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Citation: 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

Academic Editors: Dominic E.L. Ong

and Jesus Martinez-Frias

Received: 19 September 2022

Accepted: 2 November 2022

Published: 4 November 2022

Publisher’s Note: MDPI stays neutral

with regard to jurisdictional claims in

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iations.

Copyright: © 2022 by the authors.

Licensee MDPI, Basel, Switzerland.

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distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

geosciences

Article

Investigation of Virtual Bimrocks to Estimate 3D Volumetric

Block Proportions from 1D Boring Measurements

Maria Lia Napoli 1, * , Lorenzo Milan 1, Monica Barbero 1and Edmund Medley 2

1Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino,

C.so Duca degli Abruzzi 24, 10124 Torino, Italy

2Terraphase Engineering, Oakland, CA 94612, USA

*Correspondence: maria.napoli@polito.it

Abstract: Serious deﬁciencies in ground characterization, analysis and design at engineering works

can occur when working with bimrocks (block-in-matrix rocks) and bimsoils (block-in-matrix soils).

Since the 1990s, serious technical problems at engineering works performed in bimrocks/bimsoils

spurred practical research, which revealed that the behavior of these geomaterials is directly related to

the volumetric block proportions (VBPs). However, the way that VBPs can be conﬁdently and correctly

estimated remains an ongoing critical issue that still frustrates designers, contractors and owners.

Stereological techniques can be applied to overcome this challenge by inferring 3D block contents from

in situ 1D and 2D measurements, but the estimates have often been demonstrated to be erroneous.

This paper presents ﬁndings from a computer-aided reinvestigation, revalidation and extension

of Medley’s work of 1997 and subsequent researchers to provide approachable yet statistically

robust methods to limit the uncertainty associated with estimates of 3D VBPs generated from 1D

boring/scanline measurements. To this aim, a specialized Matlab code was created and virtual drilling

programs were performed through 3D computer-generated bimrock models. Supported by extensive

statistical-based investigations, a design chart is provided that updates and extends Medley’s 1999

chart relating uncertainty in estimates of VBP as a function of total boring/scanline lengths.

Keywords:

bimrocks; linear block proportion (LBP); volumetric block proportion (VBP); Matlab code;

statistical analysis; design chart; uncertainty factor

1. Introduction

The characterization of block-in-matrix formations (i.e., bimrocks and bimsoils) is

recognized as a key challenge by all geopractitioners and researchers working in the broad

ﬁeld of heterogeneous geotechnically complex formations [1–7].

Simpliﬁed approaches to characterization, such as ignoring the presence of rock blocks,

have often been adopted by geopractitioners to design engineering works in/on bimrocks

and bimsoils. However, as widely demonstrated in the literature, the presence of blocks

cannot be ignored since the strength, deformability and failure mode of these complex

geomaterials are directly related to their volumetric block proportion (VBP), when it falls

between about 20% and 75% [

8

–

15

]. Higher VBPs result in strength increases, lower

deformability and more tortuous failure surfaces [10,16–28].

Moreover, when excavating and tunneling in complex geomaterials, the presence of

rock blocks can cause damage to cutters and/or linings, face instabilities and obstructions

among several others consequences [29–32].

Information about bimrock/bimsoil block proportions is, therefore, extremely important

in order to (i) make reliable predictions of their geomechanical behavior, (ii) choose appropriate

earthwork equipment and underground excavation and support methods

[12,31,33]

and

(iii) reduce safety risks and extra costs caused by unexpected technical problems that can

occur during excavation/construction works.

Geosciences 2022,12, 405. https://doi.org/10.3390/geosciences12110405 https://www.mdpi.com/journal/geosciences

Geosciences 2022,12, 405 2 of 15

Site-scale VBP values are of paramount importance to geopractitioners and it is the

goal of this paper to provide geopractitioners with accessible means to estimate site-scale

VBPs and to understand the uncertainty (error) in those estimates.

