Download Coupled Microbial and Transport Processes in Soils

Coupled Microbial and Transport Processes in Soils
M. L. Rockhold,* R. R. Yarwood, and J. S. Selker
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ABSTRACT
1992b; MacDonald et al., 1999a, 1999b). Almost all previous work on this topic has focused on liquid-saturated
porous media systems, with relatively few attempts at
addressing more complicated unsaturated or variably saturated systems. This lack of attention to variably saturated systems may be due in part to their added complexity, or may simply be due to the fact that low nutrient
availability and competition and predation by other
microorganisms limits the growth of many soil microorganisms so that they may only occupy a small or negligible volume of the pore space. Under the high nutrient
loading conditions that might occur in applications such
as wastewater treatment or bioremediation, however,
bacteria and other microorganisms can proliferate, and
the consequent changes in soil hydraulic properties may
be significant (Rockhold et al., 2002).
Many models have been developed for describing
water flow, solute transport, and biodegradation processes in porous media. Most of these models were developed for one space dimension and are strictly applicable to saturated porous media representative of aquifer
sediments. Examples include the numerical models described by Molz et al. (1986), Widdowson et al. (1988),
Celia et al. (1989), Zysset et al. (1994a, 1994b), and Clement et al. (1997). Several multidimensional models have
also been developed for simulating transport and biogeochemical reactions in saturated porous media. Noteworthy examples include the RAFT model described by
Chilakapati (1995), and the coupling of the HYDROGEOCHEM (Yeh and Tripathi, 1991) and BIOKEMOD
models, described by Salvage and Yeh (1998). Very few
models account for possible reductions in porosity and
permeability that result from biomass accumulation, although it is well known that such effects can sometimes
be substantial, especially in the vicinity of the nutrient injection wells used in bioremediation applications (MacDonald et al., 1999a, 1999b).
The models that have been developed for simulating
water flow and solute transport processes in unsaturated
or variably saturated porous media systems such as soils
are often less sophisticated in their representations of biodegradation processes than their saturated-zone counterparts. Most of these models account for processes such as
biodegradation in terms of simple, first-order solute decay
reactions, without actually considering cell growth or substrate (nutrient) limitations on biodegradation rates. Examples include the HYDRUS-2D model (Šimůnek et al.,
1999) and the STOMP model (White and Oostrom, 1996).
Models that do not account for cell growth obviously
cannot be used to access the significance of biomassinduced changes in the hydraulic properties of unsaturated porous media. Very few attempts have been made
to directly measure or model biomass-induced changes
in the hydraulic properties of unsaturated porous media.
However, such changes can have a significant impact on
This paper reviews methods for modeling coupled microbial and
transport processes in variably saturated porous media. Of special
interest in this work are interactions between active microbial growth
and other transport processes such as gas diffusion and interphase
exchange of O2 and other constituents that partition between the
aqueous and gas phases. The role of gas–liquid interfaces on microbial
transport is also discussed, and various possible kinetic and equilibrium formulations for bacterial cell attachment and detachment are
reviewed. The primary objective of this paper is to highlight areas in
which additional research may be needed—both experimental and
numerical—to elucidate mechanisms associated with the complex interactions that take place between microbial processes and flow and
transport processes in soils. In addition to their general ecological
significance, these interactions have global-scale implications for C
cycling in the environment and the related issue of climate change.
S
oils sustain life on Earth. They are important not
only from an agronomic standpoint for supporting
the growth of plants, but also from an environmental
standpoint for mitigating many of the potentially adverse affects of surface-applied contaminants on the quality of groundwater resources. The filtering ability of soils
is due to many processes, including physical filtration,
sorption of contaminants on the surfaces of mineral grains
and soil organic matter, geochemical reactions, and biodegradation by soil microorganisms. Improved understanding of coupled microbial and transport processes
in soils is important to a variety of disciplines. In addition to their general ecological significance, these interactions have global-scale implications for C cycling in
the environment and the related issue of climate change.
As microbial biomass accumulates in porous media,
it may change the physical and hydraulic properties of
the media. These changes may then alter solute flow paths,
gas exchange, microbial growth and redistribution, and
other processes. Biomass-induced changes in the hydraulic properties of porous media have been studied
for various applications, such as enhanced oil recovery
(Jenneman et al., 1984; Raiders et al., 1986), water and
wastewater treatment (McCalla, 1946; Nevo and Mitchell, 1967; Loehr, 1977; Overcash and Pal, 1979), and bioremediation of contaminated aquifer sediments (Taylor
and Jaffe, 1990a, 1990b, 1990c; Taylor et al., 1990; Cunningham et al., 1991; Vandevivere and Baveye, 1992a,
M.L. Rockhold, Pacific Northwest National Lab., P.O. Box 999/MS
K9-36, Richland, WA 99352; R.R. Yarwood, Dep. of Crop and Soil
Sciences, Oregon State Univ., Corvallis, OR 97331; J.S. Selker,
Dep. of Bioengineering, Oregon State Univ., Corvallis, OR 97331.
Received 20 June 2003. Special Section: Colloids and Colloid-Facilitated Transport of Contaminants in Soils. *Corresponding author (Mark.
[email protected]).
Published in Vadose Zone Journal 3:368–383 (2004).
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677 S. Segoe Rd., Madison, WI 53711 USA
368
369
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water flow and solute transport (Rockhold et al., 2002;
Yarwood et al., 2002).
The presence of a second fluid phase, air, in an unsaturated porous medium creates an important pathway for
gas exchange with the atmosphere. This gas exchange,
also known as soil respiration, is crucial for the maintenance of healthy plant roots and microbial activity. Leffelaar (1987) developed a one-dimensional numerical
model to study interactions among water flow, solute
transport, microbial activity, and respiration in soil aggregates. His emphasis was on multinary gas diffusion
and anaerobiosis in soil aggregates. Hence, he did not
consider the possibility of changes in the physical and
hydraulic properties of the porous media due to biomass accumulation.
Concerns over global CO2 emissions have generated
a great deal of interest by the scientific community in
the topic of C sequestration. These concerns have also
motivated the development of a number of computer
models to describe CO2 production in soils. For example, Ouyang and Boersma (1992) developed a onedimensional numerical model to simulate the coupled
processes of water flow, heat transport, and O2 and CO2
exchange in unsaturated porous media. Their model
did not consider the effects of microbial growth or the
transport of O2 and CO2 in the aqueous phase. Šimůnek
and Suarez (1993) developed a one-dimensional numerical model called SOILCO2 to simulate water flow, heat
transport, and CO2 production and transport in soils.
Although this model contains source terms to account
for CO2 production by plant roots and microbial activity,
it neglects microbial growth, as well as the O2 transport
and consumption required to support the growth of
aerobic microorganisms and plant roots.
Several multifluid flow and transport simulators have
recently been developed or extended for use in C sequestration studies to account for multicomponent liquid- and gas-phase transport in variably saturated porous media systems using a more rigorous, fully coupled
approach (Battistelli et al., 1997; Oldenburg et al., 2001;
White and Oostrom, 2003). Most of these simulators were
designed principally for high temperature and pressure
applications in deep geologic formations. Consequently,
they do not consider microbial processes. Travis and Rosenberg (1997) and Battistelli (2003) described two threedimensional numerical simulators that consider coupled,
multifluid flow as well as biologically reactive transport
processes in variably saturated porous media. While these
simulators arguably represent the current state of the art,
they assume that microbes are strictly immobile and do
not account for pore clogging or other biomass-induced
changes in fluid-media properties.
It seems that most of the models that have been developed for simulating fluid flow and solute transport in
unsaturated or variably saturated porous media either
do not account for microbial activity at all, or they do so
in an incomplete way, by not actually accounting for microbial growth and transport, and/or by not accounting
for the possible effects of microbial growth on the physical and hydraulic properties of the porous media. The
primary objectives of this paper, therefore, are to review
the status of activity on this research topic and to highlight areas in which both experimental and numerical
research may be needed to elucidate mechanisms associated with the complex interactions that take place between microbial processes and transport processes in
soils.
This paper is organized as follows. Mass balance equations for water flow and solute and microbial transport
in variably saturated porous media are first reviewed.
