Lecture 3
Contaminant Transport
Mechanisms and Principles

BASIC DEFINITIONS
Ground surface
Vadose zone, unsaturated zone
Below ground surface
(BGS)
Capillary fringe
Water table
Saturated zone
Confining bed
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 air water solid
Contaminant concentrations:
C w
, mg/L concentration in water
C g
, mg/L or ppmv concentration in gas
C s
, gm/kg concentration in solids

PARTITIONING RELATIONSHIPS
Solid ↔ water
C s
C w
= K d
= mg/kg solid mg/L water
K d
= partition coefficient
Water ↔ vapor C g
C w
H = Henry’s Law constant
= H = mol/m 3 air mg/m 3 water

HENRY’S LAW CONSTANT
H has dimensions: atm m 3 / mol
H’ is dimensionless
H’ = H/RT
R = gas constant = 8.20575 x 10 -5 atm m 3 /mol °K
T = temperature in °K

NOTE ON SOIL GAS CONCENTRATION
Soil gas is usually reported as: ppmv = parts per million by volume
C g
(ppmv) =
C g
(mg/L) × 24,000 mL/mole molecular weight g/mole

VOLUME REPRESENTATION
Void volume, V
V
Gas volume, V g
Water volume, V
W
Solid volume, V
S
Total volume, V
T

VOLUME-RELATED PROPERTIES
Bulk density = ρ b
= mass of solids total volume
Porosity = n = θ = V
V
/V
T
Volumetric water content or water-filled porosity = θ
W
= V
W
/V
T
Saturation = S = V
W
/V
V
Gas-filled porosity = θ g
(or θ a
) = V g
/V
T
θ
W
+ θ g
= n

CONTAMINANT CONCENTRATION
IN SOIL
Total mass in unit volume of soil:
C
T
= ρ b
C s
+ θ w
C
W
+ θ g
C g
If soil is saturated, θ g
C
T
= ρ b
C s
+ n C
W
= 0 and θ
W
= n

NOMENCLATURE FOR DARCY’S LAW
Q = K i A
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 n e
, effective porosity

ADVECTIVE FLUX
Flowing ground water carries any dissolved material with it → Advective Flux
J
A
= 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
J
D
= − D
O
∂ C
∂ x in surface water !!!
D
O is molecular diffusion coefficient [L 2 /T]

DIFFUSIVE FLUX
In porous medium, geometry imposes constraints:
J
D
= −
D
O n
∂ C
∂ x
= − D * n
∂ C
∂ 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: τ =
⎛
⎜⎜
L
L e
⎞
⎟⎟
2
L = straight-line distance
L e
= actual (effective) path
τ ≈ 0.7 for sand

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 arrives first, then B, then C → mechanical dispersion
A

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:
J
M
= − D
M n
∂ C
∂ x
Dispersion coefficient, D
M
= α
L u
α
L
= 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
J
J
=
=
=
J
A nuC nuC
+ J
−
−
D
+ J
M
D
D
H
* n
∂ C
∂ C
∂ x
∂ x
− D
M n
∂ C
∂ x
D
H
= D* + D
M
= τ D
O
+ α
L 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
− ∇ • J
−
∂ J
∂ x
±
±
Sources and sinks due to reactions
S / S
S / S
(1)
(2)

TRANSPORT EQUATION
C
T
= total mass (dissolved mass plus mass adsorbed to solid) per unit volume
= ρ b
C
S
+ n C
W
= ρ b
C
S
+ n C
Note: W subscript dropped for convenience and for
Consistency with conventional notation
(3)
Substitute Equation 3 into Equation 2:
∂
(
ρ b
C
S
)
+
∂
( )
∂ t ∂ t
= −
∂
∂ x nuC − D
H n
∂ C
∂ x
± S / S
↑ no solid phase in flux term
(4)

