Desorption Processes in Reactive Transport

Environ. Sci. Technol. 2010, 44, 7996–8007

Upscaling Sorption/Desorption Processes in Reactive Transport Models To Describe Metal/Radionuclide Transport: A Critical Review ANDREW W. MILLER,* DERRICK R. RODRIGUEZ, AND BRUCE D. HONEYMAN Colorado School of Mines Environmental Science and Engineering Division, 1500 Illinois Street, Golden, Colorado 80033, United States

Received May 28, 2010. Revised manuscript received September 14, 2010. Accepted September 23, 2010.

Models are a mainstay of the environmental sciences; they allow for both deeper understanding of process knowledge and, to a limited extent, predictive capabilities of current day inputs on the future. Mathematical codes have become increasingly complex with explicit inclusion of many processes that could not be accounted for using simpler solving techniques. And yet, for metal/radionuclide transport in subsurface systems, the inclusion of smaller scale processes in a numerical solver do not always lead to better descriptions of larger scale behavior. The reasons for this are many, but included in this review are the following: unknowable conceptual model errors, discrepancy in the scale of model discretizationrelativetothescaleofthechemical/physicalprocess, and omnipresent chemical and physical heterogeneities. Although it is commonly thought that larger, more complex systems require more complex models to gain insight and predictive capability, there is little to no experimental evidence supporting this thought. Indeed, the evidence points to the fact that larger systems can be well described with simple models. To test this thought and to appreciate the incorporation of scaling behaviors into reactive transport modeling, new experiments are needed that are intermediate in scale between the more traditional bench and field scales.

1. Introduction Environmental systems exhibit a range of complexities that exist at a range of length and mass scales. Within the realm of the environmental behavior of metals/radionuclides, much work has been focused on understanding pore scale processes. In describing larger scale behavior, the results from these simplified systems must be combined to create a theory of the whole. Our intuition tells us that more complex systems (e.g., ‘larger systems’) require more complex models; however, to date no one has reviewed available literature to assess this assumption or to assess how scaling relationships are currently being considered and the relative success that has been achieved using these procedures. Through this review process three aspects will be covered including the following: how physical and chemical heterogeneities affect observed behavior of metals/radionuclides; how data from the bench scale is applied in field scale simulations; and the relationship * Corresponding author phone: (505)844-2910; e-mail: andmill@ sandia.gov. Current address: Sandia National Laboratory, Albuquerque, NM. 7996

9

ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 44, NO. 21, 2010

between the scale of measurement and the scale of application. The overarching theme between these topics is to examine the connections that have been made between bench scale behavior and characterization with field scale behavior and characterization and to evaluate the validity and efficacy of such approaches. As a tool in the geosciences, reactive transport modeling is used on a much wider array of systems and processes then will be covered here. This discussion is limited to sorption/desorption type reactions in shallow aquifer systems and does not include the following: dissolution/precipitation, colloidal transport, redox transformations, microbial processes, fracture flow, unsaturated flow domains, or diffusion dominated systems. That is not to say that the more general aspects of this paper will not apply in these systems, just that they were not explicitly considered as part of the review process. It is left for the reader to decide how to best apply these generalities in the various disciplines.

2. Processes Relating to Scaling Figure 1 illustrates a relationship between the required number of grid cells in an aquifer as a function of the grid dimensions in a synthetic numerical model domain. On the plot, the points represent a regular discretization of cubes in a 125,000 m3 aquifer. Thus for a 1 m3 grid cell, 125,000 cells are required. Also on the plot are representative scales for the different processes that can exert some control on metal/ radionuclide transport behavior. These are not intended to be exact representations; for many of these processes the exact range of possible scales will depend on specific conditions at any given field site. However, it is important to note that the physical processes related to transport are generally larger in scale relative to the chemical processes. The explicit inclusion of chemical reactions to a transport simulation nearly doubles the range of length scales involved relative to inert transport where chemical behavior can be ignored. Independent of the size range is the presence of physical and chemical heterogeneities discussed in detail below. Finally, highlighted in red on the plot is an approximation for common numbers of grid cells used in the literature. Again, the exact number that is used will often be a function of modeling goals, but the range is relatively finite ranging from ∼102 to ∼105 (Larger grids are becoming more common: 16 million grid cells (1), and even >100 million grid cells (2). Both of these simulations required massively parallel, supercomputing capabilities.). When considering scale relationships in reactive transport model (RTM) simulations, 10.1021/es101822v

 2010 American Chemical Society

Published on Web 10/13/2010

FIGURE 1. Representative scale of physical and chemical processes relating to metal/radionuclide transport. Points on the plot represent the relationship between the number of grid cells required and the size of an indivdual grid cell assuming a regular discretization of cubes in an aquifer with a total volume of 125,000 m3. The red horizontal lines represent the range of total number of grid cells commonly used in the literature. it is noteworthy that all of the chemical processes involved, and many of the physical processes involved, occur at a scale that is considerably smaller than the size of a representative grid cell, and, yet, these subgrid processes are often the controlling factors of observed behavior. Using Figure 1 as a starting point, we will summarize the chemical and physical processes controlling metal/radionuclide behavior as a function of scale. We will then discuss how chemical and physical processes interact despite the large difference in effective scales and examine how both physical and chemical heterogeneities affect these interactions. 2.1. Chemical Controls on Metal/Radionuclide Behavior. There are three major threads of knowledge relating to chemistry dependent transport of metals. The first is through spectroscopic, synchrotron, and other high energy based characterization technologies. The second is through modeling of observed macroscopic (though still at the bench scale) behavior using the modeling scheme as a heuristic device to understand chemical behavior. The third, which is not covered in this review, is molecular modeling of water/metal interactions near a mineral surface. 2.1.1. High Energy Surface/Solution Characterization. The knowledge gained from the first thread of experimental inquiry is molecular scale interactions between relatively small numbers of atoms in solution and mineral surfaces through high energy characterization techniques. These techniques also open the molecular world to a better characterization of the processes that can occur at mineral interfaces. For example, uranium(VI) reduction to uranium(IV) or other uranium species with mixed valencies (IV)(VI) due to interactions with sulfidic mineral surfaces was confirmed using several spectroscopic techniques: auger electron spectroscopy [AES], X-ray photoelectron spectroscopy [XPS], and Fourier transformed infrared analysis [FTIR] (3). The authors suggest that this surface behavior may explain precipitates formed in anoxic groundwater and may control uranium solubility. In another study, using X-ray absorption fine edge spectroscopy (EXAFS), a discrete zinc bearing precipitate formed under high surface loadings of ferrihydrite with no Fe-Zn solid solution phase (4). This precipitate was forming despite the fact that the solution was undersaturated by two or more orders of magnitude. Similar surface structures were found for a Co(II)-Al2O3 system using transmission electron microscopy (TEM), EXAFS, and XPS (5). Besides surface information, an aqueous ternary species, Ca2UO2(CO3)30, was discovered by flourescence spectroscopy (6) and later confirmed by time-resolved laser-induced spectroscopy (TRLFS (7)). Both of these studies calculated

that the ternary species would dominate most oxic groundwaters around former uranium mills and mines. Later it was found that this species is largely surface in-active and can lead to enhanced uranium migration (8-10). There are many other studies where spectroscopy has allowed for characterization of atom-atom interactions at mineral interfaces or in solution. The relevance to this review, however, is that the majority of these interactions are being measured at approximately the angstrom scale and represent an average signal for a relatively small total number of atoms. Using Figure 1, to explicitly account for all of these atom-atom interactions, 1036 grid cells would be required for the hypothetical aquifer, which is well beyond current computing abilities. A better use of this information is to anchor conceptual models for surface and aqueous complexation in a measurable reality where the known and quantifiable existence of specific surface and aqueous complexes gives credence to some of the more simplified chemical models discussed below. 2.1.2. Bench Scale/Wet Chemical Methods for Characterization. The second major thread follows a more traditional surface chemical approach where macroscopic behavior is described through chemically plausible but not necessarily real, surface interactions. The uranyl ion is well-known for forming strong complexes with inorganic constituents in solution including the ternary Ca2UO2(CO3)30 species mentioned above as well as phosphate and fluoride complexes. The addition of fluoride significantly decreases the amount of uranium sorption in batch systems and leads to higher rates of transport in columns (11). In that study the authors were able to predict uranium mobility at the column scale through the use of an RTM determined from batch parameters. Similar mobility increases have been found for other metal-anion pairings as well. Decreased metal adsorption was observed for increasing CO32- concentrations for uranium (12, 13), neptunium (14), and chromium (15), while enhanced sorption to goethite was found for increasing SO42- concentrations for lead, zinc, cadmium, and copper (16, 17). The implication is that the presence of carbonate species would lead to less sorption and faster metal migration, while the presence of sulfate would increase retardation. Specific mechanisms for these observations vary from surface site competition (15, 12, 13) to altered surface properties caused by anion sorption (16, 18) and to alterations in solution phase chemistry (14). Just as with the inorganic controls, introduced organic chemicals or natural organic matter (NOM) can have both enhancing and retarding effects on metal migration. Humic acid has been found to enhance uranium sorption at low pH values and decrease sorption at higher pH values (19). This phenomenon was also found to be mineral specific (20). In the Cu-fulvic acid-goethite system, Cu sorption can also be enhanced or reduced depending on chemical conditions (21). A conceptual model to interpret these results can be found in ref 22; in short, it relates the relative affinities of the metal and the organic matter both for each other as well as the mineral surface. In translating these behaviors to the length scales in Figure 1, the 10s of grams of sediment often used in these studies relates to approximately the cm scale in a field transport system. This requires ∼1012 grid cells in the hypothetical aquifer. In moving between the two experimental techniques, spectroscopic to bench scale, there is already a dramatic increase in both size and complexity of the experimental system. 2.1.3. Surface Complexation Models and Chemical Heterogeneity. The development of surface complexation models (SCM) has largely been driven by the chemical complexity described in the preceding sections. The goal of SCM is to explicitly account for variable surface complexes that form VOL. 44, NO. 21, 2010 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

