Analysis of hydraulic and tracer response tests within moderately fractured rock based on a transition probability geostatistical approach

Transcription

1 WATER RESOURCES RESEARCH, VOL. 40, W12404, doi: /2004wr003188, 2004 Analysis of hydraulic and tracer response tests within moderately fractured rock based on a transition probability geostatistical approach Y.-J. Park, E. A. Sudicky, and R. G. McLaren Department of Earth Sciences, University of Waterloo, Waterloo, Ontario, Canada J. F. Sykes Department of Civil Engineering, University of Waterloo, Waterloo, Ontario, Canada Received 16 March 2004; revised 8 September 2004; accepted 20 September 2004; published 9 December [1] A transition probability and Markov chain geostatistical approach is applied to synthesize the discrete permeability structure of moderately fractured rock. The approach can infuse either hard or subjective categorical information that is consistent with geological interpretations. The methodology is tested using data collected from the Moderately Fractured Rock (MFR) experiment area of the Underground Research Laboratory (URL) in southeastern Manitoba, Canada. Attributes pertaining to fracture location, frequency, and orientation along an array of boreholes intersecting the MFR experiment area, taken together with results from hydraulic response tests within packed-off intervals along the boreholes, are used to produce conditional stochastic realizations of hydraulic conductivity and effective porosity. Using the generated hydraulic conductivity and porosity realizations, we compare predicted tracer concentrations to the results of measured breakthrough data in a stochastic framework. The results show that solute migration behavior in moderately fractured rock can be successfully characterized and reasonably predicted upon careful error analysis of the results obtained from the various medium realizations synthesized from the conditional categorical descriptions of the fractured crystalline rock. INDEX TERMS: 1832 Hydrology: Groundwater transport; 5114 Physical Properties of Rocks: Permeability and porosity; 3210 Mathematical Geophysics: Modeling; 5104 Physical Properties of Rocks: Fracture and flow; KEYWORDS: fractured crystalline rock, inversion, modeling, tracer tests, transition probability geostatistics Citation: Park, Y.-J., E. A. Sudicky, R. G. McLaren, and J. F. Sykes (2004), Analysis of hydraulic and tracer response tests within moderately fractured rock based on a transition probability geostatistical approach, Water Resour. Res., 40, W12404, doi: /2004wr Introduction [2] Modeling fluid flow and solute transport in fractured rocks has been one of the most important and challenging research topics for the last few decades in hydrogeology, geotechnology, petroleum engineering, environmental science, etc. [National Research Council, 1996]. Various models, based on single continuum, double porosity, double permeability, stochastic continuum, and/or discrete fracture network conceptualizations have been developed and applied to fractured rocks, according to different simulation strategies [Long et al., 1985; Tsang and Tsang, 1987; Berkowitz et al., 1988; Schwartz and Smith, 1988; Therrien and Sudicky, 1996; Ando et al., 2003]. In any approach, the characterization and generation of heterogeneous spatial patterns of hydraulic and chemical property fields (e.g., hydraulic conductivity, effective porosity, dispersivity, sorption/reaction properties, etc.) is a crucial step for modeling flow and transport, and the success of the model relies on how closely it mimics the actual system Copyright 2004 by the American Geophysical Union /04/2004WR W12404 heterogeneity, both geologically and hydrologically. Structure imitating geostatistical methods have been one of the most popular methodologies in this context [Deutsch and Journel, 1992]. [3] The traditional geostatistical approach for estimating the spatial pattern of hydraulic properties in geologic media typically involves the development of a structural model, such as a variogram, followed by some form of data interpolation. However, statistical requirements (e.g., Gaussian distribution, stationarity and ergodicity assumptions) for the analysis are often too strict for many subsurface applications. Moreover, although they may honor the data, the hydraulic property fields produced by traditional kriging of relatively sparse data are generally too smooth and continuous, especially in fractured rocks. [4] Nonparametric indicator (i.e., categorical) geostatistical methods are becoming popular in subsurface hydrology because much geological data (e.g., facies, soil classifications, mineralization phases, etc.) is categorical [Goovaerts, 1996]. Although these procedures have the ability to represent discrete features, geological data sets rarely provide the necessary detail to implement a comprehensive structural model for the conditional simulation 1of14

2 W12404 PARK ET AL.: ANALYSIS OF TRACER RESPONSE IN FRACTURED ROCK W12404 Figure 1. Location of Lac du Bonnet batholith and Underground Research Laboratory. of stochastic hydraulic property fields. Thus incorporation of subjective geological information is inevitable and important. [5] The transition probability-based geostatistical approach, combined with three-dimensional spatial Markov chain analysis, is easy to implement, mathematically compact, and theoretically powerful [Carle and Fogg, 1996, 1997; Parks et al., 2000]. Furthermore, it is more interpretable and thus subjective geologic knowledge can be easily infused. Figure 2. Perspective view for the boreholes at the Moderately Fractured Rock (MFR) experimental area. MFR block has dimensions of m in E-W (x), N-S (y), and the vertical (z) directions, respectively [after Everitt, 2002]. 2of14

