Chemical Geology 242 (2007) 40 – 50
www.elsevier.com/locate/chemgeo
Appropriateness of equilibrium assumptions for determining metal
distribution and transport in bacteria-bearing porous media
Benjamin F. Turner ⁎, Jeremy B. Fein
Deparment of Civil Engineering and Geological Sciences 156 Fitzpatrick Hall,
University of Notre Dame, Notre Dame, IN 46556, United States
Received 6 June 2006; received in revised form 26 February 2007; accepted 3 March 2007
Editor: D. Rickard
Abstract
The fate and transport of strongly-sorbed contaminants in aquifers is tied to the mobility of the surfaces to which they are adsorbed.
Mobile particulates, including mineral colloids, natural organic matter, or microbes, can enhance the mobility of contaminants that would
otherwise be adsorbed to stationary surfaces. In this study, we compare numerical transport simulations of systems involving dissolved
metals and microbial suspensions with previously published experimental results. The transport simulations utilize a finite difference
solute transport scheme, a colloid attachment term, and a Newton–Raphson chemical speciation solver assuming thermodynamic
equilibrium in the metal distribution between solution, mobile surfaces, and stationary surfaces. While model predictions of metal
breakthrough are of mixed quality, adjustment of model equilibrium constants is adequate to provide accurate fits to observed
breakthrough curves. Therefore, the idealized behaviors apparent in experimental breakthrough curves indicate metal transport under
equilibrium sorption conditions, and hence the assumption of chemical equilibrium used in the model is appropriate. Since flow rates in
the column experiments are fast compared to most flow rates in aquifers, we conclude that the kinetics of metal redistribution are fast
enough that an assumption of chemical equilibrium would be appropriate for modeling coupled solute–colloid transport in pore
environments where water flow is controlled by advective transport under water velocity conditions typical for most aquifers.
© 2007 Elsevier B.V. All rights reserved.
Keywords: Metals; Bacteria; Aquifer; Cadmium; Transport; Equilibrium
1. Introduction
Aqueous transition metal cations (e.g. Cd2+, Cu2+,
Pb , Zn2+) adsorb strongly onto surfaces under the nearneutral pH conditions characteristic of most aquifers
(Schindler et al., 1976; Stumm, 1992). In an aquifer
setting, these surfaces can be either mobile (e.g. clays,
2+
⁎ Corresponding author. Now at Pennsylvania State University
DuBois, DuBois, PA 15801, United States. Tel.: +1 814 375 4813;
fax: +1 814 375 4784.
E-mail address: [email protected] (B.F. Turner).
0009-2541/$ - see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.chemgeo.2007.03.002
organic particulates, or microbes) or stationary, so the
mobility of the metal cations is closely linked to the
transport behavior of the particulate surfaces. In geologic
systems, a solute is redistributed among three compartments: 1) dissolved in solution; 2) adsorbed to a stationary surface; and 3) adsorbed to a mobile surface. An
important consideration when characterizing or modeling metal distibution in geologic systems is whether the
assumption of chemical equilibrium is justified.
A number of authors have investigated the appropriateness of various equilibrium assumptions in aquifer
systems with soluble adsorbates (Goltz and Roberts,
B.F. Turner, J.B. Fein / Chemical Geology 242 (2007) 40–50
1988; Brusseau et al., 1991; Ptacek and Gillham, 1992;
Kookana et al., 1994; Rahman et al., 2003; Johnson
et al., 2003), and with both soluble adsorbates and
moving particulate adsorbents (Chen et al., 1995; Saiers
and Hornberger, 1996; Pang and Close, 1999). Generally, it has been found that transport models employing
“physical nonequilibrium” better emulate field data than
traditional equilibrium advective/dispersive models
(Goltz and Roberts, 1988; Brusseau et al., 1991; Ptacek
and Gillham, 1992; Kookana et al., 1994; Pang and
Close, 1999; Rahman et al., 2003; Johnson et al., 2003).
For example, the dual-continuum model of Coats and
Smith (1956) and similar models are frequently used to
describe solute transport in aquifers. In dual-continuum
models, solute transport occurs in two regions: 1) mobile
regions where transport is governed by advection and
dispersion; and 2) immobile regions where solute is
exchanged with mobile regions by diffusion. Although
the Coats and Smith (1956) equation implies transport in
a dual-porosity medium, the equation can be reduced to a
dimensionless form that is mathematically equivalent to
transport with adsorption governed by first-order
kinetics (Nkedi-Kizza et al., 1984; van Genuchten and
Wagenet, 1989). Hence, a mathematical description of
transport by such a model can be interpreted as either
being controlled physically by a dual-porosity continuum or chemically by rate-limited sorption. In either case,
it is commonly considered that the local equilibrium
assumption (LEA), that is that solute–solid interactions
are rapid relative to advective–dispersive transport, is
not valid under these conditions (Brusseau et al., 1991).
