Numerical modeling of kinetic interphase mass transfer ... - CiteSeerX

WATER RESOURCES RESEARCH, VOL. 36, NO. 12, PAGES 3391–3400, DECEMBER 2000

Numerical modeling of kinetic interphase mass transfer during air sparging using a dual-media approach Ronald W. Falta Departments of Geological Sciences and Environmental Engineering and Science, Clemson University Clemson, South Carolina

Abstract. A dual-media multiphase flow approach is proposed for modeling the local interphase mass transfer that occurs during in situ air sparging. The method is applied to two- or three-phase flow in porous media to simulate the small gas channels that form during air sparging, allowing resolution of the local diffusive mass transfer of contaminants between the flowing gas phase and nearby stagnant liquid-filled zones. This approach provides a good match with laboratory column experiments in which dissolved trichloroethylene (TCE) is removed by air sparging, and it is shown that the simulation results are very sensitive to the nature of the local mass transfer regime. The numerical model is then applied to hypothetical field air-sparging cases involving either a dissolved plume of TCE or a TCE nonaqueous phase liquid source. In these simulations the local mass transfer appears to play a much smaller role than in the laboratory-scale tests, and significant deviations from local equilibrium simulations only occur when the mass transfer rates are reduced below the calibrated laboratory-scale values.

1.

Introduction

Recent laboratory studies have demonstrated that dissolved volatile organic compound (VOC) removal during air sparging is limited by the mass transfer of the VOC into the flowing gas phase [Braida and Ong, 1998; Hein et al., 1998; Semer and Reddy, 1998; Adams and Reddy, 1999]. Mass transfer limitations may occur at several scales because of the heterogeneous nature of gas distributions during air sparging. At a large scale the air-sparging zone is very strongly influenced by heterogeneities, which form capillary and permeability barriers to the gas flow. If the geometry and locations of these heterogeneities are well known or if the media is homogeneous, it is possible to accurately model the sparge gas flow field using a conventional multiphase flow numerical approach [Hein et al., 1997; McCray and Falta, 1997]. This type of simulator, however, cannot resolve local (sub-grid-block) diffusion-limited mass transfer effects that arise because of the millimeter- to centimeter-scale gas channels which form during sparging [Clayton, 1998; Semer and Reddy, 1998; Elder and Benson, 1999]. An illustration of these small gas channels is shown in Figure 1. Because these channels typically occur at a scale smaller than that of a numerical model grid block, compositional multiphase flow simulators that assume local chemical equilibrium between the phases may overestimate the rate of interphase mass transfer during air sparging. Figure 2 shows the effluent gas concentration measured during a recent air-sparging experiment conducted at Michigan Technological University [Hein et al., 1998]. In this experiment, water saturated with dissolved trichloroethylene (TCE) (1100 mg L⫺1) was injected into an initially dry column packed with a coarse sand. Air sparging was initiated at a Darcy velocity of 2.21 cm min⫺1. The experimental effluent gas concentration data, shown as the open circles, exhibit an initial peak, folCopyright 2000 by the American Geophysical Union. Paper number 2000WR900220. 0043-1397/00/2000WR900220$09.00

lowed by very significant tailing. An attempt to model this experiment with a conventional local equilibrium approach (solid line) using the multidimensional T2VOC multiphase flow and transport code [Falta et al., 1992, 1995] results in a poor match of the data. The local equilibrium assumption (LEA) model predicts a very rapid removal of the dissolved TCE, with virtually no tailing of the concentration at later times. This poor match is largely a result of overpredicting the rate of local interphase mass transfer in the model. While models have been developed which can account for the kinetic interphase mass transfer during sparging [Braida and Ong, 1998; Elder et al., 1999], these codes do not model the transient two- or three-dimensional multiphase flow that occurs during sparging. However, almost all of the current twoor three-dimensional compositional multiphase flow simulators assume local equilibrium between phases, and there have been only limited efforts to model kinetic interphase mass transfer in multidimensional, field-scale multiphase contaminant transport [Sleep and Sykes, 1989]. The conventional mass transfer formulation assumes that the kinetic interphase mass transfer can be modeled as a heterogeneous first-order reaction in each grid block [see, e.g., Sleep and Sykes, 1989; Gierke et al., 1992; Braida and Ong, 1998; Elder et al., 1999]. Considering a two-phase gas/aqueous (no nonaqueous phase liquid (NAPL)) system, the local rate of interphase mass transfer can be modeled by Q imt ⫽ k imta共C wH ⫺ C g兲,

(1)

where Q imt is the rate of chemical mass transfer from the water to the gas phase per unit volume of porous media (kg m⫺3 s⫺1), C w is the aqueous-phase chemical concentration (kg m⫺3), C g is the gas-phase chemical concentration (kg m⫺3), H is the dimensionless Henry’s constant for the chemical in water (assumed to be constant), and k imta is the mass transfer coefficient–interfacial area product (s⫺1). Implementing this type of mass transfer formulation in a compositional multiphase flow code is not trivial because the aqueous-phase and gasphase forms of the chemical must be treated as separate com-

