Solute transport in fractured rocks with stagnant water zone and rock matrix composed of different geological layers Model development and simulations

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1 WATER RESOURCES RESEARCH, VOL. 49, , doi: /wrcr.20132, 2013 Solute transport in ractured rocks with stagnant water zone rock matrix composed o dierent geological layers Model development simulations Batoul Mahmoudzadeh, 1 Longcheng Liu, 1 Luis Moreno, 1 Ivars Neretnieks 1 Received 1 February 2012; revised 25 January 2013; accepted 5 February 2013; published 29 March [1] A model is developed to describe solute transport retention in ractured rocks. It accounts or the act that solutes can not only diuse directly rom the lowing channel into the adjacent rock matrix composed o dierent geological layers but also at irst diuse into the stagnant water zone occupied in part o the racture then rom there into the rock matrix adjacent to it. In spite o the complexities o the system, it is shown that the analytical solution to the Laplace-transormed concentration at the outlet o the lowing channel is a product o two exponential unctions, it can be easily extended to describe solute transport through channels in heterogeneous ractured media consisting o an arbitrary number o rock units with piecewise constant geological properties. More importantly, by numerical inversion o the Laplace-transormed solution, the simulations made in this study help to gain insights into the relative signiicance the dierent contributions o the rock matrix the stagnant water zone in retarding solute transport in ractured rocks. It is ound that, in addition to the intact wall rock adjacent to the lowing channel, the stagnant water zone the rock matrix adjacent to it may also lead to a considerable retardation o solute in cases with a narrow channel. Citation: Mahmoudzadeh, B., L. Liu, L. Moreno, I. Neretnieks (2013), Solute transport in ractured rocks with stagnant water zone rock matrix composed o dierent geological layers Model development simulations, Water Resour. Res., 49, , doi: /wrcr Introduction [2] In many countries, both chemical nuclear wastes are planned to be deposited in deep geological repositories in crystalline rock. Since the rock is ractured with waterbearing ractures [Gylling et al., 1998], it is o great importance, or perormance saety assessment, to underst the eective processes governing contaminant transport through a single racture in the network o ractures [Swedish Nuclear Fuel Waste Management Company (SKB), 2011]. The aim is to predict the rate at which the escaping contaminants will be transported by lowing water rom the repositories to the biosphere, given that the canister eventually may degrade start to leak because o corrosion or any other reasons. Toward that end, many models have been developed over the years to account or both advection in the racture diusion in the rock matrix [Moreno et al., 1996], as it has been ound that the solute carried by lowing water in ractured rocks can diuse into the stagnant water 1 Department o Chemical Engineering Technology, Royal Institute o Technology, S , Stockholm, Sweden. Corresponding author: L. Liu, Department o Chemical Engineering Technology, Royal Institute o Technology, SE Stockholm, Sweden. (lliu@ket.kth.se) American Geophysical Union. All Rights Reserved /13/ /wrcr inilling pores in the porous rock matrix [Moreno Neretnieks, 1993a; Barten, 1996]. [3] In the early 1980s, Neretnieks [1980] considered a simple case in which the rock matrix is assumed to be an ininite homogeneous medium the low is considered to be evenly distributed over the entire racture. Upon this or similar cases [Tang et al., 1981; Park et al., 2001], analytical solutions to the solute concentration at the outlet o the racture have been obtained. In reality, however, the rock matrix adjacent to the racture has a more complex structure than that assumed, as it commonly consists o not only the intact wall rock but also the altered zone [Lögren Sidborn, 2010b; Pique etal., 2010]. As a result, it is anticipated that the altered parts o the rock would strongly aect the transport o solutes in ractured media because they have signiicantly dierent diusive sorptive properties than the intact wall rock [Selnert et al., 2009]. This has been shown in studies o Cvetkovic [2010], who investigated the impact o racture rim zone heterogeneity considering dierent physical chemical properties or altered rock unaltered rock. [4] The geological materials in the altered zone could, as shown in Figure 1, be made up o ault gouge, racture coating, ault breccia, mylonite, cataclasite, altered rock [SKB, 2003a]. They appear mostly to be the result o brittle reactivation o old ductile/semiductile deormation zones, while the ault breccia ault gouge originate mainly rom the movements along ault planes are thereore distributed in variable amounts proportions 1709

2 Figure 1. A sketch o a typical conductive structure o dierent rock layers, based on a conceptual model taken by SKB [2003a]. over the racture structure planes [Lögren Sidborn, 2010a, 2010b]. In spite o this knowledge, however, it is diicult to recognize all above-mentioned matrix layers with distinctively dierent geological properties in ield observations. For this reason, it is common to simpliy the structure o the altered zone somehow consider it as a mixture o some layers, as did Tsang Doughty [2003]. They have conceptualized the rock matrix as an assemblage o the intact wall rock several layers containing the altered material with an enhanced porosity, in order to well account or their dierences in geological properties. Moreno Craword [2009] have also considered the rock matrix as being composed o several geological layers in modeling solute transport in ractured rocks. They ound that the layers close to the racture surace may be important in determining the behavior o tracer retardation during site characterization. The more deeply located layers in the rock matrix may, however, have a large impact on the prediction o nuclide transport under the conditions prevailing at the timescales o perormance assessment. [5] For simplicity, however, most o the models ignored the existence o the stagnant water zone in the racture plane [Dershowitz et al., 1998; Hartley, 1998; Poteri Laitinen, 1999], in contradiction to ield observations that ground water is lowing only in a small part o a conductive racture to orm one or more lowing channels [Tsang Neretnieks, 1998]. Until recently, Neretnieks [2006] took the inluence o the stagnant water zone into account. He ound that solute diusion into the stagnant water zone adjacent to the lowing channel can substantially contribute to the retardation o solute, when it is ininitely wide. It would give rise to not only a resistance to nuclide transport but also an increase o the contact surace over which nuclides can diuse into out o the rock matrix. As a result, considerable amount o nuclides present in the lowing channel may be restrained or a long time in the rock matrix adjacent to the stagnant water zone, instead o directly leaving into the rock matrix adjacent to the lowing channel. [6] The models developed by Neretnieks [2006] Moreno Craword [2009] are both valuable in helping us to underst the important mechanisms that govern solute transport retention in ractured rocks. However, their ocuses are dierent somehow; one is on the exploration o the impacts o the stagnant water zone the other is on the understing o the roles o the rock matrix comprising dierent geological layers but only limiting it 1710

