Discrete Fracture Network (DFN) as an Effective Tool to Study the Scale Effects of Rock Quality Designation Measurements

Abstract

Rock quality designation (RQD) is a parameter that describes rock mass quality in terms of percentage recovery of core pieces greater than 10 cm. The RQD represents a basic element of several classification systems. This paper studies scale effects for RQD measurements using synthetic rock masses generated using discrete fracture network (DFN) models. RQD measurements are performed for rock masses with varying fracture intensities and by changing the orientation of the simulated boreholes to account for orientation bias. The objective is to demonstrate the existence of a representative elementary length (REL, 1D analogue of a 3D representative elementary volume, or REV) above which RQD measurements would represent an average indicator of rock mass quality. For the synthetic rock masses, RQD measurements were calculated using the relationship proposed by Priest and Hudson and compared to the simulated RQD measurements along the boreholes. DFN models generated for a room-and-pillar mine using mapped field data were then used as an initial validation, and the conclusion of the study was further validated using the RQD calculation results directly obtained from the depth data collected at an iron cap deposit. The relationship between rock mass scale and assumed threshold length used to calculate RQD is also studied.

1. Introduction

Assigning numbers to geology requires a delicate balance between the commonly held opinion that geology cannot be quantified and an engineering approach that attempts to assign precise mathematical terms to every physical quantity [1]. Rock mass classification systems can be viewed as attempts to formalize this balance regarding the ratings of several rock mass parameters. For instance, the rock quality designation index (RQD) was developed by Deere [2] to provide a quantitative estimate of rock mass quality from drill core logs. RQD is defined as the cumulative length of intact core pieces longer than 10 cm as a percentage of the total length of the core (Figure 1); RQD ratings are given in

Table 1. Rock quality designation (RQD) [4].

	Condition	RQD
A.	Very Poor	0–25
B.	Poor	25–50
C.	Fair	50–75
D.	Good	75–90
E.	Excellent	90–100

. RQD represents a fundamental parameter in several rock mass classification systems, including the Rock Mass Rating (RMR) [3] and the Rock Quality Q index [4]. More recently, Hoek et al. [5] proposed a quantification of the original Geological Strength Index (GSI) [6] that incorporates RQD. The advantage of the RQD is mainly due to its simple definition; however, the RQD index is known to be sensitive to the relative orientation of the fractures concerning the orientation of the borehole (or scanline). Shen et al. [7] introduced a method to enhance the accuracy of RQD measurements using a four-step approach involving class ratio analysis and the Confidence Neutrosophic Number Cubic Value (CNNCV). This method was applied to an open-pit slope in China, demonstrating improved stability and reliability in RQD predictions compared to traditional single-model approaches. Sánchez et al. [8] showed how integrating geostatistical techniques with considerations for directional dependence and scale effects can improve the prediction of RQD. RQD values may also depend on the engineer’s experience logging the core and the ability to distinguish natural fractures from drill- and core handling-induced fractures.

Measured RQD values are sensitive to the length of the core run [9]. The same author recommended a core run no greater than 1.5 m. While such a limited core length may be helpful in identifying poor-quality zones, it is argued that it cannot represent the rock mass as a whole. When using RQD as an input parameter in rock mass classification systems, it may be helpful to use “variable core run lengths” to estimate a more representative RQD value for the entire geotechnical/structural domain under consideration. As discussed by Palmstron [10], most core logging is performed by measuring the joints along each meter of the core. This approach might introduce measurement errors if there are alternating sections with lower and higher densities of joints. The uncertainty in predicting the mechanical behavior of a naturally fractured mass is associated with scale effects [11]. Therefore, when using RQD or any classification systems incorporating RQD for rock mass classification, it is essential to acknowledge scale effects and their impact on rock engineering design, particularly when applying classification ratings across different problem scales [12]. Using variable core run lengths would allow us to better account for the transition from small-scale anisotropic rock mass behavior to large-scale isotropic (even though heavily jointed) rock masses.

Several researchers have questioned the use of RQD in rock classification systems (Pells et al., 2017 [13]; Yang et al., 2022 [14]). In this paper, we review the RQD method and take advantage of its limitations to understand how rock mass quality may be interpreted differently as the scale of the problems varies. We introduce the concept of representative elementary length (REL) as a 1D analogue to the 3D representative elementary volume (REV) for a naturally fractured rock mass when the problem at hand is transformed into a continuum problem.

