# Inclusion-Based Effective Medium Models for the Permeability of a 3D Fractured Rock Mass

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## Abstract

Effective permeability is an essential parameter for describing fluid flow through fractured rock masses. This study investigates the ability of classical inclusion-based effective medium models (following the work of Sævik et al. in Transp Porous Media 100(1):115–142, 2013. doi: 10.1007/s11242-013-0208-0) to predict this permeability, which depends on several geometric properties of the fractures/networks. This is achieved by comparison of various effective medium models, such as the symmetric and asymmetric self-consistent schemes, the differential scheme, and Maxwell’s method, with the results of explicit numerical simulations of mono- and poly-disperse isotropic fracture networks embedded in a permeable rock matrix. Comparisons are also made with the Hashin–Shtrikman bounds, Snow’s model, and Mourzenko’s heuristic model (Mourzenko et al. in Phys Rev E 84:036–307, 2011. doi: 10.1103/PhysRevE.84.036307). This problem is characterised by two small parameters, the aspect ratio of the spheroidal fractures, \(\alpha \), and the ratio between matrix and fracture permeability, \(\kappa \). Two different regimes can be identified, corresponding to \(\alpha /\kappa <1\) and \(\alpha /\kappa >1\). The lower the value of \(\alpha /\kappa \), the more significant is flow through the matrix. Due to differing flow patterns, the dependence of effective permeability on fracture density differs in the two regimes. When \(\alpha /\kappa \gg 1\), a distinct percolation threshold is observed, whereas for \(\alpha /\kappa \ll 1\), the matrix is sufficiently transmissive that such a transition is not observed. The self-consistent effective medium methods show good accuracy for both mono- and polydisperse isotropic fracture networks. Mourzenko’s equation is very accurate, particularly for monodisperse networks. Finally, it is shown that Snow’s model essentially coincides with the Hashin–Shtrikman upper bound.

## Keywords

Fracture networks Permeability Effective medium models## 1 Introduction

Most rocks are fractured to one extent or another. Fractures that are more permeable than their host rock can act as preferential (or at least additional) pathways for fluid to flow through the rock, which is relevant in several areas of earth science and engineering, e.g. radioactive waste disposal in crystalline rock, exploitation of fractured hydrocarbon and geothermal reservoirs, or hydraulic fracturing (Bonnet et al. 2001; Neuman 2005; Salimzadeh and Khalili 2015; Tsang et al. 2015). In describing or predicting flow through fractured rock, the effective permeability of the rock, comprising rock matrix and a network of fractures, is a crucial parameter and may depend on several geometric properties of the fractures/networks, such as size, aperture, orientation, and fracture density.

It is possible to compute this effective permeability by numerically modelling flow through discrete fracture networks (DFN) and upscaling the results (e.g. Ahmed Elfeel and Geiger 2012; Lang et al. 2014). However, this numerical upscaling is computationally expensive. Considering the fact that the geometric information that is typically available on fracture networks is stochastic in nature, several realisations are required to obtain representative values for effective permeability. In addition, due to the degree of uncertainty inherent to fracture-network properties, one is often interested in quantifying the effect of this uncertainty on effective permeability by probing a multi-dimensional parameter space, e.g. when assessing the suitability of sites for radioactive waste disposal. Thus, there is a necessity for predictive analytical methods.

A prominent example of a predictive model in three dimensions is the heuristic equation proposed by Mourzenko et al. (2011). It was developed based on a vast collection of numerical simulations and theoretical arguments. Sævik et al. (2013) confirm the accuracy of the method, though it is, per definition, not applicable at low fracture densities when the background matrix has a non-negligible permeability.

The present work studies inclusion-based effective medium models which treat fractures as spheroidal inclusions embedded in a homogeneous and permeable rock matrix. Research on the use of such models to predict effective fractured-rock permeability, particularly at high fracture densities with intersecting fractures, has advanced rapidly recently (Fokker 2001; Pozdniakov and Tsang 2004; Barthélémy 2009; Sævik et al. 2013, 2014). Following the work of Sævik et al. (2013, 2014), the suitability and accuracy of some inclusion-based effective medium models are investigated here. In addition, the framework of the effective medium theory (EMT) is exploited to identify two characteristic regimes, depending on whether the overall flow through the rock mass is dominated by the fracture network or by the matrix. Due to distinctly differing flow patterns, the dependence of effective permeability on fracture density differs in the two regimes.

