SECTION 7a

Contaminant Transport

1. Sources of Contamination
2. Transport Phenomena

CEE 9870 Winter 2021 Dr. Chris Power

Groundwater Contamination
Why contaminant (or solute) transport modelling?
Environmental problems: contamination of drinking water, saltwater intrusion, chemical
discharge to surface waters
Geological questions: ion migration, ore deposition
Groundwater Contamination
Common Sources
Groundwater Contamination
How does groundwater get contaminated by industry?

• Leaking lagoons
• Leaking underground storage tanks
• Leaking landfills
• Leaking pipelines
Groundwater Contamination
Industrial Pollutants
Hazardous Organic Liquids
• Chlorinated solvents Emerging Pollutants of Concern
• PCB oils • Endocrine Disrupting Chemicals
• Creosote • Nanoparticles
• Coal tar • Pharmaceuticals
• Pesticides
• Gasoline and diesel
• Jet fuel
• Toluene
• Heating oils

Inorganic High Priority Pollutants

• Arsenic, Lead, Mercury
• Benzo(a)pyrene
• Hexavalent Chromium
• White Phosphorus
• Radionuclides
Groundwater Contamination
Waste Disposal
Groundwater Contamination
Pumping
Groundwater Contamination
Non-Aqueous Phase Liquids: Light (LNAPLs) and Dense (DNAPLs)

(Source: http://www.egr.msu.edu/igw/)
Groundwater Contamination
Field Scale DNAPL Source Zone Simulation
Groundwater Contamination
Contaminated Site Remediation

Animation
Transport Phenomena
Source Characterization: Spatial
Point Source Nonpoint Source
▪ Single polluter accountable ▪ Several polluters accountable
▪ comes from one location ▪ comes from many places
▪ Can usually be traced ▪ Difficult to trace
▪ Examples: brewery, treatment facility, ▪ Examples: bagricultural runoff, urban
industrial stack, underground storage tank runoff, motor emissions from a highway

Point Source

Line Source

Area Source

Depends on: (i) spatial scale, (ii) spatial extent of impact, and (iii) need to differentiate concentrations
at different locations
Transport Phenomena
Source Characterization: Temporal
Instantaneous: Continuous:
i.e., from a spill Constant behaviour or periodic
variation in source

vx
Qr
Cr

x= 0 x= vxt

vx

Animation Animation

Degree of temporal resolution required for source term depends on: (i) problem definition, (ii) time
scale of problem, and (iii) dynamic response characteristics of contaminant in the environment
Transport Phenomena
Transport of contaminant along with mean or bulk movement of water or air
t = t0 t = t0 + dt

Advection
Transport Phenomena
Advection
▪ Mass in - Mass out = Accumulation (𝛥 storage)
Difference in mass of contaminant entering and leaving an element in space (control volume) in time
equals the increase in mass within the element in time

J z +z J y +y
J water = Cv

y + y

y

Jx J x +x
y
z + z
z
z z
y Jy x
x x + x
x Jz
Transport Phenomena
Advection
▪ Mass in - Mass out = Accumulation (𝛥 storage)
Difference in mass of contaminant entering and leaving an element in space (control volume) in time
equals the increase in mass within the element in time

Cvz Cv y
J water = Cv

y + y

y

Cvx Cvx
y
z + z
 kg   m   kg 
 m3    s  =  m 2 s  z
z
contaminant
Cv y x
mass flux x x + x

Cvz * mass = concentration x volume

Transport Phenomena
Advection
Mass - Mass = Accumulation (𝛥 storage)
in out
[C( x ) vx ( x ) yz ] − [C( x +x ) vx ( x +x ) yz ]
+[C( y ) v y ( y ) xz ] − [C( y +y ) v y ( y +y ) xz ]
+[C( z ) vz ( z ) xy ] − [C( z +z ) vz ( z +z ) xy ]
Divide through by xyz :
 C( x +x ) vx ( x +x ) − C( x ) vx ( x )   C( y +y ) v y ( y +y ) − C( y ) v y ( y )   C( z +z ) vz ( z +z ) − C( z ) vz ( z )  C
− − − =
 x   y    z  t
The left-hand side is a derivative in the limit as 𝛥x shrinks to zero:

C C C C
3D Advection
= −v x − vy − vz
Equation t x y z
Transport Phenomena
Advection


