1 Introduction

There are numerous, well-established in-situ stress measurement techniques that are being used in practice. The knowledge of in-situ stresses is important for both engineering analysis and design. In underground excavations, the in-situ stress regime can greatly affect stability, and it is also one of the most important boundary conditions for the analysis. Existing widely used in-situ stress methods include over-coring and hydraulic fracturing (Amadei 1984). However, such methods can be expensive, laborious, and time-consuming.

In underground mining applications, in-situ stresses change with the underground activities such as drift development and stope extraction. Stress changes are not only influenced by the size and shape of underground excavations. They are also influenced by the presence of geological features such as faults, shear zones, foliations, and others. This makes it extremely difficult to predict and verify the changes of in-situ stresses even with the use of advanced numerical models (Kaiser et al. 2000). Hence, investigating the development of an effective and practical method to estimate excavation-induced stresses should prove useful in practice. The approach presented in this study and applied to underground mining, can be adapted to other engineering applications such as civil engineering shafts and tunnels in hydro power and transit projects.

There are three major categories of in-situ stress measurement. They are destructive, semi-destructive, and non-destructive methods. Hydraulic fracturing is one example of destructive methods, whereas over-coring is an example of semi-destructive methods. The diametral rock core analysis includes extracting and analyzing rock core deformations due to stress/strain relief in laboratory is considered a non-destructive approach. Strain relief occurs once the rock core is drilled off; the relief is a function of the surrounding stress field (Brady and Brown 2006). Although all approaches require high level of precision, it is relatively easier to take precision measurements in the laboratory rather than in the harsh environment of the mine site. This study adopts the non-destructive approach; it uses exploration rock cores that are drilled off primarily for the purpose of orebody definition. An experimental setup has been built to measure and analyze the diametral core deformations in laboratory. This study aims to show how rock core diametrical deformation analysis can be employed to determine the differential stress or the brittle shear ratio in the rock mass.

In the case studies presented herein, the rock samples are retrieved from depths of 1.3 to 1.8 km below surface. The influence of core size on the accuracy of the results when applying diametrical core deformation analysis is discussed Also, the relation between differential stresses and depth is investigated.

2 Diametral rock core deformation

Figure 1 illustrates the process of strain relief in a rock core upon extraction. In Fig. 1a, the core diameter is before extraction \({d}_{0}\) and is subjected to local principal stresses \({\sigma }_{\mathrm{h}}, {\sigma }_{\mathrm{H}}\) acting on a plane perpendicular to the core axis. Upon the extraction of the core, it expands due to stress relief to \({d}_{\mathrm{max}}\) and \({d}_{\mathrm{min}}\) as shown in Fig. 1b. Thus, by accurately surveying the profile of the rock core, it is possible to determine the magnitudes of the maximum and minimum diameters \({d}_{\mathrm{max}}\) and \({d}_{\mathrm{min}}\) as well as their orientation. Funato and Ito (2017) developed an analytical model based on the theory of linear elasticity for isotropic, homogenous, and continuous rock, which relates the maximum and minimum core diameters to the local principal stress field \({\sigma }_{\mathrm{h}}, {\sigma }_{\mathrm{H}}\). It is given by

$$\sigma_{\text{H}} - \sigma_{\text{h}} = \frac{E}{1 + \nu } \frac{{d_{\max } - d_{\min } }}{{d_{\text{o}} }} \approx \frac{E}{1 + \nu } \frac{{d_{\max } - d_{\min } }}{{d_{\min } }}$$
(1)

where E and \(\upnu\) are Young’s modulus and Poisson’s ratio, respectively. Since only one equation is derived for the core deformation analysis, even with known rock properties only the differential stress and principal orientation can be determined.

Fig. 1
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Schematic of rock core relief: a Rockmass under biaxial stresses; b Rock core deformation after taking out

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This study is concerned with the application of Eq. (1) to underground mining. It focuses on estimating the local principal stress differential, \({\sigma }_{\mathrm{H}}-{\sigma }_{\mathrm{h}}\) (Fig. 2) located on a plane perpendicular to the core axis. In the special case when the core axis is oriented parallel to one of the principal in-situ stresses, the stresses \({\sigma }_{\mathrm{H}}\) and \({\sigma }_{\mathrm{h}}\) will automatically be the remaining in-situ principal stresses as shown in Fig. 2.

