Calculating Bias-Free Volumetric Fracture Counts (VFCs) in Underground Works and Their Use in Estimating Rock Mass Strength and Deformability Parameters

Featured Application

Engineering assessment of rock masses for both underground and surface excavation projects.

Abstract

This paper initially provides a practical example on how to estimate a bias-free volumetric fracture count (VFC—fractures/m³) in a tunnel and incorporate it into a new and unified volumetric-based Geological Strength Index (V-GSI) chart. The quantified V-GSI chart and the methods shown in the practical example were used extensively as tools to assess rock mass conditions and assist in support determinations on the WestConnex M8 Motorway tunnel project in Sydney, Australia. The reliability of the strength and deformability estimates obtained using the V-GSI ratings while tunneling within the Hawkesbury Sandstone is demonstrated here by providing an example of deformation results obtained through 3-D finite element analysis in a single location in the tunnel. The modelling results are compared to measure convergence in the tunnel in this location, which demonstrated good correlation between predicted and observed deformation. This provides validation that the V-GSI chart and associated Hoek–Brown strength and deformability equations can be used with some confidence to determine potential deformation in underground works.

1. Introduction

One of the most difficult tasks in rock engineering is the proper characterization of the fracturing present in a rock mass for the purpose of engineering analysis [1]. Research on the rock foundations of the Katse Arch Dam in Lesotho, southern Africa, in the 1990s led to the development of a bias-free volumetric fracture count parameter (VFC—fractures/m³) that advanced the understanding and quantification of fracturing in rock [2]. Subsequently, Schlotfeldt and Carter [1] integrated the VFC parameter into a new and unified volumetric-based Geological Strength Index (V-GSI) chart. The chart provides a new approach and tool for rock engineers to calculate bias-free volumetric fracture counts and associated GSI ratings required to determine strength and deformability estimates for rock masses using the Hoek–Brown failure criteria equation [1].

The V-GSI system builds on the work of many researchers (references [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20], to name a few). For those not familiar with the development of the Geological Strength Index (GSI) chart, the following is offered as background information.

The 1980s Hoek–Brown failure criterion was expanded on in the 1990s and 2000s to aid in designing underground excavations and slopes in rock [3,4,5,6,7,8,9,10]. The criterion and associated equations, which provide a means of estimating the strength and deformability, and hence the mechanical behaviour of rock masses encountered in tunnels, slopes, and foundations, have continued to evolve over the last 40 years. A key part of the system includes the Geological Strength Index (GSI) chart, which was introduced by Marinos and Hoek in the 1990s and 2000s [3,6]. In the early versions of their GSI chart, the degree of fracturing (or structure) in a rock mass needed to be matched to schematic caricatures of jointing and descriptors in a series of horizontal rows in the chart [3]. The observed surface condition of joints or fractures in the rock mass also needed to be matched to descriptors provided in vertical columns of their chart. The graphical chart has increasingly come under criticism [1,2,9,10,11,12,13,14,15,16] because of the lack of scale on the vertical axis of the chart. Many researchers have attempted to quantify the vertical axis of the chart for this reason, but none of these approaches have gained traction in industry [1,13,14].

Hoek et al. [12] published a fully quantified chart using rock quality designation (RQD) [21] as a measure of fracturing on the vertical axis. This 2013 chart [12], too, has not gained widespread traction [1]. This is related to the well-known issues that RQD suffers from, i.e., scale and orientation bias, thus making it a less ideal parameter for inclusion in the chart [1,2,11,13,14,15,16].

The Hoek–Brown failure criterion may still be valid despite the limitations introduced with the use of RQD on the vertical axis of the 2013 chart [1]. If, for example, the dimensions of an excavation or slope on large projects in good quality rock are many times larger than the widely spaced joint sets, the Hoek–Brown failure criteria can still be applicable despite RQD being static at 100%, and therefore an insensitive scale or blockiness parameter [1,2,13,14,18]. Moreover, depending on the orientation of measurement of RQD, the results can be close to 100% in one direction but 0% in another direction for the same rock mass, even if the spacing of joints are within the sensitivity range of RQD (typically < 0.3 m) [1,14,18]. If scale and orientation biases related to RQD are not understood or considered, this can result in unreliable rock mass strength estimates if RQD alone is used in the 2013 chart. With the inclusion of the bias-free VFC parameter in the V-GSI chart (Figure 1), this overcomes many of the issues related to the use of RQD described above. The VFC parameter does not suffer from scale or orientation bias, and readers interested in more details are referred to Schlotfeldt et al. [1,2,13,15]. Nevertheless, on the assumption that an unbiased RQD can be calculated, a theoretical correlation is provided with the VFC parameter in the V-GSI chart (Figure 1), along with several other commonly used parameters.

At the core of the V-GSI chart is the inclusion of a bias-free VFC parameter (fractures/m³) on the vertical axis [1]. The VFC parameter not only extends the use of the chart to more massive rock masses (VFC < 3 where RQD is no longer sensitive—100%), but also more than adequately covers more blocky rock masses than the 2013 chart was intended for (RQD < 80%), but without the bias issues associated with RQD [1]. Validation of the VFC parameter in the V-GSI chart comes from its use to determine the degree of fracturing in the foundation rock mass on the Katse Arch Dam, Lesotho [1,2], and subsequent prediction of the moduli of deformability of the foundation rock mass using the Hoek–Brown equations [3,16,20], and comparing the results to large scale deformability testing undertaken in adits and galleries below the foundation [1].

One of the other advantages of the VFC parameter is that it does not require the running of advanced 3-D software to stochastically generate discrete fracture networks (DFNs) for rock masses to determine volumetric fracture counts. Field-collected apparent joint set spacing (Sa) data corrected to true spacing (St) data, using virtual measurement vectors on underground maps, outcrop maps and bench mapping, can be used to estimate reliable bias-free VFCs and the vertical axis ratings on the V-GSI chart. This, in turn, can be used to quantify GSI ratings and Hoek–Brown strength and deformability parameters critical for rock mechanics analyses.

