On the Influence of Sampling Scale on the In Situ Block Size Distribution

The discrete fracture network (DFN) modelling is an effective tool to obtain a high-quality characterization of the rock mass (Dershowitz and Einstein 1988; Dershowitz et al. 2004; Miyoshi et al. 2018; Elmo et al. 2021). For that, fracturing information must be obtained at an arbitrarily selected scale. The choice of this scale is subject to principles of economy, geographical accessibility, resources or database availability. Commonly chosen scales range from fracture-plane information obtained from morphostructural lineaments and geological mapping from the exposure of natural or man-made rock outcrops, to larger-scale fracture-hosting geological structures, or major discontinuities interpreted from, e.g. site geophysical surveys (Kattenhorn and Pollard 2001; Boersma et al. 2020).

The discontinuity mapping may be done in practice by on-site inspection of exposed rock surfaces, such as natural outcrops or, in a mining environment, bench highwalls. The scanline is the classical method by which lines are drawn on the outcrop surface, and orientation and dip angles of all joints intersecting the line are measured. In a mine or quarry, scanlines can be conveniently performed on the highwalls. Semi-automatic extraction of rock mass structural data from high-resolution LiDAR point clouds, or from photogrammetry models (Riquelme et al. 2014), has brought a significant advance to the knowledge of the rock mass in mining, be it in safety or in production applications. Such models of the bench faces are available nowadays from inexpensive equipment and processing software, and most mine design and modelling tools incorporate the capacity of importing such digital models.

Lu (1997) compared the in situ and the post-blast size distribution to analyse the blastability of the rock mass. Brzovic et al. (2015) demonstrated that the 3D volumetric fracture density (P32) controls the overall fragmentation of the rock mass in block caving. Wang et al. (2003) developed a software to analyse the characteristics of discontinuities in an ore body and predict the size distribution of ore fragments in block caving. Medinac et al. (2018) applied UAV technology to assess the pre- and post-blast rock fragment size distribution, as well as blast performance. Elmouttie and Poropat (2012) showed that fragmentation prediction can be made from IBSD of jointed mass with relatively small discontinuities. Junkin et al. (2019) used a rock engineering system (RES) approach to generate a fragmentation interaction matrix of controllable parameters, i.e., blast design, explosive energy and blast timing, and uncontrollable parameters, i.e. the physical and structural properties of the intact rock. Aler et al. (1996) compared the in situ block size distribution with the fragment size distribution of the corresponding muck after blasting and assessed the predominant influence of the initial structure of the rock mass on the subsequent blasting results. Latham et al. (2006) present a practical step-by-step methodology for IBSD assessment. Work initiated by White (1977) and extensively developed by Stavropoulou (2014) derived a closed-form solution that could be used to predict the IBSD in a rock mass containing three sets of joints with different spacings, based on the fracture frequency of the rock mass. However, in tectonically complex zones, it is common to find more than three fracture families, so the IBSD cannot be defined using the principal mean spacing (Latham and Lu 1999).

The present work gives an insight into the use of photogrammetry models for fracture measurement, the results of which are compared with structural data obtained from borehole image scanning. The two methods are carried out in pairs to complement the database and compensate intrinsic sampling biases due to the limitations of each technique that are used to obtain complete 3D synthetic fracture networks. Photogrammetry acquisition presents some limitations such as a lower-resolution and a time-consuming mapping phase. Besides, the quality of the photos (mainly sharpness and contrast) may compromise the significance of the resulting tracemaps. On the other hand, televiewer logging implies a high-cost technology, the acquisition is also time-consuming and presents difficulties in detecting fractures near-parallel to the borehole. The presence of dust or mud on the borehole walls could affect the reconstruction of the virtual core. Finally, both techniques require the training of the operator who performs the acquisition and post-processing phases.

