Hydraulic characterization and upscaling of fracture networks based on multiple-scale well test data

Transcription

1 WATER RESOURCES RESEARCH, VOL. 36, NO. 12, PAGES , DECEMBER 2000 Hydraulic characterization and upscaling of fracture networks based on multiple-scale well test data Auli Nicmi, Kimmo Kontio, and Auli Kuus½la-Lahtincn Communities and Infrastructure, Technical Research Centre of Finland, Espoo, Finland Antti Potcri Energy, Technical Research Centre of Finland, Espoo, Finland Abstract. Hydraulic properties and upscaling characteristics of low-permeability fractured rock are analyzed based on systematic well test data from three different measurement scales. First, tests are simulated in a large number of geological fracture network realizations, and the acceptable fracture transmissivity distribution parameters that produce the observed statistics of the two smallest measurement scales, i.e., 2-m and 10-m scales, are defined. Instead of a single value, a range of acceptable parameter values can be found to produce the observed result. Second, upscaling simulations are carried out with the calibrated networks. These indicate that the investigated system cannot be properly modeled by means of a continuum tensor presentation but would better be represented by means of "equivalent fracture" statistics. Third, the conductive characteristics of the calibrated 30-m network blocks are compared to well test results from the same scale. The results from this preliminary analysis indicate that onedimensional borehole observations interpreted with standard continuum-based methods may considerably underestimate the three-dimensional conductive characteristics of heterogeneous, noncontinuum fractured media. 1. Introduction falls between these two, that is, scales too large for fracture network models but small enough so that heterogeneity of rock During the last 15 to 20 years, great progress has been made outside the major deterministic fracture zones is still of interin developing tools and gaining an understanding of the gen- est. The range of applicability of each approach, especially the eral behavior of the hydrology of fractured rocks, both at the transition from fracture network-based approaches to continlevel of individual fractures [Moreno et al., 1990; Cvetkovic, uum-based approaches, also depends on the characteristics of 1991; Tsang et al., 1991; Tsang and Neretnieks, 1998; Painter et the rock in question. Continuum behavior is more likely to al., 1998; Cvetkovic et al., 1999] and at the level of field-scale occur in densely fractured, well-connected fracture networks fracture systems [e.g., Neuman and Depner, 1988; Neuman, with mixed fracture orientations than in sparsely fractured, 1987; Cacas et al., 1990; Dershowitz et al., 1991; Herbert et al., poorly connected, and/or strongly anisotropic systems [e.g., 1991; Long et al., 1992; Gavrilenko and Gueguen, 1998]. The Long et al., 1982]. modeling approaches for field-scale fractured media can be The support scale and the related conductivity statistics of divided into three basic categories: (1) deterministic porous stochasticontinuum models can be determined either directly medium approaches, (2) fracture network approaches, and (3) from hydraulic well test data or by means of fracture network stochasticontinuum approaches. As the relative significance modeling. In the first approach, originally proposed by Neuof local heterogeneities decreases with increasing scale, the man [1987] and subsequently applied by several others [e.g., range of applicability of these approaches depends on the scale Neuman and Depner, 1988; Follin and Thunvik, 1994; Tsang et of interest. Because of computational constraints, fracture netal., 1996], the variability in borehole hydraulic conductivity work models can usually be applied for a scale of at most a few data is introduced directly into Monte Carlo type simulation hundreds of meters. In deterministic porous medium apmodels by using the scale of field measurements as the scale of proaches the medium properties are assumed fixed and known; the model permeability zones. While this is attractive in its major fracture zones are imbedded deterministically, and the practicality, the critical question is whether the medium propremaining part of the rock is assigned averaged properties. erties are such that an equivalent continuum presentation is These approaches are applicable in the largest of scales when, for example, the effect of major fracture zones is investigated justified at the scale of the measurement. In other words, can and the effect of local heterogeneities is not of interest. The one with reasonable accuracy present the hydraulic conductivrange of applicability of the stochasticontinuum approaches ity structure of the rock with an equivalent continuum conductivity tensor. The second approach, in which the continuum properties are calculated by means of fracture network mod- 1Now at Department of Earth Sciences, University of Uppsala, Upp- eling, also allows the applicability of this assumption to be sala, Sweden. studied. This approach has been applied, for example, for Copyright 2000 by the American Geophysical Union. Paper number 2000WR /00/2000WR $09.00 Stripa data in Sweden by Herbert et al. [1991] and by Dershowitz et al. [1991] and for Fanay-Augeres data in France by Cacas et al. [1990]. La Pointe et al. [1995] looked at the continuum 3481

2 3482 NIEMI ET AL.: HYDRAULIC CHARACTERIZATION OF FRACTURE NETWORKS conductivity characteristics for the sp6 site in Sweden. Using to 1000 m. The distances between the boreholes are of the fracture network modeling for determining the continuum order of a few hundreds of meters. conductivities has the benefit of being able to take heterogeneity into account more realistically, starting from the geological characteristics. Conversely, a large number of parameters Hydraulic single-hole tests have been carried out with a fixed interval length in these boreholes; the total length of the boreholes tested was over 5000 m. Two different testing methods have to be calibrated, and immense amounts of data are and three different measurement scales have been used. The needed for this calibration. constant pressure injection method has been applied with Most of the aforementioned studies have been related to 30-m, 10-m, and 2-m packer spacings, and the so-called differsmall-scale field studies in field laboratory settings. They have ence flow method has been applied with 2-m and 10-m packer employed short distances between the boreholes, so that cross- spacings. While the constant pressure injection method is hole and tracer information has been available to validate the widely used for testing low-permeability media and described results. Furthermore, they have often concentrated on the upper hundreds of meters of the bedrock with relatively high in a number of references [e.g., National Research Council, 1996], the latter method is less common and deserves a brief transmissivities and well-connected fracture networks. Reflect- description. In the method, for each isolated test section in ing this, the continuum scales employed have been of the order of 10 m or below. An exception was the study by La Pointe et al. [1995] where support scales of the order of 50 m were also used. This study also considered greater depths with data up to 600 to 800 m. Especially in the case of site characterization for nuclear waste disposal, one is interested in the flow on large scales and at great depths. In such settings, fracture networks are less connected and of lower conductivity than in the upper parts of the rock. Interference tests and model calibration do not capture the small-scale heterogeneity observed in boreholes but rather are dominated by large-scale features. To carry out interference tests that would characterize the average "nonzone" rock, i.e., the rock outside the deterministic fracture zones, one would need very long times or, alternatively, short distances between the boreholes. These, in turn, would only cover a small part of the investigated rock. The major questions to be addressed then are how to determine the hydraulic properties of the fracture networks in the "nonzone" average rock, how much data are needed for this, and what the resulting uncertainty is in the network properties. We will address these questions by using field data from the Romuvaara crystalline rock site in Kuhmo, Finland, as the example database. More specifically, we will look at (1) the possibility of using single-hole hydraulic data from different measurement scales to estimate the hydraulic properties of turn, the steady state flow rate into or out of the borehole is measured corresponding to two different prevailing borehole pressures. Assuming a radial flow field, hydraulic conductivity is determined based on the two pressure-flow rate data pairs obtained this way. The method has been thoroughly tested and widely used especially in the Finnish waste isolation program as well as in some international waste isolation experiments [Rouhiainen, 1998; Rouhiainen and Heikkinen, 1998, 1999]. More detailed description of the method is given by Hinkkanen et al. [1996]. The rationale of the testing program was that sections found intact with larger test intervals were not retested with the smaller interval. All boreholes were systematically scanned either with 30-m or with 10-m packer spacing, and some boreholes were scanned with both. After this, 2-m tests were done in sections found conductive with the larger test intervals. The 2-m-scale difference flow measurements were done systematically in all conductive sections. Testing with the 2-m-scale constant head injection method was not quite as systematic but done more to get a comparison database for the 2-m difference flow data. An example of a borehole hydraulic conductivity profile as obtained with different measurement methods and scales is shown in Figure 1. The data on fracture geometry come from mappings of surface outcrops and investigation dikes as well as geological mappings of core samples that were fracture networks and (2) how much uncertainty and non- available for all boreholes. The fracture size distribution was uniqueness there is in such estimates. Second, we will look at the upscaling properties of the networks. More exactly, we will determined from surface observations and the position data from core samples and, to some extent, also from borehole (3) estimate the continuum versus discontinuum behavior of TV-camera observations. The fracture observations have been the upscaled networks and, (4) if applicable, determine the continuum conductivity tensor distributions that can be used, interpreted by applying an assumption of statistical homogeneity of the fracturing geometry in the investigated volume of for example, as inputs for large-scale Monte Carlo simulations. rock. It should be pointed out that surface fractures have been Finally, we will (5) compare the network-based "upscaled" subjecto erosion, and it is not clear that they are representaconductivities to those directly determined from well tests of the same scale by means of traditional well test analyses. tive of the fractures at depth, but these differences cannot be quantified without excavated underground tunnels. Because of the biased nearly vertical orientation of the boreholes, the fracture orientation data were corrected as described by Poteri 2. Data Characteristics and Laitinen [1997] Site Description and Data Collection In the present study, the main interest is in the characteris- The example data come from the Romuvaara site. This tics of the rock outside the deterministic fracture zones. Therecrystalline rock site is located in eastern Finland and has been investigated as a potential candidate for the disposal of highlevel nuclear waste. The data analyzed here do not attempt to fore data sections related to fracture zones were excluded from the hydraulic and fracture geometry databases before the statistical analyses. The question of what conductive feature is reflect all the characteristics of Romuvaara in a site character- deterministic and what is stochastic is not well defined in hyization sense but should rather be considered as representative drology of fractured rocks, and some subjectivity usually reof a low-conductivity fractured rock. The site is characterized mains in the interpretation. In the present work a conservative by relatively low overall conductivity, with a decreasing trend in conductivity with depth. The data come from altogether nine subvertical deep boreholes, ranging in depth from about 300 m interpretation was used in terms of the average rock, and features that were "uncertain" were assigned to the stochastic rather than deterministic data pool [Niemi et al., 1999].