Special emphasis is given to issues related to using a
single-phase flow approximation (i.e., the Richards equation) in conjunction with advection–dispersion–reaction
equations to model water flow and transport of biologically reactive solutes and bacteria in soils. Work related
to microbial transport in porous media is then reviewed,
and equations representing reaction rate source–sink
terms for cell attachment–detachment, growth, accumulation, substrate depletion, and by-product formation are
discussed. The role of gas–liquid interfaces is emphasized. The paper concludes with suggestions of topics
for future research.
FLOW AND TRANSPORT MODELING
Soils are three-dimensional, multiphase, multicomponent, nonisothermal, biogeochemical systems that contain a dynamic consortium of microorganisms. The reaction rates and fluid properties in soil systems are well
known to be temperature-dependent and hysteretic,
with these factors intensively studied and simulated in
previous work. For the purposes of this review, we will
focus on variably saturated porous media systems, containing a single species of bacterium, under isothermal,
nonhysteretic conditions. Our primary interests are the
processes of water flow, gas diffusion, solute and microbial transport, and other microbial processes, including
substrate consumption, cell growth, and attachment and
detachment to and from surfaces. Governing mass balance equations, constitutive relations, and other assumptions necessary for the development of numerical models that describe these processes are reviewed in the
following sections.
Water Flow
The mass conservation of water in porous media can
be expressed as
⳵
⳵t
冤 兺 (n␻ ␳ S )冥 ⫽ ⫺ 兺 (䉮F
w
␥⫽ᐉ,g
␥ ␥ ␥
␥⫽ᐉ,g
w
␥
⫹ 䉮J␥w) ⫹ ⍀
␻w␳ k k
F␥w ⫽ ⫺ ␥ ␥ r␥ (䉮P␥ ⫹ ␳␥gzg) for ␥ ⫽ ᐉ, g
␮␥
J␥w ⫽ ⫺␶␥n␳␥S␥
Mw w w
D␥䉮␹␥ for ␥ ⫽ ᐉ, g
M␥
[1]
where n is the porosity, ␻w␥ is the mass fraction of water
in phase ␥, ␳␥ is the fluid phase density, S␥ is the phase
saturation, Fw␥ is the advective flux, and Jw␥ is the diffusive-dispersive flux, and ⍀ is a source–sink term. The
superscript w denotes water, and the subscripts ᐉ and g
denote the liquid and gas phases, respectively. The term
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370
VADOSE ZONE J., VOL. 3, MAY 2004
kr␥ is the fluid relative permeability, k is the intrinsic
permeability tensor, ␮␥ is the kinematic viscosity, P␥ is
the phase pressure, g is the acceleration of gravity, and
zg is a unit gravitational direction vector. The term ␶␥ is
the phase tortuosity, M is the molecular weight, Dw␥ is
the diffusion coefficient, and ␹w␥ is the mole fraction of
water in phase ␥. (For a complete list of symbols see the
Appendix.) It is generally assumed that the principal
directions of anisotropy are aligned with the coordinate
system so that the cross components of the k tensor are
zero. If coupled, multifluid flow is considered, equations
similar to Eq. [1] can also be written to describe the
mass conservation of air and nonaqueous phase liquids
(White and Oostrom, 1996).
In the soil physics literature, water flow has traditionally been modeled using a simplified, single (aqueous)phase version of Eq. [1], known as the Richards equation
(Richards, 1931). Use of the Richards equation implies
the assumption of a continuous gas phase at constant,
atmospheric pressure. This assumption is typically justified based on the fact that the viscosity of air is about 50
times lower than that of water (Lide, 1996). Therefore,
if the gas phase in an unsaturated porous medium is
continuous, significant air flow can be caused by very
small pressure gradients, and air is assumed to maintain
atmospheric pressure, or to quickly reequilibrate to atmospheric pressure under most conditions, due to the
relatively small (assumed negligible) resistance to flow.
If atmospheric pressure is maintained, then the compressibility of air can be neglected. This assumption is
thought to be reasonable most of the time for nearsurface conditions, except when the soil becomes watersaturated, and air contained in the pore space under
the saturated region can become compressed under a
wetting front. As liquid saturation increases, the airfilled porosity of a porous medium and its relative permeability to air become reduced. Therefore, resistance
to air flow may become significant at higher water contents, and in fine-grained porous media.
Numerical solutions to Eq. [1] or to the Richards equation require constitutive relations for the volumetric
water content, ␪ᐉ ⫽ nSᐉ, as a function of capillary pressure, and for the hydraulic conductivity, K ⫽ kr␥k, as a
function of water content or aqueous-phase saturation.
These constitutive relations are frequently represented
using the well-known models of van Genuchten (1980),
Mualem (1976), Brooks and Corey (1964), and Burdine
(1953). Accumulation of bacterial cells and associated
by-products of cellular metabolism during active growth
may alter the hydraulic and transport properties of the
porous media, and these changes may need to be considered in numerical modeling (Vandevivere, 1995; De Beer
and Schramm, 1999; Rockhold et al., 2002).
Solute Transport
Transport of dilute concentrations of mobile, linearly
partitioning constituents in variably saturated porous
media system is generally modeled using equations of
the following form
⳵
(Rf␪␥Ck,␥) ⫽ 䉮 · (␪␥D · 䉮Ck,␥) ⫺ (q␥Ck,␥) ⫹ ⌳k,␥
⳵t
[2]
where Rf is a dimensionless retardation factor, C is the
mass of a particular constituent per volume of pore fluid,
D is the hydrodynamic dispersion tensor, q␥ is the Darcian flux vector, and ⌳ represents a reaction-rate source–
sink term. The subscript k is used here to refer to the constituent (i.e., k ⫽ ed for the electron donor or growth
substrate, ea for the terminal electron acceptor, O2 for
oxygen under aerobic conditions, CO2 for carbon dioxide, and m for microbes), and the subscript ␥ again refers
to the phase. If the Richards equation is used to model
water flow, then modeling the transport of constituents
that partition between the aqueous and gas phases requires additional considerations, which will be described
later. Note that bacterial chemotaxis is not accounted
for in Eq. [2] and is not considered here. This important
and fascinating topic is discussed in detail by Corapcioglu and Haridas (1984), Ford (1992), Berg (2000), Nelson and Ginn (2001), and others.
The hydrodynamic dispersion tensor in Eq. [2] is commonly expressed as (Scheidegger, 1961; Bear, 1972)
Dij ⫽ ␦ij␣T|v| ⫹ (␣L ⫺ ␣T)
vivj
⫹ ␦ijDeff
k,␥
|v|
[3]
where ␣L and ␣T are the longitudinal and transverse dispersivities, v ⫽ q␥/␪␥ is the fluid velocity, and Dk,eff␥ is an
effective diffusion coefficient for phase ␥. Effective diffusion coefficients have been defined in various ways,
such as
mol
Deff
k,␥ ⫽ Dk,␥
冢 n␪ 冣
a
␥
b
[4]
where Dk,mol␥ is the coefficient of molecular diffusion, and
a and b are empirical parameters. The second term on
the right side of Eq. [4] accounts for the tortuosity of
the diffusion path through a porous medium. This form
of the tortuosity term has been used by numerous researchers for modeling both aqueous- and gas-phase
diffusion in porous media, but with different values used
for the parameters a and b. For example, Millington
and Quirk (1960) used a ⫽ 2 and b ⫽ 2/3, while Millington and Quirk (1961) used a ⫽ 10/3 and b ⫽ 2, and
Šimůnek and Suarez (1993) used a ⫽ 7/3 and b ⫽ 2.
Moldrup et al. (2000) evaluated several data sets for gas
diffusion in repacked soils and determined that overall
the data were best represented using a ⫽ 2.5 and b ⫽
1. Models for effective diffusion coefficients that include
residual water content or irreducible saturation terms
may be more accurate for modeling aqueous-phase diffusion at low liquid saturations (Olesen et al., 1996).