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FALTA: NUMERICAL MODELING OF KINETIC INTERPHASE MASS TRANSFER

Applying Darcys law, Vg ⫽ ⫺ ⫽⫺

kk rg ␮g

冉

⭸P g ⫺ ␳ gg ⭸z

冊

kk rg kk rg 共 ␳ w ⫺ ␳ g兲 g ⬵ ⫺ ␳ g, ␮g ␮g w

(7)

so the vertical gas Darcy velocity (defined here as being positive in the z direction) in the air-sparging zone at steady state is to first approximation the gas buoyancy velocity. The gas velocity is strongly dependent on the gas saturation because of the nonlinear dependency of the gas-phase relative permeability on the saturation [k rg (S g )]. In general, k rg is a power function of S g , and it always increases monotonically with increasing S g . Equation (7) may be rearranged to solve for the gas-phase relative permeability given the gas Darcy velocity and the intrinsic permeability of the sparge system: Figure 1. Schematic diagram of local-scale gas channeling during air sparging. ponents, with separate mass balance equations. It is significant to note that this type of kinetic interphase mass transfer approach has never been applied to a field-scale multiphase flow air-sparging simulation.

2. Estimation of Gas Velocities and Gas Saturations During Air Sparging As a preface to the numerical simulation of mass transfer, it is useful to consider the relationships between gas-phase saturation, intrinsic permeability, gas-phase relative permeability, and interphase mass transfer from a conceptual viewpoint. It is possible to estimate the air-sparging gas velocity and gas saturation by considering a one-dimensional vertical gas flow at steady state. When an air-sparging system is at steady state, the water within the sparge zone is essentially static [McCray and Falta, 1996, 1997], so the water fluid pressure is hydrostatic P w ⫽ ␳ wgz

(2)

k rg共S g兲 ⫽

V g␮ g . k ␳ wg

(8)

Equation (8) can be thought of as a dimensionless sparging number which gives the gas saturation as a function of the gas Darcy velocity and the intrinsic permeability. To solve for the gas saturation, the relative permeability function must be known or estimated. Then the gas saturation in the sparge zone can by found by solving (8) implicitly. Equation (8) has some important implications. First, and most obvious, as the gas Darcy velocity in a system is increased, the gas-phase relative permeability, and hence the gas saturation, must increase. Second, for a given gas Darcy velocity a decrease in the intrinsic permeability must correspond to an increase in relative permeability and hence in gas saturation. Thus it is expected in homogeneous systems that gas saturations will be higher in finer-grained, low-permeability sediments than in high-permeability formations. This observation has been made in several laboratory- and field-scale studies [Clayton, 1998]. These conclusions have implications for the interphase mass transfer. If the sparge gas zone has a very low gas saturation, the gas is probably only moving through a few isolated gas

and the vertical water pressure gradient is ⭸P w/⭸ z ⫽ ␳ wg,

(3)

where P w is the water gage pressure, g is gravitational acceleration, ␳ w is water density, and z is the depth below the original water table. The gas pressure is related to the water pressure through the gas-water capillary pressure: P g ⫽ P w ⫹ P cgw,

(4)

⭸P g ⭸P w ⭸P cgw ⫽ ⫹ . ⭸z ⭸z ⭸z

(5)

so

If it is assumed that the gas saturation does not vary rapidly in the vertical direction or if the media has a low capillary pressure, the vertical capillary pressure gradient may be neglected, and the sparge zone gas pressure gradient becomes equal to the hydrostatic water pressure gradient ⭸P g/⭸ z ⫽ ␳ wg.

(6)

Figure 2. Comparison of numerical simulation results assuming local equilibrium with experimental data from a laboratory column air-sparging test in a 20 ⫻ 30 Ottawa sand.


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channels in the otherwise water-saturated media. This condition would be expected to lead to fairly severe local mass transfer limitations because of the small phase contact areas and the large aqueous diffusional distance. However, if the sparge gas zone has a higher gas saturation, the gas channels will be much more numerous, and closer together on average, leading to larger phase contact areas and shorter aqueous diffusional distances. Furthermore, because of the higher gas saturation the true gas velocity (for a given Darcy velocity) in these systems will be slower than in the higher-permeability systems, maximizing the phase contact times. To summarize, on average, mass transfer limitations are expected to be most severe during air sparging in high-permeability systems and are expected to be less important in lower-permeability systems [Clayton, 1998].