3 NIEMI ET AL.: HYDRAULIC CHARACTERIZATION OF FRACTURE NETWORKS 3483 Hydraulic conductivity (Log K(m/s)) (a) (b) (c) (d) (e) method: DF method: DF method: CPI method: CPI method: CPI interval: 2m interval: 10m interval: 30m interval: 10m interval: 2m OO Figure 1. Example hydraulic conductivity profiles obtained with different methods and measurement scales; difference flow measurements with (a) 2-m and (b) 10-m packer spacing and constant head injection measurements with (c) 30-m, (d) 10-m, and (e) 2-m packer spacing from Romuvaara borehole KR Fracture Geometrical Properties According to the approach presented in more detail by Poteri and Laitinen [1997], the statistical properties of the geometrical fracture data have been determined for fracture ori- entation, size, and fracture intensity. No correlation is assumed to exist between the fracture orientation and size. Orientation Table 1. Parameters of the Fitted Probability Functions of Fracture Geometrical Properties With Borehole Depths >200 m Parameter Value Open Filled Fracture orientation parameters a Mean orientation trend Mean orientation plunge Dispersion coefficient K Intensity P Size distribution b Mean Standard deviation athis is a Fisher distribution. bthis is a lognormal distribution. and size distribution have been estimated independently using borehole and outcrop data for the fracture orientations and sizes, respectively. Parameters of the orientation and size distributions have been calculated by minimizing the squared difference between the measured and simulated distributions. The spatial model of the fracturing assumes that the centers of the fractures are uniformly distributed in the modeling volume. On the basis of this assumption the fracture intensity is calculated from the mean fracture frequency along the boreholes. Only fractures classified as open or filled, and thus potentially water-conducting, were included in the analysis. As an analysis tool, the Fracman fracture network generation program by Dershowitz et al. [1996a] was used. The algorithm allows complex fracture geometries to be generated based on fracture geometric data, and it has been extensively tested and verified for this type of analyses [Dershowitz et al., 1996b]. The fracture orientations are described by means of a Fisher distribution, with parameters for mean pole trend and plunge (0, qb) and dispersion as given in Table 1. The expression for the appropriate Fisher distribution for directional data is (e.g., from Mardia [1972], as applied by Chiles and de Marsily [19931)

4 3484 NIEMI ET AL.: HYDRAULIC CHARACTERIZATION OF FRACTURE NETWORKS (b) oo 80, 60 :3 40 (c) 20 loo :3 40 (d) 20 oo A o o 60 cr 40 2O 1 KR3,KR4,KR7,KR8,KR9 (depth<200 m) DF 2m [,,,,,,,,_,,.,I,l,l,l,=,l!!!! [!... i, :,, Log T (mils} l! KR3,KR4,KR7,KR8,KR9 (depth>200 m) DF 2m 1 I ', ',,,,l,,1,i,n,1...,,,,,,,, Log T {rnt/s} ß KR3,KR4,KR7,KR8,KR9 (depth<200 m) CPI 10m I,,,,::l:1,1,1,1,1,1....,! [!,! [,,, Log T (m21s) ß KR3,KR4,KR7,KR8,KR9 (depth>200 m)cpi 10m I ', ', ', ', ', ',,,a,l,a,l,.,., : :,, : Log T {rnt/s} Figure 2. Histograms of transmissivity data obtained (a) with 2-m difference flow measurements for borehole depths <200 m (n = 375) and (b) borehole depths >200 m (n = 275) as well as (c) for data obtained with 10-m constant head injection measurements for borehole depths <200 m (n = 66) and (d) borehole depths >200 m (n = 51). This is an isotropic unimodal distribution, where 0' and (k' are independent random variables and K is the Fisher distribution parameter. Distribution is rotational symmetric around the mean orientation. This is reflected in (1) that is not dependent on the phase angle 4)' which is an independent uniform random variable in the range of [0, 2rr]. The fractures were assumed to be circular, with the radius being lognormally distributed. It was assumed that the fractures do not terminate at the intersections with other fractures. Fracture intensity is described through P32 [L-l] [Dershowitz et al., 1996a] P32 = A /l/t, (2) where./if is the total area of the fractures and Vt is the total volume of the model. In Table 1, only data below 200-m depth are shown. The division into depth intervals of above and below 200 m by Poteri and Lairinert [1997] is based on earlier studies showing a statistical difference in the hydrological data between the upper and lower parts of the rock [e.g., Niemi and Kontio, 1993]. In generating the fractures, the enhanced Baecher model [Baecher et al., 1977] was used. The fracture centers are located uniformly in space, following a Poisson process. The enhanced Baecher model allows provisions for fracture terminations and general fracture shapes [Derschowitz et al., 1996a]. The uncertainties related to geometrical fracture statistics were not investigated here, but these properties are assumed fixed. Emphasis is placed on studying the hydrological data Hydrological Data For determining the hydraulic properties of the fracture networks, the fixed-intervalength hydraulic well test data from different scales is used. The rationale for utilizing these data was as follows: (1) The data from the finest scale (2-m data) are used for obtaining an initial estimate for individual fracture transmissivities. (2) The data from the 10-m scale are used as the basis for the hydraulic fracture network calibration. Exten- sive numerical simulations are carried out with fracture net- work model to obtain the best estimate for fracture transmis- sivity distributions that would yield the observed 10-m well test results. (3) The 30-m scale constant head injection test data are used as comparison material for the conductivities determined with network simulations of the same scale. Histograms of the log transmissivities for the 2-m and 10-m data at the two depth intervals are given in Figures 2a-2d. From Figure 2 it can be observed that the portion of the nonconductive sections is significant in all of the data sets, being larger with the small test interval and in the deeper part of the bedrock. The higher number of nonconductive sections in deeper parts of the rock is due to an actual decrease of conductivity with depth. This is a well-known phenomena in this type of fractured crystalline rock where the increase in lithostatic pressure tends to close the fractures. The higher number of nonconductive sections with the smaller test inter- val, however, is due to the fact that the higher testing interval "averages out" heterogeneity. Only one fracture can create a "nonzero" hydraulic conductivity value in the 10-m-scale data, while the same fracture would create one "nonzero" and four "zero" conductivity values in the 2-m-scale data. 1 < sin O' (1) f(o', (k') = 2rr (e K- 1) exp ( cos 0'), 0-<0 '-<rr/2, O-<qb '-<2rr Initial Estimates for Fracture Transmissivity and Storage Properties The first estimate of fracture transmissivities is based on the observed fracture densities and hydraulic conductivity mea-

