## 1. Introduction

[2] In saturated fractured-rock systems, where the primary pathway for groundwater flow and solute transport is through fractures, groundwater in the matrix is considered immobile in dual-porosity conceptual models [*Tang et al.*, 1981; *Sudicky and Frind*, 1982]. Thus, although the bulk of the water travels through the fractures, a very large reservoir of water in the matrix can act to store and reduce mobility of contaminants via matrix diffusion [*Robinson*, 1994]. *Carrera et al.* [1998] presented a comprehensive study of matrix diffusion and concluded that *D*_{m} is one of the parameters that govern contaminant transport in fractured rocks. Recent field-scale tracer test interpretations by *Reimus and Callahan* [2007] highlighted the significance of fracture apertures in governing mass transfer between fractures and matrix, particularly when the field-scale fractures in which solutes flow may have larger apertures than those used in laboratory columns. Ultimately, mass transfer between fractures and matrix depends on *D*_{m}, fracture aperture, and matrix porosity. This paper addresses scaling of heterogeneous *D*_{m}.

[3] Over the years, the ability to fully characterize the parameters in the fracture-matrix mass transfer process has not kept pace with numerical and modeling expertise [*Liu et al.*, 2007]. Transport experiments are usually conducted at the sub-meter or column scale under conditions in which flow rates, tracer injections and other conditions are well controlled. Assuming relatively little heterogeneity in such experiments, analytical or semi-analytical models have been used to estimate fracture transport parameters [*Cormenzana*, 2000]. However, there remains no practical unifying theory to integrate laboratory-scale parameters in field-scale predictions for risk assessment or remedial design.

[4] Recent studies indicate that *D*_{m} estimated from the column transport experiments may not be suitable for modeling field-scale solute transport in fractured rocks. *Shapiro* [2001] reported that effective *D*_{m} in kilometer-scale systems is much greater than estimates from laboratory experiments due to complex, possibly advective, field-scale transport processes. *Neretnieks* [2002] and *Andersson et al.* [2004] estimated the effective *D*_{m} from field tracer test data at the Äspö site and obtained some values about 30 times greater than their laboratory-scale estimates, which they attributed to increased diffusion surface area in their field test. *Liu et al.* [2004] reported that the effective *D*_{m} at the field scale is generally greater than that at laboratory scales and tends to increase with the testing scale. While several potential mechanisms have been identified, they found that this interesting scale dependence may be related to rock matrix heterogeneity in fractured rock. Based on numerical experiments, *Zhang et al.* [2006] empirically determined a formula for estimating the effective *D*_{m}. However, their equation does not show dependence of the effective *D*_{m} on the spatial scales.

[5] The work we present here focuses on the spatial-scale dependence of the effective *D*_{m} in multimodal heterogeneous rocks. We start from characterization of heterogeneous matrix properties to build the covariance function of ln(*D*_{m}). Then, we derive equations to describe the relationship between the effective *D*_{m}, the statistics of *D*_{m} measurements at laboratory scales, and the domain size. Monte Carlo simulations are performed to assess the accuracy of the derived effective *D*_{m} in a synthetic example.