Scale Considerations and the Quantification of the Degree of Fracturing for Geological Strength Index (GSI) Assessments

Abstract

This paper provides research that shows that the scale and quantification of the degree of fracturing in a rock mass should and can be considered when estimating geological strength index (GSI) ratings for rock mass strength and deformability estimates. In support of this notion, a brief review is provided to demonstrate why it is imperative that scale is considered when using GSI in engineering design. The impact of scale and scale effects on the engineering response of a rock mass typically requires a definition of fracture intensity relative to the volume or size of rock mass under consideration and the relative scale of the project being built. In this research three volume scales are considered: the volume of a structural domain, a representative elemental REV, and unit volume. A theoretical framework is established that links these three volume scales together, how they are estimated, and how they relate to parameters used to estimate engineering behaviour. Analysis of data from several examples and case histories for real rock masses is presented that compares and validates the use of a new and innovative but practical method (a sphere of unit volume) to estimate fracture intensity parameters VFC or P30 (fractures/m³) and P32 (fracture area—m²/m³) that is included on the vertical axis of the volumetric V-GSI chart. The research demonstrates that the unit volume approach to calculating VFC and P32 used in the V-GSI system compares well with other methods of estimating these two parameters (e.g., discrete fracture network (DFN) modelling). The research also demonstrates the reliability of the VFC-correlated rating scale included on the vertical axis of the V-GSI chart for use in estimating first-order strength and deformability estimates for rock masses. This quantification does not negate or detract from geological logic implicit in the original graphical GSI chart.

1. Introduction

Empirically based rock mass classification systems like RMR, Q, and GSI have been used successfully for decades on projects around the world. This does not mean that these classification schemes do not have limitations. In the last 20 years, there has been significant effort to develop ‘so called’ quantified GSI charts.

Recent publications by Yang and Elmo (2022 [1], 2023 [2]) critique the quantification attempts of GSI, claiming that “attempts to quantify GSI rely upon parameters that are themselves not a quantity, and therefore it is not possible to quantify them. As a result, there cannot exist a quantified GSI or quantified GSI chart.” [1]. The authors also state “there is no need to quantify the original GSI table… since there is no apparent advantage in terms of design analysis of using any of the proposed quantification methods”.

The critique by [1,2] provided the impetus for additional research presented in this paper to show that the quantification of the degree of fracturing based on real measurements can and should be included on the vertical axis of the GSI chart.

The first part of this paper provides a reminder that the validity of the use of the Hoek–Brown (H-B) equations and relationships and the use of GSI is scale- and failure mechanism-controlled. The following are briefly reviewed in this regard:

• The historical development of scale effects in rock mechanics and why it is critical to the understanding of rock mass behaviour.
• The concept of a representative elementary volume (REV) and its relationship to the engineering behaviour of a given rock mass.
• Guidance provided in the literature that the use of GSI and the H-B strength criteria is scale- and failure mechanism-dependent.
• The development of quantified GSI charts with a focus on the volumetric GSI (V-GSI) chart proposed in Schlotfeldt and Carter (2018) [3].
• The quantification of the vertical axis on the V-GSI chart using true joint spacing on the vertical axis.
• The limitations and challenges of quantifying the horizontal axis of the V-GSI chart.

The second part of this paper presents research undertaken using data from three real rock engineering projects that provides validation that VFC (P30) and P32 parameters used in the V-GSI chart are based on measured data and the vertical axis is in fact quantifiable.

2. Scale and Scale Effects in Rock Engineering and the Use of GSI

This section provides a brief summary that describes the importance of scale and scale effects as these are critical to understanding the reasons why attempts have been made to add quantitative measures to the GSI chart.

Scale effects in rock engineering were first presented in the 1993 book called ‘Scale Effects in Rock Masses 93’ (Cunha, 1993 [4]) where it was put forward that “rock masses are basically inhomogeneous and discontinuous media”. The determination of rock, joint, and rock mass properties always involves uncertainties due to the variability of the rock material and the fracture networks. The results of laboratory and in situ tests are thus affected by both the chosen testing points and the volumes involved in the tests. The variation in the test results with the specimen size is often referred to as “Scale Effects”. The authors of [4] also asserted that an understanding of “scale effects” is required when undertaking deformability, strength, and hydraulic properties estimates, and understanding the impact of internal stresses in rock masses. At the same time the concept of representative elementary volume (REV) was also introduced in [4] and further elucidated by Oda, 1993 [5], amongst others, and this is also a key concept that to this day forms the core of the characterization and strength and deformability assessment of rock masses. The REV concept in terms of engineering behaviour is shown in Figure 1.

Priest and Hudson, 1981 [7]; Shah, 1992 [8]; Hoek, 1994 [9]; Schultz, 1996 [10]; Hoek and Brown, 1997 [11]; and Hudson and Harison, 1997 [12], (to name a few) reiterated that rock mass behaviour, and the appropriate choices of strength criteria, are scale-dependent, and this in turn can be attributed to the presence of flaws and fractures in a rock mass. In a geological framework, the authors of [10] provided the following guidance: relative scale in the context of rock masses can be defined as the “ratio or the scale of observation to the scale of fracturing” or the “dimensions at which the scale of observation greatly exceeds the block sizes or fracture spacing”. A rock unit or structural domain can be described as a rock mass when these scale criteria are met. In terms of engineering response [12], this means distinguishing between rock properties at a point and rock properties over a volume. Point properties such as density, primary porosity, intact rock permeability, and point load strength may vary spatially for a particular rock mass, but these properties are not dependent on discontinuities. On the other hand, volume properties such as the modulus of deformation, rock mass shear and tensile strength, secondary porosity, permeability, and fracture intensities of a rock mass are dependent on the presence of discontinuities and their spatial distribution within a rock mass.

In recognition of the importance of understanding geological structure in rock engineering, Marinos and Hoek, 2000 [13], and Hoek and Brown, 1997 [11], presented the graphical GSI chart that tried to capture geological structure (via caricatures and descriptions) and discontinuity condition descriptors and linked these to GSI ratings and to the H-B failure criteria equations. The aim of this paper is not to provide an extensive review of the history and development of the H-B equations or the GSI empirical system. Rather what is important to this paper, other than a reminder of the 40-year history of the development of H-B equations and 30-year history of GSI, is that GSI should not be used without understanding scale and scale effects.

Initially Shah, 1992 [8], and then Hoek, 1994 [9], followed by Hoek and Brown, 1997 [11], and many other authors subsequently recognized that the validity of the use of the H-B relationships and the use of GSI is scale- and failure mechanism-controlled. While there is no explicit scale provided in the original graphical GSI chart (Marinos and Hoek, 2000 [13]), the use of the chart was predicated on the understanding of Figure 2.

In simplistic terms it is clear from Figure 2 that if joint spacing and consequently block sizes are near or larger than the dimension of interest (e.g., tunnel span, cut slope height, or foundation dimension) then GSI should not be used or should be used with caution. If, on the other hand, the block sizes are predominantly smaller than the dimension of interest, then it is reasonable to use GSI and the H-B equations with some caveats. Along the same theme Shah, 1992 [8], discussed the limitations of the use of the H-B criterion and GSI, recommending in summary that if the span or width of an excavation is only 3× the spacing of the discontinuities in the rock mass—then another failure criterion should be used.

It is implicit in Figure 2 and from Shah’s 3× factor that in design work and in construction in or on rock, in addition to the number joint sets, the controlling engineering scale (e.g., tunnel span, cut slope height, or foundation dimension) should be compared to the characteristic degree of rock fracturing present. This requires some knowledge of the spacing and persistence of the joint sets present, which in turn should provide some indication of block sizes.

The problem with the scale criteria discussed above with respect to the original graphical GSI chart (Figure 3) is that there is no scale on the vertical structure caricatures, and it is not always intuitive to judge or understand the structure caricatures relative to engineering scale of interest. There is also no definitive process to guide this process. It is left to the user to use engineering judgement on the scale of caricatures in geological terms and the relative scale of a project. This has the potential to introduce errors and uncertainty into strength and deformability estimates.