Fig. 2
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Illustration of local stresses \({\sigma }_{\mathrm{H}}\) and \({\sigma }_{\mathrm{h}}\) to the extracted core axis

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3 Experimental setup

This experiment aims to measure the deformation of rock core sample caused by stress relief and test the elastic rock properties. The experiment can be divided into two parts, namely, rock core diameter measurement and uniaxial compressive strength (UCS) test. The method of diameter measurement is by a laser micrometer and the objective is to determine the maximum and minimum diameters of the rock core samples and provide \({d}_{\mathrm{max}}\) and \({d}_{\mathrm{min}}\) by taking multiple measurements around the circumference of the rock core. The customized test apparatus has been designed to continuously measure the rock cores over 360° rotation with diameters from 30.5 mm (AQTK) to 85 mm (PQ), which covers the range of most used drill bit sizes in mining and exploration projects (Fig. 3). The UCS tests were performed as a second part of the experiment. After determining the maximum and minimum diameters, the rock samples were prepared for the UCS test for the mechanical properties Young’s modulus and Poisson’s ratio.

Fig. 3
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Schematic of experiment for measuring the rock sample diameter

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3.1 Sensitivity analysis

This study requires the measuring instrument to have high resolution to detect small diameter differences more accurately. Sensitivity analysis is performed to investigate how the output factors can be affected by the variation of input factors, and to help determine the required resolution of the micrometer used for measuring the sample diameters.

The base case results for the sensitivity analysis are calculated from the rock core dimensions of \({d}_{\text{max}}\)= 47.61393 mm, \({d}_{\text{min}}=\) 47.59205 mm, and \({d}_{\text{0}}\)= 47.59155 mm, the elastic properties of Young’s modulus and Poisson’s ratio are 76.4 GPa and 0.174, respectively. The stress results of the base case are \({\sigma }_{\mathrm{H}}\) = 37.19 MPa and \({\sigma }_{\mathrm{h}}\)= 7.27 MPa. The sensitivity analysis results are compared with these base results. The aim of performing sensitivity analysis on the inputs \({d}_{\text{max}}\) and \(d_{\text{min}}\) is to ensure that the outputs, stress results, are within an acceptable range of accuracy. Two sensitivity analyses are performed for two resolutions namely 0.00025 mm and 0.0001 mm in input diameters. Clearly, the higher resolution of input measurements would yield higher precision of stress calculation.

Table 1 shows the results of the sensitivity analysis for the resolution of 0.00025 mm, with case 1 being the base case. To assess the influence of errors due to measuring instrument resolution, four analyses are conducted and compared to the base case (Case 1 in Table 1). For each case, the principal stresses are calculated and compared with the base case stresses. The accuracy of the results is shown in %\({\sigma }_{\mathrm{H}}\) and %\({\sigma }_{\mathrm{h}}\). As can be seen from the results, the stress estimation error varies between − 5.6% and + 6.68% for the minor principal stress and between − 1.27% and + 1.31% for the maximum principal stress.

Table 1 ±0.00025 mm resolution results
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Table 2 shows the results of sensitivity analysis for measurement resolution of 0.0001 mm, with case 1 being the base case. Like the previous analysis, four cases (No. 2 to No. 5) are compared to the base case (No. 1). As can be seen in Table 2, the stress estimation error varies between − 1.40% and + 2.67% for the minor principal stress and between − 0.27% and + 0.52% for the maximum principal stress.

Table 2 ±0.0001 mm in diameters from nominal maximum and minimum diameters
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The comparison of the results in Tables 1 and 2 indicate that a measurement resolution of 0.0001 mm in \({d}_{\mathrm{max}}\) and \({d}_{\mathrm{min}}\) will yield more accurate results than a resolution of 0.00025 mm. The comparison also reveals that the minor principal stress values are more sensitive to measurement resolution than major principal stresses. More importantly, the maximum error in the calculation of the differential stress is reduced from more than 5% in Table 1 to only less than 2.3% in Table 2. In light of these results, it can be concluded that that the resolution of the micrometer instrument should not be inferior to ± 0.0001 mm. This will ensure that the principal stress calculations will be within a margin of error of \(\pm 3\mathrm{\%}\).