3. Modeling Interphase Mass Transfer With a Dual-Media Approach An alternative method for modeling the local mass transfer during air sparging involves a dual-media formulation. This technique is fairly straightforward and can be easily implemented in existing compositional multiphase flow simulators that assume local phase equilibrium. The method is based on a dual-media formulation which is widely used for modeling both single-phase and multiphase flow and transport processes in fractured media [see, e.g., Barenblatt et al., 1960; Warren and Root, 1963; Grisak and Pickens, 1980; Pruess and Narasimhan, 1985; Zimmerman et al., 1993; Ho, 1997]. The method is also commonly used to describe single-phase solute transport in which the solute is affected by physical or chemical nonequilibrium effects because of structured soils, adsorption, or deadend pore effects [Coats and Smith, 1964; van Genuchten et al., 1974; Rao et al., 1979; Gerke and van Genuchten, 1993; Brusseau et al., 1994; Griffioen et al., 1998]. Recently, the method has also been applied to large field-scale single-phase solute transport in highly heterogeneous aquifers by Harvey [1996] and Feehley et al. [2000]. Figure 3 illustrates the so called “dual-permeability” approach used in fractured porous media [Pruess and Narasimhan, 1985]. Each overall grid block is considered to consist of two volume fractions, a volume associated with the fractures and a volume associated with the porous matrix. The two

Figure 3. Dual-permeability approach to modeling flow and transport in fractured porous media.

Figure 4. An alternate conceptual model for a dual-media system.

volumes are assigned different properties, corresponding to the respective media. Each of these domains is globally connected in a normal way, and the overall grid structure can be one-, two-, or three-dimensional with any coordinate system. The global connections retain their normal (single media) nodal distances, but the global grid block connection areas are divided into two area fractions corresponding to the average fracture area between grid blocks and to the average matrix area between grid blocks. These global area fractions are not necessarily equal to the media volume fractions, and they depend on the assumed fracture geometry inside the grid block. Locally, at the scale of a single grid block, the two domains are attached with a simple one-dimensional connection. This connection is characterized by an interfacial area (per unit total volume) between the two media ( A 12 ) and by the average nodal distance between the two media (d 1 ⫹ d 2 ⫽ d 12 ). Here the subscripts 1 and 2 refer to the fracture and matrix parts of the overall grid block and the local dual-media connection. It is important to understand that with this approach, individual fractures and matrix blocks within the grid block are not explicitly modeled. Rather, the average fracture and matrix block responses are simulated. The ratio of A 12 /d 12 determines the magnitude of the conductance between the media, and it is often called the dual-media interaction parameter. Large values of the interfacial area or small nodal separation distances correspond to rapid fluid, chemical, and heat fluxes between the media. In the limit of very large A 12 or very small d 12 , the two media remain at equilibrium with each other, and the dual-media formulation responds as if it were a normal single-media formulation with local equilibrium in each grid block. In the present work, the method is applied to porous media to simulate the effect of local gas channels that form during air sparging (Figure 4). The local (sub-grid-block) geometry is assumed to consist of two regions: a volume fraction containing coarse-grained materials and a volume fraction containing fine-grained materials. The fine-grained material is assumed to have a higher capillary pressure than the coarse-grained ma-

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Figure 5. Local mass transfer in a dual-media system.

terial for a given wetting-phase saturation. The volume fractions, the global area fractions, the interfacial area ( A 12 ), and the interfacial distances (d 1 and d 2 ) may be estimated using an appropriate conceptual model of the local two-phase flow [see, e.g., Elder and Benson, 1999] and are inputs to the numerical model. The multiple interacting continua (MINC) subroutines in the T2VOC code (see Pruess [1991] or Falta et al. [1995]) were modified to allow direct input of these quantities for this purpose. Under two-phase flow conditions the nonwetting phase (gas) will flow preferentially through the coarse-grained media, while the wetting phase (aqueous) will remain in the finegrained media. This local contrast in gas-phase saturation between the two media results in a aqueous-phase diffusional limitation to the mass transfer of a contaminant between the two media (Figure 5). The T2VOC code was modified to include aqueous-phase diffusion along with gaseous diffusion. Thus the numerical model allows a first-order resolution of the local mass transfer kinetics between the flowing gas phase and nearby stagnant water-filled zones. Compared to the usual local equilibrium approach, the dual-media approach doubles the number of equations to be solved at each time step. The two media are assumed to have the same intrinsic permeability and porosity here, so that single-phase flows of gas and water are not affected. The dual-media capillary pressure and relative permeability properties must be estimated for this method. One reasonable approach is to choose the two media properties so that the weighted averages (with local capillary pressure equilibrium) duplicate the single-media properties. Figure 6 illustrates this approach. Here the measured capillary pressure from the sand used in the Michigan Technological University column sparging experiments [Hein et al., 1998] provides the single-media property to be matched. Assuming dual-media volume fractions of 0.2 for the coarse-grained media and 0.8 for the finegrained media and assuming capillary pressure equilibrium between the media, the capillary curve parameters were adjusted until the volume-weighted average matched the experimental data. These particular curves were calculated using the two-phase van Genucten [1980] equation, and the parameter values are shown in Table 1. The two capillary pressure functions shown in Figure 6 give

