Upscaling matrix diffusion coefficients for heterogeneous fractured rocks



[1] The scale dependence of the matrix diffusion coefficient (Dm) for fractured media has been observed at variable scales from column experiments to field tracer tests. In this paper, we derive an effective Dm for multimodal heterogeneous fractured rocks using characteristic distributions of matrix properties and volume averaging of the mass transfer coefficient. The effective field-scale Dm is dependent on the statistics (geometric mean, variance, and integral scale) of laboratory-scale ln(Dm) and on the domain size. The effective Dm increases with the integral scales and is larger than the geometric mean of ln(Dm). Monte Carlo simulations with 1000 realizations of heterogeneous Dm fields were conducted to assess the accuracy of the derived effective Dm.

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 Dm 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 Dm, fracture aperture, and matrix porosity. This paper addresses scaling of heterogeneous Dm.

[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 Dm estimated from the column transport experiments may not be suitable for modeling field-scale solute transport in fractured rocks. Shapiro [2001] reported that effective Dm 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 Dm 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 Dm 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 Dm. However, their equation does not show dependence of the effective Dm on the spatial scales.

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

2. Spatial Statistics of Multimodal Dm

[6] Spatial covariance models developed from centimeter-scale measurements are important in upscaling effective parameters at larger scales. To characterize heterogeneous aquifer system, Lu and Zhang [2002] and Ritzi et al. [2004] presented a general form of multimodal correlation model of permeability. Here we apply it to modeling the covariance of ln(Dm). Heterogeneity of Dm comes from the variations of matrix physical and chemical properties within and across matrix units. Assuming a field-scale model made up of N matrix units in mutually exclusive occurrences (see Figure 1, N = 3), the distribution of matrix properties can be characterized by an indicator random variable Ik(x),

equation image

Then, the ln(Dm), denoted as Y(x), can be expressed as

equation image

where Yk(x) represents ln(Dm) within unit k. If the volume fraction of unit k is denoted as pk, the expected value of Ik is equal to pk (k = 1, 2, …, N). The composite mean MY and variance σY2 of Y(x) can be expressed as [see Ritzi et al., 2004]

equation image

where mk and σk2 denote the mean and variance of Yk(x), respectively.

Figure 1.

Heterogeneous matrix with three units created with TPROG [Carle and Fogg, 1997] by using the data listed in Table 1 (U1 = white, U2 = grey, and U3 = black). The fracture half aperture is 0.01 m and the fracture spacing is 2 m.

[7] Using indicator variables, we apply the transition probability for measuring spatial continuity of facies distributions [Carle and Fogg, 1997]. The transition probability in ϕ direction, tki(hϕ), is defined by

equation image

where hϕ is the lag distance in ϕ direction. Similar to the permeability covariance defined by Dai et al. [2005], the composite covariance CY(hϕ) of Y(x) can be represented in the term of proportion, transition probability, and the in-unit or cross-unit covariance of Yk(x) as

equation image

By assuming that the cross-covariances are negligible, i.e., Cki(hϕ) = 0 for ki [Dai et al., 2004], we can write the covariance of multimodal Y(x) in the following form:

equation image

As derived by Dai et al. [2004], we use exponential functions for transition probability and auto-covariance Ckk(hϕ),

equation image
equation image

where δki is the Kronecker delta, λI is the correlation length of the indicator variable in ϕ direction, and λk is the integral scale of Yk(x), which is a measure of spatial correlation of Yk(x), roughly the distance beyond which an attribute is considered to be uncorrelated. Substituting equations (7) and (8) into equation (6), we obtain the composite covariance function as

equation image

where λψ = λkλI/(λk + λI). This covariance function will be used to upscale the Dm from the laboratory scale to the field scale.

3. Effective Dm of Multimodal Matrix

[8] Tang et al. [1981] utilized analytical or semi-analytical solutions to model solute transport in fractured rocks, and derived an equation to represent the mass transfer coefficient as expressed in equation (10), which describes the rate at which a particular solute transfers between fractures and the rock matrix material [similarly used by Reimus and Callahan, 2007]. For heterogeneous matrix material, the effective mass transfer coefficient (CMT) at the field scale can be computed based on effective diffusion coefficient (equation image), effective matrix porosity (equation image) and effective fracture aperture (equation image) as:

equation image

[9] Taking the small-scale mass transfer coefficient as a spatial random variable, the effective field-scale mass transfer coefficient can be expressed as the volume averaging of small-scale mass transfer coefficients. We assume that Y(x) is a one-dimensional (along flow direction), second-order stationary spatial random variable. By substituting the small-scale porosity and the half aperture with their effective values equation image and equation image, we have

equation image

where L is the length of the one-dimensional domain and x is the spatial coordinate. By comparing equations (11) and (10), we obtain

equation image

which focuses this evaluation on Dm variability with the assumption that equation image and equation image have been estimated. Decomposing Y(x) as the mean MY and zero-mean perturbation Y′(x), Y(x) = MY + Y′(x), we rewrite equation (12) as a double integral in the one-dimensional domain,

equation image

where DmG = image is the geometric mean of laboratory-scale Dm and y is also a one-dimensional spatial variable. By using Taylor expansion and assuming the variance of Y(x) smaller than unity, equation (13) becomes,

equation image

If we take the expectation of equation (14) to quantify the effective Dm, then

equation image

where, CY(x,y) =〈Y′(x)Y′(y)〉 is the covariance that can be substituted using equation (9) with hϕ = ∣xy∣, so that ϕ is in the same direction as that of the one-dimensional variable. Then, we have the effective Dm as

equation image

[10] In equation (16), the effective Dm increases with the variance. If the matrix is homogeneous, the variance is 0 and the effective Dm is equal to the geometric mean, which indicates that the heterogeneity of matrix properties is the source of the scale dependence of Dm.