Understanding the above concerns and understanding the impact of scale and scale effects as summarized above provided the motivation for many authors to attempt quantifying, in particular, the vertical axis of the GSI chart. Figure 4 below provides a summary of some of the efforts up to 2013, remembering that the GSI system provides the most direct link to the H-B equations.

This research contends that if measurement of parameters like joint spacing and joint persistence are measured and used to estimate volumetric fracture counts then it is possible to include scale-based parameters on the vertical axis of the chart. Research efforts (aiming at so-called quantification on the vertical axis) to date are not intended to improve the accuracy of GSI ratings but rather simply to have a better handle on scale when using GSI and the H-B equations.

3. Theoretical Volume Considerations in Assessing P30 and P32 and Rock Strength and Deformability

The impact of scale and scale effects on the engineering response of rock mass typically requires a definition of fracture intensity relative to the volume or size of the rock mass under consideration (LaPointe et al., 1993) [22]). In this research three volume scales are considered: the volume of a structural domain, the volume of an REV, and unit volume. This section provides a brief theoretical framework that links these three volume scales together, how they used to assess and quantify P30 and P32, and how these parameters can be used to estimate engineering behaviour when using GSI.

Parameters such P30 and P32 are frequently used as inputs to determine the engineering response of a rock mass. As shown in Figure 5, P30 and P32 are defined in terms of number of fractures per unit volume and area of fractures per unit volume, respectively. Other parameters are defined in Figure 5 and are included for completeness but are not explicitly considered in this section.

3.1. Structural Domains

A structural domain in a rock mass can have any shape or size depending on the geological environment, e.g., structural domains may be fault bounded, lithology may be bounded or may result from increased fracture intensity due to stress changes with valley incision—to name a few. Depending on the geological environment, it is not always possible to define the volume of a domain. Depending on the scale of the project, several structural domains may be delineated in terms of their engineering behaviour without having an exact volume per domain. Delineation should include an understanding of geological history, the lithologies present, and some understanding of the regional stress history. Once the regional geology is understood, field work is typically required (e.g., outcrop mapping, borehole drilling, and laboratory testing) to delineate structural domains in terms of engineering response.

In more detail, to delineate and differentiate a structural domain, quantitative and qualitative parameters relevant to both the rock mass and the discontinuity surface conditions in a domain should be systematically mapped and assessed for a rock mass (Schlotfeldt, 1999 [23], and Schlotfeldt and Carter, 2018 [3]). In broad terms this should include as a minimum the following data:

• The degree of rock mass weathering and intact uniaxial compressive strengths;
• The orientations of major discontinuity sets, joint spacings for individual sets, continuity, or the persistence estimates for individual sets;
• The sidewall separation, discontinuity condition, roughness, and waviness per set.

These data should be statistically characterized to differentiate key geotechnical properties of a rock masses in adjacent or delineated structural domains at the scale of a project. While it may be an iterative process to determine the physical limits of the boundaries between structural domains, a particular domain should be relatively homogenous (in terms of the parameters defined above) within a domain but statistically different to the rock mass of a domain that may be above or below or adjacent to the domain in question, as per the authors of [3,23]. Within a delineated domain, variability will exist in terms of measured or observed parameters but when the mean or modal values or central tendencies of the data are understood and are shown to be different (not all parameters need to be different) to an adjacent structural domain, then the domain can be considered a structural domain in terms of its potential engineering behaviour.

3.2. Representative Elemental Volume (REV)

Once boundaries of a structural domain are broadly established, it becomes easier to define a representative elemental volume (REV). There is no strict protocol or clear definition in the literature on how to determine an exact volume of a REV. Neverthelss, provided the REV fits within the structural domain, it can be a cube or sphere or another shape. An REV should be the minimum volume of rock mass within a structural domain that is statistically homogeneous beyond which any sub-domains behave (in terms of strength and deformability) essentially like the whole rock mass within a structural domain ([3,5]). This concept is shown in Figure 1 in terms of engineering behaviour.

In this research P30 and P32 for an REV are established via a structural domain approach first, followed by stepping down to the minimum volume that fits in a field-determined structural domain provided P30 and P32 are relatively homogenous within both the domain and the REV of a given domain. An REV established in this manner needs to be differentiated from a computer modelling discrete facture network (DFN) approach. In both the field-determined REV and DFN approaches, estimates of P30 and P32 require a definition of the total volume under consideration. In discrete fracture network (DFN) modelling, a block volume may include a large block that can be divided into several sample-size blocks that are generated in the digital computer space. The volume of smaller sample-size blocks within the large modelling block may or may not be the same volume established in the field-assessed REV approach described above. The results in both approaches are reported in the same units as in Figure 5 and

Table 1. Nomenclature and units used to define P30 (VFC) and P32 using field REV approach, DFN approach, or unit volume approach.

Nomenclature	Definition
P30REV	Total fracture count (No.) per total volume (Number/m3) of a field established rock mass REV. Structural Domain > REV > Unit Volume
P30DFN	Fracture count (No.) per total model block volume (m3) in a computer derived DFN model
P30UNIT VOLUME or VFCUNIT VOLUME	Fracture count (No.) per unit volume (1 m3 sphere) using the new method of estimation included on the vertical axis of the V-GSI chart [3]
P32REV	Total area of fractures (m2) per total volume (m3) of a field established REV. Structural Domain > REV > Unit Volume
P32DFN	Area of fractures (m2) per total model block volume (m3) in a computer-derived DFN model
P32UNIT VOLUME	Area of fractures (m2) per unit volume (1 m3 sphere) using the new method of estimation included on the vertical axis of the V-GSI chart [3]

.

P30_REV and P32_REV is the nomenclature adopted in this paper (see Table 1) to describe the fracture size and intensity within a field-assessed REV. P30_REV is essentially the estimated number of fractures divided by the volume of the REV. P32_REV is the estimated fracture area divided by the volume of the field-assessed REV in question.

P30_DFN and P32_DFN is the nomenclature adopted here (Table 1) to describe the fracture size and intensity for an assumed block volume generated in computer space in programs such as Fracman (WSP proprietary software Version 8.0, February 2021).

3.3. Unit Volume

In the quantified V-GSI chart (Figure 6 and discussed in more detail Section 4), a unit volume is assumed to be a sphere with a radius of 0.62 m. VFC (equivalent to P30) and P32 included on the vertical axis of the V-GSI chart are estimated via a new method that only uses ‘true’ spacing data to estimate VFC. It is assumed in the V-GSI approach that joints sizes or persistence are comparable to a REV dimension and therefore are larger than the diameter of unit volume sphere (1 m³) and fully intersect the unit sphere.

P30_{UNIT VOLUME} and P32_{UNIT VOLUME} is the nomenclature adopted here (Table 1) to describe the results obtained using V-GSI method of quantification. The equations and method of quantification of P30_{UNIT VOLUME} and P32_{UNIT VOLUME} are fully described in [3] and are summarized in Section 4 below.

In the V-GSI system, the ratio of P32_{unit volume}/P30_{unit volume} (which results in m²/number fractures) gives the surface area of a single fracture intersecting a unit sphere volume of rock on the assumption that the single fracture has a persistence or size greater than the diameter of a unit volume sphere or cube of rock mass. It is also implicit in this that as the value of P32_{unit volume} and P30_{unit volume} increases, so does the summed fracture area with an increasing degree of blockiness. In the V-GSI chart (Figure 6), the ratio simply implies that one fracture with greater persistence than the diameter of a unit volume that intersects a unit volume of rock mass will have an area close to $\pi r^2$ (approximately 1.3 m²) for an assumed sphere with a radius of 0.62 m. This does not mean that the fracture area is only 1.3 m², it is only the area of intersection of a fracture in the unit sphere and, as described above, the surface area can increase up to the REV or up to the size of the structural domain.