9

7997

as a function of solution chemistry. Most SCM stem from the consideration of a mineral surface as an amphoteric functional group where cation sorption is a function of pH (23). Since the application of SCM to environmental systems, conceptual models have grown in complexity as larger numbers of fundamental behaviors are experimentally determined (e.g. refs 24 and 25 for surface precipitation). Notable additions to the amphoteric model relevant to the concept of scaling are the triple layer model (TLM 26-28) and the CD-MUSIC model (29). These two models differ mostly in how they account for surface charge. The TLM distributes the charge into the solution phase in three distinct planes moving away from the surface where each plane is physically interpreted as being a boundary between sorption mechanisms. The CD-MUSIC model distributes some of the surface charge into the mineral phase itself and also accounts for the physical structure of bound ions at the surface. Both of these models have been extensively used to describe bench scale data in single mineral phase systems. And both have been used in connecting high-energy characterization techniques to observed behavior in batch reactors. One study which transcends the atom to bench scales considered uranium sorption to imogolite and used the triple layer model to fit the data (30). Batch sorption envelopes were created for a range of chemical conditions, and surface speciation was measured using XAS. The model parameters were fit using the spectroscopically determined surface stoichiometry and macroscopic observed data from the batch reactors. A second study used the CD-MUSIC model to describe Cd, Co, Cu, Ni, Pb, and Zn sorption to both goethite and ferrihydrite (HFO) (31). Again they fit the surface complexation constants of these species around the measured stoichiometries of surface complexes. The important outcome from this paper is that using the CD-MUSIC model, the coordination of protons and a single metal to these two different solid phases were accomplished using the same set of model parameters. Both of these studies range in scale from approximately the angstrom level to tens of grams (∼1036 up to ∼1012 grid cells in Figure 1), but the more important implication is that they are scaling through different levels of surface heterogeneity. The TLM is a simpler model than CD-MUSIC in terms of the number of fitting parameters and conceptual model simplicity. However, it is still capable of explicit description of the underlying surface site heterogeneity, which causes the observed sorption behavior (30). The CDMUSIC model extends this transcendence of scale to encompass metal behavior between Fe-O minerals of variable crystallinity. The varying levels of crystallinity cause the surface environment between these two minerals to be even more heterogeneous than that of the single imogolite mineral phase, and yet the same model values described metal sorption between these two distinct solid phases. Another attempt at scaling surface chemical heterogeneity from the atom scale up to batch scale behavior is to use distributed reactivity models (32-34). Instead of specific surface reactions, assumed by both the CD-MUSIC and TLM model formulations, distributed reactivity models define surface reactions with a set of parameters to describe reactivity, generally based on KD values. The extension of this modeling construct to metal/radionuclide behavior ignores any advances that have been made based on spectroscopic data and may or may not be able to describe metal-surface interactions which are poorly described by KD based interpretations. However, distributed reactivity models may intersect with RTM in distributed rate models to describe kinetic formulations (35, 36). This is discussed below in more detail in the section relating to scaling through time. To move up on the scale of heterogeneity, and to apply SCM to transport scenarios, both the TLM and CD-MUSIC 7998

9


model are both too simple and too complex. Both models are too simple in the sense that neither model would be able to account for the surface and mineral heterogeneity associated with a natural mineral matrix while simultaneously limiting the number of fitting parameters. They are both too complex in that the inclusion of a chemical speciation calculation at that level within the framework of a RTM code would exhaust computing resources. Thus different, upscaled, conceptual models have been created to extend the SCM concept to mixed mineral systems. The Generalized Composite (GC) and the Component Additivity (CA) approaches are two competing ideas to apply SCM to RTM (37). The GC approach assumes that a natural mineral matrix is too complex to be understood and fits hypothetical reactions to observed sorption behavior. The CA approach uses sorption behavior determined through experiments on single mineral phases and then mathematically combines the behavior based on the proportions of the individual minerals present in the matrix (for fuller discussions and some applications to real data see refs 37-41). Neither of these models explicitly includes electrostatic terms, which has often been the major difference between SCM formulations. Instead, the effects of electrostatics are lumped into the complexation constant describing the sorption reaction. This exclusion of surface charge has been critically examined (42). Using previously published data (14), both an electrostatic (three fitting parameters) and a nonelectrostatic (four fitting parameters) model were able to fit batch scale sorption data. The goals of the GC modeling strategy are pragmatic in nature; sorption data collected from a range of chemical conditions is to be described with the simplest model possible. Simple, in this case, is defined by the smallest number of fitting parameters that can still describe/predict metal behavior. This more pragmatic goal of linking a simplified chemical model with transport is necessary because of computing limitations, model transparency constraints, and limitations in characterization. Currently the GC approach is more common in transport simulations (43-48). Despite the fact that the GC models are often applied at the same experimental scale as the more complex TLM and CD-MUSIC models, the GC approach is capable of describing sorption data on a far more heterogeneous sample with a limited number of fitting parameters; the GC approach can be considered an upscaled version of the TLM and CD-MUSIC models where the scale is of increasing surface heterogeneity. This is an important distinction: in order to upscale a SCM in terms of heterogeneity, a new, somewhat simplified conceptual model was necessary. Even with the required simplifications of the GC approach, good agreement to data has been observed (37, 39-41). Furthermore, the added simplicity of the GC approach allows for a semimechanistic inclusion of metal sorption as a function of solution chemistry into transport scenarios. There are limitations to the GC models in terms of extrapolation beyond a set of experimental conditions, but for the most part many field systems are bound to certain ranges of chemical compositions. Also, as with any model, decisions made by the modeler often introduce as much error as the conceptual model choice (40). 2.2. Physical Controls on Metal/Radionuclide Behavior. In this review, physical controls are defined as controls on transport that are largely related to water flow or diffusion through porous media. Many of the effects are closely related to the chemical effects mentioned above. These physical controls are separated by the scales at which they have been shown to occur: pore scale diffusion and macrodispersion. Under increasingly powerful microscopes, minerals and porous media have been found to be quite fractured. The oft invoked conceptual model divides the water filled domain

into mobile water zones between grains of porous media and immobile zones of stagnant water within the interior of the mineral grain (49-52). Within the mineral grain, diffusion is assumed to be the major transport process, while in the mobile phase physical advection is often in control. Transfer of contaminant mass from the immobile to the mobile zone is a controlling factor in contaminant tailing behavior (35, 51-53). Local advective fluxes are controlled by porous media heterogeneities both at pore and larger scales. These pore and field scale heterogeneities in water flow and subsequent transport are commonly modeled by fitting a dispersion coefficient to the data; longitudinal dispersion coefficients have been shown to be length scale dependent (54). Interestingly, however, as numerical models have become increasingly sophisticated and modeling grids have become increasingly detailed, dispersion caused by pore scale heterogeneity can be explicitly accounted for (55). Overall, the physical processes for metal/radionuclide behavior are not dissimilar to physical processes controlling any other contaminant migration. And as shown in Figure 1, these processes tend to occur at the ∼10-9m scale up to the kilometer scale. The major difference between metals/ radionuclides and organic contaminants is how the physical processes interact with chemical processes and how RTM compute both. 2.3. Combined Chemical/Physical Effects. As was seen in Figure 1, the physical and chemical processes typically occur at different scales. However, in a contaminated aquifer setting of any scale, the physical and chemical effects occur simultaneously and despite the scale discrepancy have the ability to affect each other. Many of the studies cited above either simplify the flow domain to understand the basic chemistry controlling metal transport or they simplify the chemistry to understand the physical aspects of transport. These simplifications are made at both the bench and field scales and may often confuse the “real” reason for observed macroscale behavior. This section will highlight several studies at a variety of scales where both physical and chemical heterogeneities are considered. At the column scale, Co transport behavior was found to be dependent on both the physical and chemical properties of the porous media (56). After packing a column with two sands of different hydraulic conductivities and chemical reactivities, a pulse input of radio-labeled Co was injected into the column. The Co breakthrough curve (BTC) was fairly traditional in that it had a steep rising edge and prolonged tailing behavior that persisted for about 10 pore volumes. In modeling the BTC the dual domain model previously discussed was used. Many of the parameters in the model were fit from batch sorption data or through the use of the nonreactive tracer tritiated water. The best simulation of the data explicitly included terms for the physical heterogeneity, sorption heterogeneity, rate limited mass transfer between mobile, and immobile zones and rate limited sorption. A similar connection between the chemical and physical environment and ion migration has been described at the pore scale (57). Through extensive characterization of microscale chemical heterogeneities (58) a conceptual model was created to explain the slow release of uranium from microprecipitates present in fractures deep within the porous media. The major outcome of the modeling effort was that coupled dissolution and diffusion within a fracture was slower than either process alone. This understanding may help to explain variations in kinetic data between lab and field data or in the scaling of kinetic parameters (e.g. refs 59 and 60). In contrast to the 1-D study, 2-D pore scale modeling efforts (61, 62) found that spatial distributions of reactivity under advective conditions can lead to different observed macroscopic rate behavior even though the total reactivity between simulations was the same. Furthermore, in column experi-

ments pore scale water velocities caused by particle scale heterogeneities have been linked with observed reaction rates (63), although the cause of this link has yet to be delineated (33, 64). One possibility is the interaction between aqueous and solid phases caused by advective mixing (65-68). It has been found that the more well mixed a system is, the smaller the reactive discrepancy becomes. The question still remains what, if any, effect do these pore scale processes have at even larger scales. In a lab scale experiment (60 cm2) pore-scale mineral precipitation fronts caused large alterations to the flow field (69), while in a field setting similar behavior was noted due to increasing biomass (70). In a 3-D field scale modeling effort (71) tetrachloroethene transport was simulated in a hypothetical field setting similar to that of the Borden site. Linear sorption behavior and local chemical equilibrium were invoked, and a stochastic approach to hydraulic conductivity and KD values were adopted. Despite the simplifications to the chemistry involved, “pseudo-kinetic” behavior was observed in plume displacement and dispersion. The cause of this behavior was due only to heterogeneities in the flow field caused by variations in hydraulic conductivity. In other words, in this case, porescale chemical processes were of limited importance. “Pseudo-kinetic” behavior has also been cited in a hypothetical aquifer which was physically homogeneous and chemically heterogeneous (72). So, pore-scale chemical reactions are controlling contaminant behavior; the spatial distribution of sorption parameters leads to increasing retardation factors with time and displacement distance. In a different study (73) an alternative hypothesis is presented to explain the observed macrodispersion at the Macrodispersion (MADE) site. The model that they used to describe the field data of an inert tracer did not include dispersion, but with the use of the dual domain conceptual model, better descriptions of the observed dispersion were produced. Thus confusion remains in comparing field and bench scale data and in determining methodologies to use bench data for field applications.

