Sediment and Contaminant Transport in Surface Waters - Chapter 8 (end) doc

313
8
Modeling the
Transport and Fate of
Hydrophobic Chemicals
Hydrophobic organic chemicals, such as PCBs and PAHs, often have large par-
tition coefcients, on the order of 10
3
to 10
6
L/kg or even higher. In this case,
much of the chemical is sorbed to particulate matter and is transported with it.
This particulate matter, along with the sorbed HOC, usually settles onto the
bottom of an aquatic system and forms deposits of contaminated sediments
there that can be many meters thick. At most sites with contaminated sedi-
ments, approximate calculations of the amounts of contaminants in these bot-
tom sediments and also in the overlying water lead to the conclusion that there
are orders of magnitude more contaminant in the bottom sediments than in the
overlying water. As a result, even after the cleanup of point sources of contami-
nation, these bottom sediments can serve as a major and long-lasting source
of contaminants to the overlying water. To predict sediment and water quality
over long periods of time, the ux of these contaminants between the bottom
sediments and the overlying water needs to be quantitatively understood and
modeled.
The sediment-water ux of contaminants is primarily due to sediment ero-
sion/deposition, molecular diffusion, bioturbation, and groundwater ow. Each
of these processes acts in a different way, and hence each must be described and
modeled in a different way. In general, they occur more or less simultaneously
and there are interactions among them. All these processes are continuously and
often signicantly modied by the nite rates of adsorption and desorption of the
HOC between the solid sedimentary particles and the surrounding waters. These

rates of sorption and the resulting partitioning depend on the hydrophobicity of
the chemical, that is, on K
p
. Because of this, the transport of an HOC also strongly
depends on K
p
. This is especially true for HOCs with large partition coefcients
and is a major emphasis in this chapter.
As bottom sediments erode, the contaminants associated with these sedi-
ments are transported into the water column, where they may adsorb or desorb,
depending on conditions in the overlying water relative to conditions in the bottom
sediments. Because erosion rates are highly variable in space and time, contami-
nant uxes due to erosion/deposition are also highly variable in space and time.
During calm periods and average winds, these uxes are relatively small and are
© 2009 by Taylor & Francis Group, LLC
314 Sediment and Contaminant Transport in Surface Waters
probably comparable with the uxes due to molecular diffusion, bioturbation, and
groundwater ow. However, major storms and oods can cause movement and
mixing of bottom sediments by erosion/deposition more rapidly and to depths
in the sediments much greater than that possible by these other processes. The
contaminant ux due to the erosion of particles with their sorbed contaminants
and the subsequent desorption of these contaminants into the surrounding water
would also then be much greater than the contaminant uxes due to these other
processes.
The effects of bioturbation on sediment properties and the sediment-water
ux are due to feeding and burrowing activities of benthic organisms, are quite
diverse, and depend on the amounts and types of organisms. In fresh waters,
benthic organisms disturb and/or mix the sediments down to depths of 2 to 10
cm. This does not occur instantaneously but over a period of time that depends on
the number densities of the organisms and their activities; this can be months to

years. For sea water, the depths of the disturbances due to benthic organisms are
much greater, on the order of 10 cm to as much as 1 m.
The ux of contaminants from the bottom sediments due to molecular dif-
fusion has often been considered negligible by comparison with other processes.
However, rapid erosion and deposition (as caused by oods and storms) as well
as chemical spills can cause sharp gradients in contaminant concentrations and
hence large contaminant uxes at the sediment-water interface. As will be seen
below, nite sorption rates for HOCs with large partition coefcients will exac-
erbate this effect. In addition, molecular diffusion is ubiquitous and inherently
modies and is modied by all the other ux processes. As a result, the effects of
molecular diffusion on uxes can be quite large and must be considered in calcu-
lating and predicting sediment-water uxes of HOCs.
In eld studies, groundwater ow has been shown to be a major inuence on
the sediment-water ux of HOCs in certain areas. However, these uxes are dif-
cult to measure and, in addition, models of this ux as modied by nite sorp-
tion rates have not been extensively applied or veried. As a result, the effects of
groundwater ow on the sediment-water ux of HOCs have not been well quanti-
ed. Nevertheless, the ux of HOCs due to groundwater ow can be signicant
and is a process that deserves careful consideration.
In water quality models, a common approach to modeling the contaminant
ux between the bottom sediments and the overlying water is to use the equation
q
D
h
CC
HC C
wwo
wwo



()
()
(8.1)
where q is the ux; D is a diffusion coefcient; C
w
and C
wo
are representative
contaminant concentrations in the pore waters of the sediment and in the overly-
ing water, respectively; h is the thickness of an assumed “well-mixed” or “active”
sediment layer; and H = D/h and is a mass transfer coefcient with units of
© 2009 by Taylor & Francis Group, LLC
Modeling the Transport and Fate of Hydrophobic Chemicals 315
centimeters per second (cm/s). If local chemical equilibrium in the sediments is
assumed, the above equation can be written as
q
D
h
C
K
C
s
p
wo

¤
¦
¥
³
µ

´
(8.2)
where C
s
is a representative contaminant concentration of the solids in the sedi-
ment. The parameters D, h, and H (only two of the three are independent) are
generally determined by parameterization, that is, by comparison of results cal-
culated from the water quality model with eld measurements and then modify-
ing these parameters until the model and eld results agree (e.g., see the texts by
Thomann and Mueller, 1987; Chapra, 1997). As a rst approximation, typical
recommended values for h are 10 to 15 cm, whereas a suggested value for D is
the molecular diffusion coefcient for the HOC being considered. The effects of
nite sorption rates on these uxes are generally not explicitly considered.
The difculty with Equations 8.1 and 8.2 is that they do not describe (espe-
cially in functional form) the time-dependent uxes due to sediment erosion/
deposition, molecular diffusion, bioturbation, or groundwater ow. Because these
equations do not correspond to any real process, the parameters D, h, and H are
purely empirical functions chosen to t existing data; they generally are not valid
for conditions for which they have not been calibrated, and the solutions to these
equations do not have the proper functional dependence on time. Because of this,
these equations have very little predictive capability. Equations 8.1 and 8.2 are
often called diffusion equations, but they are not; a better term would be a mass
transfer approximation.
The major processes that affect HOC transport and sediment-water ux as well
as the modeling of these processes are described in this chapter. Erosion/deposi-
tion and the subsequent transport of HOCs in the overlying water are discussed
in the following section. Section 8.2 describes the conventional one-dimensional,
time-dependent diffusion (Fickian) approximation that is often used for the sedi-
ment-water ux of nonsorbing chemicals or for sorbing chemicals when chemical
sorption equilibrium is a good approximation, that is, when sorption rates are