Cvx Cvx + (Cvx )
x

x

1D Advection C C
= −v x
Equation t x
Transport Phenomena
Advection Advection + Diffusion

Concentration
Concentration

Distance Distance

Advection + Dispersion Advection + Dispersion + Reaction

Concentration
Concentration

with decay

with sorption

Distance Distance
Transport Phenomena
Transport of contaminant along with mean or bulk movement of water or air
t = t0 t = t0 + dt

Advection

Mixing of contaminants due to gradients in contaminant concentrations

Diffusion/
Dispersion
Transport Phenomena
Diffusion/Dispersion
Mixing of contaminant at larger scales due to mixing of fluid carrying the contaminant

Turbulent diffusion or eddy diffusion:

▪ main mixing mechanism in the Shear dispersion:
atmosphere and large water bodies ▪ mixing due to velocity gradients in the
fluid (i.e., in streams and estuaries)

Hydrodynamic dispersion:
▪ mixing of contaminant as the water
encounters channels or other
heterogeneities such as soil pore
structure (groundwater)
Transport Phenomena
Diffusion/Dispersion
Dispersion accounts for the mixing

Macroscopic (hydrodynamic) dispersion

Homogeneous aquifer

Microscopic dispersion Macroscopic dispersion

Microscopic or local scale dispersion

Heterogeneous aquifers

Figure from Freeze & Cherry (1979)

Transport Phenomena
Diffusion/Dispersion
Mixing of contaminant at larger scales due to mixing of fluid carrying the contaminant

Molecular diffusion:
▪ Predominates at very small scales or in quiescent fluids
▪ Occurs through random motions of molecules

Contaminant is transported from

higher C lower C

Transport ceases when equilibrium

is reached

In this course, we use the term ‘dispersion’ in the broad sense to mean total mixing
independent of whether it is turbulent diffusion, shear dispersion, hydrodynamic
dispersion, or by molecular diffusion.
Transport Phenomena
Diffusion/Dispersion y

Molecular diffusion
z
VOLUME A VOLUME B
Ma Mb

x
• Mass transfer [kg/s] in x-direction from Volume A to Volume B:

M a→b = kMa k is a proportionality constant

• Mass transfer [kg/s] in x-direction from Volume B to Volume A:

M b→a = kMb
• Net mass transfer [kg/s]:

M = k (M a − M b )
Transport Phenomena
Diffusion/Dispersion VOL A VOL B
Ma Mb

Molecular Diffusion
▪ Flux (Jx) is net mass transfer per unit area of interface [kg/s/m2]:
(M a − M b ) (M a − M b )
Jx = k =k
A yz
▪ Convert to concentrations: Remember:
(C − Cb ) M = (C )xyz
Jx = k a xyz
yz
▪ If x = y = z: (C − C ) (C − Cb )
J x = k a 2 b x3 = k x 2 a
x x
(C − Cb )
J x = k x 2 a
x
Let kx = Dm and taking the limit as x goes to zero:
2
▪

Fick’s Equation

▪ Mass flux of a solute due to diffusion is proportional to concentration gradient

 Transport Phenomena
Diffusion/Dispersion

Mixing processes in a river

Mass injected

Depth

Vertical Transverse
x=0 mixing length
mixing length

Distance for contaminant to be well mixed is called the “mixing length”

Typically vertical mixing length is << transverse mixing length

Transport Phenomena
Diffusion/Dispersion

Mixing/dispersive processes in a river (open channel)

• Longitudinal dispersion: mixing of contaminant down the river (DL,

longitudinal dispersion coefficient)

• Transverse mixing: mixing of contaminant across the channel (Dt,

transverse mixing coefficient)

• Vertical mixing: mixing of contaminant with depth in channel (Dv, vertical

mixing coefficient)
Transport Phenomena
Diffusion/Dispersion
Transverse mixing length
• When contaminant/dye is injected into centre line of river (channel):
v xW 2
L m = 0 .1 where W = channel width
Dt
• When contaminant/dye is injected at side of river (channel):
v xW 2
Lm = 0.4
Dt
• Dt is based on shear velocity (friction along sides of channel) and turbulence