5 NIEMI ET AL.' HYDRAULIC CHARACTERIZATION OF FRACTURE NETWORKS 3485 surements at the 2-m scale. On the basis of the approach developed by Osnes et al. [1988] and implemented by Dershowitz et al. [1996a], we assume that the transmissivity of each test interval is equal to the sum of the conductive fracture transmissivities in that interval. In other words, the fractures are assumed to be independent of each other: Ti = T,j, (3) j=l h e- fn(n) = n! ' (4) Table 2. Statistics of Measured and Best Fitting Modeled 2-m Scale Fixed Interval Length (FIL) Data Along With Fracture Transmissivity Data Yielding the Fit a 2-m FIL Data Parameter Simulated Measured Mean 3.5 x 10 -s 5.9 X 10-9 Standard deviation 3.5 X x 10 -s Logt0 mean Logm std deviation Skewness Nonconductive, % where Ti is the transmissivity of the ith packer interval, t/i is the number of conductive fractures in the interval, and Tii is Fracture Parameter Value the transmissivity of the j th conductive fracture within the i th Best fitting fracture transmissivity b interval. Within any given interval the number of conductive m = fractures t/i is assumed to be a random number following a Fracture density Poisson distribution o-= fractures m - amodel assumes fractures are independent. Borehole depth is >200 m. bvalues are calculated as logm of fracture transmissivities. where h is the Poisson process rate and equal to the expected S = T, (5a) value of n. The conductive fracture frequency is given by fc = n/li, where L i is the length of the test zones. The distribution S = x/t, (5b) of fracture transmissivities T o is described by a lognormal distribution, defined with a mean and standard deviation. In s = c the numerical implementation of Fracman used here [Der- where C is a constant depending on the physical properties of showitz et al., 1996a], for a given set of the parameters describ- water and aperture characteristics of the fractures. The first ing the transmissivity distribution f(to) and conductive fracture frequency fc, the distribution of packer interval transmissivities f(t ) is found by Monte Carlo simulation, and the best fitting value is found by simulated annealing search routine. The measured and the best fitting modeled density histograms for the 2-m interval test data are shown in Figure 3. The corresponding parameter values are summarized in relation is based on the observation that in several cases shown in Table 3, the diffusivity,/ = T/S is of the order of 1, with storativity with respect to either a mean transmissivity or a fixed single-value transmissivity. The second relation is the empirical relation by Uchida et al. [1994] based on field data from the Asp6 Hard Rock Laboratory in Sweden. The third relationship comes from the definition of storativity, Table 2. Transient fracture network simulations also require infor- S = pgb(a + qb/3), (6a) mation on fracture storativities (S). Such data do not exist for where p is the density ofwater [M 3 L-3], g is acceleration due the average rock from the site. Therefore values reported in to gravity [L T-2], and b is the characteristic aperture [L], in the literature for similar sites were reviewed, and a summary is this case the aperture controlling the fracture storage propergiven in Table 3 along with the corresponding ranges of frac- ties,/3 [L M-1 r 2] is the compressibility of water, a [L M- ture transmissivities T [L2/T]. In the present study, three T 2] the compressibility of the medium, and 4> is the fracture relationships are tested for fracture storativities porosity. Assuming the medium is incompressible and setting the porosity equal to 1, that is, assuming a fully open fracture, we get Original Ti Simulated Ti Figure 3. Cumulative frequencies of measured and best fitting modeled 2-m fixed interval length (FIL) data with the Osnes et al. [1988] model used to get the first estimate for fracture transmissivities. Fractures are assumed to be independent (d > 200 m). S = pgbstorage. (6b) The aperture bstorage is the volumetric aperture controlling the storage properties and is usually larger than the hydraulic aperture controlling transmissivity. The hydraulic aperture (bhydr), in turn, can be determined from fracture transmissivity T [m 2 s- ] based on the well-known cubic law for laminar flow in open parallel-wall fractures [Witherspoon et al., 1980] pg b hydr r = (7) /x 12' Deriving hydraulic aperture from this expression, (5c) can be derived from (6). A ratio bstorage/bhydrauli c: 10 was used here, resulting a value of C = 4.4 x 10-7 [(S m-2) /3]. Obviously, in natural fractures this ratio can have a wide range of values, depending on the variability of the aperture structure of the fracture in question. A comprehensive overview and discussion concerning the meaning of different equivalent fracture apertures and their interpretation from hydraulic and

