A new approach to account for fracture aperture variability when modeling solute transport in fracture networks

Transcription

1 WATER RESOURCES RESEARCH, VOL. 49, , doi: /wrcr.20130, 2013 A new approach to account for fracture aperture variability when modeling solute transport in fracture networks Martin Larsson, 1,2 Magnus Oden, 1,3 Auli Niemi, 1 Ivars Neretnieks, 4 and Chin-Fu Tsang 1,5 Received 20 April 2012; revised 10 January 2013; accepted 23 January 2013; published 30 April [1] A simple yet effective method is presented to include the effects of fracture aperture variability into the modeling of solute transport in fracture networks with matrix diffusion and linear sorption. Variable apertures cause different degrees of flow channeling, which in turn influence the contact area available for these retarding processes. Our approach is based on the concept of specific flow-wetted surface (sfws), which is the fraction of the contact area over the total fracture surface area. Larsson et al. (2012) studied the relationship between sfws and the standard deviation ln K of the conductivity distribution over the fracture plane. Here an approach is presented to incorporate this into a fracture network model. With this model, solute transport through fracture networks is then analyzed. The cases of S ¼ 0 and S ¼ 1 correspond to those of no matrix diffusion and full matrix diffusion, respectively. In between, a sfws breakpoint value can be defined, above which the median solute arrival time is proportional to the square of sfws. For values below the critical sfws (more channeled cases), the change is much slower, converging to that of no matrix diffusion. Results also indicate that details of assigning sfws values for individual fractures in a network are not crucial; results of tracer transport are essentially identical to a case where all fractures have the mean ln K (or corresponding mean sfws) value. This is obviously due to the averaging effect of the network. Citation: Larsson, M., M. Oden, A. Niemi, I. Neretnieks, and C.-F. Tsang (2013), A new approach to account for fracture aperture variability when modeling solute transport in fracture networks, Water Resour. Res., 49, , doi: /wrcr Introduction [2] Solute transport in fractured rock has been a challenging problem [Neuman, 2005], especially due to the strong heterogeneities in rock properties. In particular, heterogeneities at different scales present a difficult parameter estimation and modeling task for regional-scale problems, such as the modeling for evaluating transport of radionuclides potentially released from underground nuclear waste repositories or study of flow and transport in deep hydrogeological systems in general. Traditionally, there are three main approaches for modeling of flow and solute transport in fractured rock. The first one is the effective porous medium approach, applicable for problems of very large scales where averaging can be done and detailed heterogeneities 1 Air, Water and Landscape Sciences, Department of Earth Sciences, Uppsala University, Uppsala, Sweden. 2 Department of Aquatic Sciences and Assessment, Swedish University of Agricultural Sciences, Uppsala, Sweden. 3 Swedish Nuclear Fuel and Waste Management Company (SKB), Stockholm, Sweden. 4 Division of Chemical Engineering, School of Chemical Science and Engineering, KTH Royal Institute of Technology, Stockholm, Sweden. 5 Earth Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California, USA. Corresponding author: M. Larsson, Air, Water and Landscape Sciences, Department of Earth Sciences, Uppsala University, Uppsala SE-75236, Sweden. (martin.larsson@hyd.uu.se) American Geophysical Union. All Rights Reserved /13/ /wrcr are not of interest. The second approach is the stochastic continuum approach [Tsang et al., 1996] also applicable for rather large scales where the fractured rock can be described as a stochastic continuum with a spatial distribution of local-scale hydraulic properties. The last and most detailed approach is the fracture network model pioneered by Long et al. [1982] where individual fractures are being modeled as discrete flow-carrying features. The scale of applicability of fracture network approach is usually comparably small, due to it being computationally intensive. The fracture network models can be set up to take into account the diffusion of solutes and radionuclides into the rock matrix and can, in principle, directly include fracturescale aperture heterogeneities. However, the computational weight has strongly limited the scale where such detailed modeling can be done. There are also hybrid methods where several approaches are combined [e.g., Cacas et al., 1990; Niemi et al., 2000; Ohman et al., 2005; Svensson et al., 2010], and they often include calculating block-scale properties with a fracture network model and then using these block-scale values in a stochastic continuum model at the larger scale. [3] In this work, we will focus on the upscaling of the effect of fracture aperture heterogeneities and its impact on solute transport in fracture networks, in particular taking into account the effect of fracture-level heterogeneity on the retarding processes of matrix diffusion and sorption. Fracture aperture heterogeneity causes the water flow in a single fracture to be channelized through channels with relatively large local flow rate compared to the surrounding 2241

2 areas [Neretnieks, 1983; Tsang and Tsang, 1989; Tsang and Neretnieks, 1998]. The channeling effect is significant; for example, in some cases only 25% of the fracture can carry 90% of the flow as shown in a numerical study by Larsson et al. [2012], and it has significant impact on solute transport retardation, as it decreases the interaction surface area where the retarding processes of matrix diffusion and sorption can occur. Advective flow dominates in fast flowing channels, giving rise to early arrival for part of the contaminant and also an early concentration peak, while the retarding processes slow down the later part of the transport so that the tracer breakthrough curve displays a long tail at late time [see, e.g., Carrera et al., 1998; Moreno and Neretnieks, 1993; Shapiro, 2001; Oden et al., 2008]. [4] The flow-wetted surface (FWS), defined as the contact area between the flowing water and the rock [Moreno and Neretnieks, 1993], is the area where matrix diffusion and sorption processes can occur. A large FWS gives a larger potential for the retarding processes, thus delaying the transport and vice versa [Neretnieks, 1980; Moreno and Neretnieks, 1993; Xu et al., 2001]. Thus, in many cases such as when investigating for a nuclear waste repository, the FWS is an important factor [Neretnieks, 1980; Elert, 1997]. Larsson et al. [2012] examined the influence of single fracture aperture heterogeneities on flow and transport channeling and subsequently on the FWS. In this article an empirical relationship showing the FWS as a function of the variance of the underlying hydraulic conductivity field is