3 W12404 PARK ET AL.: ANALYSIS OF TRACER RESPONSE IN FRACTURED ROCK W12404 [6] In this paper, we apply a transition probability-based geostatistical approach [Carle, 1999] to synthesize the discrete permeability structure of moderately fractured crystalline rock in a conditional stochastic framework which honors the data and can infuse either hard or subjective categorical information that is consistent with geological interpretations. It is not our intent to construct discretefracture networks in three dimensions using the transition probability/markov chain approach. Instead, we will use the framework, combined with inversion of hydraulic head and concentration data, to synthesize a plausible categorical type description of the salient flow and transport properties of the crystalline rock mass, conditioned upon available data, that is a workable compromise between discrete-fracture and equivalent-continuum modeling methodologies. Tracer experiments conducted at the Moderately Fractured Rock (MFR) experiment area at Atomic Energy of Canada Limited (AECL) s Underground Research Laboratory (URL) are used for the calibration and validation of the developed model. Aspects pertaining to uncertainty associated with predicting hydraulic heads and concentrations in moderately fractured rock will also be presented. 2. Study Area and Conceptual Model 2.1. Moderately Fractured Rock Experiment Area [7] As part of the Canadian Nuclear Fuel Waste Disposal Program, the URL was constructed to a depth of approximately 400 m below ground surface in the granitic rock of the Lac du Bonnet batholith, situated in southeastern Manitoba (Figure 1). Site characterization activities prior to, during and following the URL excavation produced an extensive geologic, structural, geochemical and hydrogeological database describing the hydrogeological setting within the research facility. This database has been used to develop a conceptual model for the MFR experiments [Frost et al., 1998; Everitt, 2002], which were conducted at the 240 m level of the URL as part of Ontario Power Generation s Deep Geologic Repository Technology Program [Chan et al., 2001; Vandergraaf et al., 2001; Therrien and Lemieux, 2002]. For the purpose of the MFR experiment, a moderately fractured rock block is defined as a volume of rock mass of at least 100,000 m 3 in size (a cube of approximately 50 m or more on each side (Figure 2)) having a relatively uniform distribution of intersecting permeable fractures and a fracture frequency of about 1 to 5 fractures per meter. [8] Along the 16 test boreholes around the MFR experimental area (Figure 2), about 110 intervals were packed off, and in about 100 intervals among them, hydraulic permeabilities were measured by single-borehole packer tests. Also more than 1400 intersecting fractures were logged along the test boreholes, and their geological and structural properties were analyzed [Frost et al., 1998; Everitt, 2002] Fractured Rock Classification [9] In the test boreholes, hydraulic permeabilities represent a direct measure of a hydraulic property whose support volume reflects the a priori packer installation length. On the other hand, fracture intersection logs are arguably an indirect measure of the hydraulic characteristics of the medium, but they can be converted to continuous fracture density data. Figures 3a and 3b show examples of the Figure 3. Distributions of (a) measured permeability in the seven packer intervals of MF 7 and (b) the fracture density measured for each packer interval (solid line) and for every 5 m (dashed line) along the borehole. (c) Measured permeability for the entire MFR data set showing a power law relationship with the logged fracture density in the packed-off intervals. The error bars represent 1 standard deviation. distributions of measured hydraulic conductivity and fracture density within the seven packer intervals in test borehole MF 7. The fracture density plots shown in Figure 3b are calculated from the logged fracture locations for each packer interval and also at uniformly spaced 5 m intervals along the borehole. While Ando et al. [2003] found little or no correlation between permeability and fracture density in fractured crystalline rock at a site in France, our statistical analysis of the MFR data indicates that fracture density is positively correlated to (R 2 value equals to 0.83) and has a power law relationship with the measured 3of14

4 W12404 PARK ET AL.: ANALYSIS OF TRACER RESPONSE IN FRACTURED ROCK W12404 Figure 4. Matrix of east to west (E-W) direction transition probability measurements (circles) and the Markov chain model (solid lines) for MFR rock facies. hydraulic conductivity within a packer interval described by (see Figure 3c): K ¼ 10 8:6 r 1:5 f where K is the hydraulic conductivity (m/s) measured by single-hole hydraulic test and r f (fractures/m) represents the fracture density. Because the support volume associated with the packer tests is relatively small in relation to the borehole radius, it is assumed here that the K values ð1þ 4of14 reflected by (1) are isotropic. A careful analysis of the data also revealed that there was no statistical difference between those fractures described as open, closed, or partially closed based on the core logs as they pertain to a measured hydraulic conductivity within a given packer interval. This is not surprising since core log data, which is essentially onedimensional, will not reveal three-dimensional interconnectivity in the context of a fracture network. Nevertheless, the statistically significant correlation we have observed between fracture density and hydraulic conductivity, as

