Stochastic simulation of radionuclide migration in discretely fractured rock near the Äspö Hard Rock Laboratory



[1] We study the migration of sorbing tracers through crystalline rock by combining relatively simple transport measures with particle tracking in a discrete fracture network. The rock volume is on a 100 m scale and is a replica of a thoroughly characterized site at the Äspö Hard Rock Laboratory, Sweden. Flow is driven by generic boundary conditions consistent with the natural gradient in the region. The emphasis is on the global effect of fracture-to-fracture hydraulic variability where individual fractures are assumed to be of uniform aperture. The transport measures are conditioned on two random variables: the water residence time (τ) and a parameter which quantifies the hydrodynamic control of retention (β). Results are illustrated for two radionuclides: technetium (strongly sorbing) and strontium (weakly sorbing). It is found that the assumption of streamline routing or full mixing at fracture intersections has comparatively little impact on transport. The choice of the cubic or quadratic hydraulic law (i.e., relation between transmissivity and aperture) strongly affects water residence times but has little impact on average transport since it does not affect the statistics of β. If the statistics of β are known, then the distribution of water residence time (τ) is of little importance for transport. We assess the applicability of a linearized model β = τ/bret using two different approaches to estimate the effective “retention” aperture 2bret: from transmissivity data and from fracture density and flow porosity data. Under some conditions, these conventional estimates may provide acceptable representation of transport. The results stress the need for further studies on upscaling of τ, β distributions as well as on estimating effective parameters for hydraulic control of retention.

1. Introduction

[2] The efficiency of a rock formation as a transport barrier depends not only on the fluid flow in the rock but also on the degree to which contaminants are retained in the rock due to a variety of physical and chemical processes. Open fissures or fractures in the rock provide pathways through which water and tracers may travel. Although most tracers of interest such as radionuclides have a strong tendency to sorb to mineral grains in the rock, tracers first have to diffuse from fractures into the rock matrix in order to access the extensive pool of sorption sites [Neretnieks, 1980]. Diffusion in turn depends not only on mass transfer properties of the rock matrix, but also on the structural and hydrodynamic properties of the fracture system, emphasizing the interaction between water flow, advective transport and retention processes.

[3] Flow and transport through individual fractures, as well as fracture networks, has been studied extensively over the past years both experimentally and using simulations. Studies of transport through discrete fracture networks have often been limited to nonreactive particle transport [Smith and Schwartz, 1984; Shapiro and Andersson, 1985; Andersson and Dverstorp, 1987; Cacas et al., 1990; Nordqvist et al., 1992; Berkowitz and Scher, 1997; Margolin et al., 1998]. Although understanding nonreactive transport is a prerequisite for quantifying reactive transport in fracture networks, retention of tracers adds further challenges for modeling which have received comparatively little attention in the literature.

[4] Models of retention in fracture networks are derived from models of retention in single fractures. The most basic result was obtained by Carslaw and Jaeger [1959] for heat diffusion into semi-infinite domains. Their solution was adopted by Neretnieks [1980] for crystalline fractures assuming plug flow. Tang et al. [1981] generalized the model of Neretnieks [1980] for single fractures by including dispersion. Maloszewski and Zuber [1990] extended the model of Neretnieks [1980] by including kinetic sorption in the matrix and postulating the applicability of the single fracture model to a network of fractures where effective aperture is implied. The approach of Maloszewski and Zuber [1990] was adopted by Shapiro [2001] for interpreting environmental tracer data, where instead of aperture, fracture density and flow porosity were used as effective parameters.

[5] Transport with retention in a discrete fracture network was investigated by Sudicky and McLaren [1992] in a deterministic framework. A first attempt to account for randomness of discrete fracture networks in retention modeling was made by Moreno and Neretnieks [1993] using a channel (or pipe) network model, where fluid flow was resolved on the basis of mass balance, i.e., without resolving three-dimensional flow dynamics. Cvetkovic et al. [1999] were the first to account for effects of flow dynamics on tracer retention in single heterogeneous fractures, introducing a random (Lagrangian) variable β [T/L]. Analytical results on the β statistics for fracture networks were provided by Painter et al. [1998] and Painter and Cvetkovic [2001] by extending the channel concept of Neretnieks [1980]. Results on the statistics of β in fracture networks where flow dynamics is resolved, were presented by Painter et al. [2002] using generic DFN simulations. Site-specific DFN simulations on a 1000 m scale were presented recently [Outters and Shuttle, 2000; Cvetkovic et al., 2002], where the flow field was simplified as a branching pipe network [Dershowitz et al., 1998].

[6] In this paper, we present a systematic study of radionuclide transport and retention in a discrete fracture network where flow dynamics is honored and randomness of fracture properties is accounted for. Transport is studied in a generic rock volume that is a statistical replica of one of the most thoroughly characterized crystalline rock domains to date, on approximately 100m scale at the Äspö Hard Rock Laboratory in southern Sweden [Forsmark and Rhen, 2001]. The dual-porosity modeling concept is implemented using the stochastic framework of Cvetkovic et al. [1999], here extended to fracture networks. The emphasis is on the hydrodynamic control of retention by means of two random (Lagrangian) variables, τ [T] and β [T/L]. We first summarize the conceptual framework and present relatively simple transport measures for randomly fractured porous media which take advantage of analytical solutions. The proposed measures couple hydrodynamics of fracture flow with microscopic retention processes while explicitly accounting for prediction uncertainty associated with large scale (and unobservable) fluctuations in groundwater velocity. We assess the significance of different modeling concepts for discrete fracture networks, in particular, the impact of different intersection rules (full mixing and streamline routing), and different hydraulic laws (the relationships between fracture transmissivity and aperture).

