Published online 26 May 2006
Published in Vadose Zone J 5:777-783 (2006)
DOI: 10.2136/vzj2005.0145
© 2006 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
Characterization of the Specific Solid–Water Interfacial Area–Water Saturation Relationship and Its Import to Reactive Transport
K. P. Saripalli*,
V. L. Freedman,
B. P. McGrail and
P. D. Meyer
Pacific Northwest National Lab., Richland, WA 99352
* Corresponding author (prasad.saripalli{at}pnl.gov)
Received 13 December 2005.
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ABSTRACT
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A modified rate equation for the dissolution of minerals in the unsaturated zone was proposed using a specific solid–water interfacial area (asw) varying as a function of water saturation (Sw). Vadose zone phenomena, such as infiltration, were discussed as example scenarios where such reformulation may be useful. A constitutive relationship for the asw–Sw was derived, and used to formulate an approximate model of the proposed rate equation. Implementation of this new model in the reactive transport code CRUNCH was used to evaluate the influence of a changing asw on reactive transport behavior of silica in the vadose zone. These simulations clearly demonstrate that a changing asw can have a significant impact on the silica distribution profiles in the vadose zone. This is to be expected because changes in asw are substantial in the dry Sw range, whereas they are only moderate in the relatively wet region. These findings have significant implications for contaminant transport in the vadose zone, especially in arid regions. Ignoring the effect of a changing asw is likely to result in an overestimation of the net contaminant fluxes in these cases, which leads to highly conservative risk assessments. In case of nutrient flux calculations coupled with dissolution phenomena, it may lead to an overestimation of nutrient availability. Additional theoretical and experimental verification of the proposed models would be valuable.
Abbreviations: ILAW, immobilized low activity waste
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INTRODUCTION
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MODELING reactive transport through unsaturated porous media is a demanding problem, due to the constantly changing physical and chemical conditions throughout the domain (Steefel and MacQuarrie, 1996). Even though the reactivity of solution with mineral surfaces in an unsaturated medium is a strong function of the specific solid–water contact area (asw) and water saturation (Sw), currently available geochemical kinetic formulations assume a constant mineral– water contact area (Lichtner, 1996; Rimstidt and Barnes, 1980). As a result, the presently available reactive transport codes do not adequately model the dependence of kinetics on asw, for want of information on the asw–Sw relationship and kinetic formulations that are sensitive to this relationship.
Several natural and engineered systems experience dynamic changes in asw. For example, water content of an unsaturated porous medium can change with time due to sources, sinks, and a change in the water retention characteristics of the porous medium itself. Rainfall recharge and transpiration in the root zone are well-known examples of sources and sinks of water. Such variation in water content is expected in the soils hosting the ILAW (immobilized low activity waste) glass forms at the Hanford waste site. Glass dissolution experiments in pressurized unsaturated flow systems on Hanford media revealed that water content can change significantly with time due to changes in wettability status of the glass surfaces mediated by the deposition of hydrated glass alteration phases (McGrail et al., 2001). Aquifers containing nonaqueous-phase liquids and petroleum reservoirs can experience wettability reversal (Hirasaki, 1988), which may lead to changes in asw. As such, experimental and theoretical investigations on the nature of the kinetic rate laws applicable to variably saturated soils and sediments, and development of reactive transport models including the influence of water saturation and the solid–water interfacial area on the rate laws, constitute a critical research need. No theory, models, methods, or experimental data are currently available to address this need.
We have developed a theory and constitutive relationships to predict the changes in asw (m2 m–3) as a function of Sw in unsaturated media. We defined asw to be the specific area of the solid–mobile (advective) water interface, ignoring the immobile, molecularly thin film contributions to reactive transport, which may be insignificant (Iwata et al., 1988; Bryant and Johnson, 2003). By implementing the asw–Sw relationship so developed in a simple reactive transport model, we assesses the influence of a changing asw on reactive transport in unsaturated media. Using the kinetics of silica dissolution as an analog, we then derived rate laws applicable to variably saturated soils.
