Incorporation of conceptual and parametric uncertainty into radionuclide flux estimates from a fractured granite rock mass

Abstract

Detailed numerical flow and radionuclide simulations are used to predict the flux of radionuclides from three underground nuclear tests located in the Climax granite stock on the Nevada Test Site. The numerical modeling approach consists of both a regional-scale and local-scale flow model. The regional-scale model incorporates conceptual model uncertainty through the inclusion of five models of hydrostratigraphy and five models describing recharge processes for a total of 25 hydrostratigraphic–recharge combinations. Uncertainty from each of the 25 models is propagated to the local-scale model through constant head boundary conditions that transfer hydraulic gradients and flow patterns from each of the model alternatives in the vicinity of the Climax stock, a fluid flux calibration target, and model weights that describe the plausibility of each conceptual model. The local-scale model utilizes an upscaled discrete fracture network methodology where fluid flow and radionuclides are restricted to an interconnected network of fracture zones mapped onto a continuum grid. Standard Monte Carlo techniques are used to generate 200 random fracture zone networks for each of the 25 conceptual models for a total of 5,000 local-scale flow and transport realizations. Parameters of the fracture zone networks are based on statistical analysis of site-specific fracture data, with the exclusion of fracture density, which was calibrated to match the amount of fluid flux simulated through the Climax stock by the regional-scale models. Radionuclide transport is simulated according to a random walk particle method that tracks particle trajectories through the fracture continuum flow fields according to advection, dispersion and diffusional mass exchange between fractures and matrix. The breakthrough of a conservative radionuclide with a long half-life is used to evaluate the influence of conceptual and parametric uncertainty on radionuclide mass flux estimates. The fluid flux calibration target was found to correlate with fracture density, and particle breakthroughs were generally found to increase with increases in fracture density. Boundary conditions extrapolated from the regional-scale model exerted a secondary influence on radionuclide breakthrough for models with equal fracture density. The incorporation of weights into radionuclide flux estimates resulted in both noise about the original (unweighted) mass flux curves and decreases in the variance and expected value of radionuclide mass flux.

Appendix: Fracture characterization

Numerical modeling of fluid flow in fracture dominated subsurface flow regimes requires statistical analysis of fracture data for the determination of fracture properties, such as fracture sets and their mean orientation, length, spacing and distribution, density, and permeability of individual fractures or zones (e.g., Munier 2004; Reeves et al. 2008a). Fracture characterization at the Climax stock is based on the Spent Fuel Test—Climax (SFT-C) Geologic Structure Database (Yow 1984) that consists of data describing joints, faults and shear zones (sample population n = 2, 591) that were collected during fracture mapping efforts in tunnel drifts constructed for the Climax Spent Fuel Test.

1.1 Fracture set orientation

Analysis of fracture orientation statistics, according to standard spherical statistical tests (Mardia and Jupp 2000), reveals a total of seven fracture sets (Table 3 and Fig. 10). Of the seven fracture sets, Sets 1 through 6 fit a Fisher distribution (Fisher 1953):

$$ f(x)={\frac{\kappa \cdot{\sin}(x)\cdot{e}^{\kappa\cdot{\cos}(x)}}{e^{\kappa}-e^{-\kappa}}} $$

(7)

where the divergence, x (degrees), from a mean orientation vector is symmetrically distributed $(-{\frac{\pi}{2}}\leq x \leq{\frac{\pi}{2}})$ according to a constant dispersion parameter, κ. The Fisher distribution is a special case of the Von Mises distribution, and is similar to a normal distribution for spherical data (Mardia and Jupp 2000). The extent to which individual fractures cluster around a mean orientation is proportional to values of κ (i.e., higher values of κ describe higher degrees of clustering). Values of κ for natural rock fractures range between 10 and 300 (Kemeny and Post 2003). Stochastic simulation of Fisher random deviates is based on a method by Wood (1994). The occurrence of each fracture set is governed by the prior probabilities listed in Table 3. The distribution of fracture orientation for Set 7 is assumed uniform.

