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Management and remediation of
sites for extractive industries
Subhasis GhoshalDepartment of Civil Engineering
McGill University, Canada
Universidad ORT Uruguay, Montevideo , Apri l 4-8, 2016
Organizer: Prof. Lorena Betancour, Universidad ORT Uruguay
Introduction
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Introduction
• Why do sites get contaminated?
• How do we assess contamination at a site?
• What alternatives are available to clean upcontaminated soil and ground water efficiently?
– technologies: scientific principles, designfundamentals, implementation issues
– management: choice of technology, legislation
• how clean is clean?
– technologically achievable limits vs risk to
receptors
Groundwater Contamination
• Aquifers: important source of water formunicipal, agricultural and industrial use.
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“A 2004 EPA report estimated that it will take 30 to 35 years and cost
up to $250 billion to clean up the nation’s hazardous waste sites”
Over 10,000 contaminated sites in Canada alone.Cost of clean up in billions of dollars!
Need for sustainable cost-effective solutions
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• Routine (accidental?) spills
Sources of Contamination
Underground Storage Tanks
Active efforts for site remediation in
the U.S. (and elsewhere) started only
in the early 1980’s…..
What started it?
1980 Superfund Law passed by
U.S. President Jimmy Carter
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• What are pathways of
contaminant exposure?
• What should be clean-
up levels?
• What is to cleaned up:
soil, water, source
disposal zone?
Love Canal: An in-famous contaminated site
• 70-acre industrial landfill located in Niagara Falls, New York
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Love Canal
• Originally a canal excavated in the 1890s for an unfinished
hydroelectric project
• 1942 – 1952: Hooker Chemicals and Plastics (Occidental
Chemical Corporation) disposed of 21,000 tons of hazardous
wastes in the canal – including solvents
• 1953: area was covered and the property developed, including
the construction of an elementary school
• Included in the deed transfer was a "warning" of the chemical
wastes buried on the property and a disclaimer absolvingHooker of any further liability
Love Canal
• Complaints of odors and
chemical residues began
in the 1960’s, increased in
the 1970’s, as heavy
rainfall caused the
groundwater to rise,
flooding area basements
Spring 1978
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Pathways of Contaminant Exposure• Indoor Air - Vapor Intrusion: Volatile chemicals in contaminated soil
or groundwater can migrate through subsurface soils and into indoor
air spaces of overlying buildings
Love Canal• May 1978 - EPA concluded
from basement air sampling
that vapors are a serious
health threat
• August 1978 – President
Jimmy Carter declared the
Love Canal area a federal
emergency
• More than 900 families were
evacuated
Engelhaupt,
Environ. Sci. Technol. 2008, 42, 8179–8186
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Love Canal
• 20,000-plus tons of chemicals buried at Love
Canal are there still today
• EPA deemed it too dangerous to try to remove
them
Engelhaupt, , Environ. Sci. Technol. 2008, 42, 8179–8186
Love Canal
• Site publicity directly spurred
passage of EPA’s Superfund law in
1980
• December 1995 - Occidental ordered
to pay $129 million settlement
• Residents returned to portions of the
site in late 1990’s
Engelhaupt,
Environ. Sci. Technol. 2008, 42, 8179–8186
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Love Canal
• Epidemiological study into potential health effects ongoing
– Children born at Love Canal twice as likely as children in
other parts of the county to be born with a birth defect
– Negative reproductive effects for the exposed population
over multiple generations
– The draft also reported elevated rates of kidney, bladder,
and lung cancers at Love Canal
A Canadian Example…..
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Oil Sands Mining in Alberta:
Contaminated Site Legacy in the Making?
Oil Sands: Site Contamination
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Subsurface Contamination Issues Changing?
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Contaminated Site
Management Process
The Contaminated Site Management &
Remediation Process
• Preliminary Assessment/Site Inspection
• Remedial Investigation/Feasibility Study
• Records of Decision – Remediation? Containment?
• Remedial Design/Remedial Action
• Long-term Monitoring
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• Conceptual Contamination scenario from site assessment
– Intensive site data requirements
Site Assessment
Remedial Investigations & Feasibility Studies(RI/FS)
• Conduct Field Investigations to characterize site
– Site physical characterization (surface features,geology, hydrogeology, population & land use,ecology)
– Sources of contamination
– Nature and extent of contamination
• Data analysis to establish contaminant fate andtransport – determine risk of exposure to receptors
• Treatability studies – determine remedial objectives
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Remedial Investigations & Feasibility Studies
(RI/FS)
• Data analysis to establish contaminant fate andtransport – determine risk of exposure to receptors
– Pollutant levels in soil vapour phase? in
groundwater?
