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diffusion experiment in Boom Clay at 30 °C
Physics and Chemistry of the Earth 65 (2013) 72–78
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A combined glass dissolution/diffusion experiment in Boom Clay at 30 °C Marc Aertsens ⇑, Karel Lemmens SCKCEN, Boeretang 200, B-2400 Mol, Belgium
a r t i c l e
i n f o
Article history: Available online 30 May 2013 Keywords: Boom Clay Silica Glass dissolution Diffusion accessible porosity Retardation factor Apparent diffusion coefficient
a b s t r a c t At the appropriate times, silica diffusion in clay is possibly the rate determining process for the dissolution of vitrified waste disposed of in a clay layer. For testing this hypothesis, combined glass dissolution/ silica diffusion experiment are performed. SON68 glass coupons doped with the radioactive tracer 32Si are sandwiched between two cores of humid Boom Clay, heated to 30 °C. Due to glass dissolution, 32Si is released and diffuses into the clay. At the end of an experiment, the mass loss of the glass coupon is measured and the clay core is sliced to determine the diffusion profile of the 32Si released from the glass in the clay. Both mass loss and the 32Si diffusion profile in the clay are described well by a model combining glass dissolution according to a linear rate law with silica diffusion in the clay. Fitting the experiments to this model leads to an apparent silica diffusion coefficient in the clay between 7 1013 m2/s and 1.2 1012 m2/s. Previously determined values from diffusion experiments at 25 °C are around 6 1013 m2/s (In-Diffusion experiments) and 2 1013 m2/s (percolation experiments). The maximal glass dissolution rate for glass next to clay is around 1.6 107 g glass/m2 s (i.e. 0.014 g glass/m2 day). In undisturbed clay, the measured silica concentration is around 5 mg/L. Combining these values with the previously measured (In-Diffusion experiments) product of accessible porosity and retardation factor, leads in two ways to a silica glass saturation concentration in clay between 8 and 10 mg Si/L. Another candidate for the rate determining process of the dissolution of vitrified waste disposed in a clay layer is silica precipitation. Although silica precipitation due to glass dissolution has been shown experimentally at 90 °C, extending the model with silica precipitation does not lead to much better fits, nor could meaningful values of a possible precipitation rate be obtained. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction In Belgium the disposal of vitrified radioactive waste in Boom Clay is considered. Safety studies require to estimate the release rate of radionuclides from the host glass and their transport rate in Boom Clay. Experiments have shown that many of the radionuclides dissolve congruently with the glass. Dissolution of glass next to clay is described by a model (Pescatore, 1994) according to which at large times the glass dissolution rate is controlled by the transport parameters of dissolved silica in Boom Clay: the apparent diffusion coefficient D and the product gR of the diffusion accessible porosity g and the retardation factor R. The values of these parameters have been measured by diffusion experiments (Aertsens et al., 2003; Aertsens et al., 2008) confirming silica retardation in Boom Clay. Both dissolved silica and clay are negatively charged, so initially silica sorption
⇑ Corresponding author. Tel.: +32 14 33 31 31; fax: +32 14 32 35 53. E-mail address: [email protected] (M. Aertsens). 1474-7065/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.pce.2013.05.008
on clay was not expected. Possible mechanisms explaining it have been put forward (De Cannière et al., 1998). The silica transport parameters in Boom Clay and the Pescatore model are validated by experiments where 32Si doped glass dissolves next to clay. This allows to determine not only the glass mass loss as a function of time, but also the 32Si diffusion profile in the clay. The combination of both types of measurement makes it possible to verify if the 32Si profile in the clay can indeed be coupled to the glass mass loss. If so, all parameters of the Pescatore model can be measured. Because at 90 °C silica precipitates have been observed next to a dissolving glass in contact with clay (Pozo et al., 2007), a precipitation term is added to the Pescatore model. An overview of possible precipitation mechanisms is given by Gin et al. (2001). The Pescatore model does not describe the precipitation of secondary phases (especially phyllosilicates) on the glass surface, which can form a sink for silica released from the glass (Frugier et al., 2008). This paper starts by describing the experiment. From the Pescatore model, two approximate expressions for the 32Si profile in the clay are derived. Two extensions of the Pescatore model
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M. Aertsens, K. Lemmens / Physics and Chemistry of the Earth 65 (2013) 72–78
are derived: one taking into account the finite size of the clay, and one taking into account silica precipitation in the clay. These models are used afterwards to fit the experimental data. 2. Description of the experiment SON68 glass coupons with a thickness of 2 mm and a diameter of 16 mm were doped with the radioactive tracer 32Si, leading to a specific glass activity qBq,glass = 1.24 105 Bq/g glass. The glass samples were sandwiched between two cores of humid Boom Clay with a length of typically about 2 cm each and pressed into a percolation cell with a diameter of 20 mm (see Fig. 1). After the saturation of the clay by percolation with clay water, the valves at the inlet and outlet were closed and the cells were heated to 30 °C. Possible voids around a glass coupon caused by the different diameters of the glass and the clay are unlikely because of the swelling capacity of the clay. Due to glass dissolution, 32Si is released from the glass and diffuses into the clay. After stopping the experiment, the glass weight (mass) loss (subscript WL) per surface unit QWL (unit: g glass/m2), was measured. Then the clay core was sliced to determine the 32Si diffusion profile. Therefore, the clay slices were dried, and after secular equilibrium of 32Si with its daughter 32P, in each slice the activity was measured with a Packard Auto-Gamma 5650 NaI(Tl) counter. The ratio QBq (unit: g glass/m2; Bq refers to activity):
Q Bq ¼
BBq;clay SqBq;glass
ð1Þ
with BBq,clay the total activity (unit: Bq) in clay and S the contact surface (unit: m2) provides another estimate for the glass mass loss. The mass loss QBq based on the amount of 32Si in the clay, is larger than the weight loss QWL because during the dissolution of glass next to clay, clay components are integrated in the outer part of the glass alteration layer. This process decreases the weight loss QWL, but it hardly changes the mass loss QBq based on 32Si because there is almost no 32Si in the clay next to the glass (initially the 32Si content in the clay is zero, due to glass dissolution this content slowly rises in the vicinity of the glass/clay interface). Since the mass loss QBq based on 32Si is larger than the weight loss QWL (QBq > QWL), the specific glass activity based on weight loss qBq,WL (unit: Bq/g glass), defined as
qBq;WL ¼
BBq;clay Q Bq ¼ q S Q WL Q WL Bq;glass
ð2Þ
is larger than the specific glass activity qBq,glass. In similar experiments with undoped SON68 glass, only the weight loss is measured. At the end of such an experiment, a few milliliter of Boom Clay water was pressed through the clay core for analysis. The samples were not filtered, and analyzed by ICP–AES, ICP–MS and ion chromatography, leading to a clay water composition at the end of the experiment. This clay water composition has also been determined for blank tests, where real Boom
Clay water (taken from the Underground Research Lab, where the temperature is around 13 °C) was percolated through a clay core at 30 °C. The experiments with 32Si doped SON68 were stopped after 887 days, 1227 days, 1570 and 1890 days. 3. Models The Pescatore model (Pescatore, 1994) for dissolution of glass next to clay is used to derive two approximate expressions for the 32Si concentration in the clay. Then the model is extended by taking into account the finite size of a clay core. In a second extension, a precipitation term is added. 3.1. The model in an infinite system The behavior of dissolved silica in clay is described by the diffusion equation:
@C @2C ¼ D 2 kðC C 0 Þ @t @x
with C(x, t) the concentration of silica in the pore water accessible for diffusion (unit: g Si/m3), t time, D the apparent diffusion coefficient of silica in the pore water of the clay (unit: m2/s), x the distance to the fixed glass/clay interface (where x = 0), k the precipitation rate (unit: 1/s) and C0 the background silica concentration in the clay pore water (unit: g Si/m3). Glass dissolution is assumed to be congruent and caused by the dissolution of its main component, silica. Silica is supposed to dissolve according to a linear rate law:
Cðx ¼ 0; tÞ @C ¼ gRD Jðx ¼ 0; tÞ ¼ a 1 @x x¼0 c
ð4Þ
with J(x = 0, t) the flux of dissolving silica at the glass surface (which is chosen at the origin) (unit: g Si/m2 s), a the maximal silica dissolution rate (unit: g Si/m2 s), c a constant characterizing the saturation of glass silica in water (unit: g Si/m3), g the clay porosity accessible for diffusion and R the silica retardation factor in clay. The silica concentration in the clay is C0 (C(x > 0, t = 0) = C0). According to the linear rate law (4), the net silica dissolution rate J(x = 0, t) is the difference between two processes: silica dissolution at a constant rate a and the adsorption of silica from the pore water of the clay on the glass alteration layer (with rate a C(x = 0, t)/c). The solution of (3), (4) without precipitation (k = 0) is (Pescatore, 1994)
C0 F 1 ðx; tÞ Cðx; tÞ ¼ C 0 þ c 1
ð5Þ
c
with F 1 ðx; tÞ ¼ erfc
pffiffiffiffiffiffi x x 2 pffiffiffiffiffiffi expðhx þ h DtÞerfc pffiffiffiffiffiffi þ h Dt ; 2 Dt 2 Dt
h¼
a gRDc
ð6Þ
The function F1(x, t) is dimensionless, while the unit of h is m1. Expressions (5) and (6) are used for deriving expressions for the tracer silica bulk concentration Cb,Bq(x, t) (unit: Bq/m3), measured in our experiments. In case of no adsorption of silica from the clay pore water on the glass alteration layer, the tracer silica bulk concentration Cb,Bq(x, t) is the product of the excess silica bulk concentration gR(C(x, t) C0) (unit: g Si/m3) and the specific silica activity qBq,glass/fSi (unit: Bq/g Si), where fSi = 0.21 (unit: g Si/g glass) is the silica weight fraction in the glass. Introducing the notations
cWL ¼ gR
Fig. 1. Set-up of the combined glass-dissolution/silica diffusion experiment.
