Direct numerical simulation of matrix diffusion across fracture/matrix interface

Transcription

1 Water Science and Engineering, 2013, 6(4): doi: /j.issn e-ail: Direct nuerical siulation of atrix diffusion across fracture/atrix interface Yong ZHANG 1, Eric M. LABOLLE 2, Donald M. REEVES 3, Charles RUSSELL 1 1. Division of Hydrologic Sciences, Desert Research Institute, Las Vegas, NV 89119, USA 2. Departent of Land, Air, and Water Resources, University of California, Davis, CA 95616, USA 3. Division of Hydrologic Sciences, Desert Research Institute, Reno, NV 89512, USA Abstract: Accurate descriptions of atrix diffusion across the fracture/atrix interface are critical to assessing containant igration in fractured edia. The classical transfer probability ethod is only applicable for relatively large diffusion coefficients and sall fracture spacings, due to an intrinsic assuption of an equilibriu concentration profile in the atrix blocks. Motivated and required by practical applications, we propose a direct nuerical siulation (DNS) approach without any epirical assuptions. A three-step Lagrangian algorith was developed and validated to directly track the particle dynaics across the fracture/atrix interface, where particle s diffusive displaceent across the discontinuity is controlled by an analytical, one-side reflection probability. Nuerical experients show that the DNS approach is especially efficient for sall diffusion coefficients and large fracture spacings, alleviating liitations of the classical odeling approach. Key words: fracture/atrix interface; direct nuerical siulation; transfer probability; Lagrangian algorith 1 Introduction Fractured edia are ubiquitous. Quantification of flow and transport in fractured edia reains one of the greatest challenges in hydrology, as reviewed extensively by Berkowitz (2002) and Neuan (2005). The lack of analytical solutions for containant transport through fractured rock asses otivated the developent of nuerical ethods for this purpose. Specific software suites including, for exaple, SLIM-FAST (Maxwell and Topson 2006) and DCPT (Pan et al. 2001; Pan 2002) were designed using the particle-tracking approach to capture solute dynaics through regional-scale fractured edia. Most of the current Lagrangian solvers rely on the concept of transfer probability (Liu et al. 2000) or its transient extension (Pan and Bodvarsson 2002). An intrinsic assuption or epirical quantification of the particle density or age distribution, however, is usually inevitable in building the transfer probability ethod (Hassan 2002). The applicability of an epirical ethod should be tested This work was supported by the United States Departent of Energy and the Desert Research Institute IR&D Funds. Corresponding author (e-ail: Yong.Zhang@dri.edu) Received Dec. 13, 2012; accepted Jun. 8, 2013

2 over an extensive paraeter range before practical applications, since site-specific paraeter ranges ay lead to probles not found during the developent of the ethodology. This paper proposes a direct nuerical siulation (DNS) approach to quantify solute transport in fractured edia, without any epirical assuptions. The core objective of the new ethod is to directly track solute particle dynaics, governed by a rando-walk representation of atrix diffusion, to control the exchange of ass between the fracture and atrix continua. The physical concept is otivated by the successful developent and application of the rando-walk siulation of transport across coposite porous edia with abrupt interfaces between distinct hydrofacies (LaBolle et al. 1996, 2000). Different fro the other dual-doain ass-transfer ethods entioned above, the DNS approach does not depend on the transfer probability whose calculation relies on epirical assuptions. Rather, the probability of particles oving fro one continuu to the other follows an analytical reflection law. The rest of the paper is organized as follows. In section 2, the classical transfer probability ethod is briefly reviewed and shown to be inadequate for a significant portion of paraeter space, such as that used in the transport of radionuclides in fractured rocks. The identified paraeter range failed in capturing real transport dynaics will be used to check the applicability of the DNS ethod developed in section 3. Nuerical experients are also shown in section 3, followed by discussions of the tie step criterion and the extension to discrete parallel fractures in section 4. Conclusions are presented in section 5. 2 Classical transfer probability ethod: Review and evaluation In the dual-continuu odel, which includes both the dual-porosity and dualpereability forulations, the particle transfer probability fro one continuu to the other can be calculated as the ratio of the ass entering the other continuu during a tie interval to the ass in the current continuu at the beginning of the current tie interval (Pan et al. 2001). Liu et al. (2000) proposed the following two transfer probabilities: FfΔt Qf 2Df A Pf = = + Δt (1) VC f f Vf Vf ( 1 λ ) S Ff Δt 2Df A Pf = = Δt (2) VC V( 1 λ ) S when the water flow rate fro the fracture to atrix within the block (denoted as Q f ( 3 /s)) is positive (i.e., water flows fro the fracture to atrix). Here P f (diensionless) is the transfer probability fro the fracture to atrix; P f is the transfer probability fro the atrix to fracture; F f (kg/s) is the transport rate (due to both advection and dispersion) fro the fracture to atrix; F f is the transport rate fro the atrix to fracture; V f (or V ) ( 3 ) is the liquid volue within the fracture (or atrix) continuu (defined as the grid block volue ultiplied by the porosity); C f (or C ) (kg/ 3 ) is the solute concentration in the 366 Yong ZHANG et al. Water Science and Engineering, Oct. 2013, Vol. 6, No. 4,