3.2 Measurement apparatus

Based on the sensitivity analysis, a measuring equipment with 0.0001 mm resolution is necessary for the application of diametrical core deformation analysis. Thus, a laser micrometer system (measuring and display units) from Mitutoyo Corporation was acquired (Fig. 4).

Fig. 4
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Photo of LSM-512S laser micrometer with LSM 6200 display unit

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Based on the experimental objective, the laser micrometer must have a measuring range that covers rock core diameters commonly encountered in mining exploration drilling. The range of diameters vary from 30.5 mm (AQTK) to 85 mm (PQ). Good repeatability is also important to provide reliable results in this study. The instrument and parts required for the laboratory measurement are summarized in Table 3 and illustrated in Fig. 5.

Table 3 Summary of apparatus (refer to Fig. 5)
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Fig. 5
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Apparatus of rock core diameter measurement with rock sample

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Two pairs of high-precision rollers with the same diameter are used to support and rotate the rock sample. The rollers can be either manually rotated or connected to a servo motor using a belt running at 0.7 rpm. A gap between the two pairs is left for the measuring space to allow the laser beam to be interrupted only by the core sample, see Fig. 6.

Fig. 6
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Photograph showing the gap between the two pairs of rollers

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Four pairs of brackets are fabricated for the rollers. Each pair has two aligned holes at the bottom to ensure that the roller axis is perfectly perpendicular to the laser beam and to avoid any shakiness from occurring. Two customized aluminum base plates were designed. The first plate is used as a base for the laser micrometer whereas the second plate serves as a base supporting the rollers as shown in Fig. 6. The second plate has 4 sets of threaded holes along the lines to fasten the roller brackets along any of the lines A, B, C and D shown in Fig. 7. In this way, the roller pairs can be positioned to accommodate different sizes of the rock cores, see Table 4.

Fig. 7
figure 7

Baseplates for the micrometer and rollers

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Table 4 Roller position versus rock core size
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3.3 Mechanical properties

The rock samples were prepared for the UCS test after the diameter measurement, and the test was conducted in accordance with ASTM D7012 to determine the rock mechanical properties namely uniaxial compressive strength, Young’s modulus, and Poisson’s ratio. Testing was conducted using a hydraulic servo-controlled uniaxial press and data acquisition system. Two 120 Ohm strain gauges are installed on each specimen, one horizontal and one vertical for the elastic modulus test as shown in Fig. 8.

Fig. 8
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Sample instrumentation: a Horizontal strain gauge; b Vertical strain gauge

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Based on ASTM D7012, the axial stress load must be applied between 0.5 and 1.0 MPa/s or the test duration before failure must be between 2 and 15 min. To do so, a pilot test with preferably a low loading rate of 0.2 MPa/s is needed to obtain a reasonable time estimate of the UCS. The test setup is shown in Fig. 9.

Fig. 9
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Setup of UCS test

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4 Stress analysis using rock core deformation

The application of diametrical core deformation analysis is presented with two case studies. The rock samples used in this study were collected from Fraser mine and Hoyle Pond mine. In Case study 1, rock core was extracted from Fraser mine. Rock core deformations were measured, and rock properties were tested to determine differential mining-induced stresses (Fig. 10). The results are then reported. In Case study 2, the same procedure as those used in Case study 1 is followed to implement the methodology for Hoyle Pond mine, and the relationship between differential stress and depth is investigated.

Fig. 10
figure 10

3D view of a Fraser mine level 1340 m below surface

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4.1 Case study 1—Fraser mine

The tested rock core samples from Fraser mine are obtained from three different boreholes located approximately 1340 m (4400 feet) below surface. Borehole orientations are listed in Table 5. Overburden stress can be estimated using Eq. (2). Considering a unit weight of rock of 0.028 MN/ m3, the overburden pressure for this case is estimated to be 37.5 MPa.

$$\sigma_{v} = \gamma \cdot z.$$
(2)
Table 5 Borehole orientations of case study 1
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4.1.1 Rock core deformation measurement

The samples of case study 1 were extracted from the three different boreholes by using two different sizes of drill bit: AQTK (30.5 mm) and NQ (47.6 mm). The sample deformations were measured using the laser micrometer system in Fig. 5, and the results are summarized in Table 6.

