Environ. Sci. Technol. 1998, 32, 2578-2584
Estimating Diffusion Coefficients in Low-Permeability Porous Media Using a Macropore Column DIRK F. YOUNG* AND WILLIAM P. BALL Department of Geography and Environmental Engineering, Johns Hopkins University, Baltimore, Maryland 20854
Diffusion coefficients in an aquitard material were measured by conducting miscible solute transport experiments through a specially constructed macropore column. Stainless steel HPLC columns were prepared in a manner that created an annular region of repacked aquitard material and a central core of medium-grained quartz sand. The column transport approach minimizes volatilization and sorption losses that can be problematic when measuring hydrophobic organic chemical diffusion with diffusion-cell methods or column-sectioning techniques. In the transport experiments, solutes (tritiated water, 1,2,4trichlorobenzene, and tetrachloroethene) were transported through the central core by convection and hydrodynamic dispersion and through the low-permeability annulus by radial diffusion. All transport parameters were independently measured except for the effective diffusion coefficient in the aquitard material, which was obtained by model fitting. Batch-determined retardation factors agreed very closely with moment-derived retardation factors determined from the column experiments, and no evidence of pore exclusion was found. A model with retarded diffusion was found to apply, and the effective tortuosity factor of the aquitard material was estimated at an average value of 5.1 (based on estimates that ranged from 3.7 to 7.7).
Introduction Diffusion is commonly the dominant mode of contaminant transport in low-permeability subsurface regions such as aquitards and engineered clay liners (1). For such regions, accurate measurements of diffusion coefficients are needed in order to make estimates of solute movement. For the case of volatile organic compounds, the measurement of diffusion coefficients can be especially difficult given the solutes’ volatilities and tendencies to sorb to experimental apparati. In this paper, we describe an alternative to previously used methods of measuring diffusion coefficients in low-permeability subsurface material and report results obtained using a repacked material from an aquitard. Previous means of measuring diffusion coefficients in clays and silts have been reviewed by Shackelford (2). Methods that require sectioning of the clay sample can work well for inorganic species but are problematic for highly volatile solutes because of the necessity of disturbing the sample and exposing solute to the atmosphere during sectioning and extraction of pore water (3, 4). Methods that monitor a source reservoir concentration have also been problematic for organic compounds due to sorption and losses of the compounds in the experimental apparatus (5, 6). A more * Corresponding author phone: 410-516-5434; fax: 410-516-8996. 2578
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ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 32, NO. 17, 1998
recent method (7, 8) employed a static radial diffusion cell in which solute diffuses radially into a low-permeable region from a central core. Solute was sampled by taking aliquots from the central core, requiring that the central core be replenished with the aqueous phase, thereby complicating the boundary conditions and also periodically exposing the reservoir to the atmosphere. Mixing of the central core was not performed due to the difficulties of mechanically mixing the small core volume. We have developed and applied an alternative columntype method for evaluating radial diffusion in which we pack a column with the low-permeability material for which diffusion rates are to be measured and then core the column to create a central permeable channel. The channel is backfilled with a high-permeability and nonsorbing material, through which solute is subsequently convected. The filling of the central macropore with a porous medium serves both to minimize channel volume and to provide radial mixing (dispersion) within the macropore region. As solute flows through the macropore region, solute will diffuse radially into (and subsequently out of) the low-permeability annular zone. Radial mass transfer effects in the macropore are neglected since the rate of transverse dispersion within this central core of porous material is much greater than the pore diffusion in the surrounding annulus. On the basis of subsequently measured longitudinal dispersion coefficients and an assumption that transverse dispersivity is roughly 6-24 times lower than the longitudinal dispersivity [see Fetter (9) and references therein], radial macropore disperson coefficients are roughly 50-3500 times higher than our subsequent estimates of the pore diffusion coefficients in the annular region (Dp as defined subsequently). Radial concentration gradients within the macropore should therefore be negligible. In this way, the mixing problems associated with the static radial diffusion cell (7, 8) are avoided. In addition, through proper design of the continuous flow system, exposure of solute to both the atmosphere and sorbing laboratory materials is avoided, and very precise effluent breakthrough