relatively high. The mass transfer approximation as described by Equation 8.1 or
8.2 is also discussed and compared with the diffusion approximation.
For the diffusion of sorbing chemicals with nite sorption rates, the Fickian
approximation is not valid. In this case, the basic conservation equations for the
HOC must be supplemented by a rate equation for the transfer of the HOC between
the solids and the pore water. This is done for the molecular diffusion of HOCs in
Section 8.3, where experimental and theoretical results for HOCs with a wide range
of partition coefcients are discussed; nite sorption rates are inherent in the results
and analyses. The effects of bioturbation (including nite sorption rates) on the
sediment-water ux are discussed in Section 8.4. Comparisons of the magnitudes
and time dependencies of the different sediment-water uxes, as well as a discus-
sion of the approximation of a “well-mixed” layer, are given in Section 8.5.
© 2009 by Taylor & Francis Group, LLC
316 Sediment and Contaminant Transport in Surface Waters
A simple and idealized problem of contaminant release and transport during
environmental dredging is discussed in Section 8.6; the purpose is to charac-
terize and estimate the magnitudes of various processes that affect this release
without the use of a complex model. Previously, in Section 1.2, an introduction
to the problem of water quality modeling, parameterization, and the resulting
non-unique solutions was given. In Section 8.7, this discussion is continued in the
context of a more general model of PCB transport and water quality.
8.1 EFFECTS OF EROSION/DEPOSITION AND TRANSPORT
Two HOC transport problems are discussed in this section: (1) the transport of
PCBs in the Saginaw River, including the assumption of equilibrium partitioning;
and (2) the transport of PCBs in Green Bay as affected by nite sorption rates.
8.1.1 THE SAGINAW RIVER
The transport of sediments in the Saginaw River has been modeled, and results
of these numerical calculations were presented in Section 6.4. Based on these
sediment transport calculations, the transport of PCBs in the river has also been
modeled (Cardenas et al., 1995); the specic problem was to investigate and make

preliminary estimates of the erosion, deposition, transport, and fate of PCBs from
a contaminated area in the river (Figure 6.22). Some interesting results of this
investigation are presented here.
In the calculations, it was assumed that there was equilibrium sorption of the
PCBs to the sediments with a K
p
of 2×10
4
L/kg; the nonerosional/nondepositional
sediment-water ux of PCBs was not included in order to isolate the ux due to
sediment erosion/deposition. The rst calculations were to investigate the effects
of the magnitude of ow events on the erosion and deposition of sediments and the
subsequent transport of PCBs; calculations were therefore made for ow events of
500, 1000, 1500, and 1900 m
3
/s. For reference, from 1940 to 1990, the median ow
rate was 57 m
3
/s and the maximum was 1930 m
3
/s. In these rst calculations, the
following was assumed:
1. Surcial sediments initially in the intensive study area were contami-
nated with PCBs at a level of 4 µg/g of sediment. This is a reasonable
rst approximation to the average of the PCB concentrations actually
measured at this site.
2. The thickness of this surcial layer was 10 g/cm
2
(on the order of 10 cm),
but only where the water depth was less than 3 m (Figure 6.22). This

excluded the river channel, where contaminants were generally not
found. For the contaminated area, the total amount of PCBs initially in
the sediment bed per unit of surface area was then 40 µg/cm
2
, whereas
the total amount of PCBs initially in the sediment bed was 90 kg.
3. There were sediments coming in from upstream, but they contained
no PCBs.
© 2009 by Taylor & Francis Group, LLC
Modeling the Transport and Fate of Hydrophobic Chemicals 317
During a ow event, sediments and sorbed PCBs are eroded from the sediment
bed. Some of the eroded PCBs are transported and deposited further downstream
in the river, and some are transported to Saginaw Bay. From the calculations, it
was shown that erosion of PCBs originally in the bed occurs (1) in the shallow
nearshore area to a sediment depth generally less than 1 g/cm
2
and (2) at the edge
of the channel to sediment depths as great as 30 g/cm
2
. The amount of PCBs
eroded (in µg/cm
2
) for a 1500-m
3
/s event is shown in Figure 8.1. The large erosion
of PCBs at the edge of the channel is clearly evident. During this event, a total of
28 kg PCBs were eroded and transported downstream; almost all of the eroded
PCBs were transported to the bay. A small amount (0.02 kg) was deposited in the
wide shallow part of the river, primarily near shore, with the amount of PCBs per
unit area increasing toward shore.

A comparison of the amounts of PCBs transported by the different ow
events is given in Table 8.1. The amounts transported to the bay increase nonlin-
early with the ow rate, from a small amount (0.28 kg) for the 500-m
3
/s event to
36.5 kg for the 1900-m
3
/s event. However, even for the largest ow event, only
about a third of the total PCBs in the bed are eroded. The reason for this is as
follows. Because of the currents, the highest shear stresses occur in the deepest
water, the channel, whereas the lowest shear stresses occur in the shallow, near-
shore areas. Although the shear stresses and erosion rates are high in the channel,
no PCBs are present there and therefore no erosion and transport of PCBs occur
there. Conversely, in the nearshore region, the shear stresses are low, only small
erosion occurs, and little transport of PCBs occurs. The region where the highest
erosion and transport of PCBs occurs is on the edge of the channel where PCBs
are present and where moderately high shear stresses occur. Here, the depth of
5
35
FIGURE 8.1 Amount of PCBs eroded (µg/cm
2
) in the Saginaw River for a 1500-m
3
/s
ow event. (Source: From Cardenas and Lick, 1996. With permission.)
© 2009 by Taylor & Francis Group, LLC
318 Sediment and Contaminant Transport in Surface Waters
erosion and the amount of PCBs eroded and transported depend on the shear
stress and hence on the ow rate, at least until the layer of contaminated sediment
is eroded. Thereafter, additional PCB erosion and transport is caused only by ero-

sion of surcial layers near shore.
Contaminants initially at the surface of the sediment bed will be buried by
sediments depositing during low ows. This process will reduce the subsequent
erosion and transport of the contaminated sediments by later and larger ows. To
make a preliminary investigation of this process, several calculations were made.
In these calculations, it was assumed that (1) as in the rst example, a layer of
contaminated sediments 10 g/cm
2
thick was initially present at the surface and
had a PCB concentration of 4 µg/g of sediment in the intensive study area; (2)
layers of clean sediments were deposited on top of these contaminated sediments
for different periods of time at approximately 2 g/cm
2
per year; and (3) after this
deposition, a 1500-m
3
/s ow event occurred.
The erosion and deposition of sediment for the 1500-m
3
/s event were the same
as in the above example. As far as PCB transport and fate are concerned, the dif-
ferences in the results described here are that clean overlying sediments must be
eroded rst before the contaminated sediments can be eroded and transported.
Of course, the more clean sediments that are deposited over the contaminated
sediments, the less the amount of contaminated sediments that are eroded and
transported downstream to the bay.
Calculations were made for no deposition and for deposition time periods of
1, 5, and 20 years. For these scenarios, the masses of PCBs transported to the bay
were 28.1, 26.1, 23.1, and 15.2 kg, respectively. There is little difference in PCB
transport to the bay without and with 1 year of deposition. Although the newly

deposited sediments cover almost all the shallow, nearshore area with a layer of
sediment sufcient to eliminate the erosion of contaminated sediments from this
area, little erosion occurs there, even without any deposited sediment. Almost all
the erosion of contaminated sediments occurs in a narrow region at the edge of
the channel, between the deep channel and the shallow, nearshore area. In this
region, the currents during a big event are sufcient to erode the newly deposited
sediments as well as the older contaminated sediments. Only after 5 years of
TABLE 8.1
PCB Transport in the Saginaw River for Different Flow Events
Flow Event
(m
3
/s)
Transported
to Bay
(kg)
Deposited
Downstream
(kg)
Remaining
in Bed
(kg)
500 0.28 0.02 89.8
1000 11.6 0.06 78.5
1500 28.1 0.09 62.0
1900 36.5 0.11 53.6
© 2009 by Taylor & Francis Group, LLC
Modeling the Transport and Fate of Hydrophobic Chemicals 319
deposition (approximately 10 g/cm
2