2. Transport Model

2.1. General Considerations

[7] Previous work has shown that the basic solution for Lagrangian (trajectory) representation of subsurface transport with retention is the “unit response function”, or “retention function” γ [Cvetkovic et al., 1998; Cvetkovic and Haggerty, 2002], defined in the Laplace domain as

equation image

where s is the Laplace transform variable, τ is the water residence time, equation image is the Laplace transform of the so-called “memory function” g which accounts for a general class of linear exchange processes, X(t) is the advective displacement at time t [Dagan, 1984], i is the index of a segment along a trajectory, and N is the total number of segments. The function γ is the conditional solution of transport with retention since for realistic systems τ is a random variable. In equation (1), we provide both the integral and discrete representations.

[8] Common applications require quantifying the expected tracer discharge (“breakthrough”) which is directly related to the unconditional solution h ≡ 〈γ〉. The tracer discharge from the domain, J [M/T], is then computed as

equation image

where M is the injected tracer mass, ϕ [1/T] is the prescribed injection rate, and the asterisk denotes convolution.

2.2. Retention Model

[9] Depending on the structural and retention properties of a rock formation, and the scale of the transport problem at hand, g in equation (1) may take different forms. Granitic rocks are typically of low fracture density. This fact has led to the view that in such rocks diffusion-controlled retention takes place from conducting fractures into an essentially infinite matrix [Neretnieks, 1980; Tang et al., 1981; Moreno and Neretnieks, 1993; Cvetkovic et al., 1999; Shapiro, 2001]. The question is to what extent does field evidence support this view. We briefly summarize below results from a few comprehensive tracer tests which have addressed diffusive mass transfer in fractured rock on the field scale.

[10] Tracer tests in granitic rock at Grimsel (Switzerland) were conducted in an approximately 0.05 m thick fracture zone on a 2–5 m scale [Hadermann and Heer, 1996; Heer and Smith, 1998]. Observed retention was relatively strong, taking place primarily within the fracture zone rather than in the matrix proper. The breakthrough curves were interpreted assuming unlimited diffusion, however, toward the end of the experiment, a drop in the tail occurred indicating that the fracture zone was at that point being saturated at least by the low sorbing tracers. Note that the estimated porosity 6–12% of the zone is considerably higher than 0.5% which is typical for the matrix of granitic rocks.

[11] Tracer tests at the Äspö Hard Rock Laboratory (Sweden) referred to as “Tracer Retention Understanding Experiments” (TRUE) were conducted through fractures over a 5–30 m scale [Winberg et al., 2000; Poteri et al., 2002], rather than zones as conducted in Grimsel. The measured breakthrough curves for a cocktail of tracers with widely varying retention (sorption) properties were interpreted in a consistent manner using the infinite matrix retention model; however, deviations from this model, of the type observed at Grimsel, cannot be excluded at times greater than the experimental time.

[12] Results from tracer tests at Mirror Lake (New Hampshire, USA) were carried out on a 36 m scale with nonreactive tracers with diffusivities varying within a factor of 3 [Becker and Shapiro, 2000]. Evidence of diffusion-controlled retention could not be established in these tests, and the tailing of breakthrough curves was interpreted by advective transport. It should be noted that tracers with much larger contrast in retention properties than factor of 3 are generally required for a more reliable evaluation of diffusion-controlled retention; such contrasts can in practice be achieved only with reactive tracers. The corresponding contrast in retention properties of tracers used in Grimsel and TRUE tests was over 3 orders of magnitude (between nonreactive tracer and cesium), compared to a factor of 3 for the Mirror Lake tests.

[13] Retention in fractured rocks has also been interpreted using a limited diffusion model. A series of single-well injection-withdrawal tests with nonreactive tracers were carried out in fractured dolomite in New Mexico [Meigs and Beauheim, 2001]. The dolomite was structurally quite different from typical granitic rocks, and g employed in the modeling corresponds to a diffusion-controlled exchange into spherical blocks of varying (but finite) sizes [Haggerty et al., 2001]. We emphasize that the estimated matrix porosity was very large, in the range 15–16% indicating potentially rapid diffusion into the immobile zones, compared to the slow diffusion into typical granitic rock with porosity of approximately 0.5%.

[14] On the basis of the currently limited field evidence available, it seems that several different models for diffusion-controlled retention may be appropriate for granitic and other rock types. For our current analysis of reactive tracer transport in granite of the type considered in the TRUE tests at Äspö, we shall assume g consistent with an unlimited diffusion model. More specifically, we assume that for the granitic rock at the Äspö site, the distance between fractures is sufficiently large relative to the retention properties of the tracer involved, and diffusion into the rock blocks between adjacent fractures is unlimited over the transport timescales. Further discussion on this assumption is provided in Appendix A related to the particular tracers chosen for the transport analysis.

[15] The memory function gi in equation (1) for unlimited diffusion is defined in the form

equation image

The parameters θ, De, R characterize retention: θ (dimensionless) is the porosity of the rock matrix, De [L2T−1] is the effective diffusivity into the rock matrix which accounts for the tortuosity and matrix porosity (the relationship between “effective diffusivity” De and “pore diffusivity” D is obtained through porosity as De = θD; De can be expressed using the so-called formation factor F as De = FDw where Dw is diffusivity in pure water), and R = 1 + ρ0Kd/θ is the retardation coefficient, where Kd [L3M−1] is the partitioning (distribution) coefficient, ρ 0 is the bulk density and b is the half-aperture of the i-th fracture. With equation (3), our focus is on fracture-to-fracture variability, i.e., in this study we shall neglect the internal variability of fractures.

[16] Inserting equation (3) into equation (1), we get

equation image

where B [equation image] is a parameter defined by

equation image

[17] All retention parameters in equation (4) as well as the aperture, can vary between fractures. If the retention parameters are uniform, then

equation image

[18] Equation (4) can be inverted to yield

equation image

where we have also included linear decay with λ [1/T] being the decay rate, and H() is the Heaviside step function.

