A nonequilibrium model for reactive contaminant transport through fractured porous media: Model development and semianalytical solution

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1 WATER RESOURCES RESEARCH, VOL. 48, W1511, doi:1.19/11wr1161, 1 A nonequilibriu odel for reactive containant transport through fractured porous edia: Model developent and seianalytical solution Nitin Joshi, 1 C. S. P. Ojha, 1 and P. K. Shara 1 Received 11 Noveber 11; revised 3 July 1; accepted 3 August 1; published 6 October 1. [1] In this study a conceptual odel that accounts for the effects of nonequilibriu containant transport in a fractured porous edia is developed. Present odel accounts for both physical and sorption nonequilibriu. Analytical solution was developed using the Laplace transfor technique, which was then nuerically inverted to obtain solute concentration in the fracture atrix syste. The seianalytical solution developed here can incorporate both sei-infinite and finite fracture atrix extent. In addition, the odel can account for flexible boundary conditions and nonzero initial condition in the fracture atrix syste. The present seianalytical solution was validated against the existing analytical solutions for the fracture atrix syste. In order to differentiate between various sorption/ transport echanis different cases of sorption and ass transfer were analyzed by coparing the breakthrough curves and teporal oents. It was found that significant differences in the signature of sorption and ass transfer exists. Applicability of the developed odel was evaluated by siulating the published experiental data of Calciu and Strontiu transport in a single fracture. The present odel siulated the experiental data reasonably well in coparison to the odel based on equilibriu sorption assuption in fracture atrix syste, and ulti rate ass transfer odel. Citation: Joshi, N., C. S. P. Ojha, and P. K. Shara (1), A nonequilibriu odel for reactive containant transport through fractured porous edia: Model developent and seianalytical solution, Water Resour. Res., 48, W1511, doi:1.19/ 11WR Introduction [] Fractured porous aquifers are iportant sources of water for doestic, industrial and agriculture use in both developed and developing countries. About 75% of earth s surface consists of fractured or karstic fractured rock aquifer [Pluer et al., ]. Therefore, understanding, characterizing and odeling of physical cheical interaction in fractured aquifer becoes increasingly iportant in ters of water resources developent and groundwater containation [Dietrich et al., 5]. The pereability of fractures network is greater than the pereability of porous rock therefore, fractures acts as pathways through which the both reactive and nonreactive containants can ove rapidly [Tsang and Neretnieks, 1998]. [3] The conceptual odel representing the fracture porous edia can be categorized as equivalent continuu odel (ECM) and discrete fracture odel (DFM). In the equivalent continuu odels, the transport proble is 1 Departent of Civil Engineering, Indian Institute of Technology Roorkee, Roorkee, India. Corresponding author: N. Joshi, Departent of Civil Engineering, Indian Institute of Technology, Roorkee 47667, Roorkee, India. (nitinj398@ gail.co) 1. Aerican Geophysical Union. All Rights Reserved /1/11WR1161 analyzed at the acroscopic scale and the fracture atrix syste is conceptualized into one or ore overlapping continua [Schwartz and Sith, 1988; Royer et al., ]. [4] Continuu fracture odels can be classified as single-continuu odels, dual continuu odels and triple continuu odels Coittee on Fracture Characterization and Fluid Flow [1996]. Single-continuu odels analyze the fracture atrix syste by conceptualizing it as an equivalent porous edia [Long et al., 198; Berkowitz et al., 1988]. A dual continuu odels can be subclassified into dual porosity single pereability and dual porosity dual pereability odel. A dual porosity single pereability odel assues that no flow takes place between the atrix blocks [Barenblatt et al.,196;warren and Root,1963; Zieran et al., 1993; Dershowitz and Miller, 1995; Corapcioglu and Wang, 1999] whereas, dual porosity dual pereability odels assue that both fracture and atrix contribute to the flow [Zhang and Sun, ;Xu and Hu, 5]. Triple continuu odels conceptualize the fracture atrix syste into three overlapping continua, consisting of both large and sall-scale fractures along with the porous atrix [Wu et al., 4]. Continuu odels are well suited when the scale of the proble is large, rocks have significant atrix pereability and fractures are well interconnected. [5] In the discrete fracture odels, fractures are explicitly odeled. These odels are siilar to obile-iobile odel [West et al., 4]. There are a nuber of analytical [Grisak and Pickens,1981;Tang et al., 1981] and nuerical W1511 1of19

2 [Grisak and Pickens, 198;Noorishad and Mehran, 198; Huyakorn et al., 1983] studies that use discrete fracture odels for odeling reactive containant transport through fractured porous edia. The discrete fracture odels require detailed inforation of fracture networks and their characterization; they also require large coputer storage and coputational efforts [Dershowitz et al., 4]. The Discrete fracture atrix odels are used for the study of radionuclide transport and safety assessent studies of high lever nuclear waste [Noran and Kjellbert, 199]. [6] There have been a nuber of analytical solutions given for different source conditions, transport processes and fracture/atrix doains. Bibby [1981] derived an analytical expression for solute in the fracture for the case of diffusion for a two-diensional solute ass transport in dual porosity ediu. Tang et al. [1981] gave an analytical expression for advective-dispersive transport equation including equilibriu sorption and first-order degradation coefficient with constant concentration boundary condition and neglecting transverse dispersion. Sudicky and Frind [198] and Barker [198] gave the analytical solution for cases of radioactive containant transport in a syste of discrete ultiple parallel fractures. Moreno and Rauson [1986] gave an analytical solution for a continuous injection of the constant ass flux in the fracture. There have been a nuber of analytical solutions that neglect the dispersion in the fracture [Neretnieks, 198; Grisak and Pickens, 1981]. However, Tang et al. [1981] observed that neglecting longitudinal dispersion in the fracture ight lead to significant error in case of low fluid velocity. West et al. [4] gave an analytical