3. Upscaling Geochemical Models Upscaling has been distinguished from averaging by saying that averaging predominantly relates to the mathematical formulation of a model system, while upscaling relates to both the mathematical formulation as well as to the conceptual and physical representation of the modeled system (74). The assumptions made which reduce the total number of degrees of freedom in the conceptual model represent scaling laws. In the case of SCM, the move from mechanistic chemistry to the GC approach may represent a significant scaling law. The number of degrees of freedom has been reduced by making a series of assumptions relating to the nature of the ion-surface interaction, including but not limited to the following: the surface is too complex to be truly understood, effects of electrostatics can be lumped with chemical interactions into a single complexation constant, and that observed behavior should be described by the smallest number of reactions possible. These assumptions have allowed for the advancement of RTM, but this alone is not sufficient for accurate descriptions of reactivity as a function of scale. Testing of some of these hypotheses has been carried out, but the implications of the untestable assumptions remain unknown and potentially unknowable. In the following section we will show that the extension of batch derived SCM has variable success in describing data at other scales.

4. Extension of Bench Data to Field Conditions This section will focus on scaling studies characterized by contaminants whose observed behavior is largely caused by VOL. 44, NO. 21, 2010 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

9

7999

TABLE 1. Summary of Bench and Column Scale Work Discussed in the Texta solid phase mass of batch or type (idealized vs solid phase SCM or reference column field material) used (g) sorbate isotherm (75) (56) (76) (11) (44) (77) (79) (80) a

batch column column batch column batch column batch column batch column batch column

field field idealized idealized idealized idealized idealized field field both both field field

0.1 2 1992 2 100 unspecified 122 10-50 253-266 unspecified 23-204 unspecified 31.5

U U Co U U U U Mo Mo Co Co Cs Cs

isotherm isotherm isotherm isotherm isotherm SCM SCM SCM SCM SCM SCM IX IX

3.3

experimental experimental duration duration (batch, days) (column, days)

experimental length (column, cm)

retention time (columns, h)

2

200

12

100

unspecified

333-1000

3

55-150 3.2

1.7 30.5

0.4 3.83

0.25-25

14.6 and 14.8

1.21/12.1

0.94 and 3.74 21.1

1.67/6.8

9-75

30

22.2

4.1-14.5

14.4-35.6

2.3-3.0

8.2-45

15

2.22-11.7

unspecified 12.5-42 2-250

0.67

IX ) ion exchange.

surface and aqueous reactions alone, which also include data at 1) batch to column scales and 2) bench to field scales. 4.1. Upscaling from Batch to Column Scales. Table 1 summarizes many studies, most of which contain both batch data, used to create a SCM, and column data, where the SCM is applied to transport data. To scale between these two experimental techniques, surface area of the sediment is used. What makes each of these studies unique is the choice of SCM, the mineral phase used, and the metal of concern. The simplest description of surface interactions between the contaminant and sediment is a sorption isotherm. When applied to transport data in the column experiments, the isotherm based models are not good predictors of contaminant breakthrough (56, 75, 76). Model fits improve when kinetic terms are included in the models; these kinetic terms can be assigned to chemical reactions (75, 76) or to mass transfer limitations (56). Furthermore, lowering the flow rate in a column experiment allowed the kinetic formulation to be omitted (76). A more complex descriptor of ion-surface interactions is the GC SCM, previously discussed. This approach has been applied using a nonelectrostatic formulation to describe cobalt (77) and uranium (11) transport; it has also been applied using a diffuse double layer formulation to describe molybdenum transport (44). As with the isotherm approaches kinetic parameters were required in the RTM to fill discrepancies between observed transport and batch reactor behavior (77). The explicit omission of kinetic formulations has been explored by plotting sorption isotherms derived from batch and column data (11). For similar pH the isotherms from the two scales of inquiry are coincident. For other experimental systems significant differences between column derived sorption and batch derived sorption behavior have been reported (72). These observed behaviors are attributable to groundwater flow velocities and whether the local equilibrium assumption is (11) or is not valid (76). This difference is reflected in the RTM formulations; when the local equilibrium assumption was valid there was no need for kinetic corrections from batch to column data, but several different equilibrium SCM were still considered based on experimental scale (11). The SCM parameters that were derived from column studies were capable of predicting behavior under different chemical conditions in another experimental column and predicting the behavior in batch reactors using the solid:solution ratio to scale surface interactions. However, the batch derived SCM did not give good predictions in column studies. The stated reason for this is that batch conditions are less sensitive to strong binding surface site density than are columns. The model formulation used to fit molybdenum transport data (44) was incapable of including kinetically controlled behaviors. The simulation of column data based on batch reactor behavior did not 8000

solid:sol’n ratio (g/L, batch only)

9


accurately capture molybdenum breakthrough or late time tailing behavior. The lack of model fits is attributable to mass transfer limitations as the bromide showed clear indications of nonideal transport. Another type of metal/radionuclide interaction with mineral surfaces is ion exchange. Ion exchange models describe positively charged ions interaction with the fixed negative charge present in many clay minerals (More recently, surface complexation models have also been used to describe ion exchange (78).). Largely speaking, these interactions are more prevalent with mono- and divalent species (e.g., Cs+, Sr2+, I-, Na+, Ca2+). In a pair of studies, batch scale characterization of Cs+ ion exchange with Hanford sediments (79) was used to describe Cs+ behavior in column studies (80). Ion exchange models were fit to batch data based on the cation exchange capacity (CEC) and the specific ion that was being exchanged for Cs+ (Na+, K+, or Ca2+). Depending on the exchanging ions, different numbers of surface sites with variable affinities for each of the ions were included. In the case of Ca2+, five discrete exchange sites were required for description of batch behavior. The use of these batch scale exchange parameters to describe transport data underpredicts Cs+ retardation in column experiments. However, the extent of the discrepancy depends on the specific ion pairings in the column experiments (Cs+-Na+ vs Cs+-Ca2+). To create a global ion exchange model where all of the ions are present in a single transport experiment, the batch data were refit using three exchange sites and are augmented with data from column experiments. As previously observed (11), the model fits from column data were able to describe both column and the batch data, whereas the batch data did not describe column data particularly well. In scaling information from the batch to the column scale, there is a significant jump in terms of the amount of heterogeneity. This is reflected in the total mass of sediments used between the two experimental methods and also between the elapsed experimental times in both methods (Table 1). When given by the referenced sources, the mass of sediment used in batch experiments was 1 to 2 orders of magnitude smaller than that used in columns. Using the longest experimental times, the ratio of column:batch experimental time ranges from 0.3-75. Also in moving from batch to column scale, not only does the system become more heterogeneous due to size but also the system becomes more complex due to the interplay between chemical and physical processes. Another source of heterogeneity between batch and column scale behavior relates to advective mixing. Batch reactors are generally quite well mixed, while columns are only advectively mixed. Columns also would have a low degree of mixing, as the particle size range used is generally quite small, leading to more ideal and less turbulent flow

1 800

2 914

assumed dimensionality

U

200 yrs 29.2 days 200 N (106)

uranium mill remediation uranium mill remediation (13)

reference

none

Y

720

10 days

time steps number of nodes SCM (y/n) metal(s) of concern site

Field Scale Simulations with No Supporting Data

216 yrs

simulation duration

2 37, 52 4 undeterminable 28 11 (for spatial distributions), ∼225 (for BTC) Y N Cape Cod Cape Cod (47) (48)

Mo Zn Cape Cod Cape Cod (45) (46)

Cr, Se Cr, Pb, Zn, Cu, Ni

250 400 Y Y

19 1

12 (for 11 yrs of monitoring), 52 (for 3 yrs of monitoring) 656 .18 unspecified Y Naturita (43)

U

length of model domain (m)

1 2

2 2

2 ∼1000

∼59 yrs (of contaminant migration), 15 yrs (active monitoring) 2.1 yrs ∼50 yrs (of contaminant migration), 75 year simulation 64 days 15 months

assumed dimensionality contaminant travel distance (m) experimental duration number of sampling wells (including MLS) number of sampling events SCM (y/n) metal(s) of concern site reference