By generating synthetic rock masses using a discrete fracture network approach (DFN), it is possible to characterize rock mass variability in terms of RQD and overall rock mass quality. The objective of this paper is to systematically study the influence of core run length on the measurement of RQD values, and a parametric study on core run length, fracture intensity, and orientation of the core run is conducted for two scenarios: (i) a conceptual synthetic rock mass and (ii) a synthetic rock mass generated using field data mapped in a room-and-pillar mine.

2. Discrete Fracture Network Models

DFNs are synthetic stochastic models of naturally fractured rock masses that can be generated using data collected from various sources (e.g., 1D boreholes and 2D rock exposures). High-quality drill coring can obtain relatively undisturbed rock core samples. In contrast, observations on exposed rock faces at or near the project site have the advantage of allowing direct measurements of discontinuity orientation, spacing, and fracture length.

DFN modeling has a broad range of applications in geomechanics due to its ability to accurately represent the complex geometry and connectivity of fracture networks within rock masses. Some key applications include slope stability analysis, rockburst and seismicity prediction in mining engineering, and tunnel design and stability and dam foundation analysis in civil engineering.

Elmo et al. [15] introduced a quantitative classification system incorporating a DFN approach to better capture rock mass scale and anisotropic effects and improve the reflection of the natural fracture network’s connectivity. Singh et al. [16] combined DFN and machine learning to improve the estimation of GSI. Ojeda al. [17] used DFN models to estimate volumetric fracture intensity from borehole data. Kang et al. [18] utilized a rectangular joint model to enhance the precision of simulating fracture geometries and their spatial distribution. This improved the accuracy of predicting rock mass behavior under various conditions. Khafagy et al. [19] developed a hybrid modeling method to capture the complex behavior of fluid flow and solute transport in fractured rock systems generated using DFNs in applications such as groundwater contamination. Other examples include the stability design of tunnels [20], open stope stability [21], and analysis of rock mass strength and fragmentation [22,23].

The typical process involved in the generation of a DFN model requires the definition of four primary fracture properties (

Table 2. Fracture data and derived input data for a DFN model from digitally and conventionally mapped data (modified from [25]).

Fracture Data	Source	DFN Input Data
Orientation	Boreholes, outcrops, tunnels	Orientation of fractures for every fracture set
Length	Tunnels, outcrops, lineaments	Fracture radius distribution
Terminations	Tunnels, outcrops, lineaments	Choice of the model hierarchy of the sets
Intensity	Boreholes, scanlines (P10), outcrops (P21)	Fracture intensity (P10 or P32)

), including (i) fracture orientation, (ii) fracture length, (iii) fracture terminations, and (iv) fracture intensity. The reader should refer to Elmo et al. [24] for a detailed discussion of the methodology required to generate calibrated DFN models.

Typically, in the DFN community, fracture intensity is expressed concerning a unified system of fracture intensity measures that provide an easy framework to move between differing scales and dimensions [26]. Fracture intensity is referred to as P_ij intensity, where the subscript i refers to the dimensions of the sample and subscript j refers to the dimensions of measurement. In DFN modeling, fractures are represented by circular discs or polygons with n sides. Fracture length is a critical DFN input and a key parameter for sensitivity studies, as it significantly influences block size and fracture connectivity [27].

3. Representative Elementary Volume

The concept of a representative elementary volume (REV) is a prime consideration in adopting a continuous-media methodology. It is of great importance in understanding the behavior of fractured rock masses. The definition of an REV is inherently related to scale effects, and an REV can be defined for various physical parameters. For example, Bear [28] defined the REV using the concept of porosity in rock masses according to the applicability of the theory of porous media (Figure 2). Using REV in continuum-based modeling also generates new generations of numerical simulation strategies, e.g., Xue et al. [29,30]. According to Hoek [31], rock mass strength will reach a constant value when the size of individual rock pieces is sufficiently small relative to the size of the problem being considered. Elmo and Stead [22] captured this phenomenon regarding the mechanical behavior of fractured pillars subjected to compressive loading, as shown in Figure 3. Similarly, it is reasonable to assume that a rock mass rating (e.g., RQD, RMR, Q or GSI) calculated for different problem scales would change, reaching a constant value with increasing problem scale (for a single geotechnical domain). As shown in Figure 4, as the problem scale increases, so does the sampling area used to calculate rock mass parameters. If a parameter (e.g., RQD) were to be measured concerning a relatively short core run or scanline, it would likely show large variations similar to those shown in Figure 2 for small-scale porosity.