TRANSPORT EQUATION
C
S
= K d
C by definition of K d
Assume spatially uniform n, ρ b
, K d
, u, and D
H and no S/S
(
ρ b
K d
∂ C
∂ t
= −
+ n
) ∂ C
∂ t u
ρ b
K d n
=
+
− nu
∂ C
∂ x n
∂ C
∂ x
+
+ nD
H
ρ b
K d n
∂
∂
D
H
2 x
+
C
2 n
∂ 2 C
∂ x 2
(5)
(6)

TRANSPORT EQUATION
“Retardation factor”, R d
ρ b
K d n
+ n
= 1 +
ρ b
K d n
= R d
Substituting Equation 7 into Equation 6:
(7)
∂ C
∂ t
= − u
R d
∂ C
∂ x
+
D
H
R d
∂ 2 C
∂ x 2
(8)
Effect of adsorption to solids is an apparent slowing of transport of dissolved contaminants
Both u and D
H 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
1.0
0.8
0.6
0.4
0.2
0.0
t
1 t
2 t
0
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 C .
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:
C
( )
=
C
2 o erfc
⎝
⎛
⎜
⎜
R
4 d x
R d
− ut
D
H t
⎟
⎟
⎠
⎞
Where C o
= C(x=0, t) = constant concentration at source location x = 0
Solution is a front moving with velocity u/R d

1.0
0.9
0.84
0.8
0.7
0.6
0.50
0.4
0.3
0.2
0.16
0.1
0.0
+ s x = ut/R d s
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.
x

Moving front of contaminant from constant source
10
9
8
7
6
5
4
3
2
1
0
0 t = 1 ut = 1
1 2 t = 3 ut = 3
3 t = 5 ut = 5
4 5
Distance, x
6 7
C
0
= 10 u = 1
D
H
= 0.1
R d
= 1
8
Moving front of contaminant from constant source
9

Effect of dispersion coefficient

10
9
8
7
6
5
4
3
2
1
0
0 t = 3 ut = 3
R d
= 2
1
Effect of R d
on moving front of contaminant
Effect of retardation
2 t = 3 ut = 3
R d
= 1
3 4 5
Distance, x
6 7
C
0
= 10 u = 1
D
H
= 0.1
8 9

1-D SOLUTIONS
Transport of a Conservative Substance from Pulse and Continuous Sources
Dimensions Pulse Input of Mass M
M ,
.
1-D
are instantaneous or continuous plane sources
M
M
L 2
L
M
2 T
C =
2n p
M t D x exp -
[ [
2 x = 0 t = 0 t = t
1 v
Unit Time M
.
Starting at Time t = 0
C =
.
M
2nv erfc x = 0 v
Mass input here
Front at time t
Unit Time M
.
in Steady State
C =
M nv
.
( for x > 0 ( x = 0 v to ∞
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 Pulse Input of Mass M
M , M
.
2-D
are instantaneous or continuous line sources
M
.
M
[
M
L
M
L-T
[
[[
C=
4n p t
M
D x
D exp y
[
(x-vt) 2
+ y 2
4D t
[ v t = 0 t = t
1 y x
Unit Time M
.
Starting at Time t = 0
C=
4n p
M
1/2
.
(vr) 1/2 D y exp erfc y x v
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.
Unit Time M
.
in Steady State
C=
2n p
M
1/2
.
(vr) 1/2 D y exp y x v to ∞

3-D SOLUTIONS
Transport of a Conservative Substance from Pulse and Continuous Sources
Dimensions Pulse Input of Mass M
M , M
.
3-D
are instantaneous or continuous point sources
M
.
M
[
M
L
M
T
[
[[
C = exp-
[
8n p
(x-vt)
2
+
M y 2
D x
D y
D z
+ z 2 v z y x t = 0 t = t
1
[
C=
Unit Time M
.
Starting at Time t = 0
M
.
D y
D z exp erfc z y v
Plume at time t z y
Unit Time M
.
in Steady State
C=
M
.
D y
D z exp v to ∞
Adapted from: Hemond, H. F. and E. J. Fechner-Levy. Chemical Fate and Transport in the Environment.
2nd ed. San Diego: Academic Press, 2000.