TABLE 2. Summary of Upscaling Studies from the Bench to the Field and Field Simulations

around a particle (65). Mixing at the column scale is rarely quantified, so it is difficult to understand how this effects reactions between experiments. One proxy for approaching the idea of mixing is the retention time of the columns (Table 1). In general for the columns examined here, the range of retention times is quite narrow: 0.4-22 h. Not only are retention times similar but also so are the column sizes meaning that the Darcy fluxes are also quite similar (range: 1.21-15 cm/h). So, even though the range of behaviors and scaling responses is great, the amount of advective mixing as measured by flow velocity is comparable. How the authors dealt with these heterogeneity discrepancies divides into two groups: kinetic responses (56, 75, 76) and changes in equilibrium surface models (11, 80). In both of the studies where the equilibrium SCM was altered as a function of scale, SCM fit from the more heterogeneous column systems could also describe data in the less heterogeneous batch systems. Similar behavior was observed using isotherm based modeling approaches (76). Thus it appears that either surface area and the soil:solution ratio is inadequate to scale RTM or that there is a ‘real’ scale dependency of ion behavior which is not captured using RTM. However, when the RTM is parametrized at a more heterogeneous level (column) and applied to another column or to the simpler batch systems, the SCM is adequate to describe ion behavior. This raises an interesting implication in considering how to fit a SCM for a field transport scenario; what model parametrization system is at least as heterogeneous as a field system? 4.2. Scaling from Batch to Field Scales. A common approach to modeling field scale systems contaminated with metals/radionuclides is to collect an aquifer sample, perform batch scale studies for characterization and development of a SCM, and then use this SCM directly in field simulations scaling the surface reactivity to measured values of mineral surface area (43-48). Of these studies, one occurred at the Naturita site, a former uranium mill in southwestern Colorado (43), and the rest occurred at the Cape Cod site in Massachusetts. Relevant characteristics of these studies can be found in Table 2. The Mo bench scale work previously discussed was extended to field scale behavior at the Cape Cod site (45). Besides the difference in scales, a major difference between the two works is that the bench scale SCM consisted of one surface site type and two Mo-surface reactions (44), whereas the field scale SCM uses both the single site type and a two site model with four total Mosurface reactions (45), i.e. more surface chemical heterogeneities are being explicitly included. The new site is a ‘strong’ site which has a high affinity for Mo sorption. Relatively good fits to the field data are achieved with both the one and two site SCM; however, the two site model is a closer match over longer periods of time. In model simulations, the addition of the second surface site limits lateral dispersion relative to the one site model. Similar limitations on dispersion were seen for Zn transport to Cape Cod sediments (46). From these bench to field studies it would appear that the SCM can be scaled through mineral surface area. The role of the second surface site in field scale transport may be analogous to the recalculation of SCM formulation that was required to scale from batch to column systems (11, 80). Small, unnoticeable errors in fitting SCM at the bench scale may exhibit themselves as the scale and time of transport increases. Thus the methods used to determine SCM parameters (batch vs column) and decisions made by SCM developers (one surface site vs two surface sties, how many reactions to include, etc.) appear to be critical in understanding reactive transport behavior as a function of scale. In another series of work, uranium sorption to natural sediments has been explored as a function of scale (39, 43, 81). The distinction between this work and the Mo transport

VOL. 44, NO. 21, 2010 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

9

8001

studies is that U aqueous and surface chemistry is far more complex than that of Mo; 29 equations describe surface and aqueous speciation for U (39), while only 15 are required for Mo (45). From this a hypothesis can be drawn that the complexity of U chemistry will cause more difficulty in scaling data due to local chemical heterogeneities as a function of scale. However, despite this added level of complexity, similar results were found. The SCM developed from batch scale data using a GC, nonelectrostatic approach (39) gives marginally good fits to column data (81) and good fits to field data (43). The discrepancies between model output and column data appeared to be related to chemical conditions in the column experiment; columns with a pH of 7.0 and 0.02 atm of CO2 had smaller discrepancies than columns at pH of 7.4 or 7.9 with PCO2 of 2.8 × 10-3 atm and 3.73 × 10-4 atm, respectively. The authors cite kinetic variation as a function of chemical concentrations as a possible reason for the discrepancies. At the field scale, the fits of U to the field data are very good. Hydrologic characterization of the field site showed that hydraulic conductivity was spatially uniform, while several other estimates and assumptions related to precipitative inputs, porosity, dispersivity, and source term estimates were required. All of these assumptions were tested against the nonreactive transport of Cl-, at the site as part of the ore roasting process. Since chemical and physical processes are connected, it is unclear how the assumptions related to physical characterization helped to influence the modeled U distribution. Nevertheless, the same behavior where column data are marginally well described by a SCM and field data is well described is evident. A final point is that for the model formulation used, the scaled sensitivity of the SCM terms was equal to zero for length scales up to 500 m (43). In these batch to column to field scale studies, the major scaling method is through total surface area of the mineral phase. Since the chemical behavior of the ions is controlled through surface reactions, scaling to surface area seems reasonable. However, the data show conflicting evidence on this assumption. Good correlations were found between Pb/ Zn sorbed concentrations and Fe/Al phases extracted in a selective extraction, while there were only weak correlations between Pb/Zn sorbed concentrations and particle size, a proxy for surface area (82). Thus scaling to reactive surface area is more important. In other words, only the fraction of surface area directly involved in the sorption process need be considered. The opposite has also been found experimentally; scaling uranium sorption behavior to total surface area gave better results than scaling to reactive surface area (83). Thus scaling to total surface area is more important. Separating reactive and total surface areas is complicated by poorly formed crystalline phases and surface coatings (84, 85). Obtaining the reactive surface area often requires either potentiometric titrations of the mineral surface, or saturation of the surface with the contaminant of interest, or both. However, even then there are limitations to this practice with both idealized and natural sediments (86). As SCM are usually fit to macroscopic batch data, the determined binding constants represent an averaging of contaminant sorption behavior over both the reactive and nonreactive areas on the surface. Scaling SCM through total surface area assumes that 1) the degree of surface heterogeneity/reactivity of sediment material used in SCM formulation experiments is similar to that found in the field and 2) that the water-sediment interactions and reactivity will be similar in field conditions compared to lab conditions. This latter assumption has been explicitly tested (87, 88). In both of these studies favorable comparisons were made between SCM predictions based on batch scale determinations and in situ determined KD values. The first assumption is more difficult to test, but it can be argued that the deviation between simulations based 8002

9


on batch experiments and column scale behavior is evidence that the surface reactivity is not constant as a function of scale. From the data presented here, it appears that extrapolating from batch to field scale heterogeneity via surface area appears to be an approach that gives satisfactory results. This same technique has been used to scale sorption behavior at the same experimental scale but with different minerals (i.e., scaling through increasing levels of chemical heterogeneity 89, 90). The question still remains as to why this procedure does not appear to work at extrapolating from the batch to the column scale. One possibility is that the fitting of field data smoothes any error caused by local chemical heterogeneity. In columns, where a transport model can be more accurately constrained the variability in surface behavior as a function of scale is more apparent.

5. Summary of Scaling Behavior The idea of the unknowable plays an important and unquantified role in upscaling RTM. Previously mentioned is the unknowable outcome from making certain scaling assumptions. The idea of the unknowable is also related to the level of measurable uncertainty in using RTM to move between experimental scales. For example, from Table 1 the typical batch experiment uses 10s of grams of sediment to characterize reactivity. Typical column systems use 100s of grams. And typical field sites have ∼105-108 kilograms. Thus, using a batch experiment to describe reactivity means that ∼10% of a column mass has been explicitly characterized, while only about 10-5 or 10-8% of a field scale site has been characterized. This ignores physical orientation of the chemical constituents. The idea of the unknowable means that it is impossible to determine the effects that this simplification makes on upscaling RTM. In the final section of this paper this idea is considered even more.

6. Scales of Measurement and Scales of Application Philosophically, the need for a model becomes increasingly irrelevant as the amount of characterization increases. Models are meant to be meaningful representations; however, assuming that both the physical/chemical system characteristics and processes can be measured, as the amount of characterization increases the model becomes less of a representation and instead becomes an explicit description of the system being modeled. One source of the need for scaling in transport models stems from the fact that characterization of most earth systems can only be partially accomplished. Also there is no way to know the extent of our characterization without characterizing the entire system, at which point the model loses its relevance as the system has (more than likely) been dismantled (91). Thus, the amount of characterization plays a critical role in model formulation, and it divides between spatial characterization and temporal characterization. 6.1. Characterization in Spatial Dimensions. Figure 2 compares the relative amount of characterization possible as a function of the water volume in an aquifer. The y-axis represents the total sample volume of a single sampling event in time divided by the total water volume of the aquifer presented as a percent in log-log space. For the bench scale work the sample volume was comprised of what had eluted from the effluent end of the column in between sampling events (11, 92) and what was removed from ports located as a function of distance along the column length (93). For the field scale sites (43, 45, 95) and the three studies which are not grouped (36, 47, 94), a few approximations were necessary. First, when not given, porosity of the aquifers was assumed to be 0.3. Second, the well purge volume was included as part of the sample volume, and when not given

FIGURE 2. Relative amount of characterization possible as a function of aquifer volume. Bench scale references: refs 11, 92, and 93. Field scale references: refs 43, 45, and 95. a total sample volume of 1 L was assumed for field sites. Finally, the total aquifer volume is the volume of water which was contaminated at the time of the study (For the 2-D study (47) a width of 1 m was assumed.). The most striking feature of this plot is the bimodality for traditional experimental scales. The bench scale work is characterized by having a very small aquifer volume and a high percent of characterization. The field sites are characterized by large total volumes and considerably smaller levels of characterization. The three intermediate points are included to show how certain experimental techniques can be used to help connect bench scale characterization with field scale behavior. ‘Push-pull’ tests were used to study the amount of uranium immobilization that can be expected in an in situ system (94). This is not truly a metal transport study, in that the transport involved was dominated by injection of electron donor and acceptor and then removing sample volumes after a period of incubation. However, it is included here to emphasize that new experimental techniques may be required to help understand the scaling issues associated with RTM. Another relatively unique method is the small scale field test. Chromium and selenium were injected into the Cape Cod aquifer as part of a reactive tracer test (47). The major difference between this work and others is that the scale of chromium and selenium transport was only 2.0 m. Thus the total volume is substantially smaller than other field sites. A final technique is to simply use larger versions of a bench scale apparatus. Uranium transport was explored using a column that was 15 cm in diameter and 80 cm long (36). The discrepancy in levels of characterization between experimental methods limits the total number of variables that can be accurately determined in field scale systems compared to bench scale systems. This level of characterization is related to the concept of ‘stopping rules’ (96). These rules are self- or system-imposed limits on characterization and subsequent inclusion into a conceptual model. As can be seen in Figure 2, the scale of the system changes the scale of ‘stopping rule’ implementation. Smaller systems can be characterized at smaller scales, meaning that conceptual models can explicitly include smaller scale processes. Another point relating to Figure 2 is that if a plot was constructed comparing the total number of variables that can be controlled in an experimental system vs the scale of the experimental system, the same type of relationship would hold. At smaller scales, a much higher number of variables can be controlled whereas in the field, the controllable variables are severely limited. This raises a theoretical concern about solely using field scale systems to elucidate scaling methods. Some have even raised the point that delineation