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DERIVATION OF A RATE MODEL FOR VARIABLY SATURATED SOILS
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A rate equation for the dissolution of a mineral in solute i is (Lasaga, 1981; Hegelson et al., 1984; Rimstidt and Barnes, 1980)
 | [1] |
where ni = number of moles of i released into solution, Sm = mineral surface area (cm2), 
= stoichiometric factor (taken as 1 here), k+ = forward (i.e., dissolution) rate constant (moles cm–2 s–1), aH+ = activity of hydrogen ions, t = time, Q = activity product of aqueous species involved in the reaction, 
= a power law coefficient, and Km = equilibrium constant for the precipitation–condensation reaction.
Equation [1] is popularly used in analytical and numerical models that simulate reactive transport through the vadose zone. Derivation of Eq. [1] is based on transition-state theory (Eyring, 1935), which requires that the system of heterogeneous reactions have no limits on the mass of water in which H4SiO4 is dissolved or the interfacial area between the reacting solid and the aqueous solution. As such, the reaction kinetics described by Eq. [1] require that the rate of reaction between two phases is directly proportional to the total solid–water interfacial area (A) between the phases and inversely proportional to the mass of water (M) in the system (Rimstidt and Barnes, 1980). These requirements are met in water-saturated porous media where A and M are constant; however, in unsaturated porous media, especially in variably saturated systems with temporal changes in water content due to sources (e.g., seepage in arid surface soils such as the Hanford site) or sinks (e.g., the soil root zone), neither of these requirements are likely to be satisfied. Thus, Eq. [1] may not be directly applicable, as written, for such transient conditions in unsaturated systems.
Consider a unit volume of an unsaturated sand or glass pack of volume V, containing a water of mass M and total solid–water interfacial area A. The asw is equal to A per unit volume of porous medium. Water content (
) and Sw in the unit volume are defined, respectively, as M/V and M/V
, assuming the density of soil solution to be 1, and to be porosity. Water content, saturation, and interfacial area are defined as time-dependent variables. For example, glass dissolution experiments showed oscillation in water content with time due to the dynamics of precipitation of hydrated glass alteration phases (McGrail et al., 2001). In arid soils where rainfall events are sparse, transient conditions are expected to be the norm (Hillel, 1990).
Under these transient conditions, the rate of silica dissolution can be described by the following equations (Rimstidt and Barnes, 1980):
 | [2] |
and
 | [3] |
Here, P and T refer to standard temperature and pressure, t refers to time, m represents molar concentration, and k– is the backward rate constant. Combining Eq. [2] and [3], noting M = SwV, and rearranging terms, one obtains:
 | [4] |
Noting that activity aH4SiO4 = 
H4SiO4mH4SiO4 where the activity coefficient (H4SiO4) is a constant, one writes the differential rate equation for the silica–water reaction in variably saturated systems in terms of activities as
 | [5] |
Apparent forward and backward rate constants are defined, pooling all the constants together, respectively, as
 | [6] |
 | [7] |
Upon substitution of Eq. [6] and [7] into Eq. [5], the rate of silica dissolution is obtained as
 | [8] |
In Eq. [8], A and Sw are time-dependent variables. In the recent past, modeling approaches were reported relating interfacial areas and Sw for wetting and nonwetting conditions in unsaturated porous media, focusing primarily on the immiscible fluid interfacial areas (Or and Tuller, 1999; Reeves and Celia, 1996; Saripalli, 1996; Bradford and Leij, 1997). Considering the A and Sw relationship, Gvirtzman and Roberts (1991) and Bradford and Leij (1997) suggested that the solid–water interfacial area will be a strong function of system wettability and residual saturation. A representative formulation relating A–Sw for truly water-wet systems may be given as (Saripalli, 1996)
 | [9] |
where b and n are empirical constants related to the soil texture and structure. More detailed development of the A and Sw relationship are discussed below. Equation [9] at this stage serves to illustrate the influence of a variable specific solid–water interfacial area on the kinetic rate law. Upon substitution of Eq. [9], Eq. [8] can be written in terms of water saturation as
 | [10] |
Equation [10] serves to illustrate that, when the solid–water interfacial area is not considered a constant, recognizing its dependence on water saturation, the resulting rate equation is very different from Eq. [1], which is currently routinely used in models that simulate reactive transport through the vadose zone.