solution for systes of parallel fractures, with a strip source of finite width; they also considered transverse dispersion in the fractures. Roubinet et al. [1] gave a seianalytical solution considering two-diension dispersion in fracture and two-diension diffusion in the atrix. [7] Analytical solutions are ostly liited to sipler cases and can be used as bencharks for nuerical odels. In addition, analytical solutions often consist of ultiple integrals of an oscillatory function, which is difficult to evaluate nuerically [Bodin et al., 3]. To overcoe such a difficulty, nuerical inversion of the Laplace spaced solution is widely eployed [Neville et al., ;Weatherill et al., 8;Gao et al., 1].However,incaseofnuerical odels, Weatherill et al. [8] found that, if the advective/dispersive transport in the fracture is higher than the diffusive transport in the atrix then, a very fine grid size of the order of the fracture aperture is required near the interface between fracture and atrix. Such a restriction ay lead to higher eory requireent and coputational efforts while odeling a large-scale fracture atrix syste. However, no such liitation exists in the case of analytical solutions. [8] There have been a nuber of studies that incorporate nonequilibriu sorption in a porous edia [Brusseau et al., 1989, 199; Seli et al., 1999; Leje and Bradford, 9]. Brusseau et al. [1989] developed a ulti process nonequilibriu (MPNE) odel for porous ediu where both transport and sorption related nonequilibriu process contributing to the observed nonequilibriu are considered. The odel that incorporates both transport and sorption related nonequilibriu processes yields discrete kinetic ters, each representing individual nonequilibriu processes and can be used for process level investigation of solute transport. Whereas, in case of fractured porous edia a very few studies exist that incorporate a nonequilibriu sorption in fracture/atrix. Most of the odels available for solute transport through fractured porous edia neglect the effects of sorption kinetics in fracture and atrix. Xu and Woran [1999] reported that neglecting the sorption kinetic results in errors in both the peak value of the pulse traveling in the fracture and the variance of the residence tie and can be up to several hundred percents. Coans and Hockley [199] showed that equilibration tie for sorption of Cesiu (Cs) in aquatic sedients vary over weeks in laboratory tests with illite and ontorillonite. [9] Based on experiental investigations, Neretnieks et al. [198] found that both fast and slow reactions contribute to the total retardation in the fracture. The fast reaction is on the iediately accessible fracture surface, whereas, the slow reaction is tie dependent as it increases with residence tie. Maloszewski and Zuber [199] developed a odel for reactive transport in a fracture atrix syste where the adsorption in the atrix is assued to be governed by a first-order nonequilibriu kinetic sorption. However, the tracer adsorption on fissure walls was approxiated by instantaneous equilibriu linear adsorption isothers. Berkowitz and Zhou [1996] developed an approxiate analytical solution for the dispersion of sorbing solute with first-order reaction. They investigated that a reversible surface reaction can be treated as irreversible when adsorption is very strong and desorption is relatively weak. However, at sall Daköhler nuber (ratio of reaction rate to olecular diffusion rate) the effect of surface reaction can be neglected, thus, reactive solute transport can be approxiated by nonreactive solute transport. Lee and Teng [1993] gave an analytical solution assuing nonequilibriu sorption in the atrix. [1] With the above background, the objectives of this study are (1) to propose a generalized odel which considers nonequilibriu sorption in both fracture and atrix; () to develop a seianalytical solution for the new odel; (3) to analyze the effects of various fracture and atrix sorptions and ass transfer paraeters on the breakthrough curve (BTC); (4) to test the applicability of the odel by siulating different experiental data available in the literature.. Conceptual Model [11] The physical nonequilibriu conditions (PNE) affect the transport of both sorbing and nonsorbing solutes and is evident fro large tailing and skewed breakthrough curve [Brusseau et al., 1989]. It has been widely reported that the PNE plays a critical role in the fractured porous edia, the ass transfer between the obile fracture and the iobile porous edia is described by PNE. The sorption nonequilibriu (SNE) results fro both cheical nonequilibriu and intra sorbent diffusion [Brusseau et al., 1989]. The SNE accounts for intra sorbent diffusion and rate liited interaction between the solute and sorbent. [1] The present study focuses on developing a odel that can account for both PNE and SNE in a fractured porous edia. The conceptualization of the odel is shown in Figure 1; the conceptual odel has a siilar approach to of19

3 Figure 1. Scheatic diagra of the conceptual odel. (Subscripts f and denote fracture and atrix; subscripts 1 and denote instantaneous sorbed and rate liited sorbed doain; denotes decay rate; K and k are sorption and ass transfer coefficients.) that of ulti process nonequilibriu odel developed by Brusseau et al. [1989]. The PNE is accounted by a diffusive ass transfer between the fracture and the adjacent porous atrix, as done elsewhere [Tang et al., 1981; Sudicky and Frind, 198; Maloszewski and Zuber, 199]. For SNE, the odel used a two-site conceptualization for both fracture and porous atrix. Here, at the first site (denoted by subscript 1), the sorption is assued to be governed by an instantaneous equilibriu adsorption isother whereas; at the second site (denoted by subscript ) the sorption is described by a rate-liited process, which is represented as a first-order reaction. 