ð3Þ
c fSi
C0 ; 1
c
aWL ¼
a fSi
C0 C0 ¼ aWL 1 ¼ hDcWL 1
c
c
ð7Þ
with cWL (unit: g Si/m3) the excess glass bulk concentration needed to saturate the clay with respect to the glass silica saturation c,
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M. Aertsens, K. Lemmens / Physics and Chemistry of the Earth 65 (2013) 72–78
aWL = a/fSi (unit: g glass/m2 s) the maximal (forward) glass dissolution rate and aWL (unit: g glass/m2 s) the initial glass dissolution rate, leads after insertion in Cb,Bq(x, t) = gR(C(x, t) C0) qBq,glass/fSi to C b;Bq ðx; tÞ ¼ qBq;glass cWL F 1 ðx; tÞ
ð8Þ
it will remain quite small (the altered glass contains only trace concentrations of 32Si). Mathematically, the profile Cb,Bq(x, t) is obtained by solving with the Laplace transform method (Crank, 1975) the diffusion Eq. (1) with boundary condition J(x = 0, t) = af (instead of expression (4), index f stands for forward rate), leading to:
In case of adsorption of silica from the clay on the glass alteration layer, it is assumed that the tracer silica bulk concentration Cb,Bq(x, t) is still proportional to the excess silica bulk concentration gR(C(x, t) C0), and expression (8) is replaced by:
C b;Bq ðx; tÞ ¼ C f ;1 ðx; tÞ rffiffiffiffiffiffi ! aBq;f Dt x2 x ¼ xerfc pffiffiffiffiffiffi 2 exp D p 4Dt 2 Dt
C b;Bq ðx; tÞ ¼ qBq;fit cWL F 1 ðx; tÞ
with aBq,f (unit: Bq/m2 s) a fit parameter and Cf,1(x, t) the notation for the expression on the right hand side of (14). Fitting the tracer profile with expression (14) leads to values for aBq,f and the apparent diffusion coefficient D, which do not depend on the values of the parameters used to fit the weight loss (see expression (10)). The value of the ratio aBq;f =aWL ¼ qBq;fit;f is expected to be close to the specific glass activity qBq,glass. Fitting the experiments should make clear which expression (13), (14) is the best approximation for the silica bulk concentration Cb,Bq(x, t). In the ideal case, both expressions lead to good fits and similar values for the fit parameters.
ð9Þ
Due to adsorption on the glass alteration layer, part of the clay silica concentration C0 is replaced by ‘glass silica’ containing 32Si, making the proportionality factor qBq,fit (unit: Bq/g glass), which will be estimated further on, larger than the specific glass activity qBq,glass. Because glass dissolution is assumed to be congruent and governed by silica dissolution, the glass weight loss per surface unit QWL(t) (unit: g glass/m2), being the ratio of the silica weight loss Q(t) per surface unit (unit: g Si/m2) and fSi, is given by (Pescatore, 1994):
1 fSi
Z
t
0
Jðx ¼ 0; t 0 Þdt pffiffiffiffiffiffi 1 2 pffiffiffiffiffiffi 2 ¼ aWL 2 expðh DtÞerfcðh Dt Þ 1 þ pffiffiffiffi h Dt p h D
Q WL ðtÞ ¼
3.2. The model in a finite system
0
ð10Þ
According to expression (10), the weight loss depends on two parameters: the initial glass dissolution rate aWL (unit: g glass/ pffiffiffiffi m2 s) and the combination h D (unit: 1/s0.5). Initially the weight loss is proportional to time (Q WL ðtÞ aWL t). At times t s = 1/ h2D, it becomes proportional to the square root of time pffiffiffiffiffiffi pffiffiffiffiffiffi a pffiffi (Q WL ðtÞ p2ffiffipffi c 1 Cc0 gfSiR Dt ¼ p2ffiffipffi cWL Dt ¼ p2ffiffipffi hpWLffiffiDffi t). Returning to the tracer bulk concentration profile Cb,Bq(x, t), according to mass conservation is
Q WL ðtÞ ¼
gR
Z
fSi
ðCðx; tÞ C 0 Þdx
1
F 1 ðx; tÞdx ¼
0
1
qBq;fit
Z
C b;Bq ðx; tÞdx
C b;Bq ðx; tÞ ¼
0
1 X
C f ;1 ð2Lðn þ 1Þ x; tÞ þ
n¼0
a
ffipffiffiffiffi F 1 ðx; tÞ C b;Bq ðx; tÞ ¼ qBq;fit cWL F 1 ðx; tÞ ¼ qBq;fit pffiffiffiWL h D D
a
ð15Þ
ð12Þ
1
A good fit of the tracer profile in the clay requires that R1 R1 Bq;clay S 0 C b;Bq ðx; tÞdx. Substituting Bq;clay S 0 C b;Bq ðx; tÞdx in expression (12), and taking into account expression (2) allows to estimate qBq,fit as qBq,fit qBq,WL. To test this estimate, each tracer profile is fitted simultaneously with all weight losses twice: once considering the specific activity qBq,fit as a fit parameter and once taking, apart from a fixed proportionality factor (see Section 4 for details) qBq,fit and qBq,WL equal. After introducing the notation aBq = aWL qBq,fit into expression (9),
Bq ¼ pffiffiffiffipffiffiffiffi F 1 ðx; tÞ h D D
aBq
C b;Bq ðx; tÞ ¼ pffiffiffiffipffiffiffiffi ðF 1 ðx; tÞ þ F 1 ð2L x; tÞÞ h D D
ð11Þ
0
Z
All just presented solutions assume an infinitely long clay core. This assumption is justified as long as the concentration C(x = L, t) with L the length of the clay core remains sufficiently low (much smaller than the concentration difference c C0). At larger times, the boundary condition is changed into a zero flux boundary condition at x = L. In first instance, our calculations adapt expression (13) to
As long as the product (c C0) F1(2L, t) remains much smaller than (c C0) (which is the case in the here presented experiments), expression (15) is a good approximation. Similarly, due to the zero flux boundary condition at x = L, expression (14) changes into
1
Substituting expression (9) in expression (11) leads to
Q WL ðtÞ ¼ cWL
ð14Þ
ð13Þ
In case the specific activity qBq,fit is considered as a fit parameter, four parameters are fitted simultaneously from the weight losses and one tracerpffiffiffi profile: the initial glass dissolution rate aWL , the ffi combination h D, the apparent diffusion coefficient D and the dissolution rate aBq. If qBq,fit and qBq,WL are taken equal, the parameters D, aBq and cWL are fitted. An alternative way to estimate the tracer silica bulk concentration Cb,Bq(x, t) is to neglect in the linear rate law (4) the adsorption term a C(x = 0, t)/c. For the tracer concentration in clay, this is a good approximation because initially it is zero, and also afterwards
1 X C f ;1 ð2Ln þ x; tÞ
ð16Þ
n¼0
where the function Cf,1(x, t) is introduced in expression (14). The additional term in expression (15) does not affect expression (10) for the weight loss. 3.3. Extension of the model in an infinite system with precipitation For simplicity, only the infinite system is considered. According to Crank (Crank, 1975), the solution Ck(x, t) of the diffusion Eq. (3) with a non-zero precipitation rate k can be calculated from the solution C(x, t) of the same equation without the precipitation term:
C k ðx; tÞ ¼ k
Z
t
Cðx; tÞ expðktÞdt þCðx; tÞ expðktÞ
ð17Þ
0
Substituting expression (5) into expression (17) leads for h2D – k to
pffiffiffiffiffiffi x p ffiffiffiffiffi ffi Dt þ h 2 2 Dt h Dk rffiffiffiffi ! pffiffiffiffi pffiffiffiffiffi h D pffiffiffiffi pffiffiffi x k x þ ðh D þ kÞerfc pffiffiffiffiffiffi þ kt exp D 2 2 Dt rffiffiffiffi !! pffiffiffiffi pffiffiffiffiffi h D pffiffiffiffi pffiffiffi x k p ffiffiffiffiffi ffi ð18Þ kt exp x h D k erfc þ D 2 2 Dt
C k ðx;tÞ ¼C 0 þ
c C0
2
2
h Dexpðhx þ ðh D kÞtÞerfc
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M. Aertsens, K. Lemmens / Physics and Chemistry of the Earth 65 (2013) 72–78
The amount of precipitated silica Cprec(x, t) is given by
C prec ðx; tÞ ¼ gRk
Z
t
ðC k ðx; yÞ C 0 Þdy
ð19Þ
0
Substituting expression (18) in expression (19) gives
C prec ðx; tÞ ¼
gRðc C 0 Þ 2
h Dk 2 pffiffiffiffiffiffi h D x 2 exp hx þ ðh D kÞt erfc pffiffiffiffiffiffi þ h Dt k 2 2 Dt h Dk pffiffiffiffi pffiffiffiffiffi 2 h D kt x þ kt pffiffiffi pffiffiffiffi exp 4Dt p k pffiffiffiffi pffiffiffi h D pffiffiffiffi pffiffiffi 1 x þ pffiffiffi h D þ k pffiffiffiffi pffiffiffi þ pffiffiffiffi þ 2 kt D h D k 4 k rffiffiffiffi ! p ffiffiffiffi ffi x k x erfc pffiffiffiffiffiffi þ kt exp D 2 Dt pffiffiffiffi pffiffiffi pffiffiffiffi h D pffiffiffi 1 x pffiffiffiffi pffiffiffi þ pffiffiffiffi 2 kt þ pffiffiffi kh D D 4 k h Dþ k rffiffiffiffi !! pffiffiffiffiffi x k ð20Þ x erfc pffiffiffiffiffiffi kt exp D 2 Dt
The total amount of tracer silica Ctot(x, t) is
C b;Bq ðx; tÞ ¼
qBq;fit fSi
ðgRðC k ðx; tÞ C 0 Þ þ C prec ðx; tÞÞ
ð21Þ
Substituting expression (18) in expressions (4) and (10) gives the glass weight loss per surface unit QWL(t): ! pffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffi h D 2 erfcðh Dt Þexpððh D kÞtÞ 1 þ pffiffiffi erf ð kt Þ k h Dk h Dk pffiffiffiffi ! pffiffiffiffiffi h D 1 1 pffiffiffiffiffi ð22Þ kt erf ð kt Þ þ pffiffiffiffi kt expðktÞ kt þ pffiffiffi 2 p k
Q WL ðtÞ ¼ aWL
When fitting the experimental data, it is assumed that apart from a fixed proportionality factor (see Section 4 for details) qBq,fit and qBq,WL are equal so that the four fit parameters are: the concentration cWL , the apparent diffusion coefficient D, the dissolution rate aBq and the precipitation rate k.