) is there a signicant decrease in the amount
of PCBs resuspended and transported to the bay.
As shown in these calculations, most of the erosion of contaminants occurs
at the edge of the channel; that is, the amounts of erosion/deposition vary greatly
across a river. This is signicant in that, when considering the transport and fate
of contaminated sediments and potential remedial actions, it is essential to deter-
mine the contaminant concentrations and sediment erosion/deposition rates as a
function of distance across the river and, most importantly, at the edge of a chan-
nel where the depth and ow velocities may be changing rapidly.
In this investigation, effects of variable sediment properties were not consid-
ered. However, sediment properties and hence erosion rates should vary signi-
cantly across the channel (because of the changing bathymetry, ow velocities,
and deposition of different size particles) as well as with depth (because of ood
events). Because of this, variable sediment properties should be considered in a
more realistic calculation.
8.1.2 GREEN BAY,EFFECTS OF FINITE SORPTION RATES
To investigate the effects of nite sorption rates on the transport and fate of HOCs
in surface waters, calculations were made of the transport and fate of PCBs during
storms of different magnitudes on Green Bay (Chroneer and Lick, 1997). Calcula-
tions of the hydrodynamics and sediment transport were summarized in Section
6.5 (bathymetry is shown in Figure 6.30) and were the basis for the calculations
of PCB transport presented here.
For these calculations, it was assumed that the bottom sediments of the bay
were uniformly contaminated with PCBs at a concentration of 1 µg/kg. The over-
lying water was initially free of PCBs. A moderate wind of 10 m/s from right to
left (as in Figure 6.30) for a period of 2 days then caused a resuspension of sedi-
ments and contaminants; this was followed by a low wind of 2 m/s for 12 days,
during which time the sediments deposited. As the sediments and associated con-
taminants were resuspended, the contaminants desorbed from the sediments and
dissolved in the water. This desorption was quantied by means of the model

described in Section 7.2 with the mass transfer coefcient, k, given by Equation
7.34. An average partition coefcient of 10
4
L/kg was assumed. As in Section
7.2, the diffusion coefcient for the HOC within the particle, D, was taken to be
2×10
−14
cm
2
/s. Three sizes of particles were assumed, with diameters of 4, 14,
and 29 µm and size fractions of 10, 60, and 30%, respectively. From this, the mass
transfer coefcient for each size class was calculated. Calculations were done for
nite rates of sorption and also for chemical equilibrium, the more usual assump-
tion in contaminant transport and fate calculations.
Results of these calculations are shown in Figures 8.2 and 8.3. Figure 8.2
shows the changes in C
s
and C
w
as a function of time at the location in the outer
bay denoted by a * in Figure 6.29. Subscripts s and w denote suspended sedi-
ment and water, respectively; e and ne denote equilibrium and nonequilibrium,
respectively; f, m, and c denote ne, medium, and coarse sediments; and avg
© 2009 by Taylor & Francis Group, LLC
320 Sediment and Contaminant Transport in Surface Waters
10 m/s
(a)
40
40
20

20
2
10
1
1
1
N
5
0.5
0.2
2
5
80
0 10 20 km
10
10 m/s
(b)
800
0 10 20 km
400
200
200
400
100
100
10
25
25
5
10

2
10
N
50
50
FIGURE 8.3 Concentrations of PCBs in Green Bay at the end of the 14-day event. Solid
lines are for the equilibrium calculation, whereas the dashed lines are for the nonequilibrium
calculation: (a) C
w
in ng/L, and (b) C
s
in µg/kg. (Source: From Chroneer and Lick, 1997.)
1000
3.0
2.5
2.0
1.5
1.0
C
w
(ng/L)
C
s
(µg/kg)
C
sf
C
se
C
savg

C
wne
C
sm
C
we
C
sc
0.5
0.0
900
800
700
600
500
400
300
02468
Time (days)
10 12 14
200
100
0
FIGURE 8.2 Green Bay. Contaminant concentrations as a function of time at the point
denoted by a * in Figure 6.29. Subscripts s and w denote suspended sediment and water,
respectively; e and ne denote equilibrium and nonequilibrium; f, m, and c denote ne,
medium, and coarse sediments; and avg denotes the average of all size classes. (Source:
From Chroneer and Lick, 1997.)
© 2009 by Taylor & Francis Group, LLC
Modeling the Transport and Fate of Hydrophobic Chemicals 321

denotes the average of all size classes. For the equilibrium case, C
se
=K
p
C
we
and
C
we
is therefore proportional to C
se
. As the sediments are resuspended and also
transported to this location from sites in shallower waters near shore, C
se
and C
we
increase rapidly at rst (due primarily to resuspension) and then more slowly due
to transport of sediments and contaminants from the nearshore.
This transport of PCBs is greatly modied by nite sorption rates. In this
case, C
wne
is initially much lower and C
savg
is much higher than their equilib-
rium values due to the slow desorption of PCBs from the suspended sediments
to the water. In more detail, the ne sediments (C
sf
) desorb rapidly, whereas the
medium (C
sm

) and coarse (C
sc
) sediments desorb much more slowly; the medium
and coarse sediments lose only a small fraction of their sorbed PCBs during the
14-day event. Because the ne fraction tends to stay in suspension much longer
than the medium and coarse fractions, C
savg
is dominated by the ne fraction and
approaches C
sf
as time increases. For nite rates of sorption and for the rst few
days, the sediments retain signicantly more of their PCBs as they are trans-
ported than in the equilibrium case (i.e., C
savg
>> C
se
); after deposition, the bottom
sediments that are deposited during this time would also have a much higher
concentration of PCBs. As time increases, C
sf
decreases below C
se
because of the
desorption to a low C
wne
; C
savg
decreases below C
se
for the same reason.

