9 Sediment-Water Exchange
Introduction
Sediment-water partitioning
Particle settling and deposition
Sediment erosion and resuspension
Transport equation with sediments
Contaminant transport within sediment bed
Model of sediment-water exchange
Contaminated sediment remediation
Interest in Sediments
“Geo-morphology”
Sediment as pollutant
Sediment as carrier of pollutants
Classification
φ
Class name
Diameter
(mm)
φ
Very coarse gravel
64-32
-5.5
Very fine sand
1/8-1/16
3.5
Coarse gravel
32-16
-4.5
Coarse silt
1/16-1/32
4.5
Medium gravel
16-8
-3.5
Medium silt
1/32-1/64
5.5
Fine gravel
8-4
-2.5
Fine silt
1/64-1/128
6.5
Very fine gravel
4-2
-1.5
Very fine silt
1/128-1/256
7.7
Very coarse sand
2-1
-0.5
Coarse clay
1/256-1/512
8.5
Coarse sand
1-1/2
0.5
Medium clay
1/512-1/1024
9.5
Medium sand
1/2-1/4
1.5
Fine clay
1/1024-1/2048
10.5
Fine sand
1/4-1/8
2.5
Very fine clay
1/2048-1/4096
11.5
ASCE, 1975; φ= -ln(dmm)/ln(2)
Most environmental interest in finer fractions
cohesive
Diameter
(mm)
Class name
Non-cohesive
Table 9-1 Sediment grade scale (adapted from ASCE, 1975)
MIT
Classification
SILT
SAND
Coarse
Medium
Fine
Coarse
Medium
CLAY
Fine
Coarse Medium Fine
90
70
50
30
10
10
1.0
d50
0.1
0.01
Diameter (mm)
Grain Size Distribution
Figure by MIT OCW.
0.001
0.0001
Equilibrium Partitioning
cs
Kp =
cd
Simplest model:
two phases
Partition (or distribution) coefficient:
Sorbed phase conc cs (mass contaminant
per mass solid) divided by dissolved
phase conc cd (mass contaminant per
volume solvent) in equilibrium
Units of Kp: volume/mass, e.g., cm3/g
More complicated partitioning models
Kp depends on contaminant, its concentration, concentration
of organic matter, redox potential, etc. Typical values: 101 to
102 cm3/g (hydrophilic) to 104 to 105 cm3/g (hydrophobic)
In sediment bed
Mass of dissolved contaminant/volume
= φc d
Mass of sorbed contaminant/volume
= ρ s c s (1 − φ ) = ρ s K p c d (1 − φ )
if equilibrium
Unit volume
with porosity φ
Ratio of sorbed to total mass
ρ s K p (1 − φ )
Kpρ
f =
=
ρ s K p (1 − φ ) + φ K p ρ + 1
ρ = ρ s (1 − φ ) / φ = ρ b ,d / φ
Solid-water phase ratio
ρ b ,d = ρ s (1 − φ )
Bulk (dry) sediment density
ρ b , w = ρ s (1 − φ ) + φ
(solid mass/water mass)
Bulk (wet) sediment density
Surficial sediments
ρs ~ 1.5 - 2.5 g/cm3; φ ~ 0.6 – 0.8 =>
ρb,d ~ 0.3-1 g/cm3, ρb,w ~ 1.1-1.6 g/cm3, ρ ~ 0.4-1.7 g/cm3
f =
Kpρ
K p ρ +1
ρ~ 1 => for Kp >> 1 most
contamination is sorbed to particles
While most of the contaminant is associated with solids,
the dissolved phase is very important because it is more
bio-available and amenable to sediment-water exchange
Sediment quality criteria often derived from target
dissolved phase concentrations assuming equilibrium
partitioning
In water column
“porosity” ~ 1 so
ρ = ρ s (1 − φ ) / φ ≡ [TSS ]
[TSS] ~ 1 to 100 mg/L (10-4 to 10-6 g/cm3)
f =
Kpρ
K p ρ +1
=
K p [TSS ]
K p [TSS ] + 1
=
(10 −4 to 10 −6 ) K p
(10 − 4 to 10 −6 ) K p + 1
For hydrophobic contaminants (Kp ~ 104 to 105)
concentrations in sorbed and dissolved phases can be
comparable. Hydrophylic contaminants are mostly in
dissolved phase
Non-equilibrium conditions
⎞
⎛ cs
dc d
⎜
