A significant challenge in contaminant transport modeling is to obtain a mechanistic understanding of transport parameter scaling that accurately addresses the combined influence of physical and chemical heterogeneities at different scales. In this paper, we have developed a scaling methodology to upscale matrix sorption coefficients for fractured-rock systems by characterizing both the tortuosity field (physical heterogeneity) and retardation factor field (chemical heterogeneity) in the rock matrix. We compute the effective tortuosity with a conservative tracer (e.g., tritium), and then using a sorbing tracer (e.g., uranium), we derive the equations for upscaling the sorption coefficients in a saturated, fractured rock system. The derived upscaling equations for the sorption coefficients are verified with Monte Carlo simulations, which are based on a generalized dual-porosity model to enable highly efficient and accurate numerical simulations of diffusive concentration fronts moving between the fractures and matrix material. The scientific results from this study will provide a theoretical and practical link between controlled experimental results at scales increasing from the laboratory bench to the field scale at which risk assessment and contaminant remediation are actually conducted.
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 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 , Reimus et al. , Cvetkovic et al. , Dai et al. , Liu et al.  and Frampton and Cvetkovic . A mass transfer coefficient was defined by Reimus et al.  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).
 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.
2. Transport Equations in Fractured Rock
 Fracture-matrix transport can be modeled with the one-dimensional advection-dispersion equation with one-dimensional diffusion occurring into the surrounding matrix perpendicular to the flow direction in the fracture [Reimus et al., 2003]. The controlling transport equations in fracture and matrix are: Fracture:
where Cf and Cm are the tracer concentrations in the fracture and matrix, respectively; vf is the fluid velocity in the fracture (in x direction); Df is the dispersion coefficient in the fracture; Dm is the matrix diffusion coefficient in z direction; Rf and Rm are the retardation factors in the fracture and matrix, respectively; η and ϕ are the porosity in the fracture and matrix, respectively; and b is the fracture half aperture.
Equation (2) states that the effective mass transfer coefficient (CMT) at the field scale can be computed based on the effective diffusion coefficient (m), effective retardation factor (m, for non-reactive species m = 1), effective matrix porosity (), effective fracture half aperture () and the fracture porosity. The fracture porosity in most cases can be assumed equal to 1, and the matrix diffusion coefficient can be expressed as m = D0, where is the effective matrix tortuosity, and D0 is the molecular diffusion coefficient in free water. Then, equation (2) is reformed as:
3. Effective Retardation Factor
 In heterogeneous matrix material, the tortuosity and retardation factor can be expressed as two one-dimensional (along the flow direction), second-order stationary spatial random variables, Y(x) = ln Rm and Z(x) = ln τ. The tortuosity random field represents the physical heterogeneity in the matrix, and the retardation factor random field describes the chemical heterogeneity of the matrix. If we take the measurement-scale (usually defined from column experiments) mass transfer coefficient as a spatial random variable, the field-scale mass transfer coefficient can be expressed as the volume average of measurement-scale mass transfer coefficients,
where L is the length of the one-dimensional domain and x is the spatial coordinate. By replacing the measurement-scale porosity (ϕ) and the half aperture (b) with their effective values and (which are assumed to be estimated separately), we have
 Decomposing Y(x) and Z(x) as the mean MY and MZ, and zero-mean perturbation Y′(x) and Z′(x), Y(x) = MY + Y′(x) and Z(x) = MZ + Z′(x), we rewrite (5) as a double integral in the one-dimensional domain,
where RmG = and τG = are the geometric means of measurement-scale retardation factor and tortuosity, and y is also a one-dimensional spatial variable. By using Taylor expansion and assuming the variance of Y(x) and Z(x) smaller than unity, we have,
where operator 〈 〉 represents expectation, σY2 and σZ2 are variances of log retardation factor and log tortuosity, CY (x, y) = 〈Y′(x)Y′(y)〉 and CZ (x, y) = 〈Z′(x)Z′(y)〉 are their covariances, respectively, and ωYZ (x, y) = 〈Y′(x)Z′(x) + Y′(x)Z′(y) + Y′(y)Z′(y) + Y′(y)Z′(x)〉 is the cross-covariance of retardation factor and tortuosity. We assume that the retardation factor and tortuosity are not correlated, then ωYZ (x, y) = 0. The covariance functions of the log retardation factor and log tortuosity can be assumed as:
where the lag distance h = |x - y|, λY and λZ are integral scales of log retardation factor and log tortuosity. Then, evaluating the integral in Equation (7), we have
Equation (9) represents a combined scaling effect of tortuosity and retardation factor. In order to obtain the effective tortuosity and the effective retardation factor individually, we assume the effective tortuosity is only related to the heterogeneity of the matrix, but not related to the type of solutes (e.g., conservative or reactive tracers). Then, we can use two steps to define the two effective parameters. First, we assume Rm is equal to 1 for a conservative tracer (such as tritium) to define the effective tortuosity, then
 The structure of Equation (10) is similar to that of the effective matrix diffusion equations by Dai et al.  because these two parameters are related by m = D0. Second, after defining the effective tortuosity, we can compute the effective Rm of a sorbing tracer (such as uranium) as:
Equation (11) indicates that the effective retardation factor increases with the variance of Y(x) in the matrix. If the matrix is homogeneous, the variance is 0, and the effective Rm is equal to the geometric mean, which indicates that the physical and chemical heterogeneities of matrix properties are the source of the scale dependence of retardation factor.