3 adjacent to the lowing channel. Consequently, it is o great importance to get an insight into the synchronous the overall eects o both media in retarding solute transport. Toward that end, a model is developed in this study, which combines in essence the two models developed by Neretnieks [2006] Moreno Craword [2009], respectively, based on an idealization o a basic building block o a heterogeneous rock domain. The eects o physical chemical heterogeneity, as studied by Deng et al. [2010], can then be taken into account by considering the rock matrix as a series o blocks with piecewise constant geological properties. [7] The model accounts or not only the stagnant water zone with a inite width lying in the racture plane but also the rock matrix, as schematically shown in Figure 2, composed o both the intact wall rock with a inite thickness the altered zone that may be made up o altered rock, cataclasite, ault gouge, racture coating, all at the same time. In addition, it considers that the rock matrices adjacent to both the lowing channel the stagnant water zone may have dierent structures geological properties. Despite the complexities o the system, it is shown that the analytical solution to the Laplace-transormed concentration at the outlet o the lowing channel is a product o two exponential unctions; they describe the contributions o two partial systems in parallel in retarding solute transport, respectively. As a result, the Laplace-transormed solution can readily be included in both the discrete racture network models [Dershowitz et al., 1998] the channel network models [Gylling et al., 1999] to describe solute transport through channels in heterogeneous ractured media consisting o an arbitrary number o rock units with piecewise constant properties. More importantly, the Laplace-transormed solution suggests that the large number o dierent individual parameters can be grouped into a small set o parameter groups that considerably reduce the number o independent parameters. These parameter groups have clear physical meanings that show the time constants or competing transport rates. They may be used to illustrate gain insights into when one mechanism can be expected to dominate over another mechanism, as what is to be done in a series o simulations upon a representative case where speciic data are assigned to a (large) number o individual parameters. These values are chosen rom observations o real ractures, including their coating thicknesses transport properties as well as channel stagnant water zone widths. The water low rate the low distance in the example channel are chosen to be within the range o interest or a high-level nuclear waste repository deep in ractured granite. In this manner, the discussion conclusions are only valid or the parameter values o the example but help to show how the parameter groups can be used i one wishes to seek inormation or other values o individual parameters. [8] The paper is organized as ollows. In sections 2 3, the model is ormulated in detail both conceptually mathematically. In section 4, the ocus is put on derivation o the analytical solution to the solute concentration at the outlet o the ollowing channel in the Laplace domain. Ater that, a series o simulations are made in section 5 to explore the relative importance the dierent contributions o the rock matrix the stagnant water zone in retarding solute transport in ractured rocks. In addition, they illustrate how the thicknesses, diusion, sorption properties o dierent geological layers o the rock matrix can impact the breakthrough curves o solutes how the stagnant water zone can add retardation o a solute that irst diuses into the stagnant water zone then rom there into the layers o altered rock. The conclusions are drawn in section 6 at the end o the paper. 2. Conceptual Model [9] In ield experiments observations in drits tunnels, it has been observed that water low in ractures is not evenly distributed over the whole racture surace [Abelin et al., 1983, 1985, 1991, 1994]. It lows in very narrow parts o the racture there are oten large distances between the lowing channels [Neretnieks, 2006; Neretneiks et al., 1987], due to that in variable apertures under stress large parts o the ractures are in direct contact whereas other parts are not. As a result, water low seeks out the more readily accessible pathways orming channels, leaving other parts o the racture open water illed but with Figure 2. Schematic conceptualization o a racture with a rock matrix ormed by several layers to illustrate the terms used in the text. 1711

4 much less low or even no low. The widths o the lowing channels were ound to vary rom millimeters to tens o centimeters to be spaced rom tens o centimeters over meters to many tens o meters or more [Abelin et al., 1983, 1985; Neretnieks, 2006; Neretneiks et al., 1987]. [10] In addition, it has been justiied in analysis o the tracer experiments [Abelin et al., 1991; Neretnieks, 2002] that the recovery retardation o the nonsorbing tracers could be either explained by matrix diusion i the lowing channels were wide or by diusion into stagnant water zones i they were narrow or by combinations o these phenomena. As one cannot readily dismantle the rock, the real cause is not discernable, but these similar works [Neretnieks Moreno, 2003] suggested that diusion into stagnant water zones urther transport into the rock matrix can have a very strong impact on nuclide retardation, thereore this mechanism should be better understood. [11] Based on these considerations, the basic building block o a network o intersecting ractures with distributions o lengths, orientations, positions [Moreno Neretnieks, 1993b; Gylling et al., 1999] is highly idealized, as schematically shown in Figure 3. A real low path is o course much more complex. The aperture varies not only along it but also across it. Complex low paths rather than straight channels are, thereore, expected to orm in ractures with variable apertures [Liu Neretnieks, 2005]. In this study, however, we aim to develop a sensible reasonable simpliying model by assuming the low path to be straight with constant aperture width. This allows us to emphasize highlight the most important eects in retarding solute transport in ractured media also to gain insight into dierent mechanisms that can have signiicant impacts under dierent circumstances. [12] Figure 3 illustrates that the low is assumed to take place between two smooth parallel surace separated by a distance 2b, orming a rectangular channel o width 2W. The water velocity, u, is assumed to be uniorm across the lowing channel. Likewise, the stagnant water zone next to the lowing channel is conceptualized as a rectangular cuboid with the same length as the lowing channel but dierent aperture width; they are 2b s 2W s, respectively. The rock matrix adjacent to the lowing channel is assumed to consist o n layers with dierent thicknesses o i, i ¼ 1, 2,..., n ; the layers close to the racture surace with i < n constitute the altered zone, while the last one Figure 3. Flow in a channel in a racture rom which solute diuses into rock matrix as well as into stagnant water in the racture plane then urther into the rock matrix. represents the intact wall rock. Similarly, the rock matrix adjacent to the stagnant water zone is considered to make up n s layers with dierent thicknesses o i s, i ¼ 1, 2,..., n s ; the layers close to the stagnant water zone with i < n s also constitute the altered zone, while the last one sts or the intact wall rock. [13] When applied or practical use, however, we may simply assume that the rock matrices adjacent to both the lowing channel the stagnant water zone have identical structures geological properties. In addition, the geological materials in the altered zone such as breccia, mylonite, cataclasite may also be lumped into only one layer to share the same properties. Despite these dramatic simpliications, it should be possible to choose the low parameters the geological parameters, such that pessimistic estimates can be obtained or the sake o saety perormance assessment o a deep repository or spent nuclear uel. [14] Based on the idealized geometry o the system, as shown in Figure 3, the model accounts or the act that solute can diuse directly rom the lowing channel into the adjacent rock matrix, in addition to advection through the racture. They may also at irst diuse into the stagnant water zone then rom there into the rock matrix adjacent to it. Although in Figure 3, diusion in both the stagnant water zone the rock matrix is shown only in the upward let directions, the model also considers diusion in opposite directions. In addition, linear equilibrium sorption onto both the racture surace the rock matrix is included. This implies that irst-order reactions, such as the decay o a single nuclide (but not a decay chain that involves several nuclides), can also be accounted by the model, although it has not been done explicitly. [15] The model does not, however, take advection in the rock matrix into account, since the permeabilities o the geological layers o the rock matrix are generally low. That is, it assumes that solute transport in the rock matrix results solely rom molecular diusion. In addition, the phenomenon o hydrodynamic dispersion in the lowing channel is also neglected as it is dominated in ractured media mostly by the dierences in the residence times o solute transport along dierent low [Gylling, 1997]. However, the general procedure or combining longitudinal dispersion with any retention model could be included [Painter et al., 2008]. 3. Mathematical Model [16] With the coupled 1-D approach, as was done by Hodgkinson Mual [1988] also by Neretnieks [2006], the equations o continuity describing solute transport processes can easily be ormulated or dierent regions o the system. As detailed in what ollows, this approach essentially takes a 3-D system as an assemblage o 1-D subsystems, in which solute transport is assumed to take place only in one direction. The exchange o solute between the subsystems is, then, described as a sink or source located at the interace, to couple the 1-D transport equations automatically. [17] Since the system under consideration can be conceptualized into our subsystems with dierent media dierent directions o transport, as shown in Figure 3, the equations governing the mass balance o the solute in each subsystem should be ormulated discussed one by one. 1712