3. Governing Equations 3.1. Equation for Fracture and Matrix [13] The transport process in the fracture atrix syste, can be described by two coupled one-diensional equation one for the fracture and the other for the atrix. The coupling is accounted by a diffusive ass transfer between the fracture and the adjacent porous atrix. In this study, the following processes have been considered: advection and longitudinal dispersion in the fracture, olecular diffusion fro the fracture into the atrix, adsorption and decay in the fracture and atrix. The adsorption in the fracture and atrix is assued to be governed by both an instantaneous equilibriu adsorption isother and a first-order nonequilibriu kinetic sorption. The assuptions ade while deriving the governing equations are those of Tang et al. [1981]. The governing transport equation for the fracture can be written as f f 1 f ¼v f þ C f b f C f f 1 S f 1 f S f y¼b (1a) where C f and C are solute concentrations ðm=l 3 Þ in the fracture and porous atrix, respectively. x is the spatial coordinate taken in the direction of the flow; y is the spatial coordinate perpendicular to the fracture and t is the tie variable. v f is the ean water velocity ðl=tþ in the fracture; D f is the dispersion coefficient ðl =TÞ in the fracture; b is the half fracture aperture ðlþ; is the atrix porosity; D is effective diffusion coefficient ðl =TÞ; S f 1 and S f are the ass of sorbate per unit length ðm=l Þ of the fracture for which the sorption is governed by instantaneous equilibriu linear isother and the first-order kinetic sorption, respectively. Here, subscripts 1 and are used to represent the instantaneous and rate liited sorption sites, respectively. f is the first-order degradation rate constant ð1=tþ in the liquid phase for fracture. In this study, different degradation rate coefficients are used for instantaneous and rate liited sorption sites. f 1 and f are the firstorder degradation rate constants ð1=tþ in the sorbed phase for the fracture at site 1 and, respectively. F f is the fraction of the sorbent in the fracture for which sorption is instantaneous, its value ranges between and 1. [14] The instantaneous equilibriu sorption in the fracture is given by S f 1 ¼ F f K f C f (1b) siilarly, the first-order nonequilibriu kinetic sorption in the fracture is given f ¼ k f ½ 1 F f Kf C f S f Š f S f (1c) where K and k f are the distribution coefficient ðlþ and the first-order sorption kinetic coefficient ð1=tþ in the fracture, respectively. [15] Substituting equations (1b) and (1c) in equation (1a), we get br f f ¼v f þ C f ½b f þ f 1 F f K f þ k f 1 F f Kf ŠC f þ k f S f () y¼b where R f 1 ¼ 1 þ F f K f b is the retardation factor for the equilibriu sorption site in the fracture. 3of19

4 [16] Siilarly, the equation for the porous atrix can be written C ¼ C 1 S 1 S (3a) where S 1 and S are the ass of solute adsorbed per unit ass of solid ðm=mþ in the porous atrix for which the sorption is governed by instantaneous equilibriu linear isother and the first-order kinetic sorption, respectively; is the density of the porous atrix ðm=l 3 Þ; is the first-order degradation rate constant ð1=tþ in the liquid phase for atrix; 1 and are the first-order degradation rate constants ð1=tþ in the sorbed phase for the porous atrix; and F is the fraction of the sorbent in the atrix for which sorption is instantaneous. [17] The instantaneous equilibriu sorption in porous atrix is given as S 1 ¼ F K C : (3b) [18] The first-order kinetic sorption in porous atrix is given ¼ k ½ð1 F ÞK C S Š S : (3c) [19] K is the distribution coefficient for the porous atrix ðl 3 =MÞ and k is the first-order sorption kinetic coefficient ð1=tþin atrix. [] Substituting equations (3b) and (3c) In equation (3a) we get and atrix doain. In the present study, a generalized boundary condition siilar to that used by Neville et al. [] is considered. v f C f ð; tþ:d f ð; ¼ v f C ð1 Ht ð t ÞÞ; (6a) where C is the source concentration, H is the Heaviside step function and it is used for describing step input fro tie to t. ¼ represents a constant concentration or Diritchlet type boundary condition and ¼ 1 represents a constant flux or Cauchy type boundary condition. [3] For the outflow boundary condition for the fracture, both finite and sei-infinite doains are considered here. For finite doain the boundary condition is described f ðl f ; tþ ¼ (6b) [4] For sei infinite doain the boundary condition is described by C f ð1; tþ ¼ (6c) the inflow boundary condition for atrix describing the coupling of fracture and atrix is given by C ðx; b; tþ ¼C f ðx; tþ: (6d) [5] Siilarly for the atrix, both finite and sei-infinite doain are considered. For finite doain the boundary condition is described by C ¼ ½ þ 1 F K þ k ð1 F ÞK ŠC þ k S ðx; L ; tþ ¼ [6] For sei-infinite doain (6e) where R 1 ¼ 1 þ F K is the retardation factor for the equilibriu sorption site in porous atrix. 3.. Initial and Boundary Conditions [1] A generalized for of initial conditions is considered, which assues that the fracture and atrix are uniforly containated. C f ðx; Þ ¼C f ; S f ðx; Þ ¼S f ; (5a) (5b) C ðx; 1; tþ ¼: (6f) 4. Analytical Solution in the Laplace Doain 4.1. Analytical Solution for Matrix [7] Taking the Laplace transfor of equation (3c) we get ss S ¼ k ð1 F ÞK C S k þ S; (7) k ð1 F ÞK C þ S ¼ s þ k þ : (8) C ðx; y; Þ ¼C ; (5c) [8] Taking the Laplace transfor of equation (4) we get S ðx; y; Þ ¼S : (5d) [] However, this assuption is not restrictive in nature and one can proceed with the zero values through the fracture sc C ¼ D d C R 1 dy 1 ½ þ 1 R F K 1 þ k ð1 F ÞK Š:C þ k R 1 S : (9) 4of19

5 [9] Substituting equation (8) into equation (9) and siplifying we get d " C dy sr 1 þ þ 1 F K þ k ð1 F ÞK s þ # D D D D s þ k þ k C þ D S s þ k þ þ C R 1 D ¼ : (1) [3] The solution of equation (1) is obtained by assebling copleentary solution and particular solution C ¼ E1 expðh 1 ðy bþþþe expðh 1 ðy bþþ H : (11) 1 [31] In equation (11), on the right hand side the first two ters corresponds to copleentary solution whereas, the third ter corresponds to particular solution. E 1 and E are constants depending on the boundary conditions and their expressions are given in section " H 1 ¼ sr 1 þ þ 1 F K þ k ð1 F ÞK s þ # 1= D D D D s þ k þ ; (1) ¼ k S D s þ k þ þ C R 1 : (13) D Case I-A: Sei-infinite Matrix [3] Inflow and outflow boundary conditions for the atrix with sei-infinite doain are given by equation (6d) and equation (6f), respectively. In equation (11) when y!1, for the solution to be bounded, the value of E ¼. The value of E 1 will be obtained by using the inflow boundary condition for atrix (i.e., equation (6d)) therefore, equation (11) becoes C ¼ Cf þ H1 exp ðh 1 ðy bþþ H1 : (14) [33] Taking the derivative of equation (14) ¼ C 1 H 1 C f þ : (15a) y¼b H 1 [34] For the sei-infinite atrix doain C 1 ¼1: (15b) Case I-B: Finite Matrix [35] For the atrix with finite doain, the inflow and outflow boundary conditions are given by equation (6d) and equation (6e), respectively. Using these boundary conditions the constants E 1 and E are coputed as C f þ H 1 exp ðh 1 ðl bþþ E 1 ¼ exp ðh 1 ðl bþþþexp ðh 1 ðl bþþ ; (16a) C f þ H 1 exp ðh 1 ðl bþþ E ¼ exp ðh 1 ðl bþþþexp ðh 1 ðl bþþ : (16b) [36] Substituting the values of E 1 and E in equation (11) the equation for atrix in Laplace doain is obtained. C ¼ exp ð H1 ðl yþþþexpðh 1 ðl yþþ Cf þ expðh 1 ðl bþþþexpðh 1 ðl bþþ H 1 [37] Taking derivative of equation (16c), we have H1 ¼ C 1 H 1 C f þ : (17a) y¼b H 1 [38] For the finite atrix doain C 1 ¼ exp H ð 1ðL bþþexp ðh 1 ðl bþþ : (17b) exp ðh 1 ðl bþþþexp ðh 1 ðl bþþ [39] It is to be noted that y¼b has a general for of C 1 H 1 C f þ H 1 however the C 1 values differ for atrix with sei-infinite and finite doains and is given by equations (15b) and (17b), respectively. 