4. A DFN Approach to Analyze Scale Effects in RQD Measurement

4.1. Conceptual DFN Model

Using the parameters listed in

Table 3. Parameters used to generate the conceptual rock mass.

Fracture Data	DFN Input Data
Fracture Orientation	Three near orthogonal fracture sets: dip/dip direction of 00°/000° (blue), 00°090° (grey), and 90°/000° (green), respectively. Fisher dispersion value of 80 for all sets.
Fracture Length	Log-normal distribution, mean: 3 m, standard deviation: 3 m. No truncation.
Fracture Terminations	0%
Fracture Intensity	Linear fracture intensity P10 intensity of 3, 5, 7, and 10 m−1 (for each of the three sets)

, a conceptual rock mass was generated. The model represents a homogeneous rock mass (i.e., a single geotechnical domain) with no faults or shear zones. An example of the DFN model is shown in Figure 5.

The model used for the simulation is cubic, with height, length, and width equal to 30 m. Three orthogonal boreholes passing through the center of boxes were defined to condition the fractures in the models to a given P10 value. The same boreholes were then used to measure the RQD for variable core run lengths, and another four wells along the diagonal line were defined to provide alternative measurements of RQD in each model (Figure 6).

This work generated eight models based on four classes of fracture density and two types of wells. Using a method initially proposed by Deere [2], the RQD in each well of different measuring lengths (ranging from 3 to 30 m) was calculated. The maximum and minimum RQD values were recorded for the study of the scale effect and estimation of REL.

Four off-center parallel wells were defined (Figure 7) to verify the representativeness of the centered wells used for RQD calculation.

According to

Table 4. RQD of center wells and surrounding wells.

Coordinate Axis		Average RQD of Four Surrounding Wells	Maximum RQD of Four Surrounding Wells	Minimum RQD of Four Surrounding Wells	RQD of Center Wells
	P10_3
	X	94.93%	96.06%	92.91%	93.55%
	Y	95.50%	96.29%	94.62%	94.33%
	Z	92.83%	93.49%	91.92%	94.57%
	P10_5
	X	86.77%	89.06%	84.94%	85.09%
	Y	88.09%	91.90%	84.36%	88.10%
	Z	82.23%	84.23%	80.14%	83.18%
	P10_7
	X	77.36%	79.59%	75.17%	73.70%
	Y	78.00%	81.90%	75.83%	77.80%
	Z	79.73%	82.18%	77.24%	76.30%
	P10_10
	X	63.75%	65.91%	61.67%	65.74%
	Y	69.68%	71.69%	67.48%	65.51%
	Z	59.27%	61.43%	55.29%	56.94%

, the difference between the RQD calculated using the center well and the average of the surrounding four wells is negligible (RQDs lie within a section of the same RQD Condition). Therefore, representativeness is guaranteed.

4.1.1. Core Run Lengths

For the three orthogonal and four inclined boreholes, core run lengths of 3 m, 5 m, 10 m, 15 m, and 30 m were used. Because of the stochastic nature of the DFN approach, each DFN model was generated five times, and the average RQD (for a selected core run length) for the five realizations was reported.

4.1.2. Conceptual Modeling Results

For each model with pre-defined fracture density, RQD data manipulations were performed separately for three orthogonal wells and four inclined wells. Taking the orthogonal wells as an example, maximum and minimum RQD values were recorded according to different core run lengths and an average of extremum was calculated using all three wells of the same core run length. As illustrated by Figure 8, Figure 9, Figure 10 and Figure 11, a merging trend of the maximum and minimum measured value with increasing core run length was depicted from all four models generated using different fracture densities.

For the inclined models, a similar merging trend of the maximum and minimum measured value was also depicted from all four models with different fracture densities. The trend is illustrated in Figure 12, Figure 13, Figure 14 and Figure 15.