– Potential for contaminant migration?
• Treatability studies
– Determine remedial objectives− Feasibility of remediation systems
Resistivity
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• Ground Penetrating Radar
− radio frequency waves
− Shallow contamination detected (buried objects, oil
phase, water table…)
Auger and Rotary Drilling
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Direct-Push Rig for Cone Penetration Test
Equilibrium Partitioning
of Pollutants
(Chemical Thermodynamics)
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• By understanding the potential of achemical to move from oneenvironmental compartment to another,we can evaluate the direction andmagnitude of contaminant transfer andtransformation
• Important for understanding contaminantmigration and feasible remediationalternatives
Equilibrium Partitioning
• Equilibrium partitioning varies greatlyfrom one contaminant to other and fromone environmental system to another
• Understanding of the theoreticalframework very important to developpredictive capability (modeling)
• Focus on two cases – petroleum liquids (mixture of non-ionicorganic compounds)
– arsenic (metal)
Equilibrium Partitioning
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• Compound properties governing equilibriumphase transfer behaviour of nonionic organicchemicals:
– Vapour Pressure (Pure liquid/solid phase –air partitioning)
– Aqueous Solubility (Pure liquid/solid phase – water partitioning)
– Henry’s Law Constant (Air - waterpartitioning)
– Solvent-Water Partition Coefficient
– Sorption Coefficient (Soil-WaterPartitioning)
Equilibrium Partitioning
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Vapour Pressure
• Vapour Pressure: the pressure of the vapour of a
compound at equilibrium with its pure condensed phase
(liquid or solid)
• If a compound exists in the pure solid or liquid phase, the
maximum concentration of a compound that can be
attained in the air phase is its vapour pressure
)(or)( ,, s P l P ioio
Vapour Pressure:
Pure Phase Organic Liquid – Air Partitioning
Initial State Equilibrium State
Benzene conc. in
air?
Pure Phase Liquid (say benzene)
air
- Molecule of benzene
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Vapour Pressure
Range of
values
Environmental
Organic
Chemistry
Schwarzenbach,
R., Gschwend, P.,
Imboden, D.
Vapour Pressure
Effect of
temperature
significant
Environmental
Organic
Chemistry
Schwarzenbach,
R., Gschwend, P.,
Imboden, D.
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Soil Vapor Extraction
Johnson et al., 1990, Groundwater, 28:413-429
• Aqueous solubility: the maximum conc. ofa chemical that can be attained in a water
phase
• Exact definition: abundance of the chemical per unitvolume in the aqueous phase when the solution is inequilibrium with the pure compound in its actualaggregation state (gas, liquid, solid)
Aqueous Solubility
or, at w
iat w C C
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water
Pure Phase – Water Systems
(l)iat wC ,
Initial State Equilibrium State
Pure Phase Liquid of i
water
Pure Phase Solid of i
water
water
(s)iat
wC ,
Aqueous Solubility
Range of
values
Environmental
Organic
Chemistry
Schwarzenbach,
R., Gschwend, P.,
Imboden, D.
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Aqueous Solubility
What happens
during dissolution?
ΔHs =ΔH1 +ΔH2 +ΔH3 +ΔH4
EnvironmentalOrganic
Chemistry
Schwarzenbach,
R., Gschwend, P.,
Imboden, D.
Aqueous Solubility
Effect of
temperature
Environmental
Organic
Chemistry
Schwarzenbach,
R., Gschwend, P.,
Imboden, D.
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Pumping well
Storage
Tank
Leaked Oil Dissolved contaminants
Newell, C.J. and O’Connor, J.A. American Petroleum Institute, 1998
Chemical Potential
• Chemical Potential can be used to assess the tendency of a
compound i to be transferred from one system to another
or to be transferred within a system
• At constant T, P and composition, the Gibbs free energy
added to the system with each added increment of i is
referred to as the chemical potential µi of component i
• Chemical potential of i in each phase is equal atequilibrium
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Chemical Potential
Environmental
OrganicChemistry
Schwarzenbach,
R., Gschwend, P.,
Imboden, D.
