Flow-dependence of matrix diffusion in highly heterogeneous rock fractures

Transcription

1 WATER RESOURCES RESEARCH, VOL. 49, , doi:1.12/213wr14213, 213 Flow-dependence of matrix diffusion in highly heterogeneous rock fractures Vladimir Cvetkovic 1 and Hrvoje Gotovac 2 Received 31 May 213; revised 8 October 213; accepted 25 October 213; published 2 November 213. [1] Diffusive mass transfer in rock fractures is strongly affected by fluid flow in addition to material properties. The flow-dependence of matrix diffusion is quantified by a random variable ( transport resistance ) denoted as b [T/L] and computed from the flow field by following advection trajectories. The numerical methodology for simulating fluid flow is mesh-free, using Fup basis functions. A generic statistical model is used for the transmissivity field, featuring three correlation structures: (i) highly connected non-multi- Gaussian; (ii) poorly connected (or disconnected) non-multi-gaussian; and (iii) multi- Gaussian. The moments of b are shown to be linear with distance, irrespective of the structure, after approximately 1 integral scales of ln T. Percentiles of b are found to be linear with the mean b when considering all three structures. Taking advantage of this property, a potentially useful relationship is presented between b percentiles and the fracture mean water residence time that integrates all structures with high variability; it can be used in discrete fracture network simulations where T statistical data on individual fractures are not available. Citation: Cvetkovic, V., and H. Gotovac (213), Flow-dependence of matrix diffusion in highly heterogeneous rock fractures, Water Resour. Res., 49, , doi:1.12/213wr Introduction [2] Advection by groundwater through fractures is the main transport mechanism for dissolved tracers in fractured rock. For relatively slow groundwater movement, diffusive mass transfer is another important transport mechanism, as demonstrated in the field on a variety of scales [Shapiro, 21; Reimus et al., 23; Cvetkovic et al., 27; Reimus and Callahan, 27; Cvetkovic, 21; Cvetkovic et al., 21; Cvetkovic and Frampton, 21]. A controlling parameter for tracers subject to reactions or retention that involve diffusion into the rock matrix, is the transport resistance b [T/L] [Cvetkovic et al., 1999; Cvetkovic and Frampton, 21]. Theoretical and simulation studies have shed light on the hydrodynamic control of retention [Cvetkovic, 1991; Moreno and Neretnieks, 1993; Cvetkovic et al., 1999; Cheng et al., 23; Frampton and Cvetkovic, 27; Cvetkovic and Frampton, 212; Larsson et al., 212], however, a number of outstanding issues still remain, such as the impact of heterogeneity structure, as well as the need for simple and reliable estimators. [3] Recent work focused on the transport resistance in discrete fracture networks, using numerical simulations and 1 Division of Water Resources Engineering, Royal Institute of Technology, Stockholm, Sweden. 2 Department of Civil Engineering, Architecture and Geodesy, University of Split, Split, Croatia. Corresponding author: V. Cvetkovic, Division of Water Resources Engineering, Royal Institute of Technology, Brinellv 28, Stockholm SE- 144, Sweden. (vdc@kth.se) 213. American Geophysical Union. All Rights Reserved /13/1.12/213WR14213 field data [Cvetkovic and Frampton, 212], where internal variability in fractures was neglected and the main interest was the effect of the network. Although neglecting internal fracture variability may be appropriate for dense networks, in sparsely fractured rock the internal heterogeneity of fractures will be increasingly important. [4] In this work, we present a simulation study of flow and transport in a rock fracture with three correlation structures two of which are non-multi-gaussian, with two levels of heterogeneity (high and low). Accuracy in the high heterogeneity case is ensured using a recently developed collocation method with Fup basis functions [Gotovac et al., 27]. We first focus on the statistical nature of the transport resistance b and its dependence on the correlation structure of T, and then seek simple estimators for key statistical properties of b that can be used in applications. 2. Transport Resistance [5] Consider a single rock fracture on a length scale L where a mean steady state hydraulic gradient is applied parallel to the x axis (Figure 1). The rock fracture is assumed conductive for water flow with spatially varying transmissivity T [L 2 /T]. The transmissivity can in principle be measured by hydraulic tests at any number of spatial locations, whereby the statistics of the spatially varying T can be inferred, such as the distribution and correlation scale. We shall assume that the statistics of T are known. [6] The rock matrix adjacent to the fracture is assumed to be porous and water saturated. Hence a solute particle transported along a trajectory in the fracture may be temporarily transferred and stored in the matrix, by mechanisms of diffusion and possibly sorption. 7587