Table 6 Measurement results of rock samples
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4.1.2 Orientation of principal stress

In general, if the gravity direction is marked (dashed line in Fig. 11) on the rock sample before drilling, the angle (\(\theta\)) between gravity and the major principal stresses can be determined. The principal stress directions coincide with \({d}_{\mathrm{max}}\) and \({d}_{\mathrm{min}}\) (Fig. 11). The reliability of using rock core diametral deformation to determine the principal stress orientation has recently been investigated and confirmed by Ziegler and Valley (2021). However, the rock samples used in this study had no gravity direction mark. Thus, stress directions were only labeled on the samples rather than providing their orientation with respect to gravity mark. Some sample photos labeled with \({d}_{\mathrm{max}}\)(horizontal), the directions of \({\sigma }_{\mathrm{H}}\) and \({d}_{\mathrm{min }}\)(vertical), and the direction of \({\sigma }_{\mathrm{h}}\) are provided in Table 7.

Fig. 11
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Determination of principal stress orientation from rock core deformation

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Table 7 Front view of rock samples showing the orientation of principal diameters
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As indicated in Table 7, \({d}_{\mathrm{max}}\) and \({d}_{\mathrm{min}}\) are nearly perpendicular to each other for the NQ samples. One of the AQTK samples (ID: ST46745A-DIA-51) is shown in Fig. 12. Each sample was measured three times (red lines). However, for the samples with a small diameter, such as AQTK (30.5 mm), the \({d}_{\mathrm{max}}\) and \({d}_{\mathrm{min}}\) labels from each measurement are scattered, and thus, determining the exact directions of \({\sigma }_{\mathrm{H}}\) and \({\sigma }_{\mathrm{h}}\) is difficult.

Fig. 12
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Top view of ST46745A-DIA-51 showing location of measuring results

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4.1.3 Results of Case study 1

The measured diameter values in Table 6 with rock properties that were obtained from UCS test can be directly applied to Eq. (1) to obtain the results of differential principal stress. For example, the NQ size Sudbury breccia rock core sample (ID: 441034-SDBX-84) was used to determine the differential stresses at Borehole 441034 from 84.5 m to 85 m. The principal stresses in the plane perpendicular to the borehole axis were investigated (see Plane A in Fig. 13). The measurement results are as follows: \({d}_{\mathrm{max}}=47.6139\) mm and \({d}_{\mathrm{min}}=47.5921\) mm. The elastic properties of the rock sample were obtained through UCS test, and the deformation measurement results are presented in Table 8. The differential stress for this sample is 31.3 MPa.

Fig. 13
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Schematic showing the plane of stress calculations for hole 441034-SDBX-84

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Table 8 Summary of laboratory test results (441034-SDBX-84)
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4.2 Case study 2—Hoyle Pond mine

The borehole information of rock samples provided by Hoyle Pond mine is presented in Table 9. A total of 19 samples were measured from two boreholes and extracted from 1357 and 1840 m below the surface by using NQ drill bit. The overburden pressure estimated from Eq. (2) for boreholes 25947 and 25986 is 36.6 MPa and 49.7 MPa, respectively.

Table 9 Borehole orientations of case study 2
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4.2.1 Rock core deformation measurement

The measurement results with rock types of Hoyle Pond mine rock cores are provided in Table 10.

Table 10 Measurement results of rock samples
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The average difference between \({d}_{\mathrm{max}}\) and \({d}_{\mathrm{min}}\) in Table 10 is within the range of 0.013–0.058 mm, which is higher than that in the first case. This finding indicates that the difference in principal stresses in Case study 2 is larger than that in Case study 1 (see results in Table 6). Also, it is noticed that the measured diameter results are significantly less than NQ core size (47.6 mm), which can be caused by different factors. One of them is due to material losses during the drilling process.

