REV and Three-Dimensional Permeability Tensor of Fractured Rock Masses with Heterogeneous Aperture Distributions

Abstract

This study performed a representative elementary volume (REV) and 3D equivalent continuum study of rock fractures based on fluid simulations of 3D discrete fracture networks (DFNs). A series of 3D DFNs with heterogeneous aperture distributions (the DFN-H model) and uniform apertures (the DFN-I model) were established, in which the fractures were oriented according to the geological field mapping of a high-level radioactive waste candidate site in China. The 3D DFNs of the different model sizes were extracted and rotated in a number of directions to check whether there was a tensor quality of the permeability at a certain scale. The results show that aperture heterogeneity increases the REV size and results in a necessarily larger model size to reach an equivalent continuum behavior, and this effect is more obvious when the fracture density is smaller. The shape of the 2D permeability contour is irregular, with some breaks when the model size is small. As the model size increases, its shape gradually tends to become smooth and approaches an ellipse. The shape of the permeability contours of the DFN-H model is slender compared to the DFN-I model, indicating a larger difference between the minimum and maximum values of the permeability. For the DFN-H model, there is no appropriate approximation for the equivalent permeability tensor over the studied model size range, whereas a good fit of the permeability ellipsoid is obtained for the DFN-I model, and the 3D directional permeability is calculated at this model scale. The corresponding magnitude and direction of the principal permeability are obtained, which can be viewed as the equivalent permeability tensor for the approximated continuum medium.

1. Introduction

One of the most difficult tasks in rock engineering is to accurately characterize the hydraulic properties of fractured rock masses for the purpose of engineering analysis. Due to various geological processes, a fractured rock mass comprises an intact rock matrix and discontinuities (e.g., joints, fractures, bedding planes, and faults). These discontinuities are often different in trace length, orientation, and aperture, which leads to inhomogeneity and anisotropy of the rock mass [1,2,3,4,5,6,7]. The permeability of the rock fracture is obviously larger than that of the matrix, and thus flow occurs mainly inside the connected fractures [8,9,10,11,12]. Knowledge of the detailed flow structure within fracture networks is significant with respect to characterizing the overall hydraulic response of fractured media.

The three-dimensional discrete fracture network (3D DFN) modelling approach has been applied to study flow behaviors through fractured rock masses [13,14,15]. In DFN models, fractures are represented by planar polygons or discs with different locations, orientations, sizes, and apertures. As a result, the model can represent the geometrical properties of each fracture individually, and explicitly characterize the contributions of each fracture to the flow. However, this approach requires huge computational costs when handling large-scale problems. Alternatively, the equivalent continuum method becomes attractive for a flow calculation averaged over a large domain [16,17,18,19]. In this method, the equivalent permeability is arranged to the volumetric grids to reflect the effect of fractures on the flow. Such an equivalent simplification of complex fracture systems enables an efficient calculation.

A premise for applying the equivalent continuum method is that there exists a representative elementary volume (REV) for evaluation of equivalent permeability tensors of the fracture system [20,21,22,23,24]. The REV is defined as a minimum domain size beyond which the rock mass properties remain constant. It is a fundamental concept for quantifying the scale dependency of hydraulic properties, as shown in Figure 1. However, there is no guarantee that REV size always exists for every fractured rock mass [25,26,27]. Therefore, research on the existence of the REV for a particular fracture medium is a crucial step to justify whether it is appropriate to apply the continuum method. If the REV exists, the equivalent continuum permeability tensor can be calculated from the fractured rock mass at a scale equal to or larger than the REV size.