4.. Analytical Solution for the Fracture [4] Taking Laplace transfor of equation (1c) ss f S f ¼ k f ½ 1 F f Kf C f Š kf þ f S f S f ; (18) k f 1 F f Kf C f Sf ¼ þ : (19) s þ k f þ f s þ k f þ f [41] Taking Laplace transfor of equation () and substituting the value of S f fro equation (19), we get sc f C f ¼ v f dc f R f 1 dx þ D f d C f R f 1 dx 1 ½b f þ f 1 b:r F f K f þ k f 1 F f Kf Š:C f f 1 þ k 1 f k f 1 F f Kf C f f þ A b:r f 1 s þ k f þ f s þ k f þ f þ D dc b:r f 1 dy : y¼b () [4] Substituting the y¼b fro equations (15) or (17) for a sei infinite or finite atrix doain, respectively, we get d C f dx v f dc f D f dx 1 H C f þ ¼ ; b:d f (1a) 5of19

6 k f 1 F f Kf s þ H ¼ b:r f 1 s þ b f þ f 1 F f 4 f K f þ s þ k f þ f D C 1 H 1 #; ¼ k f Sf þ R 1Cf b:d f s þ k f þ D f f þ D b:d f (1b) 1 C 1 A: (1c) H 1 [43] The solution for equation (1a) is given by where C f ¼ E3 expðh 3 xþþe 4 expðh 4 xþþ ; H 3 H 4 H 3 ¼ v f D f H 4 ¼ v f D f (a) " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 1 1 þ 4H D f ; (b) bv f " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 1 þ 1 þ 4H D f : (c) bv f [44] The constants E 3 and E 4 can be obtained using inflow and outflow boundary conditions Case II-A: Fracture Is of Sei-Infinite Extent [45] For the fracture with sei-infinite doain, the inflow and outflow boundary conditions are given by equation (6a) and equation (6c), respectively. For the sei-infinite boundary, the solution will reain bounded when E 4 ¼ and E 3 is obtained using inflow boundary condition and is given as. C f H 3H 4 E 3 ¼ v f : (3a) v f :D f H 3 [46] Substituting the value of E 3 and E 4 in equation (a) we get where C f C f H 3H 4 ¼ vf exp ðh 3 xþþ ; v f :D f H 3 H 3 H 4 C f is given by C f (3b) C ¼ ð1 exp ðst ÞÞ: (3c) s 4... Case II-B: Fracture Is of Finite Extent [47] For the fracture with finite extent, the inflow and outflow boundary conditions are given by equation (6a) and equation (6b), respectively. Using these boundary conditions the constants E 3 and E 4 are coputed as E 3 ¼ D D 1 ; (4a) E 4 ¼ D 3 D 1 ; (4b) D 1 ¼ v f :D f H 4 H3 exp H 3 L f (4c) v f :D f H 3 H4 exp H 4 L f ; D ¼ H 4 exp H 4 L f vf C f vf H 3 H 4 ; (4d) D 3 ¼ H 3 exp H 3 L f vf C f vf : (4e) H 3 H 4 [48] Substituting equation (4) in equation (a), we get D C f ¼ D 1 expðh 3 xþþ D 3 D 1 expðh 4 xþþ : H 3 H 4 (4f) 4.3. Nuerical Inversion of Solutions in Laplace Doain [49] In order to obtain the solution in tie doain, the analytical solution in the Laplace doain is to be inverted. The inversion of the Laplace transfor can be done either analytically or nuerically. However, analytical inversion of the Laplace transfored solution often leads to expressions having ultiple integrals, which ultiately needs to be evaluated nuerically [Tang et al., 1981; Sudicky and Frind, 198]. However, looking to the coplex nature of the solution obtained here a nuerical inversion of the Laplace transfored solution is adopted. There are a nuber of algoriths available for nuerical inversion; the present study uses the inversion ethod developed by de Hoog et al. [198]. The de Hoog s algorith has been used widely in the flow and transport proble for both advection and dispersion doinated cases [Neville et al., ; Gao et al., 1]. The de Hoog s algorith approxiates the inverse Laplace transfor in the for of a Fourier series. C f ðx; tþ ¼ 1 C exp ðatþ Re T! f ðx; aþ þ XM ik C f x; a þ ; T k1 (5a) where T defines the period of the approxiating Fourier series and its value is T ¼ :8t [Neville et al., ]. The second paraeter a is related to the singularities in the transfored solution and it estiated as a ¼ ln ð E rþ T : (5b) [5] The value of ¼, E r ¼ :1 and M ¼ 7 is taken as suggested by de Hoog et al. [198]. For accelerating, the convergence of the Fourier series a Quotient differences algorith is used [de Hoog et al., 198]. 5. Analytical Solution for Teporal Moents [51] Teporal oents are often used to characterize the solute transport processes in porous edia. The solute breakthrough curve (BTC) is ore convenient to easure rather than the concentration profile therefore, the teporal 6of19

7 oents can be useful in such cases. The analysis of the teporal oents can be a useful tool for understanding the ipact of various processes on the solute transport. [5] Teporal oents were coputed by using the ethod given by Aris [1956], which has been used extensively [Goltz and Roberts,1986;Valocchi,199;Srivastava et al., 4]. The teporal oents are given by " # M n ¼ ð1þ n d n C f ds n s¼ ; (6) where M n is the n th teporal oent of C f about the origin (t ¼ ). Following Kendall and Stuart [1958] the noralized first oent about origin, noralized central, second and third oent can be given as " n ¼ ð1þ n d n # ln C f ds n ; (7) where n is the n th teporal oent about the origin (for n ¼ 1) and about the ean (for n ¼ and 3). The analytical expression for concentration in the Laplace doain contains a ultiplication of several ters. When taking the logarith of the concentration each ter can be separately coputed and then added together [Srivastava et al., 4]. The ass of solute recovered at the sapling point is given by the zeroth teporal oent ðm Þ, whereas the noralized first oent by the zeroth oent, noralized central second and third oents give the ean arrival tie, variance and skewness of the breakthrough curve, respectively. [53] For fracture and atrix of sei-infinite extent, the analytical solution in the Laplace doain is given by equation (3b) where ters H ; H 3 and H 4 are given by equations (1b), (b), and (c), respectively, and assuing a zero initial concentrations leads to ¼. The expression for the oents are given as v f M ¼ ðt Þ exp ðh 3 ðþxþ; (8a) v f :D f H 3 ðþ s! 1 ¼ t :D f H = 3 ðþ H = 3 v f :D f H 3 ðþ ðþx ; (8b) 1 ¼ t :D f H == 3 ðþ :D f H = B 3 þ þ ðþ 1 v f :D f H 3 ðþ A v f :D f H 3 ðþ þ H == ðþx ; 3 :D f H == 3 ðþ 3 ¼ ðþþ þ 3 :D H = f 3 ðþh 3 ðþ v f :D f H 3 ðþ v f :D f H 3 ðþ 3 1 :D f H = 3 ðþ þ C A þ H === 3 ðþx ; v f :D f H 3 ðþ (8c) (8d) where the prie represent the differentiation of H 3 with respect to s and H 3 ðþis the value of H 3 at s ¼. The first ter on the right hand side of equation (8) represents the ð contribution of 1et s Þ v s ; the second ter that of f ðv f :D f H 3 Þ and the third ter of exp ðh 3 xþ. [54] Fro the above equations following eaningful paraeters can be deterined V eff ¼ x ; (9a) 1 D eff ¼ ; V eff (9b) 1 C s ¼ 3 1:5 ; (9c) where V eff, D eff, and C s are effective velocity, effective dispersion coefficient and the skewness, respectively. 