The effective permeability expressions obtained from the various effective medium theories (symmetric and asymmetric self-consistent, differential, Maxwell) are applied here to polydisperse fracture networks. Comparisons are also made with the Hashin–Shtrikman bounds, Snow’s equation, and Mourzenko’s equation.

## 2 Characterisation of Fractures and Fracture Networks

*R*, and the length of the short semi-axis is one half of the maximum aperture, i.e.

*h*/ 2. Then, the aspect ratio, \(\alpha \), of the fracture is

*i*is the number of fracture centres, \(N_{\mathrm {f}}\), per volume, i.e.

*L*is the length of a finite-sized cubic domain for which the permeability is to be determined. Note that, in this study, \(L>2R\). Let \(\phi _i\) be the total volume occupied by a fracture set per unit volume of rock mass. Assuming all fractures have the same aspect ratio,

*n*is the total number of fracture sets. The following definition for fracture density is used:

## 3 Upper Bounds

## 4 Dilute-Limit Solution

*K*is the effective permeability of the rock mass, \(r = \kappa ^{-1}\), and

## 5 Effective Medium Models

The various effective medium models, as derived and presented by Sævik et al. (2013), are reviewed here. These include the asymmetric and symmetric self-consistent methods, the differential method, and Maxwell’s approximation. All of the methods are presented below for a finite number of fracture sets, *n*, and their various contributions are summed. However, for a continuous distribution of fracture radii, these contributions need to be integrated. In such cases, the integration can be performed numerically.

### 5.1 Maxwell Approximation

*K*, representing the effective medium embedded in the same background rock matrix as the inclusions (see, e.g. Sevostianov 2014; Lutz and Zimmerman 2016). i.e.,

### 5.2 Asymmetric Self-Consistent Method

*K*. By also introducing fracture density, \(\varepsilon _i\), in the equation, one obtains

### 5.3 Symmetric Self-Consistent Method

### 5.4 Differential Method

*j*fracture groups. Although the predictions of the differential method depend slightly on the integration path, for cases in which there are multiple inclusion sets, a natural integration path can be chosen such that the infinitesimal amount of each fracture set added, at each step, is proportional to the final proportion of the set’s volume fraction (Sævik et al. 2013), i.e.

## 6 Heuristic Model of Mourzenko et al. (2011)

## 7 Numerical Model and Simulations

The predictions of the various approximate models are now tested against explicit numerical simulations. These use the method presented by Lang et al. (2014) to calculate fractured-rock permeability as implemented in the modelling framework CSMP++. The approach works on unstructured finite-element meshes where fractures are represented explicitly as lower-dimensional objects, i.e. surfaces embedded in a volumetric domain which represents the host rock. The full permeability tensor of the fracture–matrix ensemble is obtained by means of volume averaging of the local pressure gradients and fluxes for three independent steady-state flow simulations, and the subsequent solution of an overdetermined system of equations. Calculating the permeability tensor over a restricted sub-volume (here, 90 % of the total volume) of the flow model allows the use of arbitrary fracture-network geometries that need not be restricted by periodicity or other constraints (for details, see Lang et al. 2014).

*D*varies with fracture density and is determined at various densities by plotting the computed permeability as a function of \(\delta /R\) and extrapolating down to \(\delta /R = 0\) (see Fig. 3).

To explore the characteristics of the effective medium models, six different cases are defined here, each case posing a particular challenge to the prediction method. All of the fracture sets are isotropic and contained in a cubic domain with side lengths of 100 m. See Table 1 for a summary. Cases 1 and 2 are monodisperse networks. From the dilute-limit solution, it is clear that there is a substantial difference in the dependence of effective permeability on fracture density (or volume fraction) when the aspect ratio is much less or much greater than the permeability ratio. These two regimes, \(\alpha \ll \kappa \) and \(\alpha \gg \kappa \), are studied in Cases 1 and 2, respectively.

## 8 Comparison to Numerical Data

A comparison between the various effective medium models, the upper bound(s), Mourzenko’s model, and numerical data is achieved here by plotting \(K/K_{\mathrm {m}}\) as a function of \(\varepsilon \). For the numerical data, the range between the 5th and 95th percentiles of the 20–40 realisations are shown as a measure of spread in addition to the median value (see Fig. 4).