As an additional point of clarification, the parameter VFC (Figure 6) in the V-GSI chart is used interchangeably with P30_{unit volume} in this paper and both have the same definition and method of quantification (summarized in Table 1).

The research presented below shows that the same or similar results for P30 and P32 are obtained using an REV-defined volume, DFN block volume, or a V-GSI unit volume sphere approach, with some limitations. This provides validation that the vertical axis of the V-GSI chart can be quantified using real field measurements.

4. Attempts to Quantify GSI with an Explicit Focus on the Vertical Axis of the V-GSI Chart

Between 1997 and up to 2022 there were multiple efforts to modify, adjust, and quantify the GSI charts and the associated H-B failure criteria equations. The purpose of the research presented in this paper is not to review and compare these contributions in any detail. Readers who are interested in the various charts and associated quantification equations are referred to the authors of [1], who provide an excellent summary in this regard. Figure 4 nevertheless provides a visual summary of some of the attempts at quantification up to 2013.

4.1. V-GSI and VFC (P30_{UNI VOLUME}) and P32_{UNIT VOLUME} Estimates—A Brief Review and Discussion in the Context of the Research Presented in This Paper

A more recent attempt at quantifying the GSI chart was published by [3]. Figure 6 shows the V-GSI chart. The next sections present recent research that advances the understanding, validation, and limits of applicability of the V-GSI system that incorporates the degree of fracturing based on real measurements.

It should be noted that while the focus in this paper is on the validity of V-GSI approach, this does not mean that other quantified approaches (summaries provided in [1] and Figure 4) do not have merit or are not valid or less valid. The focus here is to show via research on real projects that scale is better considered in the V-GSI system (Figure 6) than in the original graphical chart introduced by [13] and shown in Figure 3.

In general, it is recognized that GSI and the H-B equation only provide first-order estimates of rock mass strength and deformability, and geo-mechanical modelling is typically required for engineering design. The inclusion of P30 and P32 on vertical axis of the V-GSI chart does not change this limitation but at least scale is considered.

It is acknowledged that the rating scales on both the vertical and horizontal axes of the V-GSI chart (Figure 6) are not something that can be measured directly. In this regard, [3] used results from large-scale in situ deformability testing for the foundations of the 185 m high Katse Arch Dam in southern Africa to validate the vertical axis rating scale that uses scaled VFC (P30_{UNIT VOLUME}) and P32_{UNIT VOLUME} parameters as measurements of fracture intensity. Based on ‘good’ correlation between predicted strength and deformability estimates using the V-GSI chart and H-B equations and the in situ test results, this provided validation of the vertical rating scale.

While not discussed explicitly in this paper, Schlotfeldt et al., 2022 [24], showed that predicted deformation results obtained using 3-D RS3 modelling (finite element modelling software by Rocscience—Version 4.03) that used V-GSI ratings as inputs compared well with deformation monitoring results in the crown of a tunnel during construction in Sydney, Australia. This further validates the use of scaled VFC (P30_{UNIT VOLUME}) and P32_{UNIT VOLUME} parameters on the vertical axis of the V-GSI chart.

4.2. Key Equations to Quantify VFC (P30) and P32 Estimates

Provided that potential failure mechanisms associated with longer joints or faults are understood and considered in the geological model, it is contended that scaled VFC (P30_{UNIT VOLUME}) and P32_{UNIT VOLUME} parameters as used in the V-GSI chart do have their place in engineering design.

Equation (1) can be used to estimate VFC contributions for individual joint sets in a rock mass. This is calculated as the inverse of the mean ‘true’ spacings (Sti) for each joint set present in a rock mass

$${VFC}_i\left(contribution\right)={\displaystyle \frac{1}{\mathsf{\nabla }S{t}_i}}$$

(1)

and the ‘orientation’ bias-free VFC for a rock mass is then simply the sum of the individual VFC contributions for each discontinuity set present in the rock mass, but, in general, is calculated with the use of Equation (2), irrespective of the number of ith discontinuity sets (NDS)

$$VFC=\sum _{i=1}^{i=NDS}\left({\displaystyle \frac{1}{\mathsf{\nabla }S{t}_i}}\right)$$

(2)

where $\mathsf{\nabla }S{t}_i$ implies that the distribution of the joint spacing data is log normal or exponential and therefore geometric means are used to calculate Sti.

VFC, which requires correction of ‘apparent’ spacing data to ‘true’ spacing data (described in Equation (5)) per set, can also provide reasonable estimates of P32_{UNIT VOLUME} using Equation (3)

$$P30\left(VFC\right)\to P32={\sum }_{i=1}^{i=NDS}{\displaystyle \frac{1}{{St}_i}}{\ast \pi r^2}_i={\sum }_{i=1}^{i=NDS}VFCi{\ast \pi r^2}_i$$

(3)

The 95% confidence limits on the mean VFC can also be calculated using Equation (4)

$${VFC}_{limits}=\sum _{i=1}^{i=NDS}\left({\displaystyle \frac{1}{\overline{S}{t}_i\pm 1.96\times \frac{\widehat{\sigma }S{t}_i}{\sqrt{n}}}}\right)$$

(4)

where n is the number of ‘true’ spacing observations recorded for the ith set, respectively.

One of the other advantages of the VFC parameter is that it does not require the running of advanced 3D software to stochastically generate discrete fracture networks (DFNs) for rock masses to determine volumetric fracture counts (VFCs or P30) or P32.

Field-collected apparent joint set spacing (Sa) data is corrected to true spacing (St) data, using real or virtual measurement scanline vectors on actual exposed faces or on scaled underground maps, outcrop maps, or foundation maps. St data can be used to estimate reliable ‘orientation’ bias-free mean VFCs and the associated confidence limits. These in turn can then be plotted on the vertical axis of V-GSI chart. The VFC mean and limits are then linked to correlated ratings on the vertical axis [3] (Figure 6). Once the joint conditions are assessed the overall V-GSI ratings can be assessed and H–B strength and deformability parameters can be determined.

As a reminder, St data require the use of a correction factor (CF) and the calculation of CFs is possible because the respective angular relationships between, e.g., a tunnel face, a rock cut, or foundation surface, and measurement scanlines (which are vectors when directions are measured) and discontinuities are recorded by the mapping process. Spacing correction factor (CF) per discontinuity set can be calculated with the use of Equation (5) [3]

$$CF=\left(\sin {\alpha }_n\times \cos {\beta }_n\right)\times \left(\sin {\alpha }_s\times \cos {\beta }_s\right)+\left(\cos {\alpha }_n\times \cos {\beta }_n\right)\times \left(\cos {\alpha }_s\times \cos {\beta }_s\right)+\left(\sin {\beta }_n\times \sin {\beta }_s\right)$$

(5)

CF is always the absolute value, and if <0.1, then 0.1 is set as the default value,

where

• αd = the discontinuity set mean dip direction (used to convert to pole trend);
• βd = the discontinuity set mean dip angle (used to convert to pole plunge);
• αn = the trend of pole or normal-to-discontinuity set mean dip direction;
• βn = the plunge of pole or normal-to-discontinuity set mean dip angle;
• αs = the strike or trend of scanline or measurement vector (azimuth 0–360°) on, e.g., an inclined foundation surface, tunnel heading, or sidewall;
• βs = the dip or plunge of the scanline or measurement vector (can be anywhere between 0°, horizontal, and 90°, vertical) on, e.g., an inclined foundation surface, tunnel heading, or sidewall.