=κ
− cd ⎟
⎟
⎜K
dt
⎠
⎝ p
⎛
d ( ρc s )
c s ⎞⎟
⎜
= κ cd −
⎜
dt
K p ⎟⎠
⎝
κ
κ=
Rate of increase of dissolved
phase mass
Rate of increase of sorbed phase
mass
Rate constant (t-1); e.g. (Wu and Gschwend, 1986)
0.2(1 − φ ) Dm K p ρ
φR
2
d
(c d + ρc s ) = 0
dt
R = aggregate radius, Dm =
molecular diffusivity
Total mass is conserved
Non-equilibrium, cont’d
κ/Kp
d
(c d + ρc s ) = 0
dt
Time scale for desorption from particle
= days to months => equilibrium
assumption not very good for
suspended sediments (may be OK for
stationary sediments)
Total mass is conserved
Additional Comments
Fine particles usually most important
„
„
„
„
„
Most easily resuspended
Settle most slowly
Probably have highest foc => Kp
Highest κ ~ (diameter)-2
Models often have multiple particle sizes with
individual settling velocity, tendency to
resuspend, and κ
Sediment movement
EPA, 2004
Modes of transport
Settling & deposition (non-cohesive and
cohesive)
Erosion & resuspension (non-cohesive
and cohesive)
Bed-load (non-cohesive)
Particle settling (WWT jargon)
Discrete (Type 1) (non-cohesive)
Flocculant (Type 2) (cohesive)
Hindered or zone (Type 3)
Compression (Type 4)
Focus on Types 1 and 2
Terminal velocity
Fd
Fg
Fg = ( ρ s − ρ f ) g
πd 3
6
2
1
2 πd
Fd = ρ f Cd ws
2
4
Equating, C d =
4dg ( s − 1)
3ws
2
s = ρs/ρf
Cd = f(R); R = Reynolds number = wsd/ν
Applicable to cohesive and non-cohesive sediments, but aggregate
shapes and densities are ill-defined for cohesive sediments
Spherical particles
R < 1; d < 100 µm (Stokes settling)
Fd = 3πρυdws
24
Cd =
R
g ( s − 1)d 2
ws =
18υ
R < 104 (Metcalf & Eddy, 1991)
Cd =
24
3
+
+ 0.34
R
R
More generally (0.01 mm < d < 100 mm; Dietrich, 1982)
log w* = −3.76715 + 1.92944(log D*) − 0.09815(log D *2 )
− 0.00557(log D *3 ) + 0.00056(log D *4 )
3
ws
( s − 1) gd 3
w* =
D* =
( s − 1) gυ
υ2
Spherical & non-spherical particles
10,000
Drag Coefficient, CD
1000
600
400
200
Stokes
100
60
40
ψ = 0.125
20
10
6
4
ψ = 0.220
ψ = 0.600
2
ψ = 0.806
1
0.6
0.4
ψ = 1.000
Sphere
0.2
0.1
0.001
0.01
0.1
1
2
4 6 10
100
1000
10,000
105
106
Reynolds number, based on DP
Figure by MIT OCW.
Ψ=
πD p
As
2
Brown et al. (1950)
Surface area of equivalent sphere/surface area of particle
Settling of Cohesive Sediments
Settling velocity depends on
concentration. Empirical
formula from EPA (2004):
Settling Velocity (m/d)
30
25
20
15
1.0
0.8
0.6
0.4
0.2
0.0
10
5
0
0
50
ws ( m / d ) =
0
2
4
100
6
150
Concentration (mg/L)
Figure by MIT OCW.
8
0.1 m / d CWL + 30 m / d (C COH − CWL )
C COH
10
200
CWL = wash load concentration (5
mg/L); CCOH = cohesive sediment
concentration (mg/L). Within
range of empirical observations,
e.g., Hawley (1982)
Sewage particles
Often a wide range of settling velocities
Settling Basins: discrete settling
u
ws
Inlet
Outlet
Time to settle Ts = h/ws
Hydraulic residence time Tres = V/Q = Ah/Q
Tres > Ts => Ahwsc/Qh > 1
wsc > Q/A
or
Q < Awsc
or
A > Q/wsc
Comments
Q/A = overflow rate (really a velocity); particles settling
faster will deposit
Flow capacity depends on area (not depth) => make
tanks as shallow as practical
Increase area using stacked clarifiers, inclined
plates/tubes, etc.