1
2
2
h D
2
In case of strong precipitation (h2 D/k 1), expression (22) is well approximated by Q WL ðtÞ aWL t.
Fig. 2. Fitting the glass weight losses versus pffiffiffiffitime, according to expression (9), leads to aWL ¼ 1:7 107 g glass/(m2 s) and h D ¼ 1:4 104 1=s0:5 (see Table 2). The triangles are the experimental data points for the doped glass, for which also a tracer profile in the clay is measured. The circles are the experimental data points for the undoped glass.
4. Experimental and modeling results Table 1 summarizes the mass loss results for the tracer doped glasses. Both the weight loss QWL and the mass loss QBq based on 32 Si are calculated using the real surface of a glass coupon (5.05 104 m2). Due to the very small values of the mass loss, scatter can be a problem, e.g. QWL as well as QBq have nearly the same value for the 887 and 1227 days experiments (QWL 6.5– 7 g glass/m2 and QBq 13 g glass/m2). The ratio QBq/QWL is always larger than one and increases as a function of time. Fitting the experiments to the model, the model (being one dimensional) assumes the same cross section for the glass coupon, the clay cores and the surrounding cell, while in reality the cell (clay core) cross section is a factor (2 3.14 104 m2)/ (5.05 104 m2) 1.24 larger than half the total surface of the glass coupon. Because only a fraction of 1/1.24 0.8 of the cell cross section at the glass/clay interface consists of glass, the specific glass activity based on weight loss qBq,WL averaged over the cell cross section equals the ratio qBq,WL/1.24, which is called in Table 1 the cross section averaged specific glass activity based on weight loss. Fig. 2 shows all weight losses QWL, including the undoped glasses, versus time. The fit according to expression (10) is good. Apart from the 1890 days experiment, the scatter in the measured 32Si profiles, shown in Fig. 3, is limited. Contrary to the three other experiments, where the lengths of the clay cores at both sides of the 32Si doped glass are about equal (around 2–2.2 cm), this was not the case for the 1890 days experiment (lengths of 1.3 and 3.3 cm). For this experiment, because of the different clay core lengths, and also because of the considerable difference between the measured 32Si activity in both clay cores, different symbols are used for the measured activities in both clay cores. Because of the irregularities in the tracer profile of the 1890 days experiment, no fits of the tracer profile of this experiment are presented. In the fits of Fig. 3, all three parameter fits of the mass loss data together with one tracer profile in clay, the agreement between the fit and experimental data (both profile and mass loss) is generally good. Only for the 1227 days experiment, the mass loss is not represented well with the fit parameter values used in Fig. 3. This is clear from the fitted values for both parameters determining the weightploss ffiffiffiffi (the initial glass dissolution rate aWL and the combination h D), which deviate considerably from the values obtained from the other fits (see Table 2). The major problem of the fits is that the measured bulk activity very close (always less than 2 mm) to the glass/clay interface is considerably higher than predicted by the fit. Table 2 summarizes the fit results of the weight loss and the 32Si profiles in the clay cores. Combining the fit results p (for ffiffiffiffi the initial glass dissolution rate aWL and the combination h D) from the
Table 1 Mass loss data for the four doped glass dissolution experiments. Duration Weight loss per surface unit (QWL) Mass loss based on 32Si per surface unit (QBq) Ratio QBq/QWL Specific glass activity (qBq,glass) Specific glass activity from mass loss (qBq,WL) Cross section averaged specific glass activity from mass loss
Days (g glass/m2) (g glass/m2) (Bq/g glass) (Bq/g glass) (Bq/g glass)
887 6.95 12.6 1.82 1.2E+5 2.3E+5 1.8E+5
1227 6.66 13.3 1.99 1.2E+5 2.5E+5 2.0E+5
1570 10.80 36.1 3.35 1.2E+5 4.1E+5 3.2E+5
1890 11.26 45.8 4.07 1.2E+5 5.0E+5 4.1E+5
Bulk Activity (Bq/gram clay)
M. Aertsens, K. Lemmens / Physics and Chemistry of the Earth 65 (2013) 72–78
Bulk Activity (Bq/gram clay)
76
100 887 days
50
0 0
5
10
15
20
1227 days
100
50
0
25
0
5
10
Bulk Activity (Bq/gram clay)
Bulk Activity (Bq/gram clay)
300 1570 days
200
100
0 0
5
15
20
x (mm)
x (mm)
10
15
1.3 cm clay core
300 1890 days
3.3 cm clay core
200
100
0
20
0
10
x (mm)
20
30
x (mm)
Fig. 3. 32Si profiles (expressed as the ratio of the bulk activity CBq,32 (unit: Bq/m3) and the clay density (unit: g/m3)) in the clay cores and the corresponding fits according to Pesc3 (see Table 2 for the values of the fit parameters). For the 1890 days experiment, the correspondence of the measured activities in both clay cores is less good than for the other three experiments. Also because of the higher scatter on the data, a fitted line is not shown on the figure.
Duration (days)
Fit
Fit expressions
Table 2 Parameter values obtained from fitting three glass dissolution experiments in different ways. The first column indicates what is fitted: the weight loss data (WL), the tracer diffusion profile in the clay (P) or both simultaneously (WL + P). The number(s) of the expressions used for the fittings are mentioned in the second column. Which profile is fitted is indicated in the third column. The remaining columns give the optimal parameter values: green cells contain a constant value, yellow cells the parameter values which are directly fitted and non-colored cells contain parameter values derived from the mathematical relations between the parameters combined with gR = 65 and a silica background in the clay C0 = 5 mg Si/L and the values in the green and yellow cells.