Readers interested in regression line equations for VFC ratings and VFC parameters developed for the foundation rock mass of the Katse Arch Dam are referred to [3]. The rating on the vertical axis of the V-GSI chart can be calculated with Equation (6).

$$\mathrm{V}\mathrm{F}\mathrm{C}\mathrm{R}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}=50-8.5\mathrm{L}\mathrm{N}\left(\mathrm{V}\mathrm{F}\mathrm{C}\right)$$

(6)

The overall V-GSI rating can then be estimated with the use of Equation (7).

$$V-GSIRating=1.5JCond89-8.5LN\left(VFC\right)$$

(7)

4.3. Maintaining a Geological Perspective

While Marinos and Hoek [13] are clear that the purpose of the graphical GSI system is to maintain a geological perspective that is easy to use, they do not seem to subscribe to the philosophy that no attempts should be made to improve the GSI chart or that a quantified chart can never exist, as per the authors of [1]. To quote Marinos, Marinos, and Hoek, 2007 [25], “The GSI system has considerable potential for use in rock engineering because it permits many characteristics of a rock mass to be quantified, thereby enhancing geological logic and reducing engineering uncertainty. Its use allows the influence of variables, which make up a rock mass, to be assessed and thus the behaviour of rock masses to be explained more clearly.” While this quote is somewhat ambiguous, it certainly does not advocate that no improvements or research that involves the quantification of the GSI chart should be attempted.

It is also the perspective of the authors of this paper that introducing scale on the vertical axis of the V-GSI chart does not negate or detract from the geological logic implicit in the original graphic chart. An understanding of geology and the development of a geological model should always be the first priority for any project on or in rock when using the empirical GSI system and the empirical H-B failure criteria, i.e., the use of GSI and the H-B equation derivatives “are not intended to replace and indeed, depend for their successful application on, careful thought, detailed appraisals of site geology, carefully conducted site characterization and laboratory test programmes, and detailed evaluation of the mechanics of engineering problems being addressed” (Brown, 2008 [26]). It is argued that even Dr. Hoek (one of the main proponents of modern rock mechanics) and one of the primary authors of the GSI system recognized that a sense of scale on the vertical axis is a useful addition to the original graphical GSI chart. For example, ref. [21] used rock quality designation (RQD) to provide scale on the vertical axis of the GSI chart because they recognized that it provided easily quantifiable estimates of block size. While the choice of RQD as a parameter to measure the scale or degree of blockiness is debatable (for many reasons, such as orientation bias and scale limitations), it nevertheless shows that there was recognition globally that an understanding of scale and scale effects is a necessity and that this can and should be incorporated in the GSI system.

5. Example of GSI Assessment: Graphical GSI Chart and V-GSI Chart (Without and with Scale)

To demonstrate the potential impact not explicitly understanding scale can have on a GSI assessment, this section is dedicated to a comparison of a purely qualitative (graphic) assessment of GSI (Figure 3) versus an assessment of V-GSI (Figure 6) that considers the scale of the problem.

Figure 7 and Figure 8 show that the visual assessment of a rock mass’s structural condition using the Marinos Hoek [13] graphical chart without scale can lead to GSI rating estimates that may not be correct.

Figure 7 shows a 2D DFN representation of a vertical rock face (LHS -Figure 7). A graphical GSI chart assessment without any scale is shown Figure 7 (RHS). The DFN simulation shown is from a real rock mass project (Schlotfeldt, Elmo, Panton, 2018 [27]). A photograph of the rock mass in question is also shown adjacent to the DFN block in Figure 7 to help with the visualization of the DFN model. Without knowledge of scale (the size of the blocks, the number of joint sets, the spacing of the joint sets, or the persistence of the joint sets) and by visual comparison with the graphical representation of the rock mass structure divisions, it is a reasonable assumption that the rock mass appears to be somewhere between “Very Blocky” to “Blocky” on the vertical axis as shown in Figure 7. It is known (from extensive geotechnical mapping of this outcrop) that for this rock mass the joint surface conditions are generally “Very Good” as plotted on the horizontal axis in Figure 7. This translates to a GSI rating of around 60–70 and a mean value of say 65, also shown on the GSI chart.

In Figure 8 the scale of the DFN block is shown along with the same photograph as the DFN block, as shown in Figure 7. The DFN block shown is in fact 40 m in width and 71.5 m high. The detailed mapping and statistical characterization of the measured data provided the number of joint sets along with the mean true joint spacings per set (Sti), and these are shown in Figure 8. It is also known from field mapping that sets Jo through J4 generally had measured persistences greater than 10–20 m. An ‘orientation’ bias-free VFC parameter of close to 1.9 fractures /m³ was estimated using Equations (1), (2) and (5), and this translates to a VFC scale rating of around 46 (Figure 8—RHS). Using Equation (3) it is also a reasonable estimate that P32 is likely to be just greater than 2 m²/m³ for the same rating of 46.

The same joint surface conditions apply as described for Figure 7, and this is plotted on the horizontal axis in Figure 8 and translates to a horizontal axis rating of around 40, resulting in combined V-GSI rating estimates of around 85, approximately. Equation (7) gives the same answer.

This is a scale-driven (on the vertical axis) assessment that is different to the graphical chart assessment without true knowledge of scale. Without knowledge of the scale of assessment and the spacing and persistence of individual joint sets, it would not have been possible to understand that the rock mass in question is ‘Blocky at the Scale of Interest’ as plotted on Figure 8. Figure 8 only shows the mean VFC but, with the use of Equation (4), VFC limits can be estimated and plotted on the chart.

Figure 9 shows the difference in the rock mass strength and deformability estimates for these two assessments (graphic GSI chart, lower curve, and V-GSI chart, upper curve) generated with RSData by Rocscience. Clearly the graphical assessment of GSI without scale can be misconstrued, and this misjudgment can be made by either a geologist or an engineer when scale is not fully appreciated.

The takeaways from this example that can have design implication are as follows:

• With reference to Figure 9, judging scale based on structure caricatures alone can lead to strength and deformability estimates that are conservative; e.g., the tensile strength estimate using the scaled V-GSI rating is a factor of around 4.5 times stronger than the estimate using the graphic chart estimate.
• The modulus of the rock mass from the scaled V-GSI rating (Figure 9) is around a factor of 1.5 times greater than the graphic GSI modulus estimate. Depending on the project, use of the graphic GSI chart strength and deformability estimates could lead to significantly higher support level requirements to ensure global stability than if the V-GSI estimates are adopted. The input and output generated with RSData are shown in Figure 9 (RHS), while the strength curves with increasing stress are plotted on the LHS.

While a conservative design is better than the opposite, the reverse is also possible if scale is not considered when assessing GSI ratings without considering scale:

• For example, if the graphically estimated GSI for a rock mass is considered to be Blocky or Massive with very good to good joint surface conditions (a GSI of around 70–75) without considering scale, this too can result in overly optimistic strength and deformability estimates compared to a very blocky rock mass with a VFC of say 15–20 fractures/m³ with a V-GSI of 60 to 65.
• If the graphical non-scaled GSI range is used for design, this can result in deformation and/or rock mass failures compared to the scaled V-GSI estimates. This does not mean the purpose of including scale improves accuracy, rather it can provide a better-calibrated assessment of the GSI ratings on the chart and therefore more reliable strength and deformability estimates that consider scale.

Readers interested in other examples that compare GSI ratings using graphical charts and quantified charts are referred to Bertuzzi, Douglas, and Mostyn (2016) [28].

6. A Discussion of the Qualitative Aspects (Joint Conditions) of Existing GSI Charts

For completeness, a discussion of the horizontal axis of the V-GSI chart is provided in this section of the paper given that this also formed of part of the general criticism of quantified GSI charts by [1], as discussed in the introduction.

Descriptions or qualitative assessments for Q or RMR classification systems, and used in most GSI charts, including the V-GSI chart to assess discontinuity conditions, is not true quantification. With respect to GSI, this is because to date there has been no other easy way to improve or quantify discontinuity conditions on the horizontal axis. An assessment is nonetheless a necessity—descriptive or otherwise. In the context of GSI, none of the charts to date address the real quantification of the horizontal axis.