Photogrammetry models of highwall faces were analysed for visible joint traces by ShapeMetriX 3D^® and BlastMetriX 3D^® softwares (3GSM 2010a, b). Besides, scanning of the inner surface of boreholes was made by an optical televiewer and discontinuities observed and measured by the WellCAD^® software (Advanced Logic Technology 2017). From the orientation and position of the discontinuities measured by the televiewer, and from the orientation and maps of traces measured on the photogrammetry models, discrete fracture network (DFN) models are generated using the FracMan^® suite (Golder Associates Inc 2018). From them, the distributions of volumes of the blocks formed from the fractures’ intersections (the in situ block size distribution, IBSD) are derived. The different sampling scales and sampling characteristics associated to both monitoring techniques involve a different baseline information on the discontinuities structure that encompasses fracture networks with different characteristics, ultimately resulting in apparently different IBSDs. Such differences, but also some relevant similarities, are highlighted and discussed.

The test site is El Aljibe, a quarry located in Almonacid de Toledo, Spain, that mines mylonites and other metamorphic rocks used as aggregates for track ballast, mainly for high-speed and conventional railway construction. The open pit is located within the Toledo Shear Zone that separates two discrete structural and metamorphic units of the Central Iberian massif: the Toledo Migmatite Complex unit to the north, and the Paleozoic metasediments intruded by the Orgaz granodiorite to the south (Enrile 1991). Three tectonic phases of Hercynian syn-metamorphic structure are observed in both the Migmatites Complex and the Palaeozoic metasediments, with folds tending NW–SE, E–W and N–S, respectively (Enrile 1991).

The highwall faces and boreholes of the blasts that were used for the fractures mapping were located in two blocks on the lower level of the pit, on its southwestern edge (Fig. 1, marked DS1 and DS2). The two blocks were excavated in two separate blasting campaigns. Once the joint maps were analysed, it appeared that there was a potential difference in the structural characteristics of the two blocks, likely due to the influence of a major fault, see “Fracture Domains and Discontinuity Sets”. Fracture data were collected from optical televiewer scans of the inner surface of 57 boreholes, and from photogrammetry models of the highwall faces of 12 blasts. The location of the boreholes and the highwalls is shown in Fig. 2.

Photogrammetry inspects the fractures outcropping the highwall surface, while televiewer investigates fractures that intersect the boreholes. The observation distance is some tens of metres for photogrammetry and some centimetres for optical televiewer, hence the ground sampling distance (the size on the ground of an image pixel, GSD) is greater with photogrammetry than with televiewer, even if the pixel array is denser in the photogrammetry. GSD, image quality (contrast and brightness, mainly), the capacity of the human eye, the expert skills of the operator and the capacity of the automatic recognition system influence the sampling scale (i.e. the minimum size of a morpholine that can be detected, based on an aligned sequence of pixels in the image). In our case, the sampling scales are about 10^–2 m for photogrammetry and 10^–3 m for televiewer, on average. This means that traces (i.e. an aligned recognized sequence of pixels) observed in the borehole logs may not be observable in the photogrammetric models of the highwalls.

Additional data on the orientation of discontinuities was also collected by structural sampling outside the quarry, to establish the relationship with the structural geological setting and to help adjust the distribution functions of the orientations of the families. Three outcrops of small dimensions (around 10–20 m²) were found, located on a nearby hill about 1100–1300 m west of the quarry. Among the different strategies used for outcrop sampling of the fracture orientations (Priest 1983; Rohrbaugh et al. 2002; Zhang et al. 2021), a variation of the area method (Wu and Pollard 1995) was selected and adapted to the outcrop geometry. The outcrops that constitute structural measuring stations had faces with different orientations and aspects, which reduced the possible bias in the orientation of the fractures. After visual inspection and recognition of each structural mode of the area (Enrile 1991), more than 20 orientation data were taken with a compass (Zhang et al. 2021) along the entire length of the outcrop. Orientations (dip directions and dips) were taken at: (i) several points on the same discontinuity and (ii) different discontinuity planes for each identified mode, depending on the quality of the outcrop planes (clarity in the definition of the plane to be measured), so as to assess the dispersion around the sampled mode. A total of 167 structural measurements were taken that allowed to sample surfaces with orientation different from the highwalls and the boreholes, providing complementary information which helped compensate the relative orientation sampling bias of the televiewer and photogrammetry fracture data.