of fundamental transport processes from field scale experiments may be impossible (97) or at least nonunique (98). Indeed, similar output behavior can be created from drastically different conceptual models (compare ref 71 to ref 72, and see Figure 5 in ref 47 where similar changes in breakthrough curve shape can be created by changing the SCM or the dispersion coefficient). Figure 2 is focused on the characterization of the aqueous phase; however, a similar relationship would be found between experimental scale and characterization of the solid phase material. As part of a field experiment, sediment cores are often removed for characterization of physical and chemical properties by simpler methods, such as XRD, N2 BET measurements, etc., and more extravagant methods such as SEM and TEM. Using all of these techniques the core as removed from the ground can be characterized in terms of almost any physical or chemical parameters necessary for RTM conceptual models. However, the upscaling of core characterization to a model grid scale is complicated by physical heterogeneities at the field scale. At issue is how to meaningfully relate the level of possible bench scale characterization applied to an ensemble of core measurements to the level of the field scale. Because of this limited level of characterization, new experimental methods are necessary. Ideally, these new experiments would 1) plot in the area of ref 47 and 94 in Figure 2 and 2) allow for explicit control of physical and chemical heterogeneities. These criteria are present in the larger scale bench work (36), but even larger experiments extended through all three dimensions may be necessary. Indeed, there are several reports of experiments that are taking place at this ‘intermediate’ scale. However, many of these experiments are concerned only with upscaling of flow parameters (99-101) or inert transport (102, 103), while others are focused on multiphase and organic contaminant transport (e.g. refs 104 and 105). In order to help determine scaling methods for RTM, these same types of experiments need to be completed with inorganic contaminants where local chemical and physical heterogeneities can be quantified and controlled. 6.2. Characterization in Time. Tables 1 and 2 gave a summary of experimental protocol with respect to the timing of both bench and field scale experiments, respectively. There is a very large discrepancy between an average column experiment (∼45 days as presented in Table 1) and the time scale of an average field simulation (∼decades-centuries 13, 106). With respect to RTM, this represents a significant barrier to scaling kinetic information. Ideally, rate laws relating to mineral precipitation/dissolution, adsorption/ desorption, and redox transformations can be extended through time and connected to the descriptions of mechanistic chemistry in RTM. There are two problems with this idea. The first is related to use of the semimechanistic SCM described above. Because of the semimechanistic approaches to modeling which have heretofore been applied in transport scenarios, fundamental rate laws with respect to adsorption reactions are not applicable. The second problem is related to mineral dissolution/precipitation reactions. There is a well discussed discrepancy between mineral dissolution rates as measured in laboratories compared to those estimated from field data. This discrepancy has been attributed to several reasons including the following: coupled chemical reactions (60), changes in surface reactivity (59, 107), heterogeneous grain size distributions (108), and coupled physical and chemical processes (57) as well as pore scale heterogeneities (62, 109). Without the ability to describe these rate limited reactions in a meaningful way, spatial distributions of reactive products cannot be predicted. Because of these limitations, explicit VOL. 44, NO. 21, 2010 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

9

8003

consideration of rate-limited processes has largely been limited to more empirical applications (34, 36, 57, 61). Distributed rate laws may be the best conceptual model to describe such complex systems where both physical and chemical heterogeneities contribute to spatially and temporally variable observed rates of reaction (110, 111). This method is somewhat analogous to the GC SCM approach, where the system is deemed too complex to really understand, so a hypothetical and physically meaningless representation is used instead. This approach at describing rate controlled processes has been applied to metal/radionuclide transport (34). The authors apply a distributed rate model to several column experiments with favorable results, but only a single groundwater composition was used. Since these distributed rate models are based on KD descriptions of contaminant sorption, the use of a single set of kinetic parameters for different groundwater compositions may or may not describe observed data. More recently a distributed rate law coupled to a SCM has overcome the limitations of the KD approach (36). Furthermore, similar model values are used to describe uranium transport in both 15 and 80 cm columns. It remains untested as to whether a single distributed rate function can describe data over field relevant portions of time or in multidimensional systems. In considering which kinetic formulation to use in transport scenarios we must return to Figures 1 and 2 and Tables 1 and 2. In transport scenarios we are fundamentally limited in the amount of characterization that can occur at the field scale; this is true for physical/chemical characteristics as well as temporal characteristics. In Table 2, it is remarkable that the number of sampling events for field sites range from one to several hundred over the course of, at longest, 15 years of monitoring. Despite this, the simulations range from 64 days to 216 years. If there is a rate limited process, would the limited characterization that is possible in field settings be able to detect it over the time ranges used for sampling? Extrapolation of bench data is even more complex, as the longer column experiments last about 150 days, and by their very nature solid phase cores represent a point measurement in time. In all of the field site studies presented here, kinetic formulations were not necessary to fit the observed data. It is unclear as to whether this is related to the assumptions made by the modelers, that the field sites all had conditions consistent with the local equilibrium assumption, or whether kinetic effects, if any, were below an effective ‘detection limit’ for field scale characterization.

7. Discussion In the physical hydrology literature, observations of scale dependence have led to many mathematical methods that either scale a specific parameter based on certain conditions (112-114) or scale field measurable parameters to a calculated ‘ensemble’ measurement that can be used in transport codes (110, 115-118). It has also been noted that the best scaling methodologies for determining bulk flow may not be the best scaling method for mass transport (1). Thus, each parameter in a transport code may need an independent scaling calculation. For RTM, this process would be daunting to carry out. An alternative approach has been to find the actual size of the conceptual representative elementary volume (REV, e.g. refs 119-122). At its simplest, the REV is defined to exist when increasing the sampling volume no longer changes the measured parameter. For porosity and bulk density the REV volume may range from less than 10 mL up to several hundred (122). For model variables that are not directly measurable (i.e., dispersion), finding an exact size for a REV is far more difficult and may not be possible (54). In the geochemical literature, the concept of a REV is rarely introduced; however, for upscaling RTM the size of a 8004

9


‘chemical REV’ may be important. Using the same definition as the physical hydrologists, the ‘chemical REV’ would exist when changing the size of the sample volume no longer changes the chemical behavior of the system. As the ‘chemical REV’ gets larger, more heterogeneities are included leading asymptotically to an effective chemical response. The size of the ‘chemical REV’ may be strongly dependent on the model and mode of observation chosen. For example if the GC model discussed above is considered, the ‘chemical REV’ would exist at a size where the fitted GC parameters do not change as a function of increasing sample size. This sample size would more than likely be different than the ‘chemical REV’ of, for example, dissolution/precipitation; a different model is invoked for different processes. In RTM often a myriad of chemical processes are invoked. In this case the scale of the largest ‘chemical REV’ would have to be used as the discretization cell size. It is to be expected that the size of this volume is at minimum different from that of the physical hydrologist’s REV and at maximum nonexistent. The real size may also be dependent on local flow and mixing heterogeneities. In transport simulations where chemical reactions are considered, the numerical formulation is considerably more complex (123, 124). The state of general knowledge on scaling relationships is far too limited to derive scaling equations for the multitude of variables that occur in RTM. Furthermore, many of the numerical attempts at upscaling chemical reactions in porous media are so complex that model transparency is lost; the ability to discern cause and effect relating to transport is impossible to critically evaluate (91, 125-128). This runs counter to a stated goal of RTM which is to act as “an important set of interpretive tools for unraveling complex interactions between coupled processes and the effects of multiple space and time scales in the Earth” (129). Other authors have separated the modeling goals of explanation and prediction (96, 130). The conceptual model for these two modeling goals and the subsequent numerical coding may be quite different. In explanatory models the goal is to represent a system as simply and realistically as possible such that the input is easy to gather and the output is easy to interpret. Physical meaning is less important for predictive power where calibrated values in mathematical descriptions may suffice to reproduce and predict data. So, what will an upscaled conceptual model for RTM look like? A recent DOE-led workshop considers this from a philosophic perspective (131). In that report the subsurface, in general, and contaminant transport, specifically, is compared to complex systems. Complex systems are characterized by having emergent behavior that does not result from a sum of the resultant parts (131). A basic tenet of modeling complex systems is to focus on a ‘top-down’ approach where fundamental mechanisms are largely ignored and the focus is on the emergent behavior at a larger scale. The scale at which emergent properties begin to exhibit themselves is unknown; indeed, the properties themselves are currently unknown. In order to achieve this and to account for scale in RTM, new experiments are needed at the intermediate scale where physical and chemical heterogeneities can be controlled and/or quantified. This will allow for a ‘bridge’ between bench and field data and may allow for the observation of emergent properties. Once the properties and scales of emergence have been delineated, new conceptual models can be created that will encompass physical/chemical heterogeneities as well as scale related chemical behaviors. It may very well be that our intuition is misleading us and that the behavior of more complex systems can be explained and predicted based on simple models.

Acknowledgments This material is based upon work supported by the Department of Energy under Award Number: DE-FG02-06ER64233. The manuscript benefited greatly from thoughtful considerations of three anonymous reviewers. The manuscript also benefitted from ongoing conversations between the authors and Jim Davis and Gary Curtis of the USGS, Menlo Park, CA.