Equations of the form of Eq. [2] through [4] can be
used to model the transport of solutes and microbes in
the liquid phase, as well as the transport of constituents
that partition between the liquid and gas phases, such
as O2 and CO2. With the assumption of equilibrium partitioning, the retardation factor, the effective dispersion
coefficient, and the Darcian flux used in Eq. [2] can be
defined in a manner analogous to that used by Šimůnek
and Suarez (1993):
371
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冢KMRT 冣冢␪␪ 冣
M
⫽␪D ⫹冢
␪D
K RT 冣
M
⫽q ⫹冢
q
K RT 冣
Rf ⫽ 1 ⫹
k
g
␪ᐉDE
ᐉ
[5]
ᐉ
H
k
ij
g
eff
k,g
[6]
H
Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.
qE
k
ᐉ
g
[7]
H
where Mk is the molecular weight, KH is the Henry’s Law
constant, R is the ideal gas constant, and T is absolute
temperature. If the single-phase Richards equation is
used to model water flow, dispersion in the gas phase
is assumed to be negligible, and additional assumptions
regarding gas fluxes are required.
Following Šimůnek and Suarez (1993), it may be reasonable to assume that the gas flux is zero at the lower
soil boundary, L, as well as at the lateral boundaries
of a modeled domain. Changes in liquid volume can
then be assumed to be immediately matched by corresponding changes in gas volume, such that gas is allowed
to enter or exit only through the upper boundary. For
one-dimensional, vertically oriented systems, the gas
flux into or out of a profile during a time step can be
calculated from
L
V d␪ᐉ
qg(0) ⫽ ⫺ 冮
dz
[8]
A dt
z⫽0
where V is the volume of porous media represented by
a model grid block, and A is the cross-sectional area of
the grid block normal to the direction of flow. The vertical gas flux at any location within a profile can be
estimated in a similar fashion.
For multidimensional systems, estimation of gas fluxes
will generally require the solution of fully coupled sets
of equations for the flow of both water and air, rather
than the single-phase Richards equation. Diffusion is
usually the dominant mechanism for gas-phase transport in unsaturated porous media under most natural
conditions, however, so it may not always be necessary
to explicitly consider gas advection. Gas advection may
be significant during rapid changes in barometric pressure, especially in the vicinity of wells (Massman and
Farrier, 1992), and in any type of forced pumping scenario such as air sparging or soil vapor extraction. Furthermore, if gas-phase constituents are significantly denser
than air, then a single-phase flow approximation will no
longer be adequate, and fully coupled multifluid flow
equations will be necessary to accurately model densitydriven gas advection (Lenhard et al., 1995).
Even if diffusion is the dominant mechanism for transport in the gas phase, consumption of O2 and concomitant production of CO2 by soil microorganisms may create
counter-current movement of these gases that will influence their rates of diffusion. Jaynes and Rogowski (1983)
estimated coefficients of molecular diffusion for O2 and
CO2 in air from
DO2⫺CO2,gDO2⫺N2,g
DOmol2,g ⫽
DO2⫺N2,gXCO2,g ⫹ DO2⫺CO2,gXN2,g ⫹ rDO2⫺N2,gXO2,g
[9]
and
Fig. 1. Effective diffusion coefficients for O2 in the gas phase of a
sand porous medium as a function of aqueous-phase saturation for
different values of the respiration quotient, r.
mol
DCO
⫽
2,g
DO2⫺CO2,gDCO2⫺N2,g
DCO2⫺N2,gXO2,g ⫹ DO2⫺CO2,gXN2,g ⫹ (DCO2⫺N2,gXCO2,g/r)
[10]
respectively. The terms containing D on the right sides
of Eq. [9] and [10] are binary diffusion coefficients for
each gas pair, which are assumed to be independent of
composition. The terms XO2,g, XCO2,g, and XN2,g are the mole
fractions of each constituent in the gas phase. The respiration coefficient, r (moles of CO2 produced/moles of
O2 consumed), can be calculated from the stoichiometry of the biologically mediated oxidation–reduction reaction. Equations [9] and [10] were derived from the
Stefan–Maxwell equations for steady-state, counter-current diffusion of O2 and CO2 in the ternary system O2–
CO2–N2, in which the molar flux of N2 was assumed to
be zero (Jaynes and Rogowski, 1983). In the Stefan–
Maxwell approximations, it is assumed that the effects
of Knudsen diffusion and viscous or pressure-induced
flow are negligible (Massman and Farrier, 1992). Additional assumptions associated with the development of
the Stefan–Maxwell approximations are discussed by
Hirshfelder et al. (1964) and Whitaker (1986). Note that
Eq. [9] and [10] are strictly applicable to an unbounded
gas phase and should therefore be used in conjunction
with Eq. [4] or a similar expression when used to represent gas diffusion in porous media.
Figure 1 shows an example of Deff
O2,g as a function of
aqueous phase saturation calculated using Eq. [4] and
[9] for a uniform sand with three different values of the
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372
VADOSE ZONE J., VOL. 3, MAY 2004
Fig. 2. Effective diffusion coefficients for O2 in the aqueous and gas
phases of a sand porous medium as a function of aqueous-phase
saturation.
respiration coefficient. Binary diffusion coefficients reported by Jaynes and Rogowski (1983) are used, and
tortuosity coefficients of a ⫽ 2.5 and b ⫽ 1 are assumed
(Moldrup et al., 2000). In Fig. 1, mole fractions (or partial pressures) of XO2,g ⫽ 0.21, XCO2,g ⫽ 0.000 34, and
XN2,g ⫽ 0.789 66 are represented, which are typical atmospheric concentrations of these gases. Figure 1 indicates
that the effective diffusion coefficient for O2 in the gas
phase decreases rapidly as the aqueous phase saturation
increases, due to increased tortuosity. As expected, the
effective gas phase diffusion coefficient reaches its maximum value for air-dry conditions, or at residual aqueous-phase saturation.
Figure 2 shows effective diffusion coefficients for O2
in the aqueous and gas phases (both with a ⫽ 2.5, b ⫽
1) for the same porous medium as a function of aqueous
phase saturation. Gas-phase diffusion clearly dominates
over aqueous-phase diffusion for this porous medium until
the aqueous phase saturation reaches approximately 98%.
This behavior is expected since coefficients of molecular
diffusion in air are typically about four orders of magnitude greater than in water. It should be noted, however,
that the exponents a and b in Eq. [4] may not be the
same for the aqueous and gas phases, and may also vary
with soil type.
Equations [9] and [10] were derived based on the assumption of steady-state diffusion with N2 stagnant (XN2,g
constant). If diffusion coefficients are calculated using
Eq. [9] and [10], and used in advection–dispersion–reaction equations in conjunction with the Richards equation, for consistency with the isobaric assumption (which
is implied by the use of the Richards equation), it is necessary to assume that XO2,g ⫹ XCO2,g ⫹ XN2,g ⫽ 1. This assumption neglects any contributions from water vapor
and trace gases. If nonequimolar respiration occurs, such
that XN2,g ⬆ 1 ⫺ (XO2,g ⫹ XCO2,g), changes in total gas pressure will induce advective gas fluxes in a porous medium
so that gas transport is not strictly a diffusion process.
Leffelaar (1987) and Freijer and Leffelaar (1996) used
the single-phase Richards equation to model water flow,
in conjunction with advection–dispersion–reaction equations that contained “pressure adjustment flux” terms
to ensure that isobaric equilibrium was maintained and
to compensate for possible errors associated with using
Fick’s Law to model gas transport. Their calculation of
pressure adjustment flux is strictly applicable to onedimensional systems, however, and cannot be generalized to multidimensional problems. As noted previously,
accurate estimation of gas fluxes in multidimensional
porous media systems requires the solution of fully coupled multifluid flow equations (Pruess and Battistelli,
2002; White and Oostrom, 2003). In some cases, however, such as in relatively coarse porous media at low
to moderate liquid saturations, and with respiration coefficients close to 1, Eq. [1] through [10] should provide
reasonably good approximations. Thorstenson and Pollock (1989) provided a detailed discussion of multicomponent gas transport in unsaturated porous media and
further commentary on the adequacy of Fick’s Law.
Bacterial cells are known to secrete various surfaceactive compounds that can accumulate at gas–liquid interfaces. This accumulation can result in the lowering
of surface tension and increasing gas–liquid interfacial
areas. Accumulation of these compounds at gas–liquid
interfaces can also increase the mass transfer resistance
across the interfaces. The net effect is usually a reduction
of gas–liquid mass transfer rates (Bailey and Ollis, 1986;
Schügerl, 1982). Bailey and Ollis (1986) noted that for
a variety of sparingly soluble gases, surfactant adsorption at gas–liquid interfaces resulted in an average reduction in the interphase mass transfer coefficient of
60%. Accounting for this type of mass-transfer resistance requires a kinetic modeling approach rather than
the equilibrium partitioning described above.
