Inference of field scale fracture transmissivities in crystalline rock using flow log measurements

Transcription

1 WATER RESOURCES RESEARCH, VOL. 46,, doi: /2009wr008367, 2010 Inference of field scale fracture transmissivities in crystalline rock using flow log measurements A. Frampton 1,2 and V. Cvetkovic 1 Received 3 July 2009; revised 7 July 2010; accepted 22 July 2010; published 2 November [1] Characterization of transmissivity for crystalline rock is conducted through simulation by conditioning against borehole flow rates obtained from high resolution, in situ field measurements during extraction pumping. Full three dimensional discrete fracture network simulations are carried out according to specifications obtained from site characterization data in a stochastic Monte Carlo setting. A novel method of conditioning is thereby introduced and applied using nonparametric comparison tests, which provide quantifiable measures of accuracy enabling evaluation of simulated results against field measurements. The assumption of a constitutive relationship (perfect correlation) between fracture size and transmissivity is adopted. The method is evaluated against both single and multiple realizations, various domain size, and fracture length configurations and shown to be robust for the cases considered. When the introduced method of conditioning is applied, transmissivity parameterization can be inferred to a narrow range with a quantifiable accuracy in terms of a probability value. Results indicate that elementary interpretation of transmissivity based on homogenization of a porous medium will generally underestimate transmissivity. Further implications on advective transport for natural flow conditions are briefly evaluated, indicating advective breakthrough times can be overestimated up to a factor of about 10 in the median. Citation: Frampton, A., and V. Cvetkovic (2010), Inference of field scale fracture transmissivities in crystalline rock using flow log measurements, Water Resour. Res., 46,, doi: /2009wr Introduction [2] In crystalline rock formations, flow conduits occur primarily in fractures which are often conceptualized as a network of interconnected discrete conductive features. Due to the large spatial scales involved, combined with obvious difficulties in thorough characterization of the subsurface, representation of discrete fracture networks (DFNs) is generally only possible through stochastic parameterization, commonly quantified by distributions of geometric and hydrodynamic properties. Geometric properties refer to identifying parameters for suitable distributions ascribing orientations, sizes, densities, and spatial locations for occurrence of fractures. The main hydrodynamic property of interest is transmissivity, which even though is in principle possible to relate to a distribution of (geometric) fracture aperture, the latter is seldom a measurable quantity under field conditions. This is partly due to great variability of aperture within a fracture combined with limited technical capabilities of obtaining such measurements in situ. In an extensive review study, Neuman [2005] argues that quantification of flow and transport in fractured rock requires consideration of the high degree of heterogeneity and as such should primarily be based on directly obtainable 1 Department of Water Resources Engineering, Royal Institute of Technology, Stockholm, Sweden. 2 Now at Department of Quaternary Geology and Physical Geography, Stockholm University, Stockholm, Sweden. Copyright 2010 by the American Geophysical Union /10/2009WR hydraulic information rather than solely on indirect structural information. [3] The DFN approach has been extensively applied to fractured media over a number of decades, both for applications as well as theoretical developments. Some early examples of applying three dimensional DFN simulations with in situ data collected from field measurements include flow and transport studies resulting from investigations at the Stripa mine in Sweden [Andersson and Dverstorp, 1987; Dverstorp and Andersson, 1989; Dverstorp et al., 1992]. In relation to investigations at the Fanay Augères mine in France, Cacas et al. [1990a, 1990b] simulated flow using a disc model and transport using an equivalent pipe network representation, and were able to relate results to field tracer tests. Using data from the Äspö Hard Rock Laboratory studies on flow and transport in DFNs have been conducted by Dershowitz et al. [1996], Outters [2003], Hartley et al. [2004], and Cvetkovic et al. [2004]. In a comparison study of three modeling approaches (stochastic continuum, channel network, and DFN) with specific aim toward applications for spent nuclear fuel repositories, Selroos et al. [2002] presented results indicating that the DFN approach may be favorable for representing the near repository region in terms of calculation of transport related quantities. Also, in a recent study [Cvetkovic and Frampton, 2010] show that a DFN model is capable of making predictions of water residence times (advective transport) for multiple flow paths varying between a 20 m up to an approximate 80 m travel distance, which are in reasonably agreement with transport of inert tracers obtained from the Tracer Retention Understanding Experiments [Andersson et al., 2002a, 2002b; 1of17