3 fracture (or the atrix); D f ( 2 /s) is the hydrodynaic dispersion coefficient for solute transport between the fracture and atrix; A ( 2 ) is the fracture-atrix interface area available for solute transport between the continua within the block; S () denotes the distance fro the fracture-atrix interface to the center of the atrix; Δ t is the tie step (usually sall); and λ is a diensionless shape function equal to 1/3 for the case of a layered atrix with finite thickness, in which the ter layered denotes the shape of atrix blocks (see also the sae definition used by Crank (1975) and Haggerty et al. (2000)). When there is no water flowing fro the fracture to the atrix, the transfer probabilities are siplified to 3D Pf = Δt (3) SbR R f 3D Pf = Δ t (4) 2 2 S Rθ where D = DfR, R (diensionless) is the atrix retardation coefficient, R f (diensionless) is the fracture retardation coefficient, b () is the (half) fracture aperture (so that S = B b where B () is the (half) fracture spacing), and θ (diensionless) denotes the effective atrix porosity. A detailed derivation of Eqs. (3) and (4) is shown in section 4.4. Eq. (3) shows that the transfer probability of particles fro the fracture to atrix relates also to the properties of the atrix, such as the atrix retardation coefficient R. Note, however, that it does not ean that the properties of the atrix should affect the otion of particles inside of the fracture, since the transfer probability does not drive the displaceent of particles in either doain. Siilar behavior can be observed for particles crossing a discrete interface due to discrete transport paraeters, which is a physical process of particle transport in fracture/atrix systes. The intrinsic assuption underlying Eqs. (1) and (2) (or their siplifications, Eqs. (3) and (4)) is that, as soon as particles enter each continuu, the concentration represented by the particle density can reach coplete ixing in this doain. In other words, the concentration gradient between the fracture and the atrix is ignored fro the beginning. As articulated by Liu et al. (2000), the average atrix concentration rather than the concentration at the fracture-atrix interface is used for deterining the rate of solute transport. Several studies reveal that such an assuption leads to an erratic prediction of containant dynaics for a sall dispersion coefficient (Hassan 2002; Hassan and Mohaed 2003) and a large fracture spacing (Pan and Bodvarsson 2002). To quantitatively identify the application range of paraeters, we coded the above transfer probability ethod in the software RWHet (LaBolle 2006) and then systeatically checked the nuerical results against the analytical solutions. All odel paraeters characterizing ajor fracture/atrix properties were tested, including the diffusion coefficient, fracture aperture, longitudinal dispersivity in the fracture, atrix retardation coefficient, Yong ZHANG et al. Water Science and Engineering, Oct. 2013, Vol. 6, No. 4,