2 [8] If h, D e, and R are approximately uniform (effective values), then p B5jb b ffiffiffiffiffiffiffiffiffiffiffi hd e R (2) p where j½l= ffiffiffi T Š is a material property group, and b [T/L] is the transport resistance (or the hydraulic control parameter) defined by b ð s 2ds d½xðs ÞŠ (3) [9] The variable b (3) may be interpreted as the particle advective travel time from the injection point to L, weighted by the PDF of the particle being in the proximity of the fracture wall (or rock matrix) along a trajectory. Note that being in the proximity of the fracture wall implies higher probability for the particle to diffuse into the rock matrix; hence, smaller d yields larger b, or larger mass transfer. Implicit in the twodimensional formulation for sufficiently slow advection, is that fully mixed conditions prevail, with 1/d being the uniform particle position density in the fracture transverse direction. [1] If Darcys law is applied for the x component of the flow rate q 5 TJ [L 2 /T], then ð L 2dx bðlþ5 T½XðsðxÞÞŠJ½XðsðxÞÞŠ (4) Figure 1. Examples of three realizations of each structure for simulating flow and advective transport in a rock fracture, with advecting particles distributed in space at an arbitrary time: (a) The connected Gaussian (GC) field; (b) The disconnected Gaussian (GD) field; (c) The multi-gaussian (GM) field. The blue color designates low transmissivity zones and red high transmissivity zones, with the colour spectrum in between. Only the inner simulation domain is shown. [7] Two-dimensional flow and advective transport are considered in the fracture, coupled with one-dimensional Fickian transport into the matrix. The mass balance equations are summarized in Appendix A. The tracer residence time probability density function (PDF) (A4) is controlled by two parameters: The water residence time s (A5), and B [T 1/2 ] defined by ð s p 2 ffiffiffiffiffiffiffiffiffiffiffi hd e Rds B (1) d½xðs ÞŠ where h is the matrix porosity, D e [L 2 /T] effective diffusivity for the rock matrix, R is the retardation coefficient for the rock matrix, d [L] is the spatially dependent fracture aperture and X(t) is the particle advection trajectory. For a nonsorbing tracer, R 5 1. where T is the transmissivity and J the negative x component of the hydraulic gradient; the mean hydraulic gradient J is assumed parallel to the x axis. Expression (4) can be better understood if b is compared to the water residence time s, defined in (A5). It is seen that the difference between the two is a spatially varying porosity, which in a fracture is d 5 2b. From (3) and (A5), we can define an effective half-aperture as s5b eff b ; b5xs (5) where 1/b eff x [1/L] is an effective specific surface area along the trajectory. b eff can be obtained statistically, from the correlation between b and s [Cheng et al., 23; Frampton and Cvetkovic, 27; Cvetkovic and Frampton, 212]. [11] A porous medium analogy would be the following. Given a heterogeneous hydraulic conductivity field K(x), the porosity is defined by a functional relationship h 5 h(k) (in the fracture this relationship is typically the cubic law). Analogous two variables for a porous medium are then s5 ð L hðkþdx KJ ð L dx ; b5 KJ where again, J is the spatially dependent x component of the hydraulic gradient. Note that advective transport with spatially variable porosity has been considered in the literature with typically h ln K [e.g., Dagan, 1989]. 3. Methods [12] In order to study the statistics of b (3) for highly heterogeneous rock fractures, we need to quantify fluid flow 7588

3 on the field scale, say >5m. Fluid flow in natural fractures is complex due to the complicated structure of fracture surfaces, or pore space, similar to granular porous media. In fact, analytical and simulation studies of porescale laminar flow, are relevant for gaining basic understanding in both single fractures and granular porous media [Kitanidis and Dykaar, 1997; Bayani Cardenas et al., 27; Bayani Cardenas, 28; Bayani Cardenas et al., 29; Bolster et al., 29]. Studies with analytical solutions of two-dimensional Navier-Stokes equations, for instance, reveal immobile zones as nonturbulent eddy structures [Kitanidis and Dykaar, 1997; Bolster et al., 29]. [13] Quantifying fluid flow in fractures based on aperture (or pore ) geometry may be feasible on the laboratory scale. On the field scale, however, aperture structure cannot be directly inferred or verified, and fracture transmissivity is the only measurable hydraulic property. If the flow rates are sufficiently low (e.g., in crystalline rocks), fracture flow is equivalent to a two-dimensional, heterogeneous porous medium with Darcys law applicable. In addition to transmissivity, porosity (or aperture in a fracture) is also required when computing advective transport. [14] Typically, the local aperture d and transmissivity T are related by the idealized cubic law as d T 1/3 valid for parallel plates, which leads to the so-called Reynolds lubrication equation [Snow, 1968; Brown, 1987; Zimmerman and Bodvarsson, 1996]. Limitations of the cubic law (or the Reynolds equation) for heterogeneous fractures have been investigated based on detailed flow simulations [Mourzenko et al., 1995; Zimmerman and Bodvarsson, 1996; Nicholl et al., 1999; Mourzenko et al., 21; Brush and Thomson, 23; Konzuk and Kueper, 24]. Empirical alternatives based on analysis of field observations have also been proposed [Dershowitz et al., 23; Hjerne et al., 21] and their implication for transport and retention assessed [Cheng et al., 23]. [15] The main objective of this work is to quantify the statistics of b. By relying on the flow rate q 5 TJ in (3), we do not