Previous studies have indicated that the REV and the equivalent permeability tensor of the fractured rock masses are related to geometrical properties of fractures [29,30]. Wang et al. [19] applied a 3D conceptual linear pipe fracture network model to calculate the permeability tensor of a tunnel site. They found that the permeability tensor is significantly anisotropic, in which the principal directions are consistent with the existing fracture system in the site. Rong et al. [31] indicated that the permeability anisotropy is primarily influenced by the fracture orientation, and the REV size is primarily controlled by the fracture spacing, trace length, and the number of fracture sets. Feng et al. [32] concluded that the REV size decreases exponentially as the fracture length exponent increases. The above studies assumed that flow occurs in bonds connecting the center of a fracture to the center of the intersection of two adjacent fractures, and thus that individual fractures in the model have uniformly distributed apertures. Nevertheless, rock fractures naturally have heterogeneous apertures formed by two rough surfaces [33,34,35,36,37,38,39,40,41]. The influence of aperture heterogeneity on flow behavior through 3D fracture networks has been demonstrated at both laboratory scale [42,43,44,45] and field scale [46,47,48]. Ishibashi et al. [49] stated that modeling the fracture system with a uniform aperture field potentially leads to incorrect scenarios for developing the fractured reservoirs. Therefore, the effect of aperture heterogeneity on the REV and equivalent permeability tensor of 3D fracture networks should be further investigated.

To cope with this issue, a number of 3D DFNs with heterogeneous aperture distribution and uniform aperture distribution are established, in which the fractures are oriented according to the geological field mapping of a high-level radioactive waste candidate site in China. The numerical simulation results are utilized to examine the existence of the REV and the equivalent permeability tensor. The influences of fracture density, aperture distribution, and model size on the magnitude and direction of permeability tensor are systematically analyzed.

2. Model Generation

The DFNs are generated using the geometrical parameters of a high-level radioactive waste candidate site located in Beishan, of the Gansu province, China. The geological structure of this area is mainly composed of biotite monzonitic granite. Under various geological conditions, the granite rocks with fractures are well exposed on outcrops in a region of approximately 12 km². Based on digital techniques, Guo et al. [50] identified two homogeneities—i.e., Homogeneity I and II—in this area. Homogeneity I contains two dominant discontinuity sets, and Homogeneity II contains three dominant discontinuity sets. The geometrical properties of discontinuity involving trace length, aperture, roughness, and orientation are measured using high-precision GPS-RTK equipment (Real Time Kinematic) [50]. The fracture networks in the present are generated based on the field results of Homogeneity I. Detailed statistics parameters of two dominant discontinuity sets in Homogeneity I are reported in the study of Guo et al. [50].

Since the REV is closely correlated with the number of fractures in the analyzed domain, three increasing fracture densities—i.e., P₃₂ = 0.1, 0.2 and 0.3 m²/m³—are considered. The orientations of the two discontinuity sets in Homogeneity I follow the Fisher distribution with the parameters tabulated in

Table 1. Statistics parameters of dominant discontinuity sets [50].

Dominant Set	Orientation		Fisher Constant
	Dip (°)	Dip Direction (°)
1	81.2	49	10.85
2	72	116.7	1.59

. Digital processing for the discontinuity indicates that the local aperture of fractures is non-uniform due to the surface roughness. The fracture aperture in the present study is assumed to follow the truncated normal distribution, with a mean aperture of 1.82 mm and a variance of 1.15 mm. In many previous studies, due to computational limitations, the individual fractures in the DFN model are commonly assumed to be parallel plates having an identical mechanical aperture. To clarify the effect of this simplification on fluid flow in 3D DFNs, for each DFN model with heterogeneous aperture distributions (denoted as the DFN-H model), a corresponding DFN with the same geometrical structure but an identical mean aperture (denoted as the DFN-I model) is also established for comparison.

The study’s DFNs are established using the Monte Carlo method [51,52], in which the geometrical properties utilized follow the distributions defined above. In order to avoid randomness and boundary effects, three large “parent” DFNs of 150 × 150 × 150 m are firstly established using different random numbers, as shown in Figure 2. From each parent model, smaller DFN models, with sizes varying from 5 to 50 m, are extracted for REV studies. Each smaller model is then rotated in a number of directions to check whether there exists a tensor quality of the permeability at a certain scale. Figure 3 displays an example of the rotated DFNs extracted from the large parent model. The cubes in different colors indicate the DFNs extracted at the original configuration and at the rotation of α, respectively. In total, 26 different directions—indicated by the angles of θ and φ shown in Figure 4—are needed to cover the whole 3D space at every 45°. As the magnitudes of the permeability tensor are the same in two opposite directions, only 13 directions are considered for generaing the rotated DFN models.