It is implicit in the original graphical chart (Figure 3) that GSI ratings on the horizontal axis decrease as the surface quality decreases by around 15 points per surface quality descriptor, depending to a limited degree on which structure caricature is selected on the vertical axis. In GSI charts that use 1.5 JCond89 [3,21] (see Figure 6), on the horizontal axis, ratings generally decrease by around 10 points for each rightward shift in surface quality descriptor for a given structure. Elmo and Yang revised the GSI chart [1] (provided as Figure 8, pg. 14 in ref. [1]) chart, which essentially retains the same shift of around 10 points on the horizontal axis for each shift in the surface condition descriptors. there is a.

As presented by the authors of [3], it is “recommended that, where possible, discontinuity surface conditions should be established first independently using descriptors that fit the observed discontinuity conditions for a particular project or geological environment and that the central tendency or modal frequency of observation bins or measurements be established first”. This is because, if, for example, the JCond89 system alone is used, this can result in the misinterpretation of natural trends in rock masses—rather where possible, quantitative measurements and qualitative observation should be recorded and statistically characterized fist to determine representative joint conditions—per discontinuity set preferably for a given structural domain. Assessed and representative conditions can then be fitted into JCond89 bins (where possible), and an overall rating can be obtained for JCond89 as used in the V-GSI chart.”

An example of this type of characterization that can be undertaken to establish modal frequency or central tendencies for discontinuity observation and measurement is shown in Figure 10, noting that the parameters shown are not the only parameters that can or should be assessed.

The next sections of this paper provide two case histories and some updates that demonstrate that the methods used to quantify the vertical axis on the V-GSI chart using joint spacing data and joint persistence data do provide reliable P30 and P32 estimates, which in turn can provide reasonable strength and deformability estimates for a rock mass.

7. Canal Route in Rock, Italy: Comparisons of P32 Estimates Using a New Method by Morelli (2024) [29], DFN Modelling, and the V-GSI Method

In a recent paper by [29], summary data were provided from an excavation in rock for a canal route. Morelli states that the data were collected over a stretch of about 50 m, where the rock mass was found to be fairly homogeneous—essentially in the same structural domain. According to [29], discontinuity data were sampled along 29 scanlines (16 longitudinal and 13 transverse to the canal axis), with lengths in the range 7–18 m.

Table 2. Summary discontinuity data modified from Morelli, 2024 [29].

Set	Avg. Dip (deg)	Avg. Dipdir (deg)	k-Fisher	True Spacing (m)	Size/dia. (m)
S1	41	028	23.7	LogNorm (0.93; 3.46)	Neg Exp (0.87)
S2	67	322	30.2	LogNorm (2.71; 8.72)	Neg Exp (0.73)
S3	48	230	19.3	LogNorm (1.03; 1.83)	Neg Exp (0.64)
S4	76	115	56.3	Neg Exp (4.98)	Neg Exp (0.45)

Morelli confirmed (pers comms, 2024) that the mean ‘true’ spacings per set (Sti) come from Equations (1), (2), and (5), respectively.

(provided as Table 4 in ref. [29]) provides summary statistics from the field data that identifies four major discontinuity sets and includes the mean true spacings and associated standard deviations per set, and the mean size or diameters for each set, and the distribution type.

In terms of size of the fractures of the different sets, Morelli used the methods developed by Priest, 1993 [30] and 2004 [31], to establish the distribution form and the mean size of the discontinuity planes per set, assumed as circular discs, from trace lengths obtained by scanline sampling. Mapped semi-trace lengths produced by their intersection with a scanline were corrected for sampling biases, including “trimming” or “truncation” (i.e., traces below a minimum length not observed or measured during surveys) and “curtailment” or “censoring”. The sampled REV considered by Morelli was around 8000 m³ (Morelli, pers comms). Using these data summarized in Table 2, ref. [29] estimated the P32_REV parameter with his new analytic–stochastic methodology. His method uses the true spacings and a persistence factor per set to estimate the P32_REV for non-persistent discontinuities. DFN modelling was also undertaken by Morelli using the data in Table 2 to validate his new method, resulting in a P32_DFN estimate.

It is not the intention to repeat the details of Morelli’s new method of estimating P32_REV, rather his results are used here to check if the quantification method used as a scaled parameter on the vertical axis of the V-GSI chart provides comparable P30_{UNIT VOLUME} and therefore P32_{UNIT VOLUME} estimates compared to the results of his detailed analysis and DFN modelling of the example rock mass on a real project.

In summary, Morelli provided an average estimate for P32_REV of 2.46 m⁻¹ for the given rock mass in his project using his new method (Table 2). Morelli used that same data (Table 2) to undertake DFN modelling (using a generation box with an assumed REV of 8000 m³), and this provided an estimated P32_DFN of 2.65 m⁻¹. As Morelli states—this shows a good fit between the results obtained by DFN modelling and the new method using field measured data. Morelli also quantified P32 using Chilès’s methodology (Chilès et al., 2008 [32]), and this showed a significantly higher mean P32 of 8.87 m⁻¹. Figure 11 below shows these results plotted on the scaled vertical axis of the V-GSI chart (red arrows and dashed lines for Morelli’s results and a blue arrow and dashed line for the Chilès method).

7.1. Comparison of P32_{UNIT VOLUME} Estimates Using the V-GSI Method, Morelli’s P32_REV Method, and P32_DFN Using DFN Modelling

The true spacings in Table 2 used in Morelli’s example are based on real measurements that are corrected for orientation bias. These same data can be used to quantify both VFC and P32_{UNIT VOLUME} using the V-GSI method and attain similar values compared to Morelli’s result, as shown in Figure 11.

Table 3. Summary of VFC (P32_{UNIT VOLUME}) and P32_{UNIT VOLUME} estimates using true spacing data provided by [29]—see Table 2 above (some data in Table 3 were provided independently by Morelli).

Set	n	Mean Sti	SD	1.96 ∗ SE	VFCicont (LL)	VFCicont (Mean)	VFCicont (UL)	P32UNIT VOLUME (LL)	P32UNIT VOLUME (Mean)	P32UNIT VOLUME (UL)
S1	199	0.93	3.46	0.48	0.71	1.08	2.23	0.86	1.299	2.688
S2	80	2.71	8.72	1.91	0.22	0.37	1.25	0.26	0.446	1.511
S3	153	1.03	1.83	0.29	0.76	0.97	1.35	0.92	1.173	1.632
S4	44	4.98	3	0.89	0.171	0.20	0.24	0.21	0.245	0.295
				Σ	1.85	2.62	5.07	2.24	3.16	6.13

SE—standard error, LL—lower limit, UL—upper limit, VFCicont—volumetric fracture count contribution of the ith discontinuity set.

shows the results of the quantification of VFC (P30_{UNIT VOLUME}) and P32_{UNIT VOLUME} using Equations (1) through (4) using Morelli’s data.

With reference to Table 3, a mean VFC of 2.62 fractures/m³ is estimated. The lower and upper VFC 95% confidence limits on the mean VFC are estimated at 1.85 fractures/m³ and 5.07 fractures/m³ respectively. Then using Equation (3) this translates to a mean P32_{UNIT VOLUME} of 3.16 m⁻¹. When the estimated confidence limits on spacings are considered, the lower limit P32_{UNIT VOLUME} is estimated at 2.24 m⁻¹ and an upper limit of P32_{UNIT VOLUME} is estimated at 6.13 m⁻¹, respectively. These results are also plotted in Figure 11 (green arrows and dashed lines). The mean P32_{UNIT VOLUME} and limits are not remarkably different to the range of P32 values found by Morelli using his new method, or P32_DFN estimates provided by Morelli as a check (cf Figure 11). Certainly, a similar range of P32 is obtained for all three methods of quantification (Figure 11).

The slight overestimate of the mean P32_{UNIT VOLUME} (3.16 m⁻¹) compared to Morelli’s estimate (2.46 m⁻¹) and the P32_DFN estimate (2.65 m⁻¹) is attributable to the fact that Equation (3) assumes all the fractures for each set are persistent within the unit sphere volume (1 m³). In Morelli example it is evident that the mean size of the fractures per set (Table 2) is slightly less than if the full diameter of an intersecting fracture in the unit sphere volume is assumed. Given the close fit of the P32 estimates using Morrelli’s data and the P32_{UNIT VOLUME} estimate, it is contended that Equation (3) provides a reasonable first estimate of P32 despite the slight overestimate of the mean size of the fractures. A fracture size correction factor Pi is introduced below for inclusion in the VFC estimates, where the fracture sizes are less than the diameter of a unit sphere, to address this type of limitation.

