Derivation of statistical fracture model

The basic methodology behind the DFN model, e.g. /Munier 2004/, involves a division of the background fracturing in each fracture domain into distinct sets, based on their orientations and termination relationships. Once the orientation set model is completed, the additional properties necessary to describe the statistical fracture network are calculated. A division into distinct sets has also been recognised on a larger scale for lineaments and deformation zone traces (see section 5.5.2).

In its most compact form, the geological DFN model consists of the following components.

• Orientation. This is described by the mean orientation of each identified fracture set, expressed in terms of an orientation distribution with measurement(s) of spread.

• Size. For each identified fracture set, this is described in terms of a size distribution with its parameters.

• Intensity. For each identified fracture set, the amount of fracturing is expressed in terms of fracture intensity (P32), i.e. the areal density per unit volume (m²/m³).

In addition, the scaling of intensities needs to be quantified. Commonly, a Euclidian (linear) scaling is applied. Euclidean scaling describes a model in which the number of fractures is linearly proportional to area, in the case of outcrops, or length, in the case of boreholes, and by inference, to volume in three dimensions. This means that an increase of the model volume is matched by an equal increase in the number of fractures (e.g. if the model volume is doubled, the number of frac-tures is also doubled). However, equally common in the literature is a non-linear, fractal scaling of fracture intensities. This means that an increase of the model volume by a certain factor is matched by an increase in the number of fractures by a different factor (e.g. doubling the model volume might render an increase in the number of fractures by a factor 1.8).

The scaling aspect is intimately, and intricately, related to the spatial arrangement (spatial correla-tion) of fractures. A Euclidian scaling assumes no correlation between fracture positions, i.e. the position of a fracture does not influence the position of its neighbouring fracture. However, this does not exclude the possibility of obtaining local clusters which sporadically may occur as an effect of the random process. By contrast, a fractal scaling assumes a (self-similar) correlation between fracture positions, quantified by the fractal exponent (dimension). A discussion of this issue can be found in /Bonnet et al. 2001/ and the reader is referred to /Fox et al. 2007/ for details on how this has been addressed in the current DFN modelling work.

The basic methodology behind the DFN model, e.g. /Munier 2004/, involves a division of the background fracturing in each fracture domain into distinct sets, based on their orientations and termination relationships. Once the orientation set model is completed, the additional properties necessary to describe the statistical fracture network are calculated.

DFN orientation model

The fracture set orientation model was developed primarily from the orientations, geological proper-ties and geometric relationships recorded during the detailed mapping of nine outcrops within and immediately outside the candidate area at Forsmark (section 5.2.5 and Figure 5-11). The work-flow used to construct the DFN orientation model is as follows /Fox et al. 2007/.

1. Fracture orientation sets were identified on the basis of the outcrop mapping data. A fracture orientation set is a statistically homogeneous sub-population identified through consistent fracture pole orientations and termination relationships.

2. Once outcrop sets had been assigned, qualitative analysis of the stereoplots was performed to determine if any significant geological associations could be found and, if so, could they be used as predictors for reducing uncertainty in the spatial variability of set orientations.

3. Orientation sets were quantified by assigning a spherical probability distribution to them. The orientations of fractures in a fracture set are characterised by a mean pole vector (φ, θ) and a set of concentration parameters that describe how the fracture pole vectors cluster around the mean pole. Univariate Fisher, bivariate Fisher, bivariate Normal and bivariate Bingham spherical probability distributions were fitted to the sets identified on each outcrop, both for linked and unlinked fracture traces.

4. Identified outcrop sets were assigned names based on their general strike orientations. The sets were then listed in a matrix and classified into one of two categories:

Global: A fracture orientation set visible in all or nearly all of the mapped outcrops. In the case of the EW and WNW sets, the two sets were combined into a single global set based on their mutual exclusion (any outcrop with the WNW set did not possess the EW set, and vice versa).

Local: A fracture orientation set visible only in a small sub-set of the mapped outcrops. Local sets may represent variations in local stress conditions or tectonic history that are not applicable throughout the entire model volume. A key point is that, in terms of model parameterisation, local sets do not exist throughout the entire model volume.

5. The outcrop sets identified in step 1 were then used as guides to assign borehole fractures in each fracture domain into discrete sets. It should be noted that, in contrast to past geological DFN models, hard-sectored orientations were not used to divide the fractures into sets. Rather, the generic outcrop sets (NE, ENE, NW etc) were only used as initial starting points for the orienta-tion modelling. Within each fracture domain, fracture sets were locally defined for each borehole, for multiple types of spherical probability distributions, using the software ISIS /Dershowitz et al.

1998/. All set assignments utilised Terzaghi-corrected data with a maximum correction value of 5. However, the fractures added to the data set by the Terzaghi correction were deleted after the set assignment was completed.