The inapplicability of the LEA to a given system does
not necessarily imply that chemical equilibrium between
solute and adsorbent does not exist. Rather, in a dualcontinuum medium, the diffusion-controlled exchange
of solute between mobile and immobile pores spaces
implies a relatively slow change in immobile pore
solution composition, a condition conducive to chemical
equilibrium of adsorption reactions within the immobile
pore. Thus, it is reasonable to assume that adsorbent
surfaces associated with immobile pores are in chemical
equilibrium with the solutions to which they are exposed.
If any surfaces are out of equilibrium with the solutions
to which they are exposed, they must include the surfaces
associated with the mobile pore spaces, where solute
fluxes are the highest.
In this study we investigate whether the assumption
of chemical equilibrium in mobile pore space is valid for
systems involving both bacteria and stationary mineral
surfaces as adsorbents. We compare simulation results
with previously published laboratory experimental data
of Cd and bacteria transport through sand. Since
41
columns were packed with sand grains uniform in
size, shape and mineralogy, it can be assumed that no
immobile porosity exists. Our study consists of two
main tasks: 1) to develop a coupled reactive solute and
colloid transport code that can simulate transport of a
metal that is distributed among mineral surfaces,
bacterial surfaces, and an aqueous phase; and 2) to
compare model simulations to previously-published
laboratory column experiment data. If the equilibrium
assumption is appropriate for modeling Cd transport in
this experimental system (where flow is relatively fast),
then the assumption of chemical equilibrium in mobile
pore space is expected to be appropriate in typical
aquifer settings that exhibit slower flow rates.
We compare model simulations to experimental data
reported by Yee and Fein (2002). The authors conducted
experiments of Cd2+ transport at various pH values
through columns packed with clean quartz sand or
ferrihydrite-coated quartz sand, with and without the
presence of a bacterial suspension, and they monitored
effluent concentrations of bacteria and total Cd. They
noted that the presence of bacteria can enhance or retard
the mobility of Cd, depending on the specific experimental conditions. As one of the few studies quantifying
the distribution of an aqueous transition metal cation
between aqueous, moving adsorbent, and stationary
adsorbent reservoirs in a transport setting, the Yee and
Fein (2002) data set is well suited for testing the
equilibrium assumption in a reactive transport model.
2. Modeling approach
In our study, we must account for not only physical
solute transport processes, but also transport processes
affecting the bacteria, and chemical mass transfer of
Cd between the solution and stationary and moving
surfaces. In this section, we describe the model development by first describing the mathematical modeling
of the physical transport processes. This discussion is
followed by a description of the chemical equilibrium
models incorporated into the transport model, and we
conclude with a description of our approach to model
parameterization.
2.1. Solute and colloid transport
Transport of a solute or colloids through porous
medium is governed by the onedimensional advectionreaction-dispersion equation
2 AC
AC
AC
Aq
ð1Þ
¼ DL
−
−v
At x
Ax2
Ax t At
42
B.F. Turner, J.B. Fein / Chemical Geology 242 (2007) 40–50
where C represents the concentration of a given solute
or colloid, x represents position, DL is the hydrodynamic
dispersion coefficient, and v is fluid velocity. The term
∂q/∂t is a source/sink term for the solute or colloid,
representing the mass transfer between stationary
surfaces and solution or suspension. The governing
equation is solved in this study by the finite difference
split-operator scheme, described by Appelo and Postma
(1993), in which solute equilibrium reactions are
calculated both after the advection step and after the
dispersion step, reducing numerical dispersion (i.e.
apparent dispersion produced by the solution algorithm)
and the need to iterate between chemistry and transport
(Parkhurst and Appelo, 1999). In the scheme, the
dispersion step is executed as several mixing steps
involving mixing between adjacent cells.
Deposition of colloids is governed by the following
equation (substituting for the sink term in Eq. (1)):
ACS
¼ CS a0 ð1−kCA Þ
At
ð2Þ
where CT = CS + CA, and CT, CS and CA are the total,
suspended, and attached colloid concentrations, respectively, α0 is the rate constant for attachment at zero
surface coverage, and k is a surface blocking parameter.