According to stereological principles [

34

,

35

], the real block content of a block-in-matrix

formation can be approximated by means of:

•

2D measurements (areal block proportions, ABPs), which can be obtained by examin-

ing 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 [1,3,27,36–39];

•

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 [1,2,40–45];

•

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).

A fundamental law of stereology is that PBP = LBP = ABP = VBP [

34

,

46

,

47

], but the

law holds only when there are many data, which would be extraordinary for conventional

geotechnical exploration programs. Indeed, VBPs estimated by assuming the stereological

equivalence between LBP and ABP measurements and actual 3D values have been demon-

strated to be fraught with potentially high magnitudes of error [

2

–

4

,

43

]. The errors depend

on the quantity and quality of the measurements as well as on the actual VBP and block

characteristics (i.e., shape, orientation). Hence, it is of the utmost importance to adjust the

measured estimates to accommodate this uncertainty.

Although a correct estimation of the VBP is of paramount importance, only a few

researchers have attempted to tackle this problem [

4

,

6

,

43

,

45

,

48

], by focusing on means

to apply adjustments to measured block proportions to more accurately estimate VBPs.

However, almost all of these approaches have signiﬁcant limitations.

The aim of this paper is to investigate and quantify the potential errors produced

when inferring the VBP of an in situ bimrock/bimsoil mass from ﬁeld measurements. A

statistically robust approach is used in order to overcome the limitations of the previous

studies as much as possible and to propose easy-to-use design charts to obtain appropriate

estimates of the actual block contents.

Uncertainties in VBP Estimates: An Overview

Part of our work was to validate and extend the pioneering research of Medley (1997),

who investigated uncertainty in the determination of VBPs from 1D measurements (LBP).

Because Medley’s work is over 25 years old, it is summarized below in more detail than

would be normal for a research paper.

Medley performed his research for use at the Scott Dam project in Northern Califor-

nia [

40

]: the method and ﬁndings were published by Medley [

43

,

49

,

50

]. The underlying

justiﬁcation for the research was that many bimrocks (particularly mélanges) have scale-

independent block-size distributions over several orders of magnitude. In this regard,

information collected from physical bimrock models (or laboratory specimens) is more

often applicable to site scales than is usual in geopractice when working with rock masses.

Therefore, working on centimeter-scale physical models relative to the smaller site scale

(100s of meters) of the Franciscan Complex mélange underlying the dam site was valid.

The same assumption underlies the research reported in this paper.

Four physical bimrock models were fabricated with known block-size distributions

having fractal dimension of 2.3 (typical of Franciscan Complex mélanges [

43

,

51

]). The

Geosciences 2022,12, 405 3 of 15

models had VBPs of 13%, 32%, 42% and 55%. Blocks were fabricated of clay and plasticene

(Playdoh) and were mixed with Plaster of Paris (Figure 1).

Geosciences 2022, 12, x FOR PEER REVIEW 3 of 16

(100s of meters) of the Franciscan Complex mélange underlying the dam site was valid.

The same assumption underlies the research reported in this paper.

Four physical bimrock models were fabricated with known block-size distributions

having fractal dimension of 2.3 (typical of Franciscan Complex mélanges [43,51]). The

models had VBPs of 13%, 32%, 42% and 55%. Blocks were fabricated of clay and plasticene

(Playdoh) and were mixed with Plaster of Paris (Figure 1).

Figure 1. Scanlines (blue) traced on 42% physical bimrock model (original of monochrome image of

Figure 3 of Medley’s [43] work).

Fundamental to the fabrication was the use of the characteristic (engineering) dimen-

sion, Lc (the ced of Medley, [1]). Lc 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 Lc 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 0.05Lc (6 mm) and the largest blocks (dmax) in each model were selected to be about

0.75Lc.

The models were sawn manually into ten slices, each of which was photographed,

and ten scanlines (representing model boreholes) were drawn on each image (Figure 1).

Figure 1 shows a typically messy example of a slice through a model, with many poorly

discriminated blocks. The lengths of the intercepts between the model borings and all

blocks greater than about 6 mm (the block/matrix threshold) were measured. The LBP of

each scanline was calculated as the percent proportion of the total of the block/boring in-

tercepts and the total length of the boring.