rise to a large contrast in water saturation in the two media. Figure 7 shows the dual-media water saturations as a function of the overall volume-weighted average. For example, at an overall water saturation of 0.9 (a gas saturation of 0.1) the fine-grained media has a water saturation of about 0.97, while the coarse-grained media has a water saturation of only about 0.62. A matching approach can also be used to determine the appropriate relative permeability functions. Figure 8 shows a standard single-media gas-phase relative permeability curve, calculated using one of the internal T2VOC functions (see Table 1). By adjusting the parameters in the function, the global area-fraction weighed average of the dual-media gas relative permeabilities is similar to the single-media one. Again, this weighted-average calculation assumed local capillary pressure equilibrium between the two media, a condition that may or may not exist during a multiphase flow simulation. On the basis of the above discussions it is expected that the dual-media formulation can be made to duplicate “normal” bulk porous media characteristics, while providing a mechanism for local kinetic interphase mass transfer. The fact that substantial water saturation remains in the coarse-grained media (Figure 7) is significant. Within each media, in each grid block, local equilibrium is assumed. Thus, as the gas flows through the coarse-grained region, it equilibrates with the contaminant dissolved in the pore water (or present as a NAPL) in that volume. At the same time, there is a diffusive exchange with the fine-grained region, so the model acts like a dual-region model, with local equilibrium in one region and kinetic mass transfer with the other region. The relative contributions of these regions are determined by the media volume fractions and by the water saturations in the two adjacent media. As shown earlier in Figure 5, the mass transfer between regions occurs primarily by diffusion. As the sparge gas sweeps the coarse-grained region, a concentration difference with the fine-grained region develops. The resulting diffusive flux of

Figure 6. Comparison of dual-media capillary pressures with experimental data. The center curve is a volume-weighted average of the two media curves. The dual-media capillary parameters were adjusted so that the weighted average would give a best fit with the experimental data.


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Table 1. Relative Permeability and Capillary Pressure Curves Used in the Simulations

Parameter

Equation

Gas-water capillary pressure

Pcgw ⫽

Gas-phase relative permeability

krg ⫽

Water-phase relative permeability

krw ⫽

␳w g ␣gw

冋

再冉

Sw ⫺ Swr 1 ⫺ Swr

Sg ⫺ Sgr 1 ⫺ Swr

册

冋

冊

⫺1/m

冎

1/n

⫺1

m ⫽ 1 ⫺ 1/n

n

krg ⱕ 1

Sw ⫺ Swr 1 ⫺ Swr

册

n

chemical is modeled using a first-order finite difference approach. Calling the coarse-grained volume region 1 and calling the fine-grained volume region 2, we have gas and aqueous chemical concentrations C g1 , C w1 , C g2 , and C w2 ; phase saturations S g1 , S w1 , S g2 , and S w2 ; and phase tortuosities ␶ g1 , ␶ w1 , ␶ g2 , and ␶ w2 in the two regions. The diffusive flux between the regions includes both a gas-phase diffusion component as well as an aqueous-phase diffusion component. Writing these terms as diffusive mass flows per unit total volume of porous media and using a first-order finite difference for the concentration gradient gives Q g12 ⫽ ⫺共 ␾ S g␶ gD g兲 12

共C g1 ⫺ C g2兲 A 12 d1 ⫹ d2

(9)

for the gas phase and Q w12 ⫽ ⫺共 ␾ S w␶ wD w兲 12

共C w1 ⫺ C w2兲 A 12 d1 ⫹ d2

(10)

Single-Media Equivalent

Low–Capillary Pressure Media (Media 1)

High–Capillary Pressure Media (Media 2)

␣ gw ⫽ 11.8 m⫺1 n⫽6 S wr ⫽ 0.23

␣ gw ⫽ 20.0 m⫺1 n⫽6 S wr ⫽ 0.23

␣ gw ⫽ 11.0 m⫺1 n⫽6 S wr ⫽ 0.23

S gr ⫽ 0.015 S wr ⫽ 0.23 n⫽2

S gr ⫽ 0.015 S wr ⫽ 0.23 n⫽3

S gr ⫽ 0.015 S wr ⫽ 0.23 n⫽3

S wr ⫽ 0.23 n⫽2

S wr ⫽ 0.23 n⫽3

S wr ⫽ 0.23 n⫽3

rium inside each region, the two mass flows can be combined to get the total diffusive mass flux of chemical: Q 12 ⫽ 共 ␾ S g␶ gD g ⫹ ␾ S w␶ wD w/H兲 12