6 3486 NIEMI ET AL.: HYDRAULIC CHARACTERIZATION OF FRACTURE NETWORKS Table 3. Fracture Storativity and Corresponding Transmissivity Data Used in Various Fracture Network Simulation Studies Reference T (m, rr) a S Site Derschowitz et al. [1991] Long et al. [1992] Uchida et al. [1994] Winberg [1996] La Pointe et al. [1995] 2 x 10-7, 4 x x 10-8 Stripa crown fractures 1 x 10-8, 5 x x 10-8 Stripa nonzone 2 x 10-8, 4 x x 10-8 Stripa fracture zone 1 x 10-7, 3 x x 10-8 Stripa fracture zone 1 x 10-8, 1 x x x 10 -Sb 6 X x 10-4 Stripa fracture zone 2 x x 10 -sc 1 x 10 -s S..tripa fracture zone 4 x 10-7, 7 x T ø's.a. sp6 nonzone - 8.4, 0.9 d T ø's.a. sp6 nonzone 9 x 10-7, 5 x X 10-6 Asp6 nonzone athis is a lognormal distribution. ba range of values is indicated. CA channel conductance is indicated; units are in m 3 s-. dparameters m and rr are in log w space. tracer studies is given by Tsang [1992]. Hakami [1995] documents aperture data for a number different types of fractures as measured in laboratory conditions Autocorrelation in Hydraulic Conductivity The autocorrelation structure of hydraulic conductivity can be estimated by means of variogram analysis of the well test data. This is usually expressed in terms of a so-called semivariogram [e.g., Journel and Huijgbregts, 1978] for a general ran- dom variable Z: = + h)- Z(x)]2}, (8) rather than the correlation structure within the conductive sections. The resulting variogram, determined with the model of Englund and Sparks [1991], is given in Figure 4. It shows a range of 8 to 10 m. The variogram sill captures about 85% of the corresponding sample variance (0-2 = 1.17) which can be considered a good agreement in the case of real field data. 3. Numerical Model for Fracture Network Flow Simulations In the analyses, transient groundwater flow in threedimensional networks of fractures is considered. The flow in each fracture is modeled according to the continuity equation for saturated groundwater flow in two dimensions Oh rv h = S 7 + q' (9) where V 2 is the Laplacian operator, h is hydraulic head [L], q refers external sources and sinks [L T- ], and t is time [T]. For modeling the flow in the three-dimensional networks of two-dimensional fractures, the MAFIC (Matrix Fracture Interaction Code) model [Miller et al., 1995] was used. The program uses the Galerkin finite element approximation for the numerical solution. It is interfaced with the Fracman fracture where 3/is the semivariogram, E is the expected value, x is the position vector, and h is the distance vector between two points. Here the variograms were determined from the 2-m network generation program via a specific mesh generation routine [Dershowitz et al., 1996a]. The models have been tested and verified against numerous test cases and applied to several hydraulic conductivity data. Various subsets were considered field data studies (for summary see Dershowitz et al. [1996b]). by Niemi et al. [1999], but only the main findings will be addressed here. Figure 4 shows what could be considered as the For the present study, the mesh generation routines were somewhat enhanced, and some additional test simulations most representative variogram for the 2-m scale conductive were carried out by studying the performance of the algorithms data in the average "nonzone" rock. In the underlying data all the boreholes are combined into one data set, and only variain well-defined cases where the results could be compared against analytical solutions. This was done to investigate the tion along the borehole length is observed. This was done in order to make the data sets sufficiently large to produce representative variograms. Data from both depth intervals (d < 200 m and d > 200 m) is combined for the same reason. In this type of crystalline rock, where the rock type remains the same and the decrease in overall permeability with depth is due to the continuous increase of lithostatic pressure, there is actually no particular reason to expect the autocorrelation to be especially sensitive to depth. So in this respect combining the data can be seen to be reasonable. Furthermore, quite a significant part of the data that are at the detection limit are excluded, because including them yielded variograms that reflect the extent of the intact or "nonmeasurable" sections effect of various numerical factors on the simulated results so [ [ [ [ [ Lag (m) Figure 4. Variogram of the 2-m flow meter data when data at detection limit is excluded.

7 NIEMI ET AL.: HYDRAULIC CHARACTERIZATION OF FRACTURE NETWORKS 3487 that the most optimal computational options could be selected the borehole walls of that section, representing the constant for the actual network simulations. These test simulations are head injection. As in the actual tests, an overpressure of h = presented in more detail by Niemi et al. [1999]. In the test simulations a constant-head injection test in an isolated horizontal fracture intersecting the borehole was con- 20 m (expressing the pressure in terms of hydraulic head) is used in the borehole for the duration of 15 min. The injection flow rate required to maintain this overpressure is observed. sidered. The simulated result was compared to the correspond- The upper and lower boundaries, as well as the outer bounding analytical solution by Jacob and Lohman [1952]. The discretization density, type of numerical elements, and boundary conditions at the outer boundary were varied. The results were aries, are assigned a specified head boundary condition of h = 0 m. These outer boundary conditions should obviously not affect the behavior of the test, which will be tested before the as expected, the dense meshes with quadratic elements yielding actual calibration simulations. the most accurate results. The difference between the most and least accurate cases was, however, small, the differences 4.2. Simulations for Percentage of Nonconductive from the analytical solution being by a factor of 1.01 and 1.2, respectively. For the outer boundary both no-flow and specified head boundary conditions were used. As should be the Fractures The computational effort related to simulating realistic fracture networks is significant. With the present data a typical case, this did not affect the solution. It did, however, affect the number of fractures in one cube like that shown in Plate la is computational time. The computational time in the most extensive case (quadratic elements with dense discretization and specified flux outer boundary condition) was more than 100 times that of the least extensive case (linear element, base discretization, and specified head outer boundary condition). of the order of 40,000 fractures. When discretized, this would typically result in over 420,000 numerical elements. As for each calibration run, a large number of realizations are needed; the computational effort becomes practically unfeasible. For the data in question a large portion of the fractures are of very low In stochastic fracture network simulations the computational conductivity. Excluding them from the simulation would not effort is a major limiting factor. Therefore the most timeeffective combinations for boundary conditions and discretization were chosen for later use; that is, the specified head outer boundary condition along with linear meshes will mostly be affect the result, considering the detection limit of the calibration database. To investigate how large a part of the fractures could be excluded without affecting the results, a set of test simulations was carried out. In these the well test was simuused in the later simulations. The observed factor of 1.2 is lated by progressively excluding a larger percentage of the least small in comparison to the other effects studied here. conductive fractures. This was done for four example realiza- 4. Hydraulic Calibration of Fracture Networks tions [Niemi et al., 1999]. The fracture transmissivities were assumed to follow a lognormal distribution with a mean and variance as shown in Table 2, while the fracture storativities It is well established that the geologically observable fracture were based on (5b). According to these simulations, 80% of the networks are not equal to the hydraulically active networks, least conductive fractures could be excluded without affecting and hydraulic information is needed to derive these properties. the calibration result. Therefore a 75% cutoff limit was chosen Cacas et al. [1990] used the 2.5-m lugeon test data to statistically calibrate their fracture network transmissivities. Our approach is in principle similar to theirs. However, instead of looking for only one acceptable result as done by Cacas et al., we extend the approach and look for the entire range of values that produce an acceptable agreement. The basic procedure to be used here is as follows: (1) On the basis of statistics on fracture geometry in Table 1, numerous different but statistically similar fracture networks are generated in a 30 m x 30 m for the actual calibration simulations, and only those fractures belonging to the most conductive 25% were included in the analyses. In some studies related to fractured media (e.g., long-term pressure response tests), exclusion of low-conductivity fractures that connect two large-conductivity features may affect the result and even lead to misinterpretations. In the type of injection tests we are looking at here, however, the response can be expected not to be sensitive to the lowconductivity features, which is also demonstrated by the above x 30 m cube. The initial estimate for the hydraulic properties sensitivity studies. of the networks is based on 2-m-scale hydraulic data as described in section 2.4. (2) In each network realization a tran Response of Fracture Networks During an Extended sient hydraulic well test is simulated according to the test Well Test specifications of the 10-m interval constant pressure injection The transient well tests used as a calibration basis are of tests. (3) The resulting statistics of the simulated well tests are limited duration and describe the distribution of hydraulic compared with the corresponding statistics from the measured conductivity around the borehole. Boundary effects are seldom data. (4) The distributions of individual fracture transmissivi- seen, and it has to be a requirement of the calibration simuties are adjusted until an agreement between the measured and lations as well, both from the numerical and physical point of modeled statistics achieved. The previou steps are repeated view, that the selected outer boundary conditions do not affect a number of times for a range of fracture transmissivity statis- the result. As a preliminary test, however, it is of interest to tics to find all the parameter values that produce the observed simulate the well tests long enough for the specified boundwell test statistics. aries to be reached. This is to gain confidence in the model 4.1. Model Specifications performance and to see the general shape of the injection flow curves in complex fracture networks during such an extended Plate la shows the conceptual model used in the calibration simulations using 10-m data, along with an example fracture injection experiment. A number of realizations were simulated. The constant head geometry realization. As shown in Plate la, a borehole intersects the block, and the central 10 m of the borehole are "isolated." A specified head boundary condition is imposed on injection was continued until the outer boundary was reached or, alternatively, when a specified maximum simulation time limit was reached. The results are shown in Figure 5. Several