5 3.1. Solute Transport Through the Flowing Channel [18] The 1-D transport equation or the aqueous concentration in the lowing channel, C, is o the þ b s D s b þ D1 e y¼0 1 ; (1) z¼0 where y z are the coordinates into the stagnant water zone into the rock matrix adjacent to the lowing channel, respectively, both are perpendicular to the channel. A complete list o symbols units are given at the end o the paper. [19] The initial condition is C ðx; 0Þ ¼ 0; (2) while the boundary condition or a time-dependent concentration at the inlet is given by C ð0; tþ ¼ C in ðþ: t (3) [20] The irst term at the right-h side o equation (1) describes the advection, while the last two terms describe the diusional process at the interaces between the lowing channel the stagnant water zone between the lowing channel the rock matrix adjacent to it, respectively. The hydrodynamic dispersion term is not accounted or Solute Transport Through the Stagnant Water Zone [21] Assuming that the diusion in the direction parallel to the low is negligible, the transport equation or the aqueous concentration in the stagnant water zone, C s,is given by R s 2 C s ¼ D 2 þ es ps 1 b s (4) zs¼0; where z s is the coordinate into the rock matrix adjacent to the stagnant water zone, it is perpendicular to the interace between the stagnant water zone the adjacent rock matrix. The last term at the right-h side o equation (4) describes, then, the diusional process at the interace between the stagnant water zone the rock matrix adjacent to it. [22] The initial condition is the boundary conditions are given by C s ðy; 0Þ ¼ 0; (5) C s ð0; tþ ¼ C ; ¼ 0: (7) y¼2ws [23] The processes o solute transport in the lowing channel in the stagnant water zone are thus coupled through equation (6), which describes the continuity o the aqueous concentration o the solute Solute Transport Through the Rock Matrix Adjacent to the Flowing Channel [24] Taking diusive transport linear equilibrium sorption o the solute into account, the 1-D transport equation or the aqueous concentration in the pore water o the ith layer o the rock matrix adjacent to the lowing channel, C i i ¼ D i 2 C i 2 8i ¼ 1; 2;...;n ; or X i j 1 < z < Xi j where we have deined 0 ¼ 0, the apparent diusivity D i a is deined as D i a ¼ Di e R i p "i p [25] The initial conditions are Cp i ðz; 0Þ ¼ 0 or X i j 1 (8) 8i ¼ 1; 2;...;n : (9) < z < Xi j 8i ¼ 1; 2;...;n ; while the boundary conditions are given, respectively, by C i p ðz; tþ ¼ C i 1 ðz; tþ at z ¼ Xi D i i ¼ D iþ1 p iþ1 at z ¼ Xi (10) j 1 8i ¼ 1; 2;...;n ; (11) j 8i ¼ 1; 2;...;n ; (12) where we have deined C 0 p ¼ C D n þ1 e ¼ 0. [26] When i ¼ 1, equation (11) describes the continuity o the aqueous concentration o the solute at the interace between the lowing channel the irst layer o the rock matrix adjacent to it. This couples the behavior o solute transport in the lowing channel the adjacent rock matrix Solute Transport Through the Rock Matrix Adjacent to the Stagnant Water Zone [27] Similar to equation (8), the 1-D transport equation or the aqueous concentration in the pore water o the ith layer o the rock matrix adjacent to the stagnant water zone, Cps i, can be written i ¼ D 2 Cps i 2 s 8i ¼ 1; 2;...;n s ; or X i j 1 s < z s < Xi j s (13) 1713

6 where we have deined 0 s ¼ 0, the apparent diusivity D i as has the same deinition as Di a in equation (9), except or changing the subscripts rom to s. [28] The initial conditions are Cps i ðz; 0Þ ¼ 0 or X i j 1 s < z s < Xi j s 8i ¼ 1; 2;...;n s ; (14) while the boundary conditions are given, respectively, by Cps i ð z s; t D i s Þ ¼ C i 1 ðz s ; tþ at z s ¼ Xi ps j 1 s 8i ¼ 1; 2;...;n s ; ¼ D iþ1 es at z s ¼ Xi j s 8i ¼ 1; 2;...;n s s (15) (16) where we have deined C 0 ps ¼ C s D n sþ1 es ¼ 0. [29] When i ¼ 1, equation (15) describes the continuity o the aqueous concentration o the solute at the interace between the stagnant water zone the irst layer o the adjacent rock matrix. Thereore, it couples the behavior o solute transport in the two neighboring subsystems. [30] Noticeably, the diusional process across the interace between the rock matrix adjacent to the lowing channel the rock matrix adjacent to the stagnant water zone is not accounted by the model. The reason or this is that, in most cases, these two rock matrices would have identical structures properties. Otherwise, a transition region would exist between them. The rate o solute diusion across the interace would, thereore, be small compared with that in the direction perpendicular to the geological layers o the rock matrices. As a result, the omission o the diusional process across the interace between the rock matrices would give a satisactory description o solute transport. 4. Laplace-Transormed Solution [31] The equations governing solute transport in the mathematical model can be solved by applying the Laplace transormation [Watson, 1981]. It removes the time variable, leaving a system o ordinary dierential equations, the solutions o which yield the transorm o the aqueous concentrations as a unction o space variables. [32] Since the Laplace transorm technique allows one to get the aqueous concentration in the rock matrix in the Laplace domain without obtaining the concentration proiles within the lowing channel the stagnant water zone, we should begin with the Laplace-transormed equations or solute transport through the rock matrices Solute Transport Through the Rock Matrix Adjacent to the Flowing Channel [33] To acilitate mathematical manipulations, it is wise to write the Laplace-transormed equations or solute transport through the rock matrix adjacent to the lowing channel in a vector orm. Toward that end, we deine a column vector Y ¼ y i n 1 with y i ¼ z i Xi j 1 i (17) to denote a series o local, dimensionless axes or each o the geological layers, 8i ¼ 1,2,...,n, in such a way that Y ¼ 0 (zero vector) Y ¼ 1 (ones vector) represent the two ends o the layers. [34] In addition, we deine C C p as two column vectors to describe, respectively, the aqueous concentrations in the lowing channel the rock matrix adjacent to it, i.e., C ¼ C i;1 n (18) 1 C p ¼ C i p ; (19) n 1 where i,1 is a special case (j ¼ 1) o the Kronecker delta i,j, which is 1 i i ¼ j zero i i 6¼ j. [35] Following this, the partial dierential equations or solute transport through the rock matrix adjacent to the lowing channel, i.e., equation (8) together with the initial condition o equation (10), can conveniently be written in Laplace domain in a vector orm as 8C p ¼ A C p ; (20) where 8 represents the Hadamard product, which is the entrywise product o two vectors or matrices with the same dimensions, is a column vector to document the second-order partial dierential operators in the n -dimensional Euclidean space, it is deined i 2 : (21) n 1 [36] In writing equation (20), we have deined A as a diagonal matrix, given by A ¼ a i;j 2i;j with a i;j ¼ i (22) n n qiiiiiiiii i ¼ i D s ; (23) where i D is the characteristic time or solute diusion through the ith layer o the rock matrix adjacent to the lowing channel; namely, 2 i i D ¼ : (24) D i a 1714