The distributions of C
w
and C
s
in the water of the bay at the end of the 14-day
event are shown in Figures 8.3(a) and (b), respectively. Both C
wne
<C
we
and
C
savg
<C
se
throughout the bay, by approximately a factor of two in the inner bay
and by a factor of ve or more in the outer bay.
At the end of the 14 days, the percentage of PCBs originally resuspended
and still remaining in the water (dissolved in the water plus the small amount
sorbed to the remaining suspended particles that have not yet deposited) is 27%
for the equilibrium case and only 11% for the nonequilibrium case. This per-
centage depends on the amount of sediment resuspension. At low wind speeds,
sediment resuspension is low; a higher percentage of the PCBs desorbs from the
resuspended sediments to the water and remains there as the particles settle. At
high wind speeds, the resuspension is high; a lower percentage of the PCBs des-
orbs, and most of the PCBs are therefore still sorbed to the particles and are trans-
ported with the particles as they settle to the bottom. Results for winds of 5, 10,
and 20 m/s from the northeast for 2 days are summarized in Table 8.2. Of course,
although the percentage of resuspended PCBs that remains in the water decreases
as the wind speed increases, the total amount of PCBs remaining in the water
increases because of the very nonlinear increase of sediment and contaminant

resuspension as the wind speed increases.
In these calculations, K
p
was assumed to be 10
4
L/kg, a relatively low average
value for PCBs. Because desorption rates are inversely proportional to K
p
, the
transport of PCBs with higher values of K
p
, say 10
5
to 10
6
L/kg, would differ con-
siderably from that shown here; because of the much lower desorption rates, there
would be much lower values of C
w
in the overlying water and higher values of C
s
© 2009 by Taylor & Francis Group, LLC
322 Sediment and Contaminant Transport in Surface Waters
in the overlying water and in the deposited sediments for sediments with high K
p
as compared with those with low K
p
or with equilibrium partitioning.
PCBs are generally mixtures of PCB congeners, each with a different K
p

.
These K
p
values can differ from one another by more than an order of magnitude,
and hence congener desorption rates also can differ by more than an order of
magnitude. This dependence on K
p
of the congener desorption rate and hence
transport in the overlying water also pertains to the nonerosion/nondeposition
sediment-water ux (Sections 8.3 and 8.4). Because of this as well as differing
solubilities, volatilization rates, and dechlorination rates (all of which depend on
K
p
), the relative concentrations of PCB congeners will change during transport.
This dependence on K
p
probably is a signicant contributor to the “weathering”
of PCBs (e.g., as reported for PCBs in the Hudson River (National Research
Council, 2001)).
8.2 THE DIFFUSION APPROXIMATION FOR
THE SEDIMENT-WATER FLUX
As a more accurate approximation than the mass transfer approximation of Equa-
tion 8.1 or 8.2, the vertical transport of a chemical within the sediment to the
overlying water has often been described as simple, or Fickian, diffusion. This
approximation is usually only valid for molecular diffusion of an inert, nonreact-
ing substance. Alternately, when chemical reactions are present and are fast (e.g.,
when adsorption and desorption times are small compared to diffusion transport
times in the sediments), then a quasi-equilibrium diffusion approximation can be
used. These two limiting approximations are described below and also are com-
pared with results from the mass transfer approximation as described by Equation

8.1 or 8.2.
8.2.1 SIMPLE, OR FICKIAN,DIFFUSION
The basic equations for the diffusion of an inert chemical are essentially the same
as those for heat conduction (e.g., Carslaw and Jaeger, 1959) and can be derived in
a similar manner. For the one-dimensional, time-dependent diffusive transport of
an inert chemical with concentration C(x,t), the ux is given by
TABLE 8.2
Percentage of Resuspended Contaminants in Green Bay
That Are Still in the Water at End of Event
Wind Speed(m/s)
51020
Equilibrium 76 27 13
Nonequilibrium 30 11 8
© 2009 by Taylor & Francis Group, LLC
Modeling the Transport and Fate of Hydrophobic Chemicals 323
qD
C
x

t
t
(8.3)
where D is a diffusion coefcient (cm
2
/s). The conservation equation for C (i.e.,
the time rate of change of C in an innitesimal layer due to this ux) is given by
t
t

t

t
C
t
q
x
(8.4)
These two equations can be combined to give
t
t

t
t
t
t
¤
¦
¥
³
µ
´

t
t
C
tx
D
C
x
D
C

x
2
2
(8.5)
where the second form is valid when D is not a function of x.
To illustrate the application of this equation to a specic problem, consider
the one-dimensional, time-dependent transport by diffusion of a dissolved, inert
chemical at constant concentration, C
o
, in the overlying water into clean sedi-
ment. From continuity, the chemical concentration in the sediment at the surface
is C
o
. The governing equation, initial condition, and boundary conditions can
then be written as
t
t

t
t
C
t
D
C
x
2
2
(8.6)
C(x,0)=0 (8.7)
C(0,t)=C

o
(8.8)
lim ( , )
x
Cxt
lc
 0 (8.9)
where D is assumed constant. The solution to this problem is
C C erfc
x
Dt
o

¤
¦
¥
³
µ
´
2
(8.10)
where erfc z is the complementary error function (Carslaw and Jaeger, 1959) and
zx Dt /2
. For D=1×10
−5
cm
2
/s (an approximate value for the molecular dif-
fusion coefcient for many chemicals in water), this solution is shown in Fig-
ure 8.4(a) for t = 1, 10, and 100 days. Because C is only a function of z, C can be

more economically plotted as a function of z only; this is shown in Figure 8.4(b).
© 2009 by Taylor & Francis Group, LLC
324 Sediment and Contaminant Transport in Surface Waters
It is useful to dene a decay length as the depth at which z = 1. This length
is a convenient denition for the approximate distance to which the chemical
penetrates with time. For z = 1, (1) erfc z = erfc 1 = 0.157 and C/C
o
has therefore
decreased from 1.0 at the surface to 0.157 at z = 1, and (2) x =
2Dt; that is, the
distance to which the chemical penetrates increases as the square root of time.
At the surface, the chemical ux into the sediment is given by
qt D
Ct
x
DC
Dt
o
(,)
(,)
0
0

t
t

P
(8.11)
This ux is singular as t n 0, decreases as t
−1/2

, and goes to zero as t ne. The
total mass of chemical diffused into the sediment per unit area as a function of
time, Q(t), is
Qqtdt
t

¯
0
0(,) (8.12)
For the present problem, it follows from integration of Equation 8.11 that
QC
Dt
o

¤
¦
¥
³
µ
´
4
12
P
/
(8.13)
and grows with time as t
1/2
; it is not singular as t n 0.
0
100 Days

10
1
0 1.0
C/C
0
C/C
0
5
10
15
20
x (cm)
0
0 1.0
5
10
15
20
z
(a) (b)
FIGURE 8.4 C/C
o
as a function of (a) depth at different times and (b) z = x/2(Dt)
1/2
.
© 2009 by Taylor & Francis Group, LLC
Modeling the Transport and Fate of Hydrophobic Chemicals 325
8.2.2 SORPTION EQUILIBRIUM
For hydrophobic chemicals, the effects of the adsorption and desorption of chemi-
cal between the pore waters and solid particles must be considered. As described

in Chapter 7, this sorption process is time dependent and its characteristic times
are often comparable to or longer than those for other processes of interest. When
this is true, the rate of sorption must be explicitly considered, and the problem
becomes mathematically more complex than before. In some cases, sorption rates
may be sufciently fast that the chemical sorbed to the solids is in approximate
equilibrium with the chemical dissolved in the pore water. When this occurs, a
quasi-equilibrium diffusion approximation is valid and can be used. This approx-
imation is derived as follows.
In the following, HOC transport by both molecular diffusion and bioturbation
will be considered for generality. Molecular diffusion will be approximated by a
diffusion of the HOC in the pore water with a diffusion coefcient D
m
. For simplic-
ity and as a rst approximation (see Section 8.4 for a more accurate approxima-
tion), bioturbation will be approximated as a diffusion of solids and pore water with
an effective diffusion coefcient of D
b
. The diffusion coefcient for contaminants
sorbed to solids, D
s
, is then the same as D
b
, whereas the diffusion coefcient for the
contaminant dissolved in the pore water, D
w
, is the sum of D
m
and D
b
. In general, D