The VFC parameter is relatively easily quantified with field measurements or by scaling off Sa data on maps that can be easily converted to St data per discontinuity set with the use of correction factors (CFs) obtained using dot product vector analysis [1,2,13,15]. When properly calculated, the VFC of a rock mass (within the same structural domain) is an intrinsic property that does not require the determination of block shapes and sizes to be valid and can include as many non-orthogonal joint sets as are present in the rock mass. A correlated scale is also provided with VFC and P₃₂ (area of fractures/m³) in the V-GSI chart, since this parameter is frequently used in geo-mechanical modelling programs. An approximate correlation with both RQD and block volumes is also provided on the vertical axis of Figure 1, along with parameters used in the other two major rock mass classification systems, namely RQD and RQD/Jn [22,23]. On the horizontal axis, the original surface descriptors used in the early chart [3,6] are retained along with correlated scales for Bieniawski’s 1989 [22] Joint Condition rating (JCond89) chart) and Jr/Ja (Barton et al., 1974) [23]. It should be noted that JCond89 has an applied correlation factor of 1.5, as used in Hoek et al., 2013 [12].

The VFC parameter and associated rating scale used on the vertical axis has been shown to provide reliable strength and deformability estimates at all scales, but is particularly useful in massive rock masses [1] (i.e., for rock masses where VFC is <3 fractures/m³ and where RQD tops out at 100% and is no longer sensitive; see Figure 1).

As mentioned above, this confidence comes from the application of the V-GSI system to determine the deformation moduli of foundation rock mass at the Katse Arch Dam, Lesotho Highlands [1], and by comparing these data to moduli obtained with large scale deformability testing in foundation adits.

The example provided in Section 3, from the WestConnex Motorway tunnels, demonstrates the first application of the V-GSI systems to tunnels. The purpose of this publication is to provide details on the method used to estimate the tunnel rock mass VFC and the use of the V-GSI chart to determine the strength and deformability of the tunnel rock mass. The intent of this publication is not to provide details of the V-GSI ratings for the entire M8 Westconnex Motorway project. Rather, one hypothetical example is initially provided (Section 2) to demonstrate the method used to estimate VFCs on this project in the hope that others can use the system in a tunnel environment. This is followed by providing tunnel mapping results from one actual location in the tunnel, along with the estimated VFC (Section 3). Validation of the V-GSI system for use in tunnelling comes from undertaking 3-D modelling with RS3 by Rocscience Version 4.024, (© 2022 Rocscience Inc., Toronto, ON, Canada) in the same location. Input into the model includes the V-GSI rating in this location and the assessment of the intact rock strength—all from the tunnel mapping records. The tunnel deformation predicted at this location was estimated by the modelling and is compared to real time monitoring data in the same location. This shows that the V-GSI system can be used with some confidence in underground works as well.

2. A Hypothetical Example Application of the V-GSI Chart in a Tunnel

2.1. Measurement of Apparent Joint Spacings (Sa)

The hypothetical example provided in Section 2 (Figure 2 and Figure 3) demonstrates the process of quantifying the VFC parameter for use on the vertical axis of the V-GSI chart (Figure 1) in a tunnel. Figure 2 and Figure 3 are hypothetical in the sense that the tunnel map with fracture traces are not from a real location in the tunnel, but represent a synthesis of typical fracturing seen in many locations in the tunnel, primarily to show the method of VFC quantification.

Little explanation is required for the quantification process of the horizontal axis of the V-GSI chart since JCond89, which forms part of Bieniawski’s rock mass rating (RMR) system, has been widely used for more than 30 years [18].

For this example, Figure 2 is presented as a simplified and stylized fracture trace map from an assumed, hypothetical working face in a tunnel heading excavated for the WestConnex M8 Motorway twin tunnel project in Sydney, Australia, as explained above. The tunnel profile shown in Figure 2 shows the typical dimensions of the tunnel face and the spatial location of hypothetical fracture traces (perpendicular to tunnel drive orientation), along with the projected left wall, projected right wall and projected tunnel crown appearing to the left, right and above the tunnel face, respectively, for an assumed westbound tunnel heading (two vehicle lane width). More details on the project are provided in Section 3.

In Figure 2, the mapped fracture traces of two discontinuity sets are shown: a sub-vertical tectonic joint set (green traces annotated as J1; with a mean dip and dip direction of 70°/312°), and a sub-horizontal bedding set (red traces B; with a mean dip and dip direction of 20°/195°). All orientation measurements are relative to true north.

For this example, the focus is only on the joints exposed in the working face, but it should be evident that the same sets are present in the sidewalls and crown of the tunnel, and a similar analysis presented below (Section 2.1 and Section 2.2) to calculate the rock mass VFC can be undertaken for these exposures and used in the overall analysis. In this example, the tunnel is assumed to being driven at an orientation of 061° from true north.

Figure 3 shows the concept of vector measurements of apparent spacings (Sa) for each of the two hypothetical discontinuity sets present in the rock mass for this tunnel heading. Since the working face is fully spatially defined in terms of orientation and dimensions (as are the crown and sidewalls), scaled virtual Sa vectors (colored arrows) for discontinuity sets J1 and bedding (B) can be measured between adjacent joints of the same sets because the vectors fall within the plane of the tunnel working face, with known orientation and dimensions, and therefore have both orientation and magnitude. The vectors in this case replace conversional scanline-type measurements.

Sa measurements for set J1 are scaled off the map using inserted horizontal virtual vectors that lie within the plane of the tunnel working face and are perpendicular to the direction of tunnel advance (Figure 3). Sa measurements for set B are also scaled from the map using inserted vertical virtual vectors on the tunnel face (Figure 3). Each Sa measurement falls within the plane of the tunnel working face, and as such are fully orientated in 3-D space.

Figure 3 also shows conceptually that the spacing vectors for each set do not need to be measured strictly on a scanline or sequentially, as with conventional scanline surveys, provided that individual measurements are taken in the same direction for each set. In Figure 3, horizontal spacing vectors are illustrated for the sub-vertical tectonic joint set (J1) and, in this case, trend 151° (relative to north) with 0° plunge. For the bedding set (B), Sa measurements are made using vertical vectors on the working face map. These vectors are orientated vertically downwards and therefore do not have a defined trend per se, but for further analysis a trend of 061° is selected to represent the direction the tunnel is being driven, and all the vectors have a downward plunge of 90°. It should be noted that the vectors do not have to be horizontal or vertical; they can be in any orientation that falls within the plane of the working face map.