5 W12404 PARK ET AL.: ANALYSIS OF TRACER RESPONSE IN FRACTURED ROCK W12404 Table 1. Summary of the Proportion and the Mean Length in Three Principal Directions for Each Rock Facies Category Statistics Rock Facies Category SFB SMFB MFB MHFB HFB Proportion, % Mean length, m E-W N-S Vertical noted in Figure 3c and as described by equation (1), will form the underpinning for our categorical geostatistical reconstruction/parameter inversion framework. In this study therefore the rock types are classified, with conditioning based on the observed fracture frequencies, into 5.0 m intervals with categories described as sparsely (SFB, r f < 0.5), sparsely to moderately (SMFB, 0.5 < r f < 1.0), moderately (MFB, 1.0 < r f < 1.5), moderately to highly (MHFB, 1.5 < r f < 2.0) and highly (HFB, 2.0 < r f ) fractured blocks Geostatistical Approach [10] The Markov chain model based on a transition probability approach for simulating spatial variability facilitates the integration of geological concepts and reduces reliance on the traditional empirical curve-fitting approach. Readily observable geologic attributes, including volumetric proportions, mean facies lengths, and juxtapositional tendencies, can be incorporated directly into the development of a three-dimensional Markov chain model by using transition probability estimates and by inference from geologic concepts and principles. The Markov chain model, in turn, is used in a cokriging procedure during conditional sequential indicator simulation (SIS) and simulated quenching (SQ) to generate realizations of the rock facies distribution [Carle and Fogg, 1996, 1997; Weissmann et al., 1999] Spatial Model Development [11] Geostatistical structural models (e.g., variogram or transition probability) provide estimates of how the occurrence of one categorical/continuous variable is related to another. In this study, the Transition Probability Geostatistical Software (TPROGS) [Carle, 1999] is used for the geostatistical analysis and conditional simulation of the rock facies, categorized according to the aforementioned degrees of fracture intensity. [12] The rock facies (classified into the range from SFB to HFB) in the test boreholes at the MFR block were used to construct transition probability matrices and to develop Markov chain models of spatial variability in the E-W, N-S, and vertical directions. Note that although strict stationarity could not be assured, the MFR block was found Figure 5. A geostatistical realization of the rock facies distribution at the MFR experiment area. 5of14

6 W12404 PARK ET AL.: ANALYSIS OF TRACER RESPONSE IN FRACTURED ROCK W12404 Table 2. Summary of TTn Series Tracer Experiments TT2 TT3 TT4 TT5 TT6 TT7 nonsymmetric dipole nonsymmetric dipole dipole dipole nonsymmetric dipole Flow configuration one injection/three withdrawals Injection interval MF1-4 MF9-3 MF9-3 MF9-3 MF12-5 MF12-5 Withdrawal interval(s) MF6-2, MF12-5 MF13-4/5 MF13-4/5 MF13-4/5 MF14-6/7 MF9-3, MF , 38, Distance between injection/ withdrawal intervals, m Fluid injection rate, L/min Fluid withdrawal rate, L/min 0.03, 0.27, Conservative tracer Br Br Br I I Br Tracer injection concentration, mg/l Tracer injection duration, min Test duration, days Peak concentration, mg/l NA a, 3.03, Peak arrival time, days NA a, 6.0, Velocity, m/d NA a, 6.3, a NA indicates not analyzed because of concentration lower than detection limit. to have a relatively uniform distribution of permeable fractures and a discernable trend was not observed in the distribution of the rock facies and the resulting flow and transport properties in the MFR block. An example of a transition probability matrix estimated from the MFR fracture data and the fitted Markov chain models in the E-W direction is shown in Figure 4. Diagonal elements represent autotransition probabilities within a category according to separation lags, while off-diagonal elements represent the cross-transition probability between categories. The asymptote in each diagonal element is the proportion of each rock facies within the MFR block (Table 1). For the threedimensional rock facies modeling we are concerned with here, E-W, N-S, and vertical transition probability matrices are used simultaneously in the three-dimensional reconstruction process. Table 1 summarizes the proportion and the mean length in three principal directions for each rock category, used for the three-dimensional conditional simulation of rock facies distribution Conditional Simulation [13] The three-dimensional Markov chain model was used in TPROGS to perform SIS followed by simulated quenching (SQ) to generate a geostatistical realization of the rock facies distribution at the MFR block, with each grid block being 2 m on a side, an example of which is shown in Figure 5. SIS alone does not adequately preserve the crosscorrelation structure, but SQ can produce realizations that honor the coregionalization model without adding significant computational burden [Carle, 1996; Weissmann et al., 1999]. Each realization was conditioned on the rock facies (i.e., degree of fracturing) classification in all the 16 test boreholes. For the fluid flow and tracer transport analysis, 30 different realizations were generated in three dimensions. It should be noted that the distribution of five different rock categories along each borehole are preserved in each conditional realization, but the flow and transport properties of each rock category to be determined by parameter inversion are allowed to vary from one realization to the next. 3. MFR Tracer Experiments [14] In order to characterize the solute transport behavior at the moderately fractured rock experiment area at the URL site, numerous hydraulic and tracer tests were conducted within the test boreholes (Figure 2). In this study, we will focus on a set of injection-withdrawal tracer experiments referred to as TTn, where n is a number from 2 to 7. In Table 2, the details of the tracer experiments are summarized. Tracer test TT2 was carried out between one fluid injection and three withdrawal packed-off intervals, while TT3 to TT7 were performed between one injection and one withdrawal interval as symmetric (TT3 and TT4) or nonsymmetric (TT5, TT6, and TT7) dipole flow configurations. For a nonsymmetric dipole configuration, the fluid injection rate was set to be one tenth of fluid withdrawal rate. For these tests, the distance between injection and withdrawal intervals ranged from 19 m to 57 m and steady state flow was established before releasing a pulse of nonreactive solute (Br or I) through a packed-off fluid injection interval. During the test, fluid pressures were measured by previously installed vibrating wire transducers and, for chemical analysis, the groundwater samples were taken from the withdrawal boreholes using an auto sampler 6of14