[19] The predominant focus in the literature on flow and transport in fracture networks has been on advective transport where γ → δ(t − τ). In this case we have h(t) = f(t), and the focus is on computing the water residence time density f(τ) for a given rock volume. If we consider a channel of length L and width W through a single fracture or a rock volume which preserves a constant volumetric flow rate Q, then BLW/Q, reducing equation (7) to the model originally proposed by Neretnieks [1980]. Using the channel model, we may also write B ∼ τ/bret where 2bret is an effective retention aperture [Cvetkovic, 1991] (bret is to be discussed more in a subsequent section).

[20] Assuming uniform (effective) retention parameters θ, De and Kd for crystalline rock is common in applications [Maloszewski and Zuber, 1990; Moreno and Neretnieks, 1993; Carrera et al., 1998; Shapiro, 2001]. Recent field and laboratory experiments at the Äspö Hard Rock Laboratory have provided the most detailed evidence to date on the spatial variability of matrix porosity [Andersson et al., 2002; Kelokaski et al., 2001]. The significance of this variability for evaluating results of tracer tests, and for estimating effective matrix porosity, has been discussed [Cvetkovic and Cheng, 2002]. However, systematic data on spatial variability of retention parameters is still either nonexistent (for Kd) or insufficient (for θ and De) for deciding on statistical models that capture fracture-to-fracture variability of retention properties in discrete fracture networks. By contrast, field data on hydraulic properties of fractures has been compiled over the past decades at various sites, and statistical models for fracture transmissivity are common. Although the effect of hydraulic and/or structural variability on advective transport has been addressed in a number of studies [Smith and Schwartz, 1984; Shapiro and Andersson, 1985; Cacas et al., 1990; Nordqvist et al., 1992; Berkowitz and Scher, 1997], its effect on retention in discrete fracture networks has received less attention. In our following analysis, we wish to highlight precisely this effect; hence we shall use B as defined in equation (6) in order to focus specifically on the importance of the random variable β.

2.3. Transport Measures

[21] Given the random character of transport in fracture networks, it is convenient for many applications to use compact measures of transport that capture key features of breakthrough curves. These measures (or their statistics) can be compared in sensitivity or site-selection studies. In our earlier work we introduced a measure (referred to as “containment index”) defined as 1 − equation imageγdt where equation imageγdt is the discharged mass fraction from the rock. Distribution of the containment index was computed for six radionuclides by Cvetkovic et al. [2002] using site-specific simulations of Outters and Shuttle [2000] which took advantage of a simplifying DFN modeling approach based on an algorithm which views the network as consisting of interconnected “pipes” [Dershowitz et al., 1998]. The containment index quantifies transport in an integrated sense, i.e., for all time, and provides a suitable comparative measure for a rock volume to retain radionuclides. Also, we have considered earlier the “fractional arrival time” as a compact transport measure for nondecaying tracers [Painter et al., 1998]. In this work, we introduce two new conditional measures important for safety assessment: radionuclide peak discharge, and peak arrival time.

[22] Setting the time derivative of γ(t) to zero, we compute the peak arrival time tp as

equation image

Inserting t = tp into γ(t), we get the peak discharge γp ≡ γ(t = tp) as

equation image

[23] Using the joint density f(τ, B), we can compute the distributions of tp and γp for different tracers. One of the central issues for modeling transport in fractured porous media, is separating the effects of material (retention) properties (such as matrix porosity, diffusivity and sorption coefficient), from those of flow/advection and structure. The parameter B integrates the effect of material sorption properties with that of flow/advection. However, if the material sorption properties are assumed uniform, then these effects can be separated (factorized) as given in equation (6). In this case, τ, β are random variables dependent only on the structure and flow, with a joint density f(τ, β). Given β = Bequation (6), the relationship between the densities f(τ, B) and f(τ, β) is straightforward.

[24] In the following, we shall use Monte Carlo simulations to compute the distributions of the transport measures tp and γp using τ, β as obtained from a unique set of site-specific DFN simulations.

3. Site-Specific DFN Simulations

[25] The conceptual “discrete fracture network” (DFN) model assumes that discrete fractures provide the primary hydraulic flow paths and connections. Discrete fractures are here treated as 2D features (tessellated to triangular elements). They are generated in realistic three-dimensional networks based on the structural geology and statistical information of the fracturing, and are conditioned on local measurements; full details of the DFN simulations used in this study are given by Outters [2003].

[26] A summary of the parameters that are required for constructing a DFN model is as follows.

[27] 1. For Fracture orientation distributions, fractures with similar orientations can be sorted into fracture sets. The orientation of the fracture in each set is defined by a statistical distribution.

[28] 2. For fracture size distributions, the sizes of the fractures in a fracture set are defined by their area distribution.

[29] 3. For the fracture spatial model, fracture trace maps can present different aspects that are not necessarily dependent of the fracture size or orientation. This aspect mainly depends on the spatial distribution of the fracture traces within a given surface. Fractures can be heavily clustered and regularly dispersed within the surface or can be located in small groups. A spatial analysis of a trace map permits to characterize the clustering of fractures. In the present study, the fractures are homogeneously distributed in space.

[30] 4. For the fracture set intensity, fracture set intensity can be expressed as linear frequency, trace length density or fracture area density. In this study, we use the surface density Sf (also referred to as P32) defined as the total area of fractures per unit volume of rock [m2/m3].

[31] 5. For fracture hydraulic transmissivity distributions, the hydraulic behavior of a fracture network depends mainly on the transmissivity of the fractures. The transmissivity T of a single fracture in a borehole section of 1 m length in an impermeable rock is equivalent to the conductivity of the same 1 m long borehole section in a continuous porous material. The transmissivity distributions of the conductive fractures are estimated by the statistical analysis of packer test results.

[32] 6. For the fracture transport aperture distributions, advective transport in a fracture network depends mainly on the aperture of the fractures. The fracture transport aperture is calculated as a function of the hydraulic transmissivity using a prescribed hydraulic law. The conventional assumption is the “cubic law” derived rigorously for uniform fractures. Field experience indicates, however, that for heterogeneous fractures the cubic law may not be appropriate and that the dependence of transmissivity on aperture is weaker, hence the use of the empirical “quadratic law”.