Temporal Changes in Water Retention
The water content of an unsaturated porous medium can change with time due to sources, sinks, and a change in the water retention characteristics of the porous medium itself. Common sources and sinks of water include recharge and transpiration in the root zone. In addition, researchers have found in the recent past (Demond et al., 1994; Bradford and Leij, 1997; McGrail et al., 2001) that chemical changes at the soil solid surfaces can induce alterations in the wettability status of the surfaces, such that the soil’s water retention capacity changes. These changes are reflected in the capillary moisture retention curve of the porous medium. For example, such variation in water content with time was observed in the soils hosting the ILAW glass forms at the Hanford waste site (McGrail et al., 2001), due to the precipitation of hydrated glass alteration phases.
Transient conditions in unsaturated soils are described using the Richards equation:
 | [11] |
where is porosity, h, z, and S refer to hydraulic head, depth dimension, and water source–sink term, respectively (Hillel, 1990; Campbell, 1985). Unsaturated hydraulic conductivity (K) in Eq. [11] is a function of water saturation, and can be modeled using the Brooks–Corey relation:
 | [12] |
where Ks is the saturated hydraulic conductivity and 
is the pore-size distribution index. For low rainfall rates, under unit gradient condition, Eq. [11] can be combined with Eq. [12] as
 | [13] |
Considering Sw to be spatially uniform across the simple unit volume considered in the kinetic formulation above (Eq. [1]), the derivative on the right-hand side of Eq. [13] will be equal to zero. In other words, for the purpose of demonstrating the rate law, we consider only the temporal changes in Sw. This approximation, which ignores the spatial variability in Sw and hence is somewhat unrealistic, allows one to arrive at a dissolution rate law that is sensitive to asw and yet not very difficult to implement. Under these assumptions, Eq. [13] is further simplified:
 | [14] |
which, on integration, yields an expression for the time dependence of Sw:
 | [15] |
where o is residual water content. The source term (S) in Eq. [15] can be taken as a constant recharge rate of i cm yr–1, for example, or a constant root transpiration rate of r cm yr–1. Equation [15] simply relates the effect of a time-varying water source or sink on Sw. More accurate models relating temporal changes in Sw to the source–sink term S may replace Eq. [15] in this development later. A typical capillary pressure–saturation curve describing water retention can be modeled using the Brooks–Corey (1964) function:
 | [16] |
where Sr is residual saturation, So is total saturation, and Pd is air entry pressure. If the initial, residual water content o is small and can be ignored, Eq. [15] reduces to Sw = –it. Then, combining Eq. [10] and [15], the differential rate equation for silica dissolution is obtained:
 | [17] |
It can be seen from Eq. [10] and [17] that a derivation of the kinetic rate law for mineral dissolution from first principles, considering a varying asw, can lead to a complicated formulation that is not easy to implement numerically in reactive transport models. As a first approximation, an asw– Sw formulation of the form given in Eq. [9] may be directly substituted in place of the constant interfacial area (i.e., specific surface Sm) used in the rate law represented by Eq. [1]. While this approach does not fully take into account the influence of a variable asw on the dissolution rate law, as discussed above, it is relatively easier to implement it in all existing reactive transport models simulating vadose zone processes, and represents a reasonable first approximation.
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CHARACTERIZATION OF THE SOLID–WATER INTERFACIAL AREA
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The dependence of asw on Sw itself is not well understood. One reason why the solid–water interfacial area has received little attention is a tacit assumption that soils and subsurface media are always water wet. Even when these media are unsaturated, a thin film of water (a few angstroms thick) is assumed to be covering the entire specific surface of the soil solids. As such, technically the soils are always "water wet"; however, there is considerable evidence to suggest that the molecularly thin water films coating the soil solids are unable to conduct significant fluid flow by advection (Iwata et al., 1988). As such, their role in reactive transport may be limited. It follows that the kinetic formulations for reactive transport in unsaturated media should explicitly consider the influence of the asw–Sw relationship, which omits the thin films’ contributions to asw. Several researchers have reported on modeling the dependence of specific air–water interfacial area (anw) on water content, using thermodynamic, geometric, and mixed empirical approaches (Or and Tuller, 1999; Gvirtzman and Roberts, 1991; Cary, 1994; Reeves and Celia, 1996). A brief review of air–water interfacial areas is useful here to present the development of the proposed solid–water interfacial area (asw–Sw) relationship.
