A Study of the Scale Dependency and Anisotropy of the Permeability of Fractured Rock Masses

Abstract

Affected by discontinuities, the hydraulic properties of rock masses are characterized by significant scale dependency and anisotropy. Sampling a rock mass at any scale smaller than the representative elementary volume (REV) size may result in incorrect characterization and property upscaling. Here, a three-dimensional discrete fracture network (DFN) model was built using the joint data obtained from a dam site in southwest China. A total of 504 two-dimensional sub-models with sizes ranging from 1 m × 1 m to 42 m × 42 m were extracted from the DFN model and then used as geometric models for equivalent permeability tensor calculations. A series of steady-state seepage numerical simulations were conducted for these models using the finite element method. We propose a new method for estimating the REV size of fractured rock masses based on permeability. This method provides a reliable estimate of the REV size by analyzing the tensor characteristic of the directional permeability, as well as its constant characteristic beyond the REV size. We find that the hydraulic REV sizes in different directions vary from 6 to 36 m, with the maximum size aligning with the average orientation of joint sets and the minimum along the angle bisector of intersecting joints. Additionally, the REV size is negatively correlated with the average trace length of the two intersecting joint sets. We find that the geometric REV size, determined by the joint connectivity and density, falls into the range of the hydraulic REV size. The findings could provide guidance for determining the threshold values of numerical rock mass models.

1. Introduction

Discontinuities, such as joints, bedding planes, and faults, widely develop in rock masses. These discontinuities interconnect to form a fracture network system, providing primary pathways for fluid flow [1,2,3,4,5]. As the permeability of discontinuities is several orders of magnitude higher than that of the rock matrix, the hydraulic behavior of a rock mass is mainly determined by the discontinuities [6,7]. Influenced by the discontinuities, the hydraulic properties of rock masses exhibit strong scale dependency and anisotropy [8,9,10]. The determination of the equivalent permeability of rock masses and the dominant permeability path is of great significance for dam design and stability evaluations.

Currently, the methods for determining the permeability of rock masses include laboratory experiments, in situ tests, and numerical simulations. Due to the limited sample size, rock samples in laboratory tests usually do not represent a whole rock mass containing a complex fracture network, and thus the measured values cannot reflect the true permeability [11,12]. In situ tests can measure representative permeability values at an engineering scale but are limited by time consumption and high costs [13]. Numerical simulations based on discrete fracture network models have become one of the most effective ways for solving the equivalent seepage problem in rock masses [2,14,15]. However, the permeability of rock masses has a size effect, and thus sampling a rock mass at any scale smaller than the REV size may result in incorrect property upscaling. How to obtain the permeability values representative of a rock mass remains a challenge.

A fractured rock mass is deemed to behave as an equivalent continuum when (1) there is an insignificant change in the equivalent permeability value with a slight addition to or subtraction of the sample volume, and (2) the equivalent permeability tensor exists, which predicts the correct flux when the direction of a constant gradient is changed [16]. The REV is defined as the smallest element that can represent the macroscopic properties of a fractured rock mass [17,18,19]. Evaluating the REV size of a rock mass is a critical step for determining the applicability of the continuum approach, since it determines whether a fractured rock behaves as a continuum [1,11,20,21]. Currently, various parameters of rock masses have been used to determine the REV size, including the following:

• Geometric parameters, such as joint spacing, trace length, etc. [19,22,23]. For example, Schultz [24] suggested that the REV size of a rock mass is five to ten times the average joint spacing. Zhang and Xu [25] considered that the REV size to be three to four times the maximum expected value of the joint trace length.
• Mechanical parameters, such as the shear stress, elastic modulus, and uniaxial compressive strength [19,26,27,28]. For example, Wang et al. [29] used the two-dimensional particle flow code (PFC2D) to study the size effect and directionality of the shear strength of stratified rocks and determined the REV size based on shear strength. Li et al. [28] conducted a series of uniaxial compression numerical simulation tests using PFC3D based on a synthetic rock model and estimated the REV dimensions using the elastic modulus and uniaxial compressive strength.
• Hydraulic parameters, including connectivity and permeability. For example, Min et al. [1] determined the REV size based on rock permeability and found that the permeability values calculated at the REV size have tensor characteristics. Bang et al. [30] estimated the REV size with respect to the hydraulic behavior based on a three-dimensional (3D) discrete fracture network model. Rong et al. [6] discussed the effects of fracture geometric parameters such as the trace length, spacing, aperture, orientation, and the number of fracture groups on the REV size. Wang et al. [31] studied the hydraulic properties of fractured rock masses related to the fracture length and aperture under both radial and unidirectional flow conditions and estimated the REV size.