b
is dependent on depth in the sediments; its value is a maximum at the surface and
it decreases with depth, with a characteristic length scale on the order of 5 cm for
fresh-water organisms and on the order of 10 cm or more for sea-water organisms.
With these approximations and including time-dependent, nonequilibrium
sorption as described by Equation 7.31, the one-dimensional, time-dependent
mass conservation equation for the contaminant dissolved in water (per unit vol-
ume of sediment) is
FF FR
t
t

t
t
t
t
¤
¦
¥
³
µ
´
 
C
tx
D
C
x
kC KC
w

w
w
ss pw
()( )1 (8.14)
whereas the conservation equation for the contaminant sorbed to the solids (again,
per unit volume of sediment) is
() () ()11 1
t
t

t
t
t
t
¤
¦
¥
³
µ
´
 FR FR F
s
s
ss
s
C
tx
D
C
x

RR
ss pw
kC KC() (8.15)
where K is the porosity of the sediments and has been assumed to be constant, and S
s
is
the mass density of the solid particles. The ux of contaminant between the sediments
and the overlying water due to diffusion of the dissolved contaminant is given by
qt D
C
x
t
w
w
() ( ,)
t
t
F 0 (8.16)
© 2009 by Taylor & Francis Group, LLC
326 Sediment and Contaminant Transport in Surface Waters
It is assumed that there is no ux of contaminant from the solid particles directly
into the overlying water.
Equations 8.14 and 8.15 are coupled equations and hence must generally be
solved simultaneously. To avoid this complexity, a common approximation is to
assume quasi-equilibrium (fast sorption rates) so that C
s
! K
p
C
w

. By adding Equa-
tions 8.14 and 8.15 and then using this assumption, one obtains a diffusion equa-
tion with a modied diffusion coefcient:
t
t

t
t
t
t
¤
¦
¥
³
µ
´
C
tx
D
C
x
ww
*
(8.17)
where
D
D
K
D
m

sp
b
*



¤
¦
¥
³
µ
´

1
1 F
F
R
(8.18)
For hydrophobic chemicals with large K
p
values and in the absence of biotur-
bation, D* << D
m
. If D
*
is independent of depth, then Equation 8.17 reduces to
t
t

t

t
C
t
D
C
x
ww
*
2
2
(8.19)
and has the same form as Equation 8.6. In general, the sediment-water ux is still
given by Equation 8.16, with no substitution of D* for D
w
.
Although the above three equations are reasonable approximations for many
cases, they are only valid when the assumption of quasi-equilibrium is valid, that
is, for fast reaction rates when C
s
! K
p
C
w
. As will be demonstrated in the follow-
ing sections, these equations are not valid to describe the diffusion of HOCs into
and from sediments in many, if not most, realistic conditions.
8.2.3 A MASS TRANSFER APPROXIMATION
Using the mass transfer approximation (Equation 8.2), the transfer of a dissolved
chemical at constant concentration, C
wo

, in the overlying water into a well-mixed
sediment layer of thickness h that is initially clean can be described by
()1
¤
¦
¥
³
µ
´
FR
s
ss
p
wo
h
dC
dt
D
h
C
K
C (8.20)
where C
s
(t) is the sorbed chemical concentration in the layer, K is the porosity of
the sediment, and S
s
is the mass density of a solid particle. Because of the assumed
large partition coefcient and the implicit assumption of equilibrium partitioning,
the amount of chemical dissolved in the pore water can be neglected compared

© 2009 by Taylor & Francis Group, LLC
Modeling the Transport and Fate of Hydrophobic Chemicals 327
with the amount of chemical sorbed to the solids. The term on the left-hand side is
the time rate of change of the mass of chemical per unit area in the layer, whereas
the term on the right-hand side is the ux into that layer. It is assumed that there
is no ux from the well-mixed layer into the sediments below.
The above equation can be integrated to give
C
s
=C
so
(1 − e
–kt
)(8.21)
k
D
hK
sp


2
1()FR
(8.22)
where C
so
=K
p
C
wo
. From Equations 8.20 and 8.21, the ux is then given as a func-

tion of time by
q
D
h
Ce
wo
kt


(8.23)
A decay time for q can be dened as kt* = 1 (t* is the time for q to decay to
e
−1
= 0.368 of its initial value), or
t
k
hK
D
hK
H
sp
sp
*
()
()




1

1
1
2
FR
FR
(8.24)
In the rst expression, t* ~ h
2
; in the second, t* ~ h. Either way, depending on
whether D or H is assumed constant in the modeling, it is clear that h is a crucial
parameter for the determination of t* and hence the time for natural recovery.
However, as will be seen in the following sections, the parameter h is difcult to
dene and even more difcult to accurately quantify.
The related problem of the ux of a chemical from a well-mixed layer of thick-
ness h with an initial chemical concentration C
so
into clean overlying water can be
solved in a similar manner. In this case, the solution for C
s
is given by C
so
e
−kt
; the
ux from the sediment is still given by Equation 8.23, with C
wo
=C
so
/K
p

.
As shown in Equation 8.11, the ux due to simple diffusion decays with time
as t
−1/2
. By contrast, Equation 8.23 indicates that the ux in the mass transfer
approximation decays as e
−kt
; in addition, this latter decay time, t*, depends on h,
a quantity that cannot be determined from the above equations. The two approxi-
mations are inherently different and will give widely different solutions for the
ux for large time. This will be discussed more thoroughly in Section 8.5.
8.3 THE SEDIMENT-WATER FLUX DUE TO
MOLECULAR DIFFUSION
For HOCs, the sediment-water ux due to molecular diffusion is often signi-
cantly modied by nite-rate sorption, with the amount and rate of sorption
© 2009 by Taylor & Francis Group, LLC
328 Sediment and Contaminant Transport in Surface Waters
dependent on the partition coefcient. This has been demonstrated and quantied
by experiments and theoretical modeling; some of these efforts will be described
here. The most detailed set of experiments were one-dimensional, time-dependent
experiments for hexachlorobenzene (HCB) diffusing into and sorbing to a Detroit
River sediment (Deane et al., 1999). For these sediments, the measured parti-
tion coefcient for HCB was 1.2 × 10
4
L/kg. The lengths of these experiments
were variable, but some continued for up to 512 days. Deane et al. also did triti-
ated water (THO) experiments in order to (1) illustrate the differences between
a purely diffusing (non-sorbing) chemical, THO, and a chemical that diffuses
but also strongly sorbs to the sediment, HCB, and (2) obtain parameters for pure
diffusion that were then used to more accurately interpret the HCB experiments.