共HC w2 ⫺ C g1兲 A 12. d 12

(11)

Here the sum of the effective diffusion coefficients (with the Henry’s law correction) is weighted harmonically. Recalling the traditional first-order formulation for kinetic interphase mass transfer Q imt ⫽ k imta共HC w2 ⫺ C g1兲,

(12)

it is apparent that the dual-media formulation gives a firstorder rate-limited mass transfer between the media with a mass transfer coefficient–interfacial area product equal to k imta ⫽ 共 ␾ S g␶ gD g ⫹ ␾ S w␶ wD w/H兲 12

A 12 . d 12

(13)

for the aqueous phase. Here the effective porous media diffusion coefficient ( ␾ S ␶ D) 12 is an interface average, which is calculated using a harmonic mean of the two media values, and mass flow from region 2 (fine) to region 1 (coarse) is considered positive. Using Henry’s law and assuming phase equilib-

Figure 7. The distribution of water in a dual-media grid block at capillary pressure equilibrium. The individual media water saturations are plotted as a function of the overall volume-weighted average saturation.

Figure 8. Comparison of the dual-media average (flow area weighted) gas-phase relative permeability with a typical singlemedia gas-phase relative permeability function. Note that the dual-media k rg parameters were adjusted so that the area weighted average would duplicate the single-media function.

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Therefore the current formulation can be considered a dualregion model, with local equilibrium between the flowing gas and the pore water in region 1, with first-order kinetic interphase mass transfer with the adjacent region 2. Note that this lumped mass transfer coefficient is proportional to the effective diffusivities and is inversely proportional to Henry’s constant, consistent with the observations of Braida and Ong [1998].

4. Simulation of Laboratory Column Experiments A series of laboratory column air-sparging experiments was performed at Michigan Technological University during the summer of 1998 [Hein et al., 1998]. These were performed in a vertical 5 cm diameter column, with a packed length of 28.3 cm. The column was packed with a 20 ⫻ 30 mesh Ottawa sand, having a porosity of 0.34 and an intrinsic permeability of 2.6 ⫻ 10⫺10 m2. Capillary pressure data for this sand are shown in Figure 6. TCE was dissolved in water to solubility (1100 mg L⫺1), and 128.5 mL of this solution was injected into the bottom of the initially dry column. Filling the column in this manner resulted in about 10 cm of unsaturated sand above about 18 cm of water-saturated sand. Air sparging was begun immediately after filling and continued until the effluent gas concentration was below detection limits. Experiments were performed at several gas flow rates. The 2.21 cm min⫺1 and 0.55 cm min⫺1 injection rate experiments were modeled numerically using T2VOC with the new dual-media approach. The one-dimensional (1-D) column was divided into 57 normal grid blocks, each 0.5 cm in length. A numerical simulation of the 2.21 cm min⫺1 experiment using T2VOC with a single media and 57 1-D grid blocks was previously shown in Figure 2 (see Table 1 for the capillary pressure and relative permeability functions). As noted, the normal local equilibrium modeling approach does a very poor job of matching the effluent gas concentration profile in this experiment. The dual-media grid consisting of 114 grid blocks was generated using the built-in subroutines in T2VOC. As mentioned in section 3, these subroutines were modified to allow direct entry of the dual-media volume fractions, global area fractions, the dual-media nodal separation distances, and the dual-media interfacial area per unit volume. Unfortunately, we do not have a method for estimating these dual-media parameters based on standard porous media measurements. Therefore the dualmedia geometrical parameters were estimated using a conceptual model described below and were refined slightly by attempting to match the experiments. This situation is analogous to the problem of determining a kinetic interphase mass transfer coefficient–interfacial area product for standard kinetic mass transfer modeling. Equation (8) was first used to estimate the gas saturation in the sparging experiments. Because of the high permeability it was found that the average gas saturation in the sparge zone would likely be very low, of the order of only a few percent of the total pore space. On this basis the volume fraction for the coarse-grained media was chosen to be 0.20. This 20% of the porous media was assumed to form a series of tortuous cylinders or channels, extending from the bottom of a grid block to the top of a grid block. A tortuosity of 1.3 (the tortuous cylinders are 1.3 times longer than straight cylinders would be) was chosen arbitrarily. Given the tortuosity of the low– capillary pressure media channel zones, the global area fraction of the

coarse-grained media is equal to the volume fraction divided by the tortuosity or 0.154. The 20 ⫻ 30 mesh Ottawa sand has average grain diameters in the range of 0.6 to 0.9 mm. On the basis of reports that typical gas channels are several grain diameters [Ji et al., 1993; Clayton, 1998; Elder and Benson, 1999], the average gas channel was assumed to have a diameter of about 3 mm. Note that here the “channels” are actually the coarse-grained media, and they will include both gas and water during air sparging [see, e.g., Clayton, 1998]. With the coarse-grained media geometry defined, the next step is to estimate the interfacial area and diffusion distances. Elder and Benson [1999] present experimental data and estimation procedures for these quantities, and the procedure outlined below gives similar (the same order of magnitude) estimates. The number of channels per unit volume of total porous media is calculated by dividing the total volume of the channel zones by the volume of a single channel. Considering a 1 m ⫻ 1 m ⫻ 1 m cube, number of channels ⫽ @fr/ ␲ 共d/ 2兲 2␶ ,