presented. In the present work we take the analysis further by presenting an approach to incorporate this information of fracture-level heterogeneity to a system of fractures, in a fracture network model, and thereafter analyze the effect of fracture-level heterogeneity on solute transport in fracture networks. [5] In most regional-scale models for fractured rocks, fracture-scale aperture heterogeneities are not included, due to too large computational cost. Methods to include matrix diffusion effects in fracture network modeling have been presented [e.g., Oden et al., 2008], but these are not including varying aperture in the fractures. The inclusion of matrix diffusion delays the main transport of solutes. If, however, the area of the retarding surface is overestimated by assuming the entire fracture plane to contribute to the retarding processes, the time delay on the breakthrough curve may be overestimated. In this work (section 3) we present an approach to include the effect of the FWS in a variable aperture fracture into a fracture network model, making use of the empirical formula by Larsson et al. [2012]. The solute transport is then analyzed by simulating the travel time of advective nonsorbing particles at block scale. A time delay at each fracture along the transport path of a particle is calculated to include the effect of the matrix diffusion and variable FWS. The presented method is straightforward to implement and computationally efficient. In section 4 the results of the fracture network simulations are presented for three different scenarios of the fracture networks and the effect of accounting for the variable FWS analyzed. The results can, in principle, be used to further upscale transport from block to regional scale by the method presented by Ohman et al. [2005]. [6] We may note that the matrix diffusion and sorption properties measured with field measurements, such as in two well tracer tests or single-well injection withdrawal tests, often show a large discrepancy from the properties measured in the laboratory. An example is discussed by Andersson et al. [2002]. The aim of this study is to provide an approach to evaluate whether a part of this discrepancy can be explained by aperture heterogeneity on fracture scale influencing the flow and transport in a fracture network. [7] Prior to applying the results of Larsson et al. [2012] into a fracture network analysis, we shall elaborate (section 2) the development of their relationship, alongside with a more general form of the equation, to enable users to find a more accurate relationship, if needed, for their particular applications. 2. Empirical Equation Relating Specific FWS to Aperture Variability in a Single Fracture [8] Larsson et al. [2012] investigated the transport of particles in a single fracture to examine the channeling of flow and transport as a function of aperture variability. The fracture apertures were spatially varying over the fracture plane where the hydraulic conductivity follows a lognormal distribution with a standard deviation of ln K. The specific flow-wetted surface (sfws) is defined as the fracture area traversed by particles divided by the total fracture area. It is systematically analyzed by numerical simulations. The flow and advective particle transport is calculated by applying a pressure gradient over a 2-D porous media model, in which the heterogeneous hydraulic conductivity and porosity field of the model represent variable fracture apertures. Figure 1 shows the calculated sfws (or S, as it is termed in mathematical expressions) for different fractions of particles (P) transported over the fracture under a constant pressure step across it, for a number of ln K values. In other words, S is the cumulative fraction of the total fracture area Figure 1. The sfws as a function of cumulative fraction of particles for different values of standard deviation of the fracture hydraulic conductivity distribution ( ln K ). Results of numerical calculations from Larsson et al. [2012] are shown as black lines. Red circles indicate the fit with equation (5). 2242

3 contacted by cumulative fractions of transported particles (P). The result was shown to be independent of the correlation length of the aperture/conductivity field, being only dependent on the standard deviation. In this study correlation lengths that are small compared to the fracture (up to 18% of the width of the fracture) are used. It is possible that the relationship might not hold for larger correlation lengths. For example, if the correlation length is large compared with fracture size, it is possible that there are continuously connected high conductivity areas throughout the whole fracture and thus a further channelization of the transport than presented here. From the results of Figure 1 the FWS can be determined for any percentage of the particles. In this section we also present the mathematical details in our approach to find this mathematical expression for the relationship among the sfws, the fraction of the particles having passed that surface, and the standard deviation of the hydraulic conductivity field, most notably to allow the reader to apply the data for other cases of interest. [9] The behavior of the curves in Figure 1 (which is based on the data from the study by Larsson et al. [2012]) provides the requirements (summarized in Table 1) that should be fulfilled by such an empirical relationship or equation. Specifically, the equation is required to be relatively simple for ease of use. To find an equation that describes the behavior of the sfws as a function of the fraction of particles, we search for an equation that fulfills these requirements and can reproduce the curves in Figure 1. An initial guess for a functional form is shown in equation (1), as the simplest equation which fulfills these requirements. Equation (1) was further generalized into equation (2), to allow flexibility for obtaining a better fit to the numerical data. For equations (1) (3), S is the sfws, P is the fraction of particles, and ln K is the standard deviation of the hydraulic conductivity in the fracture. The parameters and N can be varied to fit the numerical data. Expressions f( ln K )and g( ln K ) simply indicate a functional dependence on ln K. S ¼ PN ð PÞ ln K ; ð1þ Figure 2. Plot of linear regression fits in the interval ln(2 P 2 ) ¼ Linear regression gives a curve y ¼ m x þ b, where m is a function of ln K as f( ln K ), b is also a function of ln K,asg( ln K ). were tested, and in Figure 2 the plots for N ¼ 2 and ¼ 2 are shown, which was also the combination where the results over a large P interval showed a straight line behavior, approximately from P ¼ 0.42 to [12] If a linear curve is fitted to each of the curves in Figure 2, its slope and y axis intercept values can be obtained as a function of ln K (Figure 3). Therefore, the characteristics of the curves in Figure 3 give an indication of the nature of the f and g functions. [13] Examination of Figure 3, while keeping in mind the need