Diffusion-controlled tracer retention in crystalline rock on the field scale

Authors


Abstract

[1] Tracer retention is a key process for the barrier function of crystalline rock to any contaminant. Here we investigate the nature of retention mechanisms and their field-scale parametrization using results of a comprehensive tracer transport experiment in crystalline rock on the field scale (Äspö Hard Rock Laboratory, Sweden). A method for identifying dominant retention mechanisms and inferring key parameters on the site scale is presented. Taking advantage of multiple tracer tests with a wide range of sorption affinities, retention is shown to be diffusion-controlled. For the considered site, robust features of tracer migration can be reasonably well predicted within a rock volume on at least 200 m scale, by combining independent information with a simple model.

1. Introduction

[2] Water flow in the subsurface makes hydrodynamic transport by advection ubiquitous for dissolved substances of natural and anthropogenic origin. The presence of solid mineral phases and associated immobile fluid results in exchanges due to chemical processes (sorption) and/or physical processes (diffusion). An obvious consequence is retention, whereby dissolved tracers carried by water through a fractured medium are retained (or retarded) relative to groundwater flow. Since the pioneering work of Coats and Smith [1964], considerable effort has been dedicated to better understand the interplay between advection/dispersion and exchange in fractured rock [Hadermann and Heer, 1996; Becker and Shapiro, 2000; Reimus et al., 2003; Widestrand et al., 2007; Cvetkovic et al., 2007, 2010; Cvetkovic and Frampton, 2010].

[3] A key step in risk assessment of any contaminant in a fractured rock volume is site characterization. The degree of complexity and difficulty in characterizing transport properties of a rock formation will depend on how retention mechanisms are conceptualised. For instance, if retention is diffusion-controlled [Neretnieks, 1980], then characterization is relatively difficult and uncertain, as hydrodynamics and fracture aperture control retention in addition to material retention properties [e.g., Cvetkovic et al., 1999, 2004]. Whereas laboratory-scale evidence of diffusion in a crystalline rock matrix is extensive [e.g., Bradbury and Green, 1986; Hellmuth et al., 1995; Sato, 1999], direct quantitative evidence of diffusion-controlled retention on the field-scale is much more difficult to obtain and is still primarily qualitative [e.g., Cvetkovic et al., 2007].

[4] In this paper, a general methodology is presented that combines state-of-the-art analytical modelling with a compact representation of the breakthrough data from field scale tracer tests. The methodology is implemented using results from the most comprehensive tracer tests in crystalline rock to date. The overall objective of the work is to improve our basic understanding of field scale transport-retention mechanisms, and to improve procedures for their field-scale characterisation that is still one of the most challenging problems of hydrogeology [Neuman, 2005].

2. Theory

[5] The ability to discriminate between alternative retention mechanisms in a robust and quantitative manner in the field, still poses a major challenge. The critical experimental condition for meeting this challenge is conducting simultaneous tracer tests with sufficiently contrasting retention properties, preferably over several orders of magnitude. Once such experimental results are available, the conceptual task is in providing a sufficiently general yet simple modelling framework that in a consistent and unified way can take full advantage of the multiple tracer test results; such a framework is presented in the following.

[6] A dissolved (dynamically inert) tracer that enters the ground water is subject to spatially variable advection due to fluid flow, retention due to exchange processes with the immobile phase, and possibly transformation due to decay or degradation. Different transport models can be formulated for relating these mechanisms. A general expression for the tracer residence time density (or expected normalised tracer discharge) h [1/T] at the transport scale L is derived by following tracer particle (Lagrangian) trajectories and assuming a power-law distribution of exchange rates [Cvetkovic et al., 1998; Cvetkovic and Haggerty, 2002] (auxiliary material):

equation image

where T0 [T] is a characteristic retention time for a trajectory, η [−] is an exponent determining the nature of exchange, equation image−1 denotes inverse Laplace Transform, s is the LT variable, and the averaging indicated by angular brackets is over all trajectories and realisations of the rock formation with τ and T0 being random. For process discrimination, two limiting retention conceptualisations (or models) that are implicit in equation (1) will be used, namely the case with an infinitely large exchange rate, and the case of an infinite rock matrix with a very broad spectrum of exchange rates, from zero to infinity.

[7] The first model (denoted by “M1”) is based on the assumption of instantaneous exchange formally obtained by setting η = 0 in equation (1) (auxiliary material):

equation image

where a [1/T] and c [equation image] are hydrodynamic transport parameters dependent on the first two moments of the water residence time τ, and T0τAKd*; both A and Kd* ≡ Kdρ are dimensionless, with Kd [m3/kg] being the sorption coefficient and ρ [kg/m3] rock density. A possible relationship between A and physical properties of the fracture and matrix is briefly discussed in the auxiliary material.