(14)

where @fr is the volume fraction of channels, d is the assumed channel diameter (3 mm) and ␶ is the assumed channel tortuosity (1.3). Considering the surface area of each channel [␲d ␶ (1)], the dual-media interfacial area per unit total volume ( A 12 ) is calculated to be about 267 m2 m⫺3, and there would be about 22,000 channels in a 1 m ⫻ 1 m ⫻ 1 m cube of porous media. The next step is to estimate the average distance from the inside of the channel to the interface between the media and the distance from the interface to the centroid of the high– capillary pressure media (on average). The nodal distance from the interface to the average centroid of the high– capillary pressure media was estimated by considering a slice through the volume. On average, the 22,000 channels would be associated with an area of 1 m2, so the average total area associated with each channel would be about 45 mm2. Thus the channels would be separated by about 7 mm or so, and a reasonable estimate for the diffusion distance would be a few millimeters. The effective nodal distance from the 3 mm channel to the interface would be of the order of a millimeter or so. It must be emphasized here that the preceding calculations are only useful for obtaining order of magnitude estimates for A 12 , d 1 , and d 2 . Final values for these model parameters are determined by fitting experimental data (analogous to determining k imta). Figure 9 shows a dual-media T2VOC simulation of the 2.21 cm min⫺1 experiment. Here a value of 300 m2 m⫺3 was used for A 12 , d 1 was set equal to 1 mm, and d 2 was set equal to 4 mm. The other multiphase flow parameters (capillary pressure curves and relative permeability curves) were determined by matching the single-media values when possible, and the curves are shown in Figures 6 and 8 (see also Table 1). Although the numerical model overpredicts the gas concentration in the first few minutes, the overall match with the experimental data is excellent, especially compared to the earlier local equilibrium simulation. The initial overprediction of gas concentration could probably be eliminated by reducing the volume fraction of the coarse-grained media (in which local equilibrium with the water is assumed), but this type of fine tuning was not attempted. Using (13) and an average value for the gas and water saturations in the sparge zone to calculate the harmonic mean of the effective diffusivities gives an equivalent k imta value of 4.82 ⫻ 10⫺5 s⫺1. This value is in the range


Figure 9. Comparison of a numerical simulation using the dual-media mass transfer approach with experimental data from a laboratory column air-sparging test in a 20 ⫻ 30 Ottawa sand operated at a Darcy velocity of 2.21 cm min⫺1.

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Figure 11. Effect of decreasing the interaction parameter ( A/d) between the dual media in a column simulation with a Darcy velocity of 2.21 cm min⫺1. This is equivalent to reducing the mass transfer coefficient between the two media.

Using (15), a value of 4.91 ⫻ 10⫺5 s⫺1 is calculated. Essentially, the mass transfer depends on the ratio of the dual-media interfacial area to the diffusional distance in the high– capillary pressure media, A 12 /d 2 . For the results shown in Figure 9, this ratio was equal to 75,000 m⫺2. Figure 10 shows a comparison of the dual-media numerical solution with experimental data from a sparging experiment conducted at a Darcy velocity of 0.55 cm min⫺1. This simulation used the exact same parameters at the previous one except

for the lower gas flow rate. Again, the match is good although some of the early gas concentrations are overpredicted. For this case a local equilibrium numerical simulation would predict complete removal of the TCE within an hour. A series of simulations was conducted using different A 12 /d 2 ratios in order to evaluate the sensitivity of the results to the rate of mass transfer. In Figure 11 the ratio was reduced by factors of 10 and 100 to 7500 and 750 m⫺2, respectively. These changes have almost no effect on the early time data, which is dominated by the equilibrium process in the low– capillary pressure media, but they greatly restrict the rate of mass transfer at later times. It is apparent that these would be unrealistically low rates of interphase mass transfer for this experiment. Similarly, Figure 12 shows the effect of increasing the A 12 /d 2 ratio. As the ratio is increased by factors of 10 and 100 to 750,000 and 7,500,000, respectively, the rate of mass transfer greatly increases and begins to approach the pure local equilibrium simulation (equivalent to A 12 /d 2 ⫽ ⬁). It is apparent

Figure 10. Comparison of a numerical simulation using the dual-media mass transfer approach with experimental data from a laboratory column air-sparging test in a 20 ⫻ 30 Ottawa sand operated at a Darcy velocity of 0.55 cm min⫺1.