1,, and 3 are the oents obtained after correcting for the finite pulse which can be obtained by neglecting the first ter of the right hand side of the equation (8). [55] Often it is assued that bacteria, viruses and other larger sized containants/colloids do not penetrate into the rock atrix therefore, for such cases, the ass transfer into the atrix can be neglected. Assuing an equilibriu sorption with first type input boundary condition, the ass arrival is given by equation (8a) whereas the expression for the higher-order teporal oents can be siplified to ¼ R f 3 ¼ 1 D f 1 ¼ R f x v f 41 þ Df x þ v f 3x Rf þ v f V eff ¼ v f R f 41 þ 4 b f þ f K f bv f D f 4 b f þ f K f bv f D f b f þ f K f bv f 4 b f þ f K f bv f D eff ¼ D f R f ; D f 3 5 D f 3 5 D f C s ¼ 6B v f x 1 þ 4ð A bf þ f Kf ÞDf bv f 1 1= 3= 3 5 1= 1= ; (3a) ; (3b) 5= ; (3c) ; (3d) (3e) : (3f) 6. Coparison With Other Solutions 6.1. Coparison With Analytical Solutions [56] There are a nuber of analytical solutions available for the reactive transport in the fractured porous ediu. 7of19

8 Most of the odels assue a linear equilibriu sorption in the fracture atrix syste, however, the nonequilibriu odel developed in this study can incorporate both linear equilibriu and rate liited sorption in the fracture and porous atrix siultaneously. In order to evaluate the accuracy and applicability of the present sei analytical solution, it is desirable to copare the present seianalytical solution with the existing analytical solutions for the fracture atrix syste. Therefore, the present sei analytical solution is copared with the two classical analytical solutions of Tang et al. [1981] and Sudicky and Frind [198]. Since both the solutions were also derived in Laplace doain, therefore the coparison has been done analytically as given in Appendix A. It can be seen that under special cases, the present analytical solution in the Laplace doain is equivalent to the analytical solutions of Tang et al. [1981] and Sudicky and Frind [198]. 6.. Coparison With the Multi Rate Mass Transfer (MRMT) Model [57] Multi rate ass transfer (MRMT) odel conceptualizes the fracture atrix syste into obile-iobile zone, where fractures and atrix are considered as obile and iobile zone, respectively. Advection occurs through obile zone whereas ass is transferred fro the obile zone into iobile zone via diffusion/first-order ass transfer. MRMT is a particular type of continuous tie rando walk odel [Berkowitz et al., 8; Kuntz et al., 11]. MRMT odel considers nuerous types and rates of ass transfer siultaneously [Haggerty and Gorelick 1995]. MRMT odel can describe ass transfer into heterogeneous edia and is equivalent to diffusion liited ass transfer into spheres, cylinders or layers. Haggerty and Gorelick [1995] and Salaon et al. [6] gave the firstorder coefficients and capacity coefficient, which transfor MRMT odel into diffusion odel of different geoetries. [58] In this study, a discrete approxiation of the ass transfer coefficient is considered. The governing equations and analytical expression for the teporal oents for MRMT odel are given in Appendix B. Teporal oent and BTC obtained using MRMT odel and fracture atrix odel are copared in section 7. Since the fracture atrix odel and MRMT odel are conceptually different; therefore, in order to atch the BTC obtained by equilibriu fracture atrix odel, MRMT odel is used in the paraeter estiation ode. The BTC obtained by equilibriu fracture atrix odel was considered as observed data and was fitted by MRMT odel using paraeter estiation. 7. Model Analysis [59] For the successful application of the odel, it is required to analyze its behavior under a variety of situations. The purpose of this section is to evaluate the present odel with different cases of sorption and ass transfer. As it is evident fro the literature that the teporal oents have been widely used to study the behavior of solute transport processes therefore, the sae is also utilized here to reflect the sensitivity of involved processes for a wide range of odel paraeter values. One of the uses of the sensitivity analysis is ultiately to provide an insight into the use of the odel in siulation. [6] In order to differentiate the transport/sorption processes, five cases of sorption and ass transfer are considered here. An initially solute free ediu and a seiinfinite fracture and atrix doain are assued. Case 1 neglects the ass transfer fro fracture into atrix, i.e., the fracture atrix syste is approxiated as a single doain odel. In case, a local equilibriu assuption (LEA) with an instantaneous sorption is assued in both fracture and atrix. In case 3, a nonequilibriu sorption is assued in atrix, i.e., F ¼ :5 and an equilibriu sorption is assued in fracture, i.e., F f ¼ 1 which is siilar to that used by Xu and Woran [1999]; Maloszewski and Zuber [199]. In case 4, a nonequilibriu sorption is assued in both fracture and atrix F f ¼ F ¼ :5.In case 5, the fracture atrix syste is conceptualized as a obile-iobile zone and a discrete MRTM odel is used. MRMT odel considers an equilibriu sorption and liquid phase decay in both obile and iobile region. In cases 1 4, decay is considered only in the liquid phase of the fracture and atrix. The odel input paraeters for all these cases are listed in Table 1. [61] Péclet nuber (Pe) representing the ratio of dispersive to advective travel tie is an iportant paraeter and is given as Pe ¼ v f L D f : (31) [6] A higher value of Péclet nuber denotes an advection-doinated transport whereas, a lower value of Péclet nuber represents dispersion doinated transport. [63] In order to keep the parallelis with the MRMT odel, the ass transfer coefficient ½T 1 Š in the fracture atrix syste is described as in [Carrera et al., 1998]! FM ¼ D R L : (3) [64] The Daköhler nuber representing SNE in fracture and atrix is defined as the ratio of hydraulic residence tie to reaction tie, thus defines the degree of nonequilibriu [Brusseau et al., 1989] and is given as k f ¼ k f L v f ; k ¼ k L ; v f (33a) (33b) where L is the distance of interest and kf and k are the Daköhler nubers representing SNE in fracture and atrix, respectively Coparison of the BTC [65] Figure shows the BTC obtained for the cases listed in Table 1, pulse duration of 1 days was considered and the BTC was easured at distance of 1. The BTC were obtained for case 1 4 using the transport paraeters given in Table 1. However, for the case of MRMT odel the odel paraeters such as ass transfer coefficient (! ), capacity ratio ( tot ) and retardation factor of obile (R a ) 8of19

