# Investigation of Virtual Bimrocks to Estimate 3D Volumetric Block Proportions from 1D Boring Measurements

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## Abstract

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## 1. Introduction

- 2D measurements (areal block proportions, ABPs), which can be obtained by examining geological maps, mapping outcrops or photographs and/or by using digital image analyses. Specifically, the ABP can be determined as the ratio between the area of all blocks measured in a sample area and the sample area analyzed [1,3,27,36,37,38,39];
- 1D measurements (linear block proportions, LBPs), which can be obtained by analyzing exploration drilling or linear sampling traverses (scanlines) on outcrops/photographs. Specifically, the LBP can be determined from the proportion of total intercept lengths of blocks penetrated by drill cores (or scanlines) to the total length of drilling [1,2,40,41,42,43,44,45];
- 0D measurements (node or point block proportions, PBPs), which consists of the node (or point) counting technique. This is a common method in several research fields (including geology, biology and materials science) to allow the proportion of an area covered by some objects of interest to be easily determined. Specifically, a grid is i {\displaystyle i} superimposed over an image, the intersection points are counted and then divided by the total number of the points of the grid (Medley, 1994).

#### Uncertainties in VBP Estimates: An Overview

_{c}(the ced of Medley, [1]). L

_{c}is a length that signifies the scale of the problem at hand: the height of a landslide; the width of a foundation; the square root of the area of a site; the diameter of a laboratory triaxial specimen; the diameter of a tunnel, etc. Each model had an L

_{c}of about 130 mm, which was the square root of the measured plan area of the models. As per convention when studying bimrocks [1,12], the smallest blocks were about 0.05L

_{c}(6 mm) and the largest blocks (d

_{max}) in each model were selected to be about 0.75L

_{c}.

_{max}, to provide the parameter Nd

_{max}(in retrospect Nd

_{max}was poorly defined: it should have simply been L/d

_{max}= N).

_{max}, increased (i.e., with the increase in number of borings and consequent total lengths of exploration). For a sufficient linear sampling, at least equal to 10-times the length of the largest expected block (10d

_{max}), there was a tendency for the LBPs to converge to the actual 3D VBP.

_{max}axis—as shown in Figure 2b. The charts allow geopractitioners to answer the question: “If LBP is assumed to be the same as the VBP, how wrong is the assumption?”. For a given value of Nd

_{max}and total LBP, Figure 2b provides values of uncertainty as SD/VBP (standard deviation, SD, divided by VBP—essentially the Coefficient of Variation, CV). In fact, for sufficient sampling, the mean of LBPs is very close to the value of the VBP. The uncertainty provides a +/− number, which is used to adjust the LBP to VBP using the relation: VBP = LBP +/− (Uncertainty × LBP). In practice, to err on the side of safety, the uncertainty should be subtracted from the LBP for purposes of estimating the block content for use in VBP vs. strength/deformability relationships or graphs [9,14,33]. On the other hand, the uncertainty should be added to the LBP to provide a prudent (high) VBP estimate when planning engineering works in bimrocks (to avoid the economic repercussions of underestimating undesirable block contents in tunneling and excavations) [2,12,43].

_{e}, of in situ blocks necessary to use the chart the authors proposed. Another relevant and useful work was performed by Ramos-Cañón and co-workers in 2020 [6]. The authors implemented a computational algorithm to analyze the influence of the block sizes, shapes and orientations, the perforation length (block/boring intercept) and the number of boreholes on the uncertainty in VBP estimates. Specifically, 3D cubic bimrock/bimsoil samples with different dimensions and VBPs in a range 4–19% were created, using both spherical and ellipsoidal blocks. The sampling was carried out for each model, through a variable number of equidistant penetrations. The results of this study indicate that the most influential parameters are the total length of drilling and the number of boreholes, while the block characteristics and the dimension of the 3D bimrock/bimsoil domain were not found to be statistically significant. Nevertheless, the very low VBPs investigated by Ramos-Cañón et al. [6] limit the geopractice applicability of their findings.

## 2. Uncertainty in VBP Estimates from LBPs

#### 2.1. Validation of Medley’s 1997 Findings

_{c}(the ced of Medley, 1994): the smallest blocks were equal to 0.05√A = 0.05L

_{c}= 6.5 mm and the largest blocks (d

_{max}) were equal to 0.75√A = 0.75L

_{c}= 98 mm.

_{max}parameter used by Medley [2,43,49], and expresses the cumulative length of simulated drilling as a multiple of the maximum size of the largest expected block (d

_{max}).

_{max}values (i.e., 70 mm, 84 mm, 85 mm, and 95 mm, respectively, for the physical models 13%, 32%, 42% and 55% VBP) as well as slightly different heights and, therefore, different scanline lengths and resultant N values. However, as indicated above, virtual bimrock models with constant heights, scanline lengths and d

_{max}were considered in the present study.

#### 2.2. Extension of Medley’s 1997 Findings

_{max}= 0.75L

_{c}) provides a statistically viable total length of sampling and resulting LBP. That our work supported this useful “10 times rule-of-thumb” is one of the satisfying outcomes of this research.

#### 2.3. Effects of Increasing the 3D Domain Size

- The use of a larger domain allowed an increased total number of simulated boreholes (ϑ) to be analyzed and avoided short boring spacings, which would have made the LBPs of two neighboring boreholes almost identical. Hence, the increased geometry allowed for larger virtual boring spacing and more variable datasets of results to be analyzed;
- With higher ϑ values (total number of scanlines analyzed), it was possible to consider more varied borehole location distributions by adopting a greater number of randomizations (λ) for each sub-set of combined scanlines (β);
- A higher ϑ allowed the values of β considered in the analyses to be increased, to obtain results for values of N higher than 45. It is emphasized that although the value of the UF can be determined from Figure 2b [49] for values of N (Nd
_{max}) up to 100, in actuality, Medley [49] extrapolated the line trends in the interval (2, 25–45) rightwards for N of 45 (i.e., in the interval 25–45, 100). Therefore, the region of the Medley [49] graph beyond Nd_{max}~45 likely reports suspect uncertainty factors.