require the assumption of the cubic law, or the Reynolds equations. In contrast, computations of the water residence time s (A5) [e.g., Tsang and Tsang, 1989; Tsang and Neretnieks, 1998] require the cubic law (or an alternative empirical hydraulic law [e.g., Cheng et al., 23]). Likewise, when estimating the specific surface area x b/s (5) we also need to invoke a hydraulic law; for our illustrations related to x, computation of the water residence time will utilize the cubic law (B3) as an approximation Transmissivity Statistics [16] Fracture transmissivity T is the basic hydraulic property of fractured rock [NRC, 1996; Neuman, 25]. It is most readily measurable along boreholes either using pump tests or flow logs [e.g., Fransson, 22; Gustafson and Fransson, 25; Frampton and Cvetkovic, 21; Follin, 28]. Pump tests seal off sections of boreholes on a given scale using packers, ranging from say.1 m to tens of meters. Flow logs on the other hand, typically yield T values over a small scale on the order of centimeters. Field data indicate that T is correlated with fracture length when considering a wide range of scales [Dershowitz et al., 23; Hermanson et al., 25; Hartley et al., 26; Neuman, 28]. Since fracture length is commonly assumed to follow a power-law distribution [Walmann et al., 1996; de Dreuzy et al., 21], T in fracture networks then also follows a power-law distribution [Bonnet et al., 21; Hartley et al., 26; Frampton and Cvetkovic, 21]. [17] Whereas hydraulic tests or flow logs in boreholes can yield a significant amount of data for inferring T statistics of fracture networks, the corresponding data for single fractures are much more difficult to acquire. Each data value of T in a fracture requires one borehole, hence in practice, the number of data points for single fractures is very limited; this is the case even in detailed flow and transport experimental studies such as TRUE tests [Cvetkovic et al., 27] where only five T values were available for hydraulic characterization of the test fracture. Cross-hole hydraulic tests do yield additional information, broadening the available data base [Gomez-Hernandez et al., 1997; Hendricks Franssen et al., 1999], nevertheless inferred transmissivity statistics of single fractures are often highly uncertain due to insufficient data. [18] A precondition for T > on a given scale in a fracture, is an open fracture, or nonzero aperture, on that same scale. Aperture statistics have been studied extensively from fracture surface samples of various sizes (typically less than one meter), revealing self-affine (power-law) aperture distributions [Brown et al., 1986; Schmittbuhl et al., 1993; Brown, 1995; Plouraboue et al., 1995]. Other detailed studies using resin injection, for instance, reveal histograms that may be interpreted as log-normal, once the closed aperture zones are accounted for [Hakami and Larsson, 1996; Watanabe et al., 28]. [19] Irrespective of the actual distribution of the aperture inferred on small scales, however, the fact remains that T is not a local property in the same sense as the aperture. Namely, T implies a support scale, that may vary depending on how the tests are carried out, but is certainly larger than the smallest scale on which an aperture can be defined (this being a mathematical point for the aperture as a geometrical property). T values will typically imply support scales from decameters to meters. One consequence is that zero aperture areal fractions due to sealed (zero aperture) zones in a fracture such as those observed e.g., by [Hakami and Larsson, 1996, Figures 8a 8d and 8h], do not necessarily imply existence of zero transmissivity, due to a finite support scale for T. [2] Even if the aperture statistics for a fracture were known on the appropriate scale, relating aperture and transmissivity must assume some form of hydraulic law. As already noted, the theoretically derived cubic law based on the parallel plate model is of limited applicability in real heterogeneous fractures [Brown, 1987; Sillman, 1989; Zimmerman and Bodvarsson, 1996; Nicholl et al., 1999]. Alternative formulations of the hydraulic law are empirical, deduced from field observations [Hjerne et al., 21]. A rigorous alternative to the cubic law is therefore still lacking, and hence inference of T statistics from d statistics, and vice versa, is subject to uncertainty. [21] Given the uncertainty of the T distribution on the field scale, we shall use a generic statistical model for T to investigate b statistics. Specifically, a log-normal distribution of T will be assumed [e.g., Tsang and Tsang, 1989; Cacas et al., 7589