Fig. 10

Lower-hemisphere equal area projection of the poles to the mean orientation vectors for Sets 1 through 6; the remaining 15% of fractures are randomly oriented

1.2 Fracture length

Fracture length is perhaps the most important parameter for discrete fracture network investigations. The distribution of fracture length has been found in theoretical studies to control both network connectivity (Renshaw 1999; de Dreuzy 2001) and the spreading rate of solutes at the leading plume edge (Reeves et al. 2008b, c). For the Climax stock, fracture length is recorded for approximately 95% of the fractures contained in the SFT-C data base. Fracture lengths range from 0.006 to 40 m within this data set. The upper range in the data is misleading as fracture length values are restricted to drift length and orientation, i.e., the longest fracture is parallel to one of the drifts and approximately two-thirds of the total drift length (67 m). According to a maximum likelihood estimation method (Aban et al. 2006), fracture length data fit a Pareto power-law distribution (α = 1.6) that is truncated for the largest values (denoted as “TPL”) (Fig. 11). Again, the truncation in fracture length is artificial, and is a result of the drift orientation and length relative to the occurrence and orientation of fractures.

Fig. 11

Mandelbrot plot of all fracture lengths (l along with best fit Pareto power-law (PL), truncated Pareto power-law (TPL) and exponential (exp) models). Note the decay of the largest fracture lengths (i.e., distributional tail) follows a strong power-law trend (linear in log-log space) with an abrupt truncation. The truncation is artificial and results from the length and orientation of the drifts. All fracture lengths are in meters

To honor the power-law trend observed for the fracture length data set (linear trend on Fig. 11), an upper truncated Pareto model is used to randomly assign fracture lengths (Aban et al. 2006):

$$ P(L\,>\,l)={\frac{\gamma^{\alpha}(l^{-\alpha}-\nu^{-\alpha})}{1-({\frac{\gamma} {\nu}})^{\alpha}}} $$

(8)

where $L_{(1)},L_{(2)},\ldots,L_{(n)}$ are fracture lengths in descending order and L ₍₁₎ and L _(n) represent the largest and smallest fracture lengths, γ and ν are lower and upper fracture length cutoff values, and α describes the tail of the distribution. Specific values used to describe fracture lengths at the Climax stock include: α = 1.6, γ = 30 m, and ν = 1,000 m. The lower cutoff of 30 m is equal to two times the edge of a cell in the continuum grid (15 m) and the upper cutoff of 1,000 m is equal to one-third of the FCM domain in the x-direction. The purpose of assigning fracture length according to (8) is to restrict fracture length to a finite upper bound. There is no evidence, with the exception of kilometer-scale faults that bound the Climax stock, of large faults that potentially span the entire length (∼5 km) of the stock. Thus, we deem a classical Pareto power-law (denoted as “PL” in Fig. 11) as an inappropriate choice for fracture length.

1.3 Fracture density

It is not possible to directly measure the three-dimensional fracture density of a rock mass. Instead, three-dimensional density of discrete fracture networks is estimated from density measures of lower dimensions, i.e., one-dimensional fracture frequency from boreholes or tunnel drifts or two-dimensional density from outcrops or fracture trace maps. Fracture frequency in the SFT-C database, based on fracture mapping along tunnel drifts, is relatively high and ranges between 2.0 to 5.5 fractures per meter. However, the high fracture spacing in the SFT-C database is misleading as the frequency of “flowing” or conductive fractures is not considered.