9 Table 1. Paraeter Values Taken for BTC and Teporal Moent Analysis for Various Cases of Sorption and Mass Transfer Considered a Serial Nuber Paraeter Unit Case 1 Case Case 3 Case 4 MRMT 1 L Pulse duration day 1/1 1/1 b 1/1 b 1/1 b 1/1 b 3 D d 1 1.E-6 1.E-6 1.E gc c v f /v a d b K l g k day F F f K f k f day f ¼ day c a ¼ n day c! day c tot Nuber of ultirate series c 3 c R a c R n a Subscript f and denotes fracture and atrix for a fracture atrix odel. Subscript a and n denotes obile and iobile region, respectively, for a MRMT odel. A discrete approxiation of the ass transfer coefficient is considered for MRMT odel. b For Teporal oent coputation. c MRMT odel paraeters. and iobile (R n ) region were estiated using optiization algorith as discussed earlier. It is seen fro Figure that the BTC obtained by MRMT odel atches very well with the equilibriu fracture atrix odel; however, a lower concentration at sall transport tie and at tail were observed in case of MRMT odel BTC. [66] When the ass transfer between fracture and atrix is considered, (i.e., Case 5), the peak of the BTC reduces and larger tailing as copared to the single doain odel (i.e., Case 1) is observed. Considering nonequilibriu sorption in the fracture atrix syste leads to an earlier arrival tie, lower peak and larger tailing in the BTC as copared to an equilibriu fracture atrix and MRMT odel. Figure. Coparison of BTC for different cases of sorption and ass transfer at a distance of 1 and for pulse duration of 1 days. 7.. Coparison of the Teporal Moent of the BTC [67] In this section, the teporal oents of the BTC for the five cases of sorption and ass transfer as discussed earlier are presented. At the inlet of the fracture, a Dirichlet type boundary condition is considered and the pulse length t is taken as unity. Effects of various nondiensional paraeters (given by equations (31) (33)) on the ean arrival tie, variance and skewness of BTC are presented. For various fracture atrix odels, the expression of the teporal oent are given by equation (8) (3) and for MRMT odel the teporal oents are given in Appendix B Effect of Péclet Nuber [68] In presence of ass transfer, Péclet nuber significantly affects the shape of the BTC. In case of fracture atrix odel, as the Pe is increased, the solute arrival tie increases whereas the peak concentration and the tailing reduces. Figure 3a shows the behavior of ean arrival tie with Pe for different cases considered. Identical value of ean arrival tie was obtained for cases 4; therefore, only case 4 is shown here. It should be noted that the value of the Pe was varied by changing the dispersivity while keeping fracture velocity constant. The ean arrival tie increases or in other words the effective velocity decreases with an increase in Pe and at higher values of Pe (i.e., Pe > 5) it becoes asyptotic. This shows that at higher values of Pe, dispersion in fracture is not the ain transport echanis. The increase in the Pe or reduction in the dispersion coefficient would reduce the spreading of the plue therefore, the ean arrival tie increases with Pe. The decrease in the effective velocity is due to the coupling between spreading and transforation. The spreading increases the volue occupied by the plue and thus the transforation capacity. The nonequilibriu sorption in fracture/atrix does not have uch influence on the ean arrival tie of the BTC, as reported by Srivastava and Brusseau [1996]. 9of19

10 [69] Figure 3b shows the behavior of the second teporal oent or the variance of BTC with Pe and it can be seen that the variance of the BTC shows a nononotonic behavior. Identical value of variance was obtained for cases 3 4; therefore, only case 4 is shown here. At lower Pe, an increase in the variance of BTC is due to coupling between spreading and transforation whereas the decrease in the variance at higher Pe is due to reduction of the spreading in the fracture as deonstrated by case 1. In case of single doain odel (i.e., Case 1), where the spreading in the fracture is the only dispersion process, the variance of the BTC reduces onotonically with an increase in Pe and as Pe!1, the variance reduces to zero. In case of fracture atrix odels, both spreading in fracture and fracture atrix interaction accounts for the dispersion process and at higher Pe the fracture atrix interaction becoes the ain dispersion echanis. As deonstrated by the asyptotic behavior of the variance for cases 5 at higher values of Pe. [7] Figure 3c shows the effect of Pe on the skewness ðc s Þ of the BTC. It can be seen here that when ass transfer between fracture and atrix is neglected (i.e., case1) C s reduces onotonically with an increase in Pe. Whereas, when ass transfer is considered (i.e., Case 5), the behavior of C s becoes nonunifor and at higher Pe, the C s becoes asyptotic. This again shows that at higher Pe, the effect of dispersion in the fracture becoes negligible. A larger tailing is observed in case of fracture atrix odel as copared to MRMT odel thus increasing the spreading and skewness of the BTC. When nonequilibriu sorption is considered in both fracture and atrix i.e., Case 4, a higher tailing and the spreading of the BTC is observed thus resulting in an increased variance and skewness as copared to when nonequilibriu is considered only in atrix. Figure 3. (a) Behavior of ean arrival tie of the BTC with Péclet nuber for different cases of sorption and ass transfer considered (identical value of ean arrival tie was obtained for cases 4; therefore, only case 4 is shown here). (b) Behavior of variance of the BTC with Péclet nuber for different cases of sorption and ass transfer considered (identical value of ean arrival tie was obtained for cases 3 4; therefore, only case 4 is shown here). (c) Behavior of coefficient of skewness of the BTC with Péclet nuber for different cases of sorption and ass transfer considered. (Identical value of ean arrival tie was obtained for cases 3 4; therefore, only case 4 is shown here) Effect of Mass Transfer