Kinetic interphase (gas–liquid) mass transfer can be
described using traditional film models (Bailey and
Ollis, 1986), or in a similar form
冢 冣
dCk,ᐉ
Dkmol
⬀
dt
␲␶r
1/2
Agᐉ(C*k,ᐉ ⫺ Ck,ᐉ)
[11]
where Dmol
is the harmonic mean of the coefficients of
k
molecular diffusion in the liquid and gas phases, ␶r is a
characteristic time scale for diffusion across the gas–
liquid interface, which depends on its mass-transfer resistance, and Agᐉ is the specific gas–liquid interfacial surface area (Eq. [21]). The equilibrium concentration in
the liquid phase is C*k,ᐉ ⫽ KHpk, where again KH is the
Henry’s law constant, and pk is the partial pressure of
the gas. The combination of the first two terms on the
right side of Eq. [11] effectively represents a mass transfer coefficient. Eq. [11] is of a form originally proposed
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by Higbie (1935), and later adapted by Danckwerts (1970),
but modified here for interphase mass transfer in variably saturated porous media. Equations such as [11] can
be incorporated into reaction rate source–sink terms
used in Eq. [2].
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Bacterial Cell Attachment and Detachment
The processes governing bacterial cell attachment to
and detachment from surfaces are complex. Attachment
has been attributed to various physicochemical forces
including van der Waals’ forces, electrostatic interactions, hydrophobic effects, and specific adhesion (Daeschel and McGuire, 1998). Attachment of bacteria can
also be caused by physical filtration or straining (Tien
et al., 1979; O’Melia 1985; Logan et al., 1995). Bos et al.
(1999) provided an excellent review of various mechanisms for bacterial adhesion.
Numerous attempts have been made to relate various
properties of bacteria and physicochemical properties of
fluid-media systems to bacterial transport. For example,
Gannon et al. (1991) characterized the hydrophobicity,
net surface electrostatic charge, cell size, and presence
of flagella and capsule (extracellular polymeric substances or EPS) for 19 strains of bacteria, and attempted
to correlate these properties with transport of the bacteria through a water-saturated loam soil. They noted a
positive correlation between the size of bacteria and the
percentage of bacteria retained by the soil, but no statistically significant relationships were evident between
cell hydrophobicity and retention. All the strains tested
had a net negative surface electrostatic charge, but no
pattern was evident between surface charge and cell retention by the soil. Mixed results were obtained for relationships between retention of bacterial cells, capsule
formation, and presence of flagella. Gannon et al. (1991)
concluded that several physiological or morphological
properties of bacteria, interacting with properties of the
surfaces of soil particles, determine the occurrence and
extent of bacterial movement through soil. They suggested that further work would be required to define
these properties.
Mills et al. (1994) conducted batch experiments to
study the sorption of two bacterial strains with different
cell surface hydrophobicities to water-saturated, clean
quartz sand, to iron oxyhydroxide–coated quartz sand,
and to mixtures of the clean and iron oxide–coated sand.
Their data for clean quartz sand yielded linear, equilibrium adsorption isotherms whose slopes varied with the
bacterial strain used and with the ionic strength of the
aqueous solution. The greatest sorption was observed
for the highest ionic strength solutions, which is consistent with the interpretation that the electrical double
layer is compressed at higher ionic strengths, resulting
in stronger adsorption. When the iron oxyhydroxide–
coated sand was used, all of the bacteria were adsorbed
up to a threshold concentration, above which no more
bacteria were adsorbed. This irreversible, threshold adsorption was attributed to strong electrostatic attraction
between the Fe coatings and the bacterial cells, and was
modeled using a linear adsorption isotherm with a non-
373
zero intercept (Mills et al., 1994). Their results for mixtures of clean and iron oxide–coated sands were described well by a simple additive model for sorption on
the two types of surfaces.
Bacterial cell attachment to and detachment from
porous media surfaces and fluid interfaces have been
represented using both equilibrium and kinetic models.
The applicability of a local equilibrium assumption depends on environmental conditions, the type and physiological status of the cells, the mineralogical composition
of the porous media, the ionic strength of the liquid, and
the time-scale of experimental observations. For example, Bengtsson and Lindqvist (1995) conducted stirred
flow chamber and column experiments and determined
that dispersal of bacteria in their soil was controlled by
rate-limited, nonequilibrium sorption rather than instantaneous equilibrium. In contrast, Yee et al. (2000)
conducted batch experiments to study the adsorption
of a Bacillus subtilus bacterium onto the surfaces of the
minerals corundum (␣-Al2O3) and quartz as a function
of time, pH, ionic strength, and bacteria/mineral mass
ratio. Their experimental data demonstrated that adsorption of B. subtilus onto the mineral corundum is a
fully reversible equilibrium process, with equilibrium
occurring within 1 h. Adsorption of B. subtilus on quartz
was very weak, however, within the two standard deviation error bounds of their control experiments. They
assumed that cell attachment and detachment are controlled by the chemical speciation of bacterial and mineral surfaces, and successfully modeled the adsorption
of the B. subtilis on corundum using a chemical equilibrium model.
In an attempt to provide a more rigorous theoretical basis for explaining bacterial cell attachment and
transport behavior, Chen and Strevett (2001) characterized the surface thermodynamic properties of two types
of porous media (silica gel and sand) and three types of
bacteria (Escherichia coli, Pseudomonas fluorescens, and
B. subtilis) in different physiological states using contact
angle measurements with liquids of different surface
tensions. They found a strong correlation between the
total Gibbs free energies of surface interaction, calculated from contact angle measurements, and deposition
(or attachment) coefficients (Bolster et al., 1998) that
were calculated from the fraction of bacteria recovered
during column experiments in the water-saturated porous media. They also showed that deposition is correlated with physiological growth state, with stationary-phase cells being the most strongly adsorbed. In
addition, Chen and Strevett (2001) used infrared spectroscopy to show that increased deposition is correlated
with an increase in the H-bonding functional groups on
bacterial cell surfaces.
Although their calculated Gibbs free energies of surface interaction were correlated with first-order attachment coefficients, Chen and Strevett (2002) showed that
an advection–dispersion equation with simple first-order
attachment did not provide adequate descriptions of their
bacterial breakthrough curve data. Chen et al. (2003)
simulated the experiments using a model based on a
two-site (equilibrium and kinetically controlled) advec-
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374
VADOSE ZONE J., VOL. 3, MAY 2004
tion–dispersion equation (Šimůnek et al., 1999; van
Genuchten and Wagenet, 1989). Significantly improved
results were obtained relative to using an advection–
dispersion equation with simple first-order attachment.
Chen et al. (2003) showed that the parameters in the
two-site model were also strongly correlated with independently measured thermodynamic properties of the
bacterial cell and porous media surfaces.
The thermodynamic approach used by Chen and
Strevett (2001, 2002) was based on the extended DLVO
theory of van Oss (1994) that considers Lewis acid–base
interactions. One of the criticisms of applying surface
thermodynamic approaches and the DLVO theory in
general to describe bacterial adhesion in porous media
is that these approaches are typically based on the assumption of perfectly smooth, uniform, spherical particles attaching to smooth flat surfaces (Hermansson,
1999). Many bacteria and other soil microorganisms have
various heterogeneous surface structures (e.g., fimbriae,
flagella) that can facilitate attachment (Atlas and Bartha, 1993; Madigan et al., 1997; Hermansson, 1999).
Hence it could be argued that DLVO theory and its
modifications are not strictly applicable to modeling attachment during microbial transport in porous media.
Nevertheless, as shown by Chen and Strevett (2001,
2002), Chen et al. (2003), and others, surface thermodynamics have provided a useful framework for evaluating
microbial attachment in porous media.
The dynamic and adaptive nature of soil microorganisms may confound the quantification of specific mechanisms of attachment. For example, bacteria are known
to produce extracellular polymeric substances, including
polysaccharides, and other types of conditioning films
that can buffer cells from desiccation in unsaturated porous media and also promote adhesion (Williams and
Fletcher, 1996; Vandevivere and Kirchman, 1993; Chenu,
1993; Roberson and Firestone, 1992). Changes in soil
water potential are known to trigger physiological
changes in bacterial cells and to alter their metabolic
processes (Kral, 1981). Some bacteria are also known to
produce surface-active agents, or biosurfactants, during
active metabolism that can adsorb to gas–liquid interfaces, resulting in lower surface tensions (Kosaric, 1993;
Déziel et al., 1996).