D
αBq,f or αBq
α∗WL
h D0.5
γ∗WL
m2 s-1
Bqm-2 s-1
g glass m-2 s-1
s-0.5
g glass m-3
αBq,f WL P P P
10 14 14 14
1,7E-07 887 1227 1570
1,5E-12 9,6E-13 9,3E-13
1,7E-02 1,2E-02 2,4E-02 αBq
WL+P 10,13
887
1,2E-12
3,1E-02
1,6E-07
WL+P 10,13
887
1,2E-12
3,0E-02
1,7E-07
WL+P 10,13 1227
7,8E-13
2,5E-02
1,7E-07
WL+P 10,13 1227
7,3E-13
5,0E-02
2,5E-07
WL+P 10,13 1570
7,7E-13
5,4E-02
1,6E-07
WL+P 10,13 1570
7,7E-13
5,4E-02
1,7E-07
weight loss (expression (10) and Fig. 2) with the fit values for the dissolution rate aBq,f and the apparent diffusion coefficient D obtained from fitting the profiles with the constant rate expression (14) leads by expression (7) to estimates for the concentration cWL . In Table 2, the glass saturation concentration c is derived from the concentration cWL from expression (7) by taking the product gR of the porosity g and the retardation factor R as gR = 65 and the
γ
ρBq,fit,f
or ρBq,fit
mg Bq Si (g glass)-1) liter-1 ρBq,fit,f
1,4E04
1,3E04 1,2E04 1,4E04 3,6E04 1,3E04 1,3E04
τ days
612 1,0E+03 1,2E+03 1,3E+03
8 9 9
9,9E+04 7,3E+04 1,5E+05 ρBq,fit
1,1E+03
8
1,9E+05
686
1,2E+03
9
1,8E+05
751
1,4E+03
9
1,5E+05
634
8,3E+02
7
2,0E+05
92
1,4E+03
9
3,3E+05
686
1,5E+03
10
3,2E+05
678
silica background concentration in the clay C0 = 4.8 mg Si/L. Silica In-Diffusion experiments in Boom Clay at 25 °C lead to a gR value around 60–70 (leading to the estimate gR = 65), percolation experiments to a value between 100 and 150 (Aertsens et al., 2003; Aertsens et al., 2008). The silica background concentrations C0 measured in four blank tests with different durations (up to 1983 days) are 5; 5.8; 4 and 4.5 mg Si/L, resulting in an average
77
M. Aertsens, K. Lemmens / Physics and Chemistry of the Earth 65 (2013) 72–78 Table 3 Silica and boron concentrations in the percolated pore water at the glass side of the clay core at the end of the experiments. Time (days)
Sample volume (g)
Sample/pore volume
pH
B exp (mg/L)
Si exp (mg/L)
Si calc (mg/L)
1110 1148 1463 1837 2199
12.64 9.78 7.26 0.97 0.96
5.44 4.21 3.12 0.42 0.41
8.6 8.7 8.5 8.6 8.6
15.3 16.0 17.7 18.4 25.0
6.7 6.0 7.3 7.2 8.7
7.5 7.5 7.6 7.7 7.8
C0 = 4.8 mg Si/L. All weight losses have been fitted together with each of the tracer profiles (using expression (13)) twice: a four parameter fit with the specific glass activity qBq,fit as fit parameter and a three parameter fit where qBq,fit has the value of the cross section averaged specific glass activity based on weight loss (qBq,WL/1.24). It was verified that for all fits the value of (c C0) F1(2L, t) was sufficiently low. Comparing Tables 1 and 2 show that for the 887 and 1570 days experiments, a fit of qBq,fit leads indeed to qBq,WL/1.24. For the 1227 days experiment, this is not so. As already mentioned in the discussion of Fig. 3, for this experiment the three parameter fit is good for the tracer profile but not for the weight losses. The agreement between the obtained values for the apparent diffusion coefficient D with the linear rate law assumption (expression (13)) and the constant rate model (expression (14)) is good as well. The values of Table 2 slightly deviate from preliminary results (Aertsens et al., 2008) because additional data for the weight loss have become available. Next, the value of each of the three relevant fit parameters is discussed: 1. The apparent diffusion coefficient D of silica in clay The fitted values for the 1227 and 1570 days experiments nearly coincide at a value around 8–10 1013 m2/s. The value obtained at 887 days is higher (around 12–15 1013 m2/s). In all fits, the relative (fit) error on D is between 5% and 10%. This parameter has already been measured before (Aertsens et al., 2003; Aertsens et al., 2008) at a slightly smaller temperature (25 °C), leading to a value around 6 1013 m2/s (In-Diffusion experiments) and around 2 1013 m2/s (percolation experiments). 2. The initial dissolution rate aWL Apart from the three parameter fit of the 1227 days experiment (which is not very good with respect to the mass loss), all values in Table 2 are about equal: aWL 1:6 107 g=ðm2 sÞ 0:014 g=ðm2 day) with a relative fit error around 20%. This aWL value can be compared with the empirical formula (Godon et al., 2004):
aWL ¼ r0;T0 10N0 max ðpH-7;0Þ exp
Ea 1 1 R T T0
ð23Þ
with r0T0 = 1.7 g/(m2 day), N0 = 0.4, Ea = 76 kJ/mol, T0 = 373 K, R = 0.0083147 kJ/mol K1, obtained for the maximum glass dissolution rate in pure water. At pH 8.7 and 30 °C, expression (23) leads to aWL = 0.028 g/(m2 day). Substituting the values for aWL and aWL in expression (7) leads to c 2C0. Substituting the C0 = 4.8 mg Si/L in c 2C0 leads to a silica saturation concentration c 10 mg Si/L. According to Table 2 is aBq,f aBq/2. Initially, aBq,f and aBq should be equal, but due to adsorption of silica from the clay on the glass alteration layer, it makes sense that in reality the fitted value of aBq,f is smaller when experimental data obtained at larger times are included in the fit, leading to aBq,f < aBq. The average of the three values for qBq,fit,f in Table 2 is
qBq,fit,f = 1.06 105 Bq/g glass, agreeing reasonably well with the expected value qBq,glass = 1.24 105 Bq/g glass. pffiffiffiffi 3. The product h D and the concentration cWL pffiffiffiffi For the remaining fit parameter, it is possible to take h D or cWL . Again apart from the three parameter fit of the 1227 days pffiffiffiffi experiment, the values for h D in Table 2 are 1.2– 1.4 104 1/s0.5 with a fit relative error of around 33%. This high relative error is caused by the relatively small measurement times for the weight pffiffiffiffi loss. According to expression (11), accurate values for h D are only possible for measurement times much larger than s = 1/(h2D) 600–700 days. The value of the concentration cWL , in the range 1.0–1.5 103 g glass/m3 is used to derive, as confirmation, again the value of the silica saturation concentration c. Substituting fSi = 0.21, gR = 65 and C0 = 4.8 mg Si/L and the cWL values of Table 2 in expression (7) leads to c 8–10 mg Si/L (see Table 2), agreeing well with the already derived value for c from the dissolution rate. Summarizing, a consistent parameter set describing the experiments is: D = 9 1013 m2/s, C0 = 4.8 mg Si/L, c = 9 mg/L, aWL ¼ 1:6 107 g=ðm2 s) and gR = 65. Both expressions used to estimate the tracer diffusion profile in the clay lead to good fits and, apart from the rate, to similar optimal parameter values, confirming their validity. In agreement with the model, all analyses of the Boom Clay water pressed through the clay cores at the end of the experiment show an increased silica content C: C0 (4.8 mg Si/L) < C < c (9 mg/L) (see Table 3). Because of the high gR value and the relatively small time needed to press the water through the clay core, it is assumed that due to instantaneous desorption of silica, during this time interval the silica concentration does not change. Because the water is collected at the side of the clay core where the glass was, the measured water content is considered as an estimate for the silica concentration C(x = 0, t) at the glass/clay interface. Table 3 shows the calculated C(x = 0, t) values, using the parameter values mentioned in the present paragraph. The difference between the first and the last calculated value (0.3 mg/L) is smaller than the error on the measured silica concentrations, e.g. the four measurements of the background concentration C0 are respectively 5; 5.8; 4 and 4.5 mg/L. Taking into account the scatter in measurements C0, the four first measurements (1110–1837 days) lead to approximately the same value. The considerable higher value measured at 2199 days could suggest an increase, but the evidence for this is weak. The three profiles have also been fitted simultaneously with the mass losses, when the precipitation rate k is added as a fourth fit parameter (qBq,fit = qBq,WL). For the 887 days experiment, this leads to a nearly zero precipitation rate (k 1 1013 s1). Both other experiments lead to higher values (k 2 108 s1 and k 3 109 s1), but taking into account (i) the difference between these values, (ii) that for each value the fit error is considerably larger than 100% and (iii) that these fits do not fit well the points very close to the glass either, it is concluded that there is no evidence of silica precipitation in the clay. Possibly larger durations of the experiments are necessary to indicate that silica precipitation occurs or not.
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M. Aertsens, K. Lemmens / Physics and Chemistry of the Earth 65 (2013) 72–78
5. Conclusion Three glass dissolution tests with durations up to 1600 days are performed with SON68 glass in compact Boom Clay at 30 °C. These tests are described well by the Pescatore model, combining congruent dissolution according to a linear rate law with silica diffusion in clay. Fitting the profile of 32Si in the clay together with the glass mass loss as a function of time, this model leads to parameter values consistent with those found in the literature: – An apparent silica diffusion coefficient in the clay between 7 1013 m2/s and 1.2 1012 m2/s. Previously determined values from diffusion experiments at 25 °C are around 6 1013 m2/s (In-Diffusion experiments) and 2 1013 m2/s (percolation experiments). – A maximal glass dissolution rate for glass next to clay of around 1.6 107 g glass/m2 s. – The measured silica concentration in undisturbed clay is around 5 mg/L. – Two ways to derive independently the silica glass saturation concentration in clay from the present experiments and the previously measured value of the product of accessible porosity and retardation factor in In-Diffusion experiments, both lead to value between 8 and 10 mg Si/L. In agreement with the model, measurements in the clay water pressed through the clay core at the end of the experiment show an increase of the dissolved silica concentration in the clay close to the glass. Although silica precipitation due to glass dissolution has been shown experimentally at 90 °C, extending the Pescatore model with silica precipitation did not lead to much better fits, nor could
meaningful values of a possible precipitation rate be obtained. Possibly, if precipitation occurs, longer durations might be necessary to detect it. Acknowledgements Work performed as part of the program on geological disposal of high-level/long-lived radioactive waste that is carried out by ONDRAF/NIRAS, the Belgian Agency for Radioactive Waste and Fissile Materials. References Aertsens, M., De Cannière, P., Moors, H., 2003. Modelling of silica diffusion experiments with 32Si in Boom Clay. J. Contam. Hydrol. 61, 117–129. Aertsens, M., De Cannière, P., Lemmens, K., Maes, N., Moors, H., 2008. Overview and consistency of migration experiments in clay. Phys. Chem. Earth 33, 1019–1025. Crank, J., 1975. The Mathematics of Diffusion. Clarendon Press, Oxford. De Cannière, P., Moors, H., Dierckx, A., Gasiaux, F., Aertsens, M., Put, M., Van Iseghem, P., 1998. Diffusion and sorption of 32Si-labelled silica in the Boom Clay. Radiochim. Acta 82, 191–196. Frugier, P., Gin, S., Minet, Y., Chave, T., Bonin, B., Godon, N., Lartigue, J.-E., Jollivet, P., Ayral, A., De Windt, L., Santarini, G., 2008. SON68 nuclear glass dissolution kinetics: current state of knowledge and basis of the new GRAAL model. J. Nucl. Mater. 380, 8–21. Gin, S., Jollivet, P., Mestre, J.P., Jullien, M., Pozo, C., 2001. French SON68 nuclear glass alteration mechanisms on contact with clay media. Appl. Geochem. 16, 861– 881. Godon, N. et al., 2004. Dossier de référence sur le comportement à long terme des verres nucléaires, RT DTCD 2004/06. CEA, Marcoule. Pescatore, C., 1994. The dependence of wasteform leaching on migration parameters in the host medium. Radiochim. Acta 66 (67), 439–444. Pozo, C., Bildstein, O., Raynal, J., Jullien, M., Valcke, E., 2007. Behaviour of silicon released during alteration of nuclear waste glass in compacted clay. Appl. Clay Sci. 35, 258–267.