4 fracture retardation coefficient, and fracture spacing. Nuerical results show that this transfer probability ethod generates solutions that generally atch the analytical solutions, except for (1) a relatively sall free-water diffusion coefficient D 0 (the olecular diffusion coefficient for a solute in free solution), i.e., 11 D 0 < /s, (2) a relatively large fracture spacing 2B, and (3) a relatively large atrix retardation coefficient, i.e., R > 2. These paraeter values can cause a steep concentration gradient, fro the fracture/atrix interface across the atrix block, violating the well-ixed, equilibriu concentration profile assuption. Note that the influence of R on transfer probability can be cobined with the influence of fracture spacing B, since the ter SR in Eqs. (3) and (4) can be replaced by a scaled distance S = SR and B is a function of S. For the sake of siplicity, we focus below only on the influence of D 0 and 2B on particle dynaics. Several nuerical exaples are shown in Fig. 1. The odel paraeters are as follows: the fracture velocity v = 1 /d, the fracture dispersivity α L = 0, 2b = , 2B = 1, θ = 0.1, atrix tortuosity τ = 0.25, the fracture retardation coefficient R f = 1, R = 1, and the travel distance is 36. The analytical solutions shown in this figure are fro Sudicky and Frind (1982). The particle tracking results generally atch the analytical solutions for the free-water diffusion coefficient D 0 between and /s (Fig. 1(a)). If a saller D 0 is used (such as /s, as shown in Fig. 1(b)), the particle tracking solution tends to overestiate the early breakthrough curve (BTC), siilar to that observed by Hassan (2002). This iplies that Eqs. (3) and (4) underestiate the actual probability of a fracture particle being transported into the atrix. In the nuerical case we tested, only a few particles entered the atrix and then reained there for a long tie (due to the sall transfer probability fro the atrix to the fracture). Fig. 1 Coparison of nuerical solutions of classical transfer probability odel with analytical solutions using variable 0 D values The classical transfer probability ethod proposed by Liu et al. (2000) therefore is not applicable for sall diffusion coefficients. Siilar behavior is observed for the fracture spacing 2B, since the classical transfer probability in Eqs. (1) and (2) also tends to 368 Yong ZHANG et al. Water Science and Engineering, Oct. 2013, Vol. 6, No. 4,

5 overestiate the early arrivals of solute particles in the fracture where 2B is relatively large (such as 2B > 1 ). This conclusion is consistent with the result found by Pan and Bodvarsson (2002). Nuerical exaples related to a large 2B are shown further in sections 3 and 4. 3 Direct nuerical siulation ethod: Methodology developent and nuerical tests The DNS ethod is expected to siulate solute transport throughout the entire paraeter space, if it can accurately and efficiently describe the particle dynaics across the interface between the fracture and the atrix (Fig. 2). It is a specific particle tracking ethod for odeling solute transport in coposite edia where sharp contrasts exist between velocity, porosity, diffusion, dispersion, and/or retardation. These are the sae types of conditions that exist across a fracture-atrix interface. Fig. 2 Conceptual odel of DNS ethod for solute transport through single fracture/atrix syste We first consider containant transport through a single fracture within a porous atrix. The extension to parallel fractures will be discussed in the next section. The DNS approach for a single-fracture syste contains the following three ajor steps: Step 1: Calculation of independent transport coponents for each particle. Standard particle tracking schees are used to calculate the advective and dispersive displaceent of each particle during each juping event: x t + dt = x t + w D dt (5) ( ) ( ) ( ) ( ) ( ) z t + dt = z t + v z dt + w D z dt (6) where w is a unifor rando nuber with a ean of 0 and a variance of 1; D x and D z are the effective dispersion coefficients along the x and z directions (Fig. 2), respectively; v( z ) is the velocity along the fracture; and dt is the tie step. In Fig. 2, D denotes the olecular diffusion coefficient. Note that D f denotes the dispersion tensor in the fracture, which contains two coponents D x and x D z shown in Eqs. (5) and (6). It is also noteworthy that additional transport coponents can be added conveniently to particle trajectories, including advection in the atrix. Step 2: Application of the one-side reflection schee to capturing the particle dynaics across the fracture/atrix interface. Yong ZHANG et al. Water Science and Engineering, Oct. 2013, Vol. 6, No. 4,