7 W12404 PARK ET AL.: ANALYSIS OF TRACER RESPONSE IN FRACTURED ROCK W12404 [Vandergraaf et al., 2001]. An estimate of the bulk solute migration velocity, calculated from the distance between injection and withdrawal intervals and the peak arrival time, varies within an order of magnitude, ranging from 1.3 m/d to 13.3 m/d. As is typical in fractured rocks, the experimental results demonstrate that tracer transport in the test area is highly advection dominated with abrupt leading edges and extended trailing tails. 4. Numerical Simulation of Fluid Flow and Tracer Transport 4.1. Numerical Simulator and Modeling Layout [15] FRAC3DVS is a variably saturated groundwater flow and reactive solute transport simulator designed for application to discretely fractured porous media, and it can incorporate one-dimensional wells as line elements [Therrien and Sudicky, 1996, 2000; Therrien et al., 2003]. It can also simulate flow and transport in zoned-type porous media in which the material property contrasts between the various zones is large because of the advanced numerical solution strategies employed. For the flow and transport simulation of each categorical-type rock realization produced by the transition probability/ Markov chain approach, a computational domain was chosen to have a dimension 360 m 400 m 260 m in the E-W, N-S, and vertical directions, respectively. The computational domain includes all the test boreholes within the MFR block and to be large enough to accommodate far-field outer boundaries. The hydraulic gradients induced during the tracer tests are on average more than an order of magnitude higher than the pretest natural gradients in the area and are assumed to be the only hydraulic driving force acting upon the tracer transport during the injection/withdrawal experiments conducted at the MFR experiment area [Chan et al., 2001]. Thus the top and side boundaries in the model are set to be of prescribed zero drawdown. Because a very low permeability, sparsely fractured rock underlies the domain, the bottom was prescribed as a no-flow boundary. For the pumping/injection and tracer test simulations, all of the test boreholes shown in Figure 2 were represented by line elements along which each discrete packer interval was accommodated [Therrien and Sudicky, 2000]. The threedimensional computational mesh for the FRAC3DVSbased inversions described herein contains about 500,000 nodes. A total of 30 realizations constructed from the fracture-frequency-permeability data, and based on the transition probability/markov chain geostatistical model, are then used to assess the model s ability to match the breakthrough curves of the tracer tests performed in the field within the MFR block. The approach adopted in this study to estimate the flow and transport properties for prediction of the tracer experiments carried out at the Moderately Fractured Rock experimental area is summarized as follows: (1) Select rock facies categories based on fracture densities measured in the test boreholes, (2) create the Markov probability matrices for transitions between categories in three principal directions, (3) use the Markov transition matrices to generate the threedimensional conditional rock facies distributions, (4) use the steady state flow test data to calibrate the hydraulic conductivity of each rock facies category, (5) use the calibrated flow model and TT2 tracer test data to calibrate the effective porosity of each category and the dilution factor in the injection interval, (6) calculate the sum of the squared errors obtained from the flow and transport calibrations, and (7) utilize the flow and transport properties of the selected realizations for stochastic prediction of the other (TT3 to TT7) tracer experiments Parameter Calibration: Nonlinear Least Squares Optimization [16] Steady state hydraulic heads measured in the packer intervals during tracer test TT2 were used for flow calibration and the hydraulic conductivity estimation for each rock category (SMFB to HFB) was determined by a nonlinear least squares optimization procedure [van Genuchten, 1981]. The initial estimates of the hydraulic conductivity values assigned to the SMFB to HFB categories were determined from the power law relationship given by equation (1), along with an assumed initial anisotropy value equal to 1.0. The weighted squared sum of the difference between the measured and simulated heads was used as the objective function to be minimized: SSE ¼ Xnobs i¼1 2 ðh obs;i h sim;i Þw i ð2þ where SSE represents the sum of the squared error, and h obs,i and h sim,i are the measured and simulated heads and w i is the weighting at ith observation point for n obs observation packer intervals. Because the simulated heads in equation (2), determined from forward flow simulations in each rock category realization, are functions of horizontal (K h ) and vertical (K v ) hydraulic conductivities in the SFB to HFB materials, optimization targets are those hydraulic conductivities that minimize the objective SSE function. As a result, the calibrated K h and K v values estimated by inversion for each rock category for each realization differed by 1 to 3 orders of magnitude from the values initially assigned according to equation (1) (Figure 3c). In the optimization procedure, conjugate-gradient searching was repeated until the updates of all the target variables were less than 0.01 in logarithmic scale. [17] To minimize the nonuniqueness problem during flow calibration, the permeability in each rock facies was constrained under the physical condition that fracture density was positively correlated with permeability (e.g., K h,hfr K h,mfr ) and that each rock type was more permeable in the vertical direction (e.g., K v,hfr K h,hfr ) based on the observation that most fractures are vertical or subvertical in the study area. Note here that the measured hydraulic conductivity in each packed-off interval, used to quantify the relationship with fracture density in the interval, was assumed to be a lumped scalar (i.e., isotropic) value, but the calibrated hydraulic conductivity for each rock facies is a tensor whose principal (vertical) axis coincides with the geological observation that most fractures are vertical to subvertical. In Figure 6a, the observed and calibrated head changes in the test boreholes are compared for two different realizations, and Table 3 shows the calibrated flow properties for each rock category. These two realizations were chosen from amongst the 30 realizations because they were also found to provide the best fits with the measured data from the tracer tests (TT2 to TT7). 7of14