3.1. Base Case

[33] The reference case for the sensitivity analysis is constructed to emulate the hydraulic behavior of a typical Swedish granitic rock volume. In an earlier study [Outters and Shuttle, 2000], the geometric and hydraulic properties of the DFN model were based on the prototype repository and the TRUE Block Scale site at the Äspö Hard Rock Laboratory, Sweden. Since then, the DFN for the prototype repository has been updated by [Stigsson et al., 2001] who carried out an extensive analysis of pump tests which has led to an improved understanding of the fracture set definition and their hydraulic properties; their model will be used in our investigation.

[34] The size of a simulation rock volume was selected such that it is consistent on the one side with the measured data, and on the other approximately sets an upper bound for the “near-field” scale for safety assessment. The shape of the domain is a 100 m ×100 m ×100 m cube (Figure 1). Since the DFN model characteristics are stochastic, the modeling is carried out with Monte Carlo simulations. A total of 20 realizations were constructed for the base case and each of the variant cases. The DFN realizations were generated by the FracMan(c) software. A total of about 20000 three dimensional disc-shaped fractures were identified in each realization. The discrete fractures were tessellated into triangular finite elements respecting the connectivity between the fractures. Nonconnected fractures were not considered in the computations.

Figure 1.

Configuration sketch for DFN simulations.

[35] Generic boundary conditions were prescribed such as to obtain a globally unidirectional flow. The flow direction is forced from one vertical face of the model domain to the opposite face by assigning a constant head boundary condition at the faces (Figure 1). The rest of the faces are sealed by a no-flow condition. A head gradient of 0.1% (0.1 m head variation over a 100 m model length) is applied between the two fixed head boundaries. For the base case we assume 100% mixing at fracture intersections.

[36] Once the flow field is solved, inert particles are released from an area that is of square shape 50 m × 50 m and is centered in the middle of the upstream boundary (Figure 1). Particle transport was simulated within the fracture elements as particle tracking. In order to account for the effect of internal variability (dispersion) in the fractures, particles are assigned a random displacement with relatively low longitudinal and transverse dispersivities of 1 m and 0.1 m, respectively. Transverse dispersion is introduced in the simulations in order to approximately account for “meandering” due to flow variability inside the fracture; transverse dispersivity acts only in the fracture plane and not orthogonal to it.

[37] Particle transport requires the assignment of an aperture to each fracture. We used the transmissivity value of a fracture and computed an aperture based on an empirical hydraulic law (“quadratic”, or “Doe's” law) as 2b = 0.5 T1/2 where the transmissivity T has units of [m2/s] and the half-aperture b in given in meters [Dershowitz et al., 1999]. A typical realization of particle trajectories is illustrated in Figure 2. The simulation time of around 300 years allowed nearly all particles to pass through the rock volume and reach the downstream boundary. All parameter values for the base case are summarized in Table 1.

Figure 2.

Illustration of particle trajectories in one realization of the DFN.

Table 1. Summary of Simulation Input Data
 SetStrike; DipDispersionMeanSDSf (P32)Source of DataReference
  • a

    Fracture termination probability of 37%.

Orientation Fisher distribution1212.8; 83.73.96   pilot and exploratory holesStigsson et al. [2001]
Orientation Fisher distribution2126.9; 86.810.53   pilot and exploratory holesStigsson et al. [2001]
Orientation Fisher distribution317.9; 7.59.32   pilot and exploratory holesStigsson et al. [2001]
Size (radius [m]) lognormal distribution1  22 TBM-tunnelLapointe et al. [1995]
Size (radius [m]) lognormal distribution2  82 TBM-tunnelLapointe et al. [1995]
Size (radius [m]) lognormal distribution3  54 TBM-tunnelLapointe et al. [1995]
Spatial model (BART model)a      TBM-tunnelDershowitz et al. [1996, 1999]
Natural fracture intensity Sf[m2/m3]1    0.26from 1 and 3 m section pump tests in exploratory holesStigsson et al. [2001]
Natural fracture intensity Sf[m2/m3]2    0.85from 1 and 3 m section pump tests in exploratory holesStigsson et al. [2001]
Natural fracture intensity Sf[m2/m3]3    0.18from 1 and 3 m section pump tests in exploratory holesStigsson et al. [2001]
Transmissivity lognormal not correlated with size (×107) [m2/s]1, 2, 3  1.1721.8 deterministic and background fracturesDershowitz et al. [1996]

3.2. Variant Cases

[38] In order to study the influence of some key modeling assumptions on transport of sorbing tracers, we consider two variants of the base case for which the parameters and the flow/transport configuration were summarized above. Two variants (referred to in the figures as “hydraulics” and “intersections”) are defined as follows.

[39] The fracture transmissivity T and aperture in the base case are related using an empirical law, which is a “quadratic” law since flow rate is a quadratic function of the fracture aperture rather than a cubic function. The case “hydraulics” presents the conventional (uniform fracture) model of relating the fracture aperture with transmissivity using the “cubic law”, where 2b = 0.01 T1/3; again the half-aperture b is given in meters and the transmissivity T has units of [m2/s]. All other conditions and parameters are identical to the base case.

[40] The second case (“intersections”) addressed the issue of fluid mixing at fracture intersections. Here we use a revised particle tracking algorithm compared to the base case, which supports streamline routing such that there is no mixing at intersections. The details of the algorithm are described by Outters [2003]. All other characteristics and parameters are identical to the base case.

4. Results

[41] The joint density f(τ, β) is obtained for a given rock volume by considering only advective transport. First, we discuss marginal densities for τ and β, and the correlation between τ and β as obtained from Monte Carlo DFN simulations. Then, we discuss the statistics of the proposed transport measures, and compare different simulation cases.