7 Table 1. Characteristic Parameters Notation Deinition Physical Signiicance qiiiiiiiiiiiiiiiiiii MPG i MPG i ¼ D i e Ri p "i p qiiiiiiiiiiiiiiiiii MPG i s MPG i s ¼ D i es Ri ps "i ps F F ¼ x ub F s F s ¼ Ws 2 Dsbs The material property group, or the rock matrix layers adjacent to the lowing channel, measures the ease with which solute diuses into the pore water, but mostly the capacity to retain solute, within the matrix layer. A high value o it means that the matrix layer can hold a large amount o a given solute in both the dissolved the adsorbed states. The material property group or the rock matrix layers adjacent to the stagnant water zone has exactly the same explanation as material property group or the rock matrix adjacent to the lowing channel. The F -ratio is the ratio o the low-wetted surace o the lowing channel to the advection conductance (volumetric water low rate). It describes the competition between two processes governing the rate o solute transport; advection in the lowing channel diusion through the adjacent rock matrix. A high value o the F -ratio means that a large amount o solute carried by the lowing water can quickly be transported into the rock matrix. The F s -ratio is the ratio o the wetted surace o the rock matrix adjacent to the stagnant water zone (the stagnant-water-wetted surace) to the diusion conductance o the stagnant water zone. It describes the competition between two important processes governing the rate o solute transport; diusion into the stagnant water zone diusion rom the water into the adjacent rock matrix. A large value o the F s -ratio indicates that a large amount o solute diusing rom the lowing channel into the stagnant water zone can quickly be sucked by the adjacent rock matrix. ¼ x u The characteristic time or advection through the lowing channel which is actually the water residence time but can also be regarded as the solute travel time along the lowing channel without accounting or any interaction with the surroundings. s i D i Ds N MAHMOUDZADEH ET AL.: MODELING SOLUTE TRANSPORT IN FRACTURED ROCKS s ¼ Ws2 Ds i ð D ¼ i Þ 2 D i a i Ds ¼ ð i sþ 2 D i as N ¼ Dsxbs=Ws ub W The characteristic time or solute diusion through the stagnant water zone which characterizes solute travel time in the direction perpendicular to the lowing channel. The characteristic time or solute diusion through the ith layer o the rock matrix adjacent to the lowing channel. The characteristic time or solute diusion through the ith layer o the rock matrix adjacent to the stagnant water zone. The N-ratio is the ratio between the characteristic rate o diusion into the stagnant water zone the characteristic rate o advection through the lowing channel. It quantiies the raction o solute that can depart rom the lowing channel into the stagnant water zone. A high value o the N ratio indicates that the stagnant water zone has a large capability to capture the solute passing by then deliver it rom there into the adjacent rock matrix. [37] For ease o reerence, we tabulated i D in Table 1 together with other characteristic parameters with their deinitions physical signiicance. [38] The related boundary conditions then become C p Y ¼0 ¼ L C p Y ¼1 þ C ; (25) where L is a lower shit matrix, i.e., L ¼ i;jþ1 n n : (26) [39] Likewise, the other boundary conditions can be expressed as where the material property group (MPG) or dierent layers o the rock matrix adjacent to the lowing channel is deined as [Moreno Craword, 2009] qiiiiiiiiiiiiiiiiiii MPG i ¼ : (30) D i e Ri p "i p [41] It ollows that the general solution o equation (20) can be written in the orm C p ¼ K T þ S G ; (31) where K S are two diagonal matrices, deined as r 8C p Y ¼1 ¼ E r 8C p Y ¼0 ; (27) where r is the del operator generalized to n -dimensional Euclidean space; namely! r : i n 1 K ¼ S ¼ k i;j with k i;j n n s i;j with s i;j n n ¼ cosh i yi i;j (32) ¼ sinh i yi i;j : (33) [40] In writing equation (27), we have deined E as a matrix E ¼ e i;j with e i;j ¼ i MPG j n n j MPG i iþ1;j ; (29) [42] The two column vectors o constants o integration in equation (31), i.e., T ¼ðt i Þ n 1 G ¼ðg i Þ n 1, can be ound rom the boundary conditions (25) (27). However, rom the Laplace-transormed equation or solute transport through the lowing channel, it can be understood 1715

8 that the only constant o integration that will be needed is the irst element o the G vector. It may be written as with the P -parameter given by P ¼ Xn k¼1 jm kþ1 P g 1 ¼ MPG 1 C ; (34) j Y k MPG k jm j ð 1Þk tanh k i¼1 i coshði Þ; (35) where M is a matrix that characterizes solute transport properties within each layer o the rock matrix through the interace between adjacent layers, it is deined as M ¼ m i;j with n n 8 i sinh Y i sinh j i 1 cosh l j < i; l¼jþ1 >< m i;j ¼ i cosh i j ¼ i; MPG j i MPG i j ¼ i þ 1; >: 0 otherwise : (36) [43] In equation (35), M kþ1 sts or the submatrix o M, whose irst to the kth rows also columns have been deleted jm j jm kþ1 j are the determinant o M M kþ1, respectively, with the convention that jm kþ1 in the case o k ¼ n. [44] With equations (31) (34) at h, we may now proceed to evaluate the concentration gradient at the lowwetted surace o the lowing channel. First, it is noted that equation (31) 1 1 ¼ 1 sinh 1 y1 t 1 þ 1 cosh 1 y1 g 1 : (37) [45] It ollows immediately that, with the help o equation 1 ¼ 1 z¼0 1 1 ¼ y 1 ¼0 1 1 g 1 ¼ P siiiiiii s MPG 1 D 1 C : (38) a [46] This result will be used later to get the analytical solution o C ðx; sþ Solute Transport Through the Rock Matrix Adjacent to the Stagnant Water Zone [47] Following the same procedure as in the previous section, it is straightorward to show that the solution to the Laplace-transormed concentration in the pore water in the rock matrix adjacent to the stagnant water zone has exactly the same orm as that given in equation (31). The reason or this is that the Laplace-transormed equations or these regions are very similar to each other, except the dierence in the subscripts. As a result, equations (31) (38) are used with the only dierence that subscript is exchanged or s. [48] Due to these extreme similarities between the two subsystems o the rock matrices, the concentration gradient at the wetted surace o the stagnant water zone can be written immediately as, by analogy with equation 1 s zs¼0 ¼ s y 1 s ¼0 ¼ 1 s 1 s gs 1 ¼ P riiiiiii s s MPG 1 s D 1 C s : (39) as 4.3. Solute Transport Through the Stagnant Water Zone [49] Having obtained equations (38) (39) or the concentration gradient at the wetted suraces o the two rock matrices, we are now ready to work on Laplace-transormed equations or solute transport through the stagnant water zone the lowing channel, respectively. [50] To proceed step by step, it is noticed that substitution o equation (39) into Laplace-transormed equation or solute transport through the stagnant water zone, rom equations (4) (5), 2 C s D 2 sr s þ P s pii s C s ¼ 0: (40) b s [51] The solution o this equation, subjected to the Laplace transormed equations (6) (7), gives the Laplacetransormed concentration in the stagnant water zone. It can be written as with p cosh 2 iiiii piiiii s sy=ws C s ðy; sþ ¼ C p cosh 2 iiiii (41) s p ii s ¼ R s s s þ F s P s s; (42) where we have introduced two characteristic parameters s F s. [52] From equation (41), it may be shown that the concentration gradient at y ¼ p ¼ iiiii piiiii stanh 2 s C =W s : (43) y¼ Solute Transport Through the Flowing Channel [53] Now, substitution o equations (38) (43) into the Laplace-transormed equation (1) upon the initial condition o equation (2) þ sr þ P b pii 1 s þ W b s b D s W s piiiii piiiii stanh 2 s C ¼ 0: (44) 1716