The basic workflow to produce the DFN model can be summarized in four steps: (i) the identification of fracture orientation sets through the analysis of stereoplots; the parameters for the fracture set spherical probability models (mean pole vector and concentration parameters) are initially estimated from manual sectoring, used as initial condition for the computational family clustering; (ii) the identification of potential models for fracture size from bench wall surface fracture trace length data; (iii) the choice of a description of the spatial positioning of fractures (in terms of their centroids) through an assessment of the random versus clustered arrangement of discontinuities; (iv) the adjustment of the fracture number on measured linear and areal fracture densities (P₁₀ and P₂₁, respectively, shown in the abscissas of Fig. 10). The whole process was done with FracMan^® software. As a stochastic model, the DFN is built on a set of parametric statistical distributions which must be estimated with enough degree of confidence. Despite the large amount of data used here, which gives robustness to the estimates of these functions, a number of assumptions must be made when creating the model. For example, the fractures are modelled as flat discs in space, while they should present irregular and warped surfaces to ideally represent the rock mass. As the number of available models to simulate the spatial distribution pattern of the fractures is limited (Baecher, Levy-Lee, Poisson, density based), the amount of data has only allowed to define the model on stochastic point processes, rather than density based. The latter is perhaps a more desirable spatial model, considering the importance of the P₃₂ fracture density and the wide range of fracture sizes. However, it is not of great interest for the purpose of IBSD building in a context of rock blasting, as the fine end of the IBSD is not as interesting as the coarse part since fines are abundantly created from the extremely high stresses created by the blast, and are less dependent on the IBSD fine end. This is revisited in Sect. 4.

The model regions (‘region boxes’) considered for the stochastic simulation of the DFN model are 28 × 28 × 14 m for DS1 and 16 × 51 × 14 m for DS2. The planimetric dimensions are selected so that the plan surface is similar to the areas of the blocks in each domain (Fig. 1). Separate DFN models are built on each domain for each fracture set, calibrating P₃₂ (using the total number of fractures in the region box as variable) with P₁₀ and P₂₁ as target values (Miyoshi et al. 2018) for each family. The best fit of the experimental distributions of trace length are provided by lognormal distributions of fracture size. Fracture centre points are distributed in space according to the Baecher model (Baecher et al. 1977) in each of the spatial domains DS1 and DS2. Each set is generated repeatedly in FracMan^® with different numbers of fractures until the resulting mean densities P₁₀ and P₂₁, at the boreholes or highwall faces, respectively, are equal to the observed ones within a tolerance of 3%. The DFN model used is supported by the statistics of the distributions used for the orientations, fracture sizes and spatial location of each family, which would allow the generation of an infinite number of statistically similar models adjusted to the P₂₁ and P₁₀ of the highwall faces and televiewers, respectively. No attempt has been made to carry out a sensitivity analysis, or to find the impact of the DFN variability on the IBSD.

The number of simulated discontinuities required to match P₂₁ and P₁₀ is significantly different (Table 3), with the number of fractures from P₁₀ calibration being higher. The ratio between the number of fractures from P₁₀ and P₂₁ is also given in Table 3: this varies from 2.3 (Sh) to 11.8 (SvN) for DS1 and from 2.2 (Sh) to 48.7 (SvSW) for DS2. Table 3 also gives the resulting P₃₂ densities. By way of assessment of the fracture intensities, P₂₁ and P₁₀ are plotted versus P₃₂ in Fig. 10; both relations are neatly monotonic, although the latter shows a larger scatter. The information on spacing and orientations from the borehole survey (a mere fracture count along a line) is not enough to replicate the three-dimensional complexity of the fracture network. To reduce the scattering of the linear relation between P₃₂ and P₁₀ intensities (Fig. 10—right), the number of sampled boreholes should be ideally much larger than that measured in this study, so that the P₁₀ estimation is more robust for each discontinuity set in the whole rock mass. Besides, fracture lengths provided by photogrammetry improve the proportionality of P₃₂ versus P₂₁, reducing the scattering of this relation (Fig. 10—left).

Once the numbers of fractures in each family and domain are obtained (Table 3), the complete DFN models in the DS1 (Fig. 11) and DS2 (Fig. 12) domains are obtained at both the highwall (left plots) and borehole (right plots) scales.