Coefficient [71] The ass transfer coefficient significantly affects the peak and spreading of the BTC. For a fracture atrix odel, as the ass transfer coefficient is increased, the peak of the BTC reduces and the arrival tie and spreading increases. Figure 4 a shows the variation of the ean arrival tie for different values of the ass transfer coefficient. When ass transfer between fracture and atrix increases, the ean arrival tie increases or effective velocity reduces. Cases 4 show an identical behavior of ean arrival tie with the ass transfer coefficient; therefore, only case 4 is shown here. At very sall values of ass transfer coefficient, the fracture atrix odel behaves as a single region odel. At higher values of ass transfer coefficient, the MRMT odel BTC becoes syetric and any further increase in ass transfer coefficient does not affect the ean arrival tie therefore the ean arrival tie becoes asyptotic. [7] Figure 4b shows the effect of ass transfer coefficient on the variance of the BTC. For the case of fracture atrix odels, at sall values of ass transfer coefficient, the variance of the BTC becoes asyptotic. This again shows that at sall values of ass transfer coefficient, the fracture atrix odel behaves siilar to a single region odel. However, a higher asyptotic value of variance is observed for case 4, which is due to presence of rate-liited sorption in fracture. In case of MRMT odel, the variance shows a nononotonic behavior and at higher values of 1 of 19

11 ass transfer coefficient, the variance becoes asyptotic. At higher values of ass transfer coefficient, MRMT odel behaves siilar to a single region odel with the porosity equal to the total porosity of the ediu therefore, the BTC becoes syetric and thus the variance approaches an asyptotic value. Siilarly, at very sall value of ass transfer coefficient the MRMT odel also behaves as a single region odel with the porosity equal to the porosity of the obile doain. [73] Figure 4c shows the effect of ass transfer on the coefficient of skewness of the BTC. It is seen that for fracture-atrix and MRMT odel the behavior of coefficient of skewness is non-onotonic. At low values of ass transfer coefficient, the fracture atrix odel and MRMT odel behave as a single region odel. At higher values of ass transfer coefficient identical values of skewness is obtained for the case of equilibriu and nonequilibriu fracture atrix odels. It is also seen that at lower ass transfer coefficient, nonequilibriu sorption in fracture and at higher ass transfer coefficient nonequilibriu in atrix have a significant influence on the variance and skewness of the BTC Effect of Daköhler Nuber [74] At higher values of Daköhler nuber or larger ass transfer rate, the BTC with nonequilibriu sorption odel behaves siilar to an equilibriu sorption odel. Mean arrival tie of the solute essentially reains constant with the Daköhler nuber (not shown here). Figure 5 shows the variance of the BTC with different values of fracture and atrix Daköhler nuber. At higher values of Daköhler nuber (kf ¼ k > 1) variance of the BTC becoes asyptotic and identical to that obtained when equilibriu sorption is assued in fracture and atrix (case ). However, equilibriu is achieved at a saller Daköhler nuber when nonequilibriu sorption is considered only in atrix as copared to the cases when nonequilibriu sorption is considered in fracture. Larger spreading of the BTC is observed when nonequilibriu sorption is considered in both fracture and atrix as copared to the cases when nonequilibriu sorption is considered only in atrix. [75] C s shows a siilar behavior to that of variance as discussed earlier (therefore not shown here). A higher value of C s is observed when nonequilibriu sorption is considered only in atrix, as copared to the case when nonequilibriu sorption is considered in the fracture. At higher values of Daköhler nuber (i.e., kf ¼ k > 1) sorption kinetics in fracture and atrix are not so iportant therefore, local equilibriu assuption can be applied. Figure 4. (a) Mean arrival tie of the BTC with ass transfer coefficient for different cases of sorption and ass transfer considered (identical value of ean arrival tie was obtained for cases 4; therefore, only case 4 is shown here). (b) Variance of the BTC with ass transfer coefficient for different cases of sorption and ass transfer considered. (c) Coefficient of skewness of the BTC with ass transfer coefficient for different cases of sorption and ass transfer considered. 8. Model Evaluation [76] In order to evaluate the applicability of the present odel, the odel has been applied to siulate the experiental data. The experiental data were siulated using equilibriu fracture atrix odel, nonequilibriu fracture atrix odel and MRMT odel. For siulation of the BTC using MRMT odel, STAMMT-L nuerical code [Haggerty and Reeves, ] is used. STAMMT-L can incorporate both single rate and ulti rate ass transfer and considers both discrete and continuous distribution of ass transfer rate. 11 of 19

12 Figure 5. Variance of the BTC with fracture and atrix Daköhler nuber for different cases of sorption and ass transfer considered Data Set 1 [77] Grisak et al. [198] conducted laboratory investigation in which Calciu (Ca) and Chloride (Cl) were passed as tracers through a large, relatively undisturbed.76 long and.65 diaeter cylindrical saple of fractured clay loa till. [78] Assuing a sei-infinite fracture and a finite atrix extent, the nonreactive chloride data were siulated using equilibriu odel (not shown here), the flow velocity of 9.7 d 1 ; ¼ :35; fracture aperture b ¼ 4 density of the porous atrix ¼ 1:7 g c 3 and fracture spacing of 6 c was considered [Grisak et al., 198]. The adsorption of Cl in the fracture and atrix is neglected therefore, only effective diffusion coefficient (D ) and dispersivity () values were estiated using a Levenberg- Marquardt nonlinear least squares optiization algorith as D ¼.4E-6 d 1 and ¼.114. Grisak et al. [198] fitted the dispersivity value, ¼.15 using their odel. [79] Calciu data were then fitted using equilibriu and nonequilibriu adsorption odels. The value of was kept sae as estiated for Cl. The value of D for Calciu is taken as D ¼ 9:1E 7 d 1 which is obtained by ultiplying the ratio of free water diffusion coefficients of Calciu and Chloride (i.e., 38%) to the D value fitted for the chloride. Equilibriu adsorption odel paraeters K and K f were estiated as K ¼.63 l g 1 and K f ¼ :3. The estiated value of distribution coefficient corresponds to fracture and atrix retardation