6 The one-side reflection schee proposed recently by Bechtold et al. (2011) is selected for this step. This schee corrects the standard reflection ethod by splitting the tie step nonlinearly for particles across the discrete interface. It also significantly increases the coputational efficiency by transforing the reflection barrier ethod fro a two-side schee into a one-side reflection schee. Note that in the one-side reflection schee, the particle can jup freely fro the atrix to fracture, due to the one-side reflection probability defined below: D R P = f 1 θ θf Df R (7) f P f = 0 (8) if θ D R > θ D R (9) f f f where Pf is the reflection probability fro the fracture to atrix, and P f is the reflection probability fro the atrix to fracture (note that a reflection probability of 0 eans that no particles can be reflected at the interface, or, in other words, each particle can ove freely if it starts in the atrix). When the two retardation coefficients R and R f equal 1, Eqs. (7) and (8) are reduced to the forula proposed by Bechtold et al. (2011). Different fro the transfer probability in Eqs. (1) and (2), whose calculation contains assuptions, here the reflection probability has an analytical solution expressed by Eqs. (7) and (8). A unifor rando nuber U, U [ 0,1], is generated and copared to the probabilities above. If the particle is located in the fracture and U > Pf, then the particle can cross the fracture/atrix interface during the current tie step. When the particle is located in the atrix, it can ove freely in either direction. This schee is coded and verified, with soe exaples shown in Fig. 3. In the figure, the analytical solutions are fro Carslaw and Jaeger (1959), an instantaneous point source is located at x = 48, and D 1 and D 2, and θ 1 and θ 2 denote the effective dispersion coefficients and effective porosities, respectively, for each side of the coposite ediu. For the sake of siplicity, t and D shown in the figure are diensionless (note the purpose of this experient is to explore the applicability of the Lagrangian schee to capture particle dynaics across an abrupt interface). Step 3: Splitting of the tie step dt for each particle spent in the fracture and the atrix doains, and then repetition of the above steps until the final siulation tie is reached. There are two loop cycles for the three-step schee developed above. The outer iteration is for tie, and the inner iteration is for the nuber (i.e., sequence) of particles. We systeatically tested the three-step DNS ethod. One snapshot of particle positions at a specific tie is shown in Fig. 4. Advective transport in the fracture is downward with particle transfer across the fracture/atrix interface, which is shown by the vertical dashed line in Fig. 4. The particles are released initially inside the fracture at z = 0, and the horizontal 370 Yong ZHANG et al. Water Science and Engineering, Oct. 2013, Vol. 6, No. 4,

7 Fig. 3 Coparison of nuerical results of one-side reflection schee with analytical solutions at t = 5 for a coposite ediu with discrete diffusion coefficients and effective porosity Fig. 4 Snapshot (particle clouds) siulated by DNS ethod in a single fracture with downward transport and fracture/atrix particle transfer coordinate is randoly distributed. Note the coordinates in Fig. 4 are diensionless. Obviously, if the fracture aperture is sall, the particles inside the fracture cannot ake a large horizontal jup during one step of otion. This liits the coputational efficiency of the nuerical ethod. Results (Fig. 5) show that the DNS solutions generally atch the analytical solutions to a single fracture within an infinite atrix (Tang et al. 1981), if the tie step dt is properly defined. The other odel paraeters are as follows: 2b = , θ = 0.01, τ = 0.1, R = 1, R f = 1, and α L = 0.5. In all cases, particles are released at the beginning of the Yong ZHANG et al. Water Science and Engineering, Oct. 2013, Vol. 6, No. 4,

8 siulation. The solution of the classical transfer probability odel is also shown for coparison, where 2B is assued to be 100. In the cases we tested, the classical transfer probability odel overestiates the particle BTC when the fracture spacing is large. 4 Discussion Fig. 5 Siulated cuulative BTC with DNS ethod for a single fracture at depth z = versus analytical solutions 4.1 Tie step used in DNS ethod The ajor challenge of the DNS ethod is to appropriately define the tie step dt used in Eqs. (5) and (6). To liit the jup size of a single particle during one step to be less than the (half) aperture b, dt should be defined as 2 b 1 dt (10) 6 D0 Rf If there is an advective flux fro the fracture to atrix, dt should be saller than that in Eq. (10), so that the particle oves at least twice before encountering the fracture/atrix interface. We checked the criterion of Eq. (10) extensively. For exaple, for case 2 shown in 2 Table 1 and Fig. 5(c), Eq. (10) shows that dt d. In the DNS-reflective 2 siulations, we tested three tie steps: 10 3 d, 10 4 d, and 10 d. Because the three tie 372 Yong ZHANG et al. Water Science and Engineering, Oct. 2013, Vol. 6, No. 4,