The multi-dimensional spacing (MDS) calculation method in FracMan^® is chosen to obtain the synthetic IBSD (Dershowitz et al. 1998) from the DFNs. This algorithm selects random points in a defined domain and generates a series of scanlines in three orthogonal directions (N, E and H axes) from them. This could lead to unrealistic shapes of the blocks due to the independency of the spacing sampled in different directions, but this should not affect the estimation of volumes, and their equivalent size defined as the edge of a cube of equal volume. Other available algorithms have also been tested but they do not fit properly to the configuration of the study case and the characteristics of the fracture sets. The number of randomly selected points for the four models varies from 600,000 to 4,000,000. Those numbers are chosen so that sampling is performed down to the finer blocks (i.e. the sampling distance is in the range of the smaller discontinuity spacing). The boundaries of the region box where the sets are simulated are considered as fractures censoring the block size limited by the face. To obtain a significant sample of the block size population, a total of 10,000 samples are drawn. The resulting cumulative volumetric distributions of block sizes (fraction of rock volume in blocks of sizes less than the variable) are shown in Fig. 13 for each domain and measuring method. Note that the distribution is expressed here in the common way used for cumulative fragment size distributions, i.e. volume fraction (mass fraction is used for fragments) of blocks of size less than the variable (size, and not volume, is used as variable, unlike common representations of IBSD); see also Gama 1983.

The four IBSDs show different results in terms of sizes, larger for the models calibrated on P₂₁ than those calibrated on P₁₀, consistent with the higher density of fractures in the televiewer logs than in the photogrammetry surveys. Likewise, the size distributions of DS2 are finer than DS1 as the fracture density is higher in the former. Note the close collinearity of the distributions, stemming from the similar orientations of fractures in both domains, and with both measurement techniques.

Gamma, Rosin–Rammler–Weibull (RRW) (Rosin and Rammler 1933; Weibull 1939, 1951) and Gates–Gaudin–Schuhmann (GGS, a power law, Schuhmann 1940, 1960) functions were fitted to the cumulative block size distributions:b and n being the scale and shape parameters, respectively. \(\Gamma\) is the Gamma function.x₅₀ and n being the scale (median) and shape parameters, respectively.x_max and n being the scale and shape parameters (maximum size and log–log slope), respectively.

RRW and GGS have long been used to represent rock fragment size distributions from blasting and from other fragmentation processes such as starting with crushing and grinding (Harris 1968; Koshelev et al. 1971; Lundborg 1971; Larsson 1974), followed by a large number of rock fragmentation-related studies, noticeably the Kuz-Ram model (Cunningham 1983, 1987, 2005), arguably the most widely used rock fragmentation by blasting prediction formulae system that uses the RRW distribution as base formula. GGS and modifications thereof (Gaudin and Meloy 1962; Bergstrom 1966; Harris 1968) in turn are preferred for crushing and grinding comminution where a maximum size (e.g. the single-particle crusher or mill feed size) can be clearly identified (Harris 1968; Wills 1992); RRW is also largely used in particle comminution, including an upper truncated variety of it (King 2001; Tavares 2004). Contrary to RRW and GGS, the Gamma distribution is rarely used for describing rock fragmentation; an exception is the work by Faramarzi et al. (2015).

The fitted distributions are plotted in Fig. 14. The power law is fitted at fractions passing below 10^–2 m to determine the slope of the log–log quasi-linear tail of the distributions, as it does not have the upper roll feature of the Gamma or RRW. The parameters and some goodness of fit data are given in Table 4. The performance of the Gamma distribution is outstanding, providing an excellent fit across the whole size-passing range, with a median absolute relative error (MARE, see Table 4) less than 2% in all cases, and with high p values in the K–S test and low K–S statistic. This result perhaps stems from the Poisson process nature of the Baecher’s model fracture generation (Cowan et al. 2003). Conversely, the RRW does not provide a good fit in the upper or in the lower sections of the distribution, other than perhaps in the range 0.01 < P < 0.5; large errors and high K–S statistics rule out this function for representing the IBSD over the whole range in the present case. Reflecting the parallelism of the distributions, the shape parameter of the Gamma distributions varies in a tight range, between 5.03 and 5.29, and the same happens for the exponent of the power functions, between 4.32 and 4.45 (and even if the fits are not good at large, the RRW exponents also show a consistent behaviour).