Lawrence et al. (1987) studied the surface colonization behavior of a P. fluorescens bacterium and determined that it could be subdivided into the following
sequential phases: motile attachment phase, reversible
attachment phase, irreversible attachment phase, growth
phase, and recolonization phase. Harvey and Garabedian (1991) and Hendry et al. (1997, 1999) found that
it was necessary to account for bacteria that were both
reversibly and irreversibly attached at solid–liquid interfaces to match observed and simulated breakthrough
curves for bacteria in saturated porous media systems.
Ginn et al. (2002) suggested that in saturated porous
media systems, bacterial cell attachment and detachment
are related to both residence time and growth processes.
Most of the work on bacterial transport in porous
media has been done under no-growth and liquid-saturated conditions, in which bacteria adhere strictly to solid–
Fig. 3. Conceptual model of possible sorption sites for bacterial cells
in variably saturated porous media and corresponding rate coefficients.
liquid interfaces. A number of studies have also been
conducted in unsaturated porous media, however, which
have shown that some bacteria may preferentially adsorb to gas–liquid interfaces (Wan et al., 1994; Powelson
and Mills, 1998; Schäfer et al., 1998; Jewett et al., 1999).
Wan et al. (1994) suggested that sorption of bacterial
cells to gas–liquid interfaces is essentially irreversible
due to cell surface hydrophobicity. This conclusion
should be dependent, however, on the surface properties of the bacteria and fluid-media system (Docoslis
et al., 2000).
Given the adaptive nature of bacteria, and the difficulty in isolating and quantifying specific mechanisms,
unstructured models and phenomenological approaches
are generally used to represent bacterial cell attachment
and detachment processes in porous media. Given the
literature cited above, in kinetic modeling approaches
the possibility of sorption at three different types of
sites may need to be considered. These are (i) reversible
sorption at solid–liquid interfaces (and/or on other
attached cells), referred to here as s1 sites; (ii) irreversible sorption at solid–liquid interfaces (e.g., on ironoxide coatings), referred to as s2 sites; and (iii) irreversible sorption at gas–liquid interfaces. These possible
sorption sites and corresponding rate coefficients are
depicted in Fig. 3. An alternative, equilibrium approach
for modeling cell sorption at gas–liquid interfaces is
described later.
Reaction Rate Source–Sink Terms
General (kinetic) forms of the reaction rate source–
sink terms that could be used in Eq. [2] are given by
⌳ed,ᐉ ⫽ ⫺␭Yed/m␮(␪ᐉCm,ᐉ ⫹ ␳bCm,s1 ⫹ ␳bCm,s2 ⫹ ␪gCm,g)
[12]
⌳ea,ᐉ ⫽ ⫺␭Yea/m␮(␪ᐉCm,ᐉ ⫹ ␳bCm,s1 ⫹ ␳bCm,s2 ⫹ ␪gCm,g)
[13]
⌳m,ᐉ ⫽ ␭␮␪ᐉCm,ᐉ ⫺ k1␪ᐉCm,ᐉ ⫹ k2␳bCm,s1 ⫺ k3F␪ᐉCm,ᐉ ⫹
k4␪gCm,g ⫺ k5G␪ᐉCm,ᐉ ⫹ k6␳bCm,s2
[14]
where ␭ is a metabolic lag function; Yed/m and Yea/m are
yield coefficients representing the mass of electron donor or substrate (e.g., glucose) consumed and the mass
of terminal electron acceptor (e.g., oxygen) consumed,
respectively, per mass of cells generated; ␮ is the specific
growth rate of the bacterium; k1 through k6 are attachment–detachment rate coefficients; F and G are blocking
375
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functions; and ␳b is the bulk density. The terms Cm,ᐉ, Cm,s1,
Cm,s2, and Cm,g represent the mass of cells in the aqueous
phase per volume of pore liquid, the mass of cells reversibly attached to solids per mass of porous media, the
mass of cells irreversibly attached to solids per mass of
porous media, and the mass of cells attached to gas–
liquid interfaces per volume of gas phase, respectively.
Note that in Eq. [12] through [14] the same growth
rate is applied to bacteria everywhere. This does not
necessarily have to be the case. Different growth rates
could be applied to bacteria in different regions if, for
example, bacteria at air–water interfaces are thought to
have greater access to O2, and hence higher growth rates,
than those associated with solid–water interfaces.
The metabolic lag function, ␭, is an empirical means
of incorporating the time lag that is sometimes observed
before cells start to metabolize a substrate after their
initial exposure to it. This function may take a very
simple form, such as
␭⫽0
␭⫽
t ⱕ ␶L
␶ ⫺ ␶L
␶E ⫺ ␶L
␶L ⬍ ␶ ⱕ ␶E
␭⫽1
[15]
t ⬎ ␶E
where ␶ is the length of time that the cells at any given
location have been exposed to some minimum threshold
concentration of a substrate, Cmin
k,ᐉ , ␶L is the observed
time lag before the cells start to metabolize the substrate, and ␶E is the time required for the cells to reach
their exponential growth phase (Wood et al., 1995).
Strictly speaking, Eq. [15] should only be applied to
bacteria that are irreversibly attached.
The specific growth rate is usually represented by a
multiplicative Monod-type kinetics model (Megee et al.,
1970):
冢K
␮ ⫽ ␮max
ea
冣冢
Cea,ᐉ
Ced,ᐉ
⫹ Cea,ᐉ Ked ⫹ Ced,ᐉ
冣
[16]
where ␮max is the maximum specific growth rate (h⫺1),
and Kea and Ked are half-velocity (or half-saturation)
coefficients (mg L⫺1) for the terminal electron acceptor
(e.g., oxygen) and the electron donor or C source (e.g.,
glucose), respectively. Note that Eq. [16] can be easily
modified to include additional constituents that might
limit growth, such as N or P, and to account for various
types of substrate inhibition effects (Bailey and Ollis,
1986; Barry et al., 2002).
Mass balance equations for attached biomass can be
written as
⳵
(␳bCm,s1) ⫽ ␭␮␳bCm,s1 ⫹ k1␪ᐉCm,ᐉ ⫺ k2␳bCm,s1
⳵t
[17]
⳵
(␪gCm,g) ⫽ ␭␮␪gCm,g ⫹ k3F␪ᐉCm,ᐉ ⫺ k4␪gCm,g
⳵t
[18]
⳵
(␳bCm,s2) ⫽ ␭␮␳bCm,s2 ⫹ k5G␪ᐉCm,ᐉ ⫺ k6␳bCm,s2
⳵t
[19]
The maximum extent of cell sorption at gas–liquid
interfaces is assumed to correspond to monolayer cover-
Fig. 4. Gas–liquid interfacial surface area per bulk volume of a 40/50
grade of quartz sand as a function of effective aqueous-phase saturation, calculated using the Brooks–Corey water retention model
with n ⫽ 0.332, ␭bc ⫽ 8.85, hb ⫽ 26.5 cm, ␳ ⫽ 0.993 g cm⫺3, g ⫽
981 cm s⫺2, and ␴ ⫽ 72.8 mN m⫺1.
age. In a two-phase (air–water) system, the fraction of
the gas–liquid interfacial surface area available for sorption, F, can be represented using a Langmuir-type blocking function:
␪C A
F ⫽ 1 ⫺ g m,g cp
Agᐉ ⬎ 0
mcAgᐉ
F⫽0
Agᐉ ⫽ 0
[20]
where Acp is the maximum projection area of a cell,
mc is the average mass of a fully hydrated cell, and Agᐉ
is the gas–liquid interfacial surface area per volume of
porous media, which can be estimated from (Niemet
et al., 2002)
Agᐉ(Se) ⫽
冢
3␳gnhb
1
1⫺
2␴
␭bc
⫺1
冣冢
1⫺
1 ⫺ Se
1
␭bc
冣
h ⱖ hb
h ⬍ hb [21]
Agᐉ(Se) ⫽ 0
where Se ⫽ (␪ᐉ ⫺ ␪r)/(n ⫺ ␪r), and where ␪r is the residual
or irreducible water content, ␳ is the mass density of the
water, g is the gravitational constant, ␴ is the interfacial
tension at the gas–liquid interface, and hb and ␭bc are
parameters in the Brooks and Corey (1964) water retention function. Note that a number of other similar expressions have also been developed for describing interfacial surface areas in variably saturated porous media
(Leverett, 1941; Skopp, 1985; Cary, 1994; Oostrom et al.,
2001; and others).