### 8.1 Case 1

Simulation parameters for Cases 1–6

Parameter | Value | Comment |
---|---|---|

| ||

\(K_{\mathrm {m}}\) | \(10^{-14}\,\mathrm{m}^2\) | |

| 100 m | |

| ||

\(\alpha \) | \(3.75 \times 10^{-6}\) | \(\frac{\alpha }{\kappa }=0.078\) |

\(\kappa \) | \(4.8\times 10^{-5} \) | |

| 10 m | |

\(a_{\mathrm {f}}\) | \(50\,\upmu \mathrm{m}\) | |

\(K_{\mathrm {f}}\) | \(2.084 \times 10^4 \times K_{\mathrm {m}}\) | |

\(\delta /R\) | 0.3 | |

| ||

\(\alpha \) | \(3.75 \times 10^{-5}\) | \(\frac{\alpha }{\kappa }=78\) |

\(\kappa \) | \(4.8\times 10^{-7} \) | |

| 10 m | |

\(a_{\mathrm {f}}\) | \(500\,\upmu \mathrm{m}\) | |

\(K_{\mathrm {f}}\) | \(2.084\times 10^6 \times K_{\mathrm {m}}\) | |

\(\delta /R\) | 0.3 | |

| ||

\(\alpha \) | \(3.75 \times 10^{-5}\) | |

\(\delta /R\) | 0.6 | |

| ||

\(\kappa _1\) | \(1.2\times 10^{-7} \) | \(\frac{\alpha }{\kappa }=313\) |

\(R_1\) | 20 m | |

\(a_{\mathrm {f},1}\) | \(1000\,\upmu \mathrm{m}\) | |

\(K_{\mathrm {f},1}\) | \(8.33\times 10^6 \times K_{\mathrm {m}}\) | |

| ||

\(\kappa _2\) | \(3\times 10^{-6} \) | \(\frac{\alpha }{\kappa }=12.5\) |

\(R_2\) | 4 m | |

\(a_{\mathrm {f},2}\) | \(200\,\upmu \mathrm{m}\) | |

\(K_{\mathrm {f},2}\) | \(3.33\times 10^5 \times K_{\mathrm {m}}\) | |

\(\rho _{2}=\rho _{1} \left( \frac{R_{1}}{R_{2}} \right) ^{3.5}\) | Eq. (31) | |

| ||

\(\delta /R\) | 0.6 | |

\(a_{\mathrm {f}} = \frac{4}{3}R \alpha \) | Eq. (2) | |

\(K_{\mathrm {f}}= a_{\mathrm {f}}^2/12\) | Eq. (6) | |

Maximum and minimum radii are the same as for the fracture sets 1 and 2, respectively, of Case 3. | ||

\(n_{\mathrm{p}},\alpha \left\{ \begin{array}{ll} \qquad 3.5, 3.75 \times 10^{-5} &{} \qquad \qquad \hbox {Case 4} \\ \qquad 1.5, 3.75 \times 10^{-5} &{} \qquad \qquad \hbox {Case 5}\\ \qquad 1.5, 3.75 \times 10^{-6} &{} \qquad \qquad \hbox {Case 6} \qquad \end{array}\right. \) |

*K*and \(\varepsilon \), irrespective of the value of \(\varepsilon \):

### 8.2 Case 2

Here, the regime \(\alpha \gg \kappa \) is studied. As in the opposite regime (Case 1), Fig. 4b shows that the H–S upper bound and the Snow model are practically equal. The Maxwell approach tends to infinity when the denominator in Eq. (19) tends to zero, i.e. when \(\varepsilon \rightarrow \frac{27}{32} \left( \frac{4 \kappa }{\pi \alpha }+1\right) \). Hence, in this particular case, Maxwell’s equation gives meaningless predictions when \(\varepsilon \ge 0.86\). In theory, the percolation threshold for this set of randomly oriented disc-shaped fractures is at \(\varepsilon =0.244\) (Mourzenko et al. 2005). As mentioned in Sævik et al. (2013), the asymmetric self-consistent method is able to predict the percolation threshold. However, the symmetric self-consistent approach is better at capturing the general behaviour of the curve at low fracture densities. At higher densities (\(\varepsilon >0.5\)), the same type of deviation from the numerical data as in Case 1 can be observed. In this range of higher densities, the gradient of the asymmetric self-consistent curve is more accurate, though it overestimates the effective permeability. The predictions of the differential method are much lower than the numerical data. Sævik et al. (2013) state that the differential method only gives useful results when \(\alpha \le \kappa \).