As expected, the 100 LBPs obtained for each bimrock model showed much variabil-

ity, especially for bimrock models with lower VBPs. Therefore, the 1D data were statisti-

cally analyzed to investigate if a good estimation of the VBP could be obtained from the

1D measurements. Specifically: sub-sets of 2, 4, 6, 8, 10, 15 and 20 scanlines were randomly

selected (by means of a Monte-Carlo procedure) 40 times from the dataset of 100 LBPs and

combined for an overall LBP for each sub-set. This procedure is akin to drilling 40 inde-

pendent exploration campaigns at a site using 2, 4, 6, 8, 10, 15 or 20 borings each time,

where the site is assumed to be previously unexplored for each campaign.

Sampling length was characterized by dividing the total length of scanlines/bore-

holes (L) by the length of the largest block, dmax, to provide the parameter Ndmax (in retro-

spect Ndmax was poorly defined: it should have simply been L/dmax = N).

Figure 1.

Scanlines (blue) traced on 42% physical bimrock model (original of monochrome image of

Figure 3 of Medley’s [43] work).

Fundamental to the fabrication was the use of the characteristic (engineering) dimen-

sion, L

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 (dmax) in each model were selected to be about 0.75Lc.

The models were sawn manually into ten slices, each of which was photographed,

and ten scanlines (representing model boreholes) were drawn on each image (Figure 1).

Figure 1shows a typically messy example of a slice through a model, with many poorly

discriminated blocks. The lengths of the intercepts between the model borings and all

blocks greater than about 6 mm (the block/matrix threshold) were measured. The LBP

of each scanline was calculated as the percent proportion of the total of the block/boring

intercepts and the total length of the boring.

As expected, the 100 LBPs obtained for each bimrock model showed much variability,

especially for bimrock models with lower VBPs. Therefore, the 1D data were statistically

analyzed to investigate if a good estimation of the VBP could be obtained from the 1D

measurements. Speciﬁcally: sub-sets of 2, 4, 6, 8, 10, 15 and 20 scanlines were randomly

selected (by means of a Monte-Carlo procedure) 40 times from the dataset of 100 LBPs

and combined for an overall LBP for each sub-set. This procedure is akin to drilling

40 independent exploration campaigns at a site using 2, 4, 6, 8, 10, 15 or 20 borings each

time, where the site is assumed to be previously unexplored for each campaign.

Sampling length was characterized by dividing the total length of scanlines/boreholes

(L) by the length of the largest block, d

max

, to provide the parameter Nd

max

(in retrospect

Ndmax was poorly deﬁned: it should have simply been L/dmax = N).

The data scattering reduced as the sampling length, Nd

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.

The results are summarized in Figure 2a, which could be used to estimate uncertainty,

although most of the scatter points are not organized into neat trends. To ease interpretation

of the data, Medley [

49

] produced a simpliﬁed version of Figure 2a. He selected points

Geosciences 2022,12, 405 4 of 15

in the most populated portion of the graph (red box in Figure 2a) and by “eye-balling”,

assigned approximate contours through the data and then extrapolated the trend lines to

the Nd

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].

Geosciences 2022, 12, x FOR PEER REVIEW 4 of 16

The data scattering reduced as the sampling length, Ndmax, 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 (10dmax),

there was a tendency for the LBPs to converge to the actual 3D VBP.

The results are summarized in Figure 2a, which could be used to estimate uncer-

tainty, although most of the scatter points are not organized into neat trends. To ease in-

terpretation of the data, Medley [49] produced a simplified version of Figure 2a. He se-

lected points in the most populated portion of the graph (red box in Figure 2a) and by

“eye-balling”, assigned approximate contours through the data and then extrapolated the

trend lines to the Ndmax 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 Ndmax and total LBP, Figure 2b provides values of un-

certainty 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 estimat-

ing 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].