2.2. Correction Factors (CFs) and Calculation of True Spacings (St)

For both sets, J1 and B, the discontinuity traces exposed in the tunnel face are not oriented at exactly 90° to the Sa spacing measurements vectors, which are horizontal and vertical, respectively. The apparent spacing (Sa) measurements per set therefore require correction to ‘true’ spacings measurements (St) to obtain spacings that are orthogonal to the mean dip and dip direction of each set. This is simply achieved by using vector dot product analysis [1,2,15] to calculate correction factors for both sets J1 and B, respectively, given that each Sa measurement is a vector.

This calculation is possible because the respective angular relationships between the tunnel face, measurement vectors and discontinuities are recorded by the mapping process.

Spacing correction factor (CF) per discontinuity set (J1 and B in this case) can be calculated with the use of Equation (1) [1,2,15]:

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

(1)

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

where:

• α_d = discontinuity set mean dip direction (used to convert to pole trend);
• β_d = discontinuity set mean dip angle (used to convert to pole plunge);
• α_n = trend of pole or normal to discontinuity set mean dip direction;
• β_n = plunge of pole or normal to discontinuity set mean dip angle;
• α_s = strike or trend of scanline or measurement vector (azimuth 0–360°) on tunnel heading, sidewall or crown;
• β_s = dip or plunge of scanline or measurement vector (can be anywhere between 0°, horizontal, and 90°, vertical) on tunnel heading, sidewall or crown.

Table 1. Input data for the calculation of correction factors (CF) for sets J1 and B shown in Figure 3.

Discontinuity Set	Mean Vector Orientation		Mean Vector Orientation, Pole or Normal		Trend and Plunge Measurement Vectors/Scanline		Correction Factor
	αd	βd	αn	βn	αs	βs	CF
J1	312	70	132	20	151	0	0.89
B	195	20	15	70	61	90	0.94

shows the inputs and the resulting CF’s using Equation (1) for each discontinuity set shown in Figure 2 and Figure 3.

For set J1, each Sa measurement needs to be multiplied by the correction factor (CF), 0.89, to convert Sa measurements to St values. Likewise, for set B, each Sa measurement needs to be multiplied by 0.94 to convert Sa measurements to St values, as shown in Equation (2):

$$S{t}_i=C{F}_i\left(S{a}_i\right)$$

(2)

Figure 3 shows the Sa measurements scaled off the map for sets J1 and B, respectively, which are then converted to St values using Equation (2).

Table 2. Summary of true spacing (St) statistics for joint sets J1 and B.

Joint Set	Mean True Spacing (m) $\overline{\mathit{S}}{\mathit{t}}_{\mathit{i}}$	True Spacing 95% Confidence Limit (m)
J1	1.54 (1.73)	0.955
B	2.26 (2.40)	2.31

Note: Parentheses provide the apparent spacing means before the correction factors are applied.

provides a summary of the mean true spacing and 95 % confidence limits on the spacings for each set, respectively.

The confidence limits on the mean true spacings per set shown in Table 2 are calculated at the 95% level, as follows in Equation (3):

$$95\%\mathrm{Confidence}\mathrm{Limits}=1.96\frac{\widehat{\sigma }S{t}_i}{\sqrt{n}}$$

(3)

where $\frac{\widehat{\sigma }S{t}_i}{\sqrt{n}}$ is the standard error of the true spacing values, $\widehat{\sigma }S{t}_i$ is the standard deviation for the ith discontinuity set, and n = number of spacing measurements, respectively, for that set [1,2].

As can be seen in Figure 3, there are edge effects at the intersection of the tunnel heading and the sidewalls (J1) and at the crown and invert (B), and the apparent spacings are measured as “greater-than” (>) values. This means that there may be a joint from a particular set at a greater distance than measured, but it is better to include the Sa ‘edge effect’ measurement than ignore it, since ignoring it may skew the spacing statistics. For example, if there were a swarm of J1 joints in the middle of the face and none near the sidewalls, then the average spacing based on measurement in the swarm only would provide the average spacing within the swarm and not the rock mass as a whole. As can be seen in Figure 3, there are joints for set J1 mapped in the sidewalls. The orientation of the sidewalls is, however, perpendicular to the tunnel heading, and, as such, the CF for set J1 would be different, i.e., the vector measurement directions are perpendicular to the tunnel heading face. While not shown in the example, it becomes possible, however, to sum the corrected ‘edge effect’ (true) “greater-than” spacings measured on the tunnel heading and the sidewalls for two adjacent joints of a particular set to provide a single corrected ‘edge effect’ spacing value for the right-hand and left-hand sides of the tunnel, respectively.

In the example shown in Figure 3, all the discontinuities are shown as having full persistence, i.e., they have continuity longer than the dimensions of the tunnel. This is not always the case, and this effect can be partially overcome by adding more scanlines or by adding Sa measurement vectors that sample areas with less joints due to terminations.

It should be noted that it is also possible to simplify the process by estimating the joint linear density per set (λ) by counting the number of joints per metre of sampling line, e.g., the sampling line can be the width of the tunnel heading or the span of the tunnel. In the case of the tunnel, if x is the total number of discontinuities intersected by a line of total length (L) or the tunnel width, the linear density of discontinuities is approximated by x/L and the reciprocal is average spacing, i.e.:

S_average ≈ 1/λ ≈ L/x

(4)