Permeability is a key parameter in geotechnical engineering that directly affects the long-term performance of critical infrastructure projects (e.g., dams, tunnels, and underground storage) in terms of safety and durability. High permeability may lead to rock seepage and structural instability in the dam area. Affected by discontinuities, the hydraulic parameters of a rock mass have strong directivity. Accordingly, the REV sizes in different directions vary. However, the limitations of previous works are that they seldom study the scale dependency and anisotropy of complex fractured rock masses based on field measured data simultaneously. In this work, the rock mass of a dam site located in the upper reaches of the Nu River on the southeast edge of the Tibetan Plateau were investigated. A 3D discrete fracture network (DFN) model of the rock mass was established based on the field joint data using the Monte Carlo simulation. Two-dimensional (2D) models with different sizes were extracted from the 3D DFN model and then were subjected to fluid flow simulations to investigate the variations in their equivalent permeability with their sizes. The models were also rotated in 15° intervals to investigate the tensor characteristics of the directional permeability. The steady-state mean permeability in each direction was calculated. The estimated permeability ellipse was fitted, and the fitting effect was evaluated using the root mean square error. On this basis, a method for determining the REV size was proposed. The effects of joint parameters on the equivalent permeability and the REV size were analyzed. Additionally, the hydraulic REV size was compared with the geometric REV size. The findings of this study not only provide a new method for REV determinations of fractured rock masses but also provide a scientific basis for permeability and anisotropy analyses in practical engineering.

2. Discrete Fracture Network Model

2.1. Joint Mapping

Joint mapping was conducted in an exploration tunnel with a length of 200 m and a radius of 2 m, which is located at the upper reaches of the Nu River, China. The lithology is primarily Cretaceous biotite granite, mainly consisting of quartz, plagioclase, potassium feldspar, and biotite. Cretaceous mafic intrusive rocks are also outcropped in this area, presenting as dykes with widths ranging from 0.05 to 5 m [32]. In this study, the window sampling method [5,33,34,35,36] was used for joint measurements: all joints intersecting the south-side wall of the exploration tunnel with a trace length larger than 0.5 m were mapped (Figure 1a). The measured data included the joint coordinate, orientation, trace length, spacing, and aperture. Based on the degrees of unloading and weathering, the investigated rock mass was divided into three zones: a strongly disturbed zone, weakly disturbed zone, and undisturbed zone. This study selected the 80 m-long undisturbed zone as the study region, where a total of 128 joints were measured [37]. The joints were grouped into three sets based on their orientations using the fuzzy K-means algorithm [38] (Figure 1b): (1) set 1 consists of steeply dipping joints with an average orientation of 265.0°∠82.2°; (2) set 2 consists of shallow-dipping joints with an average orientation of 100.6°∠11.8°; and (3) set 3 consists of moderately dipping joints with an average orientation of 113.3°∠48.3°.

2.2. Generation of the 3D DFN Model

The discrete fracture network DFN model can be an effective simulation tool as it can largely restore realistic discontinuity geometries for in situ applications [39,40]. Typically, the parameters used to generate the DFN model can be determined from a probability distribution function derived statistically based on in situ measurements. The 3D DFN model in this paper was established from our previous study. The detailed modeling procedures can refer to our previous work [35]. The statistical joint parameters used for the fracture network modelling are listed in

Table 1. Statistical parameters of the joints used for DFN modelling [35].

Joint Set	Joint Number	Dip Direction (°)	Dip (°)	Fisher Constant	Spherical Std. (°)	Measured Trace Length			Joint Density (m−3)
						Mean (m)	Std. (m)	Distribution
1	19	265.0	82.2	10.87	14.43	1.48	0.57	Log-normal	0.030
2	35	100.6	11.8	28.40	8.75	2.19	1.57	Gamma	0.160
3	74	113.3	48.3	17.11	10.01	1.35	0.79	Log-normal	0.062

. Joints are assumed to be disks and their spatial distributions are determined by three parameters [35]: (1) orientation, (2) diameter, and (3) location. The distribution of joint orientations in each joint set is described by an empirical probability distribution, with corrections applied to the relative frequency of each measured joint direction [34]. The size distribution of joints in each fracture set corresponds to their trace length distribution, which is statistically determined from the field data. The location of joint centers is assumed to follow a uniform Poisson distribution [1,5,36]. The stereological method [41] was used to determine the mean size of joints in each set according to their trace length distributions. The volume density λ_i^V can be derived from the normal linear density λ_i^l of each joint set using the following formula [42,43]:

$${\lambda }_i^V={\displaystyle \frac{4{\lambda }_i^l}{\pi E(D^2)}}$$

(1)

where E(D²) is the second moment of the joint diameter distribution. The number of joints n_i in the simulated region for the i-th joint set is

$$n_i=V{\lambda }_i^V$$

(2)