factor of 16 and 1.3, respectively. The effective diffusion coefficient selected by Grisak et al. [198] was D ¼ 1.64E 6 d 1 and assued that reactive transport takes place in atrix only (i.e., K f ¼ ). They also concluded that the best fit appears to be at K :1 l g. [8] For the nonequilibriu transport odel, the value of D was kept sae as used for the equilibriu adsorption odel and assuing an instantaneous sorption in the atrix i.e., F ¼ 1andk ¼. Using optiization algorith K ¼ :6 l g 1, K f ¼ :5, k f ¼ 5:E 5 d 1 and F f ¼ :57 were estiated. A very low value of k f indicates that the contribution of SNE to total nonequilibriu becoes insignificant and the apparent retardation factor in the fracture as given by R f 1 ¼ 15:5 this value is siilar to that estiated by equilibriu odel. The BTC obtained using both equilibriu and nonequilibriu odels superipose over each other as shown in Figure 6. [81] Using MRMT odel the nonreactive chloride data was siulated (not shown here). The ass transfer coefficient (! ), capacity ratio ( tot ) and dispersivity () values were estiated using an optiization algorith and were found to be.35 d 1, 7.6 and.114, respectively. For a nonreactive solute, the capacity ratio is the ratio of iobile zone to obile zone porosity, i.e., n = a. The Calciu data was then siulated using MRMT odel, the value of and the ratio of porosity of iobile and obile region were kept sae as estiated earlier for Cl. The ass transfer coefficient (! ) and retardation factor of obile (R a ) and iobile (R n ) region were estiated using optiization algorith and are! ¼ :8 d 1, R n ¼ 47:81 and R a ¼ 16:6. It can be seen that for the case of MRMT odel, a higher value of fracture and atrix retardation factor is obtained as copared to fracture atrix odel. [8] Figure 6 and Table copares the BTC obtained and input paraeters used for siulation of equilibriu, nonequilibriu and MRMT odels and it can be seen that all the three odels fit the observed data very well. However, very sall difference is observed between MRMT and fracture atrix odels, an earlier arrival tie and higher peak concentration is observed with MRMT odel. [83] The perforance of the odels were evaluated by considering root ean square error (RMSE) and deterination coefficient ðr Þwhich are expressed as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 X N RMSE ¼ t C iobs C icop ; (34a) N i¼1 Figure 6. Siulation of the experiental data of Calciu [Grisak et al. 198] input paraeters for this siulation is given in Table (equilibriu and nonequilibriu siulation overlay each other). 1 of 19

13 Table. Transport Paraeters Taken for Data Siulation of Grisak et al. [198] a Serial Nuber Paraeter Unit Grisak et al. [198] Equilibriu Model Nonequilibriu Model MRMT 1 L M D d E-7 9.1E-7 9.1E gc a v f /v a d b 7 M K l g k day 1 1 F F f K f M k f day 1 5.E-5 14 f ¼ day 1 15 a a ¼ n day 1 16 a! day a tot Nuber of ultirate series 3 19 R a 16.7 R n RMSE... r a Subscript f and denotes fracture and atrix for a fracture atrix odel. Subscript a and n denotes obile and iobile region, respectively, for MRMT odel. A discrete approxiation of the ass transfer coefficient is considered for MRMT odel. b MRMT odel paraeters. r ¼ 1 X N i¼1 X N i¼1 C iobs C icop ; (34b) C iobs C iobs where C iobs and C icop are observed and coputed concentrations respectively, C iobs is the ean value of C iobs and N is the nuber of the observed concentration data at a particular observation point. Identical values of RMSE and r, i.e.,. and.98, respectively, were obtained for equilibriu, nonequilibriu, and MRMT odels as given in Table. 8.. Data Set [84] Neretnieks et al. [198] conducted an experiental investigation of the radionuclide igration in a granite cylindrical core 3 c long and c in diaeter. They obtained breakthrough curves for the nonsorbing tracers, titrated water and ligno-sulfonate olecule and for sorbing tracers Cesiu (Cs þ) and Strontiu (Sr þ). Neretnieks et al. [198] considered porosity of the rock ¼ :1; fracture aperture b ¼.18, density of the crystalline rock ¼ :65 g c 3 and dispersivity ¼ 5. Using batch sorption experients, they found that for Strontiu the distribution coefficient was K ¼ 3 6 1c 3 g 1. However, they found that for Strontiu (Sr þ), the best fit was observed at K f ¼ :1. The effective diffusivity for Sr þ D ¼ 3:6 1 9 h 1 was taken which is sae as easured by Skagius et al. [198] for the granite. Neretnieks et al. [198] in their experient S-7 considered a injection pulse of 15 in duration and flow velocity was v ¼ :414 h 1. In the present study, Sr þ data were siulated using equilibriu, nonequilibriu, and MRMT odel considering a sei-infinite fracture and atrix doain. The odel paraeters were estiated using Levenberg-Marquardt nonlinear least squares optiization algorith. [85] First, an equilibriu sorption odel was fitted to the experiental data, the effective diffusivity and volue equilibriu constant for Sr þ was taken sae as reported by Neretnieks et al. [198]. The reaining paraeters and K f were estiated as ¼ :16 and K f ¼ Neretnieks et al. [198] estiated dispersivity ¼ :5 and K f ¼ :1. It can be seen that a higher value of dispersivity was obtained with equilibriu transport odel. Siilarly, the experiental data were also siulated using the nonequilibriu odel. The value of D ¼ 3:6 1 9 h 1 and K ¼ 3c 3 g 1 were taken which are sae as used for equilibriu odel. When the effective diffusion coefficient of atrix is low, then an equilibriu sorption odel in the atrix could be adequate; therefore, the value of F ¼ 1andk ¼ are considered. Using optiization algorith ¼ :15, F f ¼ :3, K f ¼ 1: 1 3 and K f ¼ :16 h 1 were obtained. It can be seen that an identical value of K f as estiated by Neretnieks et al. [198] were obtained for nonequilibriu and equilibriu odel, the other odel paraeters are listed in Table 3. A lower value of dispersivity was estiated by nonequilibriu odel than the value estiated by Neretnieks et al. [198]. Neretnieks et al. [198] reported that both fast and slow reactions contribute to the total retardation in the fracture. The value of the fracture Daköhler nuber is obtained as kf ¼ :116. As the coputed value of kf < 1 therefore, LEA is not valid and the contribution of SNE to total nonequilibriu is significant. The retardation factor in instantaneous and rate liited sorption doain of fracture as given by R f 1 ¼ 1 þ F f K f b and ð R f ¼ 1F f ÞK f b, respectively and are R f 1 ¼ 5andR f ¼ 9:33, the total retardation factor R f 1 þ R f is It can be seen that about 65% of the retardation in the fracture occurs 13 of 19