The results for the inclined boreholes show a very similar pattern. For simplicity, the results are summarized in Figure 16 using 3D charts to show the relationships between the range of measured RQD values (for a selected core run length) and input fracture intensity P₁₀.

The following key observations and results can be highlighted:

• For every fracture intensity, the range of measured RQD values decreased with increasing core run length, and the RQD values converged to a constant value for a 30 m core run length.
• The above result is interesting, considering that the synthetic rock mass model used in the analysis represented a very homogeneous rock mass without pre-defined shear and fault zones. Despite that, for relatively short core run lengths (e.g., 3 m), the rock mass quality ranged from fair to excellent (P10 ≤ 5) and from poor to fair (P10 ≥ 5)
• The relative difference (max RQD minus min RQD) increased with increasing fracture intensity (for a selected core run length).

4.2. Room-and-Pillar DFN Model

The analysis was repeated using a DFN model from Elmo [32] using data collected at the Middleton mine (Derbyshire, UK). The mine is a classic square room-and-pillar mining operation with drift access, working mostly under a cover of about 100 m. Pillars are planned for a nominal 16 m × 16 m dimensions in a plan with rooms 14 m wide. However, completed pillars are usually smaller due to over-break. Because the rock mass quality for the Middleton mine is generally good to excellent (75 ≤ RQD ≤ 100), the original DFN model was modified by increasing the mapped intensity (the same increment was applied to all sets). All other parameters were unchanged. The objective was to simulate a poor to good rock mass (45 ≤ RQD ≤ 85) using real field data regarding fracture orientation, fracture length, and fracture terminations. The final FracMan input parameters for the DFN model in this research (see

Table 5. FracMan input parameters for the Middleton DFN model (modified from [32]).

Set	Orientation Distribution	Terminations	Radius Distribution	Volumetric Intensity
1a	Fisher Dip/Dip direction 89/308 K = 103.75	0%	Lognormal $\overline{y}=3$8.9 $y_{\sigma }$ = 9.0 Min. 2 m	4
1b	Fisher Dip/Dip direction 84/323 K = 20.75	23%	Lognormal $\overline{y}$ = 3.3 $y_{\sigma }$ = 0.6 Min. 0.25 m Max. 2.5 m	4
2a	Fisher Dip/Dip direction 86/219 K = 43	50%	Lognormal $\overline{y}$ = 3.2 $y_{\sigma }$ = 1.2 Min. 0.25 m	5.75
2b	Fisher Dip/Dip direction 89/269 K = 70.5	31%	Lognormal $\overline{y}$ = 3.2 $y_{\sigma }$ = 1.2 Min. 0.25 m	3.3
3a	Fisher Dip/Dip direction 46/193 K = 56	29%	Lognormal $\overline{y}$ = 3.7 $y_{\sigma }$ = 1.5 Min. 0.25 m	1.35
3b	Fisher Dip/Dip direction 44/016 K = 31.5	0%	Lognormal $\overline{y}$ = 3.7 $y_{\sigma }$ = 1.5 Min. 0.25 m	2.65

) were derived from the input used in Elmo’s thesis in 2006.

A 2.5-fold increase in the K value of orientation distribution (decrease dispersity) and a 5-fold increase in volumetric intensity were used. DFN fracture realization is given in Figure 17.

Modeling Results (Room-and-Pillar Mine)

RQD values were measured along three orthogonal boreholes and four inclined boreholes. Figure 18 and Figure 19 show the average RQD for orthogonal and inclined boreholes, respectively.

The results confirm the key observations made for the conceptual model. They also clearly show the existence of a representative elementary length (REL) above which RQD values assume a constant value for the rock mass. Like the previous case, the room-and-pillar DFN model represented a single homogeneous geotechnical domain without shear or fault zones. For both models, the REL was approximately 15 m (independent of the orientation of the core run). However, the definition of REL would also depend on the orientation of the core run, i.e., an REL of 10 m could be defined for the room-and-pillar DFN model concerning the orthogonal boreholes (Figure 18 vs. Figure 19).