Contents lists available at SciVerse ScienceDirect
Physics and Chemistry of the Earth journal homepage: www.elsevier.com/locate/pce
A combined glass dissolution/diffusion experiment in Boom Clay at 30 °C Marc Aertsens ⇑, Karel Lemmens SCKCEN, Boeretang 200, B-2400 Mol, Belgium
a r t i c l e
i n f o
Article history: Available online 30 May 2013 Keywords: Boom Clay Silica Glass dissolution Diffusion accessible porosity Retardation factor Apparent diffusion coefficient
a b s t r a c t At the appropriate times, silica diffusion in clay is possibly the rate determining process for the dissolution of vitrified waste disposed of in a clay layer. For testing this hypothesis, combined glass dissolution/ silica diffusion experiment are performed. SON68 glass coupons doped with the radioactive tracer 32Si are sandwiched between two cores of humid Boom Clay, heated to 30 °C. Due to glass dissolution, 32Si is released and diffuses into the clay. At the end of an experiment, the mass loss of the glass coupon is measured and the clay core is sliced to determine the diffusion profile of the 32Si released from the glass in the clay. Both mass loss and the 32Si diffusion profile in the clay are described well by a model combining glass dissolution according to a linear rate law with silica diffusion in the clay. Fitting the experiments to this model leads to an apparent silica diffusion coefficient in the clay between 7 1013 m2/s and 1.2 1012 m2/s. Previously determined values from diffusion experiments at 25 °C are around 6 1013 m2/s (In-Diffusion experiments) and 2 1013 m2/s (percolation experiments). The maximal glass dissolution rate for glass next to clay is around 1.6 107 g glass/m2 s (i.e. 0.014 g glass/m2 day). In undisturbed clay, the measured silica concentration is around 5 mg/L. Combining these values with the previously measured (In-Diffusion experiments) product of accessible porosity and retardation factor, leads in two ways to a silica glass saturation concentration in clay between 8 and 10 mg Si/L. Another candidate for the rate determining process of the dissolution of vitrified waste disposed in a clay layer is silica precipitation. Although silica precipitation due to glass dissolution has been shown experimentally at 90 °C, extending the model with silica precipitation does not lead to much better fits, nor could meaningful values of a possible precipitation rate be obtained. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction In Belgium the disposal of vitrified radioactive waste in Boom Clay is considered. Safety studies require to estimate the release rate of radionuclides from the host glass and their transport rate in Boom Clay. Experiments have shown that many of the radionuclides dissolve congruently with the glass. Dissolution of glass next to clay is described by a model (Pescatore, 1994) according to which at large times the glass dissolution rate is controlled by the transport parameters of dissolved silica in Boom Clay: the apparent diffusion coefficient D and the product gR of the diffusion accessible porosity g and the retardation factor R. The values of these parameters have been measured by diffusion experiments (Aertsens et al., 2003; Aertsens et al., 2008) confirming silica retardation in Boom Clay. Both dissolved silica and clay are negatively charged, so initially silica sorption
⇑ Corresponding author. Tel.: +32 14 33 31 31; fax: +32 14 32 35 53. E-mail address: [email protected] (M. Aertsens). 1474-7065/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.pce.2013.05.008
on clay was not expected. Possible mechanisms explaining it have been put forward (De Cannière et al., 1998). The silica transport parameters in Boom Clay and the Pescatore model are validated by experiments where 32Si doped glass dissolves next to clay. This allows to determine not only the glass mass loss as a function of time, but also the 32Si diffusion profile in the clay. The combination of both types of measurement makes it possible to verify if the 32Si profile in the clay can indeed be coupled to the glass mass loss. If so, all parameters of the Pescatore model can be measured. Because at 90 °C silica precipitates have been observed next to a dissolving glass in contact with clay (Pozo et al., 2007), a precipitation term is added to the Pescatore model. An overview of possible precipitation mechanisms is given by Gin et al. (2001). The Pescatore model does not describe the precipitation of secondary phases (especially phyllosilicates) on the glass surface, which can form a sink for silica released from the glass (Frugier et al., 2008). This paper starts by describing the experiment. From the Pescatore model, two approximate expressions for the 32Si profile in the clay are derived. Two extensions of the Pescatore model
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M. Aertsens, K. Lemmens / Physics and Chemistry of the Earth 65 (2013) 72–78
are derived: one taking into account the finite size of the clay, and one taking into account silica precipitation in the clay. These models are used afterwards to fit the experimental data. 2. Description of the experiment SON68 glass coupons with a thickness of 2 mm and a diameter of 16 mm were doped with the radioactive tracer 32Si, leading to a specific glass activity qBq,glass = 1.24 105 Bq/g glass. The glass samples were sandwiched between two cores of humid Boom Clay with a length of typically about 2 cm each and pressed into a percolation cell with a diameter of 20 mm (see Fig. 1). After the saturation of the clay by percolation with clay water, the valves at the inlet and outlet were closed and the cells were heated to 30 °C. Possible voids around a glass coupon caused by the different diameters of the glass and the clay are unlikely because of the swelling capacity of the clay. Due to glass dissolution, 32Si is released from the glass and diffuses into the clay. After stopping the experiment, the glass weight (mass) loss (subscript WL) per surface unit QWL (unit: g glass/m2), was measured. Then the clay core was sliced to determine the 32Si diffusion profile. Therefore, the clay slices were dried, and after secular equilibrium of 32Si with its daughter 32P, in each slice the activity was measured with a Packard Auto-Gamma 5650 NaI(Tl) counter. The ratio QBq (unit: g glass/m2; Bq refers to activity):
Q Bq ¼
BBq;clay SqBq;glass
ð1Þ
with BBq,clay the total activity (unit: Bq) in clay and S the contact surface (unit: m2) provides another estimate for the glass mass loss. The mass loss QBq based on the amount of 32Si in the clay, is larger than the weight loss QWL because during the dissolution of glass next to clay, clay components are integrated in the outer part of the glass alteration layer. This process decreases the weight loss QWL, but it hardly changes the mass loss QBq based on 32Si because there is almost no 32Si in the clay next to the glass (initially the 32Si content in the clay is zero, due to glass dissolution this content slowly rises in the vicinity of the glass/clay interface). Since the mass loss QBq based on 32Si is larger than the weight loss QWL (QBq > QWL), the specific glass activity based on weight loss qBq,WL (unit: Bq/g glass), defined as
qBq;WL ¼
BBq;clay Q Bq ¼ q S Q WL Q WL Bq;glass
ð2Þ
is larger than the specific glass activity qBq,glass. In similar experiments with undoped SON68 glass, only the weight loss is measured. At the end of such an experiment, a few milliliter of Boom Clay water was pressed through the clay core for analysis. The samples were not filtered, and analyzed by ICP–AES, ICP–MS and ion chromatography, leading to a clay water composition at the end of the experiment. This clay water composition has also been determined for blank tests, where real Boom
Clay water (taken from the Underground Research Lab, where the temperature is around 13 °C) was percolated through a clay core at 30 °C. The experiments with 32Si doped SON68 were stopped after 887 days, 1227 days, 1570 and 1890 days. 3. Models The Pescatore model (Pescatore, 1994) for dissolution of glass next to clay is used to derive two approximate expressions for the 32Si concentration in the clay. Then the model is extended by taking into account the finite size of a clay core. In a second extension, a precipitation term is added. 3.1. The model in an infinite system The behavior of dissolved silica in clay is described by the diffusion equation:
@C @2C ¼ D 2 kðC C 0 Þ @t @x
with C(x, t) the concentration of silica in the pore water accessible for diffusion (unit: g Si/m3), t time, D the apparent diffusion coefficient of silica in the pore water of the clay (unit: m2/s), x the distance to the fixed glass/clay interface (where x = 0), k the precipitation rate (unit: 1/s) and C0 the background silica concentration in the clay pore water (unit: g Si/m3). Glass dissolution is assumed to be congruent and caused by the dissolution of its main component, silica. Silica is supposed to dissolve according to a linear rate law:
Cðx ¼ 0; tÞ @C ¼ gRD Jðx ¼ 0; tÞ ¼ a 1 @x x¼0 c
ð4Þ
with J(x = 0, t) the flux of dissolving silica at the glass surface (which is chosen at the origin) (unit: g Si/m2 s), a the maximal silica dissolution rate (unit: g Si/m2 s), c a constant characterizing the saturation of glass silica in water (unit: g Si/m3), g the clay porosity accessible for diffusion and R the silica retardation factor in clay. The silica concentration in the clay is C0 (C(x > 0, t = 0) = C0). According to the linear rate law (4), the net silica dissolution rate J(x = 0, t) is the difference between two processes: silica dissolution at a constant rate a and the adsorption of silica from the pore water of the clay on the glass alteration layer (with rate a C(x = 0, t)/c). The solution of (3), (4) without precipitation (k = 0) is (Pescatore, 1994)
C0 F 1 ðx; tÞ Cðx; tÞ ¼ C 0 þ c 1
ð5Þ
c
with F 1 ðx; tÞ ¼ erfc
pffiffiffiffiffiffi x x 2 pffiffiffiffiffiffi expðhx þ h DtÞerfc pffiffiffiffiffiffi þ h Dt ; 2 Dt 2 Dt
h¼
a gRDc
ð6Þ
The function F1(x, t) is dimensionless, while the unit of h is m1. Expressions (5) and (6) are used for deriving expressions for the tracer silica bulk concentration Cb,Bq(x, t) (unit: Bq/m3), measured in our experiments. In case of no adsorption of silica from the clay pore water on the glass alteration layer, the tracer silica bulk concentration Cb,Bq(x, t) is the product of the excess silica bulk concentration gR(C(x, t) C0) (unit: g Si/m3) and the specific silica activity qBq,glass/fSi (unit: Bq/g Si), where fSi = 0.21 (unit: g Si/g glass) is the silica weight fraction in the glass. Introducing the notations
cWL ¼ gR
Fig. 1. Set-up of the combined glass-dissolution/silica diffusion experiment.