4.1. Marginal Distributions for τ and β

[42] Consider first the marginal density f(τ) = ∫ f(τ, β)dβ. The cumulative distribution function CDF(τ) ≡ equation imagef(τ′)dτ′ and the complementary cumulative distribution function CCDF(τ) = 1 − CDF(τ) are both illustrated in Figure 3 for different simulation cases using double logarithmic scale. The CDF provides a clear view of the early arrival, whereas the CCDF provides a view of the late arrival. In Table 2 we summarize the first two moments of τ as computed from the DFN simulations for the different cases.

Figure 3.

Complementary cumulative distribution functions (CCDF) and cumulative distribution functions (CDF) for the water residence time τ for different simulation cases.

Table 2. Summary of Simulation Results for Mean and Standard Deviation of τ and β for Base Case and Four Variant Cases
Case〈τ〉, yearsστ, years〈β〉 ×10−6, yr/mσβ ×10−6, yr/m
Base case35.0734.311.554.28
Case A (hydraulics)18.3124.061.53.87
Case B (intersections)39.6746.611.975.14

[43] The intersection law (mixing or routing) has a comparatively small effect on both the CCDF and CDF, i.e., on early and late arrival, resulting in a slight shift toward higher values if streamline routing is considered (dotted curve in Figure 3). We note that relatively small impact of the intersection rule on advective transport has been found in two-dimensional fracture networks [Park et al., 2001]. The assumed hydraulic relationship has a significant impact on the early arrival, and a comparatively small impact on the late arrival (Figure 3). Assuming cubic law as compared to the “Doe's law” (or “quadratic law”) predicts earlier arrival by approximately 1/2 order of magnitude.

[44] The first two moments of β as obtained from simulation of the different cases are summarized in Table 2. The CCDF and CDF of β are shown in Figure 4 for different simulation cases. Again, the intersection rule has comparatively small impact on global β statistics, both in the low and high range, resulting in a shift toward higher values if streamline routing is used (dotted line in Figure 4). The hydraulic relationship has no impact on β statistics, indicating that in the current simulations β is independent of the flow porosity (Figure 4). This is a direct consequence of the fact that internal aperture variability in fractures is neglected in the DFN model. The flow in a fracture is thereby effectively reduced to a channel with a constant aperture and flow rate, and with varying width.

Figure 4.

Complementary cumulative distribution functions (CCDF) and cumulative distribution functions (CDF) for the parameter β, for different simulation cases.

4.2. τ, β Correlation

[45] A direct measurement of β in the field would generally require sorbing tracers; this limits the spatial scale over which a test can be conducted in practice. One possible approach to overcoming this limitation is by an indirect estimate of the β distribution. Assuming functional dependence between τ and β, we may use estimates of τ statistics which are in principle less difficult to obtain. The basis for such an approach is the common assumption β ∼ τ (see next section). Numerical simulations in a single fracture have shown that β ∼ τm where 1 < m < 2 [Cvetkovic et al., 1999]. Recent work has provided a more comprehensive simulation set for single fractures, specifically addressing the relationship between τ and β in different flow configurations [Cheng et al., 2003].

[46] Cheng et al. [2003] summarized the various exact forms of β ∼ τm for a uniform fracture, depending on the hydraulic relationship assumed between transmissivity and aperture. If the hydraulic gradient is fixed, then β ∼ τ1.5 is applicable if cubic law is assumed, and β ∼ τ2 if the “quadratic law” is assumed. Moreover, β ∼ τ is applicable for a uniform fracture only if the aperture is assumed fixed, and the gradient or flow rate can vary (variations are here implied as a means of constructing an ensemble and generating statistics of τ, β with uniform fractures). Simulations of flow and advection in a single fracture with varying aperture showed a qualitative consistency with the exact τ, β relationships for a uniform fracture [Cheng et al., 2003].

[47] In Figure 5, DFN simulation data of τ, β are presented for the base case. Each symbol in Figure 5 represents one trajectory in one realization; 10% (or each tenth point) of the total number of approximately 18,000 data points is shown in Figure 5. A linear fit on a double logarithmic scale provides the best estimate of the correlation between τ and β, indicating power law dependence. As can be seen from Figure 5, τ and β are relatively strongly correlated, although the scatter in β for a fixed τ, extends to almost two orders of magnitude.

Figure 5.

Correlation between τ and β for different simulation cases. The thick solid and dashed lines indicate the best power law fit. Simulation τ, β data (every tenth value) is shown only for the base case.

[48] The best fit of the τ, β correlation for the base case is β = 3241 τ1.59 where perfect mixing at intersections, and the “quadratic” hydraulic law, are assumed. If streamline routing is assumed, then the correlation is almost identical as β = 3194 τ1.61 (data not shown). The best fit for the cubic law (“hydraulics”) case of the τ, β correlation is β = 17631 τ1.41 (data not show in Figure 5, only the best fit is shown), i.e., a lower exponent is obtained with the cubic law. This is qualitatively consistent with results for a single fracture, uniform as well as nonuniform, where the cubic law yields a lower exponent m in β ∼ τm, than the “quadratic law” [Cheng et al., 2003].

4.3. Peak Discharge and Arrival Time

[49] In order to illustrate the impact of hydraulic variability in a DFN on transport and retention, we shall use the proposed transport measures for two radionuclides, Technetium (99Tc) and Strontium (90Sr). The basic site-specific retention/decay data for the two tracers to be used in the calculations are summarized in Table 3. Additional parameters are the porosity and density of the rock matrix, θ = 1% and ρ0 = 2700 kg/m3, respectively. We note that the measurements of matrix porosity for the Äspö region vary over a significant range, depending if the rock is altered or unaltered; the selected value of 1% is chosen as a compromise between available data for illustration.

Table 3. Summary of Data for Sr and Tc Used in the Calculationsa
Tracert1/2, yearsDe, ×10−6 m2/yrKd, m3/kg
  • a

    The data for Sr are from laboratory experiments on Äspö (unaltered) diorite [Byegård et al., 1998; Cvetkovic et al., 2000], while the data for Tc are taken from Andersson [1999]; Kd is from Andersson's Table A.2.6.2 as the upper limit (for saline water), and De is from Table A.2.5.2 for siline and/or fresh water.