for simplicity, suggests that the function f( ln K ) could be approximated with a linear relationship, while the function g( ln K ) could be approximated with a simple quadratic equation. Therefore, the following functional form is obtained (equation (4)): ln S ¼ ln K ln 2 P 2 2 P ln K : ð4þ S ¼ PN ð P Þ f ðln K Þ exp½gð ln K ÞŠ: ð2þ [10] After some algebraic reorganization, equation (2) can be written as ln S P ¼ f ð ln K ÞlnðN P Þþgð ln K Þ: ð3þ [11] Using the numerical data, a plot of ln(s/p) as a function of ln(n P ) was prepared assuming different values of N and. If the functional form is correct, the curves will be straight lines. A number of combinations of N and Table 1. Requirements for the Functional Form For Parameter Value Required Result P ¼ 0 S ¼ 0 P ¼ 1 S 1 ln K ¼ 0 S ¼ P Increasing ln K S decreases Simple equation Figure 3. (left y axis) f ( ln K ) (circle symbols) and (right y axis) g( ln K ) (square symbols) as a function ln K. f ( ln K ) is the slope of the regressed fitted curves of Figure 2 and g ( ln K ) is the intercept of the regression curves for ln (2 P 2 ) ¼

4 [14] With the surface fitting toolbox in MATLAB the combination ¼ and ¼ was found to provide the best fitting result, leading to the following parametric form of equation (4): ln S ¼ 0:9 ln K ln 2 P 2 0:04 2 P ln K : ð5þ [15] This equation provides the relationship between the cumulative percentage of the fastest flowing particles and the sfws occupied by them, for different values of hydraulic conductivity standard deviation of a fracture aperture field. Based on its derivation, equation (5) is a good approximation for the region 0.42 < P < 0.96 and provides sufficient agreement for the remaining of the P values, so that it can used for an overall analysis. Nevertheless, if another range of P values is of particular interest for a different application (for example, if the fastest flowing/most channeled particles in region P < 0.42 are of special interest), developments as presented above can be repeated to emphasize the fit to these particular ranges. 3. Application of the sfws to Fracture Network Simulations 3.1. Fracture Network Simulations [16] In this section we present a methodology that incorporates the effect of FWS on matrix diffusion, when simulating solute transport in a network of fractures. The method is simple to implement, is computationally feasible, and allows the effect of varying fracture aperture (and thereby varying FWS) to be examined. We shall first describe the generation of the fracture networks on which the calculations of flow and particle transport are to be performed. Then, the method to include the effect of a variable FWS, due to heterogeneity of the individual fractures, is presented. Finally, the impact of the FWS on solute transport in the fracture networks is analyzed, for three different heterogeneity scenarios. [17] Multiple realizations of fracture network models are generated based on the data on statistical distributions for fracture length, orientation, density, and termination properties. These inputs are derived from geological data from the field. In our case, data from Sellafield, England, are used (Table 2). These were previously analyzed by Ohman and Niemi [2003], Ohman et al. [2005], and Oden et al. [2008], but without considering the effect of FWS. The blocks containing the fracture network are chosen to be cubic with each side of 7.5 m length. With the parameters in Table 2 in use, each block then contains about 600 fractures. A hundred realizations of these fracture networks are stochastically generated using the Fracman code [Dershowitz, 1998], all based on the same statistical input properties. The fractures in the blocks consist of four fracture sets, one of which is approximately horizontal and the other three are subvertical. Each fracture has a constant aperture and transmissivity, but the transmissivity distribution of all the fractures follows a lognormal distribution with a geometric mean (m log T )of 12.6 m 2 /s and a standard deviation ( log T ) of These data were determined based on the borehole packer test data from depths of m, as described by Ohman and Niemi [2003]. For each realization of the fracture networks, flow is calculated by imposing a specified pressure gradient first in a horizontal and then a vertical direction. Given the flow results, particle tracking is next performed to investigate the advective particle transport and travel time distribution across the block. Both the flow and the particle transport are simulated using the matrix and fracture interaction code (MAFIC) [Miller et al., 1999]. More details concerning the generation of the fracture networks and simulations without the effect of heterogeneity within the fracture plane, and thereby a variable FWS, can be found in the work of Ohman and Niemi [2003] and Ohman et al. [2005] Accounting for the Effects of Matrix Diffusion and Sorption [18] Oden et al. [2008] developed a method to include the effect of matrix diffusion when simulating flow and transport in fracture networks. The approach is based on the particle-tracking method by Yamashita and Kimura [1990]. In the approach, the advective particle travel time in each fracture is first calculated without the effect of matrix diffusion, by means of the fracture network simulation, and then adjusted by accounting for the time delay due to matrix diffusion and sorption. By assuming diffusion into an infinite rock matrix, the relative concentration (C/C 0 ) exiting a fracture can be expressed [Neretnieks, 1980; Tsang and Tsang, 2001; Moreno and Crawford, 2009]: " # Ct ðþ ðk d ¼ erfc P D e Þ 1=2 FWS C 0 ðt t w Þ 1=2 Q ; ð6þ where k d is the linear sorption coefficient (m 3 /kg), P is the rock matrix density (kg/m 3 ), D e is the effective diffusion coefficient (m 2 /s), t w is the residence time for a particle without matrix diffusion (s), t is the residence time for a particle with matrix diffusion (s), and FWS (m 2 ) is the flow-wetted surface for the water flow Q (m 3 /s). In the calculations the FWS/Q term is calculated using the equivalent expression t w /b, where b is the fracture aperture (m). The aperture is calculated from the transmissivity of the Table 2. Data on Fracture Geometry a Fracture Set Mean Dip ( ) Dip Direction ( ) Fisher Perpendicular Spacing S f (m) Total 3-D Intensity P 32 (m 2 /m 3 ) Conductive 3-D Intensity, P 32 Cond (m 2 /m 3 ) a From Ohman and Niemi [2003]. 2244