3. Flow Calculations

The steady and laminar flow through single rock fractures can be modeled via the Reynolds equation, written in [53] as

$${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{x}_f}}\left({\displaystyle \frac{\rho g{b}^3}{12\mu }}{\displaystyle \frac{\mathsf{\partial }h}{\mathsf{\partial }{x}_f}}\right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{y}_f}}\left({\displaystyle \frac{\rho g{b}^3}{12\mu }}{\displaystyle \frac{\mathsf{\partial }h}{\mathsf{\partial }{y}_f}}\right)=0$$

(1)

where x_f and y_f are the local coordinates for a fracture plane Ω_f, ρ is the fluid density, b is the local aperture, μ is the dynamic viscosity, h is the hydraulic head, and g is the gravitational acceleration. To estimate the overall flow within the fracture network, the following continuity conditions are applied on the fracture intersections S_k [8]:

$$\left\{\begin{array}{c}{h}_{k,f}=h_k, \forall f\in F_k\\ {\displaystyle \sum _{f\in F_k}{q}_{k,f}\cdot n_{k,f}=0}\end{array}\right.$$

(2)

where h_k is the hydraulic head on S_k, F_k is the set of fracture intersecting on S_k, h_k,f is the trace of the head on S_k in Ω_f, q_k,f is the flow through intersection of Ω_f, and n_k,f is the normal vector to the fracture intersection S_k in Ω_f.

To estimate the permeability in the selected direction, a unit hydraulic gradient was applied on two perpendicular faces of the model in that direction, and the other faces of the model were set to be no-flow boundaries. The directional permeability can be calculated as follows [35]:

$$K_{\mathrm{d}}\left(\mathbf{m}\right)={\displaystyle \frac{\mu }{A\rho g}}\cdot {\displaystyle \frac{\mathbf{q}\cdot \mathbf{m}}{\nabla h\cdot \mathbf{m}}}$$

(3)

where ∇h is the hydraulic gradient in the direction m, and q is the vector flux.

The following equation relates the permeability tensor K to the K_d(m) [25]:

K_d(m) = m^T·K·m

(4)

From Equation (4) the following equation can be derived:

$$K_d\left(m_i\right)=m_{i1}^2{K}_{11}+m_{i2}^2{K}_{22}+m_{i3}^2{K}_{33}+2m_{i1}{m}_{i2}{K}_{12}+2m_{i2}{m}_{i3}{K}_{23}+2m_{i1}{m}_{i3}{K}_{13}$$

(5)

where K₁₁, K₁₂, K₁₃, K₂₂, K₂₃, and K₃₃ are six independent permeability tensor components. K_d(m_i) is the ith directional permeability in the direction of m_i= (m_i1, m_i2, m_i3). When the permeability along N (N ≥ 6) directions is known, the linear equations with the six unknown quantities can be written in the following matrix form:

AX = b

(6)

with

$$A=\left[\begin{array}{c}{m}_{11},m_{12},m_{13},2m_{11}{m}_{12},2m_{12}{m}_{13},2m_{11}{m}_{13}\\ m_{21},m_{22},m_{23},2m_{21}{m}_{22},2m_{22}{m}_{23},2m_{21}{m}_{23}\\ ⋮\\ m_{N1},m_{N2},m_{N3},2m_{N1}{m}_{N2},2m_{N2}{m}_{N3},2m_{N1}{m}_{N3}\end{array}\right]$$