Technetium210 0001.31

[50] The choice of radionuclides Sr and Tc followed from the fact that they are of interest in a safety assessment context for nuclear waste repositories in crystalline rock [Andersson, 1999], and have distinctly different retention/decay properties: Sr decays rapidly but sorbs weakly, whereas Tc decays slowly but sorbs strongly. In the following, we shall assume that the transport-retention model (7) is applicable for Sr and Tc at the Äspö site; justification for this assumption is given in Appendix A.

[51] The peak arrival time tp in equation (8) is a direct measure of retention, in a mean sense, accounting for diffusion and sorption in the rock matrix. In Figures 6a and 6b we illustrate CCDF(tp) and CDF(tp) for Sr and Tc, respectively. Clearly, the retention is significantly stronger for Tc than Sr, due to relatively large Kd for Tc. For instance, it can be stated with 99.9% probability that tp will be greater than 2 × 105 years for Tc, but only 10 years for Sr. Similarly, there is only 0.1% probability that tp will be greater than approximately 20000 years for Sr, and 109 years for Tc. Assuming routing rather than perfect mixing at intersections will increase tp slightly (Figure 6), reflecting the shift of β toward higher values (Figure 4); thus tp is sensitive to β values. The hydraulic law has no impact on the upper range of tp for either Sr or Tc (Figure 7), however, it does slightly affect the lower range for Sr (Figure 7a), consistent with the shift in Figure 3.

Figure 6.

Complementary cumulative distribution functions (CCDF) and cumulative distribution functions (CDF) of the peak arrival time tp (equation (8)) for different simulation cases: (a) weakly sorbing radionuclide Sr and (b) strongly sorbing radionuclide Tc.

Figure 7.

Complementary cumulative distribution functions (CCDF) and cumulative distribution functions (CDF) of the normalized radionuclide peak discharge γp (equation (9)) for different simulation cases: (a) weakly sorbing radionuclide Sr and (b) strongly sorbing radionuclide Tc.

[52] In Figures 7a and 7b we illustrate the CCDF and CDF of the normalized peak discharge γp for Sr and Tc, respectively, for different simulation cases. For the base case, the maximum peak discharge is for Sr, the probability of γp > 0.0027 1/yr being 9% (Figure 7a). Assuming streamline routing at fracture intersections reduces this probability to about 6% (Figure 7a), i.e., the peak discharge is slightly lower in this case due to larger β (Figure 4). For Tc, the maximum peak discharge is 2.7 × 10−7 1/yr with a probability of 0.3%; assuming streamline routing again reduces the peak due to a lower β (Figure 7b). The choice of the hydraulic law has effectively no impact on the peak discharge of Sr or Tc (Figure 7).

5. Effective Parameters

[53] The conventional model for transport and retention in fractured rock is based on continuum mass balance equations (equation (B1), Appendix B). This is an advection-dispersion model, where diffusion-controlled retention in a given rock volume is assumed to take place in a “representative” fracture with effective properties. Our Lagrangian (trajectory) approach to the same problem should be viewed as a distributed version of the continuum averaged model, where the same type of diffusion-controlled retention process is accounted explicitly for each fracture along a given trajectory. Averaging based on equation (2) in effect generalizes the continuum model and provides a means of addressing the issue of effective retention parameters for the continuum model.

[54] One limitation of the continuum model (B1) is the way hydrodynamics is assumed to control retention. The key quantity in this context is the specific surface Sf, which is a prescribed parameter for a given rock volume (see, e.g., Table 1). Dividing the transport equation (B1) by nf, we obtain the key parameter in equation (B1) for the hydrodynamic control of retention, 2 Sf/nf ≡ 1/bret, where bret is an effective “retention” half-aperture for the considered rock volume.

[55] Different strategies may be taken for estimating the retention aperture 2bret from structural and hydraulic data. One approach is to use the measured transmissivity and infer bret using a hydraulic law (say cubic or quadratic). In our case, the arithmetic mean transmissivity is 1.17 · 10−7 m/s. From the quadratic law (see description of base case) used in FracMan simulations [Outters, 2003], we get bret = 0.08 mm, whereas the cubic law yields a lower value of bret = 0.025 mm.

[56] Another approach to estimate bret is to utilize the average flow rate through the system, determined in the simulations as q = 1.58[m3/yr]/2500[m2] = 0.00063 m/yr. The “measured” flow rate from the simulations is 1.58 m3/yr over the release cross-sectional area of 50 m × 50 m = 2500 m2. To infer the flow porosity nf, we recall the arithmetic average water residence time for the base case as 〈τ〉 = 35.05 yr (Table 2). The mean velocity is then estimated as U = L/〈τ〉 = 100/35.07 = 2.85 m/yr. The flow porosity and velocity are related through the flow rate and we get nf = q/U = 0.00022. To estimate an “effective” retention half-aperture bret, we use Sf = 1.29 1/m (obtained by summing individual values of Sf for the three sets of fractures given in Table 1), and obtain bret = nf/2Sf = 0.086 mm.

[57] In Figure 8 we plot the normalized expected discharge for Tc and Sr for the base case. We compare the result where full variability is accounted for using the joint density f(τ, β) as obtained from simulations, with predictions using the linear expression β = kτ, k ≡ 1/bret, where bret is estimated in three different ways discussed above. In particular, the cubic law yields bret = 0.025 mm and k = 40000 1/m, the quadratic law bret = 0.08 mm and k = 12500 1/m, and the estimate using fracture density and flow porosity, beff = 0.086 mm and k = 11600 1/m (different k values are in Figure 8 denoted by indices). We see in Figure 8 that the difference in predictions can be significant. The “quadratic law” and flow porosity/fracture density ratio yield similar values of bret, 0.08 mm and 0.086 mm, respectively; hence we consider only one case with bret = 0.08 mm and k = 12500 1/m in Figure 8. Both cubic and quadratic law yield estimates of bret which overestimate retention and yield a lower peak of expected discharge, for Sr as well as for Tc.