[8] A transport model with diffusion-controlled retention (“M2”) is obtained by setting η = −1/2 in equation (1) [e.g., Cvetkovic et al., 1999]

equation image

where Δ [1/T] is a parameter that is related to the retention time T0 by T0τ2Kd*Δ (auxiliary material).

[9] The model (3) exhibits anomalous behaviour with infinite moments and hM2(t) → t−3/2, whereas hM1(t) → 0 exponentially as t → ∞. In spite of this fundamental difference, there is a remarkable symmetry in the parametrization of (2) and (3), both of them being factorized into a single, predominantly physical parameter (Δ or A) and predominantly chemical parameter (Kd). This factorisation is applicable for moderately to strongly sorbing tracers, with roughly Kd* > 1/4. Furthermore, the physical parameter group Δ can be defined as Δ ≈ sf2De where De is effective diffusivity and sf [L2/L3] is the “active specific surface area”. Note that sorption in the matrix incorporated into (3) is at equilibrium, however, access to the sorption sites is diffusion-controlled; a simple extension to account for chemical kinetics in the matrix is possible [Cvetkovic et al., 1999].

[10] Hydrodynamic transport (i.e., transport of tagged water with Kd = De = 0) in (2) and (3) is quantified by the advection-dispersion equation in a semi-infinite domain. This model can be easily extended to include non-Fickian or even anomalous transport, for instance, as the truncated one-sided distribution [Cvetkovic and Haggerty, 2002] (auxiliary material)); however the advection-disperison equation model is considered sufficiently robust for the current discussion. The controlling parameters are a = 1/2equation imageζ2 and c = 2equation image, where equation image is the mean of τ and ζ its coefficient of variation.

3. Experiments

[11] Tracer tests considered here are from the “Tracer Retention Understanding Experiments” (TRUE) program carried out at the Äspö Hard Rock Laboratory (HRL) in southeast Sweden during 1995–2005; this is the most comprehensive and ambitious experimental undertaking for improving our understanding/characterization of field-scale retention in crystalline rock to date. The TRUE tests uniquely combined the following four important aspects: (i) Multiple scales and pathways, multiple localities and variable types of geological features tested; (ii) comprehensive site characterization and elaborate hydraulic setup ensuring almost full control of groundwater flow and tracer pathways; (iii) extensive site-specific laboratory program, including characterization of the micro-structure and sorption; (iv) cocktails of radioactive isotopes (tracers) with a wide range of sorption affinities, injected simultaneously.

[12] The TRUE tracer tests were carried out at two separate locations of the Äspö HRL, at roughly 400–500 m depth, on Euclidian length scales ranging from 5 m to 30 m. The lithology of the rock is dominated by quartz monzodiorite [Kornfält et al., 1997]. Experiments were performed using sorbing and non-sorbing tracers injected simultaneously. Eight flow paths/tests were conducted with a mixture of non-sorbing tracers and several of the following sorbing elements: 24Na+, 22Na+, 85Sr2+, 131Ba2+, 133Ba2+, 86Rb+, 134Cs+, 137Cs+, 47Ca2+ [Widestrand et al., 2007; Cvetkovic et al., 2007; Cvetkovic and Cheng, 2008; Cvetkovic et al., 2010; Cvetkovic and Frampton, 2010].

[13] A measure of water movement in the tests is given by the mean water residence time equation image, roughly corresponding to one pore volume, typically used in laboratory (column) experiments. For conditions of the TRUE tests and for the purpose of inferring retention parameters, equation image can be approximated reasonably well by the peak arrival time of non-sorbing tracers; the range of equation image estimated for the TRUE tests and length scales involved are summarized in the auxiliary material.

[14] Of the eight TRUE tests we select four for the current analysis, in which the mean water residence time was in the range 5–15 h and a flow path length in the range 5–20 m. Significantly, the four selected tests are from two separate locations approximately 200 m apart, the so-called TRUE-1 and TRUE Block Scale test volumes (auxiliary material). An important point of this study is to examine whether a unified and robust representation of transport can be obtained for both locations, thereby addressing the possibility of characterising retention properties of large rock volumes.

[15] The first and most robust measure of retention is the normalized peak arrival time θptp/equation image. We shall constrain models M1 and M2 by the following relationship A = equation imageΔ/6 such that they both yield the same approximate equation imagep = 1 + AKd* = 1 + equation imageKd*Δ/6 (auxiliary material).