9 [54] The solution o this equation, subjected to the Laplace transormed equation (3), gives the Laplace-transormed concentration in the lowing channel. It can be written as C piiiii piiiii ðx; sþ ¼ C ð0; sþexp exp N stanh 2 s (45) Table 2. Flowing Channel Properties Length Hal width Hal aperture Retardation coeicient Ground water velocity x ¼ 50 m W ¼ 0.1 m b ¼ 1.0E-04 m R ¼ 1 (no surace sorption) u ¼ 5 m/yr with p ii ¼ R s þ F P s; (46) where we have introduced three characteristic parameters,f, N. [55] Accordingly, the irst exponential unction in equation (45) describes the contribution o a partial system composed o the lowing channel the adjacent rock matrix in retarding solute transport retention, as given in equation (46) is determined solely by the parameters R,, F, P, which are all related to the lowing channel the adjacent rock matrix. By the similar token, the second exponential unction describes the contribution o a partial system composed o the stagnant water zone the adjacent rock matrix in retarding solute transport retention, with the N-ratio accounting or the raction o solute that can depart rom the lowing channel into the stagnant water zone. [56] Since equation (45) expresses the ratio o the outlet the inlet concentrations (in the Laplace domain) as a product o two exponential unctions, it can be easily extended into the case where the transport path consists o a series o piecewise constant channels, along each o which the surrounding media have constant geological properties. The result is, or L successive channels,! C ¼ C ð0;sþexp XL i i¼1 exp qiiiii qiiiii XL n o! N i i stanh 2 i s : i¼1 (47) [57] It signiies that, when applied or practical use such as in predicting the behavior o nuclide transport in ractured media rom a repository to the biosphere, it is unnecessary to calculate solute concentration at every outlet o the channels by perorming the inverse Laplace transorm o equation (45). Rather, one only needs to ollow the lowing channels along which solute travels document the involved characteristic parameters. The inormation would then be suicient to obtain solute concentration, i.e., the breakthrough curve, once at the inal end o the transport path, by use o De Hoog algorithm [De Hoog et al., 1982] to numerically transorm equation (47) back to the time domain. This would not only save computation time dramatically but also give accurate predictions, provided that the assumptions behind equation (47) are justiied to be air or the purpose o saety assessment o a deep geological repository or the spent nuclear uel. 5. Simulation Discussion [58] In this section, a series o simulations are made with the use o the inverse Laplace transorm o equation (45) in order to explore the eects o the characteristic parameters o the system in retarding solute transport in ractured media. The data used in the base case are tabulated in Tables 2 4 or the lowing channel, the stagnant water zone, the adjacent rock matrix composed o ive layers, respectively. The solute-dependent properties in MPG, such as the diusion coeicient the distribution coeicient are those or Cesium, while the geometry o the system the geological properties o the rock matrix are representative o those rom ield observations [SKB, 2006]. The available database or transport sorption properties o the rock matrix, such as extent porosity, may also be ound in SKB report [SKB, 2003b]. To simpliy the simulation without any loss o generality, however, we assume that the rock matrices adjacent to both the lowing channel the stagnant water zone have identical structures properties. [59] A step-by-step procedure is then adopted to carry out sensitivity analysis, by changing one characteristic parameter at a time. We start, however, rom a simple case, where only one matrix layer is involved in the system with the aim o exploring the eect o the intact wall rock. The rock matrix is thereupon extended to include one more layer in order to acilitate the understing o the eect o skins. Following this, the stagnant water zone is taken into account, assuming that the rock matrix consists only o an intact wall rock. The results with respect to the eects o several matrix layers are shown at the end o the section, in the interest o giving a clear picture about the indings Eect o the Intact Wall Rock [60] Consider now the simple case where there is no stagnant water zone in the racture plane, i.e., W s ¼ 0. Under this condition, equation (45) reduces to pii ðx; sþ ¼ C ð0; sþexp R s F P s : (48) C [61] In the case when all the matrix layers have the same properties, i.e., when we have just one layer as the rock matrix (intact wall rock), equation (35) or the P -parameter simpliies to P ¼ MPG m tanh m ; (49) where, henceorth, the superscript m reers to the intact wall rock as the main rock matrix. [62] The inverse Laplace transorm o equation (48) indicates that the outlet concentration would have to be o the ollowing orm C ¼ R ; F MPG m ;m D ; t : (50) Table 3. Stagnant Water Zone Properties Hal width Hal aperture Diusion coeicient Retardation coeicient W s ¼ 0.5 m b s ¼ 1.0E-04 m D s ¼ m 2 /yr R s ¼ 1 (no surace sorption) 1717