Figure 12. Effect of increasing the interaction parameter ( A/d) between the dual media in a column simulation with a Darcy velocity of 2.21 cm min⫺1. This is equivalent to increasing the mass transfer coefficient between the two media, and the curves approach the local equilibrium curve.

reported by Braida and Ong [1998] for their air-sparging experiments. Because the harmonic mean in (13) is usually dominated by the aqueous diffusion term in the fine-grained media (region 2), the equivalent mass transfer–interfacial area product can be closely approximated by k imta ⫽

共 ␾ S w␶ wD w/H兲 12A 12 . d2

(15)

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Figure 13. Hypothetical field air sparging system. that these would be unrealistically high rates of interphase mass transfer for this experiment.

5.

Field-Scale Simulations of Air Sparging

Previous field-scale multiphase flow air sparging simulations have assumed local equilibrium [Unger et al., 1995; McCray and Falta, 1996, 1997], so it is interesting to study the effect of local kinetic interphase mass transfer on field-scale simulations. A simple series of numerical simulations were performed using a hypothetical case shown in Figure 13. This example involves a homogeneous, isotropic aquifer, having the properties of a 20 ⫻ 30 mesh Ottawa sand (as in the Michigan Tech column experiments). The saturated thickness is 11 m, and the vadose zone is 10 m thick. Two scenarios are used, one in which a cylindrical volume of groundwater above the sparge point is initially contaminated with 100 mg L⫺1 TCE and one in which the cylindrical volume has a TCE NAPL saturation of 0.05. In both scenarios the contaminant is initially assumed to be uniformly distributed in both of the dual media, and the NAPL saturation in the second scenario is below residual saturation. The reader is cautioned that these cases are a simplification compared to real systems, and the presence of heterogeneities could alter the scenarios dramatically. This case was chosen because it was believed that the local mass transfer regime here would be similar to that in the laboratory experiments modeled earlier. The simulated sparge well is screened over 1 m, from 9 to 10 m below the water table, and the simulated soil vapor extraction (SVE) well is screened over 1 m from 6 to 7 m below the ground surface. Pure air is injected into the sparge well at a constant mass rate of 0.00783 kg s⫺1 (14 standard cubic feet per minute (scfm)) and gas is extracted from the SVE well at a constant mass rate of 0.0783 kg s⫺1 (140 scfm). The lower boundary is a no-flow boundary, while the upper boundary simulates atmospheric conditions with a constant gas pressure and zero TCE concentration. The system was modeled with a 2-D axisymmetric r-z model with 21 (normal) grid blocks in the vertical direction and 25 grid blocks in the radial direction (525 normal grid blocks). The radial spacing was logarithmic, out to

a maximum radius of 50 m, where a no-flow boundary was placed. This is well beyond the range of influence of the pumping system. A dual-media grid was generated with T2VOC as before, using the same parameters, for a total of 1050 grid blocks. Figure 14 shows the calculated effluent gas concentrations (from the SVE well) for the dissolved TCE case. Both the single media local equilibrium simulation and a dual-media simulation using the best fit mass transfer parameters from the laboratory column simulations ( A 12 /d 2 ⫽ 75,000 m⫺2) are shown. Surprisingly, there is very little difference in these curves, especially considering the dramatic difference seen in the earlier laboratory-scale simulations. When the interaction parameter ( A 12 /d 2 ) is reduced by a factor of 10 to 7500 m⫺2, only a modest shift in the effluent curves occurs. Figure 15 shows the cumulative recovery of TCE vapors from the SVE well. Again, the LEA simulation and the one using A 12 /d 2 ⫽ 75,000 m⫺2 are virtually the same, and only when the rate of mass transfer is restricted further do the curves change. Apparently, the rate of local interphase mass transfer seen in the laboratory (and modeled using A 12 /d 2 ⫽ 75,000 m⫺2) is fast enough to not be a limiting factor with the larger time and distance scales of this field-scale simulation. Instead, the effluent gas concentration in these simulations seems to be more dominated by larger-scale two-dimensional flow and diffusion of TCE vapors into the sparge zone and into the capture zone of the SVE well. An examination of the TCE concentration distributions in the sparge zone during these simulations shows that after about 1 or 2 days of sparging, the dual-media concentrations are nearly in equilibrium with each other and are similar to the LEA values in the same grid blocks. Furthermore, the TCE concentration in the swept (high gas saturation) regions is very small after a few days. In other words, in the field-scale simulations the dissolved TCE mass is fairly quickly removed from the zone of high gas flow, even considering the local mass transfer restriction observed at the laboratory scale.

Figure 14. Semilog plot showing the effluent gas concentrations from field-scale simulations performed using local equilibrium assumption (LEA) and with a dual-media approach. The contaminated zone initially contains 100 mg L⫺1 of dissolved trichloroethylene (TCE). The dual-media approach with A/d ⫽ 75,000 m⫺2 uses the same parameter values which gave a best fit of the laboratory column data.