Literature Cited (1) Scheibe, T. D.; Yabusaki, S. Scaling of flow and transport behavior in heterogeneous groundwater systems. Adv. Water Resour. 1998, 22, 223–238. (2) Mills, R. T.; Hammond, G. E.; Lichtner, P. C.; Sripathi, V.; Mahinthakumar, G.; Smith, B. F. Modeling subsurface reactive flows using leadership-class computing. J. Phys.: Conf. Ser. 2009, 180, 10.1088/1742-6595/180/1/012062. (3) Wersin, P.; Hochella, M. F., Jr.; Perrson, P.; Redden, G.; Leckie, J. O.; Harris, D. W. Interaction between aqueous uranium(VI) and sulfide minerals: Spectroscopic evidence for sorption and reduction. Geochim. Cosmochim. Acta 1994, 58, 2829–2843. (4) Waychunas, G. A.; Fuller, C. C.; Davis, J. A. Surface complexation and precipitate geometry for aqueous Zn(II) sorption on ferrihydrite I. X-ray absorption extended fine structure spectroscopy analysis. Geochim. Cosmochim. Acta 2002, 66, 1119– 1137. (5) Towle, S. N.; Bargar, J. R.; Brown, G. E., Jr.; Parks, G. A. Surface precipitation of Co(II)(aq) on Al2O3. J. Colloid Interface Sci. 1997, 187, 62–82. (6) Kalmykov, S. N.; Choppin, G. R. Mixed Ca2+/UO2+/CO32complex formation at different ionic strengths. Radiochim. Acta 2000, 88, 603–606. (7) Bernhard, G.; Geipel, G.; Reich, T.; Brendler, V.; Amayri, S.; Nitsche, H. Uranyl(VI) carbonate complex formation: Validation of the Ca2UO2(CO3)3(aq) species. Radiochim. Acta 2001, 89, 511–518. (8) Zheng, Z.; Tokunaga, T. K.; Wan, J. Influence of calcium carbonate on U(VI) sorption to soils. Environ. Sci. Technol. 2003, 37, 5603–5608. (9) Dong, W.; Ball, W. P.; Liu, C.; Wang, Z.; Stone, A. T.; Bai, J.; Zachara, J. M. Influence of calcite and dissolved calcium on uranium(VI) sorption to a Hanford subsurface sediment. Environ. Sci. Technol. 2005, 39, 7949–7955. (10) Fox, P.; Davis, J. A.; Zachara, J. M. The effect of calcium on aqueous uranium(VI) speciation and adsorption to ferrihydrite and quartz. Geochim. Cosmochim. Acta 2006, 70, 1379–1387. (11) Kohler, M.; Curtis, G. P.; Kent, D. B.; Davis, J. A. Experimental investigation and modeling of uranium(VI) transport under variable chemical conditions. Water Resour. Res. 1996, 32, 3539– 3551. (12) Morrison, S. J.; Spangler, R. R.; Tripathi, V. S. Adsorption of uranium(VI) on amorphous ferric oxyhydroxide at high concentrations of dissolved carbon(VI) and sulfur(VI). J. Contam. Hydrol. 1995a, 17, 333–346. (13) Morrison, S. J.; Tripathi, V. S.; Spangler, R. R. Coupled reaction/ transport modeling of a chemical barrier for controlling uranium(VI) contamination in groundwater. J. Contam. Hydrol. 1995b, 17, 347–363. (14) Kohler, M.; Honeyman, B. D.; Leckie, J. O. Neptunium(V) sorption on hematite (alpha-Fe2O3) in aqueous suspension: The effect of CO2. Radiochim. Acta 1999, 85, 33–48. (15) Zachara, J. M.; Girvin, D. C.; Schmidt, R. L.; Resch, C. T. Chromate adsorption on amorphous iron oxyhydroxide in the presence of major groundwater ions. Environ. Sci. Technol. 1987, 21, 589–594. (16) Swedlund, P. J.; Webster, J. G.; Miskelly, G. M. Goethite adsorption of Cu(II), Pb(II), Cd(II), and Zn(II) in the presence of sulfate: Properties of the ternary complex. Geochim. Cosmochim. Acta 2009, 73, 1548–1562. (17) Zhang, G. Y.; Peak, D. Studies of Cd(II)-sulfate interactions at the goethite-water interface by ATR-FTIR spectroscopy. Geochim. Cosmochim. Acta 2007, 71, 2158–2169. (18) Van Geen, A.; Robertson, A. P.; Leckie, J. O. Complexation of carbonate species at the goethite surface: Implications for adsorption of metal ions in natural waters. Geochim. Cosmochim. Acta 1994, 58, 2073–2086. (19) Lenhart, J. J.; Honeyman, B. D. Uranium(VI) sorption to hematite in the presence of humic acid. Geochim. Cosmochim. Acta 1999, 63, 2891–2901. (20) Schmeide, K.; Pompe, S.; Bubner, M.; Heise, K. H.; Bernhard, G.; Nitsche, H. Uranium(VI) sorption onto phyllite and selected

(21) (22)

(23) (24) (25) (26)

(27)

(28) (29) (30)

(31) (32)

(33) (34)

(35) (36) (37) (38)

(39)

(40)

(41) (42)

(43)

minerals in the presence of humic acid. Radiochim. Acta 2000, 88, 723–728. Weng, L.; VanRiemsdijk, W. H.; Hiemstra, T. Cu2+ and Ca2+ adsorption to goethite in the presence of fulvic acids. Geochim. Cosmochim. Acta 2008, 72, 5857–5870. Honeyman, B. D.; Ranville, J. F. Colloid properties and their effects on radionuclide transport through soils and groundwater. In Geochemistry of Soil Radionuclides; Zhang, P.-C., Brady, M. V., Eds.; SSSA Special Publication 59; Soil Science Society of America: Madison, 2002; pp 131-163. Hohl, H.; Stumm, W. Interaction of Pb2+ with hydrous gammaAl2O3. J. Colloid Interface Sci. 1976, 55, 281–288. Farley, K. J.; Dzombak, D. A.; Morel, F. M. M. A surface precipitation model for the sorption of cations on metal oxides. J. Colloid Interface Sci. 1985, 106, 226–242. Wersin, P.; Charlet, L.; Karthein, R.; Stumm, W. From adsorption to precipitation: Sorption of Mn2+ on FeCO3(s). Geochim. Cosmochim. Acta 1989, 53, 2787–2796. Davis, J. A.; James, R. O.; Leckie, J. O. Surface ionization and complexation at the oxide/water interface I. Computation of electrical double layer properties in simple electrolytes. J. Colloid Interface Sci. 1978a, 63, 480–499. Davis, J. A.; Leckie, J. O. Surface ionization and complexation at the oxide/water interface II. Surface properties of amorphous iron oxyhydroxide and adsorption of metal ions. J. Colloid Interface Sci. 1978b, 67, 90–107. Davis, J. A.; Leckie, J. O. Surface ionization and complexation at the oxide/water interface III. Adsorption of anions. J. Colloid Interface Sci. 1980, 74, 32–43. Hiemstra, T.; van Riemsdijk, W. H. A surface structural approach to ion adsorption: The charge distribution (CD) model. J. Colloid Interface Sci. 1996, 179, 488–508. Arai, Y.; McBeath, M.; Bargar, J. R.; Joye, J.; Davis, J. A. Uranyl adsorption and surface speciation at the imogolite-water interface: Self-consistent spectroscopic and surface complexation models. Geochim. Cosmochim. Acta 2006, 70, 2492–2509. Ponthieu, M.; Juillot, F.; Hiemstra, T.; van Riemsdijk, W. H.; Benedetti, M. F. Metal ion binding to iron oxides. Geochim. Cosmochim. Acta 2006, 70, 2679–2698. Weber, W. J., Jr.; McGinley, P. M.; Katz, L. E. A distributed reactivity model for sorption by soils and sediments 1. Conceptual basis and equilibrium assessments. Environ. Sci. Technol. 1992, 26, 1955–1962. Chen, W.; Wagenet, R. J. Solute transport in porous media with sorption-site heterogeneity. Environ. Sci. Technol. 1995, 29, 2725–2734. Qafoku, N. P.; Zachara, J. M.; Liu, C.; Gassman, P. L.; Qafoku, O. S.; Smith, S. C. Kinetic desorption and sorption of U(VI) during reactive transport in a contaminated hanford sediment. Environ. Sci. Technol. 2005, 39, 3157–3165. Haggerty, R.; Gorelick, S. M. Multiple-rate mass transfer for modeling diffusion and surface reactions in media with porescale heterogeneity. Water Resour. Res. 1995, 31, 2383–2400. Liu, C.; Zachara, J. M.; Qafoku, N. P.; Wang, Z. Scale-dependent desorption of uranium from contaminated subsurface environments. Water Resour. Res. 2008, 44, 10.1029/2007WR006478. Davis, J. A.; Coston, J. A.; Kent, D. B.; Fuller, C. C. Application of the surface complexation concept to complex mineral assemblages. Environ. Sci. Technol. 1998, 32, 2820–2828. Honeyman, B. D. Cation and Anion Adsorption in Binary Mixtures of Adsorbents: An Investigation of the Concept of Adsorptive Additivity. Ph.D. Dissertation, Stanford University, Stanford, CA, 1984. Davis, J. A.; Meece, D. E.; Kohler, M.; Curtis, G. P. Approaches to surface complexation modeling of uranium(VI) adsorption on aquifer sediments. Geochim. Cosmochim. Acta 2004, 68, 3621–3641. Payne, T. E.; Davis, J. A.; Ochs, M.; Olin, M.; Tweed, C. J.; Altmann, S.; Askarieh, M. M. Comparative evaluation of surface complexation models for radionuclide sorption by diverse geologic material. In Surface Complexation Modeling; Lutzenkirchen, J., Ed.; Elsevier Ltd.: Amsterdam, 2006; Vol. 11, pp 605-633. Waite, T. D.; Davis, J. A.; Fenton, B. R.; Payne, T. E. Approaches to modeling uranium(VI) adsorption on natural mineral assemblages. Radiochim. Acta 2000, 88, 687–693. Turner, D. R.; Bertetti, F. P.; Pabalan, R. T. Applying surface complexation modeling to radionuclide sorption. In Surface Complexation Modeling; Lutzenkirchen, J. , Ed.; Elsevier Ltd.: Amsterdam, 2006; Vol. 11, pp 553-604. Curtis, G. P.; Davis, J. A.; Naftz, D. L. Simulation of reactive transport of uranium(VI) in groundwater with variable chemi-