Similar concepts apply to field: deposition at river
deltas, in ponds & reservoirs, downstream from outfalls
Example of particulate phosphorus loading in reservoir
(Vollenweider plots)
Retention in lakes & reservoirs:
critical phosphorus loading
& Q
m,
Qp
wsp
p
V
dp
= m& − Qp − ws Ap
dt
In steady state
m& Q
= p + ws p
A A
m& / A
p=
Q / A + ws
V = impoundment volume
A = impoundment area
Q/A = impoundment
“overflow rate”
p = average P concentration
Phosphorous loading diagram
ws = 10m/yr
eutrophic
oligotrophic
Q/A <<ws
Q/A >> ws
Q/A = ws
After Vollenweider (1975); Chapra (1997)
Particle scavenging
Removal of marine contaminants (e.g., metals)
by natural particle settling
238U
α
234Th
h
β
(ce)
234U
234Th:
particle-reactive tracer; t1/2 = 24.1 d
w/o deposition ce = equil conc of
234Th:
Rad decay = -λce, λ = decay rate (ln2/t1/2)
Production = λce (must be same in equil)
Particle scavenging, cont’d
With deposition,
f = fraction of
f =
238U
α
234Th
(c)
h
β
234U
⎛w ⎞
f ⎜ s ⎟c
⎝ h ⎠
234Th
234Th
conc c < ce
sorbed to particles
K p [TSS ]
K p [TSS ] + 1
fws/h is effective
fws ⎞
⎛
ce λ = c⎜ λ +
⎟
h ⎠
⎝
fws (ce − c)λ
=
h
c
(ce − c)λh
ws =
cf
234Th
removal rate (t-1)
ce measured offshore;
c measured locally => ws
Particle scavenging, cont’d
Use 234Th deposition to trace
deposition of another metal (call it x),
with different partitioning
fx =
238U
α
234Th
(c)
h
β
JTh-234
234U
⎛w ⎞
f ⎜ s ⎟c
⎝ h ⎠
K px [TSS ]
K px [TSS ] + 1
(ce − c)λh
cf
J Th − 234 = fcws = (ce − c)λh
ws =
Jx
=
f x c x ws =
f x cx
(ce − c)λh
fc
Deposition
Flux of settling particles
ws C
C = sediment concentration (TSS)
Deposition rate, D
D = pws C
p = probability of depositing (sticking)
For non-cohesive sediments, p = 1
For cohesive sediments
p = (1 − τ / τ c ,d )
τ < τ c,d
0.06 < τc,d < 1.1 N/m2
τc,d = critical depositional
shear stress
(Mehta & Partheniades, 1975;
Ziegler et al, 1995)
Particle accumulation on bottom
Rate of mass approaching
interface/unit area = D
Rate of accumulation in
surface sediments
ws
h
z
d
[hρ s (1 − φ )]
dt
Equating
wo =
dh
D
=
dt ρ s (1 − φ )
If φ = 0.8, ρs = 2.5 g/cm3, ρs(1-φ) = 0.5
How to determine wo = dh/dt?
Depth of natural, accidental or
intentional marker
„
e.g., paint pigments in Fort Point
Channel
Decay of radioactive tracer
„
e.g.,
210Pb,
Fort Point Channel
Recall discussion in Chapter 4
N
Boston
Inner Harbor
18
18
Fluorescent dye
and pigment
particles released
May ’90 and July
‘91
ft
Northern Ave.
Congress St.
Summer St.
18
Cores 2,3
ft
Gillette
Dorchester Ave.
Core 1
Broadway
BOS 070
ft
Meters
100
0
Figure by MIT OCW.