Figure 4 shows the gas–liquid interfacial surface area
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376
VADOSE ZONE J., VOL. 3, MAY 2004
as a function of effective liquid saturation, given by Eq.
[21], for a uniform, 40/50 grade of quartz sand. Although
Fig. 4 indicates a nearly linear relationship, it should be
noted that Eq. [21] applies strictly to conditions following drainage from a fully water-saturated state, and does
not account for the effects of hysteresis and air entrapment, which may create much more complicated relationships between gas–liquid interfacial surface area and
liquid saturation.
The maximum extent of cell sorption at the s2 sites
is assumed here to correspond to monolayer coverage
of the fractional area of solid–water interfaces consisting
of s2 sites. This can be represented by the function
␳C A
G ⫽ 1 ⫺ b m,s2 cp
mc␰Asᐉ
[22]
where 0 ⬍ ␰ ⱕ 1 is the fractional area of solid–liquid
interfaces consisting of s2 sites, and Asᐉ is the solid–liquid
interfacial surface area per volume of porous media.
The parameter Asᐉ can be estimated from the product
of the first two terms on the right side of Eq. [21]. The
fractional area consisting of s1 sites does not necessarily
have to be explicitly accounted for in Eq. [22] if it is
assumed that bacterial cells can continue to accumulate
on top of one another, even after monolayer coverage
of all the s1 and s2 sites has been reached.
Transport equations with kinetic rate coefficients
similar to the ones described here have been proposed
for modeling the effects of sorption at gas–liquid (air–
water) interfaces on the transport of viruses (Chu et al.,
2001), bacteria (under nongrowth conditions) (Schäfer
et al., 1998; Jin et al., 2002), and colloids (Corapcioglu
and Choi, 1996). The experimental results of Wan and
Wilson (1994) and Wan et al. (1994) have been cited in
all of these studies as justification for explicitly considering sorption at gas–liquid interfaces. Most of the previously reported bacterial transport studies only considered steady flow and nongrowth conditions, however,
so complications associated with modeling attachment–
detachment and simultaneous growth at dynamic gas–
liquid interfaces were avoided. Furthermore, as noted
previously, the relative propensity for cells to attach to
gas–liquid vs. solid–liquid interfaces will depend on the
surface properties of the bacteria and fluid-media system. The nature of gas–liquid interfaces in unsaturated
porous media is also subject to some debate, as to
whether they are more representative of no-slip boundaries (e.g., the static wall of a capillary tube), or slip
boundaries (e.g., the surface of a flowing river), or some
combination of these extremes (e.g., trapped gas bubbles as well as a continuous, mobile gas phase). The
nature of gas–liquid interfaces in unsaturated porous
media obviously has some bearing on how sorption at
these interfaces should be modeled.
An alternative to the kinetic approach described above
for modeling cell sorption at gas–liquid interfaces is
to assume equilibrium adsorption. Several studies have
measured bacterial cell surface energies against air and
have found them to be less than the surface energies
(or surface tensions) at air–water interfaces (Absolom
et al., 1983; Gerson, 1993). Sorption of hydrophobic cells
and associated by-products of cell metabolism at gas–
liquid interfaces could therefore be expected to result
in some degree of surface tension lowering (Rockhold
et al., 2002). One way of accounting for sorption of
surfactants and/or cells at gas–liquid interfaces and concomitant changes in surface tension is to the use the
Gibbs adsorption equation (Adamson and Gast, 1997):
M d␴lg
⌫⫽⫺ k
RT d ln C
[23]
where ⌫ is the surface “excess” (mg cm⫺2) adsorbed to
gas–liquid interfaces, Mk is the molecular weight of the
adsorbing molecule (or cell), R is the gas constant, T is
the absolute temperature, ␴lg is the surface tension at
the liquid–gas interface, and C is the aqueous-phase
concentration. Eq. [23] can also be expressed as
⌫⫽
Mk d␥sp
C
RT dC
[24]
where the equilibrium spreading pressure is ␥sp ⫽ ␴o ⫺
␴lg, and where ␴o is the surface tension at the liquid–gas
interface for the aqueous solution in the absence of
bacteria or other surface-active compounds.
The equilibrium spreading pressure can be expressed
in terms of the familiar Langmuir- or Freundlich-type
isotherm models as (Adamson and Gast, 1997)
␥sp ⫽
ABC
1 ⫹ AC
[25]
and
␥sp ⫽ AC B
[26]
respectively, where A and B are empirical parameters.
For the Langmuir-type model, B also represents a maximum spreading pressure, B ⫽ ␴o ⫺ ␴min, where ␴min is
the minimum liquid–gas interfacial tension that would
presumably be obtained at monolayer coverage of the
interfaces by bacterial cells (or near the critical micelle
concentration for surfactants). Use of the Langmuirtype model for sorption at gas–liquid interfaces implies
a finite sorption capacity that can be estimated using
Eq. [20] and [21].
If the Langmuir-type model is used to represent sorption of cells (or surfactants) at gas–liquid interfaces,
Eq. [24] and [25] can be combined to determine a retardation factor, Rf, used in Eq. [2] as
Rf ⫽ 1 ⫹
M
冢 A␪ 冣冢RT
冣冤(1 ⫹ABAC) 冥冤1 ⫺ 冢1 2AC
⫹ AC 冣冥
gᐉ
k
2
ᐉ
[27]
Similarly, if the Freundlich-type model is used, Eq.
[24] and [26] can be combined to determine a retardation factor as
Rf ⫽ 1 ⫹
M
冢 A␪ 冣冢RT
冣AB C
gᐉ
k
2
B⫺1
[28]
ᐉ
Equations [25] through [28] are nonlinear, unless B
in Eq. [26] and [28] equals one, which corresponds to
the classical linear Freundlich isotherm model.
Equations [27] and [28] are strictly applicable to equi-
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librium adsorption at gas–liquid interfaces. If equilibrium sorption at solid–liquid interfaces also occurs, retardation effects due to sorption at these interfaces can
be accounted for in the usual way, and used in conjunction with Eq. [27] and [28]. However, it is difficult to
account for the simultaneous growth of bacteria at interfaces where equilibrium sorption has been assumed due
to the mixing of equilibrium and coupled kinetic reactions. Nevertheless, Eq. [25] through [28] can be readily
used with Eq. [2] to model microbial transport and adsorption at gas–liquid interfaces under nongrowth conditions, and/or surfactant adsorption at gas–liquid interfaces with concomitant surface-tension lowering effects.
It should be noted that both the kinetic and equilibrium
descriptions of sorption at gas–liquid interfaces given
above implicitly assume that these interfaces are effectively static, no-slip boundaries. However, gas–liquid
interfacial areas may change as a function of liquid saturation, following Eq. [21].
Surface tension lowering in porous media can result
in decreases in liquid saturation and changes in the gradients of capillary pressure. Both of these factors affect
water flow and hence solute and microbial transport.
These effects can be accounted for by scaling of capillary
pressure–saturation relations (Rockhold et al., 2002),
followed by simple updating of capillary pressures at
the end of each transport step. Changes in capillary
pressure will then create changes in liquid saturation,
permeabilities, and pressure gradients that will control
the magnitude and direction of flow velocities at the
next time step. In this loosely coupled approach, concentration-dependent changes in capillary pressure due to
surface active agents, and concomitant changes in liquid
saturation and permeability, are lagged by one time step.
This approach requires minimal additional computational effort, but may not be as accurate as some other
methods. Smith and Gillham (1994) described an alternative, iterative method that was used to simulate the
effects of surfactants on water flow and solute transport
in laboratory column experiments. Their method required
time step sizes on the order of milliseconds. Hence,
albeit more accurate, this approach may be impractical
for simulation of field-scale problems. An alternative
method, involving the solution of fully coupled multifluid flow equations, with surfactant as one of the fluids,
is described by White and Oostrom (1998).