4 199; Ewing and Jayne, 1995; Nordqvist et al., 1996; Tsang and Neretnieks, 1998; Kozubowski et al., 28], with T5T G e Y, Y being a normally distributed variable, i.e., Y! N½; r 2 Y Š and T G is the geometric mean of T. [22] Generic models of T in the literature commonly rely on a multi-gaussian correlation structure [Tsang and Neretnieks, 1998]. The geological realism of multi-gaussian structures has been questioned [Gomez-Hernandez and Wen, 1998], and alternatives that emphasize connectivity and channeling, have been considered for fractures [Tsang and Doughty, 23; Tsang et al., 28]. Many studies have indeed emphasized the significance of channeling in fractures [e.g., Tsang and Tsang, 1989; Tsang and Neretnieks, 1998; Watanabe et al., 29], which in general will be a consequence of both large variability in T [Tsang and Tsang, 1989] and its correlation structure [Cvetkovic and Shapiro, 1989; Tsang and Doughty, 23]. To capture this combined effect, a generic model for non-multi-gaussian structures will be used. [23] Three types of transmissivity correlation structures/ fields will be generated according to the approach proposed by Zinn and Harvey [23]: [24] 1. Connected Gaussian (GC) field with well connected high conductivity channels, but poorly connected low and mean conductivity zones. This type of structure is characterized by effective transmissivity greater than the geometric mean, and higher velocity variations. [25] 2. Disconnected Gaussian (GD) field with well connected low conductivity zones such that transport through low value conductivity fields become more dominant. This type of structure is characterized by effective transmissivity that is less than the geometric mean, and lower velocity variations. [26] 3. Multi-Gaussian (GM) field where extreme conductivity values are poorly connected, while mean zones are well connected. Only for this structure an effective conductivity is equal to the geometric mean; a standard multi-gaussian field (GM) is generated with the Hydrogen software [Bellin and Rubin, 1996]. [27] All the three random fields share nearly identical univariate Gaussian distributions and spatial covariance functions. Thus, all three transmissivity fields share the same basic statistics, the first and second statistical moments, but their structures differ in how the high and low ln T values are connected Flow Simulations [28] The flow equation to be solved for the exp @H exp 5 where T(x) has been normalized with the geometric mean T G and H is the normalized hydraulic head; details of the normalization for all variables are given in Appendix B. The streamline field is solved for a given r 2 Y and a given heterogeneity structure, independent of (B3). b is computed using (3) where J 52@H/@x. For some of the results, we shall compute the water residence time for which the velocity is required; the normalized x component of the velocity v x at a point x is computed as: v x ðxþ5 TJ d 52exp 2 3 YðxÞ rh where we have used dðxþ5exp ½YðxÞ=3Š in view of (B3). [29] Our recently presented simulation methodology, referred to as Adaptive Fup Monte Carlo Method (AFMCM) [Gotovac et al., 27, 29, 21], supports the Eulerian-Lagrangian formulation and separates the flow from the transport problem. It consists of the following common steps [Rubin, 23]: (1) generation of logconductivity realizations with predefined correlation structure, (2) numerical approximation of the log-conductivity field, (3) numerical solution of the flow equation with prescribed boundary conditions to produce head and velocity approximations, (4) evaluation of the transport variables for a large number of trajectories, (5) repetition of steps 2 4 for all realizations, and (6) statistical evaluation of transport variables. Further details on the simulations are given in the Appendix B. 4. Results 4.1. Moments [3] Dimensionless mean transport resistance b E½bŠ is shown as a function of dimensionless distance in Figure 2a for all cases. Whereas for low variability the dimensionless mean slope is approximately 2 for all structures (it is exactly 2 by definition for a homogeneous fracture, see Appendix B), for high r 2 Y (solid lines) the dimensionless mean slope is greater than 2 in the case of a GD structure, and significantly lower in the case of GC structure; for a GM field, the cases of low and high variability coincide. [31] Coefficient of variation of the transport resistance CV[b] is shown as a function of dimensionless distance in Figure 2b for all cases. The black dashed lines were obtained by inferring a macro-dispersivity for b using the simulated moments at x 5 4, and computing the coefficient of variation by the ADE model (see Figure 2 caption p for further details). The ADE shape of the CV 1= ffiffi x is generally preserved by the simulated CV[b]; the deviations are significant at shorter distances, up to approximately 1/4 of the domain. [32] Skewness and kurtosis of the transport resistance are shown as a function of dimensionless distance in Figures 2c and 2d, respectively, for all cases. A decreasing trend of both skewness and kurtosis with distance, generally follows the ADE model, however, the values of skewness and kurtosis deviate significantly from the ADE model for large variability and GC and GM structures. Clearly, the GC structure exhibits the strongest deviation due to highly conductive channels combined with low conducting zones. Note that for the GD structure, simulated skewness and kurtosis are consistent with the ADE model Distribution [33] The distribution of the transport resistance b is shown in Figures 3 for the three structures with r 2 Y 58at x 5 4, as the CDF and CCDF curves. In addition to the simulated values (symbols), we include in Figure 3 a comparison with the ADE model (dashed lines), where the first two moments are constrained. The ADE model significantly deviates from the simulated data for the GC and GM 759