6. Fitted orientation distributions for each fracture set in each fracture domain for borehole and outcrop data were compiled into a single data set. Only the univariate Fisher distribution fits were included in this data set. The final orientation model for each fracture set consists of:

a. A mean pole (φ,θ) of all the mean poles of the data points fitted for a single fracture domain.

For example, the mean pole orientation for the NE fracture set in domain FFM02 was

calculated by placing the fitted mean poles for the NE sets for both borehole and outcrop data set into FracSys/ISIS, and then by calculating the mean pole of the combined mean poles.

b. A univariate Fisher concentration parameter that represents the potential variation in the mean pole location (κmp). This value was calculated when the orientation of the mean pole of all the fitted sets had been assigned. The κmp value should only be used if, for a given set (NE, NW etc), the modeller wishes to simulate a variable set mean pole (i.e. a set where the average orientation varies spatially according to a univariate Fisher distribution) instead of a single fixed mean pole value.

c. An average value for the Fisher concentration parameter (κ). The average concentration parameter was calculated by computing the mean value of the individual κ values for all set fits. For example, for the NE set in fracture domain FFM02, each borehole and each outcrop has its own univariate Fisher distribution fit. The κ values from each of these individual fits were combined and the mean value calculated.

7. Parameter variability was quantified with the help of the specification of a statistical distribution for the Fisher concentration parameter κ (normal distribution), and the specification of probability distributions and 95% confidence cones for the distribution of fitted set mean poles.

Analyses conducted as part of the geological DFN parameterisation suggested that, for three of the fracture domains (FFM01, FFM03 and FFM06), four global orientation sets were consistently identified. These were the NE, NS, NW, and SH (sub-horizontal) sets. In domain FFM02, two additional sets with global scope were identified; the ENE and EW sets. These two sets exist in the other domains as local sets with limited spatial extents.

Figure 5-37 shows a stereonet in which all sets have been assigned the same amount of fractures.

The use of such a stereograph is simply to visualise the spread from the mean direction, and to visually compare this spread for different fracture sets. Figure 5-37 confirms the orientation model (e.g. Table 5-5), in that the difference in spread from the mean pole is essentially the same for all

DFN size model

The size model refers to a mathematical description of the area of the fractures. Previous analyses at Forsmark during SDM version 1.2 /La Pointe et al. 2005/ recognised that different fracture sets are likely to require different (and potentially unique) size models. Furthermore, since fracture domains have been identified that are distinguished from each other on the basis of different geological characteristics (see /Olofsson et al. 2007/ and section 5.6.1), it is reasonable to assume that the size model for the fractures differs in each fracture domain. With this reasoning in mind, size models were developed for individual fracture sets within each fracture domain.

Main alternative models

The methodology for quantifying a fracture size distribution for a fracture set in both fracture domains FFM02 and FFM03, in the main alternative models, involved fitting a scalar probability distribution based on r (the radius, in metres, of a disc-shaped equivalent-area fracture), to fracture trace length data observed in outcrop and to trace lengths derived from the interpretation of low magnetic lineaments (/Isaksson et al. 2006b/ and section 5.2.7), and the inferred intercepts of the mid-planes of deformation zones in the stage 2.2 deterministic model (/Stephens et al. 2007/ and section 5.5). The low magnetic lineaments and deformation zone traces function as a ‘pivot point’;

the radius exponent (kr) in an individual fracture domain is then determined by fitting the regional trace lengths together with outcrop data from the target domain.

As far as fracture domains FFM01 and FFM06 are concerned, the main alternative models are based on the assumption that the distributions of outcrop traces, mapped within FFM02 and FFM03, are representative also for these domains. This is a necessity, since domains FFM01 and FFM06 do not reach the ground surface. The size/intensity scaling for these models made use of the parameter r0 to match intensities observed in boreholes within FFM01 and FFM06. Thus, the fracture radius scaling exponent (kr) calculated from trace data for FFM02 is also used for domains FFM01 and FFM06.

Figure 5‑37. Representation of the DFN orientation model. For comparative purposes, all sets have been assigned the same amount of fractures.

Kamb. Exponential

Start: 2, Cl=2 Total Data: 4000

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The main alternative models at Forsmark /Fox et al. 2007/ contain two distinct size model alterna-tives, which these authors designed to encompass two end-members of a theoretical spectrum.

1. Tectonic continuum model (TCM and TCMF): This model assumes that the fracture population extends in size over a very large range in scale (Figure 5-38a). In this model, the fractures in outcrop with traces on the scale of metres are part of the same fracture population as lineaments or deformation zones with traces on the scale of kilometres. This model allows for the combina-tion of data sets at multiple scales.