Finally, the Monte Carlo method was used to generate a 3D DFN model through the Poisson process in a 100 m × 60 m × 60 m simulated region, as shown in Figure 2. An artificial sampling window with the same size as the field measurement was set in the simulated rock mass to obtain joint parameters, including the number of joints, dip direction, dip angle, spherical standard deviation, and measured mean trace length and standard deviation. The DFN model was validated by comparing the parameters of the artificial joints with those obtained from field mapping. The results in

Table 2. Comparison between the joint data obtained from field mapping and the DFN model.

Joint Set		Joint Number	Dip Direction (°)	Dip (°)	Spherical Std. (°)	Measured Trace Length
						Mean (m)	Std. (m)
1	Field	19	265.0	82.2	14.43	1.48	0.57
	Generated	20	272.5	79.0	13.08	1.26	0.51
2	Field	35	100.6	11.8	8.75	2.19	1.57
	Generated	34	89.8	8.0	9.88	2.47	1.76
3	Field	74	113.3	48.3	10.01	1.35	0.79
	Generated	73	112.1	45.3	9.85	1.65	0.75

show that the simulated parameters agree well with the field measurements, indicating that the DFN model is reliable.

3. Calculation Method for Fluid Flow in Rock Masses

Fluid flow in fractured rock masses is typically assumed to follow the linear Darcy’s law [1,2], with the governing equation expressed as follows:

$$Q=A{\displaystyle \frac{k}{\mu }}{\displaystyle \frac{\Delta P}{\Delta L}}$$

(3)

where Q is the total flow rate, A is the cross-sectional area, k is the permeability, μ is the fluid dynamic viscosity, ∆P is the pressure difference between the inlet and outlet boundaries, and ∆L is the seepage length.

For fluid flow in a fracture, it is often assumed that the both walls of the fracture are smooth, and the flow follows the cubic law [44]

$$q={\displaystyle \frac{h^3}{12\mu }}{\displaystyle \frac{\Delta P}{\Delta L}}$$

(4)

where q is the flow rate and h is the fracture aperture.

3.1. Permeability Tensor

The permeability of a fractured rock mass has tensor characteristics [16,26], which is a second-order positive definite tensor with six independent parameters:

$$k=\left[\begin{array}{ccc}{k}_{xx}& k_{xy}& k_{xz}\\ k_{yx}& k_{yy}& k_{yz}\\ k_{zx}& k_{zy}& k_{zz}\end{array}\right]$$

(5)

where k_xx, k_yy, k_zz, k_xy, k_xz, and k_yz are the six independent components of the permeability tensor. The two-dimensional form is as follows:

$$k=\left[\begin{array}{cc}{k}_{xx}& k_{xy}\\ k_{yx}& k_{yy}\end{array}\right]$$

(6)

The permeability tensor has coordinated transformation characteristics, which are a function of the position coordinates and vary with the coordinate system. As shown in Figure 3, assuming that the local coordinate system x′oy′ is formed by rotating the global coordinate system xoy counterclockwise by an angle α, the transformation relationship between them is expressed as follows:

$$k_{i^{\prime }{j}^{\prime }}=k_{ij}{\alpha }_{i^{\prime }i}{\alpha }_{j^{\prime }j}$$

(7)

where α_x′x = cosα, α_x′y = sinα, α_y′x = −sinα, and α_y′y = cosα. When the xoy coordinate system rotates counterclockwise to the x′oy′ coordinate system, the transformation relationship between the elements of each matrix is described by

$$\left[\begin{array}{c}{k}_{x\prime x\prime }\\ k_{y\prime y\prime }\\ k_{x\prime y\prime }\end{array}\right]=\left[\begin{array}{ccc}{\cos }^2\alpha & {\sin }^2\alpha & \sin 2\alpha \\ {\sin }^2\alpha & {\cos }^2\alpha & -\sin 2\alpha \\ -\frac{1}{2}\sin 2\alpha & \frac{1}{2}\sin 2\alpha & \cos 2\alpha \end{array}\right]\left[\begin{array}{c}{k}_{xx}\\ k_{yy}\\ k_{xy}\end{array}\right]$$

(8)

In the x′oy′ coordinate system, assuming that the x′ and y′ axes are aligned with the principal permeability directions k₁ and k₂, the permeability in the direction obtained by rotating the x′ axis counterclockwise by an angle β is

$$k_{\beta }=k_1{\cos }^2\beta +k_2{\sin }^2\beta$$

(9)