Figure 8.

Expected radionuclide discharge at the downstream surface of the rock volume, for the base case, using different estimation of β: (a) for Tc and (b) for Sr. “Full variability” implies that the curve was obtained by ensemble averaging of γ, using simulated τ and β. The “τ0 = 25 yr (median)” implies that the curve was obtained by neglecting variability in τ and assuming a constant value τ0.

[58] The difference between the BTC peaks obtained using estimates from cubic and quadratic law is significant (two orders of magnitude for Sr and four orders of magnitude for Tc) (Figure 8); this can be compared to the small difference between the two cases of hydraulic law shown in Figures 6–7. The choice of the hydraulic law is thus important for the estimate of bret since β, and hence transport, is strongly affected by bret through the linearized model β = τ/bret. The closest curve to the simulated case is obtained with β = τ/bret using the quadratic law for estimating bret from mean transmissivity (Figure 8). The estimate of bret using fracture density and flow porosity from the mean residence time and flow rate, with β = τ/bret, yields reasonable predictions.

[59] Finally, we address the significance of the variability in the water residence time, τ. Let τ0 be the “best estimate” of τ, say the arithmetic mean, or the median value. Then we can set f(τ, β) = δ (τ − τ0)f(β) in equation (2) to assess the significance of τ variability. In Figure 8 we plot this case, where τ0 = 25 yr is the median value from the marginal distribution ∫f(τ, β)dβ as obtained from DFN simulations. We see that for the strongly sorbing tracer the variability in τ has no impact on transport (Figure 8b). For the weakly sorbing tracer Sr, the impact of τ variability is only for the first arrival, whereas the tail part of the water residence time distribution has no impact on the expected breakthrough curve (Figure 8a).

6. Conclusions

[60] From obtained results, we draw the following conclusions.

[61] 1. Distribution of β quantifies the hydrodynamic control of retention in a discrete fracture network, and is the single most important parameter for transport predictions of tracers subject to retention. It is correlated to τ in a nonlinear manner, consistent with findings for single heterogeneous fractures [Cvetkovic et al., 1999; Cheng et al., 2003]. For the conditions considered in this study, we find β ∼ τm where m = 1.4–1.6. The correlation is stronger in the lower value range, whereas in the upper value range β can vary two orders of magnitude for a given τ. The distribution of β is significant for transport both in the upper and lower range. Any network characteristic or DFN model assumption which affects the distribution of β is likely to be significant for transport predictions.

[62] 2. The choice of the hydraulic (cubic or quadratic) law has a significant impact on τ and no impact on β; since β controls retention and thereby overall transport, the hydraulic law has little direct effect on transport measures. Indirectly, however, the choice of the hydraulic law can be critical: If the effective retention aperture 2bret is to be estimated from a given set of transmissivity data and used in β ≈ τ/bret, then bret may differ by a factor 3–4 if cubic or quadratic law is assumed, affecting transport predictions significantly. Thus there is a need to better understand the nature of bret and its estimation, as well as to relate bret to other effective apertures relevant for hydraulic characterization (“hydraulic aperture”) and advective transport (“transport aperture”) [Tsang, 1992].

[63] 3. The choice of the intersection law as full mixing or streamline routing has a comparatively small effect on both early and late tracer arrival. Assuming streamline routing results in a slight shift toward later peak arrival and lower peak values. For the conditions of our analysis, full mixing therefore appears to be a more conservative assumption than streamline routing.

[64] 4. If the distribution of β is known, then the distribution of τ is of relatively little importance for transport of reactive tracers in fractured porous media. Neglecting the variability of τ even for a weakly sorbing radionuclide Sr affected early arrival but predicted well both the peak and tailing of the expected discharge (Figure 8a); for a strongly sorbing tracer Tc, τ has a negligible impact (Figure 8b).

[65] 5. The continuum approach equation (B1) assumes a linear relationship β = τ/bret where 2beff is an effective “retention” aperture for the rock volume. For conditions comparable to the ones considered here, the simplification β = τ/bret may be applicable provided that bret is accurately estimated. Predictions with the linearized model using fracture density Sf and flow porosity nf as bretnf/2Sf, will generally overestimate retention but overall provide a reasonably accurate estimate (Figure 8). We emphasize however that if β = τ/bret is used and bret is estimated based on known Sf and nf, then accurate estimates of the τ distribution are a requirement for accurate predictions of tracer breakthrough.

[66] Fracture networks are difficult to accurately characterize in practice, and it is of interest to better understand which of the DFN characteristics are important for transport of sorbing tracers. The present methodology is well suited for further studies of DFN sensitivities for sorbing tracers, using either generic or site-specific DFN simulations. One important aspect of this sensitivity is the nonlinearity of τ, β correlation, and its consequence for the effective parameterization of β. For instance, to what extent would the τ, β nonlinear correlation be affected by different fracture densities, fracture length and transmissivity distributions? Would the relatively limited impact of the hydraulic law and intersection rule as found in this study, be influenced by varying DFN properties (such as fracture density, length and transmissivity), or by the inclusion of hydraulic variability within fractures?

[67] The approach presented in this work is general in the sense that advective transport in a fracture network is coupled with retention in a semi-analytical manner using statistics of β (or of B in the more general case) obtained along trajectories. The specific applications considered here can be modified and possibly improved in several respects. Any other DFN simulation model, which for instance includes internal fracture aperture variability, could be used for computing τ and β. If spatial variability of retention parameters has been characterized at a given site, this data could be readily included in the DFN simulations as overlapping random fields, whereby distributions of τ and B (equation (5)) are computed, rather than those of τ and β. Furthermore, alternative particle tracking algorithms for transport at fracture intersections can be used by taking advantage of more recent developments [Mourzenko et al., 2002]. For conditions where the assumption of unlimited diffusion is inappropriate, the memory function g would need to be modified. One possibility is to use the g derived by [Cvetkovic et al., 1999] which explicitly accounts for diffusion limitations, or alternatively, use a multiple-rate form of g [Cvetkovic and Haggerty, 2002] which would implicitly account for diffusion limitations. If g is such that no closed-form solution for γ is available, then transport measures would reduce to total discharged mass, whereas the response function γ would need to be obtained by numerical Laplace inversion.