Relating the change in interfacial free energy to the externally imposed capillary pressure, Leverett (1941) obtained an expression for the total interfacial area Anw created during drainage from an initially water saturated medium as
 | [18] |
where 
(dynes cm–1) is the interfacial tension at the immiscible fluid interface. Payne (1953) predicted the specific solid surface areas of glass beads from the Pc–Sw curves using Eq. [18], which compared well with the measured surface areas. Cary (1994) modeled the anw–Sw relationship by assuming a cylindrical non-wetting-phase bubble in a cylindrical pore of radius r and combining the expressions for interfacial area and water content for the pore with a functional relationship between and r based on the Young–Laplace equation. Assuming the solids of revolution formed by identical pendular rings at contact points between spheres as the basic geometric blocks that form the wetting-phase saturation and interfacial area, Gvirtzman and Roberts (1991) presented an anw–Sw relationship. Reeves and Celia (1996) combined geometric and equilibrium thermodynamic approaches via a pore-network model, and concluded that a smooth, albeit complex functional relationship does exist among Pc, Sw, and anw at the continuum scale. Modeling approaches similar to the above, reported for predicting the anw–Sw relationship, may serve as a basis for the prediction of the asw–Sw relationship as well. In this study, two alternative mathematical formulations were developed for the asw–Sw relationship, based on the approaches developed by Rose and Bruce (1949) and Cary (1994).
Rose and Bruce (1949) developed an expression for the solid surface area per unit medium volume as, relating the hydraulic radius of a porous medium (Rh = /as) to the fundamental capillary rise equation (Pc = 2/Rh), as
 | [19] |
where is porosity, is surface tension, and Pc is capillary pressure. Considering an idealized porous medium as a capillary bundle of straight, parallel tubes of varying radii, in which the largest pores empty first during capillary desaturation (Hillel, 1990), and Eq. [19], the specific solid–water interfacial area at a given Sw can be taken as
 | [20] |
In Eq. [20], is replaced by Sw, which represents the fraction of the capillaries in the capillary bundle filled with water. This substitution is justified based on an assumption that, in the idealized capillary bundle of Rose and Bruce (1949), asw is contributed by the water-wet, filled capillaries; their volume as a fraction of the total capillary bundle volume is equal to Sw. Substituting the Brooks–Corey (1964) capillary desaturation function, where is a pore-size distribution index,
 | [21] |
into Eq. [20] yields the following expression for asw in unsaturated porous media:
 | [22] |
Alternatively, following the capillary desaturation model (Cary, 1994) for an idealized porous medium, the asw–Sw relationship can be formulated by assuming a cylindrical non-wetting-phase bubble in a cylindrical pore of radius r and combining the expressions for interfacial area and water content for the pore with a functional relationship between and r based on the Young–Laplace equation:
 | [23] |
where o and m are complete and residual saturation, respectively. Here, asw,o corresponds to the specific solid–water interfacial area of the porous medium, which is in complete contact with water at 100% saturation and until air enters the medium during desaturation. This expression was obtained from the original Cary (1994) model by substituting 1/ and Pd/2 with the Campbell (1974) parameters b and ro, respectively. Also, volumetric water content w was equated to Sw, and the contribution of a "dry-end fitting factor", relevant only at irreducible water saturation, was ignored.
Shown in Fig. 1
are the two formulations for the asw–Sw relationship developed above (Eq. [22] and [23]), plotted using parameters typical for a porous sand. It can be seen that both models predict a decrease in asw with decreasing water saturation, from a maximum value equal to the specific surface of the sand to a minimum value corresponding to the irreducible water saturation. Such a decreasing trend is consistent with the physics of drainage of porous sands, where the emptying of pores results in a corresponding reduction in asw.
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Fig. 1. Specific solid–water interfacial area as a function of saturation (B-R stands for Eq. [22], derived based on the Brooks–Corey relationship).