ð3Þ
c fSi
C0 ; 1
c
aWL ¼
a fSi
C0 C0 ¼ aWL 1 ¼ hDcWL 1
c
c
ð7Þ
with cWL (unit: g Si/m3) the excess glass bulk concentration needed to saturate the clay with respect to the glass silica saturation c,
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M. Aertsens, K. Lemmens / Physics and Chemistry of the Earth 65 (2013) 72–78
aWL = a/fSi (unit: g glass/m2 s) the maximal (forward) glass dissolution rate and aWL (unit: g glass/m2 s) the initial glass dissolution rate, leads after insertion in Cb,Bq(x, t) = gR(C(x, t) C0) qBq,glass/fSi to C b;Bq ðx; tÞ ¼ qBq;glass cWL F 1 ðx; tÞ
ð8Þ
it will remain quite small (the altered glass contains only trace concentrations of 32Si). Mathematically, the profile Cb,Bq(x, t) is obtained by solving with the Laplace transform method (Crank, 1975) the diffusion Eq. (1) with boundary condition J(x = 0, t) = af (instead of expression (4), index f stands for forward rate), leading to:
In case of adsorption of silica from the clay on the glass alteration layer, it is assumed that the tracer silica bulk concentration Cb,Bq(x, t) is still proportional to the excess silica bulk concentration gR(C(x, t) C0), and expression (8) is replaced by:
C b;Bq ðx; tÞ ¼ C f ;1 ðx; tÞ rffiffiffiffiffiffi ! aBq;f Dt x2 x ¼ xerfc pffiffiffiffiffiffi 2 exp D p 4Dt 2 Dt
C b;Bq ðx; tÞ ¼ qBq;fit cWL F 1 ðx; tÞ
with aBq,f (unit: Bq/m2 s) a fit parameter and Cf,1(x, t) the notation for the expression on the right hand side of (14). Fitting the tracer profile with expression (14) leads to values for aBq,f and the apparent diffusion coefficient D, which do not depend on the values of the parameters used to fit the weight loss (see expression (10)). The value of the ratio aBq;f =aWL ¼ qBq;fit;f is expected to be close to the specific glass activity qBq,glass. Fitting the experiments should make clear which expression (13), (14) is the best approximation for the silica bulk concentration Cb,Bq(x, t). In the ideal case, both expressions lead to good fits and similar values for the fit parameters.
ð9Þ
Due to adsorption on the glass alteration layer, part of the clay silica concentration C0 is replaced by ‘glass silica’ containing 32Si, making the proportionality factor qBq,fit (unit: Bq/g glass), which will be estimated further on, larger than the specific glass activity qBq,glass. Because glass dissolution is assumed to be congruent and governed by silica dissolution, the glass weight loss per surface unit QWL(t) (unit: g glass/m2), being the ratio of the silica weight loss Q(t) per surface unit (unit: g Si/m2) and fSi, is given by (Pescatore, 1994):
1 fSi
Z
t
0
Jðx ¼ 0; t 0 Þdt pffiffiffiffiffiffi 1 2 pffiffiffiffiffiffi 2 ¼ aWL 2 expðh DtÞerfcðh Dt Þ 1 þ pffiffiffiffi h Dt p h D
Q WL ðtÞ ¼
3.2. The model in a finite system
0
ð10Þ
According to expression (10), the weight loss depends on two parameters: the initial glass dissolution rate aWL (unit: g glass/ pffiffiffiffi m2 s) and the combination h D (unit: 1/s0.5). Initially the weight loss is proportional to time (Q WL ðtÞ aWL t). At times t s = 1/ h2D, it becomes proportional to the square root of time pffiffiffiffiffiffi pffiffiffiffiffiffi a pffiffi (Q WL ðtÞ p2ffiffipffi c 1 Cc0 gfSiR Dt ¼ p2ffiffipffi cWL Dt ¼ p2ffiffipffi hpWLffiffiDffi t). Returning to the tracer bulk concentration profile Cb,Bq(x, t), according to mass conservation is
Q WL ðtÞ ¼
gR
Z
fSi
ðCðx; tÞ C 0 Þdx
1
F 1 ðx; tÞdx ¼
0
1
qBq;fit
Z
C b;Bq ðx; tÞdx
C b;Bq ðx; tÞ ¼
0
1 X
C f ;1 ð2Lðn þ 1Þ x; tÞ þ
n¼0
a
ffipffiffiffiffi F 1 ðx; tÞ C b;Bq ðx; tÞ ¼ qBq;fit cWL F 1 ðx; tÞ ¼ qBq;fit pffiffiffiWL h D D
a
ð15Þ
ð12Þ
1
A good fit of the tracer profile in the clay requires that R1 R1 Bq;clay S 0 C b;Bq ðx; tÞdx. Substituting Bq;clay S 0 C b;Bq ðx; tÞdx in expression (12), and taking into account expression (2) allows to estimate qBq,fit as qBq,fit qBq,WL. To test this estimate, each tracer profile is fitted simultaneously with all weight losses twice: once considering the specific activity qBq,fit as a fit parameter and once taking, apart from a fixed proportionality factor (see Section 4 for details) qBq,fit and qBq,WL equal. After introducing the notation aBq = aWL qBq,fit into expression (9),
Bq ¼ pffiffiffiffipffiffiffiffi F 1 ðx; tÞ h D D
aBq
C b;Bq ðx; tÞ ¼ pffiffiffiffipffiffiffiffi ðF 1 ðx; tÞ þ F 1 ð2L x; tÞÞ h D D
ð11Þ
0
Z
All just presented solutions assume an infinitely long clay core. This assumption is justified as long as the concentration C(x = L, t) with L the length of the clay core remains sufficiently low (much smaller than the concentration difference c C0). At larger times, the boundary condition is changed into a zero flux boundary condition at x = L. In first instance, our calculations adapt expression (13) to
As long as the product (c C0) F1(2L, t) remains much smaller than (c C0) (which is the case in the here presented experiments), expression (15) is a good approximation. Similarly, due to the zero flux boundary condition at x = L, expression (14) changes into
1
Substituting expression (9) in expression (11) leads to
Q WL ðtÞ ¼ cWL
ð14Þ
ð13Þ
In case the specific activity qBq,fit is considered as a fit parameter, four parameters are fitted simultaneously from the weight losses and one tracerpffiffiffi profile: the initial glass dissolution rate aWL , the ffi combination h D, the apparent diffusion coefficient D and the dissolution rate aBq. If qBq,fit and qBq,WL are taken equal, the parameters D, aBq and cWL are fitted. An alternative way to estimate the tracer silica bulk concentration Cb,Bq(x, t) is to neglect in the linear rate law (4) the adsorption term a C(x = 0, t)/c. For the tracer concentration in clay, this is a good approximation because initially it is zero, and also afterwards
1 X C f ;1 ð2Ln þ x; tÞ
ð16Þ
n¼0
where the function Cf,1(x, t) is introduced in expression (14). The additional term in expression (15) does not affect expression (10) for the weight loss. 3.3. Extension of the model in an infinite system with precipitation For simplicity, only the infinite system is considered. According to Crank (Crank, 1975), the solution Ck(x, t) of the diffusion Eq. (3) with a non-zero precipitation rate k can be calculated from the solution C(x, t) of the same equation without the precipitation term:
C k ðx; tÞ ¼ k
Z
t
Cðx; tÞ expðktÞdt þCðx; tÞ expðktÞ
ð17Þ
0
Substituting expression (5) into expression (17) leads for h2D – k to
pffiffiffiffiffiffi x p ffiffiffiffiffi ffi Dt þ h 2 2 Dt h Dk rffiffiffiffi ! pffiffiffiffi pffiffiffiffiffi h D pffiffiffiffi pffiffiffi x k x þ ðh D þ kÞerfc pffiffiffiffiffiffi þ kt exp D 2 2 Dt rffiffiffiffi !! pffiffiffiffi pffiffiffiffiffi h D pffiffiffiffi pffiffiffi x k p ffiffiffiffiffi ffi ð18Þ kt exp x h D k erfc þ D 2 2 Dt
C k ðx;tÞ ¼C 0 þ
c C0
2
2
h Dexpðhx þ ðh D kÞtÞerfc
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M. Aertsens, K. Lemmens / Physics and Chemistry of the Earth 65 (2013) 72–78
The amount of precipitated silica Cprec(x, t) is given by
C prec ðx; tÞ ¼ gRk
Z
t
ðC k ðx; yÞ C 0 Þdy
ð19Þ
0
Substituting expression (18) in expression (19) gives
C prec ðx; tÞ ¼
gRðc C 0 Þ 2
h Dk 2 pffiffiffiffiffiffi h D x 2 exp hx þ ðh D kÞt erfc pffiffiffiffiffiffi þ h Dt k 2 2 Dt h Dk pffiffiffiffi pffiffiffiffiffi 2 h D kt x þ kt pffiffiffi pffiffiffiffi exp 4Dt p k pffiffiffiffi pffiffiffi h D pffiffiffiffi pffiffiffi 1 x þ pffiffiffi h D þ k pffiffiffiffi pffiffiffi þ pffiffiffiffi þ 2 kt D h D k 4 k rffiffiffiffi ! p ffiffiffiffi ffi x k x erfc pffiffiffiffiffiffi þ kt exp D 2 Dt pffiffiffiffi pffiffiffi pffiffiffiffi h D pffiffiffi 1 x pffiffiffiffi pffiffiffi þ pffiffiffiffi 2 kt þ pffiffiffi kh D D 4 k h Dþ k rffiffiffiffi !! pffiffiffiffiffi x k ð20Þ x erfc pffiffiffiffiffiffi kt exp D 2 Dt
The total amount of tracer silica Ctot(x, t) is
C b;Bq ðx; tÞ ¼
qBq;fit fSi
ðgRðC k ðx; tÞ C 0 Þ þ C prec ðx; tÞÞ
ð21Þ
Substituting expression (18) in expressions (4) and (10) gives the glass weight loss per surface unit QWL(t): ! pffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffi h D 2 erfcðh Dt Þexpððh D kÞtÞ 1 þ pffiffiffi erf ð kt Þ k h Dk h Dk pffiffiffiffi ! pffiffiffiffiffi h D 1 1 pffiffiffiffiffi ð22Þ kt erf ð kt Þ þ pffiffiffiffi kt expðktÞ kt þ pffiffiffi 2 p k
Q WL ðtÞ ¼ aWL
When fitting the experimental data, it is assumed that apart from a fixed proportionality factor (see Section 4 for details) qBq,fit and qBq,WL are equal so that the four fit parameters are: the concentration cWL , the apparent diffusion coefficient D, the dissolution rate aBq and the precipitation rate k.