Fugacity of compound i
Ideal gas
Pure organic
liquid
Non-ideal
liquid
solution
Environmental Organic Chemistry
Schwarzenbach, R., Gschwend, P., Imboden, D.
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Fugacity
• Fleeing tendency of a compound
• Relative fleeing tendencies used forcalculation of equilibrium partitioning
oi j
i j
i j
i j f x f ,••= γ Fugacity of a compound i in phase j =
Concentration of compound
Activity coefficient= f (comfort of compound
i in phase j)
Reference phase
fugacity
Organic Liquid Mixture (Oil) – Water
Partitioning
iw
ioil f f =
Fugacity of a compound i is equal in all phases when equilibrium is attained, e.g. for
an air phase – water phase system
Initial State Equilibrium State
c1A ?
Phase 2 contains mixture of A ( ), B ( )
Phase1
(water)
Phase2
(oil)
c1B ?
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Organic Liquid Mixture (Oil) – Water Partitioning
Initial State Equilibrium State
c1A ?
Phase 2 contains mixture of A ( ), B ( )
Phase1
(water)
Phase2
(oil)
Aw
Aw
Aw
Ao
Ao
Ao
A A
f x f x
f f
....
12
γ γ =
=Compound partitioned
Phase into which it is partitioned
o: oil
w: water
c1B ?
Partition coefficient for
Oil-Water System
Aow
Aw
Aw
Aoo
Aoo f x f x
,, .... γ γ =
Aoo
Aow
Ao
Aw
wo
Ao
f
f
x
x
,
,
phasewaterinAof Conc. phaseoilinAof Conc.tCoefficienPartiton
γ
γ =
==
Initial State Equilibrium State
c1A ?Phase1
(water)c1
B ?
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Air-Water Partitioning
• Equilibrium partitioning of chemicals between the gas
phase and an aqueous phase
• Henry’s Law Constant, K H (air-water distribution constant)
phaseaqueousincompoundof abundance
phasegasincompoundof abundance
i
i K H =
essdimensionl'
⋅=mol
L
L
mol
C
C K
w
a H
Solvent-Water Partitioning• Octanol-water partition constant K ow
– Octanol is a surrogate for natural organic phases (soils, sediments,
suspended particles)
phaseaqueousincompoundof abundance
phaseorganicincompoundof abundance
i
i K ow =
essdimensionl
⋅=mol
L
L
mol
C
C K
w
org
ow
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Soil-Water Partitioning
• Why is this important?
1. Reason for contamination of the soil phase
2. Controls the rate at which contaminants move
in soil and in subsurface systems (retardation)
3. Affects other pathways (i.e., volatilization,
oxidation processes, biodegradation, etc.) –
availability reduced
4. Must be understood and quantified to carry
out effective remediation
Soil-Water Partitioning• Governing variables:
– Properties of sorbent (solid)
• hydrophobicity, specific surface area, presence of
surface charges (clays)
– Properties of sorbate (contaminant)
• hydrophobicity (K ow
), Ionic charge
– Properties of fluid medium (water, air)
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Properties of Soils• Composition highly variable
• Example: silt loam soil
Properties of Soils
• Soil Organic Matter
1. Humic materials (humus): dark brown,
yellowish polymers formed by microbial
reactions - molecular weights 10,000 or higher
• fulvic acid (soluble & extractable in base and acids)
• humic acid (soluble & extractable in base only)
• humin or kerogen (not extractable in base)
2. Non-humic materials: unaltered proteins,cellulose, etc. - biochemicals from living
organisms
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• Humic acids: Hydrophobic fraction of soils
Properties of Soils
• Linear sorption coefficient K d – Partitioning in to organic matter or mineral
surface
– Infinite number of homogeneous sorption sites
phaseaqueousincompoundof abundance
solidssoiloncompoundof abundance
i
id K =
⋅=
mol
L
kg
mol
C
C K w
sd
Solid - Aqueous Phase Partitioning
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Linear-Isotherm
Schwarzenbach et al.
• Linear correlations between log (K om or K oc) & Log K ow
Solid - Aqueous Phase
Partitioning
omoc
ococw
ococd
omomw
omomd
f f
K f C
f C K
K f C
f C K
5.0
.=.
.=.
≈
=
=
K oc = soil organic carbon distribution coefficient
K om = soil organic matter distribution coefficient
Log Koc or Log Kom= a* Log Kow + b
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Solid - Aqueous Phase
Partitioning• Kom is invariant of soil type
Schwarzenbach et al.