5 fracture using the cubic law [Witherspoon et al., 1980]. Material properties are lumped together in a factor P m, defined as P m ¼ k d P D e D ; ð7þ where D ¼ 10 9 m 2 /s is the free water diffusion coefficient. An increased value of the dimensionless factor P m results in a larger residence time due to linear sorption and diffusion into the rock matrix. Here k d is the linear sorption coefficient for the mass of microfissured rock. It includes the contaminant whichisinthewaterinthemicrofissuresaswellasthat sorbed to the solids. Thus, k d P can be expressed as k d P ¼ " P þ ð1 " P Þk 0 d s: ð7 0 Þ [19] The k 0 is the sorption coefficient based on the solid rock, s is the solid rock density, and " P is the rock matrix porosity. For a nonsorbing solute (k 0 d ¼ 0), equations (7) and (7 0 ) can be combined to give P m ¼ " P D e =D. [20] In this work the results are first shown for a constant value of P m equal to 10 7, to demonstrate the effect of varying the sfws. Later, we will also discuss the effects of varying this parameter and its interplay with the sfws value. In this approach, particle-tracking simulations are first performed to determine the advective residence times t w for all particles and for each fracture without matrix diffusion. With t w determined and the material properties group for each fracture known, the only unknown in equation (6) is the fracture travel time t when including the retardation due to diffusion or, expresses differently, the time delay t ¼ t t w. Yamashita and Kimura [1990] suggest that to describe the effect, for each fracture in the network that the particle is passing, a random number is drawn from the uniform distribution U[0, 1] and equated with C/C 0. The corresponding time t can then be determined from equation (6) by algebraic inversion. The time t is the particle residence time including both the advective time and the delaying effects of linear sorption and matrix diffusion. [21] In equation (6) the factor FWS/Q is identical to a factor WL/Q commonly used to express the FWS (for review, see, e.g., Larsson et al., [2012]), where Q is the flow rate through the fracture, W and L are the width and the length of the fracture, and WL is the area of the fracture (L 2 ). The term WL/Q is equivalent to the F factor used by Posiva [2009] and the value used by Cvetkovic et al.[1999]orthe t w /b factor, and it is the factor that describes the reaction area for the matrix diffusion and sorption as discussed in more detail by Larsson et al. [2012]. To include the effect of the FWS for each fracture, the FWS/Q factor, as obtained from the fracture network simulations, has to be multiplied by the sfws of that fracture (S), calculated by equation (5): " # Ct ðþ ðk d ¼ erfc P D e Þ 1=2 S FWS C 0 ðt t w Þ 1=2 Q : ð8þ [22] With the addition of the S factor, this equation can be used to calculate the particle residence time t including the effect of sfws area of each fracture along the particle trajectories, as described above. [23] In the network simulations, each fracture is assigned with a constant transmissivity, but it varies from fracture to fracture [ Ohman et al., 2005; Oden et al., 2008]. The effect of fracture aperture variability is examined through the sfws (equation (5)) as a function of the standard deviation of the log hydraulic conductivity within individual fractures (or transmissivity) while keeping its geometric mean equal to the assigned transmissivity. In the present network simulations the apertures are not expressed explicitly, only indirectly as the fractures are assigned different transmissivity values. In other words, when comparing two fractures, one constant transmissivity fracture and the other a heterogeneous fracture with the same geometric mean, the total flow rate through the two fractures will be the same. But the distributions of the flow within the fractures will differ, and the areas that meet the flowing water will also be different. In equation (8) the factor FWS/Q, which is equal to t w /b, is the term accounting for the surface area over the volume ratio experienced by particles within a fracture, and it controls the matrix diffusion and sorption given the advective travel time and P m value. In order to account for the heterogeneity-induced channeling, the t w /b factor, calculated with the values from the fracture network, is multiplied with the S value as done in equation (8). [24] To calculate an S value using equation (5), a fraction of particles to be considered has to be chosen. It is not obvious what the appropriate value of P should be. A suitable value of P would be within the range 0.9 P 1. In this study the 90% fraction is used, due to the fact that it is a likely recovery rate for tracer tests and that 10% slowest and least concentrated particles are probably of less interest. This is also a conservative choice for the evaluation of contaminant transport since it represents a smaller S or less retardation case. If there is an interest in the first arrival of particles a smaller value of P is recommended, e.g., P ¼ 0.1. Our interest in this work is to capture the mean behavior of the peak and median arrival time of the contaminant particles; therefore, a large value is used. When using equation (5), information of aperture (and conductivity) standard deviation is needed. There are several possible ways to determine the value of ln K. The direct method is of course by measurements of the aperture variances of core samples containing fractures. It is however often problematic to get reliable information in this way for the whole fracture network. Also, in a fracture network the connectivity between fractures is possibly not accounted for using this method, where flow is strongly affected by how well the fractures are connected with each other. Thus, it may be better to use an indirect method by applying equation (8) in a calibration study against field data. This can be done by constructing fracture networks based on the data from tracer test site and match the tracer breakthrough curves of both conservative and sorptive tracers to the breakthrough curve from the numerical study by varying the S value. In this work we use the ln K (or the sfws) as a sensitivity parameter and look at the effect of this parameter on the particle breakthrough curve Simulation Scenarios for Examining the Effect of FWS on Particle Transport in Fracture Networks [25] The effect of taking into account the sfws on solute transport was addressed by a set of simulations where this 2245

6 Table 3. Different Scenarios for Assigning the Standard Deviation for Hydraulic Conductivity for the Fractures in the Networks Scenario Subscenarios A One ln K for all fractures ln K ¼ 0.5, 1, 1.5, 2, 2.5, 3, 4 B Random ln K sampled from a ln K ¼ [0, 4], [1,3] uniform distribution C Different ln K for vertical and horizontal fracture sets vert ¼ 2, hor ¼ 0.5, 1, 2.5, 4 vert ¼ 0.5, 1, 2.5, 4, hor ¼ 2 parameter was varied. Three different scenarios for assigning fracture sfwss are being tested (Table 3). In scenario A, all fractures are assigned the same standard deviation of hydraulic conductivity, and this constant value of ln K is varied from 0.5 to 4. This range of ln K values is reasonable, especially for fractured rocks at depth [Larsson et al., 2012; Tsang et al., 2008; Winberg et al., 2003; Doughty and Uchida, 2003]. In addition to these realistic values, much lower values for sfws (corresponding to very high ln K values) were also tested to confirm that the results converged to the case with no matrix diffusion. This case that corresponds to a sfws value down to 10 4 is tested, which