In the example provided in Figure 2, the width of the tunnel heading is approximately 12.5 m and the tunnel height is approximately 8 m. L for set J1 is 6/12.5 or 0.48 joints/m, and the reciprocal is 2.08 m, which is the average apparent spacing for this set. The true mean spacing for J1 can then be approximated as 2.08 × 0.89 (CF for set J1), and the corrected true spacing is around 1.85 m for this method (c.f. Table 2). This value is similar enough to the result obtained when actual spacings are measured and corrected individually (1.54 m), but where possible, individual measurement should be used to improve the reliability of the assessment. l for set B is 3/8 or 0.38 joints /m, and the reciprocal is 2.67 m, which is the average apparent spacing for this set. The true mean spacing for B is approximately 2.67 × 0.94 (CF for set B), and the corrected true spacing is around 2.51 m. The disadvantage of this method is that it is not possible to calculate confidence limits on the average spacings. The average is also skewed somewhat because it does not consider the smaller spacing values noted in Figure 3. For the reader interested in estimating RQD, Palmstom [18] provides a rough correlation using joint volumetric counts (Jv). It should be noted that Palmstom’s Jv is not calculated in exactly the same way as the bias-free VFC; nevertheless, once VFC is calculated with Equations (5) and (6) below, it can be used to estimate RQD. However, using RQD directly (e.g., from boreholes) to estimate VFC can result in significant errors given that it is difficult to account for directional error (orientation bias) and/or when RQD is 100%.

2.3. Calculating Volumetric Fracture Counts (VFC) and Confidence Limits

As shown in Equation (5) below, the inverse of the mean true spacings for each set provides the bias-free VFC contribution for the two sets shown in Figure 3, i.e., the VFC contribution of set J1 is 0.65 fractures/m³ (1/1.54), and bedding contributes 0.44 fractures/m³ (1/2.25) to the overall rock mass fracture count in the example in Figure 3.

$$VF{C}_i \left(contribution\right)=\frac{1}{\nabla \overline{S}{t}_i}$$

(5)

where $\overline{S}{t}_i$ is the mean true spacing for the ith discontinuity set, and $\nabla$ implies the use of geometrical means or normal (Gaussian) means, depending on the spacing distribution. In this example, a normal distribution is assumed to keep the example simple and because there is insufficient data to determine an alternative spacing distribution. It is, however, well known that discontinuity sets frequently have negative exponential spacing distribution, particularly for joint sets that are tectonic in origin.

The bias-free VFC for the rock mass in the tunnel is then simply the sum of the individual VFC contributions for the two sets (0.65 + 0.44 = 1.09), but in general is calculated with the use of Equation (6), irrespective of the number of i discontinuity sets (NDS).

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

(6)

The 95% confidence limits on the true spacing data are then used to calculate the VFC contribution (cont.) for each individual set i using Equations (7) and (8) as follows:

$$Lower limit VFCi cont.=\frac{1}{\overline{S}{t}_i+1.96\times \frac{\widehat{\sigma }S{t}_i}{\sqrt{n}}}$$

(7)

$$Upper limit VFCi cont.=\frac{1}{\overline{S}{t}_i-1.96\times \frac{\widehat{\sigma }S{t}_i}{\sqrt{n}}}$$

(8)

where n is the number of spacing observations recorded for each set i. The overall lower limit and upper limit on the mean VFC is then the sum of the individual VFCi contributions found using Equations (7) and (8). In more formal terms, the 95% confidence limit on the mean VFC can be calculated using Equation (9):

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

(9)

Table 3. VFC data for the example shown in Figure 3.

Discontinuity Set	Lower Limit VFC Contribution	Mean VFC Contribution	Upper Limit VFC Contribution
J1	0.40	0.65	1.72
B	0.30	0.44	0.83
	0.702 (LL VFC)	1.09 (Mean VFC)	2.56 (UL VFC)

Note: If 1.96 × standard error is greater than or equal to the mean St for a set, then the smallest true spacing value can be substituted in place of the standard deviation ($\widehat{\sigma }S{t}_i$) to avoid a negative or zero upper limit value which is not meaningful. Alternatively, when there are too few data measurements, $\widehat{\sigma }S{t}_i$ may be approximated as the highest conceivable spacing value minus the lowest conceivable spacing value divided by Nsigma (where Nsigma is less than 6) to avoid a negative number [1]. LL—Lower Limit; UL—Upper Limit.

provides a summary of the mean VFC and confidence limits calculated for the example tunnel face in Figure 3 using Equations (1)–(9).

2.4. Vertical and Horizontal Axis Quantification on the V-GSI Chart

The overall V-GSI rating for the tunnel rock mass shown in Figure 2 can be calculated using Equation (10) [1]:

$$\mathrm{V}\text{-}\mathrm{GSI}\mathrm{rating}=1.5\times \mathrm{JCond}{89}_{}+\left(50-8.5\ln \left(\mathrm{VFC}\right)\right)$$

(10)

The latter half of Equation (10), $\left(50-8.5\ln \left(VFC\right)\right),$ provides the vertical axis rating on the V-GSI chart using the VFC data in Table 3. The former half of the equation provides the horizontal axis rating, when the Bieniawski [22] joint condition assessment table is used to calculate JCond89, and this is multiplied by the correlation factor of 1.5 on the V-GSI chart (JCond89). A correlated Jr/Ja scale is also provided in Figure 1 for practitioners that prefer the Q classification system [1,23].

In practice, it is easier to (a) plot the mean rock mass VFC and the confidence limits on the scaled vertical column, as shown in Figure 1, and (b) read off the rating in the adjacent vertical rating column included on the V-GSI chart.

This quantification is overlaid on Figure 1. The mean VFC of 1.09 fractures/m³ with 95% confidence on the mean of 0.72 fractures/m³ and 2.56 fractures/m³ results in a mean vertical axis rating of 49, and lower and upper 95% confidence ratings of 42 and 53, respectively, as shown on the vertical axis on Figure 1.

In this example, $1.5\times JCond{89}_{}$ has been assumed to have an average rating of 30 and a range of between 25 and 35 as shown in Figure 1, recalling that JCond89 includes assessments of joint persistence, aperture, roughness, infill and weathering.

The overall mean V-GSI rating of 79 is obtained at the intersection of the vertical and horizontal axes ratings (30 + 49), as shown on Figure 1. A box of associated V-GSI limits for the tunnel rock mass example is also shown on Figure 1 with a lower V-GSI limit of 67 and an upper limit of 88, respectively. These results are also summarized in

Table 4. Summary of VFC rating for the example tunnel heading shown in Figure 3.