[16] The fractional arrival time tϕQ−1(ϕ) where Q−1 is the inverse of Q(t) = equation imageq(t′)dt′/M0 (with q(t) being either hM1(t) (2) or hM2(t) (3)), will be used as a measure of tracer “breakthrough” (i.e., discharge of tracer mass at the detection point [M/T]). With tϕ, we define dimensionless relative measures equation imageI ≡ (t30%t1%)/equation image, equation imageII ≡ (t60%t1%)/equation image and θIII ≡ (t60%t30%)/equation image for each tracer and test. Quantities θp and θ ≡ [θI, θII, θIII] can now be related to sorption affinity, taking advantage of the unique span in the sorption (partitioning) coefficient Kd [L3/M] for the multiple tracers used in the TRUE tests.

[17] Measurements of Kd in the laboratory are not necessarily unique, as they will generally depend on how a test is carried out (e.g., batch or through diffusion tests), but also on the duration and fractional particle size for the batch test [Byegård et al., 1998]. Even if the absoluteKd value for a given tracer may differ depending on the test type/conditions, the relative values for different tracers tend to be constant for identical test conditions. Here we shall use Kd as obtained from batch tests, with 1–2 mm fraction, under 36 days duration, as the independent variable which defines (or “labels”) a particular sorbing tracer; these are (see Table 3-5 in Byegård et al., 1998 and Table 3 in Cvetkovic and Frampton, 2010: Na (7 × 10−6), Sr (3.6 × 10−5), Ba (1.1 × 10−3), Rb (2.8 × 10−3) and Cs (0.053), all in [m3 kg−1]. Note that the dimensionless independent variable Kd* = Kdρ with ρ = 2700 kg/m3, provides a simple means for extrapolating the test results to any tracer with given Kd.

4. Comparison

[18] In Figure 1a it is seen that θp is low for small Kd*, while attaining a relatively high and steady slope for Kd* > 2–3, directly measuring retention. The model curve (overlapping for models M1 and M2) is calibrated to yield Δ = 0.27 1/h (dashed line in Figure 1a). There are clearly deviations for low Kd* due to a relatively poor representation of early tracer breakthrough by the advection-dispersion equation model, and the fact that (3) is less accurate for low Kd*. For roughly Kd* > 1/4, the dashed model line in Figure 1a captures reasonably well the θp data.

Figure 1.

Comparison between model and experimental data: (a) Normalized peak arrival time for the tracers of the four selected TRUE tests (STT-1 red, STT-1B blue, C-1 green, P-1 black). (b) Correlation between model and experimental data, for the relative fractional measures θ [θI, θII, θIII]; the values assigned largest blue symbols (for θI of Cs) are most affected my the hydrodynamic transport, compared to θII and in particular θIII. All experimental data values in Figure 1 are given in Table S2 of the auxiliary material. For θ, the estimated mean water residence time is used for normalisation, as given in Table S1 of the auxiliary material.

[19] With De = 1.1 × 10−8 m2/h as an “average” value and sf [L2/L3] estimated as 2000–10000 1/m using numerical simulations [Cvetkovic and Frampton, 2010], we have Δ ≈ sf2De roughly in the range 0.04–1 1/h. The model curves for θp are also plotted in Figure 1a for the bounding values 0.04 and 1 1/h (thin lines). Clearly, the uncertainty range for predicting Δ based on independent information is large for modelling any individual tracer test because detailed information is limited, however, given the large scale spanned by the two test locations, the fact that the dashed curve in Figure 1a falls well within this range, is encouraging. Note that sf is uncertain due to a number of reasons, one being that the relationship between transmissivity and aperture is still not well understood for field conditions [Cvetkovic et al., 2007; Cvetkovic and Cheng, 2008; Cvetkovic et al., 2010; Cvetkovic and Frampton, 2010].

[20] Next, we plot in Figure 1b the correlation of θ [θI, θII, θIII], between the models M1 (equation (2)) and M2 (equation (3)), and the data. First, we see that for low θ (corresponding to low Kd*) both models deviate significantly from data, since low θ is dominated by hydrodynamic transport that is poorly represented by the advection-dispersion equation model. For increasing θ in the range 10 and above (corresponding to higher Kd*), the model M2 correlates reasonably well with data, while it is apparent that the correlation of model M1 with data is significantly weaker.