8 3488 NIEMI ET AL.: HYDRAULIC CHARACTERIZATION OF FRACTURE NETWORKS realizations show the development of an S-shaped curve with a straight-line slope. This straight-line slope can be used to determine the overall effective conductivity of the block according to the well test interpretation methods developed for homogeneous materials (see, e.g., de Marsily [1986] for a summary). Later, usually after a few hours, the effect of the specified head boundary condition can be seen as flattening of the curve, indicating that the injection pulse has reached the block boundaries. While some curves show an almost homo- geneous behavior before reaching the boundary, in some cases there is also a change in the slope of the curve. This indicates that the limiting conductivity varies during the experiment, because of the complex interactions between the lowconductivity and high-conductivity fractures. The hydraulic conductivities that could be determined from the "straight-line slopes" of some of the ideal S-shaped curves in Figure 5 cannot be compared to the conductivities determined frbm the experimental well tests. The former would represent larger rock blocks than the experimental data. The duration of the actual '- 2E+8 OE+O tests was only about 103 s, while the development of the Time (s) "straight-line slopes" in the simulations in Figure 5 typically Figure 5. Simulated injection flow rate (as inverse of Q) as a takes much longer than that. It is, however, important to note function of time, for example fracture network realizations. that for the time range where comparisons can be made (t < The vertical line at t s corresponds to the testing time 103 s), the general behavior of the simulated and measured in measured data. curves was similar [Niemi et al., 1999]; this similarity could be seen in the general appearance and distribution of the curves on the log 1/Q (Q represents flow) versus log time graph, including the fact that boundary effects were not observed in either data set. The curves in Figure 5 also demonstrate that a unique depact fracture storativity has on the results. This is expected as storativity determines the radius of influence of the tests and thereby the characteristics of the rock tested. Comparison of the simulated and measured values shows that assuming S cr T termination of porous medium-type "standard" conductivity and S 3X/-f both produce values below or close to the values is in many cases difficult for this type of data, and there will necessarily be subjectivity in such interpretation. These standard interpretation methods, such as the commonly known Jacob-Lohman solution [Jacob and Lohman, 1952], employ the assumption of radial flow in homogeneous porous media. The equipment detection limit (large column in Figure 7), while almost entirely failing to produce the higher values whose proportion in the measured data is about 40% (Figures 7 and 2d). The relation S X/-f also produces higher values and gives a range of values in closest agreement with the measured conductivity value is then determined from the "straight-line" data. As this relation, unlike the other two, is also based on slope of the injection flow rate versus time curve. In most of the cases in Figure 5 such "straight-line" behavior is not yet reache during the 103 S period of the actual well tests. Because of the uncertainties and subjectivity involved in the interpretation of hydraulic conductivity in the case of this type of field measurements of diffusivity, it was selected for the subsequent simulations. Even in the best fitting case, the simulated values are smaller than the measured ones. To improve the fit, the effect of increasing standard deviation of fracture transmissivities was data, this parameter was not used as a basis when comparing first tested as shown in Figure 7. For readability the simulated measured and modeled results. Instead, the total injection flow (Q tot) was used. This quantity is directly measured, unlike hydraulic conductivity that is a derived quantity, and is theredata in Figure 7 are given as Gaussian fits of the simulated histograms, while the measured data are given as the actual histogram. In these simulations the number of realizations per fore uniquely defined for both the simulated and the measured each simulation set was increased to n - 49 to be in accordata sets. With the total injection flow we mean the total dance with the number of measured observations. As can be volume of injected water during the first 15 min of the experiment. The actual target time for the constant pressure injection tests was 20 min, but as all of the tests did not actually last this long, the 15-min cutoff value was used to make the number of observations as large as possible Hydraulic Calibration of Fracture Networks In the first actual calibration study the effect of the storatseen from Figure 7, a large part of the data falls at the detection limit of the equipment. In numerical simulations, values smaller than this limit are also generated, while in field data all the small values fall into the detection limit category. Visual inspection of the results in Figure 7 shows that increasing the standard deviation indeed improves the fit in terms of the overall distribution of values by producing values both at the higher and lower ends of the spectrum in a fashion similar to ivity relation was examined. The three relations given in (5a)- that observed in the measured data. (5c) were used along with the initial estimate for transmissivities. The resulting simulated histograms, based on Search for the Range of Acceptable Values realizations each, along with the fitted Gaussian models are While a visual inspection of the simulated and measured shown in Figures 6a-6c. Figure 7 shows the corresponding distributions in Figure 7 shows a good agreement, the essential measured data. The large difference between the flow rate questions remain: (1) How good is the agreement considering distributions in Figures 6a-6c demonstrates the significant im- that both the data and the simulated results are based on a 5E+8 4E+8 ' 3E+8 OE+O 1'E+ 1E+21E+31E+41'E+51E+61'E+7

9 lo Log (Kxx) Lea (Kyy) Plate 1. (a) Example network realization used in the 10-m scale well test simulations: Model dimensions are 30 m x 30 m x 30 m; boundary conditions are h(x, y, z, t) = 20 m at the 10-m borehole injection section and h(x, y, z, t) = 0 m at the outer boundaries. (b) Diagonal components of conductivity tensors for 30 realizations (bars indicate conductivity interval based on flux values across the two opposite surfaces, red symbols indicate tensor components based on least square fitting, blue circles indicate positive definite criterion also used for the tensors, and green squares indicate both positive definite and symmetricriterion used for the tensors).

10 3490 NIEMI ET AL.: HYDRAULIC CHARACTERIZATION OF FRACTURE NETWORKS a)._z, 0.8 o a. 0.2 b)._ " I I!! li I! Log(Q)... I I! I! I I I I , Log(Q) limited number of about 50 observations in both cases? To exaggerate, is the good fit just a "coincidence" explainable by a small number of observations? (2) Are there many other combinations of fracture transmissivity parameters that would produce equally acceptable results? The measured data are only one limited sample of reality. It is not important to capture all the details of this individual sample but rather to capture what can be considered representative taking into account the number of observations on which the data are based. Furthermore, it is important to find all the fracture transmissivity distributions that produce equally good fits. To investigate this, a number of Monte Carlo simulations with different input statistics were carried out. Altogether 34 sets each with 50 realizations were simulated. In each case the fracture log transmissivity was assumed to follow a Gaussian distribution, characterized by a mean and a variance. These parameters were varied. The goodness of fit between the modeled and the measured injection flow distributions was determined by means of the bootstrapping method [e.g., Mooney and Duval, 1993]. This is a nonparametric resampling procedure to estimate a parameter from a population from which we have a sample with a given number of observations. The approach does not make any assumptions concerning the shape of the underlying distribution. The 90%, 95%, and 99% confidence intervals determined with this method are shown in Figure 8 along with the measured values which are shown as black circles. The simulated results are shown as solid curves. The accepted and rejected cumulative curves are shown in Figures 8a and 8b, respectively. For the acceptance criterion the 90% confidence interval, i.e., the innermost region in Figures 8a and 8b, was used. Accepted cases are those for which the cumulative histogram remains entirely within the 90% confidence interval, i.e., inside the innermost shaded region. Rejected cases are those for which some part of the curve falls outside this region. Varying the standard deviation had a tendency to affect the slope of the curve, while changing the mean maintained the c) ( = ( =2.6 o Log(Q) Log (Q,o,) Figure 7. Effect of fracture transmissivity standard deviation Figure 6. Effect of fracture storativity on simulated injection on simulated well tests: histogram of measured data along with flows (15-rain injection period): histograms and fitted Gaussian two simulated data sets. Simulated results are shown as Gaussmodels from 30 simulated 10-m FIL constant pressure injec- ian fits of simulated histograms, S oc /, in both cases. tion tests (a) S oc T, (b) S oc /, and (c) S oc 3 /. (Notice that the high column at Q tot ' refers to all values at or below the equipment detection limit.)