Field observations for fractured rock masses indicate that rock volumes are often intersected by only a few dominant fractures and only approximately 10% (or less) of the total fracture population contributes to flow (Dershowitz et al. 2000). This implies that 90% of fractures in the SFT-C database are not connected to the hydraulic backbone. Two specific studies at Climax provide some insight into the frequency of open fractures. In a tunnel drift for the SFT-C experiments, a series of five boreholes extending 9 to 12 meters below the tunnel drift yielded permeability values typical of unfractured granite cores (Ballou 1979). This indicates that the borehole array, which is on the scale of a grid cell, only intersects either solid rock or rock with “healed” fractures. “Healed” fractures (i.e., veins) refer to fractures containing mineral precipitates (e.g., calcite) and are, therefore, not open to flow. Several instances of healed fractures were documented in the SFT-C database by Yow (1984)—these fractures were excluded from the frequency analysis when recorded. In the permeability test conducted by Isherwood et al. (1982), only 2 out of 10 (20%) fractures in a densely fractured zone were open and had permeability values higher than the background matrix.

Given the high level of uncertainty in fracture density, this parameter is determined in the fracture continuum model during calibration (Table 1). Refer to Sect. 3.2 for additional discussion.

1.4 Hydraulic conductivity

Only a handful of field-scale hydraulic conductivity (K) measurements exist for the Climax stock (on the order of 10⁻⁷ to 10⁻¹⁰ m/s) (Murray 1980, 1981). These values describe the bulk hydraulic conductivity of the fractured stock, and most likely underestimate variability in fracture K since these estimates are based on hydraulic testing over large open borehole intervals where properties of multiple fractures are averaged. This narrow range (3 orders of magnitude) is inconsistent with other studies of highly characterized fractured granite rock masses where values of fracture K encompass 5–8 orders of magnitude (Paillet 1998; Guimerà and Carrera 2000; Andersson et al. 2002; Hendricks Franssen and Gómez-Hernández 2002; Gustafson and Fransson 2005).

Instead of relying on a handful of effective permeability measurements to parameterize a probability distribution for fracture K, the distribution of mechanical fracture apertures in the SFT-C database is analyzed. Recorded aperture values are lognormally distributed with a standard deviation of 1.05 (not shown) and this value is used to describe the variability in the fracture K distributions. This value is identical to the standard deviation of the transmissivity distribution used by Stigsson et al. (2001) at the Äspö Hard Rock Laboratory. Though rock fracture hydraulic conductivity is proportional to mechanical aperture, correlations between mechanical and equivalent hydraulic apertures are often unreliable (Bandis et al. 1985; Cook et al. 1990); therefore, mean values of K are not computed from the mechanical aperture distribution. Aperture data are used only to gauge the suitability of a lognormal distribution for fracture K and to provide an estimate of standard deviation.

To maintain a constant conceptual model for radionuclide flux estimates (refer to Sect. 3.2 for more detail), fracture K distributions were held constant at values of −7 and −8 (the original values of K are in units of meters per second prior to log transformation). Both of these mean fracture K values are within the narrow range defined by Murray (1981). The higher mean value of −7 m/s is assigned to fractures that belong to fracture set 6 (32% of the fractures) (Table 3). Fractures in this set experience the least amount of compressive stress normal to their fracture walls, suggesting that these fractures are potentially more permeable than fractures oriented at other directions to the regional stress field. The remaining fracture sets (68% of the fractures) are assigned K values according to the lower mean K of −8. A log₁₀ K standard deviation value of 1.05 is applied to all fracture sets.

1.5 Fracture porosity

The computation of cell velocity from Darcy flux values calculated in the FCM flow realizations requires values of fracture and matrix cell porosity. Constant values of 0.006 are assigned to matrix cells (Murray 1981). Equivalent porosity values of fracture cells are based on tracer test results from a similar fractured granite rock mass where equivalent porosity was found to be lognormally distributed within a range of 0.027 and 0.054 (Pohlmann et al. 2004). Given the hydraulic conductivity distribution for fractures at Climax, an empirical power-law relationship:

$$ n=0.04(K_{\rm fracture})^{0.25} $$

(9)

is used to correlate fracture cell porosity n with fracture hydraulic conductivity K _fracture, while maintaining both the range and distribution of the porosity values reported by Pohlmann (2004).