[68] Detailed DFN characterization of a site is typically performed on a smaller scale compared to the transport scale. For instance, the data from the Äspö HRL used to construct our DFN model, was confined to a 50–100 m scale. The length scale relevant for a hypothetical high-level waste repository in the Äspö region, would be one order of magnitude larger; hence transport predictions for performance assessment would require τ and β distributions on a 1000 m scale. The classical continuum model (common) provides a means for upscaling transport, if the linearization β ∼ τ is applicable and the transport is controlled by the water residence time density f(τ) as predicted by the advection-dispersion model. We have shown here that for the Äspö crystalline rock on a 100 m scale, the density f(β) rather than f(τ) controls transport of sorbing tracers, and that the correlation of τ, β is nonlinear. These findings emphasize the need for developing more general methodologies for upscaling β (or B) than that provided by the classical continuum model (B1).

Appendix A.: Assessing the Applicability of Unlimited Diffusion Assumption for Sr and Tc

[69] On the basis of geometrical arguments the average size of a rock block between adjacent fractures, ℓ [L], can be estimated as [Carrera et al., 1998]

equation image

where Sf [1/L] is the fracture density (Table 1). Summing the values of the three observed fracture sets in Table 1, we have Sf = 1.29 1/m, and ℓ ≈ 0.8 m.

[70] If a large tracer source is considered, as relevant for instance when modeling transport of environmental tracers [Shapiro, 2001], then penetration of a typical rock block by diffusion would take place from adjacent fractures. In such a case, ℓ/2 = 0.4 m would be the relevant length scale setting a lower bound for diffusion limitations. In our case, the source (from a hypothetical failed canister) is small, and hence the relevant length scale for the lower bound of diffusion limitation is the average block size of 0.8 m. This is an estimate considering all fractures, conducting and nonconducting. Density of the conducting fractures is generally lower than the density of all fractures; hence ℓ is presumably greater than 0.8 m; this would further increase the margin for the applicability of the unlimited diffusion retention model.

[71] A simple generic estimate of the penetration depth Δz [L] by diffusion into a typical rock block between adjacent fractures can be obtained by using the expression [Cvetkovic et al., 2000]

equation image

which assumes a one-dimensional diffusion process into the rock matrix in the z-direction orthogonal to the fracture plane. In equation (A1), Δt is the “contact” time of the tracer with the rock matrix. Setting Δz = 0.8 m, we can estimate the required contact time as

equation image

Using the data for Sr and Tc, we obtain Δt ≈ 200000 years for Sr, and Δt over 1010 years for Tc. These values set in effect lower bounds for the time required to penetrate the average block size of 0.8 m, since they assume constant concentration at the fracture-matrix interface, and also neglect decay.

[72] The unlimited diffusion model (7) yields a breakthrough which for Sr extends to approximately 1000 years, and for Tc to approximately 107 years (Figure 8). Note that for the same retention parameters, a limited diffusion model would result in less retention and hence shorter transport times. Comparing transport times with Δt in equation (A2) for Sr and Tc, we conclude that diffusion into the typical rock block between adjacent fractures is sufficiently slow relative to the transport time, and that assuming an unlimited diffusion model is reasonable for the considered tracers and conditions at the Äspö HRL.

Appendix B.: Continuum Transport Model

[73] Neglecting decay, the continuum model for transport and retention in fractured rock is commonly written as [Shapiro, 2001]

equation image

where C [M/L] is tracer concentration in the fractures, C* [M/L] tracer concentration in the rock matrix, q [L/T] is the volumetric flow rate per unit cross-sectional area, nf (dimensionless) is the flow porosity, Sf [1/L] is the fracture density, x [L] is the spatial coordinate in the direction of the mean flow, Dl [L2/T] is the coefficient of longitudinal dispersion, and z [L] is the spatial coordinate orthogonal to fracture surfaces. With appropriate initial and boundary conditions [e.g., Shapiro, 2001], equation (B1) can be solved to quantify tracer transport through a fractured rock volume (such as that given in Figure 1) in an average sense.

[74] In the case where retention parameters are constant, and equation (6) is applicable, the Laplace transform of h is

equation image

where s is the Laplace transform variable. For the unlimited diffusion retention model, equation image is defined in equation (4) with B = βκ from equation (6).

[75] Setting f(τ, β) = f(τ)δ(β − kτ) in equation (B2), where k ≡ 1/bret, and utilizing equation (4), we get

equation image

which is the Laplace transform of f(τ), where the Laplace transform variable is replaced by s + kκequation image. In other words, we have

equation image

[76] If we solve equation (B1) for unit pulse injection, and for the flux-averaged concentration, then h/q is equivalent to C, and

equation image


equation image

with L being the distance, and U = q/nf the mean velocity through the rock volume [Kreft and Zuber, 1978].

[77] The Laplace transform of f(τ) in equation (B5) is then

equation image

Combining equations (B4) and (B7), we get

equation image

Equation (B8) coincides with the Laplace solution for the flux averaged concentration C, obtained from equation (B1) [[Shapiro, 2001], equation (6)].

[78] We can now summarize two conditions of equivalence between the continuum model (B1) and the trajectory model (2) with equation (4): (1) β is linearized as β = kτ where k ≡ 1/bret and 2bret is an effective “retention” aperture for the considered rock volume. (2) The water residence time density f(τ) is given by the advection-dispersion model.


[79] This work was supported by the Swedish Nuclear Fuel and Waste Management Co. (SKB). The authors are grateful to two anonymous reviewers and the associate editor, for their insightful and constructive comments which helped improve the original version of the manuscript. We also thank Andrew Frampton, Royal Institute of Technology in Stockholm, Sweden, for his helpful comments.