Environ. Sci. Technol. 2004, 38, 5721-5728
Anisotropic Diffusion in Layered Argillaceous Rocks: A Case Study with Opalinus Clay L U C R . V A N L O O N , * ,† J O S E P M . S O L E R , ‡ WERNER MU ¨ LLER,† AND MICHAEL H. BRADBURY† Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland, and Institute Ciencies de la Terra, “Jaume Almera” (CSIC) Lluis Sole I Sabaris, s/nES-08028 Barcelona, Spain
Anisotropic diffusion was studied in Opalinus Clay, a potential host rock for disposal of spent fuel, vitrified highlevel waste, and long-lived intermediate-level waste in Switzerland. Diffusion parallel to the bedding was measured using a radial through-diffusion technique and diffusion perpendicular to the bedding by the classical (planar) throughdiffusion technique. The samples used were from Mont Terri (MT) and from Benken (BE). Diffusion of HTO, 36Cl-, and 22Na+ parallel and perpendicular to the bedding was studied under confining pressures of 7 MPa (MT) and 14 MPa (BE). The results indicate that diffusion parallel to the bedding is faster than diffusion perpendicular to the bedding by a factor of 4-6 for the three radionuclides, indicating that the Opalinus Clay is anisotropic. This might be explained by smaller path lengths (tortuosity) for species diffusing parallel to the fabric. The degree of anisotropy is slightly smaller for Opalinus Clay from Mont Terri than from Benken. This is due to the lower overburden pressure in Mont Terri resulting in a lower preferential orientation of the clay platelets.
Introduction The diffusion of species in rocks depends strongly on geometric parameters such as tortuosity and constrictivity. Tortuosity takes into account path lengthening because of the particulate nature of the porous medium (1), while constrictivity is concerned with pore narrowing. For aqueousphase diffusion the relationship between the effective diffusion coefficient, De, the geometric parameters, and the diffusion coefficient in water, Dw, is given by (2-4)
De )
δ‚ ‚ Dw τ2
(1)
where δ represents the constrictivity, τ is the tortuosity, and is the diffusion accessible porosity. Tortuosity and constrictivity are usually understood as purely geometric factors which, compared with a specific cross-section in free water, lengthen the diffusion pathway and reduce the diffusion cross-section, respectively. The tortuosity is defined as the effective length of diffusion through the rock divided by the * Corresponding author phone: +41-56-3102257; e-mail: luc.
[email protected]. † Paul Scherrer Institut. ‡ Institute Ciencies de la Terra. 10.1021/es049937g CCC: $27.50 Published on Web 09/28/2004
2004 American Chemical Society
FIGURE 1. Structure of a mud rock (A) during sedimentation (houseof-cards structure) and (B) after compaction (preferential orientation of clay platelets perpendicular to the load direction). Diffusion perpendicular and parallel to the fabric (C) and a sketch of anisotropic diffusion in a perfectly layered sample (D). macroscopic straight-line distance of diffusion, such that the tortuosity >1. Argillaceous rocks are mainly composed of clay platelets that have sedimented in an aqueous environment to form a mud deposit. Bennett et al. (5) reported that the clay platelets in marine deposits of low porosity are oriented preferentially perpendicular to the direction of sedimentation. During sedimentation the grains form “house-of-cards” type structures because of the electrostatic repulsion between likecharged basal planes and the attraction by oppositely charged edge and basal planes (Figure 1A). The solution composition and ionic strength have an important effect on the structure (6). After deposition, the layers of clay mud are compacted by the weight of younger, superimposed sediments (overburden pressure), resulting in water being squeezed out. During this process the particles are forced closer together and the porosity of the mud decreases. Compaction of mud has been reported to increase the preferential orientation of clay platelets (7-9). A recent study by Lash and Blood (10) gave convincing evidence for the role of gravitational pressure on the fabric of shales. They observed that mudstones recovered from pressure shadows adjacent to lateral edges of concretions are characterized by an open fabric of randomly arranged clay domainssthe cardhouse fabric. Laterally equivalent shale samples collected only 0.2 to 0.3 m distant the pressure shadows, however, reveal a low-porosity, strongly oriented shale fabric. The open clay fabric observed in the pressure-shadow mudstones was preserved by the incompressible concretions during burial. Flocculated clay beyond the shielding effect, however, collapsed during progressive burial to form the strongly oriented shale fabric. Owing to this preferentially layered structure of argillaceous rocks, tortuosity (path lengthening) is expected to be anisotropic and consequently also is the effective diffusion in such rocks (Figure 1B,C). Anisotropic diffusion has been observed in many different media; e.g. in crystals, textile fibers, and polymers in which the molecules have a preferential orientation (11, 12). When a species diffuses parallel to the fabric, tortuosity is expected to be smaller than in the case of diffusion perpendicular to the bedding. Consequently, the effective diffusion rate will be faster. VOL. 38, NO. 21, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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Although anisotropic permeability (hydraulic and thermal conductivity) has been described in the literature (9, 13-16), in geochemistry, little attention has been paid so far to anisotropic diffusion