## 1. Introduction

[2] Fractured porous aquifers are important sources of water for domestic, industrial and agriculture use in both developed and developing countries. About 75% of earth's surface consists of fractured or karstic fractured rock aquifer [*Plummer et al.*, 2002]. Therefore, understanding, characterizing and modeling of physical chemical interaction in fractured aquifer becomes increasingly important in terms of water resources development and groundwater contamination [*Dietrich et al.*, 2005]. The permeability of fractures network is greater than the permeability of porous rock therefore, fractures acts as pathways through which the both reactive and nonreactive contaminants can move rapidly [*Tsang and Neretnieks*, 1998].

[3] The conceptual model representing the fracture porous media can be categorized as equivalent continuum model (ECM) and discrete fracture model (DFM). In the equivalent continuum models, the transport problem is analyzed at the macroscopic scale and the fracture matrix system is conceptualized into one or more overlapping continua [*Schwartz and Smith*, 1988; *Royer et al.*, 2002].

[4] Continuum fracture models can be classified as single-continuum models, dual continuum models and triple continuum models*Committee on Fracture Characterization and Fluid Flow* [1996]. Single-continuum models analyze the fracture matrix system by conceptualizing it as an equivalent porous media [*Long et al.*, 1982; *Berkowitz et al.*, 1988]. A dual continuum models can be subclassified into dual porosity single permeability and dual porosity dual permeability model. A dual porosity single permeability model assumes that no flow takes place between the matrix blocks [*Barenblatt et al.*, 1960; *Warren and Root*, 1963; *Zimmerman et al.*, 1993; *Dershowitz and Miller*, 1995; *Corapcioglu and Wang*, 1999] whereas, dual porosity dual permeability models assume that both fracture and matrix contribute to the flow [*Zhang and Sun*, 2000; *Xu and Hu*, 2005]. Triple continuum models conceptualize the fracture matrix system into three overlapping continua, consisting of both large and small-scale fractures along with the porous matrix [*Wu et al.*, 2004]. Continuum models are well suited when the scale of the problem is large, rocks have significant matrix permeability and fractures are well interconnected.

[5] In the discrete fracture models, fractures are explicitly modeled. These models are similar to mobile-immobile model [*West et al.*, 2004]. There are a number of analytical [*Grisak and Pickens*, 1981; *Tang et al.*, 1981] and numerical [*Grisak and Pickens*, 1980; *Noorishad and Mehran*, 1982; *Huyakorn et al.*, 1983] studies that use discrete fracture models for modeling reactive contaminant transport through fractured porous media. The discrete fracture models require detailed information of fracture networks and their characterization; they also require large computer storage and computational efforts [*Dershowitz et al.*, 2004]. The Discrete fracture matrix models are used for the study of radionuclide transport and safety assessment studies of high lever nuclear waste [*Norman and Kjellbert*, 1990].

[6] There have been a number of analytical solutions given for different source conditions, transport processes and fracture/matrix domains. *Bibby* [1981]derived an analytical expression for solute in the fracture for the case of diffusion for a two-dimensional solute mass transport in dual porosity medium.*Tang et al.* [1981]gave an analytical expression for advective-dispersive transport equation including equilibrium sorption and first-order degradation coefficient with constant concentration boundary condition and neglecting transverse dispersion.*Sudicky and Frind* [1982] and *Barker* [1982] gave the analytical solution for cases of radioactive contaminant transport in a system of discrete multiple parallel fractures. *Moreno and Ramuson* [1986] gave an analytical solution for a continuous injection of the constant mass flux in the fracture. There have been a number of analytical solutions that neglect the dispersion in the fracture [*Neretnieks*, 1980; *Grisak and Pickens*, 1981]. However, *Tang et al.* [1981] observed that neglecting longitudinal dispersion in the fracture might lead to significant error in case of low fluid velocity. *West et al.* [2004] gave an analytical solution for systems of parallel fractures, with a strip source of finite width; they also considered transverse dispersion in the fractures. *Roubinet et al.* [2012]gave a semianalytical solution considering two-dimension dispersion in fracture and two-dimension diffusion in the matrix.

[7] Analytical solutions are mostly limited to simpler cases and can be used as benchmarks for numerical models. In addition, analytical solutions often consist of multiple integrals of an oscillatory function, which is difficult to evaluate numerically [*Bodin et al.*, 2003]. To overcome such a difficulty, numerical inversion of the Laplace spaced solution is widely employed [*Neville et al.*, 2000; *Weatherill et al.*, 2008; *Gao et al.*, 2010]. However, in case of numerical models, *Weatherill et al.* [2008]found that, if the advective/dispersive transport in the fracture is higher than the diffusive transport in the matrix then, a very fine grid size of the order of the fracture aperture is required near the interface between fracture and matrix. Such a restriction may lead to higher memory requirement and computational efforts while modeling a large-scale fracture matrix system. However, no such limitation exists in the case of analytical solutions.

[8] There have been a number of studies that incorporate nonequilibrium sorption in a porous media [*Brusseau et al.*, 1989, 1992; *Selim et al.*, 1999; *Leje and Bradford*, 2009]. *Brusseau et al.* [1989] developed a multi process nonequilibrium (MPNE) model for porous medium where both transport and sorption related nonequilibrium process contributing to the observed nonequilibrium are considered. The model that incorporates both transport and sorption related nonequilibrium processes yields discrete kinetic terms, each representing individual nonequilibrium processes and can be used for process level investigation of solute transport. Whereas, in case of fractured porous media a very few studies exist that incorporate a nonequilibrium sorption in fracture/matrix. Most of the models available for solute transport through fractured porous media neglect the effects of sorption kinetics in fracture and matrix. *Xu and Worman* [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 time and can be up to several hundred percents. *Comans and Hockley* [1992] showed that equilibration time for sorption of Cesium (Cs) in aquatic sediments vary over weeks in laboratory tests with illite and montmorillonite.

[9] Based on experimental investigations, *Neretnieks et al.* [1982] found that both fast and slow reactions contribute to the total retardation in the fracture. The fast reaction is on the immediately accessible fracture surface, whereas, the slow reaction is time dependent as it increases with residence time. *Maloszewski and Zuber* [1990]developed a model for reactive transport in a fracture matrix system where the adsorption in the matrix is assumed to be governed by a first-order nonequilibrium kinetic sorption. However, the tracer adsorption on fissure walls was approximated by instantaneous equilibrium linear adsorption isotherms.*Berkowitz and Zhou* [1996]developed an approximate 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 small Damköhler number (ratio of reaction rate to molecular diffusion rate) the effect of surface reaction can be neglected, thus, reactive solute transport can be approximated by nonreactive solute transport.*Lee and Teng* [1993] gave an analytical solution assuming nonequilibrium sorption in the matrix.

[10] With the above background, the objectives of this study are (1) to propose a generalized model which considers nonequilibrium sorption in both fracture and matrix; (2) to develop a semianalytical solution for the new model; (3) to analyze the effects of various fracture and matrix sorptions and mass transfer parameters on the breakthrough curve (BTC); (4) to test the applicability of the model by simulating different experimental data available in the literature.