In the polar coordinate system, for a point with the polar angle β and radial distance r_β, which is assumed to be 1/k_β^1/2 here, the projection of this point onto the Cartesian coordinate axes is (Figure 3)

$$\left\{\begin{array}{c}{x}^{\prime }=\cos \beta /\sqrt{k_{\beta }}\\ y^{\prime }=\sin \beta /\sqrt{k_{\beta }}\end{array}\right.$$

(10)

By transforming the above relationship into the Cartesian coordinate system, k₁ and k₂ satisfy the standard ellipse equation, which refers to the permeability ellipse [17]:

$${\displaystyle \frac{x^2}{{\left(1/\sqrt{k_1}\right)}^2}}+{\displaystyle \frac{y^2}{{\left(1/\sqrt{k_2}\right)}^2}}=1$$

(11)

The length of the two semi-axes of the ellipse is a = 1/k₁^1/2 and b = 1/k₂^1/2, and the radial distance r_β can be expressed numerically by the permeability in that direction r_β = 1/k_β^1/2. The permeability ellipse is helpful for the visualization of the permeability tensor, which provides an effective way to evaluate the calculation results of the permeability.

Once the directional REV size is determined, the steady-state permeability of the fractured rock mass in different directions is calculated. In the polar coordinate system, the polar angle θ is defined as the counterclockwise angle from the x-axis direction to the fluid flow direction, and the polar radius r = 1/k^1/2 is represented by the permeability. The models are rotated in 15° intervals to investigate the tensor characteristics of the directional permeability, resulting in 24 data points (θ_i, r_i). The least square method is used to fit these points to an ellipse, yielding the fitting equation of the permeability ellipse and the six independent components of the permeability tensor. The eigenvalues and corresponding eigenvectors of the permeability tensor are then solved to obtain the principal permeabilities and their directions, respectively. To evaluate whether the fitting of the permeability ellipse could reasonably describe the permeability characteristics of the fractured rock mass, the fitting effectiveness is evaluated by the root mean square error (RMS_Norm) [45]:

$$RM{S}_{\mathrm{Norm}}={\displaystyle \frac{2}{k_1+k_2}}\sqrt{{\displaystyle \frac{1}{N}}{{\displaystyle \sum _1^N\left[k_{tensor}(\theta )-k_{sim}(\theta )\right]}}^2}$$

(12)

where N is the number of seepage directions, which is equal to 24 in this study; k_tensor(θ) is the calculated permeability value in the θ direction; and k_sim(θ) is the fitted permeability value in the θ direction. RMS_Norm is a measure of the average difference between a statistical model’s predicted values and actual values. The smaller the RMS_Norm, the smaller the difference between the calculated and fitted permeability values, indicating a better fit.

3.2. Fluid Flow Simulation Method

This study uses the commercial software package COMSOL Multiphysics 5.6 to simulate fluid flow in the fractured rock mass. Typically, the fluid velocity in the rock matrix is low, and its permeability is much smaller than that of fractures. Therefore, the Darcy’s law interface in the porous media and subsurface flow module is used to simulate fluid flow in a rock matrix. This interface is suitable for low-velocity flows or media with low permeability and porosity, driven by pressure gradients where frictional resistance in pores is the primary influencing factor on the flow. Combining Darcy’s law and the continuity equation yields the following generalized governing equation [17]:

$${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}\left(\rho {\epsilon }_p\right)+\nabla \cdot \rho \left[-{\displaystyle \frac{k}{\mu }}\left(\nabla p\right)\right]=Q_{\mathrm{m}}$$

(13)

where ρ is fluid density, ε_p is the porosity of rock matrix, and Q_m is the mass source term.

For the calculation of fluid flow in fractures, the fracture flow interface based on the single fracture channel flow process is selected. The governing equation is derived by combining the generalized cubic law with the continuity equation [44]:

$$\nabla \cdot \left(\rho q_{\mathrm{f}}\right)=d_{\mathrm{f}}{Q}_{\mathrm{m}}$$

(14)

$$q_{\mathrm{f}}=-{\displaystyle \frac{d_{\mathrm{f}}^3}{12\mu f_{\mathrm{f}}}}\nabla p$$

(15)

where q_f is the volume flow rate per unit length in the fracture, d_f is the aperture or fracture thickness, and f_f is the roughness factor.