Properties of Soils
• Soil Organic Matter usually measured as organic carbon
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Sorption Isotherms
Freundlich Model1/K n
s f wC C =
Kf = Freundlich sorption coefficient
1/n = Freundlich exponent (measure of non-linearity)
Linearized version: log log K 1/ log s f w
C n C = +
Case 1 (n < 1)
Cw
C
s
Case 2 (n = 1)
Cw
C
s
Case 3 (n > 1)
Cw
C
s
Case 1 (n > 1) Case 2 (n = 1) Case 3 (n < 1)
Solid-Aqueous Phase Partitioning
• Irreversible sorption or Hysteresis
• Sequestration and Contaminant agingphenomena
– continuous diffusion and retention ofcompound molecules in remote andinaccessible regions within soil matrix
– The longer a contaminant is in contact with soilorganic matter, the more resistant it is toaqueous extraction
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Solid-Aqueous Phase Partitioning
• Real-world example of contaminant “aging”
Nash and Woolson (1967) Science 157, 924-927
Solid-Aqueous Phase Partitioning• Implications of contaminant aging
– Clean-up of persistent contaminants difficult, if not impossible
– What are available options for remediation and future land use?
– Reduced mobility in subsurface
– Reduced bioavailability
– Risk assessment
– Establishing meaningful regulations for clean-up
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Solid-Aqueous Phase Partitioning
• Irreversible sorption or Hysteresis
• Sequestration and Contaminant agingphenomena
– continuous diffusion and retention ofcompound molecules in remote andinaccessible regions within soil matrix
– The longer a contaminant is in contact with soilorganic matter, the more resistant it is toaqueous extraction
Equilibrium States for
Metal Pollutants
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Inorganic Contaminants
• Metals: only 30 or so used industrially – Heavy metals: metals with densities > 5.0 g/cm3
Inorganic Contaminants
Chromium (Cr)
• Cr(VI) used to control
corrosion at a natural gas
plant
• 12,000 ppb vs 50 ppb
• Water source for 18
million people
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Inorganic Contaminants
• Speciation of metals affects watersolubility, transport, toxicity,
bioavailability, and treatment
• Metals may be present as
– anions (AsO43-, CrO4
2- )
– cations (Pb2+ , Hg2+ )
– complexes (CdCl3
+)
• Solubility determined by precipitation-
dissolution, redox, acid-base chemistry
Inorganic Contaminants• Partitioning of metals in soils
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Chemical reactions determining
equilibrium speciation1. Acid – Base reactions
– Speciation represented by log C vs. pH
relationships
– Dissociation constants, Ka
2. Precipitation reactions
– Solubility product, Ksp
– Precipitates can be transported in groundwater
as particulates (colloids)
Chemical reactions determiningequilibrium speciation
3. Complexation reactions
– Complex: an ion that forms by combining
simpler cations, anions, and sometimes
molecules
– Facilitated transport – certain complexes are
soluble
4. Redox reactions
– Electrode potential controls speciation
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Complexation Reactions
• Complex:
Metal Cation + Ligand (anion or organic molecule)
• Complexation facilitates the transport of toxic metals at near
neutral pH
• Without complexation, metals are most soluble/mobile in
their free ion form which only occurs at low pH
• Calculation of the distribution of metals among various
complexes involves the solution of a series of mass law
equations and knowledge of the total metal ionconcentration in the solution
Complexation Reactions• Examples of metal complexes:
Cr 3+ + OH- Cr(OH)2+
Mn2+ + Cl- MnCl+
Fe2+ + CO32- FeCO3
Al3+ + 3OH- Al(OH)3
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Complexation Reactions
• General form of complexation reaction and the stabilityconstant
• Large values of β are associated with the stronger or more
stable complexes
[ ][ ] [ ]h H l L M
hl Lh H l MLhH lL M =↔++ β
Metal Ligand Hydrogen ion
Complexation Reactions• Speciation is controlled by pH
• Log C-pH diagram
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Redox Reactions
• Oxidation-Reduction Reactions: – chemical reactions where participating elements
change their valence state through gain or loss ofelectrons
• Reduction: electron gain, loss of positive valence