Rate Coefficients
As noted above, most of the models that have been
developed for simulating microbial transport in porous
media have used kinetic modeling approaches. Harvey
and Garabedian (1991), Hornberger et al. (1992), Deshpande and Shonnard (1999), and others assumed that
bacterial cell attachment in saturated porous media
could be represented by a first-order kinetic model and
used the model of Tien et al. (1979) to estimate attachment coefficients. In the model of Tien et al. (1979) the
attachment coefficient, k1, is estimated from
k1 ⫽
冢 冣
3 qᐉ (1 ⫺ ␪ᐉ)
␩␣c
2 ␪ᐉ
dg
[29]
377
where dg is the median grain diameter of the porous
medium (or collector), and ␩ and ␣c are the so-called
collector and collision (or sticking) efficiencies (Logan
et al., 1995; Deshpande and Shonnard, 1999). Equation
[29] is based on particle filtration theory and the earlier
work of Rajagopalan and Tien (1976). The expression
for the collector efficiency is essentially an empirical correlation function that contains a number of dimensionless
terms to describe London–van der Waals interactions,
interception, sedimentation, and diffusion. The collector
efficiency is calculated from the physical properties of
the porous media (e.g., porosity and median grain diameter), fluid properties (e.g., density, viscosity, velocity,
and temperature), and the size and density of the suspended particles (Tien et al., 1979; Logan et al., 1995).
The sticking efficiency, ␣c, is an empirical parameter.
Chiang and Tien (1985a, 1985b) developed another
empirical correlation function that modifies the model
of Tien et al. (1979) to account for a so-called “filter
ripening” effect that is frequently observed in particle
filtration systems, where the collector efficiency increases
with time as more suspended particles become attached
to the filter media. The model of Tien et al. (1979) and
the empirical correlation function by Chiang and Tien
(1985a, 1985b) do not account for detachment (reentrainment) of particles, or surface interactions resulting from
differences in charge, the presence of electrolytes, or
changes in the ionic strength of the suspending liquid.
Such surface interactions might include both particle–
particle interactions, such as flocculation or electrostatic
repulsion, and particle–collector interactions that might
occur, for example, if the electrostatic double-layer
thickness is reduced by an increase in the ionic strength
of the pore water.
In spite of the fact that Eq. [29] does not explicitly
consider the surface interactions noted above, Deshpande and Shonnard (1999) used it to study the effects of
systematic increases in ionic strength on the attachment
kinetics of a P. fluorescens bacterium in saturated sand
columns. They varied the ␩ and ␣c parameters, as well
as a third parameter, ␴c, in the filter ripening model of
Chiang and Tien (1985a, 1985b), to optimize the fit between simulated and observed bacteria breakthrough
curves for different ionic strengths of the suspending
fluid. Reasonably good results were obtained for all
cases, but somewhat better results were obtained when
the additional fitting parameter, ␴c, was used.
McDowell-Boyer et al. (1986) provide a review of various factors that may affect particle transport through
porous media and different modeling approaches. Note
that the term particle has been used rather loosely to
refer to objects of many different sizes, shapes, and compositions, including latex colloids, bacteria, and viruses.
However, particle filtration models are usually based on
the assumption of smooth, inert, and perfectly spherical
particles. Although these models account for a number
of physical factors, they generally neglect electrostatic
interactions and hydrophobic effects. Given the heterogeneous surface characteristics and nonspherical nature
of most soil microorganisms, one could question the validity of using particle filtration models (Eq. [29]) in
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378
VADOSE ZONE J., VOL. 3, MAY 2004
conjunction with advection–dispersion equations to
model microbial transport in porous media. Nevertheless, these equations have been applied successfully in
numerous cases to describe microbial transport in porous media. Murphy and Ginn (2000) and Ginn et al.
(2002) provided further discussion on this topic and
suggested alternative modeling approaches.
Rittmann (1982) developed an equation to describe
detachment as a function of fluid shear stress, using data
for a P. aeruginosa bacterium that was obtained using
a centrifuge (Trulear and Characklis, 1982). Speitel and
DiGiano (1987) suggested, however, that detachment
is much more closely related to biomass growth rate
than to the amount of biomass present, and modified
Rittmann’s model with an additional term to account
for the specific growth rate. MacDonald et al. (1999a)
used Rittmann’s model to account for detachment as a
function of fluid velocity and shear in the vicinity of
well screens for a hypothetical bioremediation scenario
involving the injection of nutrients into an aquifer.
Alternative models for bacterial cell detachment have
been proposed by Johnson et al. (1995), Ginn (1999),
and Ginn et al. (2002), who considered detachment to
be a residence time–dependent process. They did not
ascribe this time dependence to any particular mechanism. Hence these models are essentially empirical, but
provide more flexibility in fitting breakthrough curve
data. The models proposed by Johnson et al. (1995)
and Ginn (1999) require the use of particle-tracking
algorithms to track the time history of individual particles (or cells). Scheibe and Wood (2003) described a
novel approach, also based on particle tracking, to describe pore-scale exclusion processes and their effects
on bacterial transport. Using particle tracking methods
to model bacterial transport is made somewhat more
complicated if cell growth is considered, but growth was
not considered in any of the aforementioned papers.
A simplified approach for representing time-dependent detachment rates would be to use
k2 ⫽ f exp(⫺tgc)
[30]
where f and gc are empirical parameters, and t is equal
to the value of ␶ in Eq. [15], or to the time since any
location in the modeled domain is first exposed to some
minimum threshold concentration of growth substrate.
In essence, this amounts to keeping track of the time
in which local environmental conditions have been conducive to optimal growth, rather than keeping track of
the time history of individual particles (or bacterial
cells). Note that Eq. [30] has the same affect as the filter
ripening model of Chiang and Tien (1985a, 1985b).
Time-dependent rate coefficients have also been used
for modeling irreversible sorption of proteins at solid–
water interfaces. For example, Lee et al. (1999) modeled
observed decreases in protein adsorption rate using an
adsorption (or attachment) coefficient that was a power
function of time, rather than a blocking function, as
used in Eq. [18] and [19], or an exponential function of
time, of the type given by Eq. [30]. Decreases in protein
adsorption rate with time were attributed to an increasing energy barrier associated with the increase in ad-
sorbed mass as irreversible sorption sites approached
monolayer coverage (Lee et al., 1999). Changes in protein adsorption rates with time have also been attributed
to conformational or structural changes that occur in
protein molecules as they “unfold” on the surfaces to
which they attach (Daeschel and McGuire, 1998). Further discussion on protein adsorption as it relates to
virus transport in porous media is given by Thompson
et al. (1998) and Jin and Flury (2002).
In kinetic modeling approaches, adsorption of bacteria at gas–liquid interfaces has typically been assumed,
based on the results of Wan et al. (1994), to be an essentially irreversible process. It may be reasonable to
assume that sorption at gas–liquid interfaces is also a
function of both the aqueous phase concentrations of
bacteria, as well as the gas–liquid interfacial area that
is available for sorption, as suggested by Eq. [20] and
[21]. The attachment coefficient for bacteria at gas–
liquid interfaces can be represented by the parameter,
k3, which is assumed to be a constant. Detachment of
bacteria from gas–liquid interfaces can then be assumed
to be negligible until these interfaces reach monolayer
coverage, or until increases in aqueous phase saturation
reduce the gas–liquid interfacial area, causing the mobilization of cells that were previously attached. After
monolayer coverage is reached, additional growth of
cells at gas–liquid interfaces will presumably result in
instantaneous detachment and resuspension in the aqueous phase. This condition can be represented by
k4 ⫽ 0
F⬎0
k4 ⫽ ␭␮
F⫽0
[31]
The possibility for irreversible attachment of bacterial
cells at solid–liquid interfaces was also considered by
Harvey and Garabedian (1991), Hendry et al. (1997,
1999), and others. The rate of irreversible attachment
may be assumed to be proportional to the concentration
of bacteria in the aqueous phase and to the fraction of
surface area available for sorption at s2 sites, as given
by Eq. [19] and [22], where the k5 parameter is assumed
to be a constant, and the k6 parameter is represented
the same way as k4 in Eq. [31], but based on the value
of G rather than F. This treatment is consistent with
the observations of Mills et al. (1994), cited previously,
that suggest that some bacteria may become instantaneously and irreversibly adsorbed on iron-oxide coatings.