The above calculation method ensures the momentum and mass conservation in the process of seepage calculation. The continuity boundary conditions also ensure the continuity of pressure and mass flux. The fluid distribution in fractures is coupled with the rock matrix through the exchange of the flow and pressure between the upper and lower surfaces of the fractures. Changes in the fluid density and fracture aperture caused by variations in temperature and stress are not considered, and the fluid is assumed to be an incompressible medium under steady laminar flow driven only by the pressure head. These assumptions improve the computational efficiency and ensure the accuracy of the hydraulic property estimation.

3.3. Verification

In order to validate the numerical method, a series of simple fracture models, including nine single-fracture models and one double-fracture model, were selected, as shown in Figure 4. In the single fracture models, the angle α between the fracture and the horizontal direction varies from 50° to 90°, and the aperture is 5 mm. In the double-fracture model, the apertures of two fractures are 4 mm and 5 mm, respectively. The model sides are 20 m × 20 m. With head H₂ = 30 m applied above the model and head H₁ = 10 m below, the head varies linearly on both sides of the boundary. The steady-state fluid flow simulation was performed using the numerical method and the simulation results were then compared with those calculated by the theoretical method [44] and the modified equivalent permeability model proposed by Wang et al. [46].

Table 3. Comparison of the simulation results obtained by the numerical method and the method proposed by Wang et al. [46].

Fracture Model		Outlet Flow (m3/s)			Relative Difference
		Solution of Wang et al. [46]	Solution of This Work	Theoretical Solution	Wang et al. [46]	Theoretical Solution
Single-fracture model	α = 50°	0.0811	0.0777	0.0793	4.19%	2.02%
	α = 55°	0.0868	0.0831	0.0848	4.26%	2.00%
	α = 60°	0.0919	0.0879	0.0896	4.35%	1.90%
	α = 65°	0.0964	0.0920	0.0938	4.56%	1.92%
	α = 70°	0.1000	0.0954	0.0972	4.60%	1.85%
	α = 75°	0.0984	0.0980	0.0999	0.41%	1.90%
	α = 80°	0.1002	0.0999	0.1019	0.30%	1.96%
	α = 85°	0.1011	0.1011	0.1031	0.00%	1.94%
	α = 90°	0.1040	0.1015	0.1035	2.40%	1.93%
Double-fracture model		0.1424	0.1399	0.1424	1.78%	1.78%

shows the simulation results of the flow rate at the outlet for the fracture models obtained by different methods. The results show that the relative difference between the simulation results of this work and the results obtained by the two previous methods are all below 5%, indicating that the numerical method performs well in simulating fluid flows in fractured rock masses.

4. Numerical Simulation

4.1. Model Generation and Boundary Conditions

A 60 m × 60 m 2D DFN model was extracted from the 3D DFN model at x = 50 m along a plane perpendicular to the x-axis, serving as the ’master’ model. A series of square sub-models with side lengths varying from 1 m to 42 m with 1 m increments (i.e., 1 m, 2 m, 3 m, …, 42 m) were then extracted from the ’master’ model, as shown in Figure 5. In addition, each of the sub-models was rotated in a counterclockwise direction at 15° intervals with the rotation angle θ (θ = 0°, 15°, 30°, 45°, 60°, 75°, 90°, 105°, 120°, 135°, 150°, and 165°), which was defined as the angle between the fluid flow direction and the x-axis direction, as shown in Figure 6. All sub-models were imported into the software COMSOL Multiphysics to study the variation in the equivalent permeability with the model size and the directional permeability. Thus, 504 numerical simulations were performed. The boundary conditions imposed on the models are shown in Figure 7. The two lateral boundaries are constant pressure boundaries, where a 0.1 MPa/m pressure is applied to the left side and a 0.5 MPa/m pressure is applied to on the right side, yielding a constant pressure differential of 0.4 MPa/m in the flow direction. The top and bottom boundaries are impermeable boundaries.

4.2. Simulation Results

Figure 8 shows the variation in the equivalent permeability k with the model size in different fluid flow directions. When the model size increases from 1 m to 11 m, the value of k fluctuates drastically due to the existence of persistent joints in the models. As the model size further increases, the value of k begins to converge to a constant due to the absence of persistent joints. The plots in Figure 8 also indicate that the permeability of the rock mass presents strong anisotropy. This anisotropy decreases with increasing model sizes and eventually stabilizes. Figure 9 gives an example of the fluid pressure distribution in the models when θ is 0°. It seems that the permeability of the small-sized models is typically dominated by a highly permeable long persistent joint.