10 Table 4. Rock Matrix Properties Rock Zones MPG (m 2 /yr) 0.5 D (yr) Intact wall rock e06 Altered rock e04 Cataclasite Fracture coating Fault gouge [63] That is, the behavior o the breakthrough curves can depend only on the parameters R,F MPG m, m D, the independent variable t. This general result indicates already that F MPG m would have equivalent eects in retarding solute transport, as they are exchangeable. However, this is true only in the case where only one layer was involved in the solid rock, without accounting or other coatings the stagnant water zone in the racture plane. In other cases, equation (35) or the deinition o the P -parameter suggests that the F -ratio MPG m are not exchangeable in equation (48), thereore they would play dierent roles in determining the ate o solute transport, as demonstrated later. [64] Since equation (50) is too implicit to be o much value in giving an insight into the eects o the characteristic parameters, it is worth to consider some simple cases. For this reason, we notice that the hyperbolic tangent term in equation (49) would go to unity in the limiting case as m!1(equivalent to the case when we have zero concentration at ininity o the intact wall rock). As a result, combination o equations (48) (49) yields, C ðx; sþ ¼ C ð0; sþexp R s F MPG m pii s : (51) [65] For a constant source boundary with C (0, t) ¼ C 0, hence C ð0; sþ ¼ C 0 =s, the response unction, i.e., C /C 0 in the time domain, can be easily retrieved by applying the inverse Laplace transorm [Watson, 1981], to give C =C 0 ¼ erc 1 F MPG m piiiiiiiiiiiiiiiii 2 t R! or t > R : (52) [66] This is the same expression as it was obtained previously by Neretnieks (1980). [67] Likewise, or a point-source boundary, we may deine C (0, t) ¼ c 0 (t), where c 0 is not a concentration but a quantity in the unit o ML 3 T, it is the ratio between the mass injected instantaneously the volumetric low rate through the channel. This gives C ð0; sþ ¼ c 0, as a result, taking the inverse Laplace transorm [Watson, 1981] on equation (51) shows that the response unction, i.e., C / c 0 to the pulse injection is given by 1 C =c 0 ¼ piii t R or t > R :! 2 F MPG m p 2 iiiiiiiiiiiiiiiii exp4 t R F MPG m p 2 iiiiiiiiiiiiiiiii t R! (53) [68] To be noticed, it can also be obtained directly by dierentiation o equation (52) with respect to the time t. This response unction would exhibit more details than that to a step injection, as it resembles to a great extent the probability density unction, rather than the cumulative distribution unction, o the skewed normal distribution. For this reason, in this study, we will only present discuss the simulation results with respect to the breakthrough curves o a pulse injection in order to highlight the roles o the most important parameters. [69] To acilitate sensitivity analysis, two concepts are then introduced to characterize the behavior o solute transport; one is the peak value C,peak the other is the time at which the response unction reaches the peak, t peak.in the case when only one ininite matrix layer is involved in the system, it is justiied rom equation (53) that, C ;peak c 0 0:925ðF MPG m Þ 2 (54) t peak ¼ R þ 1 6 ðf MPG m Þ2 : (55) [70] Since we speciically model the sorption in even thin rapidly equilibrated surace coating layers, we set R to unity. This, together with the act that the advection time is only on the order o 10 years that the timescale we are concerned with is at least 10,000 years, as suggested by the second term on the right-h side o equation (55), indicates that the eect o R is entirely negligible. It should, however, be noted in equation (53) that R would determine the breakthrough time beore which no solute can be ound at the channel outlet. [71] By contrast, both F MPG m would play important roles in retarding solute transport. They would aect both C,peak t peak, correspondingly the spread, but not the shape o the breakthrough curve. This is shown in Figure 4, where we take the intact wall rock as the matrix layer with m D ¼ 1: years increase F -ratio continuously rom F,0 ¼ yr/m by a actor o 3 (we use the script 0 henceorth to denote the corresponding value given in the base case). [72] As discussed earlier, the F -ratio accounts or the act that when the solute is swept by the lowing water along the channel, it can also be continuously transerred Figure 4. Dependence o the response unction on F -ratio when there is no stagnant water zone the rock matrix is composed o only intact wall rock. 1718

11 rom the lowing water into out o the rock matrix. The F -ratio actually describes the ability o the rock matrix to draw o the solute rom the lowing channel can be seen as the eiciency o solute transport between the lowing water the adjacent rock matrix. It is, then, not surprising to see in Figure 4 that the higher the F -ratio, the more the solutes are retarded in the matrix due to both diusion adsorption, resulting in a large value o t peak,a lower value o C,peak, correspondingly a more tangible slope o changes. [73] The MPG characterizes, however, a dierent thing than the F -ratio does. It describes, to a large extent, the capacity o the rock matrix in restraining the solute in addition to the diiculty o solute diusion into it. A high value o MPG m indicates that the solute can quickly diuse into the rock matrix. For this reason, it is very obvious that the larger the MPG m, the longer the solute would reside in the matrix, resulting also in a larger value o t peak a lower value o C,peak o the response unction. [74] Since the MPG m the F -ratio are exchangeable in equation (50) describing the behavior o the breakthrough curve, exactly the same results as given in Figure 4 would also be obtained i MPG m is, instead, increased rom its base value MPG m ;0 by a actor o 3, while the F -ratio other parameters were kept ixed. [75] On the other h, it should be noted that equation (53) would not be valid i the m D value is not large enough so that the intact wall rock cannot be considered to have ininite thickness. The hyperbolic tangent term in equation (49) would, then, go to unity only when s!1. As a result, the diusion time, m D, would mainly inluence the downhill shape o the response unction, as shown in Figure 5. [76] Only when m D < years, the peak value o the response unction increases with decreasing the diusion time, while when m D > years, it keeps nearly the same as that obtained or an ininite matrix layer. This suggests that, given a typical value o D m a ¼ m 2 / yr, we may consider the intact wall rock to have an ininite thickness when m 1:5 m, as shown in Figure 6. It also illustrates how the tails o the response unction are inluenced by the thickness o the matrix layer Eect o the Submatrix Layer [77] When a number o thin layers exist with large MPGs in the altered zone next to the intact wall rock, the Figure 5. Dependence o the response unction on m D when there is no stagnant water zone the rock matrix is composed o only intact wall rock. Figure 6. Dependence o the response unction on rock matrix thickness when there is no stagnant water zone the rock matrix is composed o only intact wall rock. P -parameter in equation (48) cannot be expressed in a simple manner, as seen rom equation (35). However, in the case where only one submatrix layer (the geological layer made rom the altered zone as shown in Figure 2) is concerned, it can be written explicitly as P ¼ MPG m tanh m 1 þ MPG 1 MPG m 1 þ MPG m tanh m MPG 1 ð Þ tanh 1 tanh m tanh 1 : (56) [78] This expression suggests that the P -parameter is, in general, not symmetric with respect to the two layers with dierent geological properties, thereore, the exchange o the spatial sequence o the two layers would lead to dierent breakthrough curves or solute transport. Nevertheless, given that the MPGs o the two layers are identical, equation (56) would become P ¼ MPG m tanh m þ 1 : (57) [79] As a result, it turns to be symmetric in such a manner that the characteristic diusion times o the two layers are simply additive. Should the apparent diusivities in both layers be also identical, i.e., when the two layers have the same geological properties, equation (57) would reduce to equation (49). [80] In practice, however, the MPG the characteristic diusion time o the submatrix layers may be very dierent rom those o the intact wall rock; both may span several orders o magnitude, as shown in Table 4. The eect o dierent submatrix layers in retarding solute transport would, then, be expected to be quite dierent. To veriy this, we present in Figure 7 the breakthrough curves or the cases when the submatrix layer consists solely o racture coating, cataclasite, ault gouge, or altered rock, respectively. It is seen that only when racture coating is considered to be the submatrix layer, no signiicant dierence could be observed between the results obtained or the two-layer case the one-layer case. This is expected because racture coating has so small value o 1 D as to be easily penetrated by solutes, but it can only retain a marginal amount o solutes. However, the presence o thin, porous, readily 1719