(7)

$$X={\left(K_{11},K_{22},K_{33},K_{12},K_{23},K_{13}\right)}^{\mathrm{T}}$$

(8)

$$b={\left\{K_d\left(m_1\right),K_d\left(m_2\right),\cdots ,K_d\left(m_N\right)\right\}}^{\mathrm{T}}$$

(9)

When a linear regression model is applied, the estimator $K^{\mathrm{e}}={\left(K_{11}^{\mathrm{e}},K_{22}^{\mathrm{e}},K_{33}^{\mathrm{e}},K_{12}^{\mathrm{e}},K_{23}^{\mathrm{e}},K_{13}^{\mathrm{e}}\right)}^{\mathrm{T}}$ of the regression parameters $X={\left(K_{11},K_{22},K_{33},K_{12},K_{23},K_{13}\right)}^{\mathrm{T}}$ obtained from the method of least squares has the form [54]

$$K^{\mathrm{e}}={\left(A^{\mathrm{T}}A\right)}^{-1}{A}^{\mathrm{T}}b$$

(10)

The program proposed by the authors of this study was utilized to calculate the fluid flow behavior in 3D DFN models [55]. In this approach, the large parent DFNs were firstly generated with geometrical properties following the predefined distributions. The computational domain with different sizes and rotation angles was defined. The isolated fractures and the fractures outside the computational domain were deleted. Then each fracture was triangularly discretized while the discretization conformed at fracture intersections. Under this constraint, the relevant continuity conditions can be easily applied to the fracture intersections. Finally, the Galerkin method was applied to calculate the flow-through DFNs, considering the mass and flux continuity at fracture intersections. Based on the flow calculation of the model with different fracture densities, apertures, model sizes, and rotation angles, the permeability tensor as well as the REV size are estimated.

4. Results and Analysis

4.1. REV Based on Geometrical Indicators

It is known that flow calculation via 3D DFN is usually time-consuming. Geometrical indicators can be used for initial estimation of the REV size due to their efficiency. The average intersection length (L_i), defined as the ratio of total intersection length to the number of intersections, is computed to examine the existence of REV. Figure 5 shows the variation of L_i with L for DFNs of different fracture densities. For all cases, with increasing L, L_i fluctuates first and then gradually tends to stability. The variance of L_i between models generated using different random numbers decreases as P₃₂ increases. The coefficient of variance (CV) of the models with five different random numbers is calculated and plotted in Figure 6. Figure 6 shows that the value of CV decreases with increasing L, indicating that the L_i do not change much when the model size exceeds a certain value. Previous studies have shown that when the value of CV decreases to 0.2, the corresponding model size can be approximated as an REV size [21,22]. The calculated REV size is equal to 16.7 m, 10.1 m, and 8.5 m, respectively, for the DFNs with three increasing P₃₂s, implying a decreasing REV size with increasing P₃₂.

4.2. REV Based on Flow Indicators

Figure 7 displays the comparisons of preferential flow pathways along the overall flow direction of θ = 0° and φ = 90° in DFNs with different L, P₃₂, and aperture distributions. For DFN-H models, there are pronounced channeling flow effects within the individual fractures as well as within the networks. At each individual fracture, fluids are mainly concentrated in the channels with large apertures while bypassing the barriers with small apertures. Regarding the network, fluids select some transmissive fractures within the connected networks from the inlet to outlet boundaries. Increasing L or P₃₂ provides more alternative fracture connections, thus reducing the channeling flow effect at the network scale. Compared to the DFN-H models, the flow in the DFN-I models is less channelized in the individual fracture, but it shows essentially similar channeling at the network scale, which is induced by the complex topology of the fracture networks.