8 W12404 PARK ET AL.: ANALYSIS OF TRACER RESPONSE IN FRACTURED ROCK W12404 Figure 6. Comparison between the measured and the calibrated (a) heads and (b) concentrations for TT2. In Figure 6a, symbols represent two different realizations; in Figure 6b, symbols and lines are the measured and calibrated breakthrough concentrations, respectively, for two realizations. [18] For the given flow conditions and by applying the same conjugate-gradient optimization procedure with an objective function identical to (2), but using concentrations instead of heads, the effective (e.g., fracture) porosity of each rock type and the tracer injection concentration were also calibrated against the observed TT2 tracer breakthrough. We explored the use of alternative objective functions for the transport calibration process, including fitting of the temporal moments up to order five of the breakthrough curves and the matching of the peak arrival times and peak concentrations, but the results were found to be less satisfactory compared to the use of the concentration analogue of (2). Owing to uncertainties in the tracer concentration upon its delivery to the packed-off interval, which contains numerous sampling and inflation tubes, and its redistribution in the borehole injection interval upon mixing [Brouyère, 2003], the ratio between the specified and injection concentrations (C s /C 0 ) is also calibrated based on the observed breakthrough concentrations in the withdrawal zone(s) (Table 4). It should be noted that the packer intervals used for injection were designed with a volume reducer in an attempt to minimize the uncertainty resulting from tracer mixing in the well bore [Vandergraaf et al., 2001]. Sensitivity analysis performed with the model for each realization of the fractured rock medium showed that the tracer breakthrough concentrations in the withdrawal zones were relatively insensitive to the assigned dispersion properties within a reasonable range of dispersivity values (meters to tens of meters). Therefore, in the calibration and validation procedures the longitudinal dispersivity was fixed at 5 m and the transverse dispersivity was set to 0.5 m. Figure 6b shows the measured and calibrated breakthrough concentrations at two different fluid withdrawal points for two different realizations of the fractured rock mass (realization 3 and 7) that were conditionally reconstructed from the categorical permeability-fracture density relationship and using the hydraulic head and tracer breakthrough data measured for TT2 in the inversion procedure. It can be seen that there is reasonable agreement between the measurements and the model results for both realizations Calibration Error in Multiple Realizations [19] To test the calibration procedure with the transition probability-based geostatistical model, all 30 of the rock facies realizations generated with TPROGS were calibrated with respect to the head distribution and the tracer breakthrough concentrations using the head and concentration values observed during tracer test TT2. Because the flow and transport calibration procedures can suffer from nonuniqueness and stability problems, the difference between the calculated and observed values in some cases exhibited Table 3. Calibrated Horizontal (K h ) and Vertical (K v ) Hydraulic Conductivities for the Rock Facies for Two Different Realizations a SFB SMFB MFB MHFB HFB Realization K h K v /K h K h K v /K h K h K v /K h K h K v /K h K h K v /K h a Hydraulic Conductivities are given in m/s. 8of14

9 W12404 PARK ET AL.: ANALYSIS OF TRACER RESPONSE IN FRACTURED ROCK W12404 Table 4. Calibrated Effective Porosity (n e ) of the Rock Facies and the Ratio Between the Specified and Injection Concentrations (C s /C 0 ) for Two Different Realizations n e Realization SMFB SMFB MFB MHFB HFB C s /C a much larger squared sum of errors than in others (Figure 7). Frequency distributions of SSE values shown in Figures 7a and 7b indicate that most calibration SSEs deviate moderately from a modal value except for a few larger calibration SSEs. In parameter calibration, especially with a large number of parameters, it is difficult to ensure that a global minimum has been achieved with a given calibration procedure. A close inspection of the results revealed that the cases with SSE values greater than 90 could be attributed to a stagnation in the neighborhood of a local minimum in the gradient-based optimization procedure and an inappropriate hydraulic connection state among the tracer injection/withdrawal test intervals. Therefore only 11 of the 30 calibration cases in which the SSEs are smaller than 90 for both TT2 flow and transport are considered to be statistically calibrated; these 11 of the 30 realizations will be used for the predictive modeling of the other tracer experiments. [20] For the 11 successfully calibrated realizations, Figure 8 shows the calibrated horizontal and vertical hydraulic conductivities for the five different rock categories. In Figure 8, the calibrated hydraulic conductivity values differed by 1 to 3 orders of magnitude from the values estimated from the power law relationship between fracture density and measured hydraulic conductivity data. However, it is also clear in Figure 8 that the power law relationship holds between fracture density and the horizontal or vertical hydraulic conductivity values, calibrated against the measured head data. This result indicates that the geological and geostatistical conceptualization adopted in this study based on the power law relationship is preserved Predictive Modeling of Tracer Tests [21] A calibrated parameter set for a given tracer test configuration often fails to adequately simulate results obtained from a different flow configuration, especially in fractured rocks, unless the heterogeneity is modeled appropriately [Pfingsten and Soler, 2002]. In this context, a predictive comparison between the measured and simulated concentration breakthroughs for the other tracer experiments, with the model calibrated for tracer test TT2, may reveal how well the heterogeneity in the study area has been captured Dipole Tracer Tests [22] With the calibrated flow and transport parameters (hydraulic conductivity, effective porosity, and the ratio between the specified and injection concentration), dipole tracer tests TT3 and TT4 were simulated in a forward predictive model using the properties derived from the analysis of TT2. Figures 9a and 9b show that in two of the optimal realizations (realizations 3 and 7), the simulated breakthrough curves catch the extremely sharp leading edges and trailing tails of the measured concentration (related to highly advective transport through interconnected fracture networks) and that the geostatistical realizations may have successfully captured the hydraulic connectivity between the test boreholes. However, owing to the uncertainty in the distribution of the categorical rock properties between the points where the conditioning data Figure 7. Frequency distribution of the sum of the squared error (SSE) values calibrated for (a) TT2 flow and (b) TT2 transport for 30 rock facies realizations. 9of14