1
2
2
h D
2
In case of strong precipitation (h2 D/k 1), expression (22) is well approximated by Q WL ðtÞ aWL t.
Fig. 2. Fitting the glass weight losses versus pffiffiffiffitime, according to expression (9), leads to aWL ¼ 1:7 107 g glass/(m2 s) and h D ¼ 1:4 104 1=s0:5 (see Table 2). The triangles are the experimental data points for the doped glass, for which also a tracer profile in the clay is measured. The circles are the experimental data points for the undoped glass.
4. Experimental and modeling results Table 1 summarizes the mass loss results for the tracer doped glasses. Both the weight loss QWL and the mass loss QBq based on 32 Si are calculated using the real surface of a glass coupon (5.05 104 m2). Due to the very small values of the mass loss, scatter can be a problem, e.g. QWL as well as QBq have nearly the same value for the 887 and 1227 days experiments (QWL 6.5– 7 g glass/m2 and QBq 13 g glass/m2). The ratio QBq/QWL is always larger than one and increases as a function of time. Fitting the experiments to the model, the model (being one dimensional) assumes the same cross section for the glass coupon, the clay cores and the surrounding cell, while in reality the cell (clay core) cross section is a factor (2 3.14 104 m2)/ (5.05 104 m2) 1.24 larger than half the total surface of the glass coupon. Because only a fraction of 1/1.24 0.8 of the cell cross section at the glass/clay interface consists of glass, the specific glass activity based on weight loss qBq,WL averaged over the cell cross section equals the ratio qBq,WL/1.24, which is called in Table 1 the cross section averaged specific glass activity based on weight loss. Fig. 2 shows all weight losses QWL, including the undoped glasses, versus time. The fit according to expression (10) is good. Apart from the 1890 days experiment, the scatter in the measured 32Si profiles, shown in Fig. 3, is limited. Contrary to the three other experiments, where the lengths of the clay cores at both sides of the 32Si doped glass are about equal (around 2–2.2 cm), this was not the case for the 1890 days experiment (lengths of 1.3 and 3.3 cm). For this experiment, because of the different clay core lengths, and also because of the considerable difference between the measured 32Si activity in both clay cores, different symbols are used for the measured activities in both clay cores. Because of the irregularities in the tracer profile of the 1890 days experiment, no fits of the tracer profile of this experiment are presented. In the fits of Fig. 3, all three parameter fits of the mass loss data together with one tracer profile in clay, the agreement between the fit and experimental data (both profile and mass loss) is generally good. Only for the 1227 days experiment, the mass loss is not represented well with the fit parameter values used in Fig. 3. This is clear from the fitted values for both parameters determining the weightploss ffiffiffiffi (the initial glass dissolution rate aWL and the combination h D), which deviate considerably from the values obtained from the other fits (see Table 2). The major problem of the fits is that the measured bulk activity very close (always less than 2 mm) to the glass/clay interface is considerably higher than predicted by the fit. Table 2 summarizes the fit results of the weight loss and the 32Si profiles in the clay cores. Combining the fit results p (for ffiffiffiffi the initial glass dissolution rate aWL and the combination h D) from the
Table 1 Mass loss data for the four doped glass dissolution experiments. Duration Weight loss per surface unit (QWL) Mass loss based on 32Si per surface unit (QBq) Ratio QBq/QWL Specific glass activity (qBq,glass) Specific glass activity from mass loss (qBq,WL) Cross section averaged specific glass activity from mass loss
Days (g glass/m2) (g glass/m2) (Bq/g glass) (Bq/g glass) (Bq/g glass)
887 6.95 12.6 1.82 1.2E+5 2.3E+5 1.8E+5
1227 6.66 13.3 1.99 1.2E+5 2.5E+5 2.0E+5
1570 10.80 36.1 3.35 1.2E+5 4.1E+5 3.2E+5
1890 11.26 45.8 4.07 1.2E+5 5.0E+5 4.1E+5
Bulk Activity (Bq/gram clay)
M. Aertsens, K. Lemmens / Physics and Chemistry of the Earth 65 (2013) 72–78
Bulk Activity (Bq/gram clay)
76
100 887 days
50
0 0
5
10
15
20
1227 days
100
50
0
25
0
5
10
Bulk Activity (Bq/gram clay)
Bulk Activity (Bq/gram clay)
300 1570 days
200
100
0 0
5
15
20
x (mm)
x (mm)
10
15
1.3 cm clay core
300 1890 days
3.3 cm clay core
200
100
0
20
0
10
x (mm)
20
30
x (mm)
Fig. 3. 32Si profiles (expressed as the ratio of the bulk activity CBq,32 (unit: Bq/m3) and the clay density (unit: g/m3)) in the clay cores and the corresponding fits according to Pesc3 (see Table 2 for the values of the fit parameters). For the 1890 days experiment, the correspondence of the measured activities in both clay cores is less good than for the other three experiments. Also because of the higher scatter on the data, a fitted line is not shown on the figure.
Duration (days)
Fit
Fit expressions
Table 2 Parameter values obtained from fitting three glass dissolution experiments in different ways. The first column indicates what is fitted: the weight loss data (WL), the tracer diffusion profile in the clay (P) or both simultaneously (WL + P). The number(s) of the expressions used for the fittings are mentioned in the second column. Which profile is fitted is indicated in the third column. The remaining columns give the optimal parameter values: green cells contain a constant value, yellow cells the parameter values which are directly fitted and non-colored cells contain parameter values derived from the mathematical relations between the parameters combined with gR = 65 and a silica background in the clay C0 = 5 mg Si/L and the values in the green and yellow cells.