5 Figure 2. Moments of the dimensionless transport resistance b as a function of dimensionless distance x and for two values of r 2 ln T : (a) mean; (b) coefficient of variation; (c) skewness; (d) kurtosis. The dotted lines are applicable for the ADE model and are obtained by inferring a dispersivity from the simulated value of the coefficient of variation f b r b =b at x 5 4 as k b 5f 2 b4=2, where the variance of b, r 2 b, as a function of x is computed for the ADE model as 2k bx=ðb=4þ 2 (deduced from the classical expression of the travel time variance for the inverse Gaussian distribution [Kreft and Zuber, 1978]). Skewness and kurtosis for the ADE model are then obtained from the inverse Gaussian distribution with the mean as a function of distance bx=4 and variance as a function of distance 2k b x=ðb=4þ 2, where b is the simulated value. structures. Interestingly, the CDF and CCDF of b for the disconnected field GD are closely reproduced by the ADE model. [34] The deviation of the b distribution from the ADE model is significant for the GC and GM fields (Figure 3), where low b values are strongly overestimated by the ADE model. Hence, if b and r 2 b are constrained, in case of connected and multi-gaussian structures, the ADE model predicts significantly larger spreading than indicated by simulations. To further understand the fundamental statistical properties of the distribution of b, we shall consider the dependence of percentiles of the distribution on the mean. This is a way to capture the scaling of the distribution as well as its important characteristics for applications, as will be demonstrated in the next section. [35] In Figure 4, the dependence of four b percentiles (b / with four values of / specified) are shown as functions of b, for the three structures and two variances, computed at x 5 4. We find that the high variability case (blue symbols) exhibits a linear dependence on the mean, for all four percentiles that scan the distribution from 1 to 95%, covering a relatively wide range of b ; the slopes of the lines and associated parameters of the linear fit, including the correlation coefficient, are summarized in Table 1. The dotted lines in Figure 4 indicate the 1:1 slope that is applicable for plug flow. Hence, the deviation from the dotted line effectively illustrates the deviation of the actual b / value from the plug flow case with the mean b. The red symbols are for the case of low variability, in which case there is obvious convergence to the plug flow case (Figure 4). Figure 3. Cumulative distribution functions (CDFs) and complementary cumulative distribution functions (CCDFs) for the dimensionless transport resistance b at x 5 4 for r 2 ln T58. For the GD field, ADE (green dashed line) provides a close representation of the simulated CDF and CCDF, at least up to CDF 5 CCDF

6 Figure 4. Dependence of b percentile (b / ), as a function of b mean ðbþ for: (a) / 5.1; (b) / 5.1; (c) / 5.5; (d) / The three blue symbols are for the three structures and r 2 ln T58atx54, together with a linear fit; the parameters of the line, including the correlation coefficient, are given in Table 1. The dotted line is the plug flow case with slope 1:1. Red symbols are for the three structures with r 2 ln T 51. [36] The low b percentile deviates strongest, being significantly overestimated by the plug flow case (Figures 4a and 4b). For / 5.5 (median), lines and symbols are relatively close indicating that the plug flow case provides reasonable estimates of the b median, irrespective of the structure and variability (Figure 4c). The plug flow model underestimates the high percentile / 5.95 simulated slope (solid line with blue symbols) as seen in Figure 4d. With / 5.7, lines and symbols are most closely aligned, almost coinciding (not shown). Thus, we find that the slope (full line, blue symbols) is smaller relative to the plug flow case (dotted line, 1:1), for / <.7 and larger than the 1:1 line for / > Application [37] When considering tracer retention in a rock fracture or a rock block consisting of a network of fractures, a key issue is to provide a reliable estimate of the transport resistance b. Attenuation of any sorbing and degrading contaminant in fractured rock is strongly dependent on b [Cvetkovic and Frampton, 212]. A suitable measure for this process and dependence is an order of magnitude count of solute mass reduction over a given scale L, or attenuation index [Cvetkovic, 211a, 211b; Cvetkovic and Frampton, 212] where k is the decay rate. g5 1 ln 2 bj p ffiffi k (6) [38] For a heterogeneous fracture, b is a random variable and consequently so is g (6). It was shown in the previous section that the distribution of b depends on the structure as well as on the variability of T. Here we seek a simple yet general means of estimating b given an estimate of the mean water residence time. In applications, flow simulations and particle tracking are typically used for estimating the mean s, denoted by s. The question we shall address here is the following: Given s for a rock fracture, can we estimate b / where / is a percentile expressing a chosen confidence level for b? [39] In Figure 5a, the correlation between b / for four selected values of / is illustrated; note that we are Table 1. Best Linear Fit Parameters A and B in b / 5A1Bb, Applied on the Blue Symbols in Figure 4 a / A B r a Included is also the correlation coefficient, r. Note that the nonzero value of A is due to the nonlinearity for small s, which is a proxy for the distance; spatial dependence of central moments of s for instance, exhibit nonlinear behavior at short distances [e.g., Gotovac et al., 21]. 7592