500
Paint chips as markers
6m
8 cm
6 cm
Stolzenbach and Adams (1998)
Pigment surveyed with freeze corer
Dec 1993
July 1991
May ’90 to Dec ’93
14 cm/3.6 yr = 3.9 cm/yr
May 1990
Jul ’91 to Dec ’93
8 cm/2.4 yr = 3.3 cm/yr
Stolzenbach and Adams (1998)
Comments
Deposition rates of 1-4 cm/yr in FPC
(three cores)
Loading rates for all FPC sediment
sources ~ 0.14 cm/yr
Substantial import of (contaminated)
sediment
Cs-237
Fallout from bomb testing
USGS Gravity cores, Lake
Worth, TX, 2000-01
Measuring deposition with Excess
co
c
210Pb
Pb-210 particle reactive
tracer; t1/2 ~ 23 yr
c = excess concentration
Relative to (moving)
interface, steady state,
no sediment mixing
− wo
dc
= −λc
dz
λz
−
c
= e wo
co
-z
Alternatively
0
1
wo = λ ∫ c ( z )dz
co −∞
Erosion and Resuspension
Associated with bottom shear stress
τ = − ρ u ' w' = ρE z
τ = ρc f u
2
∂u
∂z
cf is bottom friction factor
Erosion of non-cohesive seds
Critical shear stress τc required to initiate
particle motion
ws > (τ/ρ)0.5 > (τc/ρ)0.5 => bedload
(τ/ρ)0.5 > ws, τc => suspended load
Re-suspension flux depends on near-bed
concentration; many formulations
EPA, 2004
Cohesive sediments
Many formulations; most apply for shear stress
above a critical erosional shear stress, τc,e
Erosion rate E (g/m2-s)
τ
E = M(
− 1) n
τ c ,e
τc,e = 0.05 to 0.3 N/m2;
M = 0.1 to 3 g/m2-s; n = 1-3
Cohesive sediments, cont’d
Erosion potential ε (g/m2)
ε=
ao
Td
m
τ
(
− 1) n
τ c ,e
Td = time (days) after deposition
τc,e = 0.1 N/m2 ; ao = 50; m = 2; n = 2.7
Erosion over specified time interval ~ 1 hr
Above parameters from Ziegler, et al., 1995 for
Watts Bar Reservoir, TN
Measurement of erosion
Linear laboratory flume
Linear flume in field
Laboratory annular flume
Portable resuspension device (Shaker;
Tsai and Lick, 1986)
Sedflume
(after McNeil
et al., 1996)
Figure by MIT OCW.
Measures erosion rates in the lab
To boat
81 cm i.d. hose
Top, Front View
2.4 m
12 cm
6 cm
Grid
Boundary layer trap
Lateral angle iron
Ravens &
Gschwend
(1999)
Measures erosion
rates in the field
Bottom, Front View
Sediment bed
test section
Angle iron
1m
Lateral angle iron
Boundary Layer Development Region
Figure by MIT OCW.
Comments
τc,e increases & E decreases with time
after deposition and depth below
sediment bed reflecting increased
strength due to compaction, armoring
from larger particles
Regions with τ > τc,e on regular or
intermittent basis exhibit erosional
tendencies
Net erosion (erosion
– deposition)
n
⎡
⎡ τ
⎤
τ ⎤
φ = E−D = M⎢
− 1⎥ − ws c ⎢1 −
⎥
⎣⎢ τ c ,d ⎦⎥
⎣⎢τ c ,e ⎥⎦
Transport Equation with Sediments
∂
∂
∂
∂c
∂
∂c
∂c ∂
∂
∂c
+ (uc) + (vc) + ( w − ws )c = ( E x ) + ( E y ) + ( E z )
∂t ∂x
∂y
∂z
∂x
∂x ∂y
∂y ∂z
∂z
z positive upward, origin at sediment interface;
c = sediment concentration; ws = settling velocity
Neglecting horizontal transport & vertical water velocity
