
Distance, x
Con
cent
ratio
n, C
(x,t)
C0 = 10u = 1DH = 0.1Rd = 1
t = 1ut = 1
t = 5ut = 5
t = 3ut = 3
Effect of retardationEffect of Rd on moving front of contaminant
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9
Distance, x
Con
cent
ratio
n, C
(x,t)
C0 = 10u = 1DH = 0.1
t = 3ut = 3Rd = 2
t = 3ut = 3Rd = 1
1-D SOLUTIONS
Mass input here
Front at time t
1-D
M
C =
ML2
MM
L2T
M exp-2n D
t = 0 t = t1
1/2 1/2
x = 0 v
to
x = 0x = 0 vv
π x xt [ [(x-vt)4D t
2
C = M erfc2nv Dx
( (x-vt t C = M
nvfor x > 0
∞
( (
M, M are instantaneous or continuous plane
sources
.
.
..2
Transport of a Conservative Substance from Pulse and Continuous Sources
Dimensions Pulse Input of Mass M Continuous Input of Mass PerUnit Time M Starting at Time t = 0
. Continuous Input of Mass PerUnit Time M in Steady State
.
Adapted from: Hemond, H. F. and E. J. Fechner-Levy. Chemical Fate and Transport in the Environment.2nd ed. San Diego: Academic Press, 2000.
Mass input here
2-D SOLUTIONS
2-DM, M are instantaneous
or continuous line sources
MML [[
M ML-T [[
t = 0 t = t Plume at time t1
to
v vv
x xy y
∞
.
. xy
C= M exp4n Dπ x xDy y[ [(x-vt)
4D t t
2 y4D t
2+ C= M exp erfc
4n (vr)1/2 1/2π xDy D tx[ [(x-r)v2D ( (r-vt
2
Transport of a Conservative Substance from Pulse and Continuous Sources
Adapted from: Hemond, H. F. and E. J. Fechner-Levy. Chemical Fate and Transport in the Environment.2nd ed. San Diego: Academic Press, 2000.
Dimensions Pulse Input of Mass M Continuous Input of Mass PerUnit Time M Starting at Time t = 0
Continuous Input of Mass PerUnit Time M in Steady State
.
.C= M exp
2n (vr)1/2 1/2π xDy[ [(x-r)v2D
.
.
3-D SOLUTIONS
3-DM, M are instantaneous
or continuous point sources
MML [[
M MT [[
v v
.
.
C= M exp erfc8n rπ xDy D tx[ [(x-r)v
2D ( (r-vt2
Transport of a Conservative Substance from Pulse and Continuous Sources
Adapted from: Hemond, H. F. and E. J. Fechner-Levy. Chemical Fate and Transport in the Environment.2nd ed. San Diego: Academic Press, 2000.
Dimensions Pulse Input of Mass M Continuous Input of Mass PerUnit Time M Starting at Time t = 0
Continuous Input of Mass PerUnit Time M in Steady State
.
.C= M exp
x[ [(x-r)v2D
.
.
v
C =M
exp-
8n D
t = 0 t = t1
3/2 3/2
to
π x Dy
x
t
[ [(x-vt)4D t
2 2
y
y 2z4D t
∞
Dz Dz 4n rπ DyDz
+ +z4D t
z y
x
z y
Plume at time t
z y