In this example, Mourzenko’s equation provides an excellent match to the numerical data, accounting well for both medium- and high-fracture-density ranges. Mourzenko et al. (2011) point out that, according to percolation theory, *K* is expected to be a quadratic function of \(\varepsilon \) at densities above but close to the percolation threshold. This statement is valid when \(K_{\mathrm {m}} \approx 0\), but also appears to hold here. At higher densities, *K* then becomes a linear function of \(\varepsilon \).

### 8.3 Case 3

There are two fracture sets with distinctly different fracture radii (4 and 20 m) and, hence, differing fracture permeabilities. This produces a significant skew in the effective-permeability distribution, as can be seen in the asymmetric spread of the values of the numerical data in Fig. 4c. While the fracture set with many small fractures leads to a relatively low median value, the large-radius fracture set is responsible for the upper outliers. In general terms, the accuracy of predictions of the effective medium models does not seem to change because of this. They tend to over- or underestimate effective permeability in much the same way they do in Case 2. However, the skewed nature of the fracture network appears to pose a problem for Mourzenko’s heuristic model, which overestimates effective permeability in a manner very similar to the asymmetric self-consistent method.

### 8.4 Cases 4–6

The comparison between the effective medium models and numerical data using a power-law distribution (Fig. 4d–f) reinforces the findings of Cases 1–3. The symmetric self-consistent model performs best at low fracture densities and eventually diverges from the data. The asymmetric self-consistent model generally overestimates effective permeability, but converges towards the correct gradient of *K* to \(\varepsilon \) at large fracture densities (this is more obvious in Fig. 4e than in Fig. 4d), where full convergence has not yet occurred. The accuracy of both self-consistent methods is unaffected by the properties of the fracture-size distribution.

Mourzenko’s heuristic model performs very well in these cases. Its accuracy appears to increase with decreasing values of the power-law exponent, \(n_{\mathrm{p}}\). This is because part of the fitting of Eq. (28) relies on the dimensionless fracture transmissivity \(\sigma '\) of the largest fracture. As \(n_{\mathrm{p}}\) decreases, the influence and relevance of \(\sigma '\), and hence the accuracy of the fit, increases.

## 9 Discussion

*no preferential flow through the fractures*when \(\alpha /\kappa \rightarrow 0\) to

*no flow through the matrix*when \(\alpha /\kappa \rightarrow \infty \).

In the following, one of the two self-consistent methods [Eq. (22) or (23)] is used to explore particular aspects of the permeability of a fractured rock mass.

### 9.1 Percolation Threshold

In the numerical data generated by Bogdanov et al. (2007) of effective permeability for fracture networks and varying values of \(\alpha /\kappa \), the percolation threshold as a distinct, discontinuous feature tends to disappear when \(\alpha /\kappa \le 7.5\). This is in agreement with the findings discussed here, although Bogdanov et al. (2007) do not consider the case \(\alpha /\kappa \ll 1\).

### 9.2 Upper Bounds

Figure 7 is a contour plot of the relative difference between the H–S upper bounds and the effective permeability as calculated using the asymmetric self-consistent method for various values of \(\varepsilon \) and \(\alpha /\kappa \). It can be seen as a measure of the fraction of fractures that contribute to flow. When \(\alpha \gg \kappa \), this difference is maximal below the percolation threshold and reduces to a constant value with increasing \(\varepsilon \). Essentially, the contribution of non-percolating clusters to overall flow is very low. At very high densities, almost all clusters are connected, and only peripheral parts of the fractures do not contribute to flow (Mourzenko et al. 2011; Leung and Zimmerman 2012).

For \(\alpha \ll \kappa \), effective permeability is consistently close to the upper bounds. Flow through the matrix is important, always connecting fracture clusters and maximising the number of fractures that contribute to flow. This suggests that, in this regime, theoretical upper bounds are reasonably accurate estimates for effective permeability.

### 9.3 Polydisperse Networks

#### 9.3.1 Smallest Fracture Radius

Polydisperse fracture networks are typically dominated by large fractures with high permeabilities. However, they also usually contain many small fractures with low permeabilities. A threshold fracture radius, \(R_{\mathrm{th}}\), may exist, for which, the contribution of smaller fractures to flow is negligible (see, e.g. Berkowitz et al. 2000; Dreuzy et al. 2001). When characterising fracture networks, it is important to know which fracture sets may be neglected within a given range of accuracy. The additive nature of effective medium methods provides a convenient framework for addressing this issue.