corresponds to the fraction of the fracture where 90% of the flow would then take place. With the controlling factor FWS/Q of equations (6) and (8) in mind the potential for matrix diffusion and sorption should decrease and go to zero. For scenario B, all fractures were assigned a ln K randomly drawn from a uniform interval. Two different uniform intervals were tested, as shown in Table 3. In scenario C the fractures in the vertical and horizontal directions were assigned different ln K values, based on the assumptions that fractures in the horizontal and vertical sets, respectively, have been created by similar geological processes. Two cases were examined, in one the vertical standard deviation was kept constant and horizontal one was varied, and in the other case, the opposite was done. All cases were simulated for both a specified horizontal and vertical hydraulic gradient across the fracture network block. Therefore, the results are obtained for both directions A Study to Verify the Model [26] Verification of a fracture network model is not straightforward, due to difficulties of finding a proper set of field data or an analytical solution to compare. The basic fracture network model used here is an established code, based on the geological data from Sellafield, United Kingdom, as described by Ohman and Niemi [2003] and Ohman et al. [2005]. The fracture networks are built so that it comply with the fracture statistics from the Sellafield site, given in section 3.1, and the flow and solute transport simulations are performed in a similar way as the well-established work of Long et al. [1982], Cacas et al. [1990], and many others. These results are shown in section 4 as the no matrix diffusion case (e.g., black curve in Figure 5). Oden et al. [2008] introduced the matrix diffusion and sorption processes in the constant-aperture fractures in the modeling, using a well-known and verified methodology that was presented by Yamashita and Kimura [1990] and Tsang and Tsang [2001]. The results are shown as the case with full matrix diffusion (e.g., the blue curve in Figure 5). The unique results presented in our present study lay in between the behavior of those two limiting cases of no and full matrix diffusion and sorption. [27] Transport in a fractured rock will experience advective dispersion, both by the contaminant particles traversing different transport paths through the different fractures, each path having a different water flow and therefore residence time. The contaminant cloud will experience further advective dispersion in the single fracture, due to velocity variations within the fracture. In the mentioned fracture network models [e.g., Cacas et al., 1990; Ohman et al., 2005; Oden et al., 2008], only the first kind of advective dispersion has been included, where the contaminant is transported through different fractures of different transmissivity, but the transmissivity/aperture in each fracture is represented by a constant value (corresponding to the geometric mean). Since the water flows are calculated using the geometric mean value, the models do not take into account the intrafracture-scale dispersion. This simplification is supported by a study of Painter [2006], who concluded that the fracture-scale advective dispersion in the apertures is relatively small relative to network dispersion and has no significant effect on the particles travel times over the fracture network. In addition to the dispersion, the transport of contaminants is affected by matrix diffusion and sorption with sfws effect, which has significant effects on the tracer breakthrough curves. Hence, a verification is done to verify the effect of variable sfws on these retarding processes. [28] The verification method is chosen as one-directional tracer injection from one side of the fracture represented by a 2-D heterogeneous transmissivity field for a time period, followed by withdrawal at the same rate. The tracer breakthrough curve during the withdrawal for a heterogeneous fracture with P m ¼ 10 3 is compared to a homogeneous fracture with a combination of S and P m. The different S values correspond to different P values calculated with equation (5). At the start of the injection period, a pulse of water with tracers of constant concentration is injected into the field over the whole fracture width, followed by injection of water for a period of time. The particle transport involves effects of both advective dispersion and matrix diffusion/sorption. In order to verify only the matrix diffusion/sorption effect, the gradient and the water flows are reversed at the end of the injection period, and the particles are followed until they return to the injection line. This is the withdrawal period. The combination of injection and withdrawal ideally cancels out the advective dispersion part of the transport, and thus, only the retardation effect of the matrix diffusion and sorption processes is studied. Using the advective travel time over each numerical element each particle has traversed, a time delay is calculated using equation (6) for the heterogeneous fracture and using equation (8) for the homogeneous fracture. Further details may be found in Larsson et al. [2013]. In Figure 4 a comparison is shown of the cumulative breakthrough curve from detailed calculation of tracer transport over a heterogeneous fracture with ln K ¼ 2.30 (which represents the ground-truth results) against the cumulative breakthrough curve for constant-aperture fracture with an S factor as shown in equation (8) (which represents the approach proposed in this paper). The comparison shows very good agreement to the S value of 0.65, which corresponds to ln K ¼ 2.30 and P ¼

7 Figure 4. Verification for the effect of matrix diffusion and sorption over a single fracture. The verification is done by comparing the result of a simulation for a heterogeneous fracture ( ln K ¼ 2.31) and with the result of simulations for a homogeneous fracture ( ln K ¼ 0) with four different S values; the comparison is performed for two different P m values for one single realization of the heterogeneous fracture. The different S values correspond to P values calculated with equation (5). The best matching curve for both P m ¼ 10 3 and 10 6 corresponds to the S value calculated with a P ¼ (equation (5)). This agreement provides a degree of confidence to the validity of our approach in the use of equation (8) for modeling the flow and transport through a network of fractures with variable apertures. It is our goal to use this kind of results with an injection withdrawal scheme in order to determine the suitable P value to use for different purposes of modeling, but it is an ongoing investigation and a subject for another article. 