The equation is solved over a given time interval by
substitution and integration:
Z CS2
Z t2
1
dC
¼
−a0 dt
ð3Þ
S
2
CS1 ð1−kCT ÞCS þ kCS
t1
adsorbed to stationary mineral grain surfaces. We use
chemical thermodynamics, assuming the system components to be in equilibrium, to calculate the distribution
of Cd between these reservoirs. We use literature values
for mineral and bacterial surface site concentrations and
Cd-surface stability constants to predict the distribution
and transport of Cd in the simulated systems, and we
compare these predictions with the experimental
observations of Yee and Fein (2002). Mathematically,
we solve the chemical equilibrium problem using a
Newton–Raphson solver similar in principle to solvers
used by MINTEQA2 (Allison et al., 1991) and
PHREEQC (Parkhurst and Appelo, 1999). The solute/
colloid transport algorithm keeps track of the partitioning of solute between solution, moving colloid surfaces,
attached colloid surfaces, and stationary mineral grain
surfaces.
The chemical speciation problem in each simulation
is parameterized to correspond to the specific experimental conditions of the experiments to which the
simulations are to be compared. The experiments of Yee
and Fein (2002) involve aqueous Cd, the bacterial
species Bacillus subtilis, and either quartz sand or
ferrihydrate-coated quartz sand in 0.1 M NaNO3 over
the pH range 3.6 to 8.3. We use the Davies equation for
calculating activity coefficients for Cd2+, which is
appropriate for the experimental ionic strength of
0.1 M (Langmuir, 1997).
Surface acidity and adsorption of Cd to B. subtilis is
described by Fein et al. (1997) using a constant
capacitance model (Table 1), with Cd adsorption
occurring primarily onto carboxyl and phosphoryl groups:
The solution is
CS2 ¼
ðR–COOÞH0 Ð ðR–COOÞ− þHþ ; K1 ¼ ð1−kCT Þ
ec −k
ð4Þ
1
1−kCT þ kCS1
ln
1−kCT
CS1
ð6Þ
½ðR−PO4 Þ− aHþ
ðR PO4 ÞH0 Ð ðR PO4 Þ− þHþ ;K2 ¼ ðR PO4 ÞH0 eFW=RT
where
c ¼ ð1−kCT Þ a0 Dt þ
½ðR–COOÞ− aHþ
ðR–COOÞH0 eFW=RT
ð7Þ
ð5Þ
and CS1 and CS2 are the suspended colloid concentrations at the beginning and end of the time interval,
respectively. Eq. (2) is solved at each mixing step in the
split-operator scheme.
½ðR OÞ− aH þ
ðR OÞH0 Ð ðR OÞ− þHþ ; K3 ¼ ðR OÞH0 eFW=RT
ð8Þ
Cd2þ þ ðR–COOÞ− Ð ðR–COOÞCdþ ; Kads1 ¼
ð9Þ
2.2. Chemical equilibrium
In the experiment of Yee and Fein (2002), Cd is
present either as an aqueous species, adsorbed to colloid
(bacterial) surfaces (either moving or stationary), or
½ðR–COOÞCdþ eFW=RT
aCd2þ ½ðR–COOÞ− Cd2þ þ ðR PO4 Þ Ð ðR PO4 ÞCdþ ; Kads2 ¼
½ðR–POÞCdþ eFW=RT
aCd2þ ½ðR–POÞ− ð10Þ
B.F. Turner, J.B. Fein / Chemical Geology 242 (2007) 40–50
Table 1
Parameters for Cd–bacteria adsorption model of Fein et al. (1997)
Parameter
Value
log K1
log K2
log K3
log Kads1
log Kads1
(R–COO)atot
(R–PO4)atot
(R–O)atot
Bacterial surface area
Capacitance
− 4.82
− 6.9
− 9.4
3.4
5.4
1.2 × 10− 4 mol g− 1
4.4 × 10− 5 mol g− 1
6.2 × 10− 5 mol g− 1
140 m2g− 1
8.0 Fm2
a
Site concentrations in mol per g of bacteria (wet weight).
where brackets indicate species concentration, and a represents the activity of the subscripted species. Data used by
Fein et al. (1997) to define equilibrium constant and site
concentration values were collected in 0.1 M NaNO3.
We test the ability of three different models to
account for adsorption of Cd to silica surfaces (Table 2).
