Advanced Geotechnical Engineering
ES4D8
Contaminated Land (Lecture 5)
Mohaddeseh Mousavi-Nezhad
Room: D211
Email:m.mousavi-nezhad@warwick.ac.uk
31/05/2016
The University of Warwick
1
Contaminant Transport Mechanisms and
Principles
Micro View of Unsaturated Zone
Contaminant concentrations:
Cw , mg/L
concentration in water
Cg , mg/L or ppmv
concentration in gas
Cs , gm/kg
concentration in solids
Partitioning Relationships
Solid
Water
Kd = partition coefficient
Water
Cs
mg / kg solid
= Kd =
Cw
mg / L water
Vapor
H = Henry’s Law constant
Cg
mol / m3 air
=H=
Cw
mg / L water
Contaminant Concentration in Soil
Total mass in unit volume of soil:
CT = rbCs + qwCw + q gCg
If soil is saturated, q g = 0
and
CT = rbCs + nCw
qw = n
Advective Flux
u
u
Flowing groundwater carries any dissolved
material with it
Advective Flux
J A = n uC
mass/area/time
=mass flux through unit cross section due to
groundwater advection
q
u= = average linear velocity
n
For transport, n is ne, effective porosity and is
needed since no flow except in pores
Darcy’s Law-Reminder
The flow per unit area is specific discharge or Darcian velocity
Hydraulic gradient
K is hydraulic conductivity
Diffusive Flux
Movement of mass by molecular diffusion (Brownian
motion)- Proportional to concentration gradient
In surface water
¶C
J D = -D0
¶x
D0 is molecular diffusion coefficient [L2/T]
Diffusive Flux
In porous medium, geometry imposes constraints:
¶C
* ¶C
J D = -t D0 n
= -D n
¶x
¶x
t is tortuosity factor
D*is effective diffusion coeffcient
Factor n must be included since diffusion is only in pores
Diffusive Flux--Tortuosity
B
A
Diffusive Flux--Tortuosity
Solute must travel a tortuous path, winding through
pores and around solid grains
Empirical expression:
æLö
t =ç ÷
è Le ø
L is straight line distance
Le is actual (effective) path
t » 0.7 for sand
2
B
A
Diffusive Flux
• Diffusion can occur when there is no hydraulic gradient
driving flow and water is static.
• Diffusion in groundwater systems is a very slow process.
• It is important in vapor transport in unsaturated zone
Mechanical Dispersion
Mechanical Dispersion
Dispersion is process of transport of solute as a
result of the fluctuation of fluid velocity through
pore space. –dispersion flux
¶C
J m = -Dm n
¶x
Dispersion coefficient, Dm = a L u
a L longitudinal dispersivity (units of length)
Advection &Mechanical Dispersion
Actual Observation of Plumes
Bromide Plume
Bromide concentration
USGS Monitoring Network
Observed Bromide PlumeVertical View
Transport Equation
Combined transport from advection, diffusion,
and dispersion (in one dimension)
J = J A + JD + Jm
¶C
¶C
J = nuC - D n
- Dm n
¶x
¶x
¶C
J = nuC - DH n
¶x
*
DH = D* + Dm = t D0 + al u
= Hydrodynamic dispersion coefficient
Transport Equation
Consider conservation of mass over control
volume (REV) of domain
REV=Representative Elementary Volume
REV must contain enough pores to get a
meaningful representation (statistical average or
model)
Transport Equation
Change in
contamina
nt mass
with time
¶CT
¶t
¶CT
¶t
Flux in less
flux out of
REV
=
- Ñ· J
=
¶J
¶x
Sources and
sinks-for
example due
to reactions
±
±
S
S
Transport Equation
CT = total mass (dissolved mass plus mass
adsorbed to solid) per unit volume
= rbCs + nCw = rbCs + nC
Note: w subscript dropped for convenience and
for consistency with conventional notation
Substitute above equation in transport
equation achieved in previous slide:
¶ ( rbCs ) ¶ ( nC )
¶æ
¶C ö
+
= - ç nuC - DH n ÷ ± S
¶t
¶t
¶x è
¶x ø
No solid phase in flux term
Transport Equation
Cs = Kd C By definition of K d
Assume spatially uniform n, K d , r b , u, DH and no S
¶C
¶C
¶C
( rb Kd + n) = -nu - nDH 2
¶t
¶x
¶x
2
¶C
u
¶C
DH
¶2 C
=æ rb K d + n ö ¶x æ rb K d + n ö ¶x 2
¶t
ç
÷
ç
÷
è
ø
è
ø
n
n
Transport Equation
Retardation factor, Rd
rb K d + n
n
=1+
rb K d
n
= Rd
Substituting in transport equation
¶C
u ¶C DH ¶ C
=2
¶t
Rd ¶x Rd ¶x
2
Effect of adsorption to solid is an apparent
slowing of transport of dissolved contaminants
Transport Equation
Retardation factor, Rd
rb K d + n
n
=1+
rb K d
n
= Rd
Substituting in transport equation
¶C
u ¶C DH ¶ C
=2
¶t
Rd ¶x Rd ¶x
2
Effect of adsorption to solid is an apparent
slowing of transport of dissolved contaminants
Solution of Transport Equation
Equation can be solved with a variety of boundary
conditions
In general, equation predicts a spreading Gaussian cloud
1-D Solution of Transport Equation
For instantaneous placement of along-lasting
source (for example, a spill that leaves a residual in
the soil), solution is:
æ R x - ut ö
C0
÷÷
C ( x, t ) = erfc çç d
2
è 4Rd DH t ø
where C0 = C ( x = 0,t ) = constant concentration at
source location x=0
Contaminant distribution
Effect of Rd on Contaminant distribution