properties. Systematic studies comparing diffusion perpendicular and parallel to the fabric for the same material and for different species are rare. The authors are aware of only one study (17) in which the diffusion of HTO and I- was measured through a piece of Opalinus Clay perpendicular and parallel to the fabric. Although they observed a larger value for diffusion parallel to the bedding, they could not exclude experimental artifacts, so that their results remain inconclusive. Most studies of natural argillaceous materials published in the open literature are on diffusion properties perpendicular to bedding (18-20) and in one case on diffusion parallel to bedding (21). In this work, the diffusion of three different species, i.e., the uncharged tritiated water (HTO), the anion 36Cl-, and the cation 22Na+ was studied perpendicular and parallel to the bedding of Opalinus Clay (OPA), a layered, argillaceous rock. Opalinus Clay was chosen because this mud rock is a potential host rock for the disposal of spent fuel, high-level vitrified waste and long-lived intermediate level waste in Switzerland (22). Opalinus Clay is known to be layered and shows hydraulic and seismic anisotropy (23). The aims of the study were (i) to demonstrate anisotropic diffusion of charged and uncharged species in a layered rock and (ii) to quantify the differences or similarities between the various species. Because the pressure on the samples had to be perpendicular to the bedding, two different setups were be used. A cylindrical geometry was the most convenient way to achieve these requirements for diffusion parallel to the bedding with a stress perpendicular to the bedding. We thus used a radial through-diffusion technique which will be presented in this work. For diffusion perpendicular to the bedding with a load also perpendicular, a one-dimensional (planar) through-diffusion method, as described in refs 24 and 25, was used.
Materials and Methods Radial Diffusion. Equipment. A sketch of the radial diffusion setup is given in Figure 2. A similar arrangement was used earlier (26, 27) for determining isotopic composition, chemistry, and effective porosities for groundwater in aquitards and for determining effective diffusion coefficients, porosity, and adsorption from in-diffusion experiments in porous geological materials. The technique developed by van der Kamp et al. (26) operated in the transient mode (in-diffusion, out-diffusion), whereas the procedure described here is essentially a steady-state method. In the through-diffusion technique, both a transient and a steady-state phase is available. The transient phase gives information on the rock capacity factor (see eq 4) or on the diffusion accessible porosity in the case of nonsorbing tracers, and the steadystate phase gives directly the effective diffusion coefficient. There is only one combination of these two parameters which can describe the flux vs time curve obtained in throughdiffusion measurements. A concentration gradient is set up across the sample in such a way that diffusion takes place from the center of the sample (high concentration region) to the outer boundary of the sample (low concentration region). Because the concentration gradient is set up to be parallel to the layering of the sample, diffusion will take place along the layering. The radial diffusion cell is schematically presented in Figure 3a. The cell comprises a sample holder, two end plates, and two cylindrical stainless steel filters. The purpose of the stainless steel filters is to ensure the integrity of the sample during prolonged contact with the pore water. A cylindrical sample with a central hole (rext ) 25.4 mm, rint ) 6.58 mm, h ) 52 mm) is used in this setup. One stainless steel filter 5722
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FIGURE 2. Radial through-diffusion setup (cross-sectional and top view). (rext ) 6.58 mm, rint ) 4.78 mm, h ) 61 mm, porosity ) 41.8 ( 0.8%, 316L stainless steel, pore diameter 10 µm; MOTT industrial division, Farmington, USA) is positioned in the central hole along the axes of the sample, and the other (rext ) 26.98 mm, rint ) 25.4 mm, h ) 61 mm, porosity ) 36.7 ( 0.6%) contacts the outer boundary of the sample. A stainless steel pin (r ) 4.78 mm, h ) 61 mm) with a spiral groove is placed in the center of the inner filter. The sample is put in the sample holder (rint ) 27 mm, L ) 61 mm), and the two end-pieces are placed in position. The wall of the sample holder has a spiral groove at the inner surface. The two end-pieces are held together by a set of 4 bolts. Packages of 12 disk springs (GR1, DIN 2093 - A16) are placed between the surface of the end plate and the head of the bolt. A well-defined load is applied to the cells by tightening the bolts. The cells are equipped with a load sensor (SENSY, Belgium) connected to a force indicator (FMD-6, Penko, The Netherlands) in order to monitor the pressure applied to the samples. The dead volumes (spiral groove, filter and tubings) are 4.8 ( 0.5 cm3 and 18.2 ( 1.8 cm3 at the inner and outer side of the cell, respectively. Theory of Radial Diffusion. Consider a hollow cylindrical sample of height h with an internal radius rint and an external radius rext. The high-concentration reservoir is the central cavity of the cylinder (from r ) 0 to r ) rint), and the breakthrough of the tracer occurs across the entire external surface (r ) rext). Consequently, the diffusion process has a cylindrical symmetry. The radial diffusion is described by
∂C 1 ∂ ∂C ) rDa ∂t r ∂r ∂r
(
)
(2)
where Da is the apparent diffusion coefficient [m2.s-1], and C is the concentration of a given tracer in the rock [mol‚m-3].