	VFC	VFC Rating	JCond89 Rating	V-GSI Rating
Lower Limit	0.70	53	35	88
Mean	1.09	49	30	79
Upper Limit	2.56	42	25	67

.

2.5. Strength and Deformability

The strength and deformability parameters for the rock mass of the example tunnel can then be estimated using the V-GSI ratings and the generalized Hoek–Brown failure criteria and associated strength equations [3,4,16,20]. The strength and deformability estimates for the example tunnel rock mass are summarized in

Table 5. Summary of strength and deformability estimates for the tunnel rock mass shown in Figure 2.

Hoek-Brown Parameter	Lower Limit	Mean	Upper Limit
mb	4.0	6.141	8.469
s	0.0256	0.097	0.264
a	0.502	0.501	0.5
Rock mass tensile strength (MPa)	−0.115	−0.284	−0.56
Rock mass uniaxial strength (MPa)	2.86	5.597	9.238
Rock mass global strength (MPa)	5.2	7.2	9.863
Rock mass modulus (GPa)	2.7	3.476	3.789
Equivalent cohesion c (MPa)	0.503	0.852	1.387
Equivalent friction angle (Phi-degrees)	51.8	53.668	54.2

Notes: 1. mb, s and a are Hoek–Brown material parameters [4] calculated with the estimated V-GSI rating overlays provided in Figure 1. 2. Other parameters calculated using software RSData Version 1.0.0.3 (© 2020 Rocscience Inc.). Input includes: Rock type—sandstone; V-GSI = 79 (limits ranging from 67 to 88, see Table 4); intact uniaxial strength (UCS) = 18 MPa; mi (material constant) = 13; D (damage factor) = 0; Ei (intact modulus) = 4 GPa. Tunnel depth is assumed to be 60 m. Equations for the H–B failure criteria and strength and deformability estimates and equivalent c and Phi estimates are well known in the literature and are not repeated here. Interested readers are referred to [3,4,16,20].

.

3. WestConnex M8 Motorway Tunnels—Using V-GSI to Determine Rock Mass Strength and Deformability in a Selected Location

3.1. WestConnex M8 Tunnels—Project Overview

The WestConnex M8 project (formerly known as the New M5 Project) consisted, in part, of twin motorway tunnels, each up to around 9 km in length, and varying from two to three lanes, respectively, for each tunnel, with the capacity to add additional lanes. The M8 Motorway opened in July 2020 and connects the M5 Motorway from Beverly Hills to a newly constructed Interchange at St. Peters in Sydney, New South Wales, Australia.

The geological and geotechnical conditions were mapped by project geologists and engineers for each tunnel heading, with an estimated 13,000 or more geological maps being generated. The majority of the excavation on the project was accomplished by roadheader. The mainline tunnels were referred to as M110 (eastbound traffic) and M120 (westbound traffic). Mining headings for each of the mainline tunnels during construction were typically split during construction, i.e., the left-hand side of the tunnel (the leading heading) would be excavated first, then supported. Excavation and support of the right-hand side (the trailing heading) would then follow (or vice versa). Following the bulk of the excavation work, a “bench” was typically excavated from the tunnel invert.

The tunnel support design consisted of several classes of rockbolts and shotcrete, designed mostly pre-construction by Aurecon-Jacobs Design Joint Venture and McMillan Jacobs Associates. Golder Associates Pty. Ltd. (now WSP Australia Pty. Ltd., Brisbane, QLD, Australia) were responsible for the geotechnical design and interpretation of the project. The construction Joint Venture consisted of CPB-Dragados-Samsung and the end client of the project being WestConnex|Transurban.

3.2. Summary of Geological and Geotechnical Conditions Encountered on the Project

The majority of the mainline tunnels were excavated within the Hawkesbury Sandstone, with the Ashfield Shale and Mittagong Formation being intersected in some areas. The Hawkesbury Sandstone is described in further detail below given its relevance to the example tunnel heading provided later in the text, i.e., the location in Figure 4 comprises the Hawkesbury Sandstone from the surface to well below the tunnel invert level.

The Hawkesbury Sandstone is a fine-to-coarse grained quartzose sandstone that is described as a fluvial fan and river deposit. It comprises sequences of sheet (cross-bedded) facies, massive facies and thin shale/laminite facies (beds) interpreted to have resulted from overbank and swamp deposits during quiescent periods. The facies are differentiated by their appearance and are the results of changes in the energy of the alluvial environment of deposition. The Hawkesbury Sandstone formation is in excess of 290 m thick and is made up of repetitions of the three facies described above [24,25].

Laboratory testing indicates the unconfined compressive strength (UCS) of fresh sandstone ranges from 10 MPa to 40 MPa but can be up to 80 MPa for thin beds (usually 1 m to 2 m thick) which may have higher quartz contents and stronger siliceous or carbonate cement. All sandstone types are generally highly abrasive, with the siliceous and carbonate cemented types having the highest abrasivity. Shale units within the Hawkesbury Sandstone, where unweathered, may have an UCS of about 10 MPa to 40 MPa (normal to bedding). Where present, the laminite beds are generally highly anisotropic with a much lower strength parallel to bedding [25].

3.3. Geotechnical Mapping Method

Following excavation of each split heading and prior to a layer of steel fibre reinforced shotcrete (SFRS) being applied to the tunnel crown (and possibly sidewalls, depending on conditions encountered), the tunnel crown and sidewalls were photographed and mapped. The tunnel sidewalls, crown and mined face were geologically and geotechnically mapped for each heading. Typical data collected while mapping included rock type(s), structure, weathering and intact rock strength; groundwater/seepage conditions; discontinuity conditions including dip, dip direction, trace length, aperture, infill type and thickness, shape, roughness and apparent joint spacings (as described in Section 2.1). Assessments were also made using the Q-system, for rock mass classification [23], the V-GSI system [1] and the Sydney Rock Mass Classification [24,25].