Vertical mixing length

• When contaminant/dye is released at mid-depth:
vx D 2 where D = channel depth
Lm = 0.25
Dz Dz = vertical mixing coefficient
• When contaminant/dye is injected at top or bottom of (channel):
vx D 2
Lm = 0.4
Dv
• Dv is based on shear velocity (friction along bottom of channel), turbulence and buoyancy
 Transport Phenomena
Diffusion/Dispersion

Mixing processes in a river

Mass injected vxW 2

Lm = 0.1
Dt
W

Depth

vx D 2
Lm = 0.25
Dz

Distance for contaminant to be well mixed is called the “mixing length”

Typically vertical mixing length is << transverse mixing length
If point of interest (observation point) is less than mixing length, then the contaminant is
still mixing (diffusing/dispersing) and we need to include that dimension
▪ e.g., if observation distance < tranverse mixing length, then need to include transverse (y)
direction
Transport Phenomena
Diffusion/Dispersion

2D (x,y) dispersion – continuous source

Transport Phenomena
Diffusion/Dispersion

2D (x,y) dispersion – continuous source

 Dispersivity
Transport Phenomena
Diffusion/Dispersion
Diffusion/Dispersion coefficient:
Hydrodynamic dispersion is typically a function of velocity

D = v + Dm
Dm = molecular diffusion coefficient (L2/T)
 = dispersivity (L)
D = dispersion coefficient (L2/T)

Diffusion/Dispersion coefficient in each direction

Dx = xvx + Dm DL = LvL + Dm
Dy = yvx + Dm DT = TvL + Dm
Dz = zvx + Dm DV = VvL + Dm

▪ L can range from 0.1 mm to 100 m depending on scale

of model (ie. laboratory vs. field) and heterogeneities
▪ T and V are typically lower than L by factor of 5 – 20 T V Dm
depending on hydrogeological conditions
(heterogeneities, layering etc.) L L
▪ L ,V, T typically have high level of uncertainty
(difficult to measure)

Example of dispersion (in MODFLOW)

Transport Phenomena
Diffusion/Dispersion
Field scale dispersion: representing heterogeneities

Animation
Transport Phenomena
Diffusion/Dispersion
▪ Mass in - Mass out = Accumulation (𝛥 storage)
Difference in mass of contaminant entering and leaving an element in space (control volume) in time
equals the increase in mass within the element in time
C C
− Dz − Dy
z y

y + y

y
C C
− Dx − Dx
x x
y
z + z
z
z
C x
− Dy x x + x
y
C
− Dz
z
Transport Phenomena
Diffusion/Dispersion
Mass - Mass = Accumulation (𝛥 storage)
in out
C C
[− D( x ) yz ] − [− D( x +x ) yz ]
x x
C C C
+[− D( y ) xz ] − [− D( y +y ) xz ] = x y z
y y t
C C
+[− D( z ) xy ] − [− D( z +z ) x y ]
z z
Divide through by xyz :
 C C   D C C   C C 
D − D  − D  D − D
 ( x +x )
x
( x )
x  − 
( y +y )
y
( y)
y  ( z +z )
z
( z )
z  = C
−   −  
  x    y    z  t
     
The left-hand side is a derivative in the limit as 𝛥x shrinks to zero:

3D Diffusion/ C  2C  2C  2C
= Dx 2 + Dy 2 + Dz 2
Dispersion Equation t x y z
Transport Phenomena
Diffusion/Dispersion
3D Diffusion/Dispersion Equation
▪ Second-order partial differential
C  2C  2C  2C
= Dx 2 + Dy 2 + Dz 2 equation (PDE)
t x y z ▪ Initial conditions and boundary
conditions required to solve equation
1D Diffusion/Dispersion Equation

C  2C
= Dx 2 ▪ Dispersion is also a random process and
t x
depends on concentration gradient
▪ In environmental applications dispersion
typically modeled same way as molecular
diffusion (flux occurs at rate directly
proportional to concentration gradient)
▪ Transport modeled by neglecting all but the
dominant mixing (diffusion/dispersion)
process
▪ Use symbol D for all forms of
diffusion/dispersion. D is globally called the
dispersion coefficient.
Transport Phenomena
Transport of contaminant along with mean or bulk movement of water or air
t = t0 t = t0 + dt