10 W12404 PARK ET AL.: ANALYSIS OF TRACER RESPONSE IN FRACTURED ROCK W12404 Figure 8. Power law relationships of the fitted horizontal and vertical hydraulic conductivities versus fracture density for the five different rock categories based on the 11 successfully calibrated realizations. are imposed, a geostatistical realization may not always capture the specific individual features related to each tracer experiment. This is especially true in fractured rock, where a narrow fracture zone can dominate the migration of the solute. Thus, in this study we will analyze the prediction uncertainty among the different realizations to test the performance of the conceptual model based on the transition probability/markov chain reconstruction framework. [23] For the 11 geostatistical realizations, each considered to be successfully calibrated according to the observed TT2 flow and transport data, TT3 and TT4 simulation results were statistically analyzed upon comparison to the measured concentrations. In Figures 9c and 9d, the ensemble mean breakthrough concentrations and the 95% confidence intervals are plotted along with the measured breakthrough concentrations. The early peak seen in tracer test TT3 (Figure 9a) is likely due to the presence of a well-connected discrete fracture between the injection and withdrawal intervals. The simulated tracer breakthrough concentrations deviate somewhat from the observed breakthrough data because the flow and transport properties were calibrated Figure 9. Comparison between the measured concentration and the simulated breakthrough behavior for dipole tracer tests (a and c) TT3 and (b and d) TT4. In Figures 9a and 9b, breakthrough curves are simulated using realizations 3 and 7, and the results in Figures 9c and 9d are from the analysis of 11 realizations. 10 of 14

11 W12404 PARK ET AL.: ANALYSIS OF TRACER RESPONSE IN FRACTURED ROCK W12404 Figure 10. Comparison between the measured concentrations and the simulated breakthrough behavior, based on the effective porosities calibrated for tracer tests (a) TT2 and (b) TT4, for tracer test TT5. Symbols represent measured concentration, and solid and dashed lines are average results over 11 realizations and their 95% confidence limits, respectively. The well bore dilution factor is recalibrated for TT5. E[SSE] represents the expectation of SSE (average SSE from 11 realizations). using the hydraulic head distribution and tracer breakthroughs from tracer test TT Nonsymmetric Dipole Tracer Tests [24] In fractured rock, it has been commonly observed that even a minor change in the induced flow field can have a major impact on tracer transport [Becker and Shapiro, 2003]. Thus the simulation of the tracer experiments having a nonsymmetric dipole flow configuration (TT5, TT6, and TT7) are here analyzed separately from the dipole flow tracer tests (TT3 and TT4) to assess our model performance. Nevertheless, it is important to compare predictions for tracer tests TT4 and TT5, which use the same injection and withdrawal packer intervals, but were conducted with dipole and nonsymmetric dipole flow configurations, respectively (Table 2). [25] The measured tracer breakthrough concentrations for TT5 are compared to the predicted breakthrough behaviors obtained from the 11 optimal realizations, for which the permeability and effective porosity values were calibrated to the data for tracer tests TT2 and TT4. The predicted simulation results demonstrated that the model calibrated for TT4 does not significantly improve the predictive performance, compared to the TT2-calibrated case, mainly because both cases tend to overestimate the measured concentrations. The expectations of the SSE values are calculated to be 117 and 404 for TT2- and TT4-calibrated cases, respectively. This is not surprising because the fluid injection rate for TT5 (0.1 L/min) is one tenth of that for TT2 or TT4 (1 L/min). Thus more dilution can be expected in the tracer injection interval for the case of TT5, but the dilution factors (C s /C 0 ) used to predict the TT5 results were calibrated using the TT2 or TT4 breakthrough data. To remove the uncertainty related to the actual injection concentration upon the release of the tracer in the packer interval, the C s /C 0 value alone was recalibrated using the tracer test TT5 concentration data but keeping the material properties as calibrated for TT2 or TT4 (Figures 10a and 10b). After recalibrating only the C s /C 0 values, the prediction SSEs for TT5 (83 or 89) become similar regardless of the calibration model (i.e., using inverted permeability and porosity parameters derived from either TT2 or TT4 data). Thus, from the results provided in Figure 10, it is evident that an appropriate calibrated injection concentration that reflects mixing in the borehole injection interval is critical for predictive modeling in moderately fractured rock at the decameter scale. This is somewhat problematic because the intricate mixing that occurs within a packed-off interval of a borehole into which a tracer is introduced cannot readily be described in a mathematical context, especially when the injection interval contains various tubes and ports used for instrumentation purposes, and volume reducers. The aforementioned C s /C 0 recalibration procedure will therefore be also applied to the predictions of the TT6 and TT7 tracer test results to account for the essentially unknown injection concentrations. [26] The measured breakthroughs for TT6 and TT7 are compared in Figure 11 to the predicted breakthrough behaviors with the transport properties calibrated according to the TT2 observations. The simulation results for TT6 show that concentration peaks at about 10 days, but the wide confidence intervals illustrated in Figure 11a, especially at early time, reflect substantial prediction uncertainty. This high uncertainty for predicting the early time concentrations could arise from the fact that a single, narrow interconnected fracture pathway between the injection and withdrawal points might control the early time behavior, which would be difficult to explicitly capture in the model [Zhang et al., 2003]. 5. Discussion 5.1. Effects of Calibration on Predictive Modeling [27] In this study, the flow and transport properties of the different rock facies, categorized by their different degrees of fracturing, were calibrated for a single tracer test and these calibrated properties were then used to predict the 11 of 14