[21] For a quantitative comparison, we take the logarithm of θ as given in Figure 1b (all data values are provided in the auxiliary material), and correlate these by the Spearman, Kendall and Pearson methods implemented in Mathematica (R). The result is summarized in Table 1 for the best fit Δ as well as for the predicted uncertainty bounds of Δ (0.04 and 1 1/h). The correlation coefficient is relatively high by all three methods in view of a consistent trend in data and model over a wide range of θ. However, it is clear that the correlation between model M2 and data is systematically higher than the correlation between model M1 and data for all methods. The correlation coefficient of log(θ) indicates a slightly higher correlation for Δ = 1 1/h, while for θp this value provides a poor representation (Figure 1a). Thus θ (quantifying the shape of the breakthrough) should obviously be considered in conjunction with θp (quantifying the position in time of tracer breakthrough) as complementary measures for inferring best estimates of Δ; this complementary aspect between θp and θ is important for a robust characterization of field-scale retention properties using results of tracer tests.

Table 1. Correlation Coefficients Obtained by Different Methods Using Mathematica (R) Software, for Relative Fractional Arrival Time Measures θ, as a Function of the Retention Parameter Δ and Retention Modela
Correlation Coefficient of log(θ)Δ = 0.04 (1/h)Δ = 0.27 (1/h)Δ = 1.0 (1/h)
M2 (Spearman)0.900.930.94
M1 (Spearman)0.670.830.85
M2 (Kendall)0.750.790.81
M1 (Kendall)0.490.650.68
M2 (Pearson)0.890.910.91
M1 (Pearson)0.780.860.89

[22] Clearly, Kd is a very simplified representation of complex chemistry, hence is prone to uncertainty, especially on the field scale. Nevertheless, Kd is the principal and from the practical point of view the most important tracer-dependent material property relevant for quantifying retention in rocks that can be determined in the laboratory. The key question for applications is estimating in-situ Kd values. Due to the fact that θp shows a clear trend with Kd* (Figure 1a), we write a scaling relationship Kd[in-situ] = χKd[reference] where the scaling parameter χ [−] is not known a-priori; for the TRUE tests χ ≈ 1 [Cvetkovic and Cheng, 2008], if Kd[reference] is the batch test value using 1–2 mm fractions as summarized earlier in this section. However, for any other site the rim zone (i.e., rock matrix adjacent to a fracture where retention takes place) would need to be characterised to some extent, both for providing independent estimates of the scaling parameter χ, and for quantifying possible trends in matrix porosity that need to be accounted for when using Δ obtained from tracer tests in transport calculations over long time scales [Cvetkovic, 2010].

5. Conclusions

[23] Based on a simple and general modelling strategy applied to the most comprehensive transport (TRUE) experiments in crystalline rock to date, we summarise our main findings as follows: (i) Normalised peak arrival time and relative fractional arrival times inferred from tracer test data, provide robust, complementary measures for characterising field-scale retention properties; (ii) these measures are shown to be comparable between the two test locations 200 m apart in the crystalline rock of the Äspö HRL, indicating that a consistent effective representation of retention properties on large scales is possible; (iii) retention in crystalline rock should be a-priori assumed diffusion-controlled where the bounding values of Δ can be estimated reasonably well based on independent information. These conclusions could be drawn only because multiple tracers with a wide range of sorption affinities were used in the TRUE tests. Diffusion-controlled retention on the field scale using non-sorbing or weakly sorbing tracers tests is generally difficult to distinguish from slow advection [Becker and Shapiro, 2000; Shapiro, 2001].

[24] As shown in this work, the uncertainty range for Δ based on information independent of tracer tests is large for predicting the outcome of any individual tracer test. This uncertainly may be further reduced by combining additional information on structural and hydrodynamic properties, combined with flow and transport modelling. The “active specific surface area” (sf [L2/L3]), for instance, is dependent on the internal heterogeneity of conducting fractures, as well as on the fracture-to-fracture flow variability [Painter et al., 1998; Cvetkovic et al., 1999; Painter et al., 2002; Cvetkovic and Frampton, 2010]; it can be measured in the field only indirectly, hence it is subject to large uncertainty. Quantifying sf is a significant challenge and a topic of active research, of importance not only for tracer retention, but also for any reaction involving diffusive exchange between fractures and the surrounding matrix [Steefel and Lichtner, 1994]. Finally, we note that over long time scales the impact of geochemical evolution on the sorption coefficient Kd would need to be explicitly accounted for in predictive modeling.

Acknowledgments

[25] Support for this study was provided by the Swedish Nuclear Fuel and Waste Management Co (SKB). The author is grateful to Jan-Olof Selroos (SKB) and Anders Winberg (Conterra AB) for helpful comments that improved the presentation of this work.

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