9 2 steps are uch less than d, the three siulated breakthrough curves are alost identical to the analytical solutions. Table 1 Two nuerical cases shown in Fig. 5 and tie step for particle tracking Case Figure b () D 0 ( 2 /s) T 1 (d) T 2 (d) 1 Fig. 5(a) Fig. 5(c) Note: T 1 = (b/6) 2 /D 0, and T 2 = (b/4) 2 /D 0. 5 For case 1, the iniu tie step defined by Eq. (10) is dt d (Table 1). In 4 the three tie steps tested in the DNS siulations (Fig. 5(a)), only the sallest one ( 10 d) is close to the criterion of Eq. (10), and hence it tends to generate a siilar result as the analytical solution. In addition, nuerical experients also iply that the iniu tie step defined by Eq. (10) can be relaxed to 2 b 1 dt (11) 4 D0 Rf Nuerical solutions generated by the iniu tie step defined by Eq. (11) are still close to the analytical solutions. It is noteworthy that the classical transfer probability ethod discussed in section 2 also requires a tie step control, as discussed by Liu et al. (2000). In particular, the criterion is that particles should not be transported out of the host continuu during one tie step. The software RWHet (LaBolle 2006) also efficiently adjusts the tie spent by each particle during each jup, given ediu property and boundary/output conditions. The coputational efficiency of the DNS ethod can be iproved by adjusting the tie step for particles reaining in the atrix continuu. If the atrix particle is far away fro the fracture/atrix interface, a relatively large dt can be assigned. We leave this exercise for a future study. In addition, the DNS approach is especially efficient for a sall diffusion coefficient, since the tie step increases with the decrease of D 0, as shown by Eq. (10) and iplied by nuerical experients. The DNS approach is also advantageous for fractured rock systes with large fracture apertures and spacings, since wide fractures lead to a large tie step and a large atrix leads to a long ean residence tie for particles, which can be accounted for conveniently using the particle tracking approach. 4.2 Extension to discrete parallel fractures The DNS ethod proposed above can be extended to capture containant transport through parallel fractures situated in a porous rock atrix. Such extension can be straightforward. If the syste is heterogeneous (for exaple, the flow velocity and dispersion coefficient are space dependent), we can directly design a periodic distribution of the Yong ZHANG et al. Water Science and Engineering, Oct. 2013, Vol. 6, No. 4,

10 fracture/atrix network. If the properties of the fracture/atrix syste are hoogeneous, we can constrain and repeat the otion of particles in a half doain of the fracture/atrix syste. The second ethod was then tested. We relocated the particle to new positions near the fracture/atrix interface whenever it exited the center of the atrix grid. One nuerical exaple is shown in Fig. 6. In the figure, the odel paraeters are as follows: 2b = , 2B = 4, θ = 0.01, τ =0.1, R = 1, R f = 1, D 0 = /s, and α L = particles are released at the beginning of the siulation. The classical transfer probability solution (Liu et al. 2000) is also shown for coparison. The sae criterion for the tie step presented in section 4.1 can be used here. For exaple, nuerical tests shown in Fig. 6 reveal that, when the appropriate tie step dt d is used, the nuerical solution is siilar to the analytical solution. This liit can be derived directly using Eq. (10), which is 0.02 d. Fig. 6 Siulated cuulative BTC with DNS ethod for parallel fractures versus analytical solutions 4.3 Coparison between different ethods The dual-doain odel expressed by Eqs. (1) and (2) can be functionally equivalent to a single-rate ass transfer (SRMT) odel, as shown in section 4.5. Such equivalence actually reveals the potential proble of the forer. In the SRMT odel, the probability of a particle aking a phase conversion does not depend on the property (including position and age) of the particle in the starting phase. This can be seen fro the SRMT equation, where the ass transfer rate is proportional to the difference of the average concentration between the two phases. The concentration gradient in each phase therefore should not affect the subsequent transition rate. The sae behavior ust be true for the dual-doain odel based on the transfer probabilities expressed by Eqs. (1) and (2), due to the sae expression of the transfer probability in the two odels. This conclusion is consistent with the intrinsic assuption for the dual-doain odel (Eqs. (1) and (2)), where the spatially varying containant concentration in the atrix continuu is replaced by its average when deriving the transfer 374 Yong ZHANG et al. Water Science and Engineering, Oct. 2013, Vol. 6, No. 4,