and the tracer flow (mol‚s-1 or Bq‚s-1) at the external surface of the cylinder is equal to
( ∂C∂r )
-2πhDe r
(5)
r )rext
The analytical solution of the cylindrical through-diffusion problem is given by (11)
A(rext,t) πC0Rh
)
2(Dat - Λ)
-
ln(rext/rint) 4
∞
J0(rint βn)J0(rext βn) exp(-Da βn2t)
n)1
βn2(J02(rint βn) - J02(rext βn))
∑
(6)
where A(rext,t) is the total amount [in mol] or activity [in Bq] of tracer which has diffused through the sample at time t and C0 is the tracer concentration [Bq‚m-3] in the high concentration reservoir. The parameter Λ [m2] in eq 6 is given by
Λ)
rint2 - rext2 + (rint2 + rext2)ln(rext/rint)
(7)
4ln(rext/rint)
The J0(x) terms are Bessel functions of the first kind of order zero, and the βn terms are the positive roots of
U0(rint βn) ) J0(rint βn)Y0(rext βn) - J0(rext βn)Y0(rint βn) ) 0 (8) and the Y0(x) terms are Bessel functions of the second kind of order zero. The first five positive roots rintβn of eq 8 for different ratios rext/rint are tabulated in ref 11, Table 5.3, p 380). At steady state, tf∞ and eq 6 reduces to
A(rext,t) )
2πC0h
(Det - RΛ)
(9)
ln(rext/rint)
Hence the total amount of tracer diffused through the sample becomes a linear function of time
A(rext,t) ) a‚t - b FIGURE 3. a: Cross-section view of the radial through-diffusion cell. b: Cross-section view of the planar through-diffusion cell.
with
a)
The apparent diffusion coefficient is defined as
Da )
De R
(3)
where De the effective diffusion coefficient [m2‚s-1] is related to the diffusion coefficient in free bulk water via eq 1, and R is the rock capacity factor and is defined as
R ) + F‚Kd
2πC0hDe ln(rext/rint)
and
b)
2πC0hRΛ ln(rext/rint)
The concentration profile of the tracer in the rock at steady state is logarithmic
(4) •
where Kd is the distribution coefficient [m3‚kg-1], and F is the bulk dry density of the rock [kg‚m-3]. For nonsorbing tracers (Kd ) 0) the rock capacity factor equals the diffusion accessible porosity, . The initial and boundary conditions are
C(rint e r e rext, t ) 0) ) 0 C(0 e r e rint, t > 0) ) C0 C(r g rext, t > 0) ) 0
(10)
C(r) ) C0
( ) rint r 1rint ln rext ln
(11)
and this relation also satisfies the boundary conditions, i.e., for r ) rint, C(r) ) C0 and for r ) rext C(r) ) 0. A typical concentration profile at steady state (eq 11) is depicted in Figure 2. The effective diffusion coefficient of the tracers in the filters is 1.2 × 10-10 m2 s-1 (28). The transport times in the filter and the rock were calculated using 〈d2〉 ) 4‚Da‚t, where VOL. 38, NO. 21, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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d is the average diffusion length [m], Da is the apparent diffusion coefficient [m2 s-1], and t is time [s]. The diffusion times of HTO calculated for the filter and OPA (Mont Terri) are 3.2 h and 76.9 h, respectively. Because HTO has the largest De value, this case represents the fastest one. The diffusion time through the filter is only 4% of that through the rock. The relative error on the porosity and De caused by the use of a filter is at maximum 4%. For 36Cl- and 22Na+, the diffusion times in the rock are longer so that the relative error on the rock capacity factor and effective diffusion coefficient caused by the use of the filter is