For each map, apparent spacing data per discontinuity set was then corrected to true spacing data, which was used to calculate a typical VFC and VFC ratings for the vertical axis of the V-GSI chart. If more than one rock type was present within the tunnel face (for example, if the lower portion of the heading was within the Ashfield Shale, and the upper portion within the Hawkesbury Sandstone, two distinct V-GSI calculations would be undertaken and plotted automatically on a V-GSI chart embedded in the mapping data sheets (see example mapping sheet and associated V-GSI chart in Figure 4). The horizontal axis was quantified using the discontinuity conditions data that can be seen in the example mapping sheet in Figure 4. The mean V-GSI rating was plotted directly in the embedded V-GSI chart (red dot) for each map with a range (red box). The maps were used to inform the level of ground support required (based on pre-defined support classes for each ground type and tunnel geometric layout).

3.4. Example Project Heading—Mapping Data and Photographs

As an example, a single heading split heading map is presented below in Figure 4 for the left-hand-side excavation of the M120 tunnel excavated in the Hawkesbury Sandstone at a discrete tunnel chainage (exact location not provided to ensure confidentially of project information). At this location, the geology comprises the Hawkesbury Sandstone solely from the surface to below the tunnel invert. For clarity, this map is referred to hereafter as the left-hand-side M120 map (LHS M120 map). The map shown in Figure 4 only shows the trailing heading, but once fully excavated (both the left and right headings were excavated), the approximate tunnel excavation dimensions were as follows: width at shoulder—13 m, full height at centerline—8 m once the lower bench was removed (height at sidewalls excavations, and height includes the subsequent “bench” excavation).

For the selected example LHS M120 map, the right-hand side heading was excavated prior to the left, as evidenced by the sidewall and void in the right-hand side of each photograph in Figure 5. Moreover, the adjacent M110 tunnel (eastbound) was excavated prior to the M120 in this location. While only one map is provided (Figure 4), the rock mass conditions for the right-hand side of the M120 were reviewed along with the information from the M110 in the same location.

The rock mass for this mined heading and for this location in general (gleaned from adjacent maps as described above) comprised fresh (i.e., unweathered) sandstone, dry to damp, with a medium to high intact rock strength, and with a layered structure (bedding). Figure 4 above presents one of the output maps generated for this location, which includes (1) a geological map of the mined heading and crown and left hand sidewall, (2) discontinuity data for three observed bedding-parallel discontinuities including the calculated true spacing (and the other data as described in Section 3.3), (3) a V-GSI rating plotted on the embedded chart generated based on this true spacing data and discontinuity condition data, and (4) a stereonet of discontinuity orientation measurements (tunneling direction also shown on the stereonet-black line). The mean VFC is estimated at 0.625 (1/1.6, i.e., 1/true mean spacing of bedding) with a rating of 54, the mean 1.5 × Cond89 is around 29 and, as shown on the map, the calculated mean V-GSI rating is around 83 (54 + 29). Jr/Ja is around 0.75 and the calculated Q’ rating is somewhere between 33.75 and 37.50. As can be seen on the scales on the horizontal axis, the lower range of 1.5 × JCond89 is around 24, which correlates to 0.75 for the Jr/Ja assessment. This correlation on the lower scale provides a convenient method of cross-checking inputs for these two parameters. If they are not close, then inputs for both parameters should be checked.

Figure 5 below shows two of the photographs attached to the trailing heading and the partially obstructed crown (bolt installation in the crown) LHS M120 map report. Note that the dark rock (minor laminated siltstone beds) in the crown and heading visible in the photographs are photos also shown in the geological map in Figure 5.

The selected example map results indicate that the ground conditions were within the expected category (conditions predicted prior to construction), and therefore the pre-designed ground support class for these conditions was installed. In this location, the support class included arrays of 10 fully grouted and tensioned double-corrosion protected, end-anchored rock bolts in the crown, each 3.40 m in length, spaced on 1.5 m centers and at 1.5 m transverse spacing. In addition, a primary/initial layer of 80 mm thickness of SFRS was applied. In addition, for this specific heading, handlebar plates were installed within the laminite zone in the crown for better adherence and structural connectivity of the SFRS. Similar conditions were also encountered on the leading (right-hand-side) excavation heading.

3.5. Finite Element Analysis of Example Tunnel Heading

Using the software RS3 by Rocscience (Version 4.024, © 2022 Rocscience Inc.), 3-D Finite Element Analysis was conducted to model tunnel convergence or deformation in the tunnel around the location of the example LHS M120 map presented in Figure 4.

Table 6. Material property input parameters used for finite element analysis.

Parameter	Value	Comments
Unit weight	0.024 MN/m3	Typical value for Hawkesbury Sandstone
Poisson’s Ratio	0.2	Typical value for Hawkesbury Sandstone
Intact Rock UCS	18.2 MPa	Estimated UCS based on project-specific conversion of point load testing near example heading (within typical range for weaker Hawkesbury Sandstone) and calibrated with UCS testing data
V-GSI	80.2	Refer to V-GSI calculation method in Section 2. VFC rating from mapping is based on 1.4 m average true spacing from face maps from both left- and right-hand-side headings (only the LHS map is shown in Figure 4), resulting in a VFC of 0.71 (Equation (6)) and a rating of 51.7. 1.5 × JCond89 is around 28.5 based on average discontinuity condition of planar, slightly rough, <1 mm coating of silt. V-GSI = 51.7 + 28.5 = 80.2. (Equation (10))
mi	13	Typical value for Hawkesbury Sandstone
D	0	Mechanical (roadheader) excavation and therefore no damage assumed
Ei	4 GPa	Typical value for the Hawkesbury Sandstone corresponding to the intact rock UCS noted above

below presents the material properties using the generalized Hoek–Brown failure criterion [22] that were selected for use in the software to represent the Hawkesbury Sandstone in the vicinity of the selected example tunnel heading. Strength and deformability estimates and associated H-B material properties are summarized in

Table 7. Summary of strength and deformability estimates for the tunnel rock mass shown in Figure 4, Figure 5 and Figure 6.