The different location of the IBSD from the two field sampling methods stems from the different number and spacing of fractures measured; this difference is explained by two main factors. (i) The photogrammetry analysis has a lower resolution in terms of GSD that was 8.5 mm for the photogrammetry and 1 mm for the televiewer. The lower limits of fracture trace length mapped from photogrammetry models are about 0.5–1 m for DS1 and 0.1–0.5 m for DS2. Thus, not all outcropping discontinuities are distinguishable and can be marked by the operator. (ii) The televiewer analysis works on a smaller scale and allows the operator to detect tiny features of the rock, even those that may not likely open upon blasting (e.g. lineation and veins) and that should hardly be considered sides of blocks.

Despite the different locations of the four IBSD, covering different size ranges, the constant log–log slopes of the tails provide evidence of multi-scale validity and a scaling invariant structure (more than two orders of magnitude) of discontinuities of the rock mass. The similar shape factor values of the Gamma distributions show that the invariant structure is preserved along the sampling range.

The fragment size distributions of the blasts carried out in each of the twelve bocks characterized were determined by on-site sieving with a mobile screen; the procedure is described and the fragmentation measured is presented in detail and discussed against blast design in a companion paper (Sanchidrián et al. 2022). The fragment size distributions are shown in Fig. 15 (eleven curves are plotted; for one blast, the coarse cut measurement could not be made so its results were not used), together with the IBSDs for the different domains, from televiewer and photogrammetry measurements; a semi-log plot is used, a common representation of IBSD and also of blasting fragment size distributions, see e.g. Latham et al. (2006) and Bamford et al. (2020). The shock waves radiating from the blastholes, and the high-pressure gases, initially in the range of GPa, crush the rock in the vicinity of the holes and open new shear cracks. Cracks propagate, branch and merge, creating new fragments much smaller than the initial blocks in the rock mass. The large generation of fine material results in cumulative size distributions as those shown in Fig. 15 (log–log slopes around or less than 1 are common). The effect of the blast is (i) to shift the size distribution to the left—towards smaller sizes—and (ii) to incorporate a large amount of fine and very fine material, reducing the overall slope of the distribution. In statistical terms, a reduction of the location parameter and an increase of the variance occur. Note the inconsistent location of the televiewer’s IBSD in DS2 that overlaps the size distributions of the fragmented rock, i.e. the coarser sizes are smaller than the larger blocks observed after blasting. This does not happen—or does to a very minor extent—for the televiewer’s IBSD of DS1, in which the maximum size is similar to the maximum blasted sizes. Televiewer logs record even tiny features of the rock, resulting in overall smaller sizes in the block construction. This feature does not favour televiewer as source of data for building the IBSD, though televiewer data has proved to be helpful in the identification of the fracture domains and the definition of the orientation sets; however, televiewer data are not sufficient for an estimation of the fracture length (hence the large scatter of the P₃₂ vs P₁₀ relation; see Fig. 10). Photogrammetry data does provide this information from the fracture maps of the highwall surface, making photogrammetry surveying essential for a description of the fractures.

The median in situ block size is commonly used as an equivalent to the mean principal joint spacing (see e.g. Latham and Lu 1999), particularly convenient when there are several joint sets. Their impact on the predictions by blast fragmentation models has been tested with the Kuz-Ram (Cunningham 1983, 1987, 2005) and xP-frag (Sanchidrián and Ouchterlony 2017) models. The rock and blasting data employed in the calculation are given in Sanchidrián et al. (2022). The results are plotted in Fig. 15 as examples for two blasts: B6 (first campaign, DS1) and B9 (second campaign, DS2). For each blast, the two IBSD medians were used: for DS1, 2.74 m and 0.72 m for photogrammetry and televiewer, respectively; for DS2, 1.84 m and 0.296 m. Note that for the Kuz-Ram model, only one distribution is plotted for blast B6; the prediction is identical to the two medians, as the joint spacing factor is defined at intervals of joints spacing, and the two DS1 IBSD medians fall in the same interval; this does not happen for DS2. The predicted curves lie generally within the bundle of measured distributions, with shapes and slopes comparable to the measured curves, the photogrammetry-derived ones yielding, as expected, coarser fragmentation than the televiewer. xP-frag seems to be more sensitive to the IBSD median and to yield finer fragment size distributions than Kuz-Ram in this case. The prediction with the televiewer IBSD is quite acceptable for B6 (this blast is in DS1 where the televiewer IBSD is, as discussed above, acceptable); for B9 the xP-frag prediction with televiewer results in a distribution with too small sizes particularly in the upper zone, and a strong bending downwards compared with the measured distributions, a possible result of the obviously incongruous position of the televiewer’s IBSD in DS2. This consistency test does not purport to be an assessment of the application of the Kuz-Ram or xP-frag models or a discussion of their merit or predictive accuracy. Although this might probably be an interesting exercise, it is beyond the scope of this paper.