2 Poteri et al., 2007; Winberg et al., 2003; Andersson et al., 2007], conducted in an approximate volume of m 3 denoted as a block scale. As such, the latter mentioned simulations can be said to be partially confirmed [Oreskes et al., 1994] against field measurable quantities. [4] In applications fracture transmissivity is preferred as the key hydrodynamic quantity for characterization of fractured media. Transmissivity is related to conductivity by integration along fracture aperture, and acts as an effective physical quantity describing the ease of fluid to flow/traverse through a fracture, essentially being represented as a porous medium. Effective fracture transmissivity can be inferred through various field tests and used to describe the average or net flow behavior of a system or network of fractures. Common approaches used to infer transmissivity in the field include evaluation of transient responses from various types of borehole pressure tests; for low transmissive formations typically slug/pulse tests are used. Interpretation of hydraulic testing of fractured rock is challenging, which generally involves transient interpretation of data in terms of heterogeneity, anisotropy as well as flow geometry [National Research Council, 1996]. Concerning the latter, Barker [1988] introduced a generalized radial flow concept, allowing nonintegral flow dimensions to be applied to fractured rock assuming isotropic conductivity fields, which has been useful in some field applications [Borgne et al., 2004]. [5] The stochastic continuum approach based on geostatistical inversion of Neuman [1987] has been applied to both unconsolidated geologic formations [Neuman et al., 2004, 2007] as well as fractured rock [Neuman and Depner, 1988], and more recently applied to investigations at the Äspö URL [Hendricks Franssen and Gómez Hernández, 2002]. Hydraulic tomography methods [Hao et al., 2008] have also been successfully applied at the Mizunami URL in Japan [Illman et al., 2009]. Neuman et al. [2004] developed a graphical method to infer statistics of local log transmissivity based on quasi steady state head data for constant rate pump tests in confined heterogeneous aquifers, and the study of Neuman et al. [2007] confirmed results with numerical simulation for random fields with exponential or Gaussian spatial correlation functions. For fractured rock, Paillet et al. [1987] measured vertical flow in multiple boreholes during pumping and inferred effective transmissivity/conductivity using a simplified flow model, representing the main effects of the system by a single horizontal fracture feature connecting all sections. A method of statistical cokriging was developed to estimate the geometry of highly conductive (major) fracture zones from steady state head data by Renshaw [1996]. [6] Flux and discharge measurements in unconsolidated soils have recently achieved significant success using a novel passive flux meter, which can provide nonintrusive, simultaneous measurements of groundwater and contaminant fluxes in aquifers [Hatfield et al., 2004; Klammler et al., 2007]. There has also been significant effort in designing and evaluating methods for measuring fracture specific flow rates, including electric conductivity logging [Tsang et al., 1990; Paillet and Pedler, 1996; Tsang and Doughty, 2003; Doughty et al., 2005] and thermal sensors [Hess, 1982; Paillet and Pedler, 1996] such as the Posiva Flow Log [Öhberg and Rouhiainen, 2000; Pöllänen and Rouhiainen, 2001; Ludvigson et al., 2002]. [7] The Posiva Flow Log (PFL) tool is particularly useful for characterization of flow in crystalline bedrock since it can measure a wide range of flow magnitudes with a small support scale. By traversing the borehole in increments less than certain section lengths, the tool is able to obtain highresolution spatial occurrence of flow rates, as well as flow magnitude and direction in terms of entering or leaving the borehole. This can then be cross examined with fracture observations, either through core log data, TV imaging, and/or electrical conductivity data, etc., in order to determine the magnitude and (binary) direction of flow associated to specific fracture observations. Additionally, the PFL tool has been extensively applied and used in the Swedish and Finnish site investigation/characterization programs for high level radioactive waste [Follin et al., 2006; Hartley et al., 2006; Follin et al., 2007; Hermanson et al., 2008]. [8] However, there are difficulties in interpretation of fracture specific flow measurements which concern the validity of assumptions used for relating flow with transmissivity. For example, preliminary observations by Niemi et al. [2000] have indicated that continuum based analysis of field measured conductivities from one dimensional borehole tests may underestimate conductive characteristics of discontinuous fractured media. Also, there is a need to reconcile transmissivity obtained from flowmeters with transmissivity obtained from classical evaluation of responses from pump tests. [9] In this study we apply site characterization data obtained from extensive field investigations of the Laxemar region of Southern Sweden to numerical three dimensional DFN simulations, and evaluate the use of flow measurements from a particular borehole of the region, with the purpose of addressing the validity of commonly used assumptions for inferring transmissivity. Also a brief analysis of how transmissivity evaluation may further impact advective transport is addressed, as this has important implications for site assessment. [10] In section 2, the formulation of the flow problem of steady state radial pumping in an aquifer/domain are stated with the assumptions which will be evaluated in this study. In section 3, the relevant aspects of site characterization data from the Laxemar region are briefly described, conceptualized, and applied to DFN modeling, as well as a presentation of the conditioning approach used. The main results in terms of flow conditioning are presented in section 4, and implications for transmissivity are discussed in section 5. Resulting potential implications on natural flow conditions and tracer transport are discussed in section 6, and the presentation is summarized with conclusions in section 7. Additionally, effects of the domain scale are evaluated in Appendix A, and effects of the fracture size distribution in Appendix B. 2. Problem Formulation [11] When used in high resolution mode the PFL tool can measure fracture specific flow rates entering or leaving a borehole (details of the PFL tool are provided in section 3). Assuming that each flowing fracture intersecting the borehole acts as an effective parallel plate, and is connected to a 2of17

3 boundary of constant head at a given distance from the borehole, and without intersecting other fractures in the fracture plane (i.e., eliminating a network of fractures), the flow entering the borehole from the fracture due to extraction pumping can be seen as flow in a confined aquifer. Thereby fracture transmissivity can be inferred from the flow rate Q entering the borehole by using the Thiem equation (also known as the Dupuit equation) [Bear, 1979], T ¼ Q log ð r 2=r 1 Þ ; ð1þ 2Dh where r 1 (m) is the radius of the borehole, r 2 (m) is the radial distance to a cylindrical boundary with constant head (reference pressure), Dh(m) is the constant drawdown in the borehole due to extraction pumping, and Q (m 3 /s) the measured fracture specific volumetric flow rate. Since r 2 is generally not known, it is frequently assigned a value based on past experience of field conditions; however since it appears as a term in the logarithm, it is not a very sensitive parameter [Bear, 1979]. [12] Clearly fractures in crystalline rock are generally not subject to these conditions. The distance to an approximate or sufficiently constant head boundary can only be assumed, and in general should vary considerably depending on hydraulic fracture properties (transmissivity and storativity), as well as connectivity of the fracture network. Even if certain flowing fractures are considered sparse in borehole observations (i.e., isolated flow measurements) the connectivity and network structure beyond the borehole within the rock mass is difficult to estimate and can certainly not be known solely by flow measurements. Also, the internal structure (heterogeneity) of fractures is poorly understood, so the assumption of an effectively homogeneous structure is verifiable in few exceptional cases only. Despite these obvious limitations, the Thiem Dupuit equation serves as a reference case for estimated or inferred fracture transmissivities directly obtainable from in situ flow measurements, and is assumed to provide indications of the order of magnitude of inherent transmissivities [Ludvigson et al., 2002], and is frequently used in applications. 3. Site Description, Conceptual Model, and Conditioning [13] As part of the Swedish site investigation program for subsurface storage of spent nuclear fuel, extensive characterization of candidate repository sites is being conducted by the Swedish Nuclear Fuel and Waste Management Co (SKB). Until recently one candidate area has been the Laxemar subregion, covering roughly 12.5 km 2 and situated slightly inland off the southeastern coast of Sweden, in the municipality of Oskarshamn, the approximate center located at Latitude N and Longitude E (Figure 1). The site characterization program is a vast investigation covering multiple scientific and technical disciplines. Here only a very brief summary of the preliminary site description data version 1.2 (which consists of data collected up until November 2004) for Laxemar is presented; for a more complete overview, refer to Swedish Nuclear Fuel and Waste Management Company [2006]. In the following section we describe the most relevant parts of the site description data from the perspective of this study, thereafter summarize the application of the data for the purpose of conducting DFN flow simulations, and finally describe the proposed generic method for inferring transmissivity from fracture specific flow rates measured in boreholes Field Data [14] To describe and model the flow of the hydrogeologic domain, surface and borehole investigations are of significant interest. In essence, surface outcrops yield information of surface fracture number frequency, often denoted by P 20 [1/m 2 ], and surface fracture trace length per given unit area, i.e., the density P 21 [m/m 2 ], as well as identification of rock mineralogy. In terms of fracture frequencies, boreholes can provide a P 10 [1/m] measure along depth together with orientations, but more importantly can also be used for various static and dynamic measurements, such as electric conductivity, flow logging, visual imaging, water sampling, etc. This way, fractures can be categorized or distinguished, e.g., if they are deemed to conduct flow or not. [15] For the data version 1.2 there are six main outcrops in the region comprising a total surface area of about 2185 m 2 and resulting in a total of about 5000 fractures. Two of the outcrops pertain to the Laxemar subregion and comprise about 777 m 2. These are used to determine fracture orientations, frequencies, lengths, and lithology. In terms of trace length, these outcrops provide scales typically from decimeters to a few meters. Larger scale features such as lineaments and ductile/brittle zones are observed through airborne electromagnetic analysis and from topographic/ terrain data and provide lengths at the scale of about 1 10 km. Such large scale features are however generally regarded as deterministic fracture zones, and mainly applied to regional scale modeling. [16] There are six core drilled boreholes for the 1.2 data version, which provide a total of about 6800 m of core and allow deep subsurface analysis, in addition to a number of shallow percussion drilled boreholes providing near surface fracturing. Data from four of the core drilled boreholes (KLX01 through KLX04) and one percussion drilled borehole (HLX15) are used here to infer three dimensional characteristics of fracturing applicable to the Laxemar subregion, providing fracture strike and dip angles as well as P 10 intensity. [17] Fractures are sorted into sets based on orientation analysis and assigned statistically significant Univariate Fisher distributions. Intensity and size distributions are determined through inverse simulation by conditioning outcrop P 20 frequencies and P 21 densities against borehole P 10 frequencies, to infer the fracture surface area density in a given volume of rock P 32 [m 2 /m 3 ][Hermanson et al., 2005], which is the primary fracture density property of interest for DFN simulation. The results of structural analysis by Hermanson et al. [2005] and later modified for hydraulic analysis (including PFL flow features) by Hartley et al. [2006] indicate that the probability density f(r) of fractures of size scale r follow (truncated) power law distributions fðþ¼ r krk 0 r kþ1 L min < r < L max ; ð2þ where k > 0 is the shape parameter, r 0 > 0 the location parameter, and L min and L max are assigned minimum and maximum length scale truncation limits, respectively. The 3of17