9

8005

(44) (45) (46)

(47)

(48)

(49)

(50) (51) (52) (53)

(54) (55)

(56) (57)

(58)

(59) (60)

(61) (62) (63) (64)

(65)

(66)

8006

9

calconditions.WaterResour.Res.2006,42,W04404,04410.01029/ 02005WR003979. Stollenwerk, K. G. Modeling the effects of variable groundwater chemistry in adsorption of molybdate. Water Resour. Res. 1995, 31, 347–357. Stollenwerk, K. G. Molybdate transport in a chemically complex aquifer: Field measurements compared with solute-transport model predictions. Water Resour. Res. 1998, 34, 2727–2740. Kent, D. B.; Abrams, R. H.; Davis, J. A.; Coston, J. A.; LeBlanc, D. R. Modeling the influence of variable pH on the transport of zinc in a contaminated aquifer using semiempirical surface complexation models. Water Resour. Res. 2000, 36, 3411–3425. Kent, D. B.; Davis, J. A.; Anderson, C. D.; Rea, B. A. Transport of chromium and selenium in a pristine sand and gravel aquifer: Role of adsorption processes. Water Resour. Res. 1995, 31, 1041– 1050. Davis, J. A.; Kent, D. B.; Coston, J. A.; Hess, K. M.; Joye, J. L. Multispecies reactive tracer test in an aquifer with spatially variable chemical conditions. Water Resour. Res. 2000, 36, 119– 134. Brusseau, M. L.; Jessup, R. E.; Rao, P. S. C. Modeling solute transport influenced by multiprocess nonequillibrium and transformation reactions. Water Resour. Res. 1992, 28, 175– 182. Neville, C. J.; Ibaraki, M.; Sudicky, E. A. Solute transport with mutiprocess nonequilibrium: a semi-analytial soution approach. J. Contam. Hydrol. 2000, 44, 141–159. Li, L.; Barry, D. A.; Culligan-Hensley, P. J.; Bajracharya, K. Mass transfer in soils with local stratification of hydraulic conductivity. Water Resour. Res. 1994, 30, 2891–2900. Griffioen, J. W.; Barry, D. A.; Parlange, J.-Y. Interpretation of two-region model parameters. Water Resour. Res. 1998, 34, 373–384. Brusseau, M. L. Transport of rate-limited sorbing solutes in heterogeneous porous media: Application of a one-dimensional multifactor nonideality model to field data. Water Resour. Res. 1992, 28, 2485–2497. Gelhar, L. W.; Welty, C.; Rehfeldt, K. R. A critical review of data on field-scale dispersion in aquifers. Water Resour. Res. 1992, 28, 1955–1974. Beckie, R. Analysis of scale effects in large-scale solute-transport models In Scale Dependence and Scale Invariance in Hydrology; Sposito, G., Ed.; Cambridge University Press: Cambridge, UK, 1998; pp 314-334. Brusseau, M. L.; Zachara, J. M. Transport of Co2+ in a physically and chemically heterogeneous porous medium. Environ. Sci. Technol. 1993, 27, 1937–1939. Liu, C.; Zachara, J. M.; Yantasee, W.; Majors, P. D.; McKinley, J. P. Microscopic reactive diffusion of uranium in the contaminated sediments at Hanford, United States. Water Resour. Res. 2006, 42. Liu, C.; Zachara, J. M.; Qafoku, O. S.; McKinley, J. P.; Heald, S. M.; Wang, Z. Dissolution of uranyl microprecipitates in subsurface sediments at Hanford Site, USA. Geochim. Cosmochim. Acta 2004, 68, 4519–4537. White, A. F.; Brantley, S. L. The effect of time weathering of silicate minerals: why do weathering rates differ in the laboratory and the field. Chem. Geol. 2003, 202, 479–506. Maher, K.; Steefel, C. I.; DePaolo, D. J.; Viani, B. E. The mineral dissolution rate conundrum: Insights from reactive transport modeling of U isotopes and pore fluid chemistry in marine sediments. Geochim. Cosmochim. Acta 2006, 70, 337–363. Li, L.; Peters, C. A.; Celia, M. A. Upscaling geochemical reaction rates using pore-scale network modeling. Adv. Water Resour. 2006, 29, 1351–1370. Li, L.; Peters, C. A.; Celia, M. A. Effects of mineral spatial distribution on reaction rates in porous media. Water Resour. Res. 2007, 43, W01419, 01410.01029/02005WR004848. Pang, L.; Close, M.; Schneider, D.; Stanton, G. Effect of porewater velocity on chemical nonequilibrium transport of Cd, Zn, and Pb. J. Contam. Hydrol. 2002, 57, 241–258. Zhang, X.; Crawford, J. W.; Young, I. M. Does pore water velocity affect the reaction rates of adsorptive solute transport in soils? Demonstration with pore-scale modeling. Adv. Water Resour. 2008, 31, 425–437. Acharya, R. C.; Valocchi, A. J.; Werth, C. J.; Willingham, T. W. Pore-scale simulation of dispersion and reaction along a transverse mixing zone in two-dimensional porous media. Water Resour. Res. 2007, 43, 10.1029/2007WR005969. Cirpka, O. A.; Kitanidis, P. K. Characterization of mixing and dilution in heterogeneous aquifers by means of local temporal moments. Water Resour. Res. 2000, 36 (5), 1221–1236. ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 44, NO. 21, 2010

(67) Li, L.; Steefel, C. I.; Yang, L. Scale dependence of mineral dissolution rates within single pores and fractures. Geochim. Cosmochim. Acta 2008, 72, 360–377. (68) Werth, C. J.; Cirpka, O. A.; Grathwohl, P. Enhanced mixing and reaction through flow focusing in heterogeneous porous media. Water Resour. Res. 2006, 42, 10.1029/2005WR004511. (69) Tartakovsky, D. M.; Redden, G.; Lichtner, P. C.; Scheibe, T. D.; Meakin, P. Mixing-induced precipitation: Experimental study and multiscale numerical analysis. Water Resour. Res. 2008, 44, 10.1029/2006WR005725. (70) Li, L.; Steefel, C. I.; Kowalsky, M. B.; Englert, A.; Hubbard, S. S. Effects of physical and geochemical heterogeneities in mineral transformation and biomass accumulation during biostimulation experiments at Rifle, Colorado. J. Contam. Hydrol. 2010, 112, 45–63. (71) Burr, D. T.; Sudicky, E. A.; Naff, R. L. Nonreactive and reactive solute transport in three-dimensional heterogeneous porous media: Mean displacement, plume spreading, and uncertainty. Water Resour. Res. 1994, 30, 791–815. (72) Espinoza, C.; Valocchi, A. J. Stochastic analysis of onedimensional transport of kinetically adsobing solutes in chemically heterogeneous aquifers. Water Resour. Res. 1997, 33, 2429–2445. (73) Harvey, C.; Gorelick, S. M. Rate-limited mass transfer or macrodispersion: Which dominates plume evolution at the Macrodispersion Experiment (MADE) site. Water Resour. Res. 2000, 36, 637–650. (74) Wood, B. The role of scaling laws in upscaling. Adv. Water Resour. 2009, 32, 723–736. (75) Barnett, M. O.; Jardine, P. M.; Brooks, S. C.; Selim, H. M. Adsorption and transport of uranium(VI) in subsurface media. Soil Sci. Soc. Am. J. 2000, 64, 908–917. (76) Gabriel, U.; Gaudet, J.-P.; Spadini, L.; Charlet, L. Reactive transport of uranyl in a goethite column: an experimental and modeling study. Chem. Geol. 1998, 151, 107–128. (77) Szecsody, J. E.; Zachara, J. M.; Chilakapati, A.; Jardine, P. M.; Ferrency, A. S. Importance of flow and particle-scale heterogeneity on Co(II/III) EDTA reactive transport. J. Hydrol. 1998, 209, 112–136. (78) Appelo, C. A. J.; Van Loon, L. R.; Wersin, P. Multicomponent diffusion of a suite of tracers (HTO, Cl, Br, I, Na, Sr, Cs) in a single smaple of Oplinus Clay. Geochim. Consmochim. Acta 2010, 74, 1201–1219. (79) Zachara, J. M.; Smith, S. C.; Liu, C.; McKinley, J. P.; Serne, R. J.; Gassman, P. L. Sorption of Cs+ to micaceous subsurface sediments from the Hanford site, USA. Geochim. Cosmochim. Acta 2002, 66, 193–211. (80) Steefel, C. I.; Carroll, S.; Zhao, P.; Roberts, S. Cesium migration in Hanford sediment: a multisite cation exchange model based on laboratory experiments. J. Contam. Hydrol. 2003, 67, 219– 246. (81) Davis, J. A.; Curtis, G. P. Application of Surface Complexation Modeling to Describe Uranium(VI) Adsorption and Retardation at the Uranium Mill Tailings Site at Naturita, Colorado; NUREG/CR-6820; U.S. Nuclear Regulatory Commission: Washington, DC, 2003. (82) Fuller, C. C.; Davis, J. A.; Coston, J. A.; Dixon, E. Characterization of metal adsorption variability in a sand and gravel aquifer, Cape Cod, Massachusetts, U.S.A. J. Contam. Hydrol. 1996, 22, 165–187. (83) Loganathan, V. A.; Barnett, M. O.; Clement, T. P.; Kanel, S. R. Scaling of adsorption reactions: U(VI) experiments and modeling. Appl. Geochem. 2009, 24, 2051–2060. (84) Arnold, T.; Zorn, T.; Bernhard, G.; Nitsche, H. Sorption of uranium(VI) onto phyllite. Chem. Geol. 1998, 151, 129–141. (85) Arnold, T.; Zorn, T.; Zanker, H.; Bernhard, G.; Nitsche, H. Sorption behavior of U(VI) of phyllite: experiments and modeling. J. Contam. Hydrol. 2001, 47, 219–231. (86) Lutzenkirchen, J.; Boily, J.-F.; Lovgren, L.; Sjoberg, S. Limitations of the potentiometric titration technique in determining the proton active site density of goethite surfaces. Geochim. Cosmochim. Acta 2002, 66, 3389–3396. (87) Payne, T. E.; Edis, R.; Fenton, B. R.; Waite, T. D. Comparison of laboratory uranium sorption data with ‘in situ distribution coefficients’ at the Koongarra uranium deposit, Northern Australia. J. Environ. Radioact. 2001, 57, 35–55. (88) Curtis, G. P.; Fox, P.; Kohler, M.; Davis, J. A. Comparison of in situ KD values with a laboratory determined surface complexation model. Appl. Geochem. 2004, 19, 1643–1653. (89) Prikryl, J. D.; Jain, A.; Turner, D. R.; Pabalan, R. T. UraniumVI sorption behavior on silicate mineral mixtures. J. Contam. Hydrol. 2001, 47, 241–253.