Fe2+ + 2e- Feo
• Oxidation: electron loss, gain of positive valence
Fe2+ Fe3+ + e-
• Mediated by microorganisms (catalysts)
Redox Reactions• Quantifying oxidation potential
• Half reaction
Ox + ne- = Red
Mass law expression: [ ]
[ ]
Red
Oxn
K e−
=
[ ]
[ ]
1/
Red
Ox
n
ne
K
−
=
Electron
activity
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Redox Reactions
• Quantifying oxidation potential
– Positive values of pe, E H : oxidizing conditions
– Negative values of pe, E H : reducing conditions
• Example: metals in sulfur systems
Possible species: SO42-, H2S, HS-, S2-
Complexes: metals sulfates or metal sulfides
Oxidizing conditions: SO42-
dominates, metals soluble
Reducing conditions: H2S, HS-, S2- dominate, metals insoluble
Redox Reactions• EH-pH diagram
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Metal Contaminants
• Reaction processes affecting transport ofmetals
– Precipitation / dissolution (low pH)
– Complexation: binding with ligands affects
solubility and sorption
– Redox conditions: determine predominance of
soluble or insoluble species
– Sorption / Ion exchange:
• Negative charges on clays, metal oxides, andhumus
• Metal ions may have greater affinity for sites
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Geogenic
Anthropogeni
c
Biogenic
Arsenicdynamics in
soil and
aquatic
ecosystems
Mahimairaja et al, 2005
Wang and Zhao, 2009
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Arsenic Sources
What do we know about As fate in the
environment?• As(V) is less mobile, less toxic and more sportive
than As (III)
• Natural oxidation of As(III) to As (V) by oxygenis slow
• Chemical oxidation involves the use of strongoxidizers like chlorine, ozone, iron chloride,hydrogen peroxide/Fe2+, permanganate andmanganese oxides, which may lead to secondaryenvironmental problems.
• Microorganism can transform As(III) to As (V)and other species
• As(V) binds to iron surfaces (eg. iron oxide, ZVI)
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As(III) and As(V)-NZVI
reactions
Ramos et al, 2009; Weilie. Y et al,2010 & 2012;
• Experiments pH 8-
11
• Eh: -198 to 158
Transport of Dissolved
Contaminants
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• Dissolved Contaminants
• inorganic or organic ions and/or
molecules dissolved in groundwater
– The term dissolved contaminant thus
excludes an oil phase or sorbed-phase
contaminants but includes molecules thatdissolve from those phases into water
Transport of Dissolved Contaminants in
Groundwater
Contaminant Transport• Contaminant Transport Mechanisms
(dissolved contaminants)
- Advection
- Diffusion
- Mechanical Dispersion
• Contaminant Transport Influenced by
- Sorption
- Reactions (chemical or biological)
Hydrodynamic
dispersion
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• Advection – Movement of contaminants along with flowing
groundwater from the source point/area, according
to the average linear velocity
– Plug flow
Mechanisms of Contaminant Transport
• Dissolved Contaminants
• inorganic or organic ions and/or
molecules dissolved in groundwater
– The term dissolved contaminant thus
excludes an oil phase (NAPL) butincludes molecules that dissolve from it
into water
Transport of Dissolved Contaminants inGroundwater
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• Diffusion• Process whereby molecules move under the influence of their kinetic
activity in the direction of the concentration gradient
• Molecular-scale process
Mechanisms of Contaminant Transport
Diffusion
Diffusion Process:
• Molecular-scale process
• causes spreading (mixing) due to theconcentration gradients and random motion
• no bulk movement (fluid velocity) needed for
diffusive transport• continues till concentration gradients becomenon-existent (system has reached equilibrium)
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One drop of red ink
Diffusion
• Fick’s First Law of Diffusion:
Mass flux is proportional to the
concentration gradient
(for fluid medium)
Fx = Mass flux [M/L2
/T]Dd= Diffusion coefficient [L2/T]
= Concentration gradient [M/L3/L]
dx
dC D F d x −=
dx
dC
Dispersion: Tendency of solute to spread out fromthe path that it would be expected to follow
Dispersion is caused by heterogeneities in the medium thatcreate variations in flow velocities and flow paths