5. Estimation of the REV Size

5.1. Estimation of the REV Size Based on Permeability

The REV is defined as the smallest element that can represent the macroscopic properties of a fractured rock mass [17,18,19]. The permeability of a rock mass shows a constant characteristic when its size is larger than the REV size. Once the rock mass size exceeds L_REV, its permeability presents both constant and tensor properties [16,44]. This study used the coefficient of variation (CV), defined as the ratio of the standard deviation to the mean, to quantitatively determine the REV size. CV has been widely used to determine the REV size of rock masses based on mechanical parameters [27,28]. Generally, when the calculated CV value is less than 0.2, the sample size is considered the REV size [45]. This study gives the estimated results of the REV size considering different threshold values of CV (i.e., CV = 0.2, 0.19, 0.18, 0.17, 0.16, 0.15, 0.14, 0.13, 0.12, 0.11, and 0.1). In this research, the REV size determined by CV refers to L′_REV, and the average permeability of the rock mass larger than the REV size refers to k′_AVG. Since CV only reflects the cumulative error of measured data but cannot measure their local fluctuations, we used the following method to correct the REV size: the minimum size corresponding to the permeability k within a 20% fluctuation range of k′_AVG was taken as the REV size and denoted as L_REV (Figure 10), and the average permeability for the models with sizes larger than L_REV was calculated as the equivalent permeability k_AVG.

The equivalent permeability k_AVG in different flow directions was plotted in the polar coordinate system to fit the permeability ellipse. The fitting of the permeability ellipse was evaluated by calculating the root mean square error (RMS_Norm). A small RMS_Norm value means that the deviation between the original values and fitted values is small. It is generally considered that the fitting of the permeability ellipse is satisfying when the value of RMS_Norm is less than 0.2 [2,5,14,45]. The L′_REV values determined by permeability according to different threshold values of CV are shown in

Table 4. Values of L′_REV determined with different CV values.

CV	L′REV (m)
	θ = 0°	θ = 15°	θ = 30°	θ = 45°	θ = 60°	θ = 75°	θ = 90°	θ = 105°	θ = 120°	θ = 135°	θ = 150°	θ = 165°
0.20	34	25	21	15	29	18	17	2	9	11	10	9
0.19	35	25	21	17	30	19	18	3	9	11	10	9
0.18	35	26	23	17	30	20	18	3	9	11	10	9
0.17	35	26	24	19	31	20	19	4	9	11	10	9
0.16	36	27	24	21	32	20	19	4	9	11	10	9
0.15	36	27	25	23	32	21	20	5	10	11	10	9
0.14	36	28	26	26	33	22	20	22	10	11	10	9
0.13	36	29	29	29	33	22	34	24	10	11	10	9
0.12	36	29	31	30	34	23	35	25	10	11	10	9
0.11	36	30	34	31	34	24	36	26	10	11	10	9
0.10	36	30	34	32	36	24	36	27	10	11	10	9

. By limiting the fluctuation range of the permeability within a 20% fluctuation range of k′_AVG, the L_REV values were obtained (

Table 5. Adjustment of L_REV considering a 20% fluctuation range.

CV	LREV (m)
	θ = 0°	θ = 15°	θ = 30°	θ = 45°	θ = 60°	θ = 75°	θ = 90°	θ = 105°	θ = 120°	θ = 135°	θ = 150°	θ = 165°
0.20	36	29	25	20	32	24	36	6	11	11	10	12
0.19	36	29	25	20	33	24	36	6	11	11	10	12
0.18	36	30	26	20	33	24	36	6	11	11	10	12
0.17	36	30	26	23	33	24	36	6	11	11	10	12
0.16	36	30	26	23	33	24	36	6	11	11	10	12
0.15	36	30	26	24	33	25	36	6	11	11	10	12
0.14	36	30	26	26	34	25	36	22	11	11	10	12
0.13	36	30	29	29	34	25	36	24	11	11	10	12
0.12	36	30	31	30	34	25	36	25	11	11	10	12
0.11	36	30	34	31	34	25	36	26	11	11	10	12
0.10	36	30	34	32	36	25	36	27	11	11	10	12

). The L_REV, k_AVG, and fitted permeability ellipse in each seepage direction were plotted in the polar coordinate system (Figure 11). The results show that the values of L_REV in the flow direction with θ = 0–105° are significantly larger than those in the flow direction with θ = 120–165°. The difference in the k_AVG values and permeability ellipses obtained by the different CV values are not significant, but CV does affect the determination of L_REV, especially in the seepage directions where the L_REV values are relatively large. Figure 12 shows the values of RMS_Norm for the fitting of the permeability ellipse under different threshold values of CV. All calculated results are less than 0.2, indicating that the fit is good. The values of RMS_Norm achieve the minimum value of 0.0617 when the CV = 0.18, suggesting that the fit of the permeability ellipse is the best. Therefore, CV = 0.18 is adopted in this study as the threshold value for determining the REV size.