(90) Bertetti, F. P.; Pabalan, R. T.; Almendarez, M. G. Studies of neptunium(V) sorption on quartz, clinoptilolite, montmorillonite, and alpha-alumina. In Adsorption of Metals by Geomedia; Jenne, E. A., Ed.; Academic Press: San Diego, 1998; pp 131-148. (91) Oreskes, N. Why Believe a Computer? Models, Measures, and Meaning in the Natural World. In The Earth Around Us: Maintaining a Livable Planet; Schneiderman, J. S., Ed.; W.H. Freeman and Co.: San Francisco, 2000; pp 70-82. (92) Kantar, C.; Honeyman, B. D. Citric acid enhanced remediation of solids contaminated with uranium by soil flushing and soil washing. J. Environ. Eng. 2006, 132, 247–255. (93) Neiss, J.; Stewart, B. D.; Nico, P. S.; Fendorf, S. Speciationdependant microbial reduction of uranium within iron-coated sands. Environ. Sci. Technol. 2007, 41, 7343–7348. (94) Senko, J. M.; Istok, J. D.; Suflita, J. M.; Krumholz, L. R. In-Situ evidence for uranium immobilization and remobilization. Environ. Sci. Technol. 2002, 36, 1491–1496. (95) Boggs, J. M.; Young, S. C.; Beard, L. M.; Gelhar, L. W.; Rehfeldt, K. R.; Adams, E. E. Field study of dispersion in a heterogeneous aquifer 1. Overview and site description. Water Resour. Res. 1992, 28, 3281–3291. (96) Beven, K. How far can we go in distributed hydrological modelling. Hydrol. Earth Syst. Sci. 2001a, 5, 1–12. (97) Bryant, S. L.; Thompson, K. E. Theory, modeling and experiment and reactive transport in porous media. Curr. Opin. Colloid Interface Sci. 2001, 6, 217–222. (98) Beven, K. On explanatory depth and predictive power. Hydrol. Processes 2001b, 15, 3069–3071. (99) Fernandez-Garcia, D.; Sanchez-Vila, X.; Illangasekare, T. H. Convergent-flow tracer tests in heterogeneous media: combined experimental-numerical analysis for determination of equivalent transport parameters. J. Contam. Hydrol. 2002, 57, 129–145. (100) Chao, H.-C.; Rajaram, H.; Illangasekare, T. H. Intermediatescale experiments and numerical simulations of transport under radial flow in a two-dimensional heterogeneous porous media. Water Resour. Res. 2000, 36, 2869–2884. (101) Barth, G. R.; Hill, M. C.; Illangasekare, T. H.; Rajaram, H. Predictive modeling of flow and transport in a two-dimensional intermediate-scale, heterogeneous porous medium. Water Resour. Res. 2001a, 37, 2503–2512. (102) Barth, G. R.; Illangasekare, T. H.; Hill, M. C.; Rajaram, H. A new tracer-density criterion for heterogeneous porous media. Water Resour. Res. 2001b, 37, 21–31. (103) Levy, M.; Berkowitz, B. Measurement and analysis of nonFickian dispersion in heterogeneous porous media. J. Contam. Hydrol. 2003, 64, 203–226. (104) Illangasekare, T. H.; Ramsey, J. L., Jr.; Jensen, K. H.; Butts, M. B. Experimental study of movement and distribution of dense organic contaminants in heterogeneous aquifers. J. Contam. Hydrol. 1995, 20, 1–25. (105) Heiderscheidt, J. L.; Siegrist, R. L.; Illangasekare, T. H. Intermediate-scale 2D experimental investigation of in-situ chemical oxidation using potassium permanganate for remediation of complex DNAPL sources. J. Contam. Hydrol. 2008, 102, 3–16. (106) Zhu, C.; Hu, F. Q.; Burden, D. S. Multi-component reactive transport modeling of natural attenuation of an acid groundwater plume at a uranium mill tailings site. J. Contam. Hydrol. 2001, 52, 85–108. (107) Van Cappellen, P. Reactive surface area control of the dissolution kinetics of biogenic silica in deep-sea sediments. Chem. Geol. 1996, 132, 125–130. (108) Lichtner, P. C.; Tartakovsky, D. M. Stochastic analysis of effective rate constant for heterogeneous reactions. Stochastic Environ. Res. Risk Assess. 2003, 17, 419–429. (109) Meile, C.; Tuncay, K. Scale dependence of reaction rates in porous media. Adv. Water Resour. 2006, 29, 62–71.

(110) Rajaram, H. Time and scale dependent effective retardation factors in heterogeneous systems. Adv. Water Resour. 1997, 20, 217–230. (111) Haggerty, R.; Harvey, C.; von Schwerin, C. F.; Meigs, L. C. What controls the apparent time-scale of solute mass transfer in aquifers and soils? A comparison of experimental results. Water Resour. Res. 2004, 40, 10.1029/2002WR001716. (112) Renard, P.; de Marsily, G. Calculating equivalent permeability: a review. Adv. Water Resour. 1997, 20, 253–278. (113) Gasda, S. E.; Celia, M. A. Upscaling relative permeabilities in a structured porous medium. Adv. Water Resour. 2005, 28, 493–506. (114) Wood, B. D.; Cherblanc, F.; Quintard, M.; Whitaker, S. Volume averaging for determining the effective dispersion tensor: Closure using periodic unit cells and comparison with ensemble averaging. Water Resour. Res. 2003, 39, 1210. (115) Cushman, J. H. On Measurement, Scale, and Scaling. Water Resour. Res. 1986, 22, 129–134. (116) Dagan, G. Statistical theory of groundwater flow and transport: Pore to laboratory, laboratory to formation, formation to regional scale. Water Resour. Res. 1986, 22, 120S–134S. (117) Neuman, S. P.; di Federico, V. Correlation, Flow, and Transport in Multiscale Permeability Fields. In Scale Dependence and Scale Invariance in Hydrology; Sposito, G., Ed.; Cambridge University Press: Cambridge, 1998; pp 354-397. (118) Selroos, J.-O.; Cvetkovic, V. Modeling solute advection coupled with sorption kinetics in heterogeneous formations. Water Resour. Res. 1992, 28, 1271–1278. (119) Baveye, P.; Rogasik, H.; Wendroth, O.; Onasch, I.; Crawford, J. W. Effect of sampling volume on the measurement of soil physical properties: simulation with x-ray tomography data. Meas. Sci. Technol. 2002, 13, 775–784. (120) Brown, G. O.; Hsieh, H. T.; Lucero, D. A. Evaluation of laboratory dolomite core sample size using representative elementary volume concepts. Water Resour. Res. 2000, 36, 1199–1207. (121) Zhang, D.; Zhang, R.; Chen, S.; Soll, W. E. Pore scale study of flow in porous media: Scale dependency, REV, and statistical REV. Geophys. Res. Lett. 2000, 27, 1195–1198. (122) Tidwell, V. C.; Wilson, J. L. Upscaling experiments conducted on a block of volcanic tuff: Results for a bimodal permeability distribution. Water Resour. Res. 1999, 35, 3375–3387. (123) Rubin, J. Transport of reacting solutes in porous media: Relation between mathematical nature of problem formulation and chemical nature of reactions. Water Resour. Res. 1983, 19, 1231– 1252. (124) Quintard, M.; Whitaker, S. Transport in chemically and mechanically heterogeneous porous media IV: large-scale mass equilibrium for solute transport with adsorption. Adv. Water Resour 1998, 22, 33–57. (125) Konikow, L. F.; Bredehoeft, J. D. Ground-water models cannot be validated. Adv. Water Resour. 1992, 15, 75–83. (126) Bredehoeft, J. D. From models to performance assessment: The conceptualization problem. Ground Water 2003, 41, 571– 577. (127) Oreskes, N.; Belitz, K. Philosophical Issues in Model Assessment. In Model Validation: Perspectives in Hydrological Science; John Wiley and Sons, Ltd.: 2001; pp 23-41. (128) Oreskes, N.; Shrader-Frechette, K.; Belitz, K. Verification, validation, and confirmation of numerical models in the earth sciences. Science 1994, 263, 641–646. (129) Steefel, C. I.; DePaolo, D. J.; Lichtner, P. C. Reactive transport modeling: An essential tool and a new research approach for the Earth sciences. Earth Planet. Sci. Lett. 2005, 240, 539–558. (130) Murray, A. B. Reducing model complexity for explanation and prediction. Geomorphology 2007, 90, 178–191. (131) U.S. DOE 2010. Complex Systems Science for Subsurface Flow and Transport: Report from the August 2009 Workshop; DOE/ SC-0123; U.S. Department of Energy Office of Science. www. science.doe.gov/ober/BER_workshops.html (accessed March 30th, 2010).

ES101822V


9

8007

Desorption Processes in Reactive Transport

Recommend Documents