11 probability (as shown by Eq. (12) and Eq. (18) in Liu et al. (2000)). This assuption can be applicable for a relatively large olecular diffusivity and sall atrix blocks, since a large diffusive strength can push particles to arrive quickly to the internal region of the atrix continuu and approach the concentration equilibriu. When the diffusion is relatively slow and the resultant concentration gradient at the fracture/atrix interface reains high for a long period, the assuption can cause a serious underestiation of the transfer probability. In addition, when the fracture spacing is relatively large, the concentration near the fracture/atrix interface can be uch larger than the average concentration, and the solute transport rate can be underestiated seriously. To correct the assuption underlying Eqs. (1) and (2), Pan and Bodvarsson (2002) proposed a particle age-dependent transfer probability, and Hassan and Mohaed (2003) used a travel distance-dependent transfer probability (note that Eqs. (1) through (4) define a tie-independent transfer probability if the tie step Δ t reains constant). Additional epirical assuptions, however, are still needed in these corrections in order to calculate the updated transfer probability, as articulated by Pan and Bodvarsson (2002) and Hassan and Mohaed (2003). In contrast to these corrections, the DNS ethod proposed in this study does not rely on any assuptions. 4.4 Derivation of transfer probabilities The transfer probabilities expressed in Eqs. (1) and (2) are linked to properties of the fracture and atrix. Neither Liu et al. (2000) nor the other studies (such as Pan and Bodvarsson (2002) and Hassan (2002)) gave the final forula for the above two transfer probabilities. The recent work of Maxwell and Topson (2006) did provide further expression of transfer probabilities related to atrix/fracture properties, but they did not show the procedure in deriving the final expression. Here we derive the final transfer probabilities based on Eqs. (1) and (2). As will be clearly explained, the final transfer probabilities differ slightly fro those provided by Maxwell and Topson (2006). Further coparisons between the nuerical and analytical solutions were then conducted to check the reliability of these new forulas. For descriptive siplicity, we first ignore the influence of atrix and fracture retardation coefficients on transfer probabilities. As explained by Liu et al. (2000), such influence can be added conveniently by adjusting the transport paraeters (velocity, the dispersion coefficient, and porosity) for nonreactive transport. All the variables used in transfer probabilities of Eqs. (1) and (2) can be related to fracture/atrix properties. For exaple, the water volue in the fracture within each grid ( V f ) can be defined as (Eq. (D3) in Pohlann et al. (2004)) V = θ dxdydz (12) f f where dx, dy, and dz () are the grid sizes along the x, y, and z directions, respectively. Siilarly, V is defined as (see for exaple, Eq. (D6) in Pohlann et al. (2004)) S V = d d d x y z B θ (13) Yong ZHANG et al. Water Science and Engineering, Oct. 2013, Vol. 6, No. 4,

12 The contact area between the fracture and atrix, A ( 2 ), is defined as dx A= d y d z B (14) Substituting Eqs. (12) and (13) into Eq. (1), we obtain the probability of a fracture particle being transported into the atrix during the tie interval Δ t : Qf 3Df P f t t θfddd xyz SBθ f (15) Substituting Eqs. (13) and (14) into Eq. (2), we obtain the reverse transfer probability: 3Df Pf = Δ t 2 S θ (16) Eqs. (15) and (16) are the sae as those provided by Maxwell and Topson (2006) (Eq. (B23)), if there is no retardation. If there is no advection in the fracture, the ratio of Eqs. (15) and (16) is Pf θ SB V = = Pf θf Vf (17) which is consistent with the conclusion drawn by Hassan (2002), who found that the ratio between the two probabilities is equivalent to the ratio between the volue of water in the atrix and the volue of water in the fracture. We now consider the case of sorption, where the effective dispersion coefficient and porosity can be written as (see also Eq. (30) in Liu et al. (2000)) Df = D R (18) θ = θr (19) b θf = θfrf = Rf B (20) where the variables with a superscript denote the original paraeter values without retardation. For exaple, the effective dispersion coefficient D can be calculated by (Liu et al. 2000) D = Dτθ 0 (21) Substituting Eqs. (18) and (20) into Eq. (15) results in the final equation for the transfer probability P f : Qf 3D Pf = Δ t + brfdx dy dz B SbRRf Δ t (22) which siplifies to Eq. (3) if Q f = 0. Siilarly, by substituting Eqs. (18) and (19) into Eq. (16), we obtain the other transfer probability expressed in Eq. (4), which is the sae as the one used by Maxwell and Topson (2006). The transfer probability fro the fracture to atrix, however, is slightly different fro the following one used by Maxwell and Topson (2006) even when Q f = 0 : 3D Pf = Δ t SbR. (23) Nuerical experients (Fig. 7) show that the Eq. (22) is valid for a broad range of 376 Yong ZHANG et al. Water Science and Engineering, Oct. 2013, Vol. 6, No. 4,