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The slope of the asw–Sw curves is flat in the high water saturation range, because drainage in this range corresponds to emptying of larger pores, which represent a large volume of water but a small surface area. In contrast, the slope of the curves is steep in the lower water saturation range, because drainage in this range corresponds to emptying of smaller pores, which represents a small volume of water but a large surface area. The two models agree reasonably in a majority of the water saturation range; however, Eq. [23] predicts that asw becomes zero near very low water saturations, which is unlikely because water at the irreducible water saturation is in contact with a finite fraction of the specific surface, as Eq. [22] appears to predict. For the final implementation in a reactive transport model, we use only Eq. [23] and not Eq. [22], which is based on relatively more unrealistic assumptions. An important assumption underlying these predictions is that the contribution of the molecularly thin water films (covering the surfaces of the drained pores) to asw is negligible for the purposes of reactive transport. This assumption is plausible, considering the fact that fluids cannot flow through such films (Hunt and Ewing, 2003). This approach for determining the solid–water interfacial area is not experimentally validated, and does not consider the influence of wettability or residual saturation; however, it is possibly the first report of a mathematical model for the asw–Sw relationship, which allows us to assess the influence of varying asw–Sw on reactive transport. It needs to be verified using dissolution experiments in unsaturated media at varying degrees of unsaturation.
Influence of Varying Specific Solid–Water Contact Area on Reactive Transport
To investigate the influence of a changing solid–water interfacial area on reactive transport, the formula for asw (Eq. [23]) was incorporated into kinetic rate laws and implemented in a reactive transport simulator, CRUNCH (Steefel, 2001; Steefel and Yabusaki, 1996). The CRUNCH simulator implements a rate law that is based on transition state theory (Lasaga, 1981; Aagard and Helgeson, 1982), and is defined as
 | [24] |
where km is the rate constant, asw is the specific solid–water contact area, Kmeq is the equilibrium constant, Q is the ion activity product, defined by
 | [25] |
al is the activity of a species making up the mineral, and l represents the stoichiometric coefficients. The activity of species l is raised to a power p, an empirical coefficient that affects the dissolution rate. Coefficients M and n are two positive numbers that are usually determined experimentally. The term sgn(log Q/Km) gives the sign of the expression, negative for undersaturation and positive for supersaturation. The last term in Eq. [24] is the far-from-equilibrium rate term. For quartz dissolution, the closer the dissolved silica is to equilibrium with quartz, a slower dissolution rate (rm) results.
Although CRUNCH is capable of simulating reactive transport, hydrodynamic flow calculations are not performed in CRUNCH. As such, constant flow velocities and saturations were fixed independently of any flow calculations. Hence, any changes in unsaturated hydraulic conductivities were not reflected in the transport simulations.
Unsaturated transport was simulated for a hypothetical one-dimensional column experiment for quartz dissolution. The column was simulated as 2 m in length, and was slightly undersaturated with respect to quartz at 100°C. In five unsaturated transport simulations, water saturation was fixed at 0.2, 0.4, 0.6, 0.8, and 1.0 and the reactive surface area formulation given in Eq. [23] was implemented (Table 1). Trends in the asw–Sw relationship predicted by both Eq. [22] and [23] agree closely in this range (Fig. 1) and the use of Eq. [22] for the simulations is not likely to yield significantly different results. For these asw simulations, reactive surface areas were both temporally and spatially invariant. Specifically, the asw values derived from Eq. [23] (shown in Table 2) were substituted in Eq. [24]. Equation [17], which is meant to illustrate how the fundamental dissolution rate equation would change due to a varying asw–Sw relationship, is currently not implemented in CRUNCH, which uses Eq. [24] as the standard rate equation. Eventually, implementation of modified rate equations (similar to Eq. [17]) that explicitly consider the dynamics of changing asw–Sw in reactive transport models may be useful.
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Table 1. Quartz reactive surface areas for the baseline and varying solid–water interfacial area (asw) simulations.
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At the beginning of each simulation, a water flux that was depleted with respect to silica entered at the top of the column and continued for 10 d. A constant Darcy velocity of 0.002 cm s–1 was fixed for the entire column. All input parameters for the column appear in Table 2. A baseline simulation was performed first, assuming the column to be fully saturated and the reactive surface area determined using the two-thirds power law defined in CRUNCH:
 | [26] |
where V is the mineral volume fraction and the subscript i represents initial values. Because this formula accounts for changes in mineral volumes, the reactive surface area for the baseline simulation was temporally and spatially variant. This meant that the reactive surface area of quartz was reduced as dissolution proceeded (Table 1). Note that the formula in Eq. [26] does not account for changes in reactive surface area as a function of water saturation.