Advection

Mixing of contaminants due to gradients in contaminant concentrations

Diffusion/
Dispersion

Advection +
Diffusion/Dispersion
Transport Phenomena
Advection + Diffusion/Dispersion
▪ Mass in - Mass out = Accumulation (𝛥 storage)
Difference in mass of contaminant entering and leaving an element in space (control volume) in time
equals the increase in mass within the element in time
C C
Cvz − Dz Cv y − Dy
z y

y + y

y
C C
Cvx − Dx Cvx − Dx
x x
y
z + z
z
z
C x
Cv y − Dy x x + x
y
C
Cvz − Dz
z
 Transport Phenomena
Advection + Diffusion/Dispersion
Mass - Mass = Accumulation (𝛥 storage)
in out
C C
[C( x ) vx ( x ) − Dm ( x ) yz ] − [C( x ) vx ( x ) − Dm ( x +x ) yz ]
x x
C C C
+[C( y ) v y ( y ) − Dm ( y ) xz ] − [C( y ) v y ( y ) − Dm ( y +y ) xz ] = x y z
y y t
C C
+[C( z ) vz ( z ) − Dm ( z ) xy ] − [C( z ) vz ( z ) − Dm ( z +z ) xy ]
z z
Divide through by xyz :
 C C   C C C   C C 
− Dm ( y +y ) − C( y ) v y ( y ) + Dm ( y )
 C( x +x ) vx ( x +x ) − Dm ( x +x ) x − C( x ) vx ( x ) + Dm ( x ) x   ( y +y ) y ( y +y ) − Dm ( z +z ) − C( z ) vz ( z ) + Dm ( z )
v
y   ( z +z ) z ( z +z )
C v
y z z  = C
 − − 
 x   y   z  t
 
  
  

The left-hand side is a derivative in the limit as 𝛥x shrinks to zero:

3D Advection + C C  2C C  2C C  2C
Diffusion/Dispersion = −vx + Dx 2 − v y + D y 2 − vz + Dz 2
Equation t x x y y z z
Transport Phenomena
Advection + Diffusion/Dispersion
3D Advection + Diffusion/Dispersion Equation

C C  2C C  2C C  2C
= −vx + Dx 2 − v y + D y 2 − vz + Dz 2
t x x y y z z

1D Advection + Diffusion/Dispersion Equation

C C  2C
= −v x + Dx 2
t x x
2D Advection + Diffusion/Dispersion Equation
C C  2C  2C Advective velocity only in the
= −vx + Dx 2 + Dy 2
t x x y (longitudinal) x-direction

Reminder: 1D Equations for Other Transport Phenomena

C C
= −vx 1D Advection Equation
t x

C  2C
= Dx 2 1D Diffusion/Dispersion Equation
t x
Transport Phenomena
Advection + Diffusion/Dispersion

Determining whether it is advection, diffusion/dispersion, or both:

vL v = velocity
▪ Peclet Number Pe = L = length
D D = dispersion/diffusion

▪ If Pe << 1 (e.g., Pe < 0.1), dispersion-dominated. This can occur in lakes and reservoirs.

▪ If Pe >> 1 (e.g., Pe > 10), advection-dominated. This often occurs in rivers.

▪ If Pe ~ 1 (e.g., 0.1 < Pe < 10), advection + dispersion. Therefore, no approximation of

equation, and full equation must be used.

Pe is dependent on zone of interest: for larger distances (L), Pe is large and

advection dominates
Transport Phenomena
Advection + Diffusion/Dispersion
Advection Advection + Diffusion

Concentration
Concentration

Distance Distance

Advection + Dispersion Advection + Dispersion + Reaction

Concentration
Concentration

with decay

with sorption

Distance Distance
Transport Phenomena
Transport of contaminant along with mean or bulk movement of water or air
t = t0 t = t0 + dt

Advection

Mixing of contaminants due to gradients in contaminant concentrations

Diffusion/
Dispersion

Advection +
Diffusion/Dispersion

Reaction: sorption or decay

Advection +
Diffusion/Dispersion
+ Reaction
Transport Phenomena
Advection + Diffusion/Dispersion + Reaction
3D Advection + Diffusion/Dispersion + Reaction Equation

C C  2C C  2C C  2C
= −vx + Dx 2 − v y + Dy 2 − vz + Dz 2 − S Sink/source term
t x x y y z z (reaction)

1D Advection + Diffusion/Dispersion + Reaction Equation

C C  2C
= −v x + Dx 2 − S
t x x

S: rate solute mass is lost or added to the solution phase (mass of solute per unit
volume per unit of time, M/L3/T)

Homogeneous reactions – occur in the aqueous phase (e.g., first order degradation)

Heterogeneous reaction – involve a solid phase or surface (e.g., sorption, mineral

dissolution)
Transport Phenomena
Advection + Diffusion/Dispersion + Reaction
First order decay: reaction takes place within the aqueous phase

S = kC First order rate constant

k (T-1) Why does unit of k mean?