Lawrence et al. (1987) suggested, however, that cells
first become reversibly attached before they become
irreversibly attached, which could possibly represent a
different mechanism, such as the production of conditioning films by the bacteria.
By inspection of Eq. [18] and [19], it can be seen that
the rate parameters k3 and k4 are somewhat redundant
with the rate parameters k5 and k6, and may therefore
be unnecessary. Although they can potentially be used
to represent different mechanisms, their net effect is
indistinguishable under most conditions, unless sorption
of cells at gas–liquid interfaces also causes other effects,
such as surface tension lowering. If this is the case,
then sorption at gas–liquid interfaces might be better
379
Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.
www.vadosezonejournal.org
represented using the Gibbs adsorption equation, as
described above.
It may be of interest to note that some earlier solute
transport studies also observed more apparent sorption
or incomplete solute displacement at lower liquid saturations (Nielsen and Biggar, 1961). However, this behavior was attributed to a larger fraction of water in the
porous media being contained in relatively slow moving
or stagnant regions that acted as sinks to ionic diffusion.
These observations led to the development of the mobile–immobile water concept that is frequently used in
solute transport modeling (Coats and Smith, 1964; van
Genuchten and Cleary, 1979; van Genuchten and Wagenet, 1989). Use of the mobile–immobile water concept
may lead to reductions in simulated peak concentration
values that may be similar to the results obtained by
Wan et al. (1994) for bacteria when liquid saturation
was reduced in their glass micromodel and column
experiments. However, the mechanisms to which this
phenomenon is attributed are different. The mobile–
immobile water model generally leads to extended tailing of breakthrough curves, due to slow mass transfer
of solutes out of immobile water regions after the main
solute pulse has passed, while irreversible sorption tends
to simply reduce peak concentrations without significantly affecting the higher moments of a breakthrough
curve.
SUMMARY AND CONCLUSIONS
We have discussed coupled microbial and transport
processes in soils. These coupled processes and interactions include (i) the effects of substrate limitations, biomass growth and accumulation, and by-product formation on the physical and hydraulic properties of variably
saturated porous media, and the associated effects on
water flow and solute and microbial transport and (ii)
the ability of gas-phase diffusion, which is strongly dependent on aqueous-phase saturation as well as biomass
concentration distributions, to provide O2 as a terminal
electron acceptor in aerobic, biologically mediated oxidation–reduction reactions in soils. For one reason or
another, such interactions have been largely neglected
in both experimental work and numerical modeling of
transport processes in soils. The lack of consideration
of these types of interactions in many experimental and
modeling studies may be partly responsible for the difficulty in reconciling some lab and field data, and in reproducing field data for reactive transport problems using
model parameters obtained from lab studies.
Two main topics emerge as warranting further research: (i) the development of novel experimental methods for observing and measuring coupled biogeochemical processes in variably saturated porous media systems,
to elucidate underlying mechanisms and emergent phenomena that may be associated with these process interactions, particularly in heterogeneous systems, and (ii)
the development of more computationally efficient and
accurate numerical methods for simulating these coupled processes, their interactions, and feedback mechanisms. Novel experimental methods involving biolumi-
nescent bacteria, translucent porous media, and lighttransmission chambers, such as those described by Yarwood et al. (2002) and Weisbrod et al. (2003), may
provide new insights into the interactions between microbial processes and transport processes in unsaturated
porous media. Although the Richards and advection–
dispersion–reaction equations may be applicable under
some conditions, the strongly coupled nature of the processes, and various feedback mechanisms, will ultimately
require the use of fully coupled, multifluid flow and multicomponent reactive transport equations, such as those
described by Pruess and Battistelli (2002) and Battistelli
(2003), that allow for more accurate representations of
the coupled processes for a wider range of conditions.
Ginn et al. (2002) provided additional suggestions for
future research related to microbial transport in the subsurface.
This work has focused primarily on processes associated with the growth and transport of a single bacterial
species in variably saturated porous media, and methods
for modeling these processes. It should be recognized,
however, that all naturally occurring biogeochemical
processes in the subsurface are influenced by interacting
microbial consortiums, rather than single species (Atlas
and Bartha, 1993; Barry et al., 2002; Battistelli, 2003).
Further research is warranted toward developing a better understanding of the complex processes and interactions associated with the spatial and temporal dynamics
of microbial communities under variable fluxes of water,
energy, and nutrients, especially in heterogeneous, unsaturated porous media. Topics of special interest include
the dynamics of mixed microbial populations growing on
multiple substrates (e.g., diauxic growth), and chemotaxis in response to substrate limitations. Interdisciplinary research that combines the traditional fields of study
of soil physics, chemistry, and microbiology (or microbial ecology) should prove to be especially fascinating.
APPENDIX
a
A
Acp
Agᐉ
b
B
C
dg
D
Dk,mol␥
Dk,eff␥
f
F
Fw␥
g
gc
G
exponent in numerator of tortuosity function
empirical parameter in isotherm models
average cell projection area (m2)
gas–liquid interfacial surface area/bulk volume
(m2 m⫺3)
exponent in denominator of tortuosity function
empirical parameter is isotherm model
solute or bacterial concentration (kg m⫺3)
median grain diameter (m)
diffusion–dispersion tensor (m2 s⫺1)
coefficient of molecular diffusion, (m2 s⫺1)
effective diffusion coefficient, (m2 s⫺1)
empirical parameter in model for detachment
coefficient
Langmuir-type blocking function for air-water
interfaces
advective flux of water in phase ␥ [kg/(m2 s)]
gravitational constant (m s⫺2)
empirical parameter in model for detachment
coefficient
blocking function for solid–liquid interfaces
380
h
hb
Jw␥
Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.
K
k1–k6
kr
KH
Kea
Ked
mc
Mk
N
pk
qᐉ, qg
r
R
Rf
S␥, Se
t
T
v
Xk
Y
␣c
␣L, ␣T
␦
␥sp
⌫
␩
␪ᐉ, ␪g
␭
␭bc
⌳
␮, ␮max
␮f
␰
␳
␴, ␴0
␶
␶L
␶E
␶r
⍀
VADOSE ZONE J., VOL. 3, MAY 2004
soil water pressure head (m)
air-entry pressure (m)
diffusive–dispersive flux of water in phase ␥ [kg/
(m2 s)]
intrinsic permeability (m2)
attachment–detachment rate coefficients (s⫺1)
relative permeability or hydraulic conductivity
(m s⫺1)
Henry’s Law constant [kg/(m3–atm)]
half-velocity coefficient for terminal e-acceptor
(kg m⫺3)
half-velocity coefficient for electron donor
(kg m⫺3)
average mass of bacterial cell (kg)
molecular weight of adsorbing molecule or cell
(kg mol⫺1)
porosity (m3 m⫺3)
partial pressure of gas component k, (atm)
Darcian fluxes of liquid and gas phases (m s⫺1)
respiration coefficient (mol mol⫺1)
gas constant [(L atm)/(mol K)]
solute retardation factor
absolute and effective fluid saturations (m3 m⫺3)
time (s)
absolute temperature (K)
fluid velocity (m s⫺1)
mole fraction (partial pressure) of gas component k
yield coefficient (kg kg⫺1)
collision or sticking efficiency
longitudinal and transverse dispersivities, (m)
Kronecker delta function
equilibrium spreading pressure (mN m⫺1)
surface “excess” adsorbed to gas-liquid interfaces (kg m⫺2)
collector efficiency
volumetric liquid (water) and gas (air) contents
(m3 m⫺3)
microbial metabolic lag,
Brooks and Corey function parameter
reaction rate source–sink term (kg s⫺1)
specific and maximum specific growth rates,
(s⫺1)
fluid viscosity [kg/(m s)]
fractional area of solid-liquid interfaces consisting of s2 sites
density of aqueous solution (kg m⫺3)
gas–liquid interfacial tension with and without
cells or surfactants (mN m⫺1)
exposure time of cells to substrate (s)
lag time before cells start to metabolize substrate (s)
time required to reach exponential growth
stage (s)
time-scale for diffusion across gas-liquid interface (s)
fluid source sink term (kg s⫺1)
ACKNOWLEDGMENTS
We would like to thank Markus Flury, Brian Wood, and
an anonymous reviewer for their helpful critiques of this paper.
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