7 hydraulic gradient J 5:1. The transmissivity field is variable but a detailed characterization is not available, and the correlation structure is unknown. Based on prior experience, say r 2 Y < 8 where r2 Y 58 is used as an upper bound, and I Y 5 1 m. The normalization constant for b is then b 5I Y =T G J 532; yr=m. [43] Consider now 239 Pu as a tracer, with K d 55m 3 =kg ; D e 51: m 2 =yr and half-life t 1=2 524; yr. The matrix porosity h in Swedish crystalline rock for instance, is typically.5 with a density of 27 kg/m 3. A simple formula for the attenuation index, or order-of-magnitude count g (7), requires the decay rate for 239 Pu, which is k5ln 2=t 1=2 52: p 1=yr. With j5 ffiffiffiffiffiffiffiffiffiffiffi hd e R where R511ð12hÞK d q=h; g (7) is plotted in Figure 6 as a function of dimensionless s, obtained from g / 5 1 ln 2 x p /sb j ffiffi k (7) Figure 5. Estimators for percentiles of the transport resistance: (a) dependence of b / on the dimensionless mean water residence time s ; (b) dependence of the slope in a) (that can be interpreted as an effective specific surface area x (5)) as a function of percentiles. primarily interested in the low values of b (i.e., low percentiles /), since the low limit is most relevant for risk assessment. Also, we are primarily concerned with higher variability as presumably more realistic. It is seen in Figure 5a that the three structures yield a straight line with a slope dependent on /, where data at 6 distances from the source are plotted; for high variability the lines cover a relatively wide range of s, whereas for low variability s is bounded by approximately 5. The slope of the lines for high variability is one possible definition of an effective active specific surface area x (5), denoted by x / and applicable for a given confidence level /. [4] In Figure 5b, the dependence of x / on / is shown. The lowest value of / used in the computations is.1. Clearly, for low / (i.e., higher confidence that b > b / ), x / is lower, the normalized lower bound being around.35. [41] Figure 5 provides a novel means of estimating an upper bound of b for a rock fracture, that is general with respect to structure and conservative with respect to the level of variability, given the mean water residence time s. Once s is measured or computed, we select the confidence level 1 2 / depending on the application, and use an effective value of the specific surface area, x / from Figure 5b. The transport resistance with the selected confidence level b / is then obtained as b / 5sx /. [42] To illustrate the application of x / (Figure 5), consider a rock fracture with T G m 2 /s and a mean where / indicates the chosen percentile; the value of x / is obtained from Figure 5b. The constants x and s are defined in (B5) and (B4), respectively. [44] It is seen in Figure 6 that higher confidence implies lower attenuation in the considered fracture. For instance, with median confidence, and a value of say s515 (similar as was found in the computations at x 5 4 for the GD field), the attenuation index is around 8, meaning that ultimately 1 28 of the 239 Pu initial unit mass will be released from the fracture. 6. Discussion [45] As noted earlier, s and b are analogous random variables which is why they are relatively strongly correlated, e.g., in fracture networks [Cvetkovic and Frampton, 212]. Because of the analogy, the statistics of b may be used as a proxy for the statistics of advective travel time, although the magnitude as well as the units, are different from those of s. [46] Irrespective of the structure, the basic physical mechanism at work is macro-dispersion by heterogeneous advection. For low percentiles, the hydrodynamics of flow constrains the advective movement in the high-velocity Figure 6. Attenuation index g (7) as a function of the mean water residence time for 239 Pu, for different percentiles. 7593

8 Figure 7. Asymptotic part of the complementary cumulative distribution function (CCDF) for the dimensionless transport resistance b, for the large variability case with r 2 ln T 58. Dashed lines are obtained with the ADE model using first two moments as explained in the caption of Figure 2. regions, relative to the behavior of Brownian motion that is unconstrained by the flow. In other words, the Brownian motion implies more rapid initial propagation of small mass factions (or channelized transport ) than what actually happens due to the combined effect of structure and flow dynamics. [47] The high percentiles (slow transport), on the other hand, drag on longer than predicted by the ADE model, implying extended tailing. This is illustrated in Figure 7 where the effect of low velocity zones is clearly visible. Thus, the ADE model predicts more rapid flushing of the tracer through the fracture, than what we find in the simulations. The simulated distribution of b for high variability exhibits power-law, but not anomalous features (Figure 7). Note that the second central moment (or CV) of b is relatively large due to the prolonged tail apparent in Figure 7. [48] The low velocity zones in a single fracture with high T variability, are delaying the tracer stronger than implied by Brownian motion, whereas the high velocity zones