Shown in Fig. 2
are the total silica concentration distributions for both the baseline and asw simulations at the end of the 10-d period. Similar trends are noted for all six of the simulations. At 10 d, the system has approached a steady state, where the total silica concentrations at the top of the column are 25% higher than at the bottom of the column and are close to an equilibrium concentration of 8.4 x 10–4 mol kg–1. Because the influent solution is further from equilibrium than the resident fluid in the column, the system at the top of the column is also further from equilibrium than the system at the bottom. Since the rate formulation in CRUNCH is based on transition state theory, the further the system is from equilibirium, the faster the reaction rate (see Fig. 3
). Hence, a higher concentration of dissolved silica is found at the top of the column (distance = 0 m) than at the bottom (distance = 2 m).
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Fig. 2. Final concentration distributions after 10 d for the quartz dissolution column for both the variable solid–water interfacial area (asw) and baseline simulations. Sat refers to the degree of water saturation. (On the y axis, distance = 0 corresponds to column inlet).
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Fig. 3. Final dissolution rates after 10 d for the quartz dissolution column for both the variable solid–water interfacial area (asw) and baseline simulations. More highly negative numbers indicate faster rates of dissolution. Sat refers to the degree of water saturation. (On the y axis, distance = 0 corresponds to column inlet).
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For the five asw simulations, the influence of asw can be seen in the concentration profiles in the upper part of the column. At low saturations, where the reactive surface area is smaller and the rates of reaction slower, the total dissolved silica concentrations also are lower. Although higher concentrations may be expected due to lower moisture contents, the rate of dissolution is also slower, which releases less dissolved silica into the system. At higher saturations, the increase in the reactive surface area results in higher total concentrations of silica. Although specific surface areas in the baseline column simulation were not exactly equal to 100 m2 m–3 as in the asw fully saturated simulation, the difference in the specific surface area was too small (
0.3 m2 m–3) to impact the total silica concentration in the column. Hence, the baseline and fully saturated asw lines coincide in Fig. 2.
The baseline and fully saturated ai lines also coincide in Fig. 3, which shows the spatial distribution of quartz dissolution rates at the end of the 10-d period. In the upper part of the column, dissolution rates are slower than at the bottom of the column due to the far-from-equilibrium rate formula given in Eq. [24]. For regions where the aqueous–solid phase reactions are closer to equilibrium, slower rates of quartz dissolution result. Very little distinction among the five asw exist at the top of the column, although slower reaction rates are associated with higher saturations. At the bottom of the column, once again, faster dissolution rates are associated with lower concentrations. Although not easily discernible in Fig. 2, lower saturations are also associated with smaller total silica concentrations at the bottom of the column. This result is again consistent with the far-from-equilibrium reaction rate formula, which states that faster dissolution rates result in systems further from equilibrium.
At the end of the 10-d simulation, the total solute flux for both the baseline and fully saturated asw columns was 655 mol d–1. Using a variable solid–water interfacial area for saturations of 0.2, 0.4, 0.6, and 0.8, the resulting total solute fluxes were 549, 591, 621, and 638 mol d–1, respectively. For the wettest case, a saturation of 0.8 leads to a 3% overestimate in the total contaminant flux relative to the fully saturated cases. For the driest case, however, a saturation of 0.2 translates into a 16% overestimate in the total contaminant flux out of the column.
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CONCLUSIONS
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These simulations clearly demonstrate that a changing asw can have a significant impact on the cumulative fluxes of silica dissolution in the vadose zone. This is to be expected, because, as shown in Fig. 1, changes in asw are rapid in the dry Sw range, whereas they are only moderate in the relatively wet region. These findings have significant implications for contaminant transport, especially in arid regions. Ignoring the effect of a changing asw is likely to result in an overestimation of the net contaminant fluxes in these cases, which leads to highly conservative risk assessments. In the case of nutrient flux calculations coupled with dissolution phenomena, it may lead to an overestimation of nutrient availability. More complete numerical implementation and experimental verification of the proposed models are necessary for their validation.
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TOP
ABSTRACT
INTRODUCTION