_{c}, was kept constant at about 130 mm to maintain the same minimum (0.05L

_{c}) and maximum (0.75L

_{c}) limits of the block-size distribution used by Medley [43]. Moreover, the relative frequencies of the dimensional classes used in the analyses with the smaller domains were also preserved. Rather than work with the fixed geometry of the small-domain models shown in Figure 1 and Figure 5, the geometry of the large domain models was generalized by adopting multiples of L

_{c}. The dimensions of the large domains (LD) were H × B × L = L

_{c}× L

_{c}× 10L

_{c}and were, thus, about 8-times longer and 10% less wide than the small-domain (SD) models analyzed previously. By way of example, Figure 8 illustrates the 32% VBP bimrock model created for the extended investigation.

## 3. Application Example: VBP of the Mélange in Foundation of Scott Dam, California

_{max}) and the LBP was 40%. From Figure 2b, the uncertainty was, thus, about 0.2. Hence the VBP was 40% +/− (0.2 × 40)% or the range 32% to 48%. The 32% value would be applicable for use in assigning strength values. In the case of the dam site, 31% VBP was adopted based on other considerations (i.e., site investigations and interpreted lithologic logs, assembled by considering the lithology of the cuttings and other information from the driller’s logs) [12,40].

## 4. Results and Discussion

_{max}, or 0.75L

_{c}) provides enough data for dependable LBPs for use in identifying the uncertainty in estimates of VBP.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Scanlines (blue) traced on 42% physical bimrock model (original of monochrome image of Figure 3 of Medley’s [43] work).

**Figure 2.**(

**a**) Uncertainty (CV) in estimates of the VBP from 1D measurements as a function of the total sampling length (expressed as Nd

_{max}or the multiple N of the length of the largest block, d

_{max}) and the measured LBP (modified from [49]); (

**b**) portion of Figure 2a (red box), drafted as a design aid, in which trend lines were very approximately mapped through the data points in Figure 2a (modified from [49]).

**Figure 3.**Uncertainty, expressed with a coefficient of variation (CV) vs. relative lengths of scanlines, L/D

_{e}, obtained with the analytical and numerical solutions by Lu et al. [45] and compared to Medley’s [43] findings. In this graph, L

_{b}indicates the linear fraction of blocks, L is the total length of the scanline and D

_{e}indicates the equivalent diameter of the block (modified from [45]).

**Figure 4.**Uncertainty in the VBP estimate from 2D measurements, as a function of the total investigation surface (expressed as multiples, β, of the area of engineering interest, A

_{c}= L

_{c}

^{2}) and block contents measured (ABP) (modified from [4]).

**Figure 6.**Uncertainty in the VBP estimate from LBPs as a function of the total sampling length, N, obtained in this research. Medley’s (1997) [43] results are also shown by way of comparison.

**Figure 7.**Influence of λ (maximum number of randomizations) on the uncertainty in the VBP estimate from LBPs, as a function of the total sampling length, N. Dashed trend lines from Medley [49] chart (Figure 2b) shown for comparison with λ = 40 trend lines from this study. (Note: symbols on trend lines are line markers and not data points).

**Figure 9.**Uncertainty in the VBP estimate from LBPs, when a larger 3D domain is analyzed (SD: small domain, LD: large domain).

**Figure 10.**Recommended chart superseding Figure 2b from Medley [49]: Uncertainty in the VBP estimate from LBPs, as a function of the total sampling length. Since the trajectories for the 50% and 60% VBPs are superimposed, the trend for VBP = 70% is assumed to be also co-incident. (Note: symbols on trend lines are line markers and not data points).

**Table 1.**Specifications of the linear fittings shown in Figure 10. UF is the uncertainty factor, c1 and c2 are constants and R

^{2}is the coefficient of determination.

VBP [%] | Fitting Equation | c1 [-] | c2 [-] | R^{2} [-] |
---|---|---|---|---|

10 | UF = −c1∙ln(N) + c2 | 0.1940 | 0.9491 | 0.925 |

20 | 0.1303 | 0.6386 | 0.926 | |

30 | 0.1075 | 0.5266 | 0.927 | |

40 | 0.0771 | 0.3777 | 0.928 | |

50 | 0.0558 | 0.2735 | 0.925 | |

60 | 0.0558 | 0.2726 | 0.923 |

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Napoli, M.L.; Milan, L.; Barbero, M.; Medley, E. Investigation of Virtual Bimrocks to Estimate 3D Volumetric Block Proportions from 1D Boring Measurements. *Geosciences* **2022**, *12*, 405.
https://doi.org/10.3390/geosciences12110405

**AMA Style**

Napoli ML, Milan L, Barbero M, Medley E. Investigation of Virtual Bimrocks to Estimate 3D Volumetric Block Proportions from 1D Boring Measurements. *Geosciences*. 2022; 12(11):405.
https://doi.org/10.3390/geosciences12110405

**Chicago/Turabian Style**

Napoli, Maria Lia, Lorenzo Milan, Monica Barbero, and Edmund Medley. 2022. "Investigation of Virtual Bimrocks to Estimate 3D Volumetric Block Proportions from 1D Boring Measurements" *Geosciences* 12, no. 11: 405.
https://doi.org/10.3390/geosciences12110405