13 fracture retardation coefficients. Eq. (23) is equivalent to Eq. (22) only if there is no sorption in the fracture. In Fig. 7, the odel paraeters are as follows: v = 1 /d, α L = 0, 2b = , 2B = 1, θ = 0.1, τ = 0.25, R = 1, and D 0 = /s, and the travel distance is 36. Fig. 7 Siulated BTC in fracture continuu using Eq. (16) versus analytical solutions fro Sudicky and Frind (1982) 4.5 Coparison to single-rate ass transfer odel The dual-doain odel based on Eqs. (1) and (2) can be linked to the SRMT odel, or a two-state Markov process. Below we show that, when (1) olecular diffusion is the only echanis for ass transfer between fractures and the atrix, and (2) particles in the atrix reain stagnant until they probabilistically return to fractures, the classical dual-continuu odel discussed in section 2 is functionally equivalent to the SRMT odel. The linear non-equilibriu ass transfer equation for the SRMT odel (Haggerty et al. 2000) is given as CIM = α ( CM CIM) (24) t where α (s 1 ) is the first-order rate coefficient, and C M and C IM (kg/ 3 ) denote the concentrations in the obile and iobile phases, respectively. The transport of obile solute is driven by the advective and dispersive fluxes: CM CIM + β = ( vcm) + D CM (25) t t x x x where β (diensionless) is the capacity coefficient, v (/s) is the velocity along the coordinate x, and D ( 2 /s) is the dispersion coefficient. Assuing that a particle is originally located in the fracture, the probability of finding it in the atrix at a later tie t is equal to the noralized ass (increase) in the atrix. This leads to the relationship ( 1+ β ) αt β βe Pf = (26) 1+ β When the tie t in Eq. (26) is short, one obtains the approxiation P βαδ t (27) f Yong ZHANG et al. Water Science and Engineering, Oct. 2013, Vol. 6, No. 4,

14 Here, t is replaced by Δ t, representing the tie interval. The capacity coefficient can be approxiated by the ratio of water volues in different phases, according to Haggerty et al. (2000) and Zhang et al. (2007). The rate coefficient can be approxiated by the reciprocal of the tie scale for diffusion in the atrix blocks, or the haronic ean of the density function of rate coefficients (see Haggerty et al. (2000)). Considering also the influence of the effective atrix porosity and a layered atrix, we obtain 2 1 θ S 3D Pf θ Δt Δt (28) θf 3D SBθf which is the siplified version of Eq. (1), when there is no advective flux fro the fracture to the atrix. Adding the influence of retardation will result in Eq. (3). Siilarly, Eq. (4) can be built following the above arguent. 5 Conclusions A direct nuerical siulation ethod was proposed and validated in this study for its ability to overcoe the probles of the classical approach in quantifying solute transport through a single fracture or discrete parallel fractures in a porous rock atrix. The classical transfer probability approach proposed by Liu et al. (2000) is liited to certain ranges of paraeters, due to an intrinsic assuption of an equilibriu concentration profile in the atrix blocks in building the transfer probability. Subsequently, this ethod fails in describing ass transfer for paraeter cobinations that violate this assuption, including sall diffusion coefficients, relatively large fracture spacings, and/or oderate atrix retardation. The new DNS ethod directly tracks the particle dynaics across the fracture/atrix interface, where no assuption is needed. Nuerical experients show that the DNS ethod is ore coputationally efficient for sall diffusion coefficients and large fracture spacings. However, it can be tie-consuing for sall fracture apertures or large diffusion coefficients, since the tie step size ust be sall enough for particles to jup at least once in a fracture. Future studies are needed to iprove the coputational efficiency of the DNS ethod for all ranges of paraeters, by, for exaple, assigning different tie steps for particles in different continua. Acknowledgeents This work was supported by the United States Departent of Energy, and was also partially supported by the Desert Research Institute IR&D funds. Any opinions, findings, conclusions or recoendations do not necessarily reflect the views of the funding agencies. References Bechtold, M., Vanderborght, J., Ippisch, O., and Vereecken, H Efficient rando walk particle tracking algorith for advective-dispersive transport in edia with discontinuous dispersion coefficients and water contents. Water Resources Research, 47(10), W [doi: /2010wr010267] 378 Yong ZHANG et al. Water Science and Engineering, Oct. 2013, Vol. 6, No. 4,