It is important to note that the TCM is a coupled size-intensity model, i.e. it describes both the distribution of fracture sizes and their intensity (P32). It is not possible to separate the two components without invalidating the model. The scaling exponent kr has been calibrated against the distribution of trace lengths in outcrops within FFM02, while the minimum radius r0 has been calibrated on the basis of both outcrops within FFM02 and borehole data from each domain (FFM01, FFM02, FFM03 and FFM06). /Fox et al. 2007/ proposed two variants of the TCM model; one assuming Euclidean size-intensity scaling (TCM), and one assuming fractal size-intensity scaling based on the fractal mass dimension. (TCMF)

2. Outcrop scale model (OSM) in combination with tectonic fault model (TFM): This model is a composite size model that does not assume a single coupled size-intensity relationship (Figure 5-38b). In this specific parameterisation, fractures have been hypothesised to belong to different populations (joints and faults) with different intensities at different scales. This model has the following two components.

a. OSM – The outcrop scale model is a size model that is based solely on matching the sizes and intensities of fractures recorded as outcrop traces to the intensities in borehole data.

Fundamentally, it treats fracture traces exposed in outcrop as joints.

b. TFM – The tectonic fault model is a size model designed to be used in conjunction with the OSM. The TFM has been fitted solely to the lengths and intensities (P21) of the deformation zone traces inside the regional and local model volumes, and to the high-resolution, ground magnetic lineaments inside the north-western part of the candidate area. The fundamental hypothesis is that these structures represent faults, rather than joints. The TFM uses an rmin

value of 28 m, which is the radius of a fracture that will most likely produce a trace length of 50 m. This lower size limit is highly uncertain. As for the OSM, the TFM is valid up to a maximum size (rmax) of 564 m.

The TCM models imply that there are fewer overall fractures in FFM01 and FFM06, relative to FFM02, but that they are larger. The OSM + TFM model implies that there are more small fractures in FFM01 and FFM06 than FFM02, as well as fewer fractures overall. Together, the models cover a wide range of parameter space. Clearly, the size model parameterisation of the fracture sets in FFM01 and FFM06 are more uncertain than in FFM02 and FFM03, largely due to the lack of trace length or other size data from these domains.

Size

Area normalised number

TCM

Size

Area normalised number

OSM + TFM

OSM TFM

a b

Additional alternative model

Bearing in mind the geological understanding of the site (see sections 5.5.2 and 5.6.1), with the effects of different geological processes at depth compared with the surface and near-surface realm, it is possible that the fracture size distributions deduced from outcrops, especially in fracture domain FFM02, are different from those in fracture domains FFM01 and FFM06. For this reason, an alternative concept was discussed within the framework of uncertainty assessment in /Fox et al. 2007/.

In this alternative concept, rather than tuning r0 to match the intensity in boreholes, r0 was fixed to equal the borehole radius and, instead, the scaling parameter kr of the power-law distribution was tuned to simultaneously match intensity in the boreholes and the intensity of lineaments and defor-mation zones (see /Fox et al. 2007, section 5.1.1, p. 213/ for details), i.e. the alternative approach utilises borehole fracture frequency data, whereas the main alternative models utilise surface fracture trace data in the determination of the fracture size distribution. This approach forms the foundation to the additional alternative model here referred to as the “r0-fixed” model alternative.

The approach to keep r0 fixed to equal the borehole radius has also been employed in the hydrogeo-logical DFN modelling work (see section 8.5). However, the approach used in the hydrogeohydrogeo-logical DFN modelling to determine the intensity of larger-scale structures is different (see chapter 8).

Heterogeneity in the density of lineaments and deformation zones

Since the low magnetic lineaments and the surface traces of deformation zones were assumed to have an homogeneous density in space and these structures were not subdivided into fracture domains (see section 5.6.2), the same sets of trace lengths for these structures were used in the parameterisation of all fracture domains in all model alternatives in the geological DFN modelling.

This methodology is appropriate for a case where the lineament and deformation zone density is roughly isotropic across all domains. However, at Forsmark, there is a clear variation in the occur-rence of lineaments and steeply dipping deformation zones with WNW-ESE to NW-SE strike in different volumes (see section 5.5.4); the intensity of these structures is much higher outside relative to that inside the tectonic lens where domains FFM01, FFM02, FFM03 and FFM06 are situated.

Furthermore, gently dipping deformation zones are also less common in the target volume in the north-western part of the candidate volume, where domains FFM01, FFM02 and FFM06 are present.

Thus, a consequence of this assumption is that the geological DFN model parameterisation for these domains shown in section 5.6.4 will tend to overestimate the intensity of NW-striking fractures in the MDZ size range 28 m to 564 m.