4 Figure 1. The Laxemar subregion, showing the location of six core drilled boreholes (KLX series) and several percussion drilled boreholes (HLX series). Fracture and pump test data from boreholes KLX01 KLX04 combined with surface analysis of the two outcrops ASM and ASM are used as basis for obtaining the stochastic structural and hydraulic parameters of a local DFN representation. Also, high resolution flow measurements obtained during pumping of borehole KLX04 is used; the KLX04 borehole is located roughly in the center of the subregion and is dominated by a fairly uniform mineralogical rock type known as Ävrö granite. (Image source: Swedish Nuclear Fuel and Waste Management Company.) flow through the system will be dominated by small fractures if k > 2, by large fractures if k < 1, and 1 < k <2is considered a crossover scale [de Dreuzy et al., 2001]. The minimum observable fracture trace length sets the location parameter, which is typically of the order of 0.5 m for outcrops, and k is fitted through inverse modeling of fracture intensities. [18] Scaling assumptions for fracture size have been reported in field observations [Hermanson et al., 2005] even though underlying physical causes being an active field of research [Bonnet et al., 2001]. Additional investigations involve finding constitutive relationships between fracture properties such as size, density, permeability and/or transmissivity, and have achieved certain degree of success [Neuman, 2008] within the framework of nonstationary multimodal representations of random fields [Neuman, 2003]. Some evidence has been observed for relating fracture size with permeability in geologic media [Walmann et al., 1996; Neuman, 2008] and recent studies at the Äspö Hard Rock Laboratory have provided indications that a fracture size and transmissivity correlation may be applicable [Dershowitz et al., 2003]. [19] A significant challenge is distinguishing between flowing and nonflowing fractures; the latter can be cases where mineral deposits have developed and visually sealed previously open fractures, or cases where fractures appear open (i.e., have voids) but do not seem to conduct measurable quantities of flow, i.e., can be interpreted as being disconnected from the hydraulic system or dead end features. Thus fractures which are deemed not to be open and not to be conductive (nonflowing) are removed from the statistical sample, and hence slightly alter the fracture statistics/distributions. [20] With the above analysis various levels of DFN representation are obtainable. First, a DFN representation consisting of all observed fractures, thereafter a DFN consisting of all fractures deemed to be open, and finally a DFN of all fractures deemed to be conductive conduits for flow. The last set, i.e., of conductive or flowing fractures, is obtained by combining the open DFN with hydrodynamic measurements, such as the PFL tool. The distributions and parameters used to represent the DFN properties of flowing fractures, called the hydro DFN model, are summarized in Table 1, which are obtained from Hartley et al. [2006] and applicable to the 4of17

5 Table 1. Structural Properties of the Stochastic DFN Representation From Preliminary Site Investigation Data Applicable to the KLX04 Borehole at the Laxemar Subregion (Sweden) a Fracture Set Label S A S B S C S d S f Strike description ENE WSW N S NW SE Horizontal NW SE Orientation trend (deg) Orientation plunge (deg) Fisher dispersion coefficient Length min L min (m) Length max L max (m) Length exponent k Density P 32 (m 2 /m 3 ) a The Univariate Fisher distribution is used for describing orientations, and a truncated power law distribution is used for fracture size. For these blockscale simulations, the density of fractures is approximated to be uniform throughout the model domain. Laxemar subregion in the vicinity of borehole KLX04 for the preliminary site description version 1.2. [21] Extraction pumping is conducted during about 7 days such that steady state conditions prevail and a series of PFL flow measurements at various support scales along the borehole length are conducted [Rouhiainen and Sokolnicki, 2005], producing essentially discrete flow values as shown in Figure 2. Furthermore, these flow values can be correlated to borehole core log and TV imaging observations enabling inference of fracture specific flow rates. The interpretation of these flow values to transmissivity using equation (1) is shown in Figure 3 (cross markers). [22] The PFL tool can be used in several modes. When used in the so called high resolution mode (from which the data in this study originates), then the support length of the device is 1 m, and a sequence of overlapping measurements are conducted with 0.1 m intervals. Thus the effective support scale is 0.1 m, which in most cases is sufficient to associate flow values with fracture observations. In our study however, we only use flow values (below the depth z = 300 m) and disregard the association with fractures. The typical range of measurable values is approximately m 3 /s to m 3 /s, but depends on various in situ conditions and can vary locally along sections of the borehole. A detailed description and discussion of the limitations of the device is provided by Rouhiainen and Sokolnicki [2005]. [23] Although not used in this study, single hole pressure injection tests are also conducted with the so called Pipe String System (PSS) utility, which performs several pump tests at various support scales (100 m, 20 m and 5 m), including constant head tests with a transient response phase and slug/pulse tests, the choice of which being determined in situ depending on the magnitude of the initial pressure response observed in each packed off section. Transmissivity of the section can then be inferred either using the Moye equation [Moye, 1967; Rahm and Enachescu, 2004] for the steady state phase, and/or using type curve analysis with various methods for transient phases, depending on the type of response observed. For the KLX04 borehole, the steady state responses yield lower transmissivities when compared against the transient responses [Follin et al., 2006]. Transmissivities along borehole depth obtained using the PSS utility are shown in Figure 3 for the three support scales (solid lines), and as can be seen the highresolution PFL transmissivities (crosses) are generally lower. A similar observation was found at the Apache Leap Research Site in central Arizona, USA, using single hole pneumatic injection tests conducted at various scales [Illman, 2004]. There is also an evident depth trend, where transmissivity generally decreases with increasing depth. Depth trends in conductivity are well known and commonly observed for various formations and scales [Manning and Ingebritsen, 1999; Niemi et al., 2000]. Figure 2. Flow rates obtained from PFL measurements in borehole KLX04 during steady state extraction pumping. (a) Measured flow rates at essentially discrete instances along the depth of the borehole, with an effective support scale of about 0.1 m. (b) The empirical distribution (cumulative and complement of the cumulative) of the flow rates shown above (dashed) and the distribution of flow rates below depth 300 m only (solid). The filtered data are used in this study to reduce near surface effects, such as increase in fracture density. 5of17