∂c ∂
∂
∂c
− ws c = ( E z )
∂t ∂z
∂z
∂z
Boundary conditions
∂c
− ws c − E z
=0
∂z
− ws c − E z
at surface (z = h)
∂c
= (α − 1) ws c
∂z
at bottom (z = 0)
Surface and bottom BCs
z
h
ws c
dc
− Ez
dz
0
α=0 α=1 α>1
α denotes relative amount of erosion
c
Vertical sediment distribution
Under steady state
dc
− ws c = E z
dz
Logarithmic velocity profile in channel; turbulent
diffusivity = viscosity
c( z ) ⎡⎛ h − z ⎞⎛ a ⎞⎤
= ⎢⎜
⎟⎜
⎟⎥
ca
z
h
a
−
⎠⎝
⎠⎦
⎣⎝
w s / κu *
ws κu*
Rouse number
Depth-average Ez (= 0.07u*h)
ws h
Pe =
Ez
Peclet number ~ 6 ws/κu*
T=tws/h
Pe=200
T = tV/h
0
Pe = Vh/D = 200
.025
T = .1
.2
z
h
Z/h
.2
.4
1.4 1.3
0
1.2
1.0
1.1
20
.8
40
80
C/Co (%)
20
100
Vh/D = 20
.025
T = .1
.2
.4
.8
1.0
1.51.4
1.3 1.2
0
20
1.1
1.0
.9
40
20
40
60
80
100
0.02
C/Co (%)
Vh/D = 0.02
0
1.2 1.0 .8
.6 .5
.7
1.1 .9
.4
.3
.2
.025
T = .1
.6
.8
80
2
100
1.0
0
20
40
60
C/Co (%)
Pe = Vh/D = 2.0
0
No erosion
(α=0)
Constant Ez
.6
60
C/Co (%)
1.0
0
Z/h
.5
1.8
1.7
1.6
.3
.4
.3
.4
.7
T = .4
.2
.2
.8
c(z/h,T,Pe)
.2
.8
60
.6
T = 3.0
2.0
.6
.7
0
Z/h
.025
.1
Z/h
.6
.9
Vh/D = 0.2
.4
.5
.6
T = tV/h
0
.2
.3
.4
.8
T = 1.6
T = 1.5
1.0
0.2
0
80
100
0.002
Vh/D = 2.0 x 10-8
ws h
Pe =
Ez
.025
.2
Z/h
.2
T=1
.4
.4
.2
.6
Z/h
.6
.3
1.9
1.6
.8 2.2 1.8 1.4 1.2 1.0
.9 .8
1.5
T = 3.0 2.0
1.3 1.1
1.7
1.0
0
20
40
.7
.6
60
C/Co (%)
.5
.4
.8
80
1.9
100
1.0
0
2.1
2.5
3.0
1.61.4 1.2 1.0
1.8
1.5 1.3 1.1
1.7
.8
.9
.6
.7
T = .4
.3
.5
2.0
20
40
60
80
100
C/Co (%)
Dhamothran et al (1981)
Figure by MIT OCW.
Pe < 0.2 =>
well-mixed)
Pe > 100 =>
stratified
Application: Settling basin & river
100 mg/L
40 mg/L
L = 50m, W = 6m, h =
4m, Q = 0.2m3/s
h = 1 m,
u = 0.3 m/s
Pe = wsh/Ez
Ez = 0.07u*h; assume u* ~ 0.05u => Ez = 0.0035uh
Pe = ~ 300ws/u
ws = 10-2 to 10-6 m/s (Table 9.2)
Often a wide range of settling velocities
Focus on Basin
100 mg/L
40 mg/L
L = 50 m, W = 6 m, h
= 4 m, Q = 0.2 m3/s
Basin
h = 1 m,
u = 0.3 m/s
u = Q/hW = 0.0083 m/s
Pe = ~ 300ws/u (second column of Table 9.3)
wsc = Q/A = Q/LW = 7.7x10-4 m/s
(faster settling particles theoretically removed)
Pe for settling basin and river
Ws
(m/s)
10-2
Pe = wsh/Ez
(Basin)
340
Pe = wsh/Ez
(River)
10-3
34
10-4
3.4
0.1
10-5
0.34
0.01
10-6
0.034
0.001
Focus on River
100 mg/L
40 mg/L
L = 50 m, W = 6 m, h
= 4 m, Q = 0.2 m3/s
River
h = 1 m,
u = 0.3 m/s
Pe = ~ 300ws/u (third column of Table 9.3)
Comments
In basin, turbulence insufficient to mix
particles that settle (Pe > 30)
In river, turbulence sufficient to mix
particles that don’t settle in basin (Pe <
0.1) (river can be treated as well
mixed)
In basin, τb = ρu*2 = 0.07 N/m2 < τc,e
In river, τb = ρu*2 = 0.22 N/m2 <~ τc,e
(possible resuspension)
Vertically well-mixed conditions
Pe < 0.2
3-D equation
∂c ∂
∂
∂c
− ws c = ( E z )
∂t ∂z
∂z
∂z
Vertical integration
h
⎡
∂c ⎤
∂c ⎡
∂c ⎤
= ⎢ ws c + E z ⎥ − ⎢ ws c + E ⎥
∂z ⎦ bot
∂t ⎣
∂z ⎦ surf ⎣