Example of decay rate (in MODFLOW)

Advection +
Diffusion/Dispersion
+ Decay
Transport Phenomena
Advection + Diffusion/Dispersion + Reaction
Sorption: surface reaction where solute is adsorbed to solid surfaces thereby
delaying its transport (retardation)
▪ Adsorption: adhesion of solute to the solid phase
▪ Desorption: release of solute from the solid phase

Sorption depends on e.g.,:

▪ Concentration and characteristic of contaminant
▪ Soil type, mineralogy
▪ Chemical composition of water (pH of water)

▪ Decreases concentration of solute in aqueous phase until sorption capacity is

reached
▪ No net loss of solute (its transport is just retarded)

Advection +
Diffusion/Dispersion
+ Sorption
Transport Phenomena
Advection + Diffusion/Dispersion + Reaction
Sorbed vs. Non-sorbed Species
▪ Sorption slows transport of (retards) the contaminant
▪ Sorbing contaminants will eventually ‘get there’
▪ Some compounds irreversibly adsorbs to soil

Concentration
profile
Transport Phenomena
Advection + Diffusion/Dispersion + Reaction
Sorption isotherms
▪ Concentrations of solute in adsorbed phase relative to dissolved phase is given
by an isotherm

▪ Linear isotherm: qe = adsorbed concentration (mass of adsorbed species per mass of dry
aquifer material (-) )
qe = KdC C = dissolved concentration (M/L3)
Kd = distribution coefficient (L3/M)

▪ Freundlich isotherm:

qe = KdC1/ n qe

1/n can range from 0.5 to 2 and qe Freundlich

depends on the solute species,
soil type, aqueous chemistry etc. isotherm
Transport Phenomena
Advection + Diffusion/Dispersion + Reaction
Sorption in the Transport Equation
qe = adsorbed concentration (mass of adsorbed
C  2C C
= Dx 2 − vx −S species per mass of porous media)
t x x b = bulk density of porous media
ne = effective porosity
C  2C C b qe
= Dx 2 − vx −
t x x ne t
qe qe C
= •
t C t
For a linear isotherm:
qe
= Kd
C
C  2C C b C
= Dx 2 − vx − Kd Example of sorption (in MODFLOW)
t x x ne t
Sorption animation
Transport Phenomena
Advection + Diffusion/Dispersion + Reaction
When is reaction important relative to advection and diffusion/dispersion?

Damkohler number: (reaction rate vs transport rate)

kLc
• For comparing advection with reaction: Da =
vx

kLc 2
• For comparing dispersion with reaction: Da =
Dx

• If the Damkohler number is very small (< 0.01), then chemical reaction can
be neglected compared to transport

• For larger distances, chemical reaction will be more important

Transport Phenomena
Advection + Diffusion/Dispersion + Reaction

Observation well

What will the concentration

vs. time look like at these
observation wells?
Transport Phenomena
Advection + Diffusion/Dispersion + Reaction
Breakthrough Curve: Sorption of Contaminants

Which contaminant is most retarded (adsorbed)?

Transport Phenomena
Advection + Diffusion/Dispersion + Reaction
Breakthrough Curve: Sorption of Contaminants

Breakthrough long tail

curve

Concentration
profile
Transport Phenomena
Advection Advection + Diffusion

Concentration
Concentration

Distance Distance

Advection + Dispersion Advection + Dispersion + Reaction

Concentration
Concentration

with decay

with sorption

Distance Distance
Transport Phenomena
Concentration Profiles and Breakthrough Curves
Transport Phenomena
Concentration Profiles and Breakthrough Curves
Transport Phenomena
Concentration Profiles and Breakthrough Curves