12 Figure 7. Comparisons between breakthrough curves when there is no stagnant water zone the rock matrix is composed o an altered layer the intact wall rock. accessible coatings can considerably delay the early arrival o the solute. It has the same eect as rapid surace sorption; the peak is not aected much but the breakthrough curves move to longer times. This can have an important impact on some decaying nuclides, as a result, we cannot simply ignore the existence o these submatrix layers whenever they are involved in the rock matrix. [81] In other cases, the submatrix layer would play an important role, either in signiicantly delaying the peak time t peak, as in the cases o cataclasite ault gouge, or also in strongly weakening the peak value C,peak, as in the case o altered rock. Notably, the peak sequence in Figure 7 is not determined by either MPG 1 or 1 D, but by qiiiiiii MPG 1 1 D ; it is about 0.02, 0.20, 0.54, 2.69 or racture coating, cataclasite, ault gouge, altered rock, respectively. The larger this value, the lower the peak is correspondingly the larger the peak time t peak is. This is reasonable, because equations (9), (24), (30) together show that MPG i qiiiiiii i D ¼ i Ri p "i p ; (58) which is, thereore, the total amount o the solute that the matrix layer can contain, per cross-sectional area, per concentration o the dissolved solute. [82] In addition, comparison o the data used the results obtained or the cases o cataclasite altered rock seems to show that 1 D would be the key parameter, rather than MPG 1, in measuring the capacity o the submatrix in restraining the solute. This is, however, not true. As we discussed earlier, or a given solute, it is the MPG that quantiies mostly the capacity o the matrix layer in retaining the solute while the diusion time only describes the available space in containing it. The reason or the signiicant dierence between the response unctions o the two qiiiiiii cases is actually due to the big dierence in MPG 1 1 D. This quantity would be 13 times greater in the case o altered rock than that o cataclasite. [83] To demonstrate the similarity the dierence between the eects o MPG 1 1 D in retarding solute transport, we then present in Figures 8 9 the dependence o the response unctions on MPG 1 1 D, respectively. Figure 8. Dependence o the response unction on MPG 1 when there is no stagnant water zone the rock matrix is composed o altered rock intact wall rock. [84] It is seen that they do play similar roles in aecting the behavior o response unctions when both are increased rom the data used in the base case or altered rock by a actor o 3, respectively; the larger the values, the longer the solute would be retained in the rock matrix. However, detailed analysis suggests that the main dierence in the change o the response unctions is not the peak value C,peak but the peak time t peak ; it increases rom years dramatically to years in Figure 8, while only slightly to years in Figure 9. This illustrates that solute retention in the submatrix layer is more sensitive to the magnitude o MPG 1, rather than that o 1 D, although the parameter qiiiiiii group MPG 1 1 D is more appropriate in characterizing the capacity o the submatrix layer in containing the solute Eect o the Stagnant Water Zone [85] When the stagnant water zone in the racture plane is accounted or, part o the solute may, at irst, diuse rom the lowing channel into the stagnant water zone then rom there into the rock matrix adjacent to it. An additional space becomes, then, available or solute transport retention. However, as an intermediate buer, the stagnant water zone also limits the transer o solute directly rom the lowing channel into out o its adjacent porous media, especially when it has a large volume or solute storage. As a result, in cases with a narrow channel a wide stagnant water zone, the latter may well dominate the Figure 9. Dependence o the response unction on 1 D when there is no stagnant water zone the rock matrix is composed o altered rock intact wall rock. 1720

13 retardation o solute transport. To explore this the other possible eects o the stagnant water zone, the complete orm o equation (45) together with equations (46) (42) have to be used. [86] For illustration purposes also or simplicity, however, we shall still consider the case where the rock matrix consists only o the intact wall rock. The P -parameter is then given by equation (49) while the P s -parameter is deined in the same way, except or changing the subscripts rom to s. As a result, the inverse Laplace transorm o equation (45) indicates that the outlet concentration would have to be o the ollowing orm C ¼ R ; F MPG m ;m D ; R piiiiiiiiiiiiiiiiii s s ; N ; F s MPG m s ; m Ds ; t : (59) [87] This general result indicates already that F s MPG m s would have equivalent eects in retarding solute transport, as they are exchangeable. This is true, however, only in the case where only one layer was involved in the rock matrix without accounting or other coatings. In other cases, they would appear in dierent places in the equation, as it can be understood rom the deinition o the M s matrix. On the other h, equation (59) shows that the inluence o the rock matrix adjacent to the stagnant water zone would be dierent rom that adjacent to the lowing channel, as the power o F s MPG m s is not taking the same value as that o F MPG m, i.e., unity, but 1/2 instead. The physical reason behind this is that the solute can directly get into contact with the rock matrix adjacent to the lowing channel but not directly with that adjacent to the stagnant water zone, which acts as a diusion-controlled buer. Only through this buer can part o the advected solute penetrate into be restrained in the rock matrix adjacent to the stagnant water zone. As a result, the partitioning between these two paths o solute transport, as determined by the N ratio, also plays an important role in determining the behavior o the breakthrough curves. [88] Since the solution given in equation (59) is too implicit complex to be o much value in giving an insight into the eects o the important parameters, it is wise to consider the simplest case where the intact wall rock is assumed to have an ininite thickness. Under this condition, the s -unctions used in equation (45) can be explicitly written as ¼ R s þ F MPG m s ¼ R s s s þ F s MPG m s pii s (60) p ii s : (61) [89] In the limiting case as s! 0, i.e., when the stagnant water zone is very limited in width, s would go to nearly zero so that linearization o the hyperbolic tangent term in (45) is acceptable. As a result, equation (45) simpliies to C ðx; sþ ¼ C ð0; sþexp exp ð 2Ns Þ: (62) [90] The analytical solution to the response unction o a pulse injection becomes, then, available with the use o equations (60) (61). It would be in the same orm as that given in equation (53), i.e., with C =c 0 ¼ piiiii exp or >0 (63) ¼ t R 2NR s s (64) ¼ F MPG m þ 2NF s MPG m s : (65) [91] Thereore, analogous to equations (54) (55), the peak value C,peak the peak time t peak can be determined as C ;peak c 0 0:925ðF MPG m þ 2NF s MPG m s Þ 2 (66) t peak ¼ R þ 2NR s s þ 1 2: 6 F MPG m þ 2NF s MPG m s (67) [92] Not surprisingly, when s ¼ 0, these results would reduce to those given in equations (53) (55). As discussed earlier, the eect o R is entirely negligible in determining the behavior o the breakthrough curves. By the same token, the diusion time R s s can also be saely deleted rom equations (64) (67). [93] In addition, it is noted that when the diusion time s is short, the N-ratio the F s -ratio would be exchangeable in equations describing the outlet concentration in both the Laplace the time domain. They would, then, have equivalent eects in retarding solute transport. Moreover, equations (66) (67) show that the ratio o NF s /F can be used as an eective quantity to measure the contribution o the rock matrix adjacent to the stagnant water zone in determining the ate o solute transport, under the typical assumption that the rock matrices adjacent to both the lowing channel the stagnant water zone have identical or at least similar structures properties. In more general cases, we know rom previous discussions that the diusion rates into the rock matrix adjacent to the stagnant water zone the lowing channel may be characterized by Q s ¼ NF s D m es =m s Q ¼ F s D m e =m, respectively. The partitioning between these two paths o solute transport can thus be determined by Q s Q ¼ N F sd m es =m s F D m e =m : (68) [94] It would reduce to NF s /F in the case that the rock matrices adjacent to both the lowing channel the stagnant water zone are identical. Thereore, the ratio o Q s /Q is more general superior to the ratio o NF s /F in measuring the ability o the rock matrix adjacent to the stagnant water zone in transerring the solute. 1721