11 ß ß ß ß ß NIEMI ET AL.: HYDRAULIC CHARACTERIZATION OF FRACTURE NETWORKS 3491 slope but moved the overall position of the curve in the horizontal direction. In Figure 9 all the tested mean and standard deviation combinations are shown on one graph. Inspection of the results shows that a range of acceptable mean values can be found along with a range of acceptable standard deviation values. The preliminary transmissivity estimates based on the Osnes et al. [1988] approach produced clearly too small flow rates. The simulated well test curve corresponding to this preliminary estimate is seen in Figure 8b as the curve in the 3.6 '. 3.4 Ro calibration, used parameters for Log (T) 4 -i :... e: : i : ' i : i :...'... :.. :...'...:... '... ß. : : : ß : : : 9 ee e. ß <;) : : : : : ß. : i 6o6e i i : ß ß ; : : '. :... ;... < o. 6. <)..6.e...e. 6..o a) Ro calibration -acce =ted CDF$ -o 3.2 ß : :... ß... '., ß ß... l l ;... :... ß ß :... :... ;... ' i... :......, 2.6 ' ' ' ; ; ' ; ' ; ' Mean Figure 9. Accepted (solid circles) and rejected (open circles) combinations for fracture transmissivity mean and standard deviations. Results are based on stochastic fracture network simulations of 10-m-scale injection tests. Acceptance through comparison with measured distribution by means of bootstrapping criteria (90% level) as shown in Figure 8 is indicated. b) a.s -a -2.S -2 Log(Q ) Ro calibration - rejected CDFs uppermost left corner of the graph. This could reflect the fact that in the Osnes analysis the fractures are independent, which is not a realistic presentation of a true network system. Such an assumption may underestimate the true variability of the fracture conductivitiesince a network can be expected to smooth out the local heterogeneities. 5. Upscaling of Network Properties The number of fractures in the previous network simulations is already at the limit of reasonable computational effort. Methods are needed to "upscale" this behavior, that is, to find simplified presentations that reproduce the observed flow of the detailed network. This is needed both for general site understanding and as direct input for large-scale models. If the upscaling of fracture networks to an "intermediate" scale (i.e., the scale of variability of large-scale models) shows continuumlike behavior that can be adequately described by means of continuum conductivity tensors, distributions of these tensors can be used as input for large-scale stochastic continuum Monte Carlo simulations. If the data do not meet the contin- uum criteria, other approaches are needed such as approaches for determining "equivalent discontinuum" properties Log(Q o ) Figure 8. Cumulative probabilities of measured injection flows of 10-m constant pressure injection tests (circles) along with the 90%, 95%, and 99.9% confidence intervals obtained with the bootstrapping method (shaded areas) and with cumulative probabilities based on simulated tests: (a) accepted and (b) rejected cases Properties of Hydraulic Conductivity Tensor and Criteria for Continuum Behavior at the Intermediate Scale For anisotropic porous medium, Darcy's law can be written as Oh (lo)

12 3492 NIEMI ET AL.: HYDRAULIC CHARACTERIZATION OF FRACTURE NETWORKS where q i are the components of the specific discharge or the simulated from three different directions. The direction is var- Darcy velocity vector, K o are the components of the hydraulic ied so that each pair of opposing faces of the cube, in turn, is conductivity tensor, and Oh/Oi are the components of the hydraulic gradient vector. For a homogeneous porous medium the hydraulic conductivity tensor has to be symmetric with respect to the diagonal (i.e., Kxy = Kyx, Kzx = Kxz, and Ky z -- Kzy ), and it can be transformed into a diagonal form by a rotation of the coordinate axes. Anisotropic porous media folassigned the maximum head difference. For hydraulic properties the calibrated values from section 4.4 were used. More precisely, values falling in the middle of the acceptable region in Figure 9 are selected, with a mean of log transmissivity m = and standard deviation tr = 3.6. As these simulations are steady state, no values are needed for low this concept. Graphically, the tensor plots as an ellipsoid in fracture storativities. The same 30 network realizations are three dimensions and as an ellipse in two dimensions, with the major and minor axes corresponding to directions of minimum used as before, but the inactive fractures to be removed are readjusted according to the new problem specifications. In the and maximum permeability. When looking at whether a frac- calibration simulations the fractures that were not connected ture network behaves as a continuum, the properties of the conductivity ellipse have been used as a criterion [Long et al., 1982; Cacas et al., 1990; National Research Council, 1996]. The conductivity ellipse is then determined by simulating the steady state flow through a network block and by determining the to the borehole were removed, as they did not influence the flow. Here they are not excluded. However, isolated fractures that had no connection to any of the boundaries, including connections through other fractures, are excluded. Such fractures are completely isolated in terms of the overall block effective hydraulic conductivity based on the one-dimensional conductivity and do not influence the solution physically. They form of Darcy's law according to the imposed hydraulic gradient and the observed total flow. The direction of the main flow is varied with respec to the network and the conductivity (or more precisely 1/x/K); for each direction is plotted as a function of the rotational angle. The closer to a perfect ellipse this plot is, the closer to a continuum the system behaves. There is no guarantee that fractured media will behave like this. Long et do, however, create problems for the iterative solution procedure. With these specifications the flow field is simulated for 30 network realizations altogether, with three simulations corresponding to the three main directions of flow for each. The Darcy flux inside the simulation cube (q) can be determined based on the observed flow rates through the six faces of the cube (Q') according to the equation al. [1982] looked at different fracture systems in two dimensions and found that a perfect ellipse was, indeed, often not aq = Q', (11) found for realistic fracture systems. The factors favoring the where the Darcy flux is q = {qx, qy, qz}, Q' = {Q /A, continuum-type behavior were high fracture density and mixed Q 2/A 2... Q 6/A 6 }, with A being the cross-sectional areas of directions of the fracture orientations. Cacas et al. [1990] the block surfaces and Q i being the flow rates through them. looked at the permeability ellipse of fracture network data A is the coefficient matrix A = {aik}(j = 1,..., 6 and k = from Fanay-Augeres mine and found relatively well behaving x, y, z), where a x, a y, and a z represent the components of ellipse-like shapes. The Fanay-Aurgeres data were character- the unit vector normal to surface j. ized by a relatively high overall conductivity and well- When simulation directions are selected to coincide with x, connected networks and came from relatively shallow depths y, and z, the (11) has three unknowns (the components of of a few hundred meters. It is of interest to note, however, that vector q) and six equations corresponding to flow through each the model failed to reproduce the lack of hydraulic intercon- of the block surfaces. This means that, for example, for the nection observed at the site. The reason for this has been component qx we have two estimates corresponding to the two proposed to be the fact that the network model was based on flow rates Q i through the two opposite faces of the cube perapparent rather than conductive fracture geometry [National pendicular to x direction. If the medium were a perfect homo- Research Council, 1996]. geneous porous medium, these should be equal to each other. Another criterion for continuum behavior, which actually In the case of heterogeneity and spatially variable qx inside the follows from the ellipse/ellipsoid requirement, is that the flow cube, the two values differ. How much they differ can be taken coming in through one face of a block is equal to that going out as a measure of deviation from the homogeneous porous methrough the opposite face. The three-dimensional networks dium behavior. studied here are computationally extensive. Therefore, as a Knowing the components of the Darcy flux based on (11), starting point, we use this criterion to evaluate the continuum- the components of the permeability tensor can be solved from like character of the networks. This allows the use of somewhat (10). If no assumptions are made concerning the symmetric fewer simulation directions per realization than would be re- properties of the tensor, (10) has nine unknowns and three quired when determining the full ellipsoid. equations. Therefore a minimum of three sets of simulations, corresponding to three different sets of hydraulic gradient, is 5.2. Simulations for Block Conductivities needed to solve the nine components of the conductivity tensor. The objective of these simulations is to look at the validity of the continuum behavior of the network blocks and to deter Continuum Characteristics and Tensor Properties mine the equivalent continuum conductivities that would re- Plate lb shows the calculated diagonal components of the produce their hydraulic behavior. The calibrated 30-m fracture conductivity tensors for the 30 realizations. These are marked network blocks were used in the simulations. The steady state with three different symbols, depending on the mathematical flow through the blocks is simulated by imposing constant criteria used in their determination. These will be discussed specified head boundary conditions on opposite sides of the later. More importantly, Plate lb also shows the conductivities cube along with linearly varying specified head boundaries determined based on the flow through each individual face of along the other four sides. Consequently, the external potential the simulation block. According to the notation of Long et al. gradient is as constant as possible, as required by Darcy's law [1982] we call this conductivity K a, i.e., hydrauliconductivity and continuum approximation. For each realization the flow is in the direction of the imposed gradient,