Figure 8 shows the calculated permeability in the direction of θ = 0° and φ = 90° for DFN models of different model sizes, fracture densities, and aperture distributions. The results show that, for all cases, K varies significantly when L is small, which is induced by the influence of the random numbers used to generate the DFN models. As L exceeds a certain value, K becomes almost constant, and this phenomenon is more obvious with increasing P₃₂. Although the variation of K decreases significantly when L exceeds a certain value, the difference between the minimum and maximum values of K is still about 4 times greater for the DFN-H model with P₃₂ = 0.1 m²/m³. This difference gradually decreases with increasing P₃₂. The DFN-I model usually has a larger K compared to the DFN-H model, indicating that the aperture heterogeneity reduces the permeability of fracture networks.

The CV of K for the models with five different random numbers is calculated and plotted in Figure 9. With increasing L, the CV decreases rapidly at first and then gradually tends to stability. The calculated CV for the DFN-I and DFN-H models with P₃₂ = 0.1 m²/m³ is larger than 0.2 over the whole range of L, indicating that the REV does not exist within the calculated model size range. For DFN-H models with P₃₂ = 0.2 and 0.3 m²/m³, the calculated REV size is 41.1 m and 12.2 m, respectively, when CV = 0.2. Compared to the calculated REV size based on the geometrical indicator of L_i, the flow indicator of K indicates an obviously larger REV size, especially when P₃₂ is small. For DFN-I models with P₃₂ = 0.2 and 0.3 m²/m³, the calculated REV size is 23.4 m and 11.6 m, respectively, when CV = 0.2. Comparisons between the estimated REV size of DFN-I and DFN-H models indicate that the heterogeneous aperture distribution increases the REV size, and this effect is more obvious when P₃₂ is smaller.

4.3. Permeability Anisotropy

To calculate the magnitude and direction of the permeability tensor, Figure 10 shows the 2D directional permeability contours in the x-z plane (rotation along the y axis) for DFNs with different model sizes, fracture densities, and aperture distributions. Note that the rotation angle is decreased to 30° to fit the shape of the permeability contour precisely. For all cases in Figure 10, the shape of the permeability contour is irregular, with some breaks when L is small. With an increase in the L, its shape gradually tends to be smooth, approaching an ellipse, and the direction of the minimum or maximum magnitude of K remains almost constant. When P₃₂ = 0.1 m²/m³, as shown in Figure 10a,b, the shape of the permeability contour is irregular, even for the model with a larger L. The maximum and minimum permeabilities fall into the orientation region of 90°~150° and 330°~30°, respectively. As the P₃₂ increases from 0.2 m²/m³ to 0.3 m²/m³, the magnitude of K expands while its shape remains almost constant. The maximum permeability changes to a location in the range of 60°~120° and a minimum permeability in the range of 330°~30°. The effect of fracture density on the shape of the permeability contour shows a similar tendency to that of the model size. The permeability contour shape of the DFN-H model is slender compared to the DFN-I model, indicating a larger difference between the minimum and maximum value of K. To help quantitatively estimate the goodness of fit for an ellipse, a measure, RMS, is calculated according to the equation

$$RMS={\displaystyle \frac{2}{\left(K_1+K_2\right)}}\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _1^N{\left[k_g\left(\alpha \right)-{\overline{k}}_{ij}\left(\alpha \right)\right]}^2}}$$

(11)

where k_g(α) is the estimated permeability along every rotated direction, K₁ and K₂ are the major and minor principle permeabilities, respectively, and $\overline{k}$_ij(α) is the rotated average permeability. As indicated by Öhman and Niemi [56], an RMS < 0.2 is viewed as an acceptable fit for the fracture networks, implying the applicability of an equivalent continuum approximation for DFNs.

Figure 11 shows the calculated RMS for DFNs with different fracture densities, model sizes, and aperture distributions. When P₃₂ = 0.1 m²/m³, the RMS exhibits a large variation greater than 0.3 for both DFN-I and DFN-H models at all model sizes up to 50 m, which indicates that there is no appropriate approximation for the equivalent permeability tensor over the studied model size range. As P₃₂ increases, the RMS of the DFN-I and DFN-H models shows different decreasing tendencies. For the DFN-H model, although the RMS value decreases, it is still larger than 0.24. In contrast, for the DFN-I model, a good approximation of an ellipse can be obtained when L = 26.1 m and 18.6 m (RMS = 0.2) for P₃₂ = 0.2 and 0.3 m²/m³, respectively. Comparisons between the DFN-I and DFN-H models show that the heterogeneous aperture distribution increases the flow anisotropy of DFNs, thus leading to a larger model size to reach an equivalent continuum behavior.