These results and analyses by means of numerical models are described rst.
For further understanding of the sediment-water ux of HOCs due to molecu-
lar diffusion, additional experiments were later done with two different sediments
and with HOCs that had a range of K
p
values from approximately 10 L/kg to
5×10
4
L/kg (Lick et al., 2006b); these results and analyses are presented next.
For remediation purposes, sediment-water uxes need to be predicted over long
periods of time, up to 100 years. On the basis of the experimental work and analy-
ses, numerical calculations were made and are used here to illustrate the char-
acteristic behavior of HOC diffusional uxes over these long periods of time.
For a more general understanding of the molecular diffusion of HOCs, related
problems with different boundary and/or initial conditions than those above are
also discussed. In particular, results of desorption experiments are presented that,
when compared to the results of adsorption experiments, demonstrate the revers-
ibility of the process of molecular diffusion with nite sorption rates.
8.3.1 HEXACHLOROBENZENE (HCB)
8.3.1.1 Experiments
In the experiments described here, HCB diffused from the overlying water into
an initially clean sediment. The procedure was such that replicate sediment col-
umns (more accurately called patties because of their small length-to-diameter
ratio) were rst formed, the upper surfaces of these patties were then exposed to
water with a high concentration of dissolved HCB, the HCB diffused into each
patty, and the amount of HCB in each patty was measured as a function of depth
and time. Radio-labeled HCB was used, and concentrations were measured using
liquid scintillation counting. The experimental procedure is only briey summa-
rized here; Deane et al. (1999) should be consulted for the details.
The sediments used were ne-grained sediments (median particle size of

15 µm) from the Detroit River; the organic carbon content was 3.2%. To form the
sediment patties, cylindrical dishes were constructed with an inner diameter of
1.5 cm and a depth of 1 cm. The bottom of each patty was supported by a move-
able piston. The piston could be moved at 1-mm increments; 1-mm slices of the
patty could then be taken to determine the HCB concentration as a function of
© 2009 by Taylor & Francis Group, LLC
Modeling the Transport and Fate of Hydrophobic Chemicals 329
depth. Replicate patties were sacriced at different time intervals; this allowed
the determination of HCB concentration as a function of depth at different time
intervals. Patties were placed in a 500-mL glass jar with a Teon-lined lid along
with a magnetic stir bar. The jars contained 300 mL HCB-saturated water and
a stainless steel source jar containing solid HCB. The HCB source jar and stir
bar kept the water well mixed and held the dissolved HCB concentration near
saturation over the duration of the experiments. In this way, the upper surfaces
of the patties were exposed for long periods of time to water containing HCB at
concentrations near saturation.
THO experiments were performed with the same sediments and in a similar
manner except that tritiated water was used as the diffusing chemical. Because
THO does not sorb to sediments, the quantity measured in the experiments was
the THO concentration in the pore water, C
w
. HCB has a large K
p
and hence sorbs
strongly to the solids in the sediments; the amount of HCB dissolved in the water
is negligible by comparison with that sorbed to the solids. Because of this, the
quantity measured was the concentration of the HCB sorbed to the solids, C
s
.
8.3.1.2 Theoretical Models

Although Fickian diffusion is a valid approximation for chemicals diffusing into
sediments in some limiting cases (as described in the previous section), the actual
transport of either tritiated water or HCB is more complicated than this. Consider
rst the simpler case of the diffusion of tritiated water. Preliminary compari-
sons of experimental results with theoretical models demonstrated that, in order
to describe this transport accurately, it was necessary to assume that the water-
lled pores in the sediments could be divided into two compartments (Coates and
Smith, 1964; Van Genuchten and Wierenga, 1976; Nkedi-Kizza et al., 1984; Har-
mon et al., 1989): (1) the main channels where vertical diffusion occurs and (2)
less accessible side pores into which a chemical can diffuse from the main chan-
nels but cannot diffuse vertically in these side pores. Mass conservation equations
(per unit volume of sediments) to describe this process can be written as
FF Fv
C
t
vD
C
x
vk C C
ww
ww w1
1
1
2
1
2
121
t
t


t
t
() (8.25)
FFv
C
t
vk C C
w
ww w2
2
121
t
t
 () (8.26)
where C
w1
= chemical concentration in compartment 1 (the main channels);
C
w2
= chemical concentration in compartment 2 (the side pores); k
w
= mass trans-
fer coefcient (cm/s) for transport of the chemical from compartment 1 to com-
partment 2; and v
1
and v
2
= fractional volumes of sediments (solids and water) in
compartments 1 and 2, respectively.
For the case of HCB, or more generally HOC transport, the time-dependent

sorption of the HOC to the organic matter in or on the solid particles must be
© 2009 by Taylor & Francis Group, LLC
330 Sediment and Contaminant Transport in Surface Waters
considered. For pure molecular diffusion and subsequent sorption in a single com-
partment, the conservation equations reduce from Equations 8.14 and 8.15 to
FF FR
t
t

t
t
 
C
t
D
C
x
kC KC
w
w
w
ss pw
2
2
1()( )
(8.27)
() ()( )11
t
t
  FR FR

s
s
ss pw
C
t
kC KC (8.28)
where it is assumed that D
w
is constant.
In the most general case of molecular diffusion with sorption, it is assumed
that (1) HCB in solution diffuses through the main channels and then into the
side pores; (2) in each compartment, HCB in solution sorbs to the solids in that
compartment; and (3) three different size classes of sediments exist, each with
a different mass transfer coefcient, k
i
, and size fraction, f
i
. The resulting mass
conservation equations are then
FF F
F
v
C
t
vD
C
x
vk C C
w
w

w
ww w1
1
1
2
1
2
121
1
t
t

t
t


()
())()vfkCKC
s
i
ii si p w1
1
3
11
R

£

(8.29)
FF FRv

C
t
vk C C v fk
w
ww w s
i
i2
2
121 2
1
3
1
t
t
   

£
()()
iisi pw
CKC()
22
 (8.30)
() () (11
1
1
11

t
t
  FR FR

si
si
si i si p w
fv
C
t
fvk C K C
11
) (8.31)
() () (11
2
2
22

t
t
  FR FR
si
si
si i si p w
fv
C
t
fv k C K C
22
) (8.32)
where C
si1
is the chemical concentration in the i-th component of the solids in
compartment 1, and C

si2
is the chemical concentration in the i-th component of
the solids in compartment 2.
In the present case of the time-dependent ux of a dissolved HOC in the over-
lying water into clean bottom sediments, the boundary and initial conditions are
that C
w
(0,t)=C
wo
=constant, C
w
(x,0) = 0, and C
s
(x,0) = 0; for a sediment of depth
h, vC
w
(h,t)/vx = 0; and for a sediment of innite depth, C
w
(x,t) n 0 as x ne.
8.3.1.3 Diffusion of Tritiated Water
For the diffusion of THO into Detroit River sediments, experimental results for
the normalized concentration, C
w
/C
wo
, are shown as a function of depth and at
© 2009 by Taylor & Francis Group, LLC
Modeling the Transport and Fate of Hydrophobic Chemicals 331
different times in Figure 8.5. The results are qualitatively similar to those expected
for pure Fickian diffusion. Because the sediment thickness is only 1 cm, the triti-