Two of the models involve a two-site non-electrostatic
model (Schindler et al., 1976; Brady, 1992), and the
third uses a one-site double layer model (Benyahya and
Garnier, 1999). The models of Schindler et al. and Brady
describe Cd adsorption as follows:
Cd2þ þ ðuSi−OÞH0 Ð ðuSi−OÞCdþ þ Hþ ;
fðuSi−OÞCdþ gaHþ
KSiCd1 ¼ 2þ Cd
ðuSi−OÞH0
ð11Þ
Cd2þ þ 2ðuSi−OÞH0 Ð ðuSi−OÞ2 Cd0 þ 2Hþ ;
ðuSi−OÞ2 Cd0 a2Hþ
KSiCd2 ¼ 2
Cd2þ ðuSi−OÞH0
ð12Þ
where braces indicate surface species concentration in
mol kg− 1. The authors also describe proton exchange
with the silica with the reaction
ðuSi−OÞH0 Ð ðuSi−OÞ− þHþ
subject to the mass balance constraint
½ðuSi−OÞtot ¼ ðuSi−OÞH0 þ ½ðuSi−OÞ− :
ð13Þ
ð14Þ
Instead of using a conventional mass action expression,
however, the authors use an empirical expression to
account for the concentration of (≡Si–O)−:
1
½ðuSi−OÞ ¼ 10logpH=bþc
a
−
ð15Þ
43
where a, b, and c are empirical constants (Table 2).
Schindler et al. (1976) and Brady (1992) both determined
stability constants and site concentrations under conditions of 1.0 M NaClO4 background electrolyte, 25 °C,
using amorphous silica “Aerosil 200,” and assuming a
surface potential of zero.
Although the models of Schindler et al. (1976) and
Brady (1992) have been widely cited, both models exhibit
important limitations that impede their application beyond
the specific experimental conditions under which the
models were parameterized: 1) Cd2+ solution concentrations are used in the mass action expressions instead of
activities; 2) surface species concentrations in the mass
action expressions are expressed as relative to adsorbent
mass rather than solution volume or mass of water; and 3)
an unconventional empirical expression is used to account
for [(≡Si–O)−] rather than a conventional mass action
expression. To address the first limitation and to extend the
applicability of the model to the experimental conditions of
Yee and Fein (2002), we convert the mass action
expressions to include Cd2+ activity rather than concentration using an activity coefficient of 0.2504 for Cd2+ at
1.0 M calculated using the Truesdell–Jones equation. The
expression of surface species concentrations as mol per g of
adsorbent is particularly problematic in Eq. (12), making
the expression meaningless when applied to adsorbents
with different (site concentration)/(adsorbent mass) ratios
than those used by Schindler et al. (1976) or Brady (1992).
When the surface site concentrations are given as mol per g
of adsorbent, the value of the ion activity product is
effectively dependent on the specific surface area of the
adsorbent, rather than on the amount of surface species
available to solution. This is not problematic in Eq. (11)
because the stoichiometry results in the surface species
Table 2
Parameters for Cd–silica adsorption models
Parameter
Schindler
et al. (1976)
Brady
(1992)
Benyahya and
Garnier (1999)
log KSi1
Original
log kCd1
Original
log kCd2
Corrected
log kCd1
Corrected
log kCd2
(≡Si–O)tot
NA
−6.09
NA
− 7.0
− 7.71
− 5.57
−14.20
− 12.8
NA
−5.49
− 6.4
NA
−11.9
− 10.5
NA
9.06 × 10− 6
mol m− 2
1.45
0.1923
−5.349
7.25 × 10− 6
mol m− 2
1.45
0.1930
− 5.31
3.9 × 10− 6
mol m− 2
NA
NA
NA
a
b
c
44
B.F. Turner, J.B. Fein / Chemical Geology 242 (2007) 40–50
concentration dimensions canceling out. However, in Eq.