13 NIEMI ET AL.: HYDRAULIC CHARACTERIZATION OF FRACTURE NETWORKS 3493 Kai--Qi/(AhAi) i= 1..., 6, (12) where Ah is the imposed hydraulic gradient. For each main flow direction, there are two faces yielding two values for this estimate. These are shown as the ends of the grey bars in Plate lb. The closer the values are to each other, the shorter the bar is. For continuum approximation to be valid, the two values of K a should be equal. Inspection of the K a values in Plate lb shows that for the present data the K a conductivities differ in most cases quite significantly, often by orders of magnitude. This indicates that the behavior of the cubes cannot be well presented with a continuum tensor, and the use of a continuum tensor is not justified. For demonstrative purposes these are, however, also given in Plate lb. Values indicated with red are determined based on the three simulations and (10) and (11) without any constraints for the characteristics of the resulting matrix. The blue symbols correspond to the corresponding values when the matrices were in addition required to be positive definite, and the green square symbols correspond to the case where they were required to be symmetric as well as positive definite. Inspection of the results in Plate lb indicates the following: (1) There appears to be no clear anisotropy; that is, the conductivity values are of the same order of magnitude in all directions. (2) In most cases, there is very little difference between the diagonal terms of the three different types of matrices, and the different symbols fall on top of each other. (3) The calculated values of the diagonal terms are always very close to the upper one of the two K a values. This is an artefact From of both the solution method and the fact that the two values of Well Tests the Darcy flux corresponding to the two opposite faces of the cube differ so much. The set of the six equations (equation (11)) was solved with the method of least squares. In doing so, one searches for the solution by minimizing the sum of squared differences. For example, in the case of Darcy flux in the x direction the quantity (qx - Qxl/Axl) 2 + (qx- Qx2/Ax2) 2 is minimized. When doing so with values that differ even by orders of magnitude, the higher value is automatically overemphasized. An example of a conductivity ellipsoid corresponding to a continuum-like behavior with a small difference between the... ".... m ;.: : :.. e : ¾... r... '" Figure 10. Conductivity "ellipsoid"(1/ K) for realization number (a) 4 and (b) 29 in Plate lb. (Notice that the Presen - tation in Figure 10b is partly "truncated" as othemise 1/ K would become ve large when K becomes ve small.) 6. Comparison of Fracture Network-Based Conductivities With Conductivities It is of interest to compare the results of the network simulations to the well test results of the same scale. In this case these are the constant-pressure injection tests with 30-m fixedinterval testing length. From these data the hydraulic conductivities have been determined with different standard interpretation formulae, using the method by Jacob and Lohman [1952], the Horner method [Uraiet and Raghavan, 1980], and the method by Moye [1967]. All of these standard interpretation methods assume continuum behavior with radial or radial- to-spherical flow geometry. In "classical" stochastic continuum analysis [Neuman, 1987; Gomez-Hernandez and Gorelick, 1989; Tsang et al., 1996] such data are used directly as the basis for generating input conductivity distributions for Monte Carlo type simulations. The information from the well tests can be seen as repre- K a is given in Figure 10a. This is realizationumber 4 in Plate lb. An example with large deviation from this behavior, i.e., realization number 29 in Plate lb, is given in Figure 10b. Only three simulation directions were used. It is possible senting a one-dimensional sample of the rock conductive properties along the borehole. It is of interest to see how well it that these happen to present especially heterogeneous direccorresponds to the conductivities and resulting flow rates tions not representative of the medium. Then choosing a finer through two-dimensional surfaces in heterogeneous systems. angle of rotation would result in a more ellipsoid-like appear- In homogeneous systems these values are equal to one anance in the heterogeneous realizations. This is, however, quite other. unlikely as we already have 3 x 30 samples of the K a perme- Although the present data do not appear to be well approxabilities in the selected scale, the vast majority of them indiimated by a continuum conductivity tensor, the Ka values in cating heterogeneous behav. ior. From the above results it is Plate lb can be taken as samples of some type of "block apparent that the least square approximation used to average interface" conductivity in the 30-m scale. Conductivity then the Darcy flux through the cube is also not suitable. Another refers simply to a proportionality coefficient linking the imaveraging method could be chosen that does not so apparently posed hydraulic gradient, flow rate, and surface area. These overemphasize the effect of higher flux. The overall large dif- values were plotted as cumulative histograms by taking the Ka ference between the K a values at oppositends of the cube, values for each computational direction as an independent however, indicates that the continuum approximation is not a sample. For conductivity perpendicular to the x direction all 60 good presentation for the data and scale in question in general. values (both ends of the bars) in the first column in Plate lb are Therefore alternative averaging methods were not investigated combined into one data set. The resulting cumulative histofurther. gram is shown in Figure 11a along with the corresponding

14 3494 NIEMI ET AL.' HYDRAULIC CHARACTERIZATION OF FRACTURE NETWORKS (a) 0 '1,,, I I I [ I -8 Log (K); m/s - Kg (xx) -- all tests I [! I I -4 cumulative histogram based on the 30-m well test data. A comparison of the two curves in Figure 11a indicates that the conductivities determined from the well tests are much lower. This can be seen also in the statistics of Table 4. In Table 4 the statistics for the block interface conductivities are given for all three directionshown in Plate lb. All directionshow clearly higher statistics than the well test data. Figure l lb also shows the same comparison when the 30-m well test data only come from boreholes KR3 and KR4. These are the only boreholes from which both 10-m data used for calibration and 30-m data have been systematically measured. As can be seen, the behavior is very similar to that in Figure 1 la, the discontinuity in the middle of the well-test curve being an artefact due to a small number of data points rather than any statistical difference. In Figures 12a and 12b the databases from boreholes KR3 and KR4 are compared to the total databases in the case of both the 10-m data and the 30-m data. I urn (lata -I ' (b) I ' ' ' I ' ' ' I ' ' I I 1._1,,,, ' (a) t Kg (xx) - KR3 and KR4 0.2 all KR3 and KR ,,, I, Log (K); m/s Figure 11. Comparison of cumulative histograms of simulated 30-m block interface conductivities (K a values based on flux across simulation cube surfaces) and 30-m scale well test results: (a) block interface values versus data from boreholes KR2, KR3, KR4, and KR5 and (b) block values versus data from boreholes KR3 and KR Log (K)' m/s 30 rn data (b) 0.6 Table 4. Cumulative Percentiles of Log-Conductivities for 30-m Scale: Comparison of Well Test Conductivities and Block Interface Conductivities According to Results in Plate lb Cumulative Percentile Well Test Results b Kxx c Kyy c Kzz c : all I It : KR3and KR4 I:l 0!-I,,,, I,,,, I,,,, 1,,,, I... I,,,, I-I Log (K)' m/s allere d > 200 m, with no fracture zones. bthese are 30-m scale constant head injection test data from bore- Figure 12. Comparison of cumulative histograms of well test holes KR2, KR3, KR4, and KR5 based on the Moye [1967] interpre- conductivities from different borehole databases: (a) 10-m tation formula [Niemi, 1994]. data from boreholes KR3 and KR4 versus KR3, KR4, KR7, CBlock interface conductivities Kg are based on flux in the direction KR8, and KR9 and (b) 30-m data from boreholes KR3 and of imposed gradient ("ends" of the bars in Plate lb). KR4 versus KR2, KR3, KR4, and KR5.