12 W12404 PARK ET AL.: ANALYSIS OF TRACER RESPONSE IN FRACTURED ROCK W12404 Figure 11. Comparison between the measured concentration (circles) and the simulated mean breakthrough behavior (lines) for tracer tests (a) TT6 and (b) TT7. Note that the presence of residual tracer from prior tracer tests was not accounted for in the simulations. breakthrough data obtained from the other tracer tests. Owing to computational limitations, the model could not be calibrated using all of the data from the various tracer experiments in a simultaneous fashion. After removing the uncertainty related to the concentration C s in the injection interval as it is affected by well bore mixing, the calibrated models showed similar levels of prediction SSEs (see e.g., Figure 10). This implies that there is no significant difference in the predictive capability of the various structural models even if the flow configuration for the tracer tests used for forward prediction differed from the flow configuration of the tests used for calibration. This is an encouraging result because tracer transport is known to be highly sensitive to the configuration of the induced flow conditions, as well as the injection/withdrawal locations, in fractured rocks. Nevertheless, a question arises: To what degree does a better calibrated realization (smaller calibration SSE) improve the predictive capability (prediction SSE) for a range of different flow conditions with different injection/withdrawal locations? We will explore this question in section Measuring Predictive Performance [28] Table 5 shows the absolute and normalized calibration and prediction SSEs for all of the TTn series tracer experiments for the 11 successfully calibrated realizations, where the flow and transport properties were calibrated for tracer test TT2. Note, however, that the dilution factors (C s /C 0 ) are recalibrated for each tracer test. The general trend in Table 5 indicates that the absolute SSEs are largest for TT4 and smallest for TT6. This is because the measured concentrations for TT4 are much larger than those for the other cases, whereas the measured concentrations for TT6 are typically much smaller. To analyze the predictive performance amongst the various medium realizations, the SSEs for a given tracer test were therefore normalized by the average SSE values for each test as shown in Table 5. The normalized SSE results provided in Table 5 indicate Table 5. Normalized and Absolute SSEs for TTn Series Tracer Tests Based on TT2-Calibrated Transport Parameters With the Dilution Factor Calibrated for Each Test Normalized Sum of Squared Error and Absolute SSE a Realization TT2 TT3 TT4 TT5 TT6 TT7 Average Standard Deviation (32) 1.11 (203) 1.08 (476) 0.65 (54) 0.94 (2.0) 3.59 (264) 1.36 (172) 1.10 (181) (28) 0.32 (59) 0.54 (241) 0.92 (76) 1.46 (3.1) 0.72 (53) 0.78 (76.7) 0.39 (84.4) (19) 0.61 (112) 0.99 (437) 0.52 (43) 0.52 (1.1) 0.30 (22) 0.57 (106) 0.23 (167) (59) 1.21 (221) 0.95 (422) 0.79 (65) 2.86 (6.1) 0.22 (16) 1.26 (132) 0.90 (162) (31) 0.14 (26) 0.66 (292) 0.52 (43) 0.33 (0.71) 0.42 (31) 0.48 (70.6) 0.23 (109) (87) 0.49 (90) 0.81 (360) 1.69 (140) 0.44 (0.93) 2.02 (149) 1.28 (138) 0.79 (122) (81) 2.82 (513) 1.82 (805) 1.85 (153) 0.70 (1.5) 0.42 (31) 1.61 (264) 0.89 (324) (19) 0.30 (54) 1.10 (488) 0.44 (36) 1.92 (4.1) 0.88 (65) 0.85 (111) 0.61 (186) (35) 0.79 (144) 0.82 (361) 0.73 (60) 0.94 (2.0) 0.48 (35) 0.77 (106) 0.16 (134) (22) 2.76 (502) 0.65 (288) 1.61 (133) 0.80 (1.7) 1.44 (106) 1.30 (176) 0.83 (190) (20) 0.44 (80) 1.58 (698) 1.29 (107) 0.11 (0.24) 0.52 (38) 0.74 (157) 0.56 (268) Average 1.00 (39) 1.00 (182) 1.00 (443) 1.00 (82.7) 1.00 (2.1) 1.00 (73.6) 1.00 (137) 0.61 (175) a Absolute SSEs are in parentheses. 12 of 14