6 Figure 3 6of17

7 3.2. Simulations [24] The stochastic DFN description is implemented by conducting multiple Monte Carlo realizations in order to obtain ensemble statistics, each representing an equally probable realization drawn from the ensemble. For each realization, fractures are generated in a given domain by sampling from the distributions provided in Table 1 and according to the densities of each set. This fracture domain is significantly larger than the model domain (for which boundary conditions are assigned) in order to avoid a slight decrease in density in the vicinity of fracture domain boundaries. [25] Simulations are carried out with the CONNECTFLOW software package [Hartley and Holton, 2008], using the NAPSAC module [Hartley et al., 2008] for DFN modeling only, i.e., flow through the rock matrix is neglected. To simulate extraction pumping, the lateral sides of the domain are assigned a constant reference pressure (0 m head), and the simulation borehole is assigned the approximately constant drawdown ( 5.8 m head) used in the field during pumping, which results in an approximate radially converging hydraulic gradient. The groundwater pressure distribution and flow equation is solved throughout the domain assuming steady state, single phase flow with constant density through the network of connected fractures; conservation of mass implies rq ¼ 0 where Q (m 3 /s) is the volumetric flux of water. Darcy s law is used for obtaining flow assuming each fracture can be represented as a porous media slab with hydraulic conductivity K (m/s) and cross sectional area A (m 2 ), then Q = AKrF. By integrating conductivity over the fracture opening (aperture 2b), this is expressed in terms of the fracture width w and transmissivity T =2bK as Q ¼ wtrf: A finite element mesh is imposed on each fracture surface in order to approximate connections between fractures. Then the pressure head field F [m] is calculated for nodes along each line of intersection of fractures in the network by solving the resulting system of linear pressure equations using a Pre Conditioned Conjugate Gradient iterative solver. Convergence of each solution is ensured by using a low convergence criterion and monitoring the absolute and relative mass flux balances through the domain. ð3þ ð4þ 3.3. Soft Conditioning [26] The objective of the approach adopted for conditioning is to match the distribution of PFL flows with the distribution of simulated borehole flows, occasionally referred to as soft conditioning in order to distinguish it from conditioning against specific/hard data, as in the terminology of [Rubin, 2003]. This way the problem is a simplified inverse problem, where we attempt to match the distribution of simulated borehole flows g(q) to the distribution of PFL flows f(q) by varying the distribution of fracture transmissivity. In principle, transmissivity can vary greatly and be dependent on other physical properties, hence certain model assumptions are introduced in order to reduce the dimensionality of unknowns and make the inverse problem tangible. [27] Recent investigations in this region have suggested a correlation between fracture transmissivity and size [Dershowitz et al., 2003]. In this study a perfect correlation between transmissivity and size is adopted, since this represents one extreme of possible relationships (e.g., the opposite of an uncorrelated assumption), and also has the advantage of simplifying and reducing model parameters. Thus transmissivity T [m 2 /s] is related to the size of the fracture as T ¼ m L ; ð5þ L min where L [m] is the fracture length scale defined as the square root of the fracture s surface area, L min [m] is the smallest fracture length scale from the truncated power law distribution (cf. Table 1), and m [m 2 /s] and a [ ] are the model intercept and exponent parameters respectively. The intercept parameter has units of transmissivity and sets the minimum transmissivity assigned by the smallest fracture size. To facilitate presentation, a dimensionless intercept parameter m* m 10 8 is defined, and results discussed in terms of simulation cases with various values of m*. Then the intercept and exponent parameters of the transmissivity model are systematically varied to find ranges of values which provide improved goodness of fit. [28] In order to quantify and evaluate the similarity between f(q) and g(q) the two way Kolmogorov Smirnov and Kuiper tests are used, both which are nonparametric statistical tests for comparing empirical cumulative distributions. The Kolmogorov Smirnov test statistic [Kolmogorov, 1933] is based on the maximum absolute deviation between the two empirical distributions, is invariant under reparameterization of the variable, but is not considered to have good power. However it is most sensitive to the part of the cumulative distribution around the median, which means it is suitable to identify linear transformations (translational shifts) between data sets, but to a lesser extent identifies differences in spread [Press et al., 1992]. The Kuiper test statistic [Kuiper, 1962] is the sum of the maximum deviation above and below the two empirical cumulative distributions, is invariant under cyclic transformations, and equally sensitive to differences along the entire range of the distributions, hence more suitable to identify differences in spread [Press et al., 1992]. Both tests evaluate the significance level, that is the probability, for the null Figure 3. Transmissivity data for Laxemar borehole KLX04 inferred from in situ PFL flow measurements and PSS pressure injection tests. The PFL transmissivities (crosses) are inferred from flow measurements using equation (1). Transmissivities are also inferred using classical pressure injection tests with 100 m (solid blue), 20 m (solid red), and 5 m (solid green) support scales (packed off section lengths along the borehole). Also shown are six assumed regional deformation zones (DZ1 DZ6) applicable to the preliminary site description model v1.2. (Image source: Swedish Nuclear Fuel and Waste Management Company.) 7of17