4.3. Comparison with Empirical Formulae for the Determination of RQD

Priest and Hudson [33] defined a relationship between RQD and discontinuity (fracture) spacing measurements made on core or from surface exposures for different rock types (the majority of sedimentary rock types). The relationship is given in the Equation below:

RQD = 100 e^−0.1λ (0.1 λ + 1),

(1)

where λ represents the mean fracture frequency (equivalent to the linear P₁₀ intensity used in DFN modeling).

The results are presented in

Table 6. Comparison of simulated vs calculated RQD.

	Simulated λ
Orthogonal boreholes (OB)	4.0	6.4	9.2	13.3
Inclined boreholes (IB)	2.7	5.0	7.0	10.1
	Calculated RQD
Empirical Formulae (OB)	93.8	86.2	76.3	61.6
Empirical Formulae (IB)	96.9	90.9	84.3	73.1
Conceptual DFN (OB)	94.1	85.4	75.9	62.7
Conceptual DFN (IB)	91.8	77.6	66.7	48.8

Note: (i) In the DFN model, λ represents the number of fractures intersecting the unit length of the borehole. (ii) The simulated RQD results refer to a core length of 30 m. (iii) The simulated RQD results refer to input P₁₀ values of 3, 5, 7, and 10, respectively. Note that the input P₁₀ values differ from the measured λ, as the latter represents the number of generated fractures intersecting the borehole and not just the defined input frequency along that borehole for one of the three sets.

. There is a good agreement between simulated RQD values and those obtained using the equation for orthogonal boreholes. The difference between the simulated RQD values and those calculated using the equation for inclined boreholes could be explained considering the orientation of the fracture sets and the fact that the equation was primarily developed from data collected for sedimentary rock types. It is safe to assume that for sedimentary rock types, there would be a much better agreement between fracture frequency (spacing) and RQD along core orientations that are normal to the orientations of the main fracture sets.

5. Case Study of Iron Cap Deposit

The analysis was repeated using the depth data collected for an undisclosed mine location in Canada. The deposit extends approximately 1200 m SW-NE (along strike), 700 m NW-SE, and 700 m vertically and consists of strong, moderately fractured rock. The strength of the rock mass does not change significantly within the deposit, and thus, fracture frequency variation determines the rock quality.

In this case, DFN models were not generated, as fracture depth data were available from three boreholes (IC-10-014, IC-10-015, and IC-10-016), each of which has a length of more than 200 m. For each borehole, RQD calculations were made using depth data obtained from the boreholes using different core run lengths (3 m, 5 m, 10 m, 15 m, and 30 m, as for the conceptual model). The average of each core run length was calculated as the final calculation result.

The RQD calculation results directly obtained from the depth data are shown in Figure 20, Figure 21 and Figure 22.

Even though the rock mass quality had little range from good to excellent (75 ≤ RQD ≤ 100), the results also confirm the critical observations made for the DFN conceptual model and show the existence of a representative elementary length (REL) of approximately 30 m for borehole IC-10-014 and 15 m for boreholes IC-10-015 and IC-10-016.

6. The Relationship between Rock Mass Scale and Assumed Thresholds

The bias associated with the assumed threshold length for RQD could strongly influence RQD [32]. Hypothetically, a rock mass with a uniform fracture spacing of 9 cm would yield an RQD of 0%, while a rock mass with a uniform spacing of 11 cm would yield an RQD of 100%.

For example (Figure 23), let us assume a rock mass made of a collection of uniform bricks; in one direction, the dimension of the bricks is slightly longer than the RQD threshold value (10 cm), while a somewhat shorter dimension (less than 10 cm) is defined in a direction normal to the first one. The RQD would be either 0% or 100% for the rock mass under consideration depending on the sampling orientation.

The example above is hypothetical, but shows the limitations of arbitrarily using a cut-off threshold value. A threshold value of 10 cm is universally adopted in the assessment of RQD, and one significant limitation of the RQD definition is its dependency on the arbitrarily selected threshold length [35,36,37]. Thus, different thresholds of RQD determination were used for the study of scale effect and REL, and comparisons were made among different thresholds.

RQD values of different thresholds (20 cm and 30 cm) were measured for the Middleton mine along three orthogonal and four inclined boreholes. The average RQDs for orthogonal and inclined boreholes are given in Figure 24, Figure 25, Figure 26 and Figure 27, respectively. The results confirm the key observations made for the conceptual model, and the REL is approximately 15 m.