Hoek–Brown Parameter	Mean
mb	6.41
s	0.111
a	0.501
Rock mass tensile strength (MPa)	−0.315
Rock mass uniaxial strength (MPa)	6.051
Rock mass global strength (MPa)	7.6
Rock mass modulus (GPa)	3.53
Equivalent cohesion c (MPa)	0.927
Equivalent friction angle (Phi-degrees)	53.6

Parameters calculated using software RSData Version 1.0.0.3 (© 2020 Rocscience Inc.). Input parameters to RSData provided in Table 6. The same parameters are calculated automatically in RS3.

(the data provided in Table 7 are automatically calculated in RS3 using the input from Table 6).

The objective of this exercise was to compare the measured convergence from monitoring prisms in the crown of the tunnel in this location against predicted deformation from of the 3-D modelling. The tunnel geometry used to build this model is described in Section 3.4. The tunnel in this location was approximately 63 m below ground surface (measured to the tunnel invert) and this depth of cover was used to estimate the overburden stress. No groundwater was applied for modelling purposes, even though the map describes the rock mass as “dry to damp” as the tunnels were designed as draining structures. It is assumed that the groundwater table had already been drawn down in this location by excavation of the adjacent parallel tunnel (which was excavated in advance of the selected example heading presented). Based on project experience, a stress field in the model was defined based on Equations (11) and (12):

$${\sigma }_H=2.0\times {\sigma }_v+2.5\left(\mathrm{MPa}\right)$$

(11)

$${\sigma }_h=1.6\times {\sigma }_v \left(\mathrm{MPa}\right)$$

(12)

where ${\sigma }_v$ is the vertical (lithostatic) stress, ${\sigma }_H$ is the primary horizontal stress, trending 90° in model space (i.e., perpendicular to the direction of tunnel drive), and ${\sigma }_h$ is the secondary horizontal stress, trending 0° in model space (i.e., parallel to the direction of tunnel drive), all expressed in MPa. Note that the addition of 2.5 MPa in Equation (11) represents an assumed “locked-in” stress based on experience working in the Hawkesbury Sandstone.

Of specific interest in Table 6 is the V-GSI rating, which was calculated using representative values for both the left-hand-side and right-hand-side maps of the example heading, not just that assessed for the map shown in Figure 4. There is only one discontinuity set (bedding-parallel discontinuities) with an average true spacing (St) of around 1.4 m in both the M120 and M110 in this location with an associated VFC of 0.7 factures/m³ (1/1.4), and therefore a V-GSI rating of around 80.2 using the method described in Section 2 above.

Using the parameters for the Hawkesbury Sandstone presented in Table 6, and the ground support discussed in Section 3.4, 3-D finite element analysis was conducted with three excavation stages (in chronological order based on mapping records). These stages are presented in Figure 6 below:

• Excavation of the mainline M110 (eastbound) tunnel and installation of ground support in this tunnel;
• Excavation of the leading right-hand-side heading of the M120 (westbound) tunnel and installation of ground support in this leading heading;
• Excavation of the left-hand-side M120 trailing heading (the heading shown in the example map presented in Figure 4), and installation of ground support in this heading.

Figure 6 shows the “vertical displacement (the “z displacement”—displacement in the z axis of the model) around the excavation perimeter for each stage. Note that tunnel support is hidden from view for better viewing of the excavation geometry, but in Figure 6d, bolts and shotcrete are shown within one tunnel of the model (for the final stage, when excavation and support have been added for all sections of the tunnel).

While excavation/ground support in this section of the tunnel was carried out typically in split headings of around 5 m length, the model was simplified to reduce excavation stages such that an advance of 50 m (the entire longitudinal span of the model in 3-D space) was excavated at once. Ground support (bolts and SFRS) were installed within the same excavation stage within the model. Moreover, as the M110 tunnel was not the subject of modelling efforts, it was simplified to be excavated in one stage (the first stage), although in reality would also have been excavated as a split heading. The effects of these simplifications are discussed further in Section 4.

Figure 7, Figure 8 and Figure 9 present various model outputs viewed on a two-dimensional section plane of the model, cut perpendicular to the tunnels. The LHS M120 example heading presented in Figure 4 is shown on the left in each screen capture in these figures. Ground support is again hidden in these figures, but is accounted for in the model. Figure 7 shows the vertical displacement in the rock mass around the tunnels (the “z displacement”—displacement in the z axis of the model), Figure 8 shows the total displacement (i.e., the absolute value of displacement) around the tunnels, and Figure 9 shows the principal effective stress in the rock mass around the tunnels. Note that, as no water table has been applied and the rock mass is assumed to be drained, the total stress is equivalent to the effective stress (shown as effective stress in Figure 9). Figures showing yielded elements and yielded support have also not been shown, as no yielded elements or support yielding were recorded in the modelling results.

Figure 7 effectively shows the same results as shown Figure 6c,d, but on a more visible scale. Of particular note for later discussion is the maximum downward vertical displacement at three locations in the crown (negative values indicate downward displacement). The maximum downward vertical displacement in the crown in Figure 7 is approximately 2.2 mm. Figure 9 indicates that the principal stress is concentrated in the tunnel crowns.

3.6. Comparison of Measured Convergence and Modelled Displacement

As tunnelling progressed, prisms were installed for convergence monitoring at specified intervals and at locations within the tunnel sidewalls and crown. Typical installation locations included one prism on each sidewall, and three prisms in the crown. Prisms were only installed once the support was installed for safety reasons. Of interest to our selected example are the three survey stations located near the tunnel crown. Convergence was tracked by assuming zero displacement at installation and relative movement over time at these stations.

A query line was generated within the 3-D model, within the section plane shown in Figure 7, Figure 8 and Figure 9 and across the crown of the excavation perimeter. The vertical displacement across this query line was graphed initially assuming no ground support, and then with ground support as described above (Figure 10).

The approximate locations of the three survey stations located in the tunnel crown are also presented in Figure 10. As illustrated in Figure 10, the maximum downward vertical displacement at the station near the crown centerline is around 2.2 mm when no support is installed, and 2.1 mm when the support is installed. Left and right of the centerline, downward vertical displacements are approximately 1.8 mm and 2.1 mm, respectively, with support installed.