(12), the surface species concentration dimensions do not
cancel out due to the stoichiometry of the adsorption
reaction. Hence, in order to apply the model to our system,
we convert the mass action expressions to forms that can be
applied to systems other than the experimental systems of
Schindler et al. (1976) and Brady (1992):
KSiCd1 ¼
½ðuSi−OÞCdþ aHþ
aCd2þ ðuSi−OÞH0
ð16Þ
We model the adsorption of Cd to ferrihydrite using
the two-site double layer model of Dzombak and Morel
(1990):
þ
ðuFes OÞH0 aHþ
s
0
þ
uFe O H2 Ð ðuFe OÞH þ H ; KFe1 ¼
FW=RT
ðuFes OÞHþ
2 e
S
ð20Þ
−
uFeS O H 0 Ð uFeS O þHþ ; KFe2 ¼ − uFeS O aH þ
ðuFes OÞH0 eFW=RT
ð21Þ
KSiCd2
ðuSi−OÞ2 Cd0 a2Hþ
¼
2
aCd2þ ðuSi−OÞH0
ð17Þ
where brackets, again, represent concentrations in terms
of moles of sites per L of solution. Corrected log K values
are reported in Table 2. Although the practice of using
dimensions of moles of sites per solid mass for surface
species in mass action expressions was advocated by
Stumm (1992), the practice of using site concentrations of
moles of sites per to solution volume as used by Dzombak
and Morel (1990) is less problematic given reaction
stoichiometries as represented by Eq. (12). The unconventional method of estimating [(≡Si–O)−] used by
Schindler et al. (1976) and Brady (1992) (Eq. (15)) cannot
be used in commonly used speciation calculators such as
MINTEQA2 and PHREEQC. However, since we utilize a
custom-built speciation solver in our transport code, we
utilize this expression directly to account for (≡Si–O)−
concentrations. Another limitation inherent in Eq. (15) is
that [(≡Si–O)−] varies only with pH, regardless of the
amount of Cd adsorbed. This assumption is only appropriate for systems where [(≡Si–O)Cd+]≪ [(≡Si–O)]tot
and [(≡Si–O)2Cd0] ≪ [(≡Si–O)]tot, which is the case for
the experimental system of Yee and Fein (2002).
Benyahya and Garnier (1999) developed their model
using silica gel in a 0.05 M NaNO3 background
electrolyte. Their double layer model is described as
ðuSi−OÞH0 Ð ðuSi−OÞ− þHþ ; KSi1 ¼
½ðuSi−OÞ− ½ðuSi−OÞH 0 aH þ eFW=RT
ð18Þ
Cd
2þ
þ
þ
þ ðuSi−OÞH Ð ðuSi−OÞCd þ H ;
KSiCd1 ¼
0
þ
½ðuSi−OÞCd aHþ e
aCd2þ ðuSi−OÞH0
FW=RT
ð19Þ
where F is Faraday's constant, Ψ is surface potential, R
is the ideal gas constant, and T is absolute temperature.
Model parameters are given in Table 2.
ðuFew OÞH0 aHþ
w
þ
w
0
þ
ðuFe OÞH2 Ð ðuFe OÞH þ H ; KFe1 ¼
FW=RT
ðuFew OÞHþ
2 e
ð22Þ
ðuFew OÞH0 Ð ðuFew OÞ− þHþ ; KFe2 ¼ ½ðuFew OÞ− aHþ
ðuFew OÞH0 eFW=RT
ð23Þ
ð uFes O ÞH0 þ Cd2þ Ð ðuFes OÞCdþ þ Hþ ;
uFeS O Cdþ aHþ eFW=RT
KsFeCd ¼
aCd2þ uFeS O H0 Cd2þ
ð24Þ
ðuFew OÞH0 þ Cd2þ Ð ðuFew OÞCdþ þ Hþ ;
KwFeCd ¼
½ðuFew OÞCdþ aHþ eFW=RT
aCd2þ ðuFew OÞH0 Cd2þ
ð25Þ
where parameters are tabulated in Table 3. Dzombak
and Morel (1990) determined model parameters under
varied background electrolyte strengths and using
synthetic ferrihydrite.
We use the bacterial surface area and site concentrations tabulated in Table 1. Mineral surface site
Table 3
Parameters for Cd–ferrhydrite adsorption model of Dzombak and
Morel (1990)
Parameter
Value
log KFe1
log KFe2
log KsFeCd
log KwFeCd
(≡FesO)(a)
tot
(≡FewO)(a)
tot
7.29
−8.93
0.47
−2.90
0.094 × 10− 4 mol g− 1
3.7 × 10− 5 mol g− 1
a
Based on values used by Dzombak and Morel (1990): weak and
strong site concentrations of 0.2 and 0.005 mol (mol Fe)− 1, 89 g
ferrihydrite (mol Fe)− 1, and a specific surface area of 600 m2g− 1.
B.F. Turner, J.B. Fein / Chemical Geology 242 (2007) 40–50
Table 4
Column and transport parameters
Mineral surface areaa
Mineral mass in columna
Pore volumed
Mineral mass per solution volumed
Porosityd
De
vf
Experimental lag timeg
45
Table 5
Parametersa for colloid attachment models
0.028 m2g− 1b or 0.064 m2g− 1c
10 g
4.08 mL
2450 g L− 1
0.52
1.0
0.086 cm s− 1
142 s
a
Values reported by Yee and Fein (2002)Yee and Fein (2002).