4. Results 4.1. All Fractures With Same Heterogeneity Statistics (Scenario A) [29] In scenario A, a single standard deviation of the hydraulic conductivity is applied to all fractures in the entire domain, and several different ln K values are tested. The combined breakthrough curves from a hundred realizations are shown in Figure 5. The results show how the decreasing sfws (S) changes the breakthrough curve from a behavior similar to the case of full matrix diffusion over the whole fracture surface (slow transport), to behaving similarly to the case with no matrix diffusion at all (fast transport). In all cases, no qualitative differences were found between the horizontal and the vertical gradient results, as can be seen in Figures 5a and 5b. Therefore, only the results from the horizontal gradient simulations will be discussed in more detail. The overall slopes of the cumulative breakthrough curves increase systematically with decreasing S, with curves corresponding to high values of S exhibiting long tails due to a part of the mass being retarded, while the highly channeled cases show significantly less retardation. Figure 5. Simulated cumulative breakthrough through fracture networks for different values of sfws (S) for(a)a horizontal gradient and (b) a vertical gradient. The No Diff curves shows the cumulative breakthrough curve for simulations without matrix diffusion and sorption included, and M Diff shows the curve of the simulations with matrix diffusion and sorption for the full fracture surface (scenario A). Figure 6 and Table 4 show the median arrival time and the slope of the cumulative breakthrough curve at this median value as a function of the sfws. Figure 6 shows these values as a function of the lumped factor S (P m ) 1/2 ; however, for these simulations a constant P m value has been used. It can be seen that the dependencies are approximately linear over the interval of S varying from 0.1 to 1. For the strongly channeled cases of S < 0.1, instead, both the median transit time (t m¼0.5 ) and the slope become relatively constant and close to the results with no matrix diffusion (filled square and circle on the left y axis in Figure 6). Figure 6. (left axis) Logarithm of median travel time and (right axis) slope of the cumulative breakthrough curve as a function of logarithm of S (P m ) 1/2. The filled squares and circles on the left- and right-hand axes correspond to (right axes) the extreme cases of full matrix diffusion over the fracture surface and (left axes) no matrix diffusion, respectively. In the simulations a constant P m value of 10 7 has been used (scenario A). 2247

8 Table 4. Slope of the Cumulative Breakthrough Curve and Arrival Time for 50% of the Mass for a Horizontal Gradient, as a Function of S and ln K Value Case/S ln K Slope of CDF t m¼0.5 Matrix diffusion 2.0E S ¼ 1 2.0E S ¼ E S ¼ E S ¼ E S ¼ E S ¼ E S ¼ E S ¼ E S ¼ E 4 94 S ¼ E 3 48 S ¼ E 3 19 S ¼ E S ¼ E S ¼ E No matrix diffusion 4.2E In other words, there appears to be a breakpoint close to the S value of 10 1 ¼ 0.1, above which the logarithm of the median mass transit time (t m¼0.5 ) and the slope of the cumulative breakthrough curve can be estimated with a simple linear relationship with respect to the logarithm of S. In particular, for median transit time t m¼0.5, we have log 10 t m¼0:5 3:5 þ 2log 10 S for S > 10 1 and P m ¼ 10 7 so that t m¼0:5 / S 2 : ð9þ ð10þ [30] The proportionality in equation (9) can be understood by examining equation (8), where the factor S is divided by a factor including the square root of the residence time including matrix diffusion. An increase in S will be equaled by the same increase in the squared residence time including matrix diffusion for each fracture, and therefore, a similar result is seen for block-scale transport. [31] The behavior of the breakthrough curve is dependent not only on the S value, but also on the material properties of the rock, in the form of the lumped factor of (k d p D e ) 1/2 S in equation (8), which can also be expressed in terms of the material property group term P m as (P m D) 1/2 S. Figure 6 shows the median travel time and the slope of the cumulative breakthrough curve as a function of this factor. If this lumped factor is constant, the method will yield the same result regardless of the individual values of the P m, D, and S. The factor P m describes the capacity of the rock material for sorption and diffusion, a larger value causing a larger retardation on the breakthrough curve. In this study a material property group value P m of 10 7 is used. As an example, for a given fracture network, a combination of P m ¼ 10 7 and S ¼ 0.1 ( ln K ¼ 5.5) will yield the same result as the combination of P m ¼ and S ¼ 0.5 ( ln K ¼ 2.3). This means that for P m ¼ the breakpoint above which the equations (9) and (10) hold is S ¼ 0.5 (or ln K ¼ 2.3), and reversibly below the breakpoint the system behaves similar to the no matrix diffusion case. Figure 7 shows the combinations of S and Figure 7. Line indicating the different combinations of S and P m that result in the same breakthrough curve. The line shows the breakpoint (lower limit) for the validity of equation (10) such that above the line the median travel time is proportional to the square of the sfws, while below the line it is approximately constant (scenario A). P m that will produce the same breakthrough curve; these combinations can be seen as a straight line in Figure 7. In Figure 7, for combinations of the S and P m values falling on the area above the straight line, equation (10) holds, while for values below the curve the behavior is close to the case without matrix diffusion. [32] The median travel time is the arrival time for the 50% of cumulative mass. The arithmetic mean of the travel times is not used in this study, because it is dominated by a few very slow particles with very large travel times. In other words, it is largely influenced by the behavior of the long tail of the breakthrough curve. The cumulative mass at the arithmetic mean is not constant and vary considerably. For moderate heterogeneity case, the arithmetic mean of the travel times for a conservative tracer can be estimated by the factor V/Q, where V is the volume of the domain, and Q is the flow rate through it. However, it is not a suitable measure in this study, due to the influence of tail mentioned. [33] The particle frequency (number of particles arriving through the fracture network per unit of time) is a measure that qualitatively shows the behavior of concentration, averaged over a time period (Figure 8). The particle frequencies, for the case of full matrix diffusion and S values close to one, show a relatively continuous decrease over time with no clear peak arrival. The cases with no matrix diffusion and small S values are showing higher initial values, are first increasing showing a peak around 1 10 years, and are after this period decreasing fast, with a faster decrease for smaller S values. As shown earlier for the cumulative breakthrough curves, a large S value close to one has a particle density curve very similar to the curve with matrix diffusion for the full fractures. On the other hand, the particle density curve for very small S values is qualitatively equal to the simulations without matrix diffusion. For S ¼ 10 4 the particle density curve is overlaying the curve for no matrix diffusion until a time of 400 years, and the amount of mass arriving later than 400 years corresponds to only 0.06% of the total mass. Three features in the concentration breakthrough curves shown in Figure