=0
Vertically well-mixed conditions,
cont’d
No resuspension (α = 0)
dc
wc
=− s
dt
h
c = co exp(− ws t h)
co = initial depth-averaged concentration
ws/h = first order removal rate, κs
Partially-mixed conditions sometimes
analyzed using κs > ws/h (because near
bottom concentrations are greater than co)
Multiple size fractions
First order settling of different
size fractions can resemble
second order settling:
wc
dc
= −∑ s i
dt
h
i
co
c=
1 + Btco
ws/h = 0.3x10-5s-1
1.0x10-5s-1
0.3x10-5s-1
Ave
2nd O settling (Bco = 3x10-4s-1)
Contaminant transport within &
across the sediment bed
Porewater advection (GW
movement; sediment
compaction; wave or bedform
induced pressures; biomixing
J a = φuc d
Contaminant transport within &
across the sediment bed
Porewater advection (GW
movement; sediment
compaction; wave or bedform
induced pressures; biomixing
J a = φuc d
Porewater diffusion
J d = −φD' dc d dz
D' = φDm
Contaminant transport within &
across the sediment bed
Porewater advection (GW
movement; sediment
compaction; wave or bedform
induced pressures; biomixing
Bulk sediment motion
(“turbulence”)
J b = − Db d (c d + ρ s c s ) / dz
J a = φuc d
Porewater diffusion
J d = −φD' dc d dz
D' = φDm
Sediment Profile Imaging
Benthic fauna mix
dissolved oxygen and
other sediment
characteristics
(oxygen rich areas
are light colored)
Note feeding tubes
near surface
EPA, 2006
Measuring bioturbation with
co
c
234Th
Th-234 particle reactive
tracer (c); t1/2 ~ 24.1 day
Relative to (moving)
interface, steady state,
including sediment mixing
d 2 cs
dc
− wo
= Db
− λc
2
dz
dz
For Dbλ >> wo
-z
d 2 cs
0 = Db
− λc
2
dz
c
= exp(− λ Db z )
co
Comments
Db correlates with wo (reflecting flux of
organic matter)
Coastal sediments: Db = 10-7 to 10-6
cm2/s
Deep sea sediments: 10-9 to 10-8 cm2/s
DDT on Palos Verdes Shelf (WE 9-4)
DDT commonly used
pesticide until 1970s
(Silent Spring).
~ 1700T discharged by
LACSD’s White Point
outfall (60m depth) (also
agricultural run-off)
~100T (p-p’-DDE) still
buried in sediment
Issues of environmental
racism
EPA Superfund Site
(Montrose Chemical Co.)
Vertical Profiles
700
600
500
400
300
200
100
0
Concentration vs depth
(USGS; Lee, 1994)
1981
1989
0
20
40
60
80
Depth in cm
Vertical distribution of porosity (open
squares) and bioturbation coefficient
(closed squares)
1
0.8
0.6
0.4
0.2
0
φ
Depth in cm
49
45
41
37
33
29
25
21
17
13
9
5
Db/Dbo
1
ppm
Core 8C 1981 (solid) to 1989 (open)
Exponential distribution
of porosity and
bioturbation (latter based
on worm density)
Issues
Contamination slowly decreasing. But is it biodegradation or surface loss?
Will natural sedimentation cap contaminants?
Decreasing since WWTP upgrade; introduce
clean sediments from flood control reservoirs?
Current strategy of institutional controls (public
outreach, fish monitoring, etc.) Is this enough?