The issue of lineament and DZ heterogeneity also applies to the global sub-horizontal (SH) set in all model alternatives. It is not possible to identify lineaments from the high-resolution ground magnetic data set as sub-vertically or sub-horizontally dipping. Therefore, it is also not possible to assign lineaments to the SH set. As a consequence, the radius exponent (kr) for the SH set for the TCM/TCMF and OSM/TFM models is based only on the trace data from outcrops and the surface traces of SH deformation zones in the regional model volume. There is no available information on the size of lineaments within the MDZ size range (< 564 m). In addition, the significant heterogene-ity in the spatial distribution of SH-dipping deterministic deformation zones, with a concentration in the southern half of the local model area largely inside domain FFM03, is important. Therefore, it is also possible that the intensity of MDZ-size structures in the SH set is over- or under-predicted by the geological DFN modelling work; there is not sufficient data available to determine which (over- or under-prediction) is the case at Forsmark.

DFN intensity model

The stage 2.2 DFN model at Forsmark presents fracture intensity estimates in several forms.

1. As a single matching intensity value in the coupled size-intensity models. These models are based on the arithmetic mean P32 intensity in borehole data for a given fracture set within a given fracture domain.

2. As a set of descriptive statistics for each fracture set, by fracture domain. Statistics include the arithmetic mean, standard deviation, median, quartiles and percentiles. No assumption is made about the form of the distribution.

3. As a gamma distribution, where applicable (see /Fox et al. 2007/ for more details).

4. As a “lithology correction factor”, based on the bedrock lithology.

Fracture set intensities depend significantly upon the linear fracture intensity data (P10) collected in the cored borehole logs. P32 values were calculated for individual borehole sections at multiple intervals (6 m, 15 m and 30 m); the resulting values were then combined by set and domain. Outcrop fracture intensity (P21) was used to bound the coupled size-intensity models, and as a validation tool for the final model parameterisation.

Multivariate statistical analyses suggest that fracture intensity varies in part due to local variations in lithology. The base rock for all DFN simulations (mean P32 described in the first bullet item) is assumed to be medium-grained, metamorphosed granite to granodiorite (SKB code 101057).

A “lithology correction factor” can be applied to generate DFN models in other rock types.

Fracture intensities were computed for both sealed and open fractures; no distinction between these two types of fractures was carried out in the geological DFN model. However, global and local sets were treated differently in terms of model intensity parameterisation. Global sets were hypothesised to exist everywhere within the model volume. For this reason, intervals with no fractures were considered as a part of the spatial distribution of the global sets, and were included in the intensity statistics and gamma distribution calculations. By contrast, local sets were hypothesised to represent truly local phenomena and, for this reason, intervals without fractures in the local set were not considered as a part of the spatial distribution of the local fractures. Therefore, zero intervals in the local fracture set data were omitted before the intensity statistics and gamma distributions were calculated.

DFN spatial model

The spatial model describes how many fractures occur in a specific volume of rock at a specific location in the modelled fracture domain. As such, the model may depend upon depth, rock type, the influence of geological processes, the volume of interest and other factors. It may also differ by fracture set. The assessment of the spatial variation and the mathematical description of this variation were based on analysis of the scaling properties of the fractures, in combination with multivariate statistical analyses to identify any statistically significant relations between mappable geological parameters and fracture intensity variations.

The mass dimension, which models how fracture intensity changes as a function of scale, was computed for the traces in outcrop and for the fracture locations in boreholes. This analysis produced a set of data for each analysis consisting of the scale and the average fracture intensity at that scale, with intensity measured as P21 for outcrop traces and P10 for borehole data. The results were dis-played on graphs with doubly logarithmic axes. This methodology makes it visually apparent as to whether the scaling behaviour is a power law, i.e. Euclidean, and, thus, consistent with the tectonic continuum hypothesis, or some other mathematical form, which might have other implications for the spatial model. The key parameter in these mass dimension plots is whether the data conform to a straight line in the doubly logarithmic display, which would support the tectonic continuum hypothesis, whether it is better modelled by two or more straight line segments, implying different characteristic intensities at different scales, or whether it fails to conform to a straight line over any portion of the data record. For portions of the data that do conform to straight line segments, the slope of the line that approximates the data describes the scaling model.

The analyses in /Fox et al. 2007, section 4.3.2/ show that, for the global fracture sets in the geo-logical DFN model, fracture locations are best represented by a Euclidean scaling at model scales of 30 m and larger. Mild fractal clustering (mean fractal dimension of 1.9) was observed at scales less than 30 m. However, the 95% confidence interval surrounding D is ±0.23. This suggests that it may be difficult to distinguish between the natural variability inherent in a Poisson model follow-ing Euclidean scaling and a modestly fractally-clustered model. For this reason, /Fox et al. 2007, section 7.4/ recommended the use of Euclidean scaling (TCM) at all scales.

5.6.4 DFN models