D
αBq,f or αBq
α∗WL
h D0.5
γ∗WL
m2 s-1
Bqm-2 s-1
g glass m-2 s-1
s-0.5
g glass m-3
αBq,f WL P P P
10 14 14 14
1,7E-07 887 1227 1570
1,5E-12 9,6E-13 9,3E-13
1,7E-02 1,2E-02 2,4E-02 αBq
WL+P 10,13
887
1,2E-12
3,1E-02
1,6E-07
WL+P 10,13
887
1,2E-12
3,0E-02
1,7E-07
WL+P 10,13 1227
7,8E-13
2,5E-02
1,7E-07
WL+P 10,13 1227
7,3E-13
5,0E-02
2,5E-07
WL+P 10,13 1570
7,7E-13
5,4E-02
1,6E-07
WL+P 10,13 1570
7,7E-13
5,4E-02
1,7E-07
weight loss (expression (10) and Fig. 2) with the fit values for the dissolution rate aBq,f and the apparent diffusion coefficient D obtained from fitting the profiles with the constant rate expression (14) leads by expression (7) to estimates for the concentration cWL . In Table 2, the glass saturation concentration c is derived from the concentration cWL from expression (7) by taking the product gR of the porosity g and the retardation factor R as gR = 65 and the
γ
ρBq,fit,f
or ρBq,fit
mg Bq Si (g glass)-1) liter-1 ρBq,fit,f
1,4E04
1,3E04 1,2E04 1,4E04 3,6E04 1,3E04 1,3E04
τ days
612 1,0E+03 1,2E+03 1,3E+03
8 9 9
9,9E+04 7,3E+04 1,5E+05 ρBq,fit
1,1E+03
8
1,9E+05
686
1,2E+03
9
1,8E+05
751
1,4E+03
9
1,5E+05
634
8,3E+02
7
2,0E+05
92
1,4E+03
9
3,3E+05
686
1,5E+03
10
3,2E+05
678
silica background concentration in the clay C0 = 4.8 mg Si/L. Silica In-Diffusion experiments in Boom Clay at 25 °C lead to a gR value around 60–70 (leading to the estimate gR = 65), percolation experiments to a value between 100 and 150 (Aertsens et al., 2003; Aertsens et al., 2008). The silica background concentrations C0 measured in four blank tests with different durations (up to 1983 days) are 5; 5.8; 4 and 4.5 mg Si/L, resulting in an average
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M. Aertsens, K. Lemmens / Physics and Chemistry of the Earth 65 (2013) 72–78 Table 3 Silica and boron concentrations in the percolated pore water at the glass side of the clay core at the end of the experiments. Time (days)
Sample volume (g)
Sample/pore volume
pH
B exp (mg/L)
Si exp (mg/L)
Si calc (mg/L)
1110 1148 1463 1837 2199
12.64 9.78 7.26 0.97 0.96
5.44 4.21 3.12 0.42 0.41
8.6 8.7 8.5 8.6 8.6
15.3 16.0 17.7 18.4 25.0
6.7 6.0 7.3 7.2 8.7
7.5 7.5 7.6 7.7 7.8
C0 = 4.8 mg Si/L. All weight losses have been fitted together with each of the tracer profiles (using expression (13)) twice: a four parameter fit with the specific glass activity qBq,fit as fit parameter and a three parameter fit where qBq,fit has the value of the cross section averaged specific glass activity based on weight loss (qBq,WL/1.24). It was verified that for all fits the value of (c C0) F1(2L, t) was sufficiently low. Comparing Tables 1 and 2 show that for the 887 and 1570 days experiments, a fit of qBq,fit leads indeed to qBq,WL/1.24. For the 1227 days experiment, this is not so. As already mentioned in the discussion of Fig. 3, for this experiment the three parameter fit is good for the tracer profile but not for the weight losses. The agreement between the obtained values for the apparent diffusion coefficient D with the linear rate law assumption (expression (13)) and the constant rate model (expression (14)) is good as well. The values of Table 2 slightly deviate from preliminary results (Aertsens et al., 2008) because additional data for the weight loss have become available. Next, the value of each of the three relevant fit parameters is discussed: 1. The apparent diffusion coefficient D of silica in clay The fitted values for the 1227 and 1570 days experiments nearly coincide at a value around 8–10 1013 m2/s. The value obtained at 887 days is higher (around 12–15 1013 m2/s). In all fits, the relative (fit) error on D is between 5% and 10%. This parameter has already been measured before (Aertsens et al., 2003; Aertsens et al., 2008) at a slightly smaller temperature (25 °C), leading to a value around 6 1013 m2/s (In-Diffusion experiments) and around 2 1013 m2/s (percolation experiments). 2. The initial dissolution rate aWL Apart from the three parameter fit of the 1227 days experiment (which is not very good with respect to the mass loss), all values in Table 2 are about equal: aWL 1:6 107 g=ðm2 sÞ 0:014 g=ðm2 day) with a relative fit error around 20%. This aWL value can be compared with the empirical formula (Godon et al., 2004):
aWL ¼ r0;T0 10N0 max ðpH-7;0Þ exp
Ea 1 1 R T T0
ð23Þ
with r0T0 = 1.7 g/(m2 day), N0 = 0.4, Ea = 76 kJ/mol, T0 = 373 K, R = 0.0083147 kJ/mol K1, obtained for the maximum glass dissolution rate in pure water. At pH 8.7 and 30 °C, expression (23) leads to aWL = 0.028 g/(m2 day). Substituting the values for aWL and aWL in expression (7) leads to c 2C0. Substituting the C0 = 4.8 mg Si/L in c 2C0 leads to a silica saturation concentration c 10 mg Si/L. According to Table 2 is aBq,f aBq/2. Initially, aBq,f and aBq should be equal, but due to adsorption of silica from the clay on the glass alteration layer, it makes sense that in reality the fitted value of aBq,f is smaller when experimental data obtained at larger times are included in the fit, leading to aBq,f < aBq. The average of the three values for qBq,fit,f in Table 2 is
qBq,fit,f = 1.06 105 Bq/g glass, agreeing reasonably well with the expected value qBq,glass = 1.24 105 Bq/g glass. pffiffiffiffi 3. The product h D and the concentration cWL pffiffiffiffi For the remaining fit parameter, it is possible to take h D or cWL . Again apart from the three parameter fit of the 1227 days pffiffiffiffi experiment, the values for h D in Table 2 are 1.2– 1.4 104 1/s0.5 with a fit relative error of around 33%. This high relative error is caused by the relatively small measurement times for the weight pffiffiffiffi loss. According to expression (11), accurate values for h D are only possible for measurement times much larger than s = 1/(h2D) 600–700 days. The value of the concentration cWL , in the range 1.0–1.5 103 g glass/m3 is used to derive, as confirmation, again the value of the silica saturation concentration c. Substituting fSi = 0.21, gR = 65 and C0 = 4.8 mg Si/L and the cWL values of Table 2 in expression (7) leads to c 8–10 mg Si/L (see Table 2), agreeing well with the already derived value for c from the dissolution rate. Summarizing, a consistent parameter set describing the experiments is: D = 9 1013 m2/s, C0 = 4.8 mg Si/L, c = 9 mg/L, aWL ¼ 1:6 107 g=ðm2 s) and gR = 65. Both expressions used to estimate the tracer diffusion profile in the clay lead to good fits and, apart from the rate, to similar optimal parameter values, confirming their validity. In agreement with the model, all analyses of the Boom Clay water pressed through the clay cores at the end of the experiment show an increased silica content C: C0 (4.8 mg Si/L) < C < c (9 mg/L) (see Table 3). Because of the high gR value and the relatively small time needed to press the water through the clay core, it is assumed that due to instantaneous desorption of silica, during this time interval the silica concentration does not change. Because the water is collected at the side of the clay core where the glass was, the measured water content is considered as an estimate for the silica concentration C(x = 0, t) at the glass/clay interface. Table 3 shows the calculated C(x = 0, t) values, using the parameter values mentioned in the present paragraph. The difference between the first and the last calculated value (0.3 mg/L) is smaller than the error on the measured silica concentrations, e.g. the four measurements of the background concentration C0 are respectively 5; 5.8; 4 and 4.5 mg/L. Taking into account the scatter in measurements C0, the four first measurements (1110–1837 days) lead to approximately the same value. The considerable higher value measured at 2199 days could suggest an increase, but the evidence for this is weak. The three profiles have also been fitted simultaneously with the mass losses, when the precipitation rate k is added as a fourth fit parameter (qBq,fit = qBq,WL). For the 887 days experiment, this leads to a nearly zero precipitation rate (k 1 1013 s1). Both other experiments lead to higher values (k 2 108 s1 and k 3 109 s1), but taking into account (i) the difference between these values, (ii) that for each value the fit error is considerably larger than 100% and (iii) that these fits do not fit well the points very close to the glass either, it is concluded that there is no evidence of silica precipitation in the clay. Possibly larger durations of the experiments are necessary to indicate that silica precipitation occurs or not.
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M. Aertsens, K. Lemmens / Physics and Chemistry of the Earth 65 (2013) 72–78
5. Conclusion Three glass dissolution tests with durations up to 1600 days are performed with SON68 glass in compact Boom Clay at 30 °C. These tests are described well by the Pescatore model, combining congruent dissolution according to a linear rate law with silica diffusion in clay. Fitting the profile of 32Si in the clay together with the glass mass loss as a function of time, this model leads to parameter values consistent with those found in the literature: – An apparent silica diffusion coefficient in the clay between 7 1013 m2/s and 1.2 1012 m2/s. Previously determined values from diffusion experiments at 25 °C are around 6 1013 m2/s (In-Diffusion experiments) and 2 1013 m2/s (percolation experiments). – A maximal glass dissolution rate for glass next to clay of around 1.6 107 g glass/m2 s. – The measured silica concentration in undisturbed clay is around 5 mg/L. – Two ways to derive independently the silica glass saturation concentration in clay from the present experiments and the previously measured value of the product of accessible porosity and retardation factor in In-Diffusion experiments, both lead to value between 8 and 10 mg Si/L. In agreement with the model, measurements in the clay water pressed through the clay core at the end of the experiment show an increase of the dissolved silica concentration in the clay close to the glass. Although silica precipitation due to glass dissolution has been shown experimentally at 90 °C, extending the Pescatore model with silica precipitation did not lead to much better fits, nor could
meaningful values of a possible precipitation rate be obtained. Possibly, if precipitation occurs, longer durations might be necessary to detect it. Acknowledgements Work performed as part of the program on geological disposal of high-level/long-lived radioactive waste that is carried out by ONDRAF/NIRAS, the Belgian Agency for Radioactive Waste and Fissile Materials. References Aertsens, M., De Cannière, P., Moors, H., 2003. Modelling of silica diffusion experiments with 32Si in Boom Clay. J. Contam. Hydrol. 61, 117–129. Aertsens, M., De Cannière, P., Lemmens, K., Maes, N., Moors, H., 2008. Overview and consistency of migration experiments in clay. Phys. Chem. Earth 33, 1019–1025. Crank, J., 1975. The Mathematics of Diffusion. Clarendon Press, Oxford. De Cannière, P., Moors, H., Dierckx, A., Gasiaux, F., Aertsens, M., Put, M., Van Iseghem, P., 1998. Diffusion and sorption of 32Si-labelled silica in the Boom Clay. Radiochim. Acta 82, 191–196. Frugier, P., Gin, S., Minet, Y., Chave, T., Bonin, B., Godon, N., Lartigue, J.-E., Jollivet, P., Ayral, A., De Windt, L., Santarini, G., 2008. SON68 nuclear glass dissolution kinetics: current state of knowledge and basis of the new GRAAL model. J. Nucl. Mater. 380, 8–21. Gin, S., Jollivet, P., Mestre, J.P., Jullien, M., Pozo, C., 2001. French SON68 nuclear glass alteration mechanisms on contact with clay media. Appl. Geochem. 16, 861– 881. Godon, N. et al., 2004. Dossier de référence sur le comportement à long terme des verres nucléaires, RT DTCD 2004/06. CEA, Marcoule. Pescatore, C., 1994. The dependence of wasteform leaching on migration parameters in the host medium. Radiochim. Acta 66 (67), 439–444. Pozo, C., Bildstein, O., Raynal, J., Jullien, M., Valcke, E., 2007. Behaviour of silicon released during alteration of nuclear waste glass in compacted clay. Appl. Clay Sci. 35, 258–267.