ated water rapidly saturates the sediment and reaches an approximate steady state
in about 8 days. A numerical calculation was rst made with the assumption that
the diffusion was Fickian, that is, by means of Equation 8.6. The results were quali-
tatively correct but did not agree well with the experimental results for all times. Much
better agreement was found if it was assumed that the diffusion was governed by Equa-
tions 8.25 and 8.26, that is, diffusion into the less accessible pores was signicant.
However, although the agreement was improved for longer times, the calculated results
for the rst few hours were still not in good agreement with the experimental results.
This was probably an experimental artifact due to the initial placement of the sediment
column into the receiving water; this placement inevitably caused small convection
currents in the overlying receiving waters that may have caused a slight convective
penetration of THO into the sediment column. To correct for this in the calculations,
it was assumed that the initial conditions for the calculation were those given by
the experimental data at 1 hour. The calculations were then continued by means of
Equations 8.25 and 8.26. Results of this latter calculation are shown along with the
experimental results in Figure 8.5. There is excellent agreement between the calcu-
lations and experimental data for all time. For this calculation, it was assumed that
D=6×10
−6
cm
2
/s, k
w
=5×10
−6
/s, and v
1
=v
2
= 0.5. The value of D is approximately

the same as that typically found for dissolved chemicals in water.
8.3.1.4 HCB Diffusion and Sorption
For the experiments with HCB diffusing from the overlying water into clean sedi-
ments, the measured HCB concentrations on the solids, C
s
, as a function of depth
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1.0
Core Section (1-mm thicknesses)
C
w
/C
w0
8d
4d
2d
6h
12h
1d
1h
FIGURE 8.5 Experimental and theoretical results for the diffusion of THO into consoli-
dated Detroit River sediment. Concentration of THO in the pore water is normalized by
the overlying water concentration. (Source: From Deane et al., 1999. With permission.)
© 2009 by Taylor & Francis Group, LLC
332 Sediment and Contaminant Transport in Surface Waters

with time as a parameter are shown in Figure 8.6. Due to sorption, the diffusion
of HCB into the interior is much slower than that of THO. Despite the length of
the experiments (512 days), (1) signicant changes in C
s
are limited to a few milli-
meters near the sediment-water interface, and (2) measured values of C
s
/C
wo
near
the surface are generally less than 0.1 of their equilibrium value at the surface
(where C
s
/C
wo
in equilibrium should equal K
p
, i.e., 1.2 × 10
4
L/kg).
Results of a numerical simulation (which necessarily included nonequilibrium
sorption) are shown as the solid lines in Figure 8.6. Parameters assumed were
D=6×10
–6
cm
2
/s; k
i
=1.6×10
−8

, 8×10
−9
, and 8 × 10
−10
/s; and f
i
= 1/3 for the
three size classes with i = 1, 2, and 3. The agreement with the experimental results
is quite good. In this calculation, the assumptions of three size classes of sediment
aggregates (equivalent to diameters of 9, 120, and 400 µm as shown below) as
well as diffusion into primary and secondary pores were made. For a calculation
with only one size class (120 µm), the agreement was good for intermediate to
large times but was not quite as good for small times. A calculation that ignored
diffusion into secondary pores modied the results, but not signicantly.
Because of experimental limitations, the chemical concentration of HCB in
the pore water, C
w
, could not be measured. However, this quantity was calculated;
values of C
s
and K
p
C
w
(both normalized with respect to C
wo
) are shown in Fig-
ure 8.7 as a function of depth at different times from 4 to 512 days. Both C
s
and

K
p
C
w
monotonically increase with time and decrease with depth. It can be seen
that (1) C
s
and K
p
C
w
are quite different and are therefore not in local chemical
equilibrium with each other, or even close to equilibrium, even after 512 days; and
(2) C
w
is almost independent of time and is only a function of distance.
0
0
12345678910
Depth (cm)
C
s
/C
w0
(L/kg)
200
400
600
800
1000

1200
1400
4 d
16 d
32 d
64 d
128 d
256 d
512 d
32 d
64 d
128 d
512 d
256 d
FIGURE 8.6 Experimental and theoretical results for the molecular diffusion of HCB
into consolidated Detroit River sediments. C
s
/C
wo
is shown as a function of depth at differ-
ent times from 4 to 512 days. (Source: From Deane et al., 1999. With permission.)
© 2009 by Taylor & Francis Group, LLC
Modeling the Transport and Fate of Hydrophobic Chemicals 333
When the entire 1-cm column is saturated, C
w
should equal C
wo
and C
s
/C

wo
should equal K
p
throughout the sediment column. From the experimental results,
it is quite clear that C
s
/C
wo
is far from saturation everywhere. In fact, by means of
the model, it can be demonstrated that attainment of 90% of saturation throughout
the 1-cm sediment column will take approximately 300 years, or about 10
5
days.
This is to be compared with about 8 days for THO saturation (Figure 8.5).
In this calculation, the value of D was taken to be 4 × 10
−6
cm
2
/s, that is,
approximately the same as that for THO in the previous calculation and similar
to that for many dissolved chemicals in water. It has been suggested that col-
loids contribute signicantly to HOC transport (Thoma et al., 1991). From the
present results, the effect of colloids on transport seems to be minimal because
the diffusion coefcient necessary for calibration is not signicantly greater than
those for dissolved chemicals in water.
Although the average disaggregated size of the Detroit River sediments was
15 µm, the effective sizes for sorption can be quite different because, during con-
solidation, the sediment particles form into larger aggregates of particles. This
can be shown as follows. For each size class, an effective mass transfer coefcient
is given by Equation 7.34, and therefore

k
D
d
i
i

0 0165
2
.
(8.33)
12000
11000
10000
9000
8000
7000
6000
5000
K
p
= 12000 L/kg
D
m
= 3 × 10
–6
cm
2
/s
4000
3000

2000
1000
0
0 0.1 0.2 0.3 0.4 0.5
Depth (cm)
K
p
C
w
/C
wo
C
s
/C
wo
C
s
/C
wo
and K
p
C
w
/C
wo
(L/kg)
0.6 0.7 0.8 0.9 1
FIGURE 8.7 Calculated results for the molecular diffusion of HCB into consolidated
Detroit River sediments. C
s

/C
wo
and K
p
C
w
/C
wo
are shown as functions of depth at times of 4,
16, 32, 64, 128, 256, and 512 days. (Source: From Deane et al., 1999. With permission.)
© 2009 by Taylor & Francis Group, LLC
334 Sediment and Contaminant Transport in Surface Waters
From previous experiments and analyses with Detroit River sediments, the value
of the diffusion coefcient within the particle, D, is approximately 2 × 10
−14
cm
2
/s
(Lick and Rapaka, 1996). The above equation then gives a relation between the
effective diameters for sorption, d
i
, and the mass transfer coefcients assumed in
the numerical calculations. It follows from the above equation and the assumed
values of k
i
that the effective particle diameters for the three size classes are 9,
120, and 400 µm. The sediments in the smallest size class are then comparable
to the disaggregated sediment particles, whereas the sediments in the larger size
classes show the effects of aggregation during consolidation and have much larger
sorption equilibration times.