4. Effective Sorption Coefficient
 The effective sorption coefficient can be computed from the effective retardation factor from the equation:
where, d is the effective sorption coefficient, and ρ is the dry density of the matrix material. By replacing the effective retardation factor in (12) with (11), we obtain an expression for the effective sorption coefficient,
 A synthetic field-scale heterogeneous matrix system is built to investigate the scale dependence of Kd, and the corresponding statistical parameters are listed in Table 1. The diffusion coefficient and retardation factor represent uranium (a sorbing tracer) and are abstracted from Wolfsberg et al. . Using (13) and the data listed in Table 1, we plot the effective Kd vs. the integral scale in curve A of Figure 1, which shows that the effective Kd increases with the increasing integral scales. When the integral scale is 300 m, the corresponding effective Kd is 4.7 cm3/g, which is about 20% larger than its geometric mean (3.95 cm3/g). Curve B of Figure 1 shows that the effective Kd decreases with the ratio of domain size and integral scale. This result indicates that if the integral scale is a constant, when the domain size increases, the effective Kd decreases approaching the geometric mean of the sampled sorption coefficients in the field.
Table 1. Statistical Parameters for a Synthetic Heterogeneous Matrix Systema
Mean of Y(x) or Z(x)
Variance of Y(x) or Z(x)
Integral Scale (m)
D0 = 6.64 × 10−10 m2/s, ρ = 2.5 g/cm3, ϕ = 0.2 and L = 1000 m.
1.78 × 10−11
2.05 × 10−11
 Furthermore, if λY/L → 0, which means the field is not correlated or Y(x) is totally randomly distributed, (13) is approximated as:
On the other hand, if λY/L is sufficiently large, (13) is approximated as:
These two equations represent the minimum and the maximum of the effective Kd, which are illustrated in Figure 1.
5. Monte Carlo Simulations
 Monte Carlo simulations are conducted to assess the accuracy of the upscaling equations of the sorption coefficients in fractured rocks with the generalized double porosity model (GDPM) [Zyvoloski et al., 2008]. The GDPM numerical model has a length of 1000 m, a fracture spacing of 2 m, and a half aperture of 0.001 m. The model has 1001 fracture nodes (uniform resolution in direction of flow Δx = 1 m) and 10010 matrix nodes (each fracture node connects to 10 matrix nodes perpendicular to the flow direction with variable spatial spaces from 0.001 to 0.4 m). The inflow water has a constant injection rate of 0.012 kg/s. In the inflow water, the solute concentration of uranium is normalized to 1.
 There are two spatial random variables, tortuosity Z(x) and retardation factor Y(x) to represent the physical and chemical heterogeneities of the matrix. The heterogeneous fields of Z(x) and Y(x) were generated with a Gaussian random field generator [Lu and Zhang, 2004]. We generated 5000 realizations with the statistical data listed in Table 1. The quality of the generated fields was checked by comparing the covariance calculated from the generated realizations with the analytical, exponential covariance model. The comparison shows that the realizations match the specified mean, variance, and integral scale. Then, the generated Z(x) and Y(x) are converted to Dm and Kd for GDPM models.
 For each Monte Carlo simulation, we use the numerical simulator of GDPM [Zyvoloski et al., 2008] to solve Equations (1a–b) and to compute the concentration breakthrough at the outflow fracture node. The evolutions of concentration mean and variance with the number of simulations are also computed until the solution of the Monte Carlo simulations converges (Figure 2). Figure 3 shows that the concentration breakthrough simulated with the effective Kd calculated from Equation (13) matches well to the mean concentration after 5000 Monte Carlo simulations, while with the geometric mean of Kd the concentration is overestimated. This result indicates that the derived effective sorption coefficient is an accurate estimate for the field-scale modeling.
6. Summary and Conclusion
 The physical and chemical heterogeneities of matrix properties are a source of the scale dependence of sorption coefficients. The major factors affecting sorption coefficient heterogeneity include matrix porosity, tortuosity, mineral facies and rock units. In this paper, we take the mass transfer coefficient as a lumped spatial random variable to incorporate the variation of all these factors and upscale the sorption coefficient from the laboratory or measurement scale to the field scale.
 The effective sorption coefficient is dependent on the geometric mean, variance, integral scale, and domain size. Its value increases with the integral scale and is greater than the geometric mean. The Monte Carlo simulations with 5000 realizations of heterogeneous fields demonstrate that the derived effective sorption coefficient is an accurate estimation of Kd for field-scale transport modeling in saturated fractured rocks. The effective sorption coefficient is derived under the condition that the variance is smaller than unity. Further work is needed to identify the maximum variance that is applicable for the first-order perturbation method. The next extension of this effort will be to incorporate the more complex physical and chemical heterogeneity such as multimodal mineral facies to test how the multimodal heterogeneity affects the scaling of the sorption coefficients.
 The reported research was supported by Los Alamos National Laboratory's Directed Research and Development Project (20070441ER). We are grateful to Kay H. Birdsell for her constructive comments on this paper.