9 Figure 8. (a) Particle frequencies as a function of transit time for a large range number of S values and (b) frequencies for smallest S values at early times (scenario A). are of interest. First, at early times all curves in Figure 8 show very similar particle frequencies, regardless of S value (if the matrix diffusion is included). It is indicating that the transport through the fracture network is dominated by advective processes, even if the matrix diffusion is included in the simulation. Second, the longtime tails show the following dependence log 10 C ¼ const ð3=2þlog 10 t or C / t 3=2 : ð11þ [34] The t 3/2 dependence is typical for long-tailed breakthrough curve dominated by diffusion processes and can be expected for the longtime tail behavior of the tracer breakthrough curves for dual-continuum tracer transport [Konosavsky et al., 1993; Rumynin, 2011]. Third, the curves for S 10 3 have a change of slope at about log 10 t 3(see Figure 8b), the early part of the tail displaying approximately a 4 power law slope and after t 10 3 years it converges to a 3/2 slope. The most likely explanation to this is that in these highly channelized cases the early behavior is clearly dominated by fast advective flow with essentially no matrix diffusion, while at late times the matrix diffusion effects become evident. In the less channeled cases with larger S values (e.g., the curve for S ¼ 0.1 in Figure 8b), there is no change in slope, possibly because the transport is dominated by the diffusive processes at an earlier time. For the simulations without matrix diffusion, the tail of the breakthrough curve is displaying a slope steeper than 4 on the logarithmic scale, which is steeper than the 2 power law tail found by Becker and Shapiro [2003]. The steep slope possibly indicates that the fracture network has a number of fairly well-connected flow paths and that the advective transport occurs predominantly through them. [35] The results presented here are averaged over the 100 realizations simulated for each case. The variations among realizations are relatively large and originate from the different configurations of the fractures in the network block. For the simulations with ln K ¼ 2 the standard deviation ( tm ) for the cumulative mass arrival time of 0.25 is 210 years (t m¼0.25 ¼ 165), 0.5 is 2400 years (t m¼0.5 ¼ 2200 years), and 0.75 is 28,000 years (t m¼0.75 ¼ 31,000 years). There are realizations where most of the mass arrives fast, due to a small number of highly conducting and connected fractures. Large early outflow concentrations can be found in about 15 of the 100 realizations, where some very wellconnected fractures are providing a short and fast path through the block. Other realizations that are predominantly not well connected have much longer transit times. The variability arises from the fracture network realizations, not from the method to include the fracture aperture variability and its effect on the matrix diffusion and sorption presented here. The variability in the fracture network simulations for the case without matrix diffusion is further discussed by Ohman et al. [2005]. The corresponding standard deviation values for the simulations without matrix diffusion are much smaller (e.g., tm¼0.5 ¼ 3.3 years) and for the simulations with matrix diffusion and sorption for the full fracture are larger than the above stated values (e.g., tm¼0.5 ¼ 750 years), but in relation to its cumulative arrival time the values are in the same order of magnitude. This variability will average out in larger flow domains where particles flow in multiple fractures, a situation which is usually of interest in actual predictive simulations, such as those done by Oden et al. [2008] and Ohman et al. [2005]. Therefore, for eventual application to flow and transport of a regional scale, we conclude that it is the average behavior that is the most important result. 2249

10 Figure 9. Cumulative particle breakthrough through the networks as a function of time for two different random intervals of ln K, compared to a case with ln K ¼ 2 (horizontal gradient case; scenario B) Fractures With Randomly Sampled Heterogeneity Statistics (Scenario B) [36] In scenario B every fracture is randomly assigned a conductivity standard deviation ( ln K ) from a uniform distribution. Two different uniform intervals are considered (Table 3) and compared to the case with a single standard deviation equal to the mean of the interval ( ln K ¼ 2) being applied to all of the fractures. The comparison shows no practical difference between the two different distributions and the constant ln K case, as can be seen in Figure 9. In other words, a randomly assigned variability appears to be averaged out during the transport through the network, with the alternating large and small S values producing a similar result than the corresponding mean value of S. The practical importance of this result is that it is not necessary to consider the fracture-to-fracture variability if the mean transport behavior is of interest and if there is no reason to believe that there are some directional differences or differences related to the fracture sets in the heterogeneity characteristics of the individual fractures Different Aperture Statistics for Fracture Sets With Different Orientations (Scenario C) [37] For scenario C, ln K (and the S value) is assigned differently depending on whether the fracture belongs to a horizontal or vertical fracture set (Table 3). The result is different depending on whether the standard deviation in the horizontal or in the vertical direction is varied, as can be seen from the cumulative breakthrough in Figure 10. For these fracture network models (properties shown in Table 2), the vertical fractures are about three times more abundant than the horizontal fractures, and thus, it follows that the breakthrough curves vary more when varying the ln K in the vertical rather than the horizontal direction. Figure 10 shows simulations for a horizontal gradient; the simulations with a vertical gradient show a similar behavior. 5. Summary and Conclusions [38] Three major conclusions can be drawn from this study. The first conclusion is a method that has been developed to include efficiently the effect of matrix diffusion and linear sorption for tracer transport through fracture networks, where individual fractures are heterogeneous, exhibiting various degrees of flow channeling caused by the variability in their aperture distributions. This has not Figure 10. Cumulative particle breakthrough through the fracture networks for cases with (a) constant vertical and varying horizontal ln K and (b) constant horizontal and varying vertical ln K (horizontal pressure gradient case; scenario C). 2250