7. Conclusions and Recommendation

Numerical techniques such as DFN modeling simulate the discontinuity of rock mass at scales ranging from each rock block to the scale involved in the entire engineering structure. DFN code like FracMan allow for direct two- and three-dimensional modeling of the physical properties of rock mass with definable fracture length, spacing, direction, etc., and therefore provides a much more representative alternative than the empirical characterizations and rock mass parameter estimation approaches through core logging. RQD has proved helpful in rock engineering as it gives a rough representation of the actual quality of rock masses. Through RQD measurement, the quality of the rock masses is assessed with simple tools with low operating costs, but inevitable limitations. Although these limitations have been addressed, RQD is still directly and indirectly (a parameter in some rock mass rating systems) used in many geotechnical engineering applications without correction. This paper studies the importance of the data characterization process for RQD determination through different measuring lengths and directions. In this way, based on the synthetic rock mass of 30 m, the influence of measuring length and sampling direction is given. The definition of REL and REV would depend on the variation in geotechnical domains. Note that the current analysis results apply to relatively homogeneous geotechnical domains, as both shear and fault zones should be treated separately.

For the primary goal of the study of the scale effect, in this work, eight models were developed based on four classes of fracture density and two types of wells. RQD in each well of different measuring lengths was calculated using a method initially proposed by Deere in 1963 to determine the scale effects in RQD with varying lengths ranging from 3 to 30 m. For intervals of each defined length (i.e. 3 m, 5 m, 10 m and 15 m), a maximum, a minimum, and an average value of the calculated RQD were recorded. The results indicate that, with the increase in the measuring interval, the variance in the calculated RQD of the synthetic fractured rock mass linearly decreases. The size of the REL of the fractured rock mass in terms of RQD was assumed to be ten times the recommended core run length by Deere (1988) [9]. The estimated REL would represent the dimension at which the rock mass could be considered a continuum medium, and its properties would be defined using an equivalent continuum approach.

When comparing all the models, the largest difference in the measured RQD extremum, which was calculated at an interval of 3 m, exists in the rock mass with the highest fracture intensity. The measured extremum difference is most sensitive to changes in core run length, followed by changes in fracture intensity, and then fracture orientation due to changes in the well drilling direction.

Different thresholds of RQD determination were also used to study the scale effect. A comparison between rock mass scale and assumed threshold (10 cm vs. 20 cm vs. 30 cm) was made. The results show that greater REL occurs for longer thresholds.

RQD represents a limited part of rock mass quality, and some inherent limitations result from its simple measuring manipulation. According to Deere in 1988 [9], the calculation of the RQD should be based on the actual drilling-run length used in the field, but the laboratory core length is no more than 1.5 m. This methodology could serve as a reference for the quantification of RQD with a range and, thus, better predict the characteristics of the rock masses studied in engineering projects.

This study proposed a method to define fracture sets by exclusively using linear discontinuity frequency data. Due to the constant length of the inserted well, higher frequency results in more fractures, whereas dispersity remains unchanged. In future studies, the spacing and persistence of fractures can be defined by constructing models with areal or volume density. There is a need to extend the current work to DFN models with more complex fracture sets and to compare actual RQD measurements along boreholes to the simulated RQD values measured by the associated DFN models. Moreover, a pre-defined upper limit of the size (30 m side length cube) of the rock mass when studying the existence of the REV is quite different from most of the previous studies.

Based on the work carried out in this paper, the potential of DFN models for fragmentation determination and evaluation was confirmed. DFN modeling shows great applicability to RQD assessment and the study of the factors influencing the measured extremum. The parameters investigated here were limited to the most general cases (i.e., three orthogonal discontinuity sets) of ideally simplified rock mass. Confidence in the conclusions could be increased by assessing different model generation types, fracture sizes, distribution types, etc., particularly in models with non-orthogonal joint sets defined based on accurate data without modification. A mathematical relation between fracture intensity, measured interval, and range of RQD might be established by attestation of large amounts of RQD calculations obtained from laboratory core logging.

There may remain limitations in generating DFN models, since rock masses consisting of complex fracture sets require specific spatial generation models.