data can be obtained. These data are then fitted with a solute transport model in which the effective diffusion coefficient through the immobile region is the single fitted parameter. Conceptually, similar methods have been used to determine diffusion coefficients in polymer films (10-12). In this paper, we describe the application of this macropore column method to the measurement of diffusion coefficients for sorbing and nonsorbing solutes through aquitard material obtained from the subsurface coring of a site at Dover Air Force Base, DE. Measurements were made on repacked material that closely mimicked the porosity and visual consistency of the in-situ material. While this approach has been advantageous to us in terms of our ability to independently measure sorption properties and physical parameters (and to thereby test some of our modeling assumptions and method accuracy), we recognize that repacked columns may not fully reflect some of the more subtle aspects of in-situ structure. In this regard, we believe that tests with the macropore column using intact cores should also be possible and would be a valuable future extension of the method. Meanwhile, we offer the subsequently reported results as a demonstration of the method and as a single set of accurate results on our well-defined and homogeneous sample. S0013-936X(97)01132-2 CCC: $15.00
1998 American Chemical Society Published on Web 07/28/1998
The boundary condition at the column inlet for square wave and continuous inputs is
vmcm(0,t) - DH
∂cm (0,t) ) vmcinput(t) ∂x
(6)
where cinput(t) is the input concentration as a function of time (kg/m3). The boundary condition at the outlet is
∂cm (L,t) ) 0 ∂x
(7)
where L is the column length (m). The goal of this work was to arrive at a measurement for the effective pore diffusion coefficient (Dp in eq 3) through the pore fluid of the aquitard material in the column annulus. The pore diffusion coefficient is related to the diffusion coefficient in free solution by
Dp ) Dm/χ
FIGURE 1. Diagram of macropore column construction.
Modeling As shown in Figure 1, the macropore system consists of two regionssan annulus of relatively low hydraulic conductivity and a central core of comparatively high hydraulic conductivity. The model equations for solute transport through a macropore with radial diffusion into a surrounding matrix have been given by van Genuchten et al. (13). For the case without sorption in the macropore and linear sorption isotherms in the annular region, the transport equation is
( )(
)
∂cm a a + FaKd ∂ca ∂2cm ∂cm b2 + 2-1 ) DH 2 - vm ∂t m a ∂t ∂x a ∂x where a is the radius of the macropore (m); b is the outside radius of the annulus (m); ca(x,t) is the average aqueous concentration in the annular region at position x and time t (kg/m3); cm(x,t) is the aqueous concentration in the macropore at position x and time t (kg/m3); DH is the hydrodynamic dispersion coefficient within the macropore (m2/s); Kd is the local linear sorption distribution coefficient in the annular region (m3/kg); t is time (s); vm is the velocity of fluid in the macropore (m/s); x is axial distance (m); m is the local macropore region porosity; a is the local annular region porosity; and Fa is the local annular region bulk density [kg/m3]. The average concentration in the annular region is given by
ca )
2 b - a2 2
∫ rc b
a
a
dr
where r is the radial coordinate (m). The concentration distribution in the annular region is governed by Fickian diffusion, as described by
(
)
( )
a + FaKd ∂ca 1 ∂ ∂ca r ) Dp a ∂t r ∂r ∂r
(3)
where ca(x,r,t) is the local aqueous concentration in the annular region at axial position x, radial position r, and time t; and Dp is the effective pore diffusion coefficient in the annular region. Boundary conditions of the annular region are
ca(x,a,t) ) cm(x,t)
(4)
∂ca (x,b,t) ) 0 ∂r
(5)
(8)
where Dm is the molecular diffusion coefficient in bulk solution (m2/s) and χ is a tortuosity factor. The tortuosity factor accounts for all factors that tend to reduce solute diffusion through the porous media such as the tortuous nature of the diffusion path, dead end pores, and steric hindrance (2, 14). As described in further detail subsequently, the above equations were incorporated into a CrankNicolson finite difference approximation scheme and solved numerically in a Fortran computer program. With independent measurement of flow characteristics, solute partitioning coefficients, and physical parameters of the porous media, Dp is left as the single fitting parameter of the modeling simulations.
Materials and Methods Solids. Solids used in this study include 40-60 mesh Ottawa sand (Fisher Scientific; Pittsburgh, PA) and material from an aquitard underlying an unconfined aquifer at Dover Air Force Base (DAFB), DE (15). The DAFB material was obtained from a depth of between 47 and 49 ft below ground surface in subsurface cores identified as PPC-5 and CPC-5 [see Figure 1 in Ball et al. (16)]. The material was air-dried, homogenized, and stored in plastic bags prior to use. This material has been previously identified as “orange silty clay loam” (OSCL; 15, 16) and is relevant to an ongoing investigation of contaminant diffusion in the DAFB aquitard (17). The material comprises 35% clay-sized particles (