8 hypothesis that the data sets are drawn from the same underlying distribution. Thus small values of the probability show that the two data sets are significantly different. 4. Flow Distributions [29] A series of block scale 3D DFN simulations are conducted, configured according to data obtained from field site characterization, and results against PFL flow measurements are assessed, as discussed in section 3. Initially the domain of size m 3 is investigated and values for parameterization of transmissivity are assigned as m* = 1, 2, 3, 4 and 5 with a = 1. For this domain size each realization typically generates of the order of fractures, with about half removed after connectivity analysis identifying isolated or dead end features, which do not influence pressure and flow calculations (since the rock matrix is not considered). One DFN realization colored by transmissivity is shown in Figure 4a, and a cross section plot showing the pressure field (drawdown) in Figure 4b. Additional details of the simulation configuration are shown in Table 1. Figure 4. One realization of the three dimensional simulation domain of size with discrete fractures for the case m* = 3 colored by (a) transmissivity and (b) cross sections showing drawdown. Each realization typically results in about fractures, but approximately half are removed after connectivity analysis. The line traversing the center is the simulated sink representing borehole KLX04. To replicate extraction pumping in field conditions, the lateral sides of the domain are assigned a constant reference pressure, set to 0 m head, and the simulation borehole is assigned the effective drawdown of approximately 5.8 m head Statistical Comparison Tests [30] Results from applying the Kolmogorov Smirnov (K S) and Kuiper (Kp) comparison tests, introduced in section 3.3, are shown in Figure 5 (left). The intercept parameter is varied as m* = 1, 2, 3, 4, 5 (rows) with the exponent parameter initially set to the constant a = 1.A total of 20 fracture network realizations are generated and the flow field is numerically calculated, as discussed in section 3.2. Then for each realization the distribution of simulated flow values in the borehole representation are compared with the field flow values obtained from PFL measurements using the K S (squares) and Kp (circles) tests. In addition to this, the set of simulated flow values is accumulated as each new realization is performed, and this consecutively increasing set of flow values for a given number of realizations is compared against the field flow values using the K S (lines with squares) and Kp (lines with circles) tests. [31] Note that for individual realizations (dots), the probabilities of both test statistics can vary between near null to near unity between realizations, and that the probabilities can vary significantly between the two tests. This scatter illustrates the great variability inherent between the equally probable fracture network realizations, and clearly indicates the importance of conducting multiple realizations in order to obtain ensemble/statistically converged results. This observed variability originates from the distributions used to describe the DFN, and ultimately due to the inherent uncertainties and natural variation in field data. Figure 5. Inferring fracture transmissivity by conditioning distributions of simulated borehole flows against field flow measurements. (top to bottom) Transmissivity parameterization is varied as m* = 1, 2, 3, 4, 5 with constant a = 1. (left) Results from applying the Kolmogorov Smirnov (squares) and Kuiper (circles) tests, both on single realizations (single markers) and on the accumulating ensemble of realizations (lines with markers). (middle) Empirical CD and CCD of borehole flows obtained from field measurements (solid) and from the simulation ensemble (dashed). (right) Empirical CD and CCD of transmissivities obtained by applying equation (1) to field flows (solid) and to the simulation ensemble (dashed), as well as from actual values used in the simulation ensemble (dash dotted). 8of17

9 Figure 5 9of17

10 [32] For the cumulative ensemble (lines with markers), there is a certain degree of variability/spread in probability both between the two tests and as the ensemble increases. There is a tendency for this spread to decrease somewhat as the number of realizations increase, but for the 20 realizations conducted here it is difficult to determine a stable asymptote. However, we note that the primary usefulness of the comparison tests are to reflect the test scores (probability values) between simulation cases for a consistent number of realizations, which is a fair comparison, since the K S and Kp tests are based on empirical samples of data. [33] The highest probability is obtained for the scenario with m* = 3 (Figure 5, left column, third row), which we shall consider as the approximate best model parameter for the fracture transmissivity and size correlation under this simulation scenario. Further, the K S and Kp statistics can be used to quantify the accuracy of the match. For the m* = 3 case the probability at the ensemble level of 20 realizations is approximately 0.97 and 0.86 for the K S and Kp tests, respectively. [34] For the other cases the probabilities at the ensemble level of 20 realizations are significantly lower; m* = 1 (Figure 5, left column, first row) and m* = 5 (fifth row) both have very low probabilities, of order 10 6 and 0.05, respectively. The case with m* = 2 (second row) has approximate asymptotic probabilities of 0.1 and 0.3 for the K S and Kp tests, respectively, and m* = 4 (fourth row) of 0.2 and 0.5, respectively Evaluation of Flow Rates [35] The distribution of flow values, shown as the cumulative distribution (CD) and the complement of the cumulative distribution (CCD), obtained from PFL field measurements and from simulation at the ensemble level of 20 realizations for the same five cases of m* are shown in Figure 5 (middle). The PFL distribution (solid) consists of 76 flow observations for depths z < 300 m obtained during pumping of borehole KLX04 (cf. Figure 2). For this simulation configuration there is on average 0.32 fracture intersections per meter of borehole, hence the distribution of simulated flows (dashed) has about 640 values at the ensemble level of 20 realizations, which produces a fairly smooth empirical distribution for the densities shown (down to 0.01). [36] The probabilities obtained from applying the statistical comparison tests on these distributions correspond to the last values shown by the markers at the end of the lines in Figure 5 (left). As can be seen, there is good visual correspondence between the ensemble of the simulated flows and the field measured PFL flows for the m* = 3 case (Figure 5, middle column, third row), in agreement with the results obtained from the K S and Kp tests (Figure 5, left). The cases m* = 1 and 2 (first and second rows) slightly underestimated the measured flows, and the cases m* = 4 and 5 (fourth and fifth rows) slightly overestimate them. [37] For m* = 3 most of the distribution agrees with field data and especially the median values agree well; only the low density part of high flows (i.e., the tail) is somewhat overestimated. Even though these high flows are less than about 5% of the values obtained from simulation, they are sufficient for the Kp test to yield a slightly lower probability value than the corresponding K S value (cf. last markers of the lines in the left column, third row). This is consistent with the applicability of the respective tests, since the Kp test is more sensitive to deviations in spread of distributions, and the K S more so to the regions of higher density and translations of medians. [38] The case m* = 2 should suffice as a lower bound of flow values, since if these parameters are chosen then the simulated flow will in general be lower than that measured in the field. For a candidate upper bound, the case of m* =4 should serve sufficiently well, since even if certain parts of the low densities of the CD and CCD curves may coincide with the field measurements, the general behavior of these simulated flows will be greater than the PFL flows. Alternatively if a greater safety margin is desired, e.g., considering the limited sample size of PFL flows, then clearly the cases with m* = 1 and m* = 5 should serve as more certain lower and upper bounds, respectively. [39] In summary it seems possible to assign a sensitivity range as well as to quantify the uncertainty for the parameters of the chosen transmissivity model, by selecting upper and lower bounds and using the obtained probability values. For this configuration it seems that m* = 3 with a = 1 is the approximate best model parameterization, which has corresponding probability p ^ Assigning the sensitivity range 2 < m* < 4 provides a probability value p ^ 0.1, or alternatively the even safer error margin of range 1<m* < 5 with corresponding p ^ [40] The effect of widening the range of m* should be interpreted as increasing the tolerance of the model, but at the cost of decreasing the probability of exactly matching the measured field flow values Sensitivity of Parameterization [41] The exponent parameter a in equation (5) controls the spread of the resulting transmissivity distribution, such that for a < 1 the spread is less than the spread of the fracture length distribution, for a > 1 it is greater, and for a = 1 the spreads are equal, i.e., a linear correlation assumption. However, the exponent parameter also causes a secondary shift in the location of the median; thus both parameters should be conditioned simultaneously. [42] To further evaluate the sensitivity of the results obtained in Figure 5, the m* and a parameters are varied as m* {2, 3, 4} and a {0.5, 0.75, 1, 1.25, 1.5}. As previously, simulated borehole flows obtained from the ensemble of 20 realizations are compared with PFL flows using the K S and Kp tests; results for each combination of cases are shown in Table 2. For the K S test, the cases with (m*, a) = (4, 0.75), (3, 1.25) and (2, 1.25) provide reasonable probability values, as well as (m*, a) = (3, 1.25), (4,1) and (2, 1.25) for the Kp test. The highest probabilities for both tests are clearly obtained for m* =3,a = 1, confirming this indeed seems to be close to a best match for the correlated model assumption. [43] Analysis of the corresponding distributions of simulated flows compared with PFL flows indicate that increasing a increases the spread, primarily at the tails of the distributions, but also causes a slight shift of the median toward higher flows; likewise, decreasing a decreases the spread of flows (results not shown). However, the change in flow distribution is hardly noticeable visually. Thus there is a fine sensitivity inherent in the comparison tests, since even 10 of 17