7.2. Consideration of a Persistence Correction Factor (Pi) in P32 Estimates for V-GSI

Table 4. Pi persistence correction method results providing adjusted VFC’ and adjusted P23’ estimates.

Set	Mean Sti	Pi	$\sqrt[3]{{\mathit{P}}_{\mathit{i}}}$	${\mathit{S}\mathit{t}}_{\mathit{i}}^{\mathbf{\prime }}$	VFC’	Pi × r2	P32’
S1	0.93	0.49	0.79	1.18	0.85	1.21	1.03
S2	2.71	0.34	0.70	3.87	0.26	1.21	0.31
S3	1.03	0.27	0.65	1.59	0.63	1.21	0.76
S4	4.98	0.13	0.51	9.76	0.10	1.21	0.12
				ΣVFC’	1.84	ΣP32’	2.22

below shows the results using a suggested new method that considers the estimated means’ size (disc diameters) and spacing means per joint set as a correction factor for P32 estimates using field-measured data. Morelli’s data (Table 2) are used again to validate Equations (8) and (9) below. In essence the Pi factor is used to adjust the true spacings of joint sets that have less persistence than the diameter of a unit volume to correct the overestimate of the VFC parameter. The Pi ratio is the mean field-assessed area per joint set over the maximum area possible in a unit volume sphere. The adjusted spacings are then used to recalculate VFC and P32 for use in the V-GSI chart. The geometry used in this factor is already explicitly considered in Equations (1)–(3) and (5). Pi comes from concepts proposed by Cai et al. (2004 [17]) and Morelli and is adapted here for the ith set of non-persistent joints, such that “equivalent” [true] spacings (Sti’) [can be] “adjusted” based on the mean true spacing of the ith joint set (Sti) and the mean persistence of fractures, as per Equation (8)

$${St}_i^{\prime }={\displaystyle \frac{{St}_i}{\sqrt[3]{P_i}}}$$

(8)

where Pi is the persistence factor that can be considered the ratio of the mean fracture area (assumed to be a disc) for the ith discontinuity set (from Table 2) divided by the maximum disc area for a 1 m³ unit volume sphere, as used in the unit volume approach in the V-GSI chart as follows:

$$P_i={\displaystyle \frac{FieldAssessedMeanDiscarea(\pi {r^2)}_i}{MaximumDiscArea(\pi {r^2)}_{forunitspherevolumewherer=0.62m}}}$$

(9)

As a point of clarification, the denominator in Equation (9) does not imply that every field-measured disc has a constant area—rather on average most joints of consequence will intersect the unit sphere and each disc per set that fully intersects the theoretical unit volume will have an area close to 1.2 to 1.3 m². This is not to be confused with the actual size of the joints.

As can be seen in Table 4 below, the estimated ratio Pi varies between 0.5 and 0.13. When the ratio Pi from Equation (9) is used in Equation (8), the Pi-adjusted true spacings per set give an adjusted Sti’ for the ith discontinuity set for “equivalent persistence planes”.

For example, the mean true spacing for Set 1 increases from 0.93 m to 1.2 m. Using these adjusted spacings and estimating the VFC contribution per set for an adjusted VFC’ of 1.8 fractures/m³ and then using Equation (3) provides an adjusted (reduced) P32 estimate of 2.2⁻¹ compared to 3.16⁻¹ (Table 3).

This is a difference of 0.24 between the adjusted P32’ using the Pi persistence correction factor and Morelli’s method, which is in virtually the same order of the difference between Morelli’s P32 using his new method and his P32_DFN estimate of around 0.19. Assuming this level of accuracy is required for P32 when using the V-GSI chart, the Pi method outlined above can be adopted and included on the vertical axis of the V-GSI chart. Pi therefore provides a method to include joint persistence.

Figure 11 also shows the adjusted mean P32’ plotted on the vertical axis of the V-GSI chart (just below the upper green arrow plot of P32_{UNIT VOLUME}).

The inference from the result above is that if, in general, the mean trace length or persistence is observed or assessed to be less that the diameter of the unit sphere then Pi and the resulting Sti’ correction suggested above provide a reasonable adjustment to both the P30 and P32 estimates—which in turn adjusts the VFC plot on the vertical axis of the V-GSI chart (in this case the adjusted VFC’ is 1.84 fracture/m³ compared to 2.61 fractures/m³ without the adjustment (cf. Table 3 and Table 4).

8. Katse Arch Dam, Lesotho: Comparisons of P32 Estimates Using REV Approach, V-GSI Method, and DFN Modelling

The VFC parameter, which is equivalent to P30, forms the core of the quantified vertical axis of the V-GSI chart [3], and was pioneered during the excavation of the foundation keyway for the 185 m high Katse Arch Dam (Figure 12—photo). The dam is a key structure in a series of dams and tunnels that make up the Lesotho Highlands Water project in southern Africa.

Quantitative data (e.g., joint spacing, trace length or continuity measurements, and joint sidewall separation) and qualitative or descriptive data (e.g., rock mass and joint degree of weathering, joint sidewall separation, joint infill, and roughness and waviness) were systematically mapped on the exposed foundation surface during excavation [3].

8.1. Structural Domains and VFC and P32_{UNIT VOLUME} Estimates for the Foundation Keyway of the Katse Dam

The foundation rock mass at the Katse Dam was shown to be divisible into twelve structural domains—six domains per flank (Figure 13). Individual domains essentially correlate across both flanks of the dam foundation, thus translating into six spatially distinct, near-horizontal domains across both flanks. Individual domains are relatively homogenous within the domain but are statistically different to the rock mass of the domains above and below.

In general, the Katse foundation rock mass, which comprises basalt flows, shows a fairly low degree of fracturing. Two relatively more fractured Domains (than elsewhere in the foundation) are recognizable on both abutments between EL 1900 and EL 1980 approximately (Figure 13). These two Domains are referenced as left flank (LF) Domain III and right flank (RF) Domain III, respectively. RF Domain III is present between EL 1974 and EL 1894 (Figure 13) but has a minimum dimension that is limited to approximately 15 m horizontally into the foundation rock mass and discussed in more detail in Section 8.2.

Figure 13 and distils the degree of fracturing per domain. In the case of RF Domain III (

Table 5. Estimation of VFC and P32_{UNIT VOLUME} estimates for RF Domain III using a unit volume approach. LL—lower limit; UL—upper limit.

Set	n	Mean Sti	SD	1.96 × SE	VFCicont (LL)	VFCicont (Mean)	VFCicont (UL)	P32UNIT VOLUME (LL)	P32UNIT VOLUME (Mean)	P32UNIT VOLUME (UL)
Fc	98	3.3	1.3	0.26	0.28	0.30	0.33	0.34	0.37	0.40
Jh	141	1.7	1.6	0.26	0.51	0.59	0.70	0.61	0.71	0.84
Jv1	108	5.2	1.7	0.32	0.18	0.19	0.20	0.22	0.23	0.25
Jv3	94	5.4	1.7	0.34	0.17	0.18	0.20	0.21	0.22	0.24
Jsr	86	2	1.7	0.36	0.42	0.50	0.61	0.51	0.60	0.74
				Σ	1.57	1.77	2.04	1.89	2.14	2.46

), the mean VFC is 1.76 fractures/m³ with 95% confidence limits on either side of the mean VFC of 1.6 and 2.0 fractures/m³, respectively, from five discontinuity sets including a stress relief set (Jsr). The VFC estimates are calculated using Equations (1), (2), and (4) (Section 4.2). The P32_{UNIT VOLUME} estimates were estimated using Equation (3), resulting in a mean P32_{UNIT VOLUME} estimate of 2.14⁻¹ and 95% confidence limits on either side of the mean of 1.9⁻¹ and 2.5⁻¹, respectively, for RF Domain III.