• in individual pore channels molecules travel at differentvelocities at different points in the channel due to dragexerted on the fluid by the roughness of the surface
• difference in pore sizes along the flow paths - differentvelocities in different pore channels
• variable path lengths, branching and tortuosity of pores
Mechanical Dispersion
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Mechanical Dispersion
• Mechanical dispersion: caused by any combination of the
following…
• Turbulence of fluid
• Fluid shear
• Porous medium
• Boundary effects
1-D Models
x
C v
C D
t
C x x
∂
∂−
∂
∂= )(
2
2
∂
∂
Rate of
change of
concentration
with time at
any x
Dispersion Advection
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Analytical Solutions to Mass Transport
Equation
1-D Models
t xv
x D x x
∂
∂=
∂
∂−
∂
∂ CCC2
2
Instantaneous source in 1-D:
( ) ( )
−−=
t D
t v x
t D
M
x
x
x4
exp4
tx,C2
π
where M = injected mass per unit cross-sectional area
Initial concentration C(x,0) = 0
Boundary conditions:
C(0,t)=Co, 0<t≤t0; C(0,t)=0, t> t0
C(∞,t)=0, t≥0
Contaminant Transport• Pulse or Instantaneous Source
Advection only
Advection +
dispersion
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Continuous Supply of Contaminants
Constant vel
water Co
Contaminant
reservoir
t=t1 t=t3t=t2
Packed
Porous media
(sand)
1-D Contaminant Transport
C vs. x plot
Continuous Supply – No Sorption
Constant v
Advection
0 1.00.5
C/Co
X ( m )
Constant vCo
Contam
reservoir
Advection &
Dispersion
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Advection only
Contaminant Transport
• Continuous Source
Advection +
dispersion
Sorption Effects
• R = retardation factor
sorption
sorptionno
d
b
v
v
K n R =
+=
ρ
1
Contaminant Transport
t x R
v
x R
D x x
∂
∂=
∂
∂−
∂
∂ CCC2
2
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sorption
0 1.00.5
C/Co
X ( m )
Co
1-D Contaminant Transport
C vs. x plot
Contaminant
reservoir
no sorption
Contaminant TransportContinuous Supply (with sorption)
Co
Constant v Constant v
t = t1 t = t1
t = t1 t = t2
Groundwater flow
Without sorption
t = t0 t = t1 t = t2
t = t0
With sorption
Contaminant TransportInstantaneous Supply
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• Plume Spreading: Separation of contaminants in aplume
– Gasoline spill (BTEX and MTBE)
Contaminant Transport
source
xylenes
benzene
MTBE
1) Rate of release (solubility)
2) Sorption
3) Biodegradation
• Natural Gradient Field Tests
– Tracers injected into shallow sand and
gravel unconfined aquifers
– Purpose: to assess the validity of
assumptions made in modeling
contaminant transport
Case Studies
1. Canadian Forces Base Borden, Ontario
2. Otis Air Base, Cape Cod, Massachusetts
Contaminant Transport
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• Canadian Forces Base Borden
– 12 m3 of chloride injected
into aquifer
– 5000 observation points
monitored over 2 years
– Data analyzed to determine
advection and dispersion
Contaminant Transport
• Canadian Forces Base
Borden
• Results
– Dispersivities vary
with scale
Contaminant Transport
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• Cape Cod, Mass.
– 7.6 m3 of tracer solution injected into
aquifer
– Bromide used as conservative tracer
– Lithium, molybdate, and fluoride used as
reactive tracers
– 9840 sampling points monitored over 3
years
– Data analyzed to describe movement
and behavior of plume in saturated zone
Contaminant Transport
Contaminant Transport• Cape Cod, Mass.
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Newell, C.J. and O’Connor, J.A. American Petroleum Institute, 1998
Newell, C.J. and O’Connor, J.A. American Petroleum Institute, 1998
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Interphase-Mass Transfer
Rates of chemical transfer from
one phase to another
Oil-Water Mass Transfer
Interface
Water
Distance
Oil
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Film Theory: Oil-Water
Given that
wowoil t
t wwoil
t ot
K C
C
k k K K
C K
C K J
W oil ow
wo
1
*
1
1
where
/
,/
,
//
/
+=
−∗=
=
Oil-side Boundary Layer Mass
Transfer Limitation
Interface
Water
Distance
Oil
C w,eq
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Water-Side Boundary Layer Mass
Transfer LimitationInterface
Water
Distance
Oil
C w,eq
)()(2
2
t eqt x x C C V
A K
x
C v
C D
t
C −−
∂
∂−
∂
∂=
∂
∂
Rate of
change ofDispersion Advection Mass transfer