11 Table 2. Sensitivity of Transmissivity Model Parameters a and m* in Terms of Probabilities Obtained From the Comparison Tests for the Ensemble of 20 Realizations a m* =2 m* =3 m* =4 KS Probabilities a = e 7 1.8e 4 5e 3 a = e a = a = a = Kp Probabilities a = e 5 1.9e 3 8.4e 3 a = a = a = a = a Both the K S and Kp probabilities are maximum for m* =3,a =1. with a minor change in m* and a and hence only a minor change in flow distribution, the probability values can change significantly. [44] Whereas it may be possible to obtain a higher probability say in the range 3 < m* < 5, 0.75 < a < 1.25, it seems unlikely a better match would deviate far from m* =3,a = 1. Other combinations, especially for a = 0.5 and a = 1.5 generally produce less agreeable matches regardless of m*. Thus there is no immediate strong indication that alternate parameterization, where a may be significantly above or below unity should be expected, even though the soft conditioning approach applied here is in general nonunique. [45] Furthermore, in Appendix A an analysis of the method with respect to domain scale is conducted and shown to be robust both in terms of flow boundary conditions assigned as radial hydraulic gradients of various magnitudes, as well as in terms of network connectivity provided a sufficiently large ensemble set of simulation data is acquired, as is inherent to stochastic approaches. Also, in Appendix B the sensitivity of the approach is analyzed in terms of fracture size truncation limits. 5. Inference of Transmissivity Parameters [46] The resulting fracture transmissivities obtained from conditioning of flow and the use of a correlated fracture size model can be compared with transmissivities inferred from PFL flow measurements using the Thiem Dupuit assumptions, as discussed in section 2. [47] The distribution of PFL transmissivities inferred from flow measurements using the Thiem Dupuit equation (1) is shown by solid lines in Figure 5 (right). This is compared with the resulting distribution of actual fracture transmissivities used in simulation (dash dotted) for each of the five cases considered. As can be seen, there is significant difference for the best case simulation m* = 3 (third row), especially for low and intermediate values. The other bounding cases also show significant differences, but the best agreement between inferred PFL transmissivities and simulated transmissivities lies in the vicinity of cases m* =1 and m* = 2 (first and second rows). The lower cutoff in the distribution of simulated fracture transmissivities (dashdotted) is due to the minimum fracture size threshold used in this simulation configuration. [48] Thiem Dupuit transmissivities obtained from simulated flows, i.e., using the Thiem Dupuit equation on the flow values obtained from simulation (similar to the way in which PFL transmissivities are obtained), are shown as dashed curves in Figure 5 (right). Whereas the simulation configuration is cubic in this setup, the radial distance r 2 to the boundary of constant head used in equation (1) is here approximated by the shortest distance to a lateral side of the domain, which in this case is 50 m. In the case of the field data, the distance to a boundary of constant head, if indeed possible to be approximated by a constant radial distance, is not known and is simply an assumed value ( 20 m). Since the Thiem Dupuit equation relates transmissivity as proportional to flow, these distributions (i.e., dashed curves in Figure 5, right) are proportional to the distribution for flow (i.e., dashed curves in Figure 5, middle). [49] Therefore it is interesting to note that for the case m* = 3 (Figure 5, right column, third row), there is very good visual agreement between transmissivity distribution inferred from PFL flows (solid) against those inferred from simulated flows (dashed). In fact when comparing these two distributions, both the K S and Kp tests give a reasonably high probability value of about 0.6 for the ensemble of 20 realizations. [50] Thus it is clear that using PFL transmissivities inferred with equation (1) as input for block scale simulations of the Laxemar domain would likely result in underestimation of transmissivity, when compared to transmissivities obtained through conditioning to measured flows; in this case by a factor of about 2 in the median and of about 5 at the 0.1 density level (low transmissivities) of the distribution. High transmissivities however seem to agree reasonably well, at the 0.9 density level of the distribution Thiem Dupuit values are underestimated by a factor of about 1.5 only. Naturally, these simulations have been conducted with several assumptions which could influence results. One significant aspect is the range of obtainable transmissivities, which is controlled by the fracture size truncation limits combined with the intercept parameter m*. In the case m* = 3 the transmissivity m = m 2 /s is assigned to the minimum fracture size (which is set to L min = 1 m), and then the maximum possible simulated transmissivity is m 2 (since L max = 1000 m). [51] The above values can be compared to the minimum and maximum transmissivities obtainable by use of the Thiem Dupuit equation (1), which are directly related to the measurement limits of flow rate using the PFL tool. Assuming a borehole radius of r 1 = m, a radial distance of r 2 = 20 m to a boundary of constant head, and a drawdown of D h = 5.8 m, we have that the least and greatest values of transmissivity in KLX04, inferred from the least and greatest values of measured flow rates, are approximately m 2 /s and m 2 /s, respectively. For the threshold values due to the technical measurement limits of the PFL tool, and assuming the same physical conditions, we have minimum and maximum transmissivities of approximately m 2 /s and m 2 /s, respectively [Hermanson et al., 2002]. [52] Thus the range of PFL transmissivities inferred from flow measurements in borehole KLX04 by using the Thiem Dupuit assumption (equation (1)) are somewhat larger than those obtained through simulation by conditioning to flow, particularly extending toward low transmissivity ranges (i.e., 11 of 17