Possible future capping. Will this work? (2000
pilot capping failed.)
Sediment Fate Processes
ppm
Core 8C 1981 (solid) to 1989 (open)
Lee, 1994 (USGS)
700
600
500
400
300
200
100
0
1981
1989
0
20
40
Depth in cm
60
80
Deposition of clean
sediment (deposition
velocity w in cm/yr)
Biological mixing (Db
in cm2/yr)
Biodegradation (1st O
rate λ in yr-1)
Release to surface (k
in cm/yr)
J = kρ s (1 − φ )cso
Mass Transport in Sediments
∂c s
⎫
∂
∂ ⎧
∂
+ [ w(1 − φ ) ρ s c s ] = ρ s
(1 − φ ) ρ s
[(1 − φ )c s ]⎬ − λ (1 − φ ) ρ s c s
⎨ Db
∂t ∂ς
∂ς ⎩ ∂ς
⎭
Boundary conditions
∂c s
Db
= (k + wo )c s
∂ς
at
ς =0
cs = 0
at
ς =∞
ζ = depth below
(moving) sediment bed
Use observations to calibrate unknown
parameters w, Db, λ and k
Simplification
∂c s ' ' ∂
∂ −ς / L ∂
+
( wo c s ' ' ) = Dbo
[e
c s ' ' ] − λc s ' '
∂t
∂ς
∂ς
∂ς
where
c s ' ' = (1 − φ )c s /(1 − φ o )
wo = w(1 − φ ) (1 − φ o )
Db = Dbo e −ς / L
Spatial Moments
∞
M i = ∫ c s ς i dς
∞
M i ' = ∫ cs e
0
−ς / L
ς dς
i
0
∞
M i ' ' = ∫ c s ' ' ς dς
dM o ' '
= −kc so − λM o ' '
dt
D
dM 1 ' '
− wo M o = Dbo c so − bo M o ' ' '−λM 1 '
dt
L
i
0
∞
M i ' ' ' = ∫ c s ' ' e −ς / L ς i dς
0
4 moment equations
in 4 unknowns
2 Dbo
dM 2 ' '
− 2wo M 1 = 2 Dbo M o ' ' '−
M 1 ' ' '−λM 2 ' '
dt
L
dM 3 ' '
3Dbo
− 3wo M 2 = 6 Dbo M 1 ' ' '−
M 2 ' ' '−λM 3 ' ' '
dt
L
wo = 1.7 cm/yr, Dbo = 44 cm2/yr, λ = 0.03 yr-1, k = 9.6 cm/yr.
Surface loss λ M o ' ' and degradation loss k c so
comparable; times scale of each ~ 30 yrs
Sediment water exchange model
-H
1 Flushing
Cd0
Cd1
-ZW
cd2 = cs2/Kp 2 Water-side diffusion
ZS
3 Sorption kinetics
4 Bio-mixing
L
cdL = csL/Kp
Z
Figure by MIT OCW.
Chen, 1993
Steady state
Includes bioturbation,
pore-water diffusion and
sorption kinetics, but no
resuspension, deposition
or bio-degradation
Colloidal transport
included but not
described here
Applied to PAH’s in
Boston Harbor
Sediment water exchange model
d 2 cd
κ
+
0 = ( Db + D ' )
(c s − K p c d )
2
Kp
dz
d 2 cs
κ
0 = Db ρ
+
( K p cd − cs )
2
Kp
dz
dissolved
sorbed
Boundary Conditions
cd = cd1
c d K p = c s = c sL
dc s dz = 0
at z = 0
at z = L
Approximate Solution
cd − cd1
1 − e − rz + εrz
=
c sL K p − c d 1
1 + εrL
cs − K p cd1
c sL − K p c d 1
=
1 + ε (rz + e
1 + εrL)
− rz
-H
cd0
)
φ ( Db + D' )c s∞ / K p + (1 + εrL)( Dm / δ w )c d 0
cd1 =
(1 + εrL)( Dm / z w ) + φ ( Db + D' )r
cd1
-zw
zs
4 bio-mixing
cdL=csL/Kp
r = κ /( Db + D' )
z
( Db + D ' )
ρK p Db
cd2=cs2/Kp 2 water-side diffusion
3 sorption kinetics
L
ε=
1 flushing
Flux to surface
Dissolved phase concentration
in equilibrium with csL
c sL / K p
J=
z
τ
2.2 R
L
+ w +
+
H Dm (1 − φ )[( Db + D' ) Dm ρ s K p ]1 / 2 (1 − φ ) ρ s Db K p
1
2
3
4
Denominator: 4 “resisters” in series: 1)
flushing, 2) water-side diffusion, 3)
sorption kinetics, 4) bio-mixing
Parameters