According to 2D analysis on the permeability contours, shown in Figure 10 and Figure 11, the equivalent permeability for the analyzed fracture networks with P₃₂ = 0.2 m²/m³ and 0.3 m²/m³ can be approximately equal to the permeability tensor of DFN-I models estimated at L = 30 m and 20 m, respectively. In contrast, for the DFN-H model, the acceptable equivalent approximation does not exist over the studied model length range. As a result, the 3D directional permeability of the DFN-I models—both with L = 30 m and P₃₂ = 0.2 m²/m³ and L = 20 and P₃₂ = 0.3 m²/m³—are estimated, and the results along with fitted ellipsoids are displayed in Figure 12. The dots representing the actual permeability are close to the boundary of the fitted ellipsoid, indicating a satisfactory equivalent continuum behavior at the calculated REV size. The corresponding magnitudes and directions of the principal permeabilities of DFN-I models with different P₃₂s are tabulated in

Table 2. Magnitudes and directions of principal permeabilities of DFN-I models.

P32 (m2/m3)	L (m)	K1		K2		K3
		Magnitude × 10−11 m2	Direction	Magnitude ×10−11 m2	Direction	Magnitude × 10−11 m2	Direction
0.2	30	6.959	(−0.486, 0.502, −0.716)	5.221	(0.443, −0.564, −0.697)	3.452	(−0.754, −0.655, 0.051)
0.3	20	6.447	(−0.090, 0.251, 0.964)	5.701	(−0.681, 0.690, −0.243)	3.579	(0.727, 0.678, −0.109)

. The estimated permeability tensor can be viewed as the equivalent permeability tensor for the approximated continuum medium.

5. Conclusions

This study conducted a systematic, numerical simulation to investigate REV and directional permeability based on two kinds of 3D DFNs, with heterogeneous apertures and uniform apertures. The REV sizes of the two kinds of DFN models were determined based on both geometrical and flow indicators. The effects of fracture aperture and density on the REV size and the permeability tensor (both magnitude and direction) were analyzed. The following conclusions were obtained:

• Both the average intersection length and the permeability of DFNs generated with different random numbers vary significantly when the model size is small. As the model size exceeds a certain value, their magnitudes become almost constant, and this phenomenon is more obvious with increasing fracture density.
• The calculated REV size based on the flow indicator of the permeability is obviously larger than that based on the geometrical indicator, especially when the fracture density is small. Aperture heterogeneity increases the REV size, and this effect is more obvious when the fracture density is smaller.
• The shape of the 2D permeability contour is irregular, with some breaks, when the model size is small. As the model size increases, its shape gradually tends to be smooth and approaches an ellipse, with an almost unchanged direction of the minimum or maximum permeabilities. Compared to the DFNs with uniform apertures, the shape of the permeability contours of DFNs with heterogeneous apertures is slender, indicating a larger difference between the minimum and maximum values of the permeability.
• There is no appropriate approximation of the equivalent permeability tensor over the studied model size range for the DFN-H models in this study. In contrast, for the DFN-I model, a good fit for the permeability ellipsoid is obtained, and the 3D directional permeability is calculated. The corresponding magnitude and direction of the principal permeability can be viewed as the equivalent permeability tensor for the approximated continuum medium.

Note that the effect of contact between the upper and lower surfaces on fluid flow is not considered in this study, which is mainly due to the difficulty of meshing the poorly predisposed geometry around the contact. In further studies, we will try to obtain a robust mesh for 3D DFNs and to quantify the contact effect on the REV and the permeability tensor.