12 Figure 6. Advective travel time distributions obtained for transmissivity parameterization m* = 0.1 (dashed) corresponding to Thiem Dupuit transmissivities inferred from PFL and m* = 3 (solid) corresponding to flow conditioned transmissivities. The effect of the sensitivity range of the flow conditioned transmissivities, with m* = 2 (dash dotted) and m* = 4 (dashed double dotted). The distributions are obtained from a single realization of the simulation domain m 3, with boundary conditions set to replicate a natural hydraulic gradient of the region. About 1000 particles are released on the inflow boundary surface at x = 50 m and traverse the fracture network by following the advective flow field downstream until exiting through the outflow boundary at x =50m. first part of the CD); high transmissivity values (tails of the CCD) which are most significant for transport modeling generally agree well. 6. Implications for Flow and Transport Modeling [53] Fractures of high transmissivity have a major significance on the bulk movement of subsurface flow in crystalline bedrock, which may also impact the motion and fate of contaminants. On the other hand, fractures of low and intermediate transmissivity can contain transport retention qualities which help retard the spread of contaminants, even though they may not significantly contribute to the net fluid flux. Therefore investigating the full spectrum of plausible fracture transmissivities plays a significant role in assessing the transport and retention properties of fractured rock. [54] From comparison of the results of transmissivities obtained in this study, i.e., through soft conditioning of field measurements of fracture specific flow, against transmissivities inferred using the Thiem Dupuit equation, we see there is a significant difference in the distributions obtained (cf. Figure 5, right). In particular, Thiem Dupuit transmissivities tend to underestimate conditioned transmissivities for intermediate and low ranges. For the median of the distribution, transmissivity is underestimated by a factor of about 2, and at the 0.1 level, i.e., low transmissivities by a factor of about 5. High transmissivities generally agree reasonably well, only with a slight tendency of underestimation. The impact of these transmissivity estimates are evaluated in terms of flow and advective transport for possible long term conditions, represented by a weak linear hydraulic gradient. The reference case configuration of m 3 with the hydraulic gradient rh = h/ x = is used, and no flow on all other sides of the domain. [55] To enable comparison, the distribution of PFL transmissivities is approximated to a fracture transmissivity and size correlation model by setting the minimum PFL transmissivity as the model intercept parameter m, and using the previously obtained best value for the exponent parameter, a = 1. This corresponds to a worst case scenario, since the PFL and conditioned transmissivity distributions have maximum separation for low densities (cf. solid versus dash dotted curves in Figure 5, right column, third row). Also, this does not make full use of PFL transmissivity data, since only a single value from the distribution is used. The minimum PFL transmissivity inferred from these flow measurements using the Thiem Dupuit equation is m 2 /s thereby yielding m p * = 0.1. This is compared with the best case conditioned transmissivity m* = 3 with sensitivity range m* = 3 ± 1. Note that m = m 2 /s approximately corresponds to the median of the PFL transmissivity distribution (intersection of solid curves in Figure 5, right column, third row). For simplicity, and since we are interested in a relative comparison of change in transmissivity parameterization, we consider one realization only (where the same seed is used to generate the DFN, so that the same structural representation is used). [56] Since the difference in volumetric flux for the two cases is directly proportional to the change in transmissivity, the fraction m*/m p * indicates that the case of PFL transmissivities has 30 times smaller flux than the case of flowconditioned transmissivities. However, the effects on transport are related to the fluid velocity in fractures, which in turn depends on the relationship between aperture and transmissivity; here we apply the cubic law assumption. The characteristic time for flow through a fracture (e.g., traversing the entire domain L) can be estimated as t =2bL/ (T rh) 2b/T; that is, travel time scales as aperture 2b over transmissivity. Under the cubic law assumption T (2b) 3, hence time scales as t (2b) 2 or equivalently t T 2/3. Thus a 30 fold increase in transmissivity implies the characteristic travel time is increased by a factor of about 9.7. Estimates of the effect on the distribution of travel time through the network system can be obtained through particle tracking. About 1000 inert and massless particles are released on the inflow boundary surface by flux weighting and allowed to traverse the fracture network downstream. The resulting arrival time distributions on the outflow boundary are shown in Figure 6 for the m* = 3 (solid) cases and m* = 0.1 (dashed). As can be seen, the median breakthrough time is decreased by about 1 order of magnitude, however the spread in arrival times obtained (based on the 1000 particle sample) is perhaps more significant, ranging between about years (m* = 3) or years (m* = 0.1). Thus a notable impact of transmissivity parameterization on nonreactive transport is observed, hence the effect on reactive tracer transport is likely to be equally or more significant, which is however beyond the scope of this study. 7. Conclusions [57] In this work a method for transmissivity characterization based on conditioning of the distribution of field 12 of 17