*x*-axis) is calculated with Eq. (5) for

*all*fracture radii.

#### 9.3.2 Characteristic \(\langle \alpha /\kappa \rangle \)

*K*.

## 10 Previous Work Dealing with the Parameter \(\alpha /\kappa \)

Nonetheless, when Bogdanov et al. (2003) gradually increase \(\sigma '\) from 10 to \(10^4\), they also find an increasingly more distinct percolation threshold (as discussed in Sect. 9.1). Their investigation is limited to fracture densities of \(\varepsilon <1\). Bogdanov et al. (2003) seem to assume that the effective permeability is only linear with fracture density at very low fracture densities below the percolation threshold or when \(\kappa >1\). As pointed out in Sect. 8.1, if \(\alpha \ll \kappa \), this linearity also holds at high fracture densities even when \(\kappa \ll 1\).

Sævik et al. (2014) consider the parameter \(\lambda ^{-1} = \alpha /\kappa \), varying it between 10 and \(10^4\), with \(\varepsilon < 5\).

## 11 Conclusions

In predicting the effective permeability of a three-dimensional fractured rock mass, two distinct regimes can be distinguished, depending on the relative size of two small, dimensionless numbers: the aspect ratio of the spheroidal fractures, \(\alpha \), and the permeability ratio, \(\kappa \). When \(\alpha /\kappa \ll 1\), effective permeability is linearly dependent on fracture density without a distinct percolation threshold. With increasing \(\alpha /\kappa \), this relationship becomes increasingly nonlinear at low and intermediate fracture densities, but remains linear at high fracture densities. The ratio \(\alpha /\kappa \) can therefore be interpreted as a measure of the relative contributions of the fracture network and matrix to the overall flow, similar to the flux ratio discussed by Paluszny and Matthai (2010), and Nick et al. (2011). For polydisperse fracture networks, a characteristic value of \(\langle \alpha /\kappa \rangle \) can be determined by weighting the individual values of \(\alpha /\kappa \) of the various fracture sets with fracture volume while averaging.

Comparison to explicit numerical simulations of mono- and polydisperse isotropic fracture networks (with fracture-size dependent fracture permeabilities) shows that the self-consistent effective medium methods are generally capable of predicting effective permeability over a wide range of conditions. While the symmetric self-consistent method is particularly accurate at low fracture densities, the asymmetric self-consistent method predicts the correct asymptotic behaviour (cf. Sævik et al. 2013). The differential method is only useful when \(\alpha /\kappa <1\). In that regime, even the Hashin–Shtrikman and Snow upper bounds appear to give good approximations. Maxwell’s approximation is only reliable at very low fracture densities.

These effective medium models have been shown to be powerful tools. They are valid at both low and high fracture densities. When considering polydisperse fracture networks, the type and characteristics of the fracture-size (as well as aperture and fracture-permeability) distribution do not affect the applicability or accuracy of these models. These results are perhaps surprising, since intuitively, fracture intersections would seem to be a crucial factor in controlling the effective permeability, and yet these models do not explicitly contain fracture intersections as a parameter. Moreover, they are all based on the problem of a single “inclusion”, for which the concept of intersection is not even meaningful.

It should be noted that the heuristic model proposed by Mourzenko et al. (2011) is good at predicting effective permeability. It is very accurate for monodisperse fracture networks, but suffers a slight loss of accuracy for polydisperse networks, that appears to depend on the attributes of fracture-size distribution. The present work has in effect shown a link between predictions of methods based on inclusions-in-a-matrix and methods based on fracture networks.

Finally, for the case of zero matrix permeability, the well-known approximation of Snow, which is based on network considerations rather than a continuum approach, is shown to essentially coincide with the upper Hashin–Shtrikman bound, thereby proving that the commonly made assumption that Snow’s equation is an “upper bound” is indeed correct.

## Notes

### Acknowledgments

We acknowledge the Natural Environment Research Council, Radioactive Waste Management Limited, and Environment Agency for the funding received for this project through the Radioactivity and the Environment (RATE) programme. This work is also partially funded by the European Commission TRUST collaborative project, 309067.

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