Quartz sand.
c
Ferrihydrite-coated sand.
d
Based on specific gravity of 2.65, and column length and diameter of 10 cm and 1 cm, respectively (porevolume = 0.25π · length ·
diameter 2 MminG− 1ρ−w1, where Mmin is mineral mass, G is specific
gravity, and ρw is the density of water).
e
Value fitted to experimental breakthrough curves.
f
Based on reported flow rate of 0.035 mL s− 1.
g
Based on experimental breakthrough curves.
b
concentrations are estimated using the mineral surface
areas in Table 4 and site densities in Tables 2 and 3.
System
α0
k
Quartz, pH 7.6
Ferrihydrite, pH 5.7
Ferrihydrite, pH 6.6
0.008
0.028
0.033
14.3
2.6
4.3
a
Values obtained by fitting simulated bacterial breakthrough curves to
observed experimental curves.
simulated bacterial breakthrough curves reasonably
represent the bacterial transport behavior (Fig. 1).
Although colloid attachment parameters had to be
determined through a fitting procedure using experimental results, we stress that the chemical equilibrium model
parameters (Tables 1–3) were obtained directly from the
literature sources cited, yielding independent predictions
of the experimental systems and, thus, the comparison
between calculated and observed Cd distributions is a
valid and rigorous test of the assumption of chemical
equilibrium that is a foundation of the modeling approach.
3. Results and discussion
2.3. Parameterization of model
Transport simulation parameters (Table 4) were
derived directly from reported values where possible.
Based on the average breakthrough of two reported
experimental conservative tracer curves, an experimental
lag time of 142 s (corresponding to the travel time through
tubing external to the column) was added to simulation
output. Our simulated tracer breakthrough curve, using a
dispersion coefficient value of 1.0 cm, is reasonably
representative of the two reported curves (Fig. 1). Colloid
attachment parameters α0 and k (Table 5) were obtained
for each experiment by matching simulated bacterial
breakthrough curves with experimental curves. The
Fig. 1. Comparison of model simulations with experimental
breakthrough curves for conservative tracer and bacteria. Solid lines
represent simulations. Experimental data from Yee and Fein (2002).
In Figs. 2–5, we compare simulated Cd breakthrough
curves to experimental curves in systems with and
without bacteria using surface complexation models
from the literature. In systems without bacteria,
agreement between model-predicted Cd breakthrough
and experimental observation is of mixed quality. For
the quartz system (Fig. 2), there is good agreement at pH
3.6, while all models at pH 8.3 predict faster
breakthrough (and less adsorption) than observed. At
pH 3.6, the simulation is insensitive to adsorption model
parameters since adsorption is negligible and hence Cd
breakthrough is virtually conservative. At pH 8.3, the
models substantially under-predict adsorption, reflecting the sensitivity of the transport simulation to
adsorption model parameters for this system at this
pH. For the ferrihydrite system (Fig. 3), the model
predicts a slightly faster breakthrough, indicating that
the speciation model significantly under-predicts adsorption of Cd. However, the shapes of model-predicted
and experimental breakthrough curves are similar for the
ferrihydrite system. In each abiotic system where
adsorption is significant, the models under-predict
adsorption.
In systems containing bacteria, the observed breakthrough of aqueous and total Cd are reasonably well
predicted in both the quartz (at pH 7.6: Fig. 4) and
ferrihydrite (at pH 5.7 and 6.6: Fig. 5) systems. In these
systems, the observed aqueous Cd breakthrough is
characterized by a retarded sinusoidal curve that
46
B.F. Turner, J.B. Fein / Chemical Geology 242 (2007) 40–50
Fig. 2. Comparison of model simulations with experimental
breakthrough curves for systems involving Cd and quartz sand
without bacteria at (a) pH 3.6 and (b) pH 8.3. Symbols indicate
experimental data, and solid lines indicate simulations. The models
predict identical breakthrough at pH 3.6. Cd concentration is
7.9 × 10− 6 mol L− 1. Experimental data from Yee and Fein (2002).
plateaus at a steady-state concentration. The concentration of total Cd (aqueous Cd plus Cd adsorbed to
moving cells) increases as the effluent concentration of
bacteria approaches C0. Changes in pH and mineralogy
are accounted for reasonably, and relatively minor
discrepancies in retardation time and/or plateau concentration likely result uncertainties in the parameters of the
adsorption models. In the quartz system, the reasonable
agreement between simulated and experimental breakthrough curves are due to the ability of the Cd–bacteria
adsorption model of Fein et al. (1997) to predict
adsortion, and the ability of the fitted colloid transport
submodel to match adsorption. The Cd–quartz adsorption models are relatively insensitive to uncertainty at
pH 7.6 because Cd adsorption onto Si sites is low.