13 measured fracture specific flow rates through the use of nonparametric statistical comparison tests is developed and evaluated. The simulations are configured according to available field characterization data of the Laxemar region in Sweden. The method proposed is sufficiently general to be applicable to other investigation sites and data sets containing groundwater flow measurements, and is particularly useful for discrete fracture specific flows. The method is evaluated using two particular statistical comparison tests, but could equally well be applied with other types of tests for empirical data. From the basis of the results and analysis conducted in this study, our main findings are summarized as follows. [58] 1. The proposed method directly honors field measurements without additional homogeneity assumptions or assumptions on boundary and domain configuration. This is shown to be feasible at least if using a constitutive relationship between fracture transmissivity and size. [59] 2. The proposed method can also provide limiting bounds of transmissivity parameterization, which is preferred to obtaining specific best matches. Quantification of the uncertainty of each bound is obtained in terms of a probability value. [60] 3. The obtained parameter range is shown to be robust in terms of simulation domain scale, which impacts both the number of fractures in the system as well as assigned boundary conditions, here in terms of the magnitude of a radial hydraulic pressure gradient in the system. [61] 4. For this particular data set and simulation configuration, the obtained range for transmissivity parameterization is shown to be relatively narrow. [62] 5. Characterization of fracture transmissivity using the Thiem Dupuit equation is shown to significantly underestimate values when compared against flow conditioned transmissivities. Differences in transmissivity exhibited between steady state analysis and transient evaluation are thereby to some extent qualitatively reconciled. [63] The specific transmissivity distributions obtained are obviously only applicable for the particular data set and relate to the simulation configurations considered, which contain several simplifications. [64] First, the choice of constitutive relationship between fracture transmissivity and size is likely to impact results from a conditioning approach. Here the perfect correlation assumption is adopted both due to its simplicity (only two parameters) and as it represents one extreme case of correlation models. This however does not account for the significant variability observed in available data [Dershowitz et al., 2003], and is unable to capture effects of internal variability, which must exist in particular for larger fractures. The main challenge lies in obtaining correlation data from field investigations, which is mainly due to difficulties in measuring (or inferring) fracture size in situ. Semi correlated models with a perturbation around the perfect correlation assumption have been suggested [Hartley et al., 2006], but require further theoretical basis and support from field data. Uncorrelated models may also be plausible. [65] Furthermore, one may account for trends in heterogeneity, as it is generally observed that flow and fracture frequency decreases with depth. Thus different transmissivity parameters would likely be obtained if depth trends were accounted for. Similarly, additional boreholes in the region could be used to analyze regional trends in flow and transmissivity. Also data from classical hydraulic evaluation, such as transient tests carried out in packed off sections in the same or neighboring boreholes, could be incorporated, for example to evaluate (confirm or disprove) conditioned results. Appendix A: Scale Dependence of Conditioning [66] The conditioning approach is evaluated with respect to the size of the simulation domain to provide indications on the scalability of the DFN, both in terms of asymmetry inherent in the DFN (cf. Table 1) and effects of the magnitude of the radial hydraulic gradient. [67] The following five size configurations are considered: m 3, m 3, m 3, m 3, and m 3 each with five cases of transmissivity parameterization m* = 1, 2, 3, 4, 5 with a =1. The comparison tests are applied for the ensemble with 20 realizations; the results of each combination of domain size and transmissivity parameterization for the K S tests are shown in Figure A1a and for the Kp tests in Figure A1b. [68] For the small domain of size m 3 (downward pointing triangles) the highest probability of about 0.4 is obtained for m* = 2 for both the K S and Kp tests. It is however likely that a higher probability would be obtained in the 1 < m* < 3 region, and since the probability at m* = 3 is slightly higher than at m* = 1, it is reasonable to assume a better match could be obtained in the 2 < m* <3 region. Similar results are obtained when increasing the vertical extent, i.e., for the domain of size m 3 (delta symbols), except that the probability for the m* = 2 case is slightly lower for the K S test ( 0.3), and for the m* = 1 case the values are somewhat higher. Furthermore, since the m* = 3 case slightly lower, this may indicate a better match could be obtained in the 1 < m* < 2 region instead. [69] The difference between the K S and Kp tests for the peak case m* = 2 should indicate that when compared to the small cubic domain, the flow distribution in the vertically elongated domain is possibly shifted around the median but the overall spread of the distribution is similar. The increase in probability of the elongated domain for m* = 1 combined with the decrease for m* = 3 indicates the best match is shift toward lower m*, i.e., toward lower transmissivities. This in turn indicates that flow rates are increased for the transition in domain from the cubic m 3 to the vertically elongated m 3. [70] Possible explanations for this should be combinations of the changing distance between the top and bottom no flow boundaries, as well as network artifacts such as the increased connectivity of larger systems, or the vertical anisotropy inherent in the fracture network. Near vertical fractures in the small horizontal domain with elongated vertical extent have a greater probability of conducting flow since they are more likely to make direct connections between the lateral boundaries and the borehole. For a domain with a smaller vertical extent however, such fractures may be more likely to connect to the borehole through the top or bottom no flow boundaries, thus only conducting flow via other fractures if connected to the lateral bound- 13 of 17

14 value only from about 0.97 to This should indicate the spread of the distribution of flows in the m 3 case is deviating more than in the m 3 case, but the median only slightly altered. Also, the m 3 case with m* = 3 has almost the same probability of about 0.88 for both tests, indicating the goodness of fit is essentially unaltered between the m 3 and m 3 cases. Figure A1. Results from applying the (a) Kolmogorov Smirnov and (b) Kuiper tests on the ensemble of all realizations for five cases of transmissivity parameterization m* (with a = 1) under the simulation domain size configurations m 3 (triangles), m 3 (delta symbols), m 3 (circles), m 3 (squares), and m 3 (diamonds). aries. The power law length distributions of the five fracture sets are similar, with the exception of set S f which generally consists of smaller features (cf. length exponent parameters in Table 1), whence the variation in transmissivity between sets should be similar. Of the five sets, three have significant near vertical dip (sets S A, S B and S C ) and thus a significant density of fractures in the network have a preferred near vertical direction, as shown in the crosssection plots of one of the realizations for the m 3 domain in Figure A2. Therefore, this anisotropy, which is primarily due to a greater fracture density rather than length or transmissivity, should contribute to increased flow whenever the domain is extended in the vertical direction. [71] The larger domains of sizes m 3 (circles), m 3 (squares), and m 3 (diamonds) all have similar behavior, now with best match in the vicinity of m* = 3. The Kp probabilities for m* = 2 and m* = 4 are slightly higher than the K S values, but the converse occurs at m* = 3. In particular, the most notable change occurs when increasing the vertical extent from 100 m to 500 m (circles to squares), where the Kp probability is reduced from about 0.86 to 0.57, but the K S Figure A2. Cross section of fractures colored by set (cf. Table 1) in the (x, z) plane for one realization of the reference case domain of size m 3 (a) showing sets S A (green), S B (magenta), and S C (blue), which have a significant near vertical dip and (b) showing sets S d (red) and S f (cyan), which have a significant near horizontal dip. 14 of 17