8.2. P30_REV and P32_REV Estimates for the Foundation Keyway of the Katse Dam Using the REV Approach

Table 6. Summary P32_REV statistics for RF Domian III, Katse Dam Foundation rock mass.

Set	Mean Sti	Mean Size (m)	Disc Area (m2)	Disc Area (m2) Contribution per Joint Set
Fc	3.3	15	176.70	803.25
Jh	1.7	13	132.70	1171.17
Jv1	5.2	5	19.63	56.64
Jv3	5.4	13	132.73	368.70
Jsr	2	13	132.73	995.49
			ΣDisc area (m2)	3395.44
			P32REV	1.92

LL and UL are the lower and upper limit 95% confidence limits for P32 and are calculated to be 1.7 and 2.1 respectively.

below shows the discontinuity sets, the assessed mean true spacing of (Sti), and the mean size of each discontinuity set in RF Domain III [3]. The mean sizes shown in Table 6 are not precise trace lengths but are typically close to or slightly less than the mid-point of the observed field trace measures that are grouped by length bins, i.e., approximately 5 m, where the observation bin modal frequency high was 3–10 m, or 13 m when the observation bin modal frequency high of the observation bin was 10–20 m, and 15 m when the modal frequency high of the observation bin was not fully observable but was recorded as around 20 m. Sensitivity analysis can be undertaken within the range of the observation bins (one iteration is provided here). It is also noted that, for this example, potential sampling biases using semi-trace length methods were not undertaken to establish the distribution form and the mean size as developed by [30]. Nevertheless, given that the exposed foundation surfaces and sidewalls provided multiple observation windows with varying orientations for mapping, it is considered reasonable to use the trace length bin approach for fracture size estimates [23]. For this exercise, the fractures are assumed to be discs.

Figure 14 (RHS) is a projected section through the right flank of the keyway (when facing downstream) that shows the original ground level on the right side of the valley and the final foundation line. This figure is presented to graphically show how the REV dimensions and volume were established for RF Domin III. The REV dimension shown in Figure 14 (LHS) is an approximate (horizontal) distance from the foundation surface (black line) to the red dashed line, which is the approximate location beyond weathering, and valley-side stress relief fracturing (Jsr) is no longer present further into the valley side rock mass, noting that the other sets present in this domain are shown in Table 6.

The field-fitted REV shown in Figure 14 (red circle LHS enlargement) is considered to be a sphere with an approximate diameter of 15 m and therefore has a radius of approximately 7.5 m. The REV sphere volume of RF Domain III is therefore around 1767 m³.

Other than set Jv1, the mean size of the trace lengths per set is on average close to the diameter of the REV and a single-disc area intersecting the REV sphere can be estimated using the assumed mean disc size (Table 6) per set as follows:

$${\mathit{S}\mathit{i}\mathit{n}\mathit{g}\mathit{l}\mathit{e} \mathit{D}\mathit{i}\mathit{s}\mathit{c} \mathit{A}\mathit{r}\mathit{e}\mathit{a}}_{\mathit{i}}=\mathit{\pi }.\left({\left({\displaystyle \frac{{\mathit{M}\mathit{e}\mathit{a}\mathit{n} \mathit{D}\mathit{i}\mathit{s}\mathit{c} \mathit{S}\mathit{i}\mathit{z}\mathit{e}}_{\mathit{i}}}{2}}\right)}^2\right)$$

(10)

Then the fracture area contribution in the REV for the ith set can be estimated as

$${\mathit{A}\mathit{r}\mathit{e}\mathit{a}}_{\mathit{C}\mathit{o}\mathit{n}\mathit{t}\mathit{r}\mathit{i}\mathit{b}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n} \mathit{i}}={\displaystyle \frac{\mathit{R}\mathit{E}\mathit{V} \mathit{D}\mathit{i}\mathit{a}\mathit{m}\mathit{e}\mathit{t}\mathit{e}\mathit{r}}{{\mathit{S}\mathit{t}}_{\mathit{i}}}}.{\mathit{S}\mathit{i}\mathit{n}\mathit{g}\mathit{l}\mathit{e} \mathit{D}\mathit{i}\mathit{s}\mathit{c} \mathit{A}\mathit{r}\mathit{e}\mathit{a}}_{\mathit{i}}$$

(11)

where ${\displaystyle \frac{\mathit{R}\mathit{E}\mathit{V} \mathit{D}\mathit{i}\mathit{a}\mathit{m}\mathit{e}\mathit{t}\mathit{e}\mathit{r}}{{\mathit{S}\mathit{t}}_{\mathit{i}}}}$ gives the number fractures for the ith set in the REV and the REV diameter (field assessed) is assumed to be a vector orientated perpendicular to the orientation of the ith set (vectors since REV diameters have magnitude and direction per set).

Then P32_REV can be estimated as

$${\mathit{P}32}_{\mathit{R}\mathit{E}\mathit{V}}=\sum {\displaystyle \frac{{\mathit{A}\mathit{r}\mathit{e}\mathit{a}}_{\mathit{C}\mathit{o}\mathit{n}\mathit{t}\mathit{r}\mathit{i}\mathit{b}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n} \mathit{i}}}{{\mathit{V}\mathit{o}\mathit{l}\mathit{u}\mathit{m}\mathit{e}}_{\mathit{R}\mathit{E}\mathit{V}}}}$$

(12)

Then using Equations (10) through (12), P32_REV for RF Domain III can be approximated as 1.92 m⁻¹, as shown in Table 6.

Obviously, the areas of fractures away from the centre of the REV sphere are theoretically smaller than the full diameter of the sphere but, since the assumed average persistence for the joint sets present are close (other than Jv1) to the diameter of the sphere, it is considered reasonable to use a constant disc area (133 m²) to estimate the area contribution for the sets. Alternatively, a cube with sides equal to 15 m could have been considered for the REV and this would have yielded a slightly larger surface area per set fracture, but in both cases the average persistence for sets FC, Jh, Jv3, and Jsr is close to the maximum REV disc area and, since the exact surface area per joint cannot be determined, it is reasonable to assume that the surface area per fracture (133 m² in this case) is a constant for the given REV.

As shown in Table 6, Jv1 (a sub-vertical joint set) is the only set that has an assumed mean trace length that is noticeably less than the diameter of the REV. In this case the area of a single Jv1 intersecting the REV is estimated to be 19.6 m². The total area contribution of this set in the REV is estimated as 56.6 m², which reflects the fact that fractures of Jv1 are not fully persistent in the REV. Assuming a case where there is more than one set with mean trace lengths that are less than the REV diameter, it may be better to use a Pi approach as discussed in Section 7.2 above.

Comparing Table 5 and Table 6 shows a good fit for the unit volume approach and the REV approach, i.e., the mean P32_{UNIT VOLUME} is estimated at 2.14⁻¹ compared to the mean P32_REV, which is estimated at 1.92⁻¹, respectively.

8.3. P32_DFN Estimates for the Foundation Keyway of the Katse Dam Using DFN Modelling

As additional validation, a simple DFN was prepared for Katse RF Domain III using the mean parameters shown in

Table 7. RF Domain III input value for simple DFN development.