Figure 16 shows the computed effluent gas concentrations from the case where residual TCE NAPL is present above the sparge point. Both the LEA single media simulation and dualmedia simulations are shown. As in the previous case, the simulation performed using the best fit dual-media interaction parameter from the laboratory-scale simulations ( A 12 /d 2 ⫽ 75,000 m⫺2) is essentially the same as the LEA simulation. However, when the A 12 /d 2 ratio is reduced by a factor of 10, a moderate shift in the effluent concentrations occurs. The larger effect seen here compared to the dissolved TCE case is due to the overall concentration of the source. In this NAPL case the contaminant mass is over 700 times larger than in the dissolved case, so local rate limitations in the source zone are felt over a longer timescale. The cumulative removal of TCE is shown in Figure 17. The LEA and dual-media simulation with A 12 /d 2 ⫽ 75,000 m⫺2 produce essentially the same recovery curve, while the case with A 12 /d 2 ⫽ 7500 m⫺2 requires a substantially longer remediation period. During the period from 50 days to about 125 days the dual-media simulation with A 12 /d 2 ⫽ 75,000 m⫺2 actually shows a slightly higher recovery than the LEA simulation. This result is due to small differences in the multiphase flow at the fringe of the sparge zone. The bottom outer edge of the contaminated cylindrical volume (Figure 13) receives very little gas flow, so the NAPL tends to persist in this region. In this location the average gas saturation in the dual-media simulation was slightly higher than in the LEA simulation (0.0295 versus 0.0225) leading to a somewhat higher average relative permeability and gas flow rate for the dual media simulation. Because the rate of local mass transfer was fast, the increased gas velocity lead to faster removal of the NAPL in the fringe area in this dual-media simulation. The field-scale simulation results suggest that the mass transfer limitations caused by the small millimeter-scale gas channels observed in the laboratory may not be rate-limiting at the field scale. Instead, it appears that larger-scale channeling, at the scale of centimeters or more, would be required before the mass transfer was substantially limited compared to local equilibrium. It appears likely that the kinetic interphase mass transfer coefficient for air sparging is a scale-dependent parameter that depends both on the physical scale of the sparging

Figure 15. Recovery of the dissolved TCE by air sparging in the field-scale simulations. The initial TCE mass was 1.78 kg.

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Figure 16. Semilog plot showing the effluent gas concentrations from field-scale simulations performed using local equilibrium and with a dual-media approach. The contaminated zone initially contains a TCE dissolved nonaqueous phase liquid (DNAPL) saturation of 5%. The dual-media approach with A/d ⫽ 75,000 m⫺2 uses the same parameter values which gave a best fit of the laboratory column data. regime, as well as on the modeling approach and resolution. These hypotheses can only be confirmed through further experiments, and they highlight the need for carefully controlled field-scale tests of air sparging.

6.

Conclusions

The following conclusions can be drawn: 1. It is possible to adapt an existing three-dimensional local equilibrium multiphase flow and transport simulator to model

Figure 17. Recovery of the TCE DNAPL by air sparging in the field-scale simulations. The initial TCE mass was 1278 kg.

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the problem of locally rate-limited mass transfer by using a dual-media approach. This method is completely analogous to the technique used in fractured porous media. Because the dual-media approach is simply a mesh generation step, it could be readily adapted to other multiphase flow and transport codes that use the integral finite difference method. 2. The dual-media approach allows for a fraction of the system to be at local phase equilibrium, with first-order ratelimited transport with the other fraction. This model achieved a good match with laboratory column air sparging data, and the column scale simulations were very sensitive to the rates of local mass transfer. 3. The dual-media approach involves a number of parameters that cannot be determined by traditional porous media measurements. Although methods are available for estimating these parameters, this must be considered a weak point. The situation is analogous to that of determining mass transfer coefficients. 4. It appears that for field-scale sparging in homogeneous systems, the overall system response is not dominated by the local interphase mass transfer regime seen at the laboratory column scale. Instead, the rate-limited mass transfer may become important when the sparge gas channels are separated by centimeters or more. The apparent kinetic interphase mass transfer coefficient for air sparging is probably scaledependent, reflecting both the physical scale and degree of heterogeneity of the sparging volume, as well as the modeling approach and resolution. Acknowledgments. This work was supported by the U.S. Department of Energy Federal Technology Center through contract DEAR021-96MC33082 to Michigan Technological University with a subcontract to Clemson University. The experiments discussed here were performed at Michigan Tech by Andrea Wolfe, Gretchen Hein, Neil Hutzler, and John Gierke. The author would like to thank Wilson Clayton and Cliff Ho for their helpful comments on this manuscript.

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