11 been done in a computationally feasible way before. It is accomplished through the concept of sfws, which is the fraction of the total fracture surface area available for diffusion into and sorption onto the rock matrix. Larsson et al. [2012] presented a functional relationship (equation (5)) that relates the sfws to the ln K of the fracture. Thus, knowing the hydraulic conductivity standard deviation of the fracture allows the determination of its sfws, i.e., the surface available for matrix diffusion and sorption. In this paper, an approach is presented to incorporate this expression into a comprehensive fracture network analysis, allowing the inclusion of the effect of a smaller sfws area into solute transport calculations in a numerically undemanding way. By including the effect of the sfws into the fracture network simulations, its influence on matrix diffusion in fracture networks with different levels of heterogeneity and flow channeling can be studied. The presented results are the average of 100 realizations for each case, among which there is large variability. About 15% of the realizations contain a few very well-connected fractures that provide a fast pathway through the fracture block, giving raise to the large variability, but this variability is dependent on the chosen block size. The average results are considered to be a better representation of the behavior over the rock domain. The full range of cases was considered, ranging from the case with full matrix diffusion over the entire fracture surface to the case with no matrix diffusion. For very small values of sfws, the breakthrough curves become increasingly similar to the case with no matrix diffusion. For S ¼ 1, on the other hand, the breakthrough curve is identical to the result of the simulation including full matrix diffusion. [39] The second major conclusion is that there are two different ranges of results depending on the value of a lumped factor S (P m ) 1/2, whether it is greater or less than a breakpoint of For large sfws values (for the presently analyzed data with P m ¼ 10 7, these are values of S > 0.1, which corresponds to a ln K value up to approximately 5.5), the median tracer arrival time is found to be proportional to the square of S. This is a quality derivable from equation (8) where the argument has the ratio S/t 1/2. This has the same effect on a single fracture (as it is applied for equation (8)) as on the block-scale transport. Such an observation may be important for the detailed transport calculations to the investigation of potential leakage of radionuclides from a nuclear waste repository. For values below the breakpoint of the lumped factor S (P m ) 1/2 ¼ , the change in median arrival time is much slower. And it corresponds essentially to S ¼ 0 results. A low sfws value means that the majority of the transport occurs in just a small fraction of the fracture, with small active areas for matrix diffusion. As pointed out, the actual value of this lumped parameter depends on both the P m value and the S value, where the P m value is a material property value that describes the potential of the rock matrix for diffusion and retardation. For example, a P m value of 10 7 gives a breakpoint for S value of 0.1 (the case discussed above), whereas for P m ¼ the breakpoint S value is 0.5 or ln K ¼ 2.3. The results of this analysis are summarized in Figure 7, showing the combination of P m and sfws values, above which there is a quadratic relationship between the median travel time and the sfws, and below which the median arrival time is relatively constant and close to the case with no matrix diffusion. In our example, when we compare the full matrix diffusion case with a case with an aperture distribution corresponding to ln K ¼ 4 (which corresponds to S ¼ /4), the median travel time decreases to approximately one sixteenth of the median travel time of the full matrix diffusion case. These results show that not including the effect of fracture-scale aperture heterogeneity on matrix diffusion and sorption, when analyzing the tracer breakthrough curve for a tracer migration field test, will result in an estimation of the matrix diffusion and sorption properties that is smaller than the one measured in the laboratory. This observation has been found in many field-scale tracer test programs. [40] Examining the breakthrough curve for particle densities (a measure showing concentration variation over time) the advective transport processes dominate for the early arriving mass, since all the simulations regardless of matrix diffusion properties are similar. For late arrival times the curves for simulations including matrix diffusion (from matrix diffusion for the full fractures to very small P m values) display a typical 3/2 slope on the logarithmic scale, which is typical for transport processes dominated by diffusion. [41] The third major conclusion is that the detailed data on the aperture heterogeneity are not needed for simulations of transport through fracture networks, but an average value is sufficient to use in modeling. This is important for field applications, where it is necessary to find the representative value for the examined domain. If the fractures were assigned a random sfws from a uniform distribution, we find that the breakthrough curves (the mean of multiple realizations) are essentially the same as if all the fractures are assigned a single sfws value equal to the mean of the distribution. In other words, the effects of the variable FWS are averaged out when the particles are traversing alternating fractures of high and low sfws. This observation may suggest that we can use a simplified model (with much simpler data requirements) when assessing the effects of fracture aperture variability on flow and transport in 3-D networks, where we do not need details of sfws values on the fractures, but only the mean value. There was, however, somewhat of an effect when different fracture sets (with different orientations) were assigned different heterogeneity characteristics. But even here the averaging effect of the fracture network is quite effective, and the variability is much less than the variability between the base cases of different overall sfws values. [42] Acknowledgments. This work has been financed by the Swedish Research Council Formas (grant ), which is gratefully acknowledged. Discussions with and assistance from Christine Doughty of Lawrence Berkeley National Laboratory, Berkeley, are very much appreciated. The authors thank the Associate Editor and the anonymous reviewers of Water Resources Research for their careful review and constructive comments. The last author would also like to acknowledge the partial support of the JAEA-LBNL binational collaborative project under U.S. Department of Energy contract DE-AC02-05CH11231 with Lawrence Berkeley National Laboratory. References Andersson, P., J. Byegård, and A. Winberg (2002), Final report of the TRUE block scale project, Technical Report TR-02-14, Swed. Nucl. Fuel and Waste Manage. Co., Stockholm, Sweden. 2251