are channelizing the tracer less rapidly than implied by the Brownian motion, for a fixed (relatively) large CV in the GM and GC fields. By contrast, for the GD field with better connected low transmissivity zones, the balance between the high and low velocity advection appears consistent with the ADE model (Figure 3, green symbols and dashed curve). It should be emphasized that the GC field implies a low mean b, whereas the GD field implies a large mean b, with the mean b for the GM being in between (Figure 2a). These properties result in an important feature, namely that the percentiles of b show a linear dependence on mean b for the three structures; advantage was taken of this linearity to correlate b percentiles with mean water residence time (Figure 5), providing a potentially useful tool for applications (Figure 6). 7. Summary and Conclusions [49] In this paper, two-dimensional simulations of water flow and advective particle transport in a highly heterogeneous rock fracture were presented; the accuracy of the simulations is ensured by the AMCFM tool. A generic statistical representation of transmissivity is used in order to study the statistics of the flow-dependent transport resistance b, a key variable for diffusive mass transfer. The domain of the fracture is 4 integral scales of ln T with a multi-gaussian structure (GM), a connected non-multi- Gaussian structure (GC) and a disconnected non-multi- Gaussian structure (GD). The two levels of heterogeneity considered are with r 2 ln T 51 and r2 ln T 58. [5] Based on the obtained results, we draw the following main conclusions: [51] 1. Moments of b are linear with distance, i.e., they depend on distance in a way that is qualitatively consistent with the ADE model, after approximately 1 integral scales of ln T (Figure 2). Skewness and kurtosis are larger for the GC and GM structures compared to the ADE model, whereas with the GD structure these measures are consistent with the ADE model. [52] 2. The distribution of b strongly depends on the structure and level of variability. For high variability in GC and GM fields, the distribution of b deviates significantly from the Brownian motion (Figure 3), due to the large velocity contrast associated with low and high transmissivity zones. [53] 3. Distribution of b for a GD structure is consistent with the ADE model even for large variability (Figure 3); it appears that for this structure, the presence and correlation of high and low velocity zones are balanced in such a manner that transport is consistent with Brownian motion, and the b distribution is reproduced with the ADE model. [54] 4. Percentiles of b were shown to be linearly dependent on the mean b when considering all three correlation structures with high variability. Further analysis of advective transport e.g., with different T distributions and more general correlation structures, is needed to better understand the limitations of the result in Figure 4. [55] 5. Percentiles of b can also be related to the mean water residence time s for high variability with the three correlation structures. This is potentially useful in simulations of transport through discrete fracture networks, where typically fracture T is assumed uniform, and computations yield the mean water residence time of individual fractures [e.g., Frampton and Cvetkovic, 211]; Figure 5 can then be used to estimate b / for each fracture in cases where T statistics of individual fractures are not available. Appendix A: Transport Model [56] The mass balance equations for two-dimensional, advection-dominated transport in a fracture with onedimensional Fickian diffusion into the rock matrix, are written as [e.g., Cvetkovic et al., 1v rc f 5 D e b m j y5b 2kC f (A1) 5 D 2 C m 2 2kC m (A2) where v5t J=2b T J=d is the spatially variable fluid velocity in the fracture with T being the transmissivity and J the negative hydraulic gradient vector, C f [M/L 3 ] is the dissolved concentration in the fracture, C m [M/L 3 ] is the concentration in the rock matrix, d 2b [L] is the fracture aperture, h [ ] is the matrix porosity, D e [L 2 /T] is the 7594

9 effective diffusivity in the matrix, R is the retardation coefficient in the rock matrix, and k [1/T] is the decay rate. The parameters h, D e, R, and b may all be functions of the twodimensional position vector x, i.e., spatially variable. [57] Equations (A1) and (A2) can be transformed onto a trajectory using the water residence time s [T] as the independent variable, following the methodology of Cvetkovic and Dagan [1994]. Then, (A1) 5 D m j y5b 2kC f (A3) [58] For injection in the flux of unit mass, assuming continuity at the fracture-matrix interface and zero concentration in the rock matrix at infinity, the system (A3) and (A2) can be solved [Cvetkovic et al., 1999] to yield the probability density function of the solute particle residence time, h, at x 5 L in the Laplace domain as hðt; LÞ5L 21 p hexp 2ss2B ffiffi s i (A4) which generalizes the original matrix diffusion model [Carslaw and Jaeger, 1959; Neretnieks, 198]; in (A4) s is the Laplace Transform variable and L 21 denotes inverse Laplace Transform. The water residence time between x 5 and x 5 L is defined by s ð L dv v x ½XðsðvÞÞŠ 5 ð L 2d½XðsðxÞÞŠdx T½XðsðxÞÞŠJ½XðsðxÞÞŠ (A5) [59] If the material parameters h, D e and R are approximated as uniform, effective