## 1. Introduction

[2] Spatial variations of the reactive transport parameters in porous media have been studied at variable scales ranging from column experiments to field tracer tests [e.g., *Ginn*, 1999; *Xu et al.*, 1999; *Yeh*, 2000; *Davis et al.*, 2004; *Dai and Samper*, 2006; *Dai et al.*, 2006; *Robinson et al.*, 2007]. Scale dependence of the transport parameters such as retardation factors in porous media has been addressed by *Bellin et al.* [1993], *Rajaram* [1997], and *Fernàndez-Garcia et al.* [2005]. Using a Lagrangian approach, *Rajaram* [1997] defined the effective retardation factors in temporal and spatial domains, and derived the theoretical expressions for these effective factors by assuming the spatial correlations between log conductivity and log retardation factor fields. His results indicate that retardation factors vary with scale, and at large temporal and spatial scales, the effective retardation factor approaches the arithmetic mean of the random retardation factor field. *Fernàndez-Garcia et al.* [2005] obtained similar conclusions of the scale dependence of the effective retardation factors in the three-dimensional physically and chemically heterogeneous porous media. More studies on the scaling of the retardation factors in porous media have been performed by *Cvetkovic and Dagan* [1994], *Chao et al.* [2000], *Andersson et al.* [2004] and *Samper and Yang* [2006].

[3] When dealing with sorption in fractured rock, we obtain a substantially different conceptual model than for porous media. In saturated fractured rock systems where the primary pathway for groundwater flow is through the fractures, the matrix material is saturated with groundwater that is considered immobile in the dual-porosity conceptual model [*Tang et al.*, 1981; *Sudicky and Frind*, 1982]. Although the bulk of the water travels through the fracture, the matrix can act as a reservoir to store contaminants temporarily via matrix diffusion and sorption processes. If a contaminant sorbs onto the matrix material for a period of time, its transport rate in the fractures is effectively retarded [*Robinson*, 1994]. Studies on the scaling of transport parameters in fractured rock have been conducted by *Berkowitz and Scher* [1998], *Reimus et al.* [2003], *Cvetkovic et al.* [2004], *Dai et al.* [2007], *Liu et al.* [2007] and *Frampton and Cvetkovic* [2007]. A mass transfer coefficient was defined by *Reimus et al.* [2003] to describe the rate at which a particular solute transfers between fractures and the rock matrix material when both diffusion and sorption are involved. The mass transfer coefficient depends on the matrix diffusion coefficient, retardation factor, fracture aperture, and matrix porosity. Therefore, the mass transfer coefficient is a lumped parameter that describes a critical component of reactive contaminant transport in fractured rock systems (advection and dispersion in the fractures being the other components).

[4] The scaling of the retardation factor of a chemical species is related to the variability of the sorption coefficients in the rock mineral facies [e.g., *Allen-King et al.*, 2006; *Zavarin et al.*, 2004]. Without an applicable upscaling rule, the parameters controlling sorption under field conditions usually are approximated by the values derived from column experiments. Because the parameters are spatially heterogeneous at various scales, characterization of the physical and chemical heterogeneities that control sorption processes is required. Then, the upscaling equations are derived to upscale the transport parameters from the measurement scale to field or modeling grid scales. In this study, we develop an upscaling methodology for modeling sorption in fractured rock at the field scale. Monte Carlo simulations are performed to demonstrate the accuracy of the derived upscaling algorithms for the effective sorption coefficient.