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.