Figure 13a shows that the REV size is smallest (L_REV = 6 m) in the seepage direction of θ = 105°, which is 3–4 times the mean joint trace length. Additionally, the REV size is the largest (L_REV = 36 m) for θ = 0° and 90°. In the seepage directions of θ = 0–90°, the value of L_REV is no less than 20 m, whereas the value of L_REV does not exceed 12 m in the seepage directions of θ = 105–165°. Due to the influence of the joints, the rock mass permeability exhibits strong anisotropy, with the value of k ranging from 4.71 × 10⁻¹⁶ to 2.55 × 10⁻¹⁵ m² (Figure 13b). The long and short principal axes of the permeability ellipse are at θ = 48° and θ = 138°, respectively (Figure 13c). Figure 13 also shows that the rock mass permeability is relatively smaller and the REV size is relatively larger in the seepage directions of θ = 0–90° compared with the seepage directions of θ = 105–165°. Therefore, it seems that the directional REV size is negatively correlated with the directional permeability of the rock mass.

5.2. Estimation of the REV Size Based on the Anisotropy Index

The anisotropy index (AI_p) was introduced to quantitatively describe the anisotropy of the permeability of the fractured rock mass:

$$A{I}_{\mathrm{p}}={\displaystyle \frac{k_{\max }}{k_{\min }}}$$

(16)

where k_max and k_min represent the maximum and minimum permeability values in different directions.

Figure 14 shows the variation in the AI_p with the model size. When the model size reaches 11 m, the AI_p value drops by nearly three orders of magnitude. Beyond this size, the AI_p values remain below 10, indicating a dramatic change in permeability. To verify the applicability of the AI_p for estimating the REV size, the RMS_Norm was used for comparison (Figure 14). The RMS_Norm is less than 0.2 when the size of the rock mass reaches 11 m, which is consistent with the REV size determined by the AI_p. Figure 15 shows the variation in the permeability k with the model size. The average permeability k_AVG reaches the minimum value of 4.71 × 10⁻¹⁶ m² at θ = 45° and it reaches the maximum value of 2.55 × 10⁻¹⁵ m² at θ = 150°. The permeability exhibits a variation pattern similar to a cosine function with a periodicity of π, reflecting the directional properties of the directional permeability.

6. Discussion

6.1. Influence of the Joint Geometry on the Permeability and REV Size

The statistical parameters of the joints in the 2D ‘master’ DFN model (Figure 5a) are shown in Figure 16. Here, the orientation of a joint in a 2D plane is defined as the counterclockwise angle from the x-axis direction to the joint trace. Based on this definition, the average orientation of the joint sets S1, S2, and S3 is 80.5°, 137.5°, and 171.9°, respectively. Notably, the L_REV values at θ = 90°, 135°, and 180° are 36 m, 11 m, and 36 m, respectively, which are all the local maxima. Meanwhile, the bisector angles of each pair of the joint sets (i.e., S1 and S3, S1 and S2, and S2 and S3) are 36.2°, 109.1°, and 154.8°, respectively, while the L_REV values at θ = 45°, 105°, and 150° are 20 m, 6 m, and 10 m, respectively, which are all the local minima. Therefore, it could be concluded that the local maximum REV size appears in the direction of the average orientation of each joint set, while the local minimum REV size appears in the direction of the angle bisector of the average angle between each pair of the joint sets. We also find that the smallest value of L_REV (6 m) appears in the angle bisector direction between the joint sets S1 and S2 with larger joint lengths, whereas the largest value of L_REV (36 m) appears in the average orientation direction of the joint set S3 with a smaller joint length. This indicates that a larger joint length leads to a smaller value of L_REV. The fracture groups with larger average trace lengths are easier to intersect with the fractures distributed in other directions, which in turn form more persistent joints along that direction, greatly facilitating the connectivity of the fracture network. As the model size increases, the seepage process in this direction will reach the steady state more quickly, thus making the REV size smaller than the other directions.

6.2. Comparison Between the Geometric and Hydraulic REV Sizes

The geometric parameters are also important indicators for determining the REV size of rock masses. This study used the geometric parameters (i.e., joint connectivity and intensity) to estimate the geometric REV size. Joint connectivity refers to the number of joint intersections per unit area, while the joint intensity here refers to the number of joints per unit area (P₂₀) and the total joint length per unit area (P₂₁) [47]. The coefficient of variation (CV) was used to determine the geometric REV size. A CV value of 0.05 was selected as the threshold value for the REV size estimation.