Descrepancy in the total Cd breakthrough curves in the
quartz system between 250 and 750 s is largely due to a
minor descrepancy between simulated and observed
bacterial breakthrough (Fig. 1).
A sensitivity analysis of the ferrihydrite system at pH
6.6 (Fig. 6) reveals that the steady-state plateau in the
Fig. 3. Comparison of model simulations (solid line) with experimental
breakthrough curves (open circles) for systems involving Cd and
ferrihydrite-coated quartz sand without bacteria at (a) pH 5.7 and (b)
pH 6.6. Cd concentration is 7.9 × 10− 6mol L− 1. Experimental data
from Yee and Fein (2002).
concentration of dissolved Cd is controlled by adsorption to bacterial cells, and that retardation in Cd breakthrough is controlled by Cd adsorption to ferrihydrite.
Since the adjustment Cd–mineral equilibrium constants
affects the time of initial breakthrough of Cd (Fig. 6a), it
Fig. 4. Comparison of model simulations with experimental
breakthrough curves at pH 7.6 for systems involving Cd, quartz
sand, and bacteria. The models predict identical breakthrough at this
pH. Cd concentration is 7.9 × 10− 6 mol L− 1. Bacterial concentration
1.0 g L− 1. Experimental data from Yee and Fein (2002).
B.F. Turner, J.B. Fein / Chemical Geology 242 (2007) 40–50
47
dissimilar substrate materials (amorphous silica rather
than quartz) and dissimilar ionic strengths (1.0 and
0.05 M rather than 0.1 M) used in their respective
experimental studies compared with those used in the
experiments by Yee and Fein (2002). Also, the Cd–
silica adsorption model of Benyahya and Garnier (1999)
is a one-site model that fits poorly with the data from
which it was derived, resulting in an adsorption
envelope with respect to pH that is much too narrow.
In contrast, the two-site models of Schindler et al.
(1976) and Brady (1992) fit their respective data sets
very well. While the above models underpredict
adsorption in the experimental system investigated
here, the log K values can be adjusted so that accurate
predictions of the pH 8.3 abiotic quartz system are made
(Fig. 7). Using these adjusted K values to generate new
predictions of the Cd transport behavior results in
negligible differences in the predicted breakthrough
curves for the abiotic quartz experiment at pH 3.6, and
minimal differences in the predicted breakthrough curves
for the pH 7.6 quartz experiments that contained bacteria
Fig. 5. Comparison of model simulations with experimental
breakthrough curves at (a) pH 5.7 and (b) pH 6.6 for systems
involving Cd and ferrihydrite-coated quartz sand, and bacteria. Cd
concentration is 7.9 × 10− 6 mol L− 1. Bacterial concentration 1.0 g L− 1.
Experimental data from Yee and Fein (2002).
is apparent that retardation of breakthrough is controlled
by adsorption to stationary mineral surfaces. Likewise,
since the adjustment of Cd–bacteria equilibrium constants affects the steady state value of dissolved Cd
(Fig. 6b), it is clear that the plateau is controlled by
adsorption to bacteria. Therefore, the under-prediction of
the aqueous Cd plateau in Figs. 4 and 5 is due to a slight
over-prediction of Cd adsorption to the bacterial cells.
The qualitative agreement between the idealized behaviors inherent in the simulations and the measured
breakthrough curves indicate Cd transport under essentially equilibrium conditions. Therefore, the assumption
that chemical equilibrium controls Cd distributions in the
transport experiments is appropriate. The published
equilibrium models of Fein et al. (1997) and Dzombak
and Morel (1990) used here yield reasonably accurate
predictions of Cd transport for the Cd–bacteria–
ferrihydrite system.
The poor prediction of Cd adsorption onto quartz in
the absence of bacteria at pH 8.3 by the models of
Schindler et al. (1976), Brady (1992) and Benyahya
and Garnier (1999) can be explained largely by the
Fig. 6. Sensitivity of simulation output to chemical equilibrium
constants for system involving Cd and ferrihydrite-coated quartz sand,
and bacteria. Log K values for (a) Cd–ferrihydrite and (b) Cd–bacteria
adsorption mass action expressions were adjusted by the values
indicated. Cd concentration is 7.9 × 10− 6mol L− 1. Bacterial concentration 1.0 g L− 1.