14 [95] Having discussed this, it is critical to know when the linearized approximation holds in order to apply these results appropriately or practical use. Toward that end, comparison between the numerical results the approximate solutions are made upon the data used in the base case but extending the thickness o the intact wall rock rom 1 to 10 m. It is seen in Figure 10 that when F s MPG m s < 10 the approximation solution agrees excellently with the numerical result; the smaller the value o F s MPG m s the better the agreement, while the agreement is still acceptable when F s MPG m s is in the range between This analysis also suggests that the N-ratio the F s -ratio might play very dierent roles in determining the ate o solute transport when F s MPG m s >> 20, not to mention the case where the rock matrix is thin. [96] It is noted that, given a typical value o MPG m s ¼ 0:0057 ðm 2 =yr Þ 0:5 or the intact wall rock, F s MPG m s ¼ 10 would give F s a value o roughly yr/m, corresponding to a hal width o 7.5 cm o the stagnant water zone with b s ¼ 0.1 mm or the solute with D s ¼ m 2 /yr, thereore s ¼ years. This indicates that the approximate solutions obtained above would well be applicable in cases when s is smaller than 0.18 years. The stagnant water zone would, then, have a negligible eect o buer so that the solute departing rom the lowing channel can directly diuse into the rock matrix adjacent to the stagnant water zone without any pause. [97] When s is ar larger than 0.18 years, it is anticipated that the N-ratio, the F s -ratio, or the MPG m s would play dierent roles in retarding solute transport, since then equations (63) (67) would not hold even i the rock matrix is thick enough. To demonstrate this, we present in Figures the dependence o the response unctions on F s N, respectively, upon the base case (with s ¼ years m s ¼ 1:0 m) as described earlier. [98] In Figure 11, the F s -ratio is increased continuously rom F s,0 ¼ yr/m by a actor o 3, while the other characteristic parameters are all kept at ixed values as in the base case. In particular, MPG m s ¼ 0:0057 N ¼ 6.3. It is seen that the peak value C,peak decreases Figure 11. Dependence o the response unction on F s -ratio when there is stagnant water zone the rock matrix is composed only o the intact wall rock. slightly, while the peak time t peak increases considerably, with increasing the F s -ratio. This is in signiicant contrast to what was ound previously in Figure 4 to changes in the F -ratio. Nevertheless, it is expected, as equations (66) (67) already indicate that F s MPG m s would dominate the ate o solute transport i it is ar larger than F MPG m.in the cases shown, the product F MPG m has a value o roughly 565 while F s MPG m s is in the range between , meaning that the stagnant water zone has a considerable buer capacity in this example. As a result, the ratio o Q s /Q increases rom 5.0 to 8.66 in Figure 11, indicating that more than 83% o the solute would directly depart rom the lowing channel then diuse into out o the rock matrix adjacent to the stagnant water zone in all cases studied. The dominant role o the rock matrix adjacent to the stagnant water zone in retarding solute transport in ractured rocks should, thereore, take main responsibility to the sensitivity o the results shown in Figure 11 on the F s -ratio. In addition, it should also take ull charge to the act that a signiicant decrease in the peak height C,peak a dramatic increase in the peak time t peak are observed when comparing Figure 11 with Figure 6 or the case where the stagnant water zone is absent. [99] Since the MPG m s has an equivalent eect as the F s - ratio in retarding solute transport, as discussed earlier, exactly the same results as given in Figure 11 would also be obtained i MPG m s is, instead, increased rom its base Figure 10. Comparisons between the numerical results (lines) the approximate solution (markers) or dierent F s MPG m s when there is stagnant water zone with very limited width the rock matrix is composed only o the intact wall rock with an ininite thickness. Figure 12. Dependence o the response unction on N-ratio when there is a stagnant water zone the rock matrix is composed only o the intact wall rock. 1722

15 value MPG m s;0 by a actor o 3, while the F s-ratio other parameters were kept ixed. This is true, no matter whether the stagnant water zone is narrow or wide. The same conclusion cannot, however, be drawn or the N-ratio. As can be seen in Figure 12, where the N-ratio is increased rom the value used in the base case, N 0 ¼ 6.3, by a actor o 3 while keeping other characteristic parameters ixed at values as in the base case, the results only resemble those o Figure 11 in appearance. The degree to which the peak value C,peak decreases, correspondingly the peak time t peak increases, with increasing the N-ratio is slightly dierent rom those ound in Figure 11. [100] In order to underst this dierence, we may consider another limiting case where the stagnant water zone has a very large width, while the intact wall rock is still assumed to be ininitely thick. Under this condition, s would become large enough such that the hyperbolic tangent term in equation (45) would go close to unity. As a result, it becomes C piiiii ðx; sþ ¼ C ð0; sþexp exp N s ; (69) where s are also given by equations (60) (61), respectively. [101] This allows us to write, by ollowing the method o Tang et al. [1981], the analytical solution to the response unction o a pulse injection as C ¼ 8c 0 p iii with R Z 0 exp 2 t R N Z 1 p iiiiiiiiiii R s s=4 s exp 2 2 s 4 N 2 R s s = 2 dd (70) F MPG m ¼ p 2 iiiiiiiiiiiiiiiiiiiiiiiii (71) t R N 2 F s MPG m s s ¼ q iiiiiiiiiiiiiiiiiiiiiiiiiiiiiii : (72) N 2 R s s = 2 [103] Since in Figure 12, s ¼ years; it is ar larger than the critical value o years when the stagnant water zone can be considered to cause no eect, but not large enough so as to make the stagnant water zone to act as a purely diusion-controlled buer. The results shown in Figure 12 corresponds, then, the transition region between the two limits o the stagnant water zone, thereore the power o N should be somewhere between 1 2. This explains why the exchangeability or the equal importance o the F s -ratio the N-ratio was not observed in Figures [104] To explore, then, when the stagnant water zone can be considered to be ininitely wide, we show in Figure 13 the dependence o the response unctions on the diusion time s, which would inluence both the N-ratio the F s -ratio, as seen in Table 1. It is clear rom the simulation results that when s is in the range between 5 15 years, the peak value C,peak o the response unction decreases appreciably, while the peak time t peak increases considerably, with increase in the diusion time. When s > 15 years, the response unctions remain nearly identical. This together with the analysis given earlier suggest that when s < years the stagnant water zone can saely be seen as i it does not exist, while when s > 15 years, the stagnant water zone can be regarded to have an ininite width. Thereore, or the present dimensions parameter values, the stagnant water zone would have a hal width o at least m when it becomes a purely diusion-controlled buer, given a typical value o D s ¼ m 2 /yr Overall Eect [105] In most cases, solute transport in ractured rocks would take place in such a system that consists o not only the lowing channel, the stagnant water zone, the intact wall rock, but also the altered zone that may be composed o altered rock, cataclasite, ault gouge, racture coating, all at the same time. This, or even more complex, system would certainly make it hard to clariy the eects o dierent components in retarding solute transport. However, based on sensitivity analysis we have made, it is anticipated that each o the components would play a similar role as we discussed beore in determining the ate o solute transport even when everything is put together into the system. To illustrate this, we compare in Figure 14 the breakthrough curves obtained or dierent cases where the [102] Thereore, it is now the product o N 2 F s MPG m s that determines the contribution o the rock matrix adjacent to the stagnant water zone to the breakthrough curve o solute transport, inluencing its moments, instead o NF s MPG m s in equation (63) or the case where the stagnant water zone has a negligible eect o buer. The power o N characterizes, then, the buer capacity o the stagnant water zone. When it is unity, the stagnant water zone would give no retardation o solute transport so that it can quickly be equilibrated by the solute diusing into it. When it is 2, however, the stagnant water zone would become diusion controlled so that a concentration proile o the diusion-driven solute develops along its width; only the part o solute that are very close to the mouth would quickly be penetrated into the adjacent rock matrix then be hardly released. Figure 13. Dependence o the response unction on s when there is stagnant water zone the rock matrix is composed only o the intact wall rock. 1723