4.2.2 Results of case study 2

Like case study 1, the test results can be applied to Eq. (1) to calculate differential principal stress. In this case, the rock type from the same borehole remains the same, and thus, the results between different samples are comparable. The NQ size rock cores belonging to the basaltic komatiite type extracted from Borehole #25986 from 43 to 51 m (10 samples) were used to conduct differential stress analysis. Figure 14 illustrates the planes for stress investigation. The elastic properties are E = 83.3 GPa and ν = 0.13. As shown in Table 11, the results are promising. Differential stress exhibits a trend of increase with depth (Fig. 15). Differential stress variation on Fig. 15 can be caused by nearby mining activities. The location of borehole with respect to mine workings was not disclosed by the mine.

Fig. 14
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Illustration of investigation stress planes (Borehole #25986)

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Table 11 Summary of differential stresses (Borehole ID: 24986)
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Fig. 15
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Differential stress changes with depth

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4.3 Practical application of differential stress

As the Canadian mining operations continue to reach deeper deposits, in-situ stresses become higher, the risk of violent brittle failure or rockburst of hard rock becomes high. The differential stress can be used to determine the maximum shear stress expressed as:

$${\text{Maximum}}\,{\text{shear}}\,{\text{stress}} = \frac{{\left( {\sigma_{1} - \sigma_{3} } \right) }}{2}$$
(3)

where \({\sigma }_{1}\) and \({\sigma }_{3}\) are the major and minor principal induced stresses.

Also, combining the differential stress with UCS of an intact rock sample (\({\mathrm{UCS}}_{\text{intact}}\)) it is possible to obtain the brittle shear ratio (BSR), which is a commonly used as a strainburst instability indicator in hard rock mines.

$${\text{BSR}} = \frac{{\left( {\sigma_{1} - \sigma_{3} } \right) }}{{{\text{UCS}}_{{{\text{intact}}}} }}$$
(4)

The rock damage level associated with ranges of the BSR was developed by Castro et al. (2012) (Table 12).

Table 12 Potential rock damage based on BSR
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In Canadian mines, the vertical stress is normally smaller than the two horizontal stresses based on the Canadian shield stress database. As shown in Fig. 16, in the case of a sample retrieved far from the free face, which means \({\sigma }_{z}\) is greater than \({\sigma }_{y}\), then the \({\sigma }_{\text{H}}\) and \({\sigma }_{\text{h}}\) illustrated in Fig. 2 will naturally become \({\sigma }_{1}\) and \({\sigma }_{3}\). Thus, it is practical to use diametral core deformation to determine the differential stress and calculate BSR for the instability analysis of brittle rock mass.

Fig. 16
figure 16

Illustration of principal stresses on the rock core. a Rock core subject to the 3-direction in-situ stresses; b The cross-sectional view of rock core with principal stresses

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5 Discussion and conclusions

Using diametral core deformation to conduct stress analysis requires high precision of the measuring unit and the proper experimental design since every 1 µm change of the rock diameters will affect 1%–3% of the stress results. The two case studies are carried out applying the diametral core deformation analysis to estimate the differential stresses. It is shown that it is feasible to measure and use diametral rock core deformation to investigate the stresses in underground mining. The first case study reveals that the larger the rock core, the better the repeatability of the deformation measurement. That is, large cores with large diametric deformations are likely to lead to more accurate stress estimates. The second case study shows a trend of increasing differential stress with depth; the variation of differential stress may be caused by nearby mining activities. Finally, it is shown that the diametral core deformation also can be employed to determine the strainburst potential of brittle rock in terms of maximum shear stress and the brittle shear ratio.

Although diametral rock core deformation analysis provides a simple and practical method to determine the differential stresses, the following should be noted.

  1. (1)

    This study considers rock has a perfect linear elastic and isotropic behavior, which is not always the case.

  2. (2)

    The rock diametral deformation and fracture caused by stress release can be affected by the mineral components (Gao et al. 2021). The principal stresses cannot be determined with proposed methodology if the deformation is greatly influenced by the cracks (Funato and Ito 2017).

  3. (3)

    This method can be adopted to determine the principal stresses acting on the plane that parallel to the mining face. Thus, the core orientation has to be known when conducting the analysis.