48
B.F. Turner, J.B. Fein / Chemical Geology 242 (2007) 40–50
(data not shown). While the quantity of adsorption can be
matched in this way, the shape of the breakthrough curves
are dependent on the model. This effect appears to be
controlled by the adsorption isotherm implied by the
model (Fig. 8). The more convex shape of the isotherm
implied by the Benyahya and Garnier (1999) model
apparently causes adsorbed Cd to more quickly reach a
steady-state concentration than the less convex isotherms,
resulting in a steeper breakthrough curve than what was
observed experimentally (Fig. 7).
As discussed above, the sensitivity of the Cd transport
simulation to chemical equilibrium model parameters is
dependent on the speciation of Cd in the system. Where
the concentration of adsorbed Cd is negligible, for
example, the predicted transport behavior is not sensitive
to the adsorption model. According to a speciation
diagram calculated for the Cd–quartz–bacteria system
(Fig. 9a), adsorption of Cd to quartz is probably small at
pH 7.6. Hence the transport simulation for this system is
relatively insensitive to uncertainty in the Cd–quartz
adsorption models at pH 7.6 in the presence of bacteria,
but highly sensitive at pH 8.3 without bacteria.
The Cd–bacteria and Cd–ferrihydrite models appear
to describe Cd adsorption in these systems with reasonable accuracy. In both the absence and presence of
bacteria, the observed breakthrough of aqueous Cd
exhibits behavior characteristic of transport under
Fig. 8. Adsorption isotherms predicted by models in systems with Cd
and (a) quartz or (b) ferrihydrite-coated sand. Values given in the
legend are adjustments added to log K values for the respective
models. Total adsorbent concentration is 2450.0 g L− 1 in each case.
Fig. 9. Closed-system equilibrium distribution of Cd in systems with
bacteria and (a) quartz or (b) ferrihydrite-coated sand. Cd–quartz
sorption model used is the Schindler et al. (1976) model with log K
values adjusted by +1.8. Cd concentration is 7.9 × 10− 6mol L− 1
Bacterial concentration 1.0 g L− 1.
Fig. 7. Comparison of adjusted model simulations with experimental
breakthrough curves for Cd and quartz sand system without bacteria at
pH 8.3. Symbols indicate experimental data, and solid lines indicate
simulations. Values given in the legend are adjustments added to log K
values for the respective models. The models predict identical breakthrough at pH 3.6. Cd concentration is 7.9 × 10− 6 mol L− 1. Experimental data from Yee and Fein (2002).
B.F. Turner, J.B. Fein / Chemical Geology 242 (2007) 40–50
equilibrium conditions. Idealized equilibrium transport
behavior involving the adsorption of a solute to a
stationary solid, as illustrated by our simulation results,
is characterized by a retardation of breakthrough
followed by a sinusoidal breakthrough curve shape.
The flow rate of these experiments (74 m day− 1) is
very high compared to most groundwater flow rates. If
the assumption of chemical equilibrium is appropriate
for experimental systems such as those examined by Yee
and Fein ((2002), then it should also be appropriate for
mobile pore spaces in slower moving ground water
systems. The assumption of chemical equilibrium in a
dynamic system greatly simplifies the modeling process. Since adsorption and desorption rates for other
divalent metal cations such as Co2+, Ni2+, Cu2+, Zn2+
and Pb2+ are similar to that of Cd2+ (Jeon et al., 2003,
2004; Borrok and Fein, 2005), chemical equilibrium
models are likely to yield reasonable predictions for the
distributions of these metals in geologic systems as well.
49
suggests that transport of Cd in the experimental systems
occurs under essentially equilibrium conditions. Equilibrium of adsorption reactions appears to control Cd
distributions in systems with or without bacteria, using
quartz or ferrihydrite substrates, and over a range of pH
conditions. Our findings indicate that the assumption of
chemical equilibrium is appropriate for column experiments similar to those used for comparison. Since flow
rates in the column experiments (74 m day− 1) are fast
compared to most flow rates in aquifers, we conclude
that the kinetics of metal redistribution are fast enough
that an assumption of chemical equilibrium would be
appropriate for modeling coupled solute–colloid transport within mobile pore spaces in most aquifers.
Acknowledgements
Funding for this work was provided by a NSF
Environmental Molecular Science Institute grant (NSFEAR-0221966).
4. Conclusions
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