Varia
ble
Definition
Value(s)
Dm
Aqueous solution diffusivity
D’
Aqueous solution diffusivity
corrected for porosity
0.8x10-5
cm2/s
0.5x10-5
Db
Bioturbation coefficient
10-7, 10-6, 10-5
cm2/s
Kp
Solid-water partition coefficient
L
Biologically active depth
101 to
106cm3/g
5 cm
φ
Porosity
0.8
ρs
Sediment density
2.5 g/cm3
Sorbed concentration at z = L
10-6 g/g
Characteristic aggregate radius
0.01 cm
Water-side boundary layer
thickness
Hydrodynamic residence time of
overlying water
0.06 cm
Depth of overlying waterbody
6m
Desorption rate constant
Eq (9.5)
R
H
5 day
Db = 10-5 cm2/s
1.0
D1/D
0.9
Fractional Resistance
0.8
0.7
D2/D
0.6
0.5
D3/D
0.4
0.3
0.2
D4/D
0.1
0.0
1
2
3
4
log (Kp in cm3/g)
5
6
Db = 10-6 cm2/s
1.0
D1/D
0.9
D2/D
Fractional Resistance
0.8
0.7
D3/D
0.6
0.5
0.4
D4/D
0.3
0.2
0.1
0.0
1
2
3
4
log (Kp in cm3/g)
5
6
Db = 10-7 cm2/s
1.0
D1/D
0.9
D2/D
Fractional Resistance
0.8
D3/D
0.7
0.6
0.5
0.4
0.3
D4/D
0.2
0.1
0.0
1
2
3
4
log (Kp in cm3/g)
5
6
Comments
Water side bl (2) controls for large Kp & Db
Bioturbation (4) controls for small Kp & Db
(Resistance on side with smallest equilibrium
concentration)
Desorption not limiting factor
Longest clean-up times for high Kp (nearly a
century for benzo(a) pyrene (Kp ~ 105) in
Boston Harbor)
Dealing with Contaminated Sediment
Natural attenuation (Let it sit)
„
„
„
If evidence of natural recovery (deposition, biodegradation)
Or if other options problematic
Combined w/ active monitoring & inst controls
Capping (Cover it up)
„
„
With clean sediment
In situ or in confined aquatic disposal (CAD) cells
Dredging (Remove it)
„
„
„
Environmental (remove contamination)
Maintenance (keep harbors/channels open)
Improvement (make harbors/channels deeper)
Boston Harbor Navigation
Improvement Project
Deepen to 38-40’
(versus maintenance
or environmental)
1.7x106 yd3 clay
(MBDS)
1.1x106 yd3 silt (CAD
cells)
Confined Aquatic Disposal Cells
Figure by MIT OCW.
Dredge buckets
Environmental
Clam shell
CAD Challenges
E
5 0 6 0 50
W
Core M4-5
g
n
i
h
rto
N
Core M4-4
Core M4-2
5 0 5 9 50
5 0 5 8 50
5 0 5 7 50
717350
717450
717550
717650
E a s tin g
Hitting target
717750
717850
Subbottom line 6-003 from cell M4 (OSI 1999), annotated at bottom showing location of cores,
fluidized mud layer (above red dashed line), sand zone (between red and blue dashed lines),
and approximate bottom of cell (green dashed line). Note reversal of East and West.
Cell M4 - Post-Cap Sub-Bottom Profile
Verifying CAP integrity
Waiting for sufficient
consolidation
Additional Issues
Containing dredged and capping material
(during descent & upon impact)
Time of disposal (environmental windows
to allow migrating fish passage)
Residual silt (should you “rake all the
leaves?”)
Open cells (exposure to uncapped
material)