Discontinuities were monitored in the inner walls of 57 boreholes with optical televiewer and in 3D models of 12 bench faces reconstructed through photogrammetry. Five fracture sets were identified and characterized in terms of spatial distribution, density parameters (P₁₀ and P₂₁), and trace length statistical parametric distributions. Two different fracture domains were defined in terms of fracture density, related to the distance from a major extensional fault, the density of fractures being higher in the zone closer to the fault. In both domains, the density of fractures was consistently higher for the televiewer observations than for the photogrammetry ones.

A DFN model was created for each fracture domain in which distributions of orientation and size were determined for each fracture set. Fractures were generated in space on a Baecher distribution, at a density such that the calculated densities P₁₀ and P₂₁ matched the borehole and surface measured ones, respectively. Four fracture models were created, two for each of the domains, one from televiewer data and another from photogrammetry data. Intensity parameters were verified through their mutual relationships (Fig. 10). The multi-dimensional spacing (MDS) algorithm (Dershowitz et al. 1998) was adopted to sample the population of simulated blocks from the calibrated DFN models to calculate the IBSD. The FracMan^® software suite was used in this process.

Gamma, Rosin–Rammler–Weibull (RRW) and the power law Gates–Gaudin–Schuhmann (GGS) distributions were tested to represent the IBSD obtained. The performance of the Gamma distribution is found to be outstanding, while the RRW performs poorly. The GGS, fitted only to the fines range (less than 10^–2 m fraction passing), shows a good fidelity. The Gamma distribution has been frequently applied to describe discontinuity spacings (Huang and Angelier 1989; Gross 1993; Bonnet et al. 2001; Stavropoulou 2014; Tan et al. 2014; Massiot et al. 2015; Bamford et al. 2017), but not, to the authors knowledge, to describe IBSD. Conversely, a number of previous studies, e.g. Lu (1997), consider RRW, together with GGS and log-linear equations to describe simulation results of the IBSD. Aler et al. (1996) also used RRW distribution to fit the IBSD.

The four IBSD (one from each discrete fracture network built) involve different locations, covering different size ranges, those deriving from higher density of fractures yielding smaller block sizes. The constant log–log slopes of the tails provide evidence of multi-scale validity and a scaling invariant structure (more than two orders of magnitude) of discontinuities of the rock mass. Besides, the similar shape factor values of the Gamma distributions imply that the invariant structure is preserved along the sampling range.

The photogrammetry sampling scale involves that not all outcropping discontinuities are distinguishable, so they cannot be marked by the operator. For its part, televiewer logs sampling tends to record even tiny features of the rock, resulting in overall smaller block sizes in the DFN to the point, in one case, that the maximum sizes of the IBSD from televiewer are significantly smaller than the larger fragments observed after blasting. This indicates that some of the discontinuities observed in the televiewer logs are not activated by the blast either because they are too small, or too tight, or just not even joints but other planar features such as banding, foliation, etc. These features, common in stress metamorphic rocks as the mylonite in this work, do not represent discontinuities or weakness surfaces. This probably rules out the IBSDs determined at the televiewer’s borehole scale unless very expert judgement is exercised to censor non-jointing planar features. Conversely, IBSDs from photogrammetry traces were capable of consistently replicating the scaling characteristics for smaller block sizes obtained from borehole DFN models, providing a lower dispersion in the P₃₂ versus P₂₁ scatter plot. This makes photogrammetry a more complete, standalone surveying technique. It is also faster and easier to apply than the televiewer logging. If photogrammetric models are dense enough, automatic or software-assisted joint detection algorithms could contribute to the acquisition of a proper and inexpensive fracture description of the rock, upon which an in situ block size distribution can be built.