An additional data series is shown on Figure 10, showing an additional scenario that was modelled where ${\sigma }_v$ = ${\sigma }_H$ = ${\sigma }_h$. With the stress fields defined as such, the vertical displacement is shown to increase significantly. This is shown to illustrate the model’s sensitivity to changes in the input stress conditions, and this is an important consideration since the exact horizontal stress conditions were not fully understood or measured.

The convergence data recorded over time at this station can therefore be compared to the modelled vertical displacement near the three survey stations located in the tunnel crown. This comparison is presented in Figure 11 and Figure 12 and indicates that the modelled vertical displacement (2.1 mm to 3.75 mm is within the range of values of the measured displacement data (1 mm upwards to 5 mm downwards) at this station. These results are discussed further in Section 4.

4. Discussion

In the finite element example provided in Section 3 above, a single V-GSI rating value (80.2) was used in the modelling. In the examples provided in Section 2, the best practice method of quantifying the V-GSI rating is to include a range or confidence limits for the mean V-GSI rating where possible. This would require running the model three times using the upper bound value, mean value and lower bound values. This approach would provide a sensitivity check on the range of deformation for comparison with the monitoring data.

As can be seen in Figure 11 and Figure 12, the movement in the crown of the tunnel ranges from 1 mm of upward movement to a maximum of 5 mm of downward movement over a period of time. The support is installed before monitoring starts. The prism installation accuracy levels are typically in the 1 mm or less range, and therefore the monitoring data are indicative rather than absolute values. The finite element modelling predicts just greater than 2 mm in the crown. This level of deformation is also indicative rather than absolute, and depends on the degree of confidence of the input parameters when setting up the models. As can be seen in Figure 10, for example, deformation in the crown is quite sensitive to horizontal stress inputs, with deformation increasing up to 3.75 mm for ${\sigma }_v$ = ${\sigma }_H$ = ${\sigma }_h$ conditions. Despite the indicative nature of the monitoring data and modelling results, there is a reasonable level of correlation between the two data sets, and they both approximate the real deformation at this location in the tunnel.

It was standard practice to determine V-GSI rating for each heading during the M8 tunnel excavation, and on occasions where monitoring results indicated deformation threshold exceedances (shown in Figure 11 and Figure 12), it was possible to undertake modeling to determine if revised or additional ground support was required. The V-GSI rating provided key input data to this type of modeling.

While only one example of 3-D finite element modeling is shown in this paper, it demonstrates that the V-GSI ratings established via the method presented in Section 2 can be used with some confidence to determine tunnel rock mass conditions, and hence performance.

The calculation of a VFC for a rock mass, and its application to the V-GSI, does have limitations that need to be acknowledged. For example, discontinuity sets that are sub-parallel to the working heading should be sampled in the crown and sidewalls where possible, along with corrected true spacing measurements, and these data should be included in the VFC estimates to avoid underestimating this parameter. The VFC value can also be easily overestimated if intensely fractured areas of rock are exposed in a heading associated with a joint swarm and the spacing within the swarm are used to determine the VFC contribution of a particular set without considering areas of rock mass with less fracturing between swarms. Persistence of discontinuities (trace lengths), if not adequately understood, can also result in overestimating VFCs. This can be partially overcome by sampling where joint terminations are noted and larger spacings are present between joints that are farther apart.

Bedding-parallel discontinuities are common within the Hawkesbury Sandstone, and as tunnelling progresses, these may occasionally be encountered (but not seen or mapped) a short distance above the tunnel crown. The VFC contribution from bedding will accommodate this—as explained in Section 2—because ‘greater than’ (>) spacing measurements are included on the assumption that there is a bedding plane located just above the crown. Nevertheless, it is important to note that if the bedding plane is open, has clay gouge infill and the horizontal stress is high, the bed above the crown can flex, acting as a beam, and result in increased downward displacement. Under these conditions the installation of additional ground support may be required, particularly in the case of high horizontal stresses. This phenomenon was monitored through the use of regular endoscopic surveys in drillholes in the tunnel crown. The modelling presented in Section 3 is a continuum method and cannot account directly for the flexural displacement of beds. Nevertheless, the modelled vertical displacement, while slightly less, are comparable to the monitored displacement, even though it is known through endoscope monitoring that there was a partially open bedding plane 3.3 m above the crown in the location modeled.

Simplifications to the model are described in Section 3.5. These simplifications are expected to slightly reduce the magnitude of displacement in the modelled rock mass at the excavation crown. This is because, in reality, the rock mass would relax slightly before support elements are installed, and this may result in slightly more deformation that in the model. As described in Section 3.5 above, support in the model is installed instantaneously, and more load is picked up by the support. The difference is nevertheless expected to be insignificant, and this effect is shown in Figure 10, which shows several modelling scenarios including one with no support installed and one with support installed. These scenarios show <0.1 mm difference between supported and unsupported ground for a model length of 50 m. The length of the model (50 m) is not expected to influence the magnitude of displacement. This is likely since there is little yielding of the rock mass due to its strength relative to the field stresses.

5. Conclusions

Vertical displacement data for the tunnel crown in a discrete location in the WestConnex M8 Motorway tunnel (see Figure 4) was compared to modelled tunnel deformation. Key input data to the model included the use of a V-GSI rating calculated using the tunnel mapping records and the methods and equations presented in Section 2 to determine a bias-free VFC for the tunnel rock mass. The results demonstrate good correlation between predicted and observed deformation in the crown, and this demonstrates that the V-GSI chart and Hoek–Brown equations [1,3,16] can be used with some confidence to determine potential deformation in underground works. This is particularly useful for assessment of rock masses in tunnels that are blocky or massive at the scale of interest (i.e., where the volumetric fracture count may be less than 3 fractures per m³, and where RQD is insensitive at 100%); conditions that are common for the Hawkesbury Sandstone in Sydney.

Good correlation between predicted and measured deformation provides further indication that the V-GSI chart is a reliable tool for the bias-free VFC assessments of rock masses and strength and deformability estimates for underground works. The V-GSI chart was originally developed based on work undertaken for the Katse Arch Dam by Schlotfeldt and Carter [1]. This work, also validated by comparing predicted and measured deformation moduli, showed that the methods can be used in both underground and surface works.