The variation in the geometric parameters of the fractured rock mass with model size is shown in Figure 17. When the model sizes are 20 m, 10 m, and 7 m, the CV values for joint connectivity, P₂₀, and P₂₁ are all less than 0.05. Therefore, the geometric REV sizes based on joint connectivity, P₂₀, and P₂₁ are 20 m, 10 m, and 7 m, respectively. This observation reveals an interesting phenomenon: (1) the geometric REV size estimated by joint connectivity (20 m) is approximately equal to the lower limit of the REV size in the directions with relatively low permeability (i.e., θ = 0°, 15°, 30°, 45°, 60°, 75°, and 90°); (2) the geometric REV size estimated by P₂₁ (7 m) is similar to the lower limit of the REV size in the directions with relatively high permeability (i.e., θ = 105°, 120°, 135°, 150°, and 165°); and (3) the geometric REV size estimated by P₂₀ (10 m) is approximately equal to the REV size determined by the AI_p. In fact, the hydraulic properties of a rock mass are influenced by various joint parameters, such as the joint density, length, aperture, orientation, etc. The REV size based on geometric parameters could not reflect the variation and anisotropy of hydraulic properties, leading to incorrect characterization and property upscaling.

Fractured rock masses, due to their complex internal structures, often exhibit significant hydraulic anisotropy, making the variation in REV size in different directions non-negligible. Traditional homogeneous assumptions or isotropic models fail to accurately describe the permeability characteristics of fractured rock masses in many cases, leading to considerable deviations in seepage analysis and engineering calculations. The newly proposed permeability-based method for estimating the directional REV size of fractured rock masses provides a more targeted approach for characterizing rock mass permeability. In dam design, accurately assessing the permeability of the bedrock has profound implications for the design of anti-seepage systems, the development of seepage monitoring plans, and the prediction of long-term operational safety. Therefore, a reasonable estimation of the REV size of fractured rock masses, especially the directional REV size under anisotropic conditions, is crucial for improving the accuracy of permeability predictions, optimizing seepage control measures, and enhancing dam safety assessments.

However, for real natural fractured rock masses, further research is needed. First, two-dimensional models inherently underestimate the connectivity and anisotropy of three-dimensional fracture networks. As a result, the 2D approach may lead to an underestimation of permeability, thereby affecting the estimation of the REV size. This limitation is particularly evident in rock masses with significant variations in fracture dip angles or high fracture densities. Second, the geometric shape and surface roughness of fractures influence the fluid distribution, leading to channeling and anomalous transport phenomena, which in turn affect permeability predictions. Therefore, although this study provides an effective 2D method for estimating the directional REV size of fractured rock masses, further refinements are necessary for application to real natural fractured rock masses. Future research could integrate three-dimensional DFN models, fracture roughness models, and a geometric morphological analysis to explore how these factors interact with fracture network connectivity, thereby improving the accuracy and reliability of directional REV assessments.

7. Conclusions

This paper investigates the scale dependency and anisotropy of the hydraulic properties of a fractured rock mass. A 3D DFN model was built using the joint data obtained from a dam site in southwest China. A total of 504 2D sub-models with sizes ranging from 1 m × 1 m to 42 m × 42 m were extracted from the DFN model and then were used for equivalent permeability tensor calculations based on the finite element method. We proposed a new method for estimating the directional REV size of fractured rock masses based on permeability. The steady-state average permeability in each direction was calculated and the permeability ellipse was fitted.

The results show that the rock mass permeability exhibits strong anisotropy, with the value of k ranging from 4.71 × 10⁻¹⁶ to 2.55 × 10⁻¹⁵ m². The long and short principal axes of the fitted permeability ellipse are θ = 48° and θ = 138°, respectively. The hydraulic REV size L_REV is smallest (6 m) in the seepage direction of θ = 105°, which is 3–4 times the mean joint trace length. Additionally, the REV size is the largest (L_REV = 36 m) when θ = 0° and 90°. We also find that the smallest value of L_REV (6 m) appears in the bisector direction between the joint sets S1 and S2 with larger joint lengths, whereas the largest value of L_REV (36 m) appears in the average orientation direction of the joint set S3 with a smaller joint length. This indicates that a larger joint length leads to a smaller value of L_REV. A comparison between the geometric and hydraulic REV indicates that the geometric REV size falls into the range of the hydraulic REV size.

However, the 2D model used in this research inevitably underestimates the connectivity and permeability of the realistic 3D fracture network system [1,48,49,50]. Moreover, aperture changes induced by stress variations are not within the scope of this work. Future research is recommended to utilize a 3D DFN model considering aperture changes to study fluid flow in fractured rock masses.