8.3.2 ADDITIONAL HOCS
Experiments and theoretical analyses were done to broaden the HCB investiga-
tion so as to include additional HOCs with a wide range of K
p
values and with
diffusion into two different sediments (Lick et al., 2006b). Experiments were
done with three HOCs (a tetrachlorobiphenyl, TPCB; a monochlorobiphenyl,
MCB; and pentachlorophenol, PCP) in Detroit River sediments (organic content
of 3.2%) and ve HOCs (HCB, pyrene, phenanthrene, naphthalene, and benzene)
in Lake Michigan sediments (organic content of 1.8%). The procedures to do this
were similar to those described above.
8.3.2.1 Experimental Results
The HOCs, K
p
values, and the duration of each experiment, T, are given in Table 8.3.
The K
p
values range from 46,000 to 11.5 L/kg. Results for three representative HOCs
TABLE 8.3
Parameters for Molecular Diffusion Experiments
Chemical
K
p
(L/kg)
Length of
Experiments,
T (days)
Molecular
Weight
D

w0
(10
–6
cm
2
/s)
Detroit River Sediments (3.2% o.c.)
TPCB 46000 256 292 4.0
HCB 12000 512 285 4.0
MCB 1200 64 113 6.35
PCP 1000 64 266 4.14
Lake Michigan Sediments (1.8% o.c.)
HCB 9400 96 285 1.12
Pyrene 3700 64 202 1.36
Phenanthrene 2300 64 178 1.44
Naphthalene 80 32 128 1.68
Benzene 11.5 96 78 2.16
© 2009 by Taylor & Francis Group, LLC
Modeling the Transport and Fate of Hydrophobic Chemicals 335
are shown in Figure 8.8 for TPCB, K
p
= 46,000 L/kg, Detroit River sediments (Fig-
ure 8.8(a)); MCB, K
p
= 1200 L/kg, Detroit River sediments (Figure 8.8(b)); and
naphthalene, K
p
= 80 L/kg, Lake Michigan sediments (Figure 8.8(c)). The results
for these HOCs as well as for the other HOCs not shown here are qualitatively simi-
lar to those for HCB (Figure 8.6); that is, (1) signicant changes in C

s
are limited
to a few millimeters near the sediment-water interface and (2) values of C
s
/C
wo
are
generally much less than their equilibrium value of K
p
at the surface. In addition,
500
400
300
200
100
0
0 1 2 3 4 5 6
Depth (mm)
C
s
/C
wo
(L/kg)
Day 2
Numerical day 64
Day 4
Day 8
Day 16
Day 32
Day 64

Day 128
Day 256
7 8 9 10
(a)
350
300
250
200
150
100
50
Numerical day 64
Day 4
Day 16
Day 16
Day 32
Day 32
Day 64
Day 64
0
0 1 2 3 4
Depth (mm)
C
s
/C
wo
(L/kg)
5 6 7 8 9 10
(b)
FIGURE 8.8 Experimental and theoretical results for the molecular diffusion of HOCs

into consolidated Detroit River sediments. C
s
/C
wo
is shown as a function of depth at differ-
ent times. (a) TPCB, K
p
= 46,000 L/kg. Experimental results are shown at 2, 4, 8, 16, 32,
64, 128, and 256 days; calculated results are shown at 64 days. (b) MCB, K
p
=1,200 L/kg.
Experimental results are shown at 4, 16, 32, and 64 days; calculated results are shown at
64 days. (Source: From Lick et al., 2006b. With permission.)
© 2009 by Taylor & Francis Group, LLC
336 Sediment and Contaminant Transport in Surface Waters
the results show that as K
p
decreases, (3) the chemical diffuses more rapidly into
the interior, and (4) values of C
s
/C
wo
tend to approach their equilibrium value of K
p
at the sediment-water interface more rapidly.
Comparison of the results for HOCs with comparable K
p
values in the two
different sediments shows that the concentrations are somewhat higher and the
HOC has diffused further into the sediments for the Detroit River sediments than

for the Lake Michigan sediments. The differences are not unexpected because the
two sediments are somewhat different with different amounts and possibly qual-
ity and size of organic matter and with somewhat different particle size distribu-
tions. However, quantitative reasons for these differences are not known.
8.3.2.2 Theoretical Model
In the modeling of HCB diffusion as described above, one-dimensional, time-
dependent diffusion with a nite rate of sorption of HCB between the solid
particles and pore water in the sediment was assumed. The general features of
the model used in the calculations shown here are essentially the same except
that (1) the diffusion coefcient and porosity are assumed to vary with depth (this
can be signicant near the sediment-water interface) and (2) molecular diffusion
occurs in the water in the main channels only; that is, the presence of secondary
pores is ignored (this latter process was shown to have a minor effect in the above
calculations for HCB).
35
30
25
20
Numerical day 32
Day 1
Day 1
Day 2
Day 2
Day 4
Day 4
Day 8
Day 8
Day 16
Day 16
Day 32

15
10
5
0
0 1 2 3 4 5
Depth (mm)
C
s
/C
wo
(L/kg)
6 7 8 9 10
(c)
FIGURE 8.8 (CONTINUED) Experimental and theoretical results for the molecular
diffusion of HOCs into consolidated Lake Michigan sediments. C
s
/C
wo
is shown as a func-
tion of depth at different times. (c) Naphthalene, K
p
= 80 L/kg. Experimental results are
shown at 1, 2, 4, 8, 16, and 32 days; calculated results are shown at 32 days. (Source: From
Lick et al., 2006b. With permission.)
© 2009 by Taylor & Francis Group, LLC
Modeling the Transport and Fate of Hydrophobic Chemicals 337
With these approximations, the one-dimensional, time-dependent conserva-
tion equation for the contaminant dissolved in the water (per unit volume of sedi-
ment) is
FF FR

t
t

t
t
t
t
¤
¦
¥
³
µ
´
 
£
C
tx
D
C
x
fk C
w
w
w
i
si i si
() (1KKC
pw
) (8.34)
whereas the conservation equation for the contaminant sorbed to each size frac-

tion of the solids (again per unit volume of sediment) is
() () ( )11
t
t
  FR FR
si
si
si i si p w
f
C
t
fk C K C (8.35)
The ux of contaminant between the sediment and the overlying water is again
given by
qD
Ct
x
w
w

t
t
F
(,)0
(8.36)
The spatial variations in the diffusion coefcient and porosity are represented by
DD D e
ww w
x
 


01
1
1
(
)G
(8.37)
FF F
G
 

01
1
2
()e
x
(8.38)
where D
w0
, D
w1,
K
0
, K
1
, H
1
, and H
2
are constants and are to be determined from the

experiments. The variations of D
w
and K with depth for different sediments are
described below.
8.3.2.3 Numerical Calculations
Based on the above theoretical model, numerical calculations were made for
each HOC. In the calculations, several parameters are present that need to be
determined. In particular, the effective molecular diffusion coefcient, D
w
, and
the mass transfer coefcients, k
i
, are needed for each chemical and sediment.
These can be determined by a comparison of the numerical calculations and
experimental results and an adjustment of the parameters in the model until there
is agreement between the two.
However, for a uniformly valid analysis, the variations of these parameters
from one chemical to another are not arbitrary but are constrained by theory. The
basic procedure to determine these parameters was therefore to rst adjust the
coefcients for HCB for a particular sediment so as to obtain good agreement
between the calculated and experimental results. For the other HOCs and the same
sediment, the variations in the diffusion and mass transfer coefcients were then
determined from basic theory as follows. The coefcient D
w
was varied so as to be
© 2009 by Taylor & Francis Group, LLC