15 Berkowitz, B Characterizing flow and transport in fractured geological edia: A review. Advances in Water Resources, 25(8-12), [doi: /s (02) ] Carslaw, H. S., and Jaeger, J. C Conduction of Heat in Solids. Clarendon: Oxford University Press. Crank, J The Matheatics of Diffusion. 2nd ed. New York: Oxford University Press. Haggerty, R., McKenna, S. A., and Meigs, L. C On the late-tie behavior of tracer test breakthrough curves. Water Resources Research, 36(12), [doi: /2000wr900214] Hassan, A. E Coent on Deterination of particle transfer in rando walk particle ethods for fractured porous edia by H. H. Liu et al. Water Resources Research, 38(11), W1221. [doi: / 2000WR000132] Hassan, A. E., and Mohaed, M. M On using particle tracking ethods to siulate transport in single-continuu and dual continua porous edia. Journal of Hydrology, 275(3-4), [doi: /s (03) ] LaBolle, E. M., Fogg, G. E., and Topson, A. F. B Rando-walk siulation of transport in heterogeneous porous edia: Local ass conservation proble and ipleentation ethods. Water Resources Research, 32(3), [doi: /95wr03528] LaBolle, E. M., Quastel, J., Fogg, G. E., and Gravner, J Diffusion processes in coposite porous edia and their nuerical integration by rando walks: Generalized stochastic differential equations with discontinuous coefficients. Water Resources Research, 36(3), [doi: /1999wr900224] LaBolle, E. M RWHet: Rando Walk Particle Model for Siulating Transport in Heterogeneous Pereable Media, User s Manual and Progra Docuentation. Davis: University of California. Liu, H. H., Bodvarsson, G. S., and Pan, L Deterination of particle transfer in rando walk particle ethods for fractured porous edia. Water Resources Research, 36(3), [doi: / 1999WR900323] Maxwell, R. M., and Topson, A. F. B SLIM-FAST: A User s Manual. Liverore: Lawrence Liverore National Laboratory. Neuan, S. P Trends, prospects, and challenges in quantifying flow and transport through fractured rocks. Hydrogeology Journal, 13(1), [doi: /s ] Pan, L. H., Liu, H. H., Cushey, M., and Bodvarsson, G DCPT v1.0: New Particle Tracker for Modeling Transport in Dual-Continuu Media. Liverore: Lawrence Liverore National Laboratory. Pan, L. H User Manual (UM) for DCPT v2.0. Liverore: Lawrence Liverore National Laboratory. Pan, L. H., and Bodvarsson, G Modeling transport in fractured porous edia with the rando-walk particle ethod: The transient activity range and the particle transfer probability. Water Resources Research, 38(6), W0901. [doi: /2001wr000901] Pohlann, K., Pohll, G., Chapan, J., Hassan, A. E., Carroll, R., and Shirley, C Modeling to Support Groundwater Containant Boundaries for the Shoal Underground Nuclear Test, Report DOE/NV/ Las Vegas: Desert Research Institute. Sudicky, E. A., and Frind, E. O Containant transport in fractured porous edia: Analytical solutions for a syste of parallel fractures. Water Resources Research, 18(6), [doi: / WR018i006p01634] Tang, D. H., Frind, E. O., and Sudicky, E. A Containant transport in fractured porous edia: Analytical solution for a single fracture. Water Resources Research, 17(3), [doi: / WR017i003p00555] Zhang, Y., Benson, D. A., and Baeuer, B Predicting the tails of breakthrough curves in regional-scale alluvial syste. Ground Water, 45(4), [doi: /j x] (Edited by Yun-li YU) Yong ZHANG et al. Water Science and Engineering, Oct. 2013, Vol. 6, No. 4,