Set	Dip	Dip Direction	Fracture Radius (m)	Spacing (m)	P10 (1/m)	P10 Well Plunge	P10 Well Bearing
Fc	2.4	86.3	7.5	3.3	0.30	87.6	266.3
Jh	2.4	86.3	6.5	1.7	0.59	87.6	266.3
Jv1	86.8	31.4	2.5	5.2	0.19	3.2	211.4
Jv3	76.7	67.7	6.5	5.4	0.19	13.3	247.7
Jsr	47.6	70.3	6.5	2.0	0.50	42.5	250.3

. Rather than using statistical variation in joint orientation and size, these parameters were kept constant in the DFN model to illustrate a direct comparison for the field-assessed REV and the unit volume approach (Table 5 and Table 6). Table 7 also shows the linear fracture frequency (P10—Figure 5) which is the number of fractures per unit length of a scanline. The mean P10 is taken as the inverse of the mean fracture spacing Sti for each joint set for a scanline or “well” drilled perpendicular to the mean orientation of the joint set (c.f. Mean Sti in Table 5 and Table 6). The “well” plunge and bearing are shown in Table 7 for completeness.

Even with constant fracture lengths and orientations, a variation in P32 can be observed across a sample rock mass. A 120 m × 120 m × 120 m total volume sample box is shown as Figure 15, with the colours in this figure showing the variation in the calculated P32_DFN values for the discrete fractures that fall within each sample grid block. Each grid block is 12 m × 12 m × 12 m in size (i.e., the overall rock mass sample is divided into further sample boxes), and is intended to represent an equivalent REV sphere volume of approximately 1767 m³. For each 120 m × 120 m × 120 m volume with smaller sample boxes, 10 realizations or randomly generated fracture networks were developed to assess variability in P32_DFN in the DFN model. Realization 8 is shown in Figure 15.

The variation in the P32_DFN shown in Figure 15 and Figure 16 can be considered analogous to variation in the natural rock mass. However, in this DFN model the variation in P32_DFN occurs only as a result of the discrete location of the fractures generated for each joint set.

Figure 16 illustrates that assuming a log normal distribution for P32, the mean P32_DFN calculated for this rock mass is 1.81 m⁻¹ for realization 8. When compared to the results from Table 5 and Table 6 above, the range in P32_DFN is from 1.2 m⁻¹ to 2.5 m⁻¹ (shown Figure 16) and the mean P32_DFN = 1.81 m⁻¹, it is evident that the results are comparable to the Unit Volume and REV approaches for RF Domain III. It is noted that introducing statistical variation in joint lengths and orientations in the DFN model using distribution-applied input parameters will introduce additional variation over and above that shown in Figure 16. Constant parameters were used to simplify the analysis.

8.4. Comparison of P32 Using the Unit Volume Approach, REV Approach, and DFN Modelling Estimates for a Domain in the Foundation Keyway of the Katse Dam

Figure 17 below is provided as a visual summary that allows for comparison of the three different approaches discussed above. See Table 5 and Table 6.

The close fit shown in Figure 17 provides reassurance that the VFC and P32 parameters on the vertical axis of the V-GSI chart are indeed validated and quantified measures.

8.5. Additional Discussion on the Katse Arch Dam Case History

Provided that the potential failure mechanism associated with the interconnectedness of the longer joints is understood at the scale of the problem, the V-GSI method provides a reasonable means for evaluating the strength and deformability of the rock mass using P30 and P32. Potential failure mechanisms need to be understood before assuming the H-B equations are valid and before assuming the V-GSI method is applicable for all situations (Berwick and Elmo (2025) [33].

As a gut check with respect to the scale of the structure and the spacing of the joint sets for the Katse example (see Table 5 and Table 6), the ratio of 5 discontinuity sets divided by the mean VFC for RF Domin III (5/1.78) equals an average joint spacing of 2.8 m for all sets present in RF Domian III. The assessed REV diameter is 15 m and therefore 15/2.8 means that the dimension of the field-assessed REV at Katse is 5.3 times the average spacing of all the sets. The foundation widths in RF Domain III range between 40 m and 60 m and clearly the dimensions of the dam at this location greatly exceed the average spacing of the joint sets. The H-B failure criteria and equations can, therefore, be used for strength and deformability estimates. In the case of RF Domain III, this translated to an average modulus of deformability, with a mean UCS = 120 MPa and a mean V-GSI rating of 65 of 16.6 GPa, as shown in Figure 14.

8.6. The Correlation of V-GSI Rating Scale (On the Vertical Axis) with the Rock Mass at Katse Dam

As stated in this paper, validating the rating scale on the V-GSI chart requires correlation of rock mass strength and deformability estimates using V-GSI vertical axis ratings and those obtained with either in situ testing or deformation monitoring. In the case of large dams like the Katse Arch Dam example provided above, one of the most important parameters in the design and performance of dams founded on rock is the rock mass modulus of deformability (E*).

At several locations in the foundation keyway, E* data provided an opportunity to correlate deformability estimates obtained using the V-GSI system and those obtained via large-scale deformability testing. Figure 18 below shows the results of this correlation for the rock mass, where V-GSI E* estimates and in situ E* testing results that were obtained in domains that were beyond the influence of weathering and stress relief fracturing (Jsr) [3].

This plot shows a strong correlation was achieved using V-GSI ratings and H-B equation estimates for E* and the in situ results demonstrated by a high r-squared goodness of fit. This provides confidence that the qualification of the degree of fracturing (VFC) used in the V-GSI chart and the associated rating scale can provide reliable first estimates of strength and deformability.

9. Conclusions

The above research and analysis and discussion provide evidence and validation that scale can and should be considered on the vertical axis of GSI charts. In this discussion the V-GSI chart is predominantly discussed, but similar conclusions can be drawn with other GSI charts with scaled vertical axes. Scale and scale effects are an inherent part of rock mechanics and rock engineering, and as demonstrated above, the inclusion of scale is feasible and a necessary progression for the GSI system.

In conclusion, research in this paper shows that:

• P30 and P32 parameters estimated via DFN modelling, a field-defined REV approach, or the unit volume approach used on the scaled vertical axis of the V-GSI, provide comparable results even though the methods of calculation are different. This is an important finding because it provides validation that VFC (P30) and P32 parameters included on the vertical axis of the V-GSI chart, which is based on measured data, are in fact quantifiable.
• Provided the size of the fractures is considered, then the inclusion of VFC (P30) and P32 as measures of increasing fracture intensity on the V-GSI chart can provide reliable first estimates of strength and deformability for use in engineering design.
• The quantification of the vertical axis of the V-GSI chart provides a rational basis for comparing the ratio of the scale of a project or the scale of interest (a structure to be built on or in rock) to the scale or intensity of fracturing in a rock mass, a necessary requirement for the use of GSI.

This is not to say that DFN modelling should not be undertaken as a means of checking or calibrating the quantification of P30 and P32 on the V- GSI chart. It is also not to say that the V-GSI cannot be improved on. It is an empirical system.

Parameters comparable to those used in DFN modelling are also being used on the vertical axis in the V-GSI chart. It is recommended that future research in this regard should answer the following questions:

• Why is it acceptable to use joint orientation, joint set spacing, joint set fracture frequency, and joint set persistence or size estimates to undertake DFN modelling that includes scale, but it is claimed that the same parameters cannot and should not be used to quantify GSI charts?
• Why is it acceptable to quantify P30 or P32 from the same measured data using DFN methods for use in coupled DFN–mechanical models but it is not acceptable to quantify the vertical axis of GSI charts using the same parameters? Is it because P30 or P32 values are in turn related to GSI ratings on the vertical axis (a necessity to link to the H-B equations that need to be calibrated) or is it because a DFN model is visual?

Where data from in situ strength and large-scale deformability testing exists for real-world rock engineering projects, it is also recommended that additional back analyses be undertaken to determine if the associated rating scales (vertical and horizontal axes) on the V-GSI or other quantified charts can be better calibrated that may in turn lead to more reliable strength and deformability estimates. It is also recommended that research be undertaken that compares results using coupled DFN–mechanical models with results from finite element continuum models or geo-mechanical models that use V-GSI ratings. It is also recommended that additional research should be undertaken to ascertain the limitation of the V-GSI system for highly anisotropic rock masses.

As a final note, the use of quantified GSI charts does not exonerate a designer from understanding the geology and the variability of a given rock mass or from understanding potential failure mechanisms that cannot be captured by GSI and the H-B equations alone [33].