values, then where hðt; LÞ5L 21 p hexp 2ss2bj ffiffi s i j p ffiffiffiffiffiffiffiffiffiffiffi hd e R (A6) (A7) and b [T/L] is the transport resistance defined in (3). Note that substituting the Laplace Transform variable s with s 1 k yields the case with a decaying tracer, which was the basis for deriving the attenuation index g [Cvetkovic, 211a]. Appendix B: Numerical Simulations [6] The AFMCM methodology [Gotovac et al., 29] is based on Fup basis functions with compact support (related to the other localized basis functions such as splines or wavelets) and the Fup Collocation Transform (FCT), which is closely related to the Discrete Fourier Transform. It provides a multiresolution representation of any signal, function or set of data using only a few Fup basis functions and resolution levels on nearly optimal adaptive collocation grids, that are capable of resolving all spatial and/or temporal scales and frequencies. Fup basis functions and the FCT are presented in detail in Gotovac et al. [27]. Other improved Monte Carlo (MC) methodology aspects are: (a) the Fup Regularized Transform (FRT) for data or function (e.g., log-conductivity) approximations in the same multiresolution way as FCT, but computationally more efficient, (b) Adaptive Fup Collocation method (AFCM) for approximation of the flow differential equation, (c) particle tracking algorithm based on the Runge-Kutta-Verner explicit time integration scheme and FRT, and (d) MC statistics represented by Fup basis functions. All aforementioned MC methodology components are summarized in Gotovac et al. [29]. [61] We consider a two-dimensional steady state and uniform-in-the-average flow field in a single fracture, imposing the following flow boundary conditions: Left and right boundaries are prescribed a constant head, while the top and bottom are no-flow boundaries. Transport simulations are performed in the inner computational domain to avoid nonstationary influence of the flow boundary conditions. The tracer mass is injected instantaneously using an in flux injection mode [Kreft and Zuber, 1978; Demmy et al., 1999]. [62] The isotropic GM field (Figure 1c) consists of a regular grid defined by eight blocks per integral scale and a negative exponential correlation function exp (2v/I Y ), with I Y as the integral scale. The connected (Figure 1a) and disconnected fields (Figure 1b) were generated through a transformation of the GM field in four steps: [63] 1. The absolute value of the GM field (zero mean, unit variance) is calculated. This transform shifts extreme values to become high values, and values originally close to the mean become low values. [64] 2. The histogram of the values in the field is converted back to a univariate Gaussian distribution by mapping the CDF value at each point to a standard normal CDF. [65] 3. The block size of the field is increased such that the integral scale matched that of the original GM field; this provides the final disconnected field (Figure 1b). [66] 4. The connected field (Figure 1a) is then generated from the disconnected field by reflecting the values of the disconnected field around the mean. [67] In order to match statistics of the GM field with new non-gaussian fields, it is important to include the relationship between their correlations: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qðvÞ 2 1qðvÞarcsin qðvþ q ðvþ5 p22 (B1) where q ðvþ and q(v) are correlation functions of ln T for the non-multi-gaussian and multi-gaussian fields, respectively, with a lag or separation v. Thus, the correlation function for the transformed variable is dependent on the correlation function of the original variable at the same separation distance v implying an independence of the correlation at other separation distances. Note that the correlation function for the transformed variable is always less than that of the untransformed variable. [68] Finally, we adapt the above mentioned procedure such that a multi-gaussian field is generated by numerically inverting (B1). The following relationship is applied for computing values of the disconnected field (Figure 1b) with the same statistics as a multi-gaussian field (Figure 1c) used in Gotovac et al. [29, 21]: Y 5 ffiffi p 2 erf 21 Y 2erf p ffiffiffi 21 2 (B2) where Y are absolute values for the GM field obtained from 7595

10 inversion of (B1) using zero mean and unit variance. The connected field (Figure 1a) is generated from the disconnected field by reflecting the Y values around the mean simply by multiplying its value with 21. Disconnected and connected fields also share the same statistics. The values of Y are finally multiplied by standard deviation of the logtransmissivity in order to obtain the desired heterogeneity level for the ln T fields. [69] The cubic law assumption, which originates from the parallel plate concept and corresponds to the case of perfectly smooth fractures, relates aperture and transmissivity as d5 12Tl 1=3 T 1=3 q w g c (B3) where l 5.1 [kg/ms] is fluid viscosity, q w the water density, and g the gravitational constant. [7] The integral scale of Y ln (T/T G ), I Y, will be used as the normalization length for the distance x. [71] The water residence time is normalized with s defined by and I Y s T 2 3 G c 1 3J x 51=ðT G =cþ 1=3 is used for normalizing x b/s (5). 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