[11] To further investigate the scale dependence of Dm, we set up a synthetic field-scale heterogeneous matrix system with three units (Table 1 and Figure 1). Using equation (16) and the data listed in Table 1, we plot the effective Dm vs. the integral scale of unit 1 (U1) in Figure 2, which shows that the effective Dm increases with the increasing integral scales. Additional numerical experiments also show that the effective Dm is positively correlated to the integral scales of units 2 and 3, and the indicator correlation length.

Figure 2.

Effective Dm versus integral scale of U1.

Table 1. Mean and Variance of Yk(x), Proportions, and Integral Scales of the Units
UnitskpkmkDmkG, m2/sσk2λI, mλk, mλψ, m

4. Effective Dm of Bimodal and Unimodal Matrix

[12] In equation (16), if N = 2, Y(x) follows a bimodal distribution and the expression of the effective Dm becomes,

equation image

If N = 1, Y(x) follows a unimodal distribution, equation (16) can be simplified as,

equation image

Furthermore, if λ/L → 0, which means the field is not correlated or Y(x) are totally randomly distributed, equation (18) is approximated as:

equation image

which is a first-order approximation of Zhang et al. [2006, equation (10)]. On the other hand, if λ/L is sufficiently large, equation (18) is approximated as:

equation image

[13] Assuming that in equation (18) the mean of unimodal Y(x) is −22.6 ln(m2/s), the variance is 0.88, and the domain size is 1000 m, we plot effective Dm as a function of the integral scale in Figure 3a. The effective Dm increases with the integral scale. For comparison, the effective Dm computed with equations (19) and (20), which correspond with the cases that λ → 0 and λ is sufficiently large, are also illustrated in Figure 3a (assume L is constant). When λ → 0, the effective Dm is 1.86·10−10 m2/s and is greater than the geometric mean (1.53·10−10 m2/s). When λ = 300 m, it is 2.01·10−10 m2/s, and when λ is sufficiently large, it is 2.21·10−10 m2/s. Figure 3b shows that the effective Dm decreases when the ratio L/λ increases.

Figure 3.

Effective Dm versus (a) the unimodal integral scale and (b) the ratio of domain size and integral scale.

5. Monte Carlo Simulations

[14] To assess the accuracy of the effective Dm, we conducted Monte Carlo simulations for conservative tracer transport in unimodal fractured rocks with the generalized double porosity model (GDPM [Zyvoloski et al., 2003]). The GDPM numerical model has a length of 1000 m, a fracture spacing of 2 m, and a half aperture of 0.01 m. The model has 1001 fracture nodes (constant spatial space 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 0.01 to 0.4 m). At the first fracture node (point A in Figure 1), the water injection rate is constant at 0.0116 kg/s. In the injection water, the solute concentration is normalized to 1. For the purpose of this demonstration, the only spatial random variable in the simulations is Dm.

[15] The heterogeneous fields of unimodal Y(x) were generated with a Gaussian random field generator [Zhang and Lu, 2004]. We generated 1000 realizations with a mean Y(x) of −22.6 ln(m2/s), variance of 0.88, and integral scale of 300 m. The quality of the generated Y(x) 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 Y(x) are converted to Dm for GDPM models.

[16] During the Monte Carlo simulations, we compute the mean, variance, and the 95% confidence interval of the concentration breakthrough at the last fracture node (point B in Figure 1) after each simulation, and check the evolution of concentration variance and mean with the number of simulations until the solution of Monte Carlo simulations converges. Figure 4 shows that the concentration breakthrough simulated with the effective Dm computed by equation (18) matches well to the mean concentration after 1000 Monte Carlo simulations, while with the geometric mean of Dm the concentration is overestimated. This result indicates that the derived effective Dm is an accurate estimate of Dm for the field-scale modeling.

Figure 4.

Computed concentration breakthroughs from the effective Dm, geometric mean and Monte Carlo simulations, as well as the concentration bounds of the 95% confidence intervals.

6. Discussion and Conclusion

[17] The heterogeneity of matrix properties is the source of the scale dependence of Dm, which comes from the variations of matrix physical and chemical properties within and across matrix units. The covariance of Y(x) can be used to characterize the matrix heterogeneity with transition probability of the multimodal matrix units and the covariance of Y(x) within each unit. The major factors affecting Dm heterogeneity include matrix porosity, tortuosity, solute charge, and temperature. In this paper we take Dm as a lumped spatial random variable to incorporate the variation of all these factors and upscale Dm from the laboratory scale to the field scale.

[18] The effective Dm 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. Monte Carlo simulations with 1000 realizations of heterogeneous matrix diffusion fields demonstrate that the derived effective Dm is an accurate estimation of Dm for the field-scale transport modeling in the fractured rocks. The effective Dm is derived under the condition that the variance is smaller than unity. However, the first-order perturbation might give accurate estimates of effective Dm for variance as large as 4, as discussed by Dai et al. [2004] for deriving macrodispersion equations. 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 influence of other processes affecting mass transfer such as spatial variations in aperture and matrix porosity.


[19] The reported research was supported by Los Alamos National Laboratory's Directed Research and Development Project (20070441ER). We are grateful to Bruce Robinson and Kay Birdsell for their constructive comments on the manuscript of this paper.