13 W12404 PARK ET AL.: ANALYSIS OF TRACER RESPONSE IN FRACTURED ROCK W12404 Figure 12. Comparison between the measured (symbols) and the simulated (lines) concentration breakthroughs with realization 7 for tracer tests TT2 to TT7. that the prediction of all tracer tests obtained with realization 7 is the most stable (smallest standard deviation) and accurate (closest mean) amongst the 11 realizations. In Figure 12, the measured breakthrough concentrations are compared to the simulated breakthrough curves for all six tracer tests for realization 7. As can be seen, the results obtained with realization 7, all are in reasonable agreement with the concentration data measured for all of the tests for the various flow configurations and injection/withdrawal rates. This implies that this particular realization is relatively robust in its predictive capability, at least within what we perceive to be an acceptable level of performance from a pragmatic perspective for predicting solute transport in moderately fractured crystalline rock. Nevertheless, it should also be noted here that the model s predictive performance was assessed on the basis of the difference between the concentrations obtained from each of the 11 calibrated realizations (for TT2) and the actual measured values for all tracer tests. While the use of an ensemble mean prediction might yield a significant prediction error for a given tracer test, we note that the error is effectively distributed across all tracer tests for which the comparisons are made. 6. Summary [29] A geostatistical approach was applied to generate a distribution of discrete rock facies, according to the frequency of observed fractures, for the MFR block located at the AECL s URL. The rock facies classifications ranged from sparsely to highly fractured blocks (SFB to HFB) and were based on logged fracture information and permeability measurements in the test boreholes. Rock facies categories occurring within the test boreholes were then used to construct three-dimensional Markov chain models of transition probability, representing the permeability and porosity structure of the crystalline rock. Geostatistical conditional simulations were next performed with sequential indicator simulation (SIS) and simulated quenching (SQ) to construct realizations of the crystalline rock for groundwater flow and tracer transport simulations of the various tracer tests performed in the rock. [30] During the flow calibration, physical constraints based on hydrogeological observations (i.e., greater presence of vertical fractures that enhance vertical permeability) were used to minimize some of the nonuniqueness and stability problems commonly encountered in optimization and calibration procedures. Transport parameters (effective porosity) and the uncertain specified concentration value in the tracer injection interval (well bore dilution) were also calibrated using concentration data from the tracer experiments. For validation purposes, a series of calibrated models were then tested by comparison of results from the calibrated model with data collected from several other tracer experiments having different injection-withdrawal configurations. Calibration and validation results show that a realistic geostatistical conceptualization of the hydraulic connection state between the test boreholes can be developed for moderately fractured rock and can be successfully applied for predictive modeling via multiple stochastic realizations and a careful error analysis. It was also demonstrated that a transition probability/markov chain approach to modeling categorical rock facies in three dimensions is a viable methodology when combined with inverse modeling to 13 of 14

14 W12404 PARK ET AL.: ANALYSIS OF TRACER RESPONSE IN FRACTURED ROCK W12404 construct a workable model of flow and transport in fractured crystalline rock. [31] We would stress that the MFR experimental area has a relatively homogeneous fracture distribution and has been characterized intensively and extensively. Thus a relatively small but workable number of conditional realizations were required for the stochastic analysis. While transport prediction uncertainties can be appreciable for a given level of data conditioning and calibration, the approach used in this study also allows data worth issues to be explored in a systematic manner. [32] Acknowledgments. This research was supported by Ontario Power Generation (OPG), the Natural Sciences and Engineering Research Council of Canada (NSERC) and Canadian Water Network grants to E. A. Sudicky and J. F. Sykes, and by the Postdoctoral Fellowship Program of Korea Science and Engineering Foundation (KOSEF) to Y.-J. Park. Additional support for the work was also provided by funding through a Canada Research Chair (Tier I) awarded to E. A. Sudicky. The authors would especially like to thank Mark Jensen and Andre Vorauer from OPG for supporting the project on their critical reviews, and Graham Fogg from University of California, Davis, for supplying the TPROGS code. References Ando, K., A. Kostner, and S. P. Neuman (2003), Stochastic continuum modeling of flow and transport in a crystalline rock mass: Fanay- Augères, France, revisited, Hydrogeol. J., 11, Becker, M. W., and A. M. Shapiro (2003), Interpreting tracer breakthrough tailing from different forced-gradient tracer experiment configurations in fractured bedrock, Water Resour. Res., 39(1), 1024, doi: / 2001WR Berkowitz, B., J. Bear, and B. Carol (1988), Continuum models for contaminant transport in fractured porous formations, Water Resour. Res., 24, Brouyère, S. 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(mclaren@uwaterloo.ca; yj2park@sciborg.uwaterloo.ca; sudicky@sciborg. uwaterloo.ca) J. F. Sykes, Department of Civil Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. 14 of 14