15 NIEMI ET AL.: HYDRAULIC CHARACTERIZATION OF FRACTURE NETWORKS 3495 Any differences in Figures 12a and 12b are small in comparison to the differences in Figure 11. This is to be expected, as it has been earlier confirmed that the data from the various boreholes do not differ statistically from each other. This has been tested both for the 30-m scale and 10-m scale data [Niemi, 1994; Niemi et al., 1999]. Finally, it should be pointed out that the 30-m scale well test values referred to are based on the steady state Moye [1967] interpretation equation. This has been shown, at least in the case of the data in question, to produce too high rather than too low conductivity values in comparison to the other standard interpretation methods [e.g., Niemi, 1994]. Even though preliminary in nature, the result in Figure 11 may have important implications. If the difference observed is characteristic of heterogeneous networks in general, it indicates that using continuum-based analyses for clearly noncontinuum data underestimates the flow-carrying capacity of the medium. It appears that the difference observed in Figure 11 is related to the observation geometry. The borehole representing a one-dimensional sample of the medium can be more sensitive to the sampled low conductivities and somehow underestimate the overall conductivity in comparison to the twodimensional reality where more pathways can be formed. To see what kind of estimates could be made based on borehole observations, we can look at the statistics summarized in Table 4. The cumulative percentiles selected correspond to the critical percentiles of a normal distribution; that is, 50% corresponds to G(m); 16% corresponds to G(m - tr), and 84% corresponds to G (rn + tr)). 7. Conclusions The hydraulic characteristics and upscaling properties of low-permeability fractured rocks have been analyzed, based on multiple-scale field data from the Romuvaara crystalline rock s te. It is well accepted that geologically observable fracture networks do not coincide with hydrologically active networks, as only a small portion of the observed fractures and/or fracture surface areas are water-conducting. The characterization of a low-conductivity medium is complicated by, for example, the fact that interference tests are usually dominated by largescale features, providing very little information concerning the average "nonzone" rock. Systematic single-hole data from three different measurement scales were used for estimating the hydraulic properties of the networks. A large number of network realizations were generated corresponding to fracture statistics from the site. Next, the hydraulic properties of the networks were determined by calibrating the networks against the hydraulic data. Starting from an initial estimate for fracture transmissivity distribution based on 2-m test interval data, the 10-m constantpressure well tests were simulated in a number of network In comparison to the initial estimate based on 2-m data, the mean transmissivity had to be decreased and the standard deviation increased to produce a realistic spread of the measured 10-m well test statistics. This may be a reflection of the underlying assumption of the preliminary analysis that fractures were assumed to be independent of one another. Such an assumption may underestimate the true variability of the network conductivitiesince a network can be expected to smooth out local heterogeneities. Then, if such "averaged" heterogeneity is assigned to individual fractures, too low a standard deviation may result. Another possible reason can be autocorrelation in fracture conductivities. In the simulations, fracture conductivities are assumed to be noncorrelated. The preliminary variogram analyses have indicated that there might be systematic autocon'elation in conductivities between neighboring data values, with autocorrelation ranges of the order of 8 to 10 m. Including this type of autocorrelation structure is likely to produce both more pronounced low-conductivity sections and high-conductivity pathways and thus capture the extreme values in the measured data better than when correlation is not considered. Upscaling characteristics of the calibrated 30-m scale network blocks were also investigated. If continuum behavior were valid, continuum conductivity tensors could be determined and used as input for Monte Carlo type stochasticontinuum simulations. Altogether 30 realizations were studied, and for each realization, flow was simulated from three orthogonal directions, with the main gradient in the x, y, and z directions, respectively. The results showed that in most cases, the hydraulic conductivities in the direction of the imposed gradient (Ka) differed significantly when determined based on the observed flow through the opposite ends of the simulation cube. This means that the Darcy flux differs with location inside the cube, and the behavior cannot be properly presented with a continuum tensor. The fact that noncontinuity was observed in most realizations and in all directions indicates that this is an actual characteristic of the rock and not due to an unfortunate selection of the simulation flow directions. For this type of data, instead of equivalent continuum statistics, one should find the statistical characteristics of "equivalent fractures" for the scale in question. These statistics could then be sampled as input for larger-scale Monte Carlo models. Regardless of the validity of continuum tensor presentation, the results of the 30-m fracture network blocks give an estimate of the two- and three-dimensional conductive character- istics of the rock at this scale. The K a values can be taken as samples of "block interface" conductivities, with "conductivity" then meaning simply a proportionality coefficient linking the imposed hydraulic gradient, flow rate, and surface area. These can be compared to the 30-m well test results interpreted by means of standard interpretation formulae. This comparison indicates that the values determined from the well tests are realizations. Assuming a lognormal distribution for fracture much lower, This result, even though preliminary in nature, is transmissivities, a large number of possible combinations of potentially important. It is important, for example, when apmean and standard deviation were tested, and the goodness of plying stochastic continuum simulations directly to well test fit between the measured and simulated well test statistics was data as is done in pure stochasti continuum analyses. Then determined. The result showed that a range of acceptable care should be taken that continuum criterion is, indeed, fulfracture transmissivity distribution parameters could be found filled at the scale in question. It appears that the difference is that produce the measured well test statistics. In the accepted related to the observation geometry. The borehole represents range the mean of log transmissivity varied between and a one-dimensional sample of the medium, and the well tests -15.3, and standard deviation varied between 4.0 and 3.2, with are controlled by the small-scale heterogeneity in the small the increase in standard deviation compensating for the de- circle around the well. Then the low-conductivity values next to crease in mean. the borehole will cut down the conductivity significantly, while

16 3496 NIEMI ET AL.: HYDRAULIC CHARACTERIZATION OF FRACTURE NETWORKS for the block conductivities there is more room for the flow to go around the local low-conductivity points and more pathways can be formed. Further studies with different fracture charac- teristics are recommended to investigate and possibly to further validate this observation and its implications in different fracture systems. Acknowledgments. The work leading to this paper has been supported by POSIVA Oy, Finland; for this support we wish to express our gratitude. We also acknowledge Aimo Hautojfirvi from POSIVA Oy for interesting comments and discussions. We wish to thank Chin-Fu Tsang from Lawrence Berkeley National Laboratory and Jane Long from Mackay School of Mines for their review and valuable comments on the manuscript. Thanks go to Tiina Vaittinen from VTT for her help with some of the graphical presentations. 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