Figure 2. (a) Uncertainty (CV) in estimates of the VBP from 1D measurements as a function of the

total sampling length (expressed as Ndmax or the multiple N of the length of the largest block, dmax)

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]).

The chart in Figure 2b has been used by geopractitioners for about 25 years, despite

being based on limited data generated from very simple, crudely fabricated physical mod-

els with inevitable inherent experimental errors.

Since Medley’s [43,49] contribution, few other studies have been published that pro-

vide a means for assessing the uncertainty error in VBP estimates. Tien et al. [48] devel-

oped an analytical solution to quantify the uncertainty in estimates of VBPs of a bim-

rock/bimsoil using scanline measurements in samples with monodisperse circular blocks

and validated it by means of 2D numerical models. More recently, Lu et al. [45] extended

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]).

The chart in Figure 2b has been used by geopractitioners for about 25 years, despite

being based on limited data generated from very simple, crudely fabricated physical models

with inevitable inherent experimental errors.

Since Medley’s [

43

,

49

] contribution, few other studies have been published that pro-

vide a means for assessing the uncertainty error in VBP estimates. Tien et al. [

48

] developed

an analytical solution to quantify the uncertainty in estimates of VBPs of a bimrock/bimsoil

using scanline measurements in samples with monodisperse circular blocks and validated

it by means of 2D numerical models. More recently, Lu et al. [

45

] extended the work of Tien

et al. [

48

]. The authors analyzed 2D bimrock/bimsoil models with polydisperse circular

inclusions and used CT scan images from artiﬁcial bimrocks/bimsoils to validate the ana-

lytical solution of Tien et al. [

48

]. The results obtained by the authors (Figure 3) were similar

to those obtained by Medley [

43

], despite that Lu et al. [

45

] made simplifying assumptions

about Medley’s block sizes, block size distributions and other experimental quantities.

Geosciences 2022,12, 405 5 of 15

Geosciences 2022, 12, x FOR PEER REVIEW 5 of 16

the work of Tien et al. [48]. The authors analyzed 2D bimrock/bimsoil models with poly-

disperse circular inclusions and used CT scan images from artificial bimrocks/bimsoils to

validate the analytical solution of Tien et al. [48]. The results obtained by the authors (Fig-

ure 3) were similar to those obtained by Medley [43], despite that Lu et al. [45] made sim-

plifying assumptions about Medley’s block sizes, block size distributions and other exper-

imental quantities.

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

L/De, obtained with the analytical and numerical solutions by Lu et al. [45] and compared to Med-

ley’s [43] findings. In this graph, Lb indicates the linear fraction of blocks, L is the total length of the

scanline and De indicates the equivalent diameter of the block (modified from [45]).

However, the findings of Lu et al.’s [45] study seem of limited use for geopractition-

ers because, in practice, it is very difficult to accurately define the equivalent diameter, De,

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 bore-

holes 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 influ-

ential 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.

Napoli et al. [4] developed a 3D statistical approach to determine the uncertainties

associated with VBP estimates from 2D measurements; similarly to Medley [43] and Lu et

al. [45], a graph was developed to determine the uncertainty factor to adjust the initial

ABP measured as a function of the size of the outcrop area investigated (Figure 4).

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 Deindicates the equivalent diameter of the block (modiﬁed from [45]).

However, the ﬁndings of Lu et al.’s [

45

] study seem of limited use for geopractitioners

because, in practice, it is very difﬁcult to accurately deﬁne the equivalent diameter, D

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 imple-

mented 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.

Napoli et al. [

4

] developed a 3D statistical approach to determine the uncertainties

associated with VBP estimates from 2D measurements; similarly to Medley [

43

] and Lu

et al. [

45

], a graph was developed to determine the uncertainty factor to adjust the initial

ABP measured as a function of the size of the outcrop area investigated (Figure 4).

Geosciences 2022, 12, x FOR PEER REVIEW 6 of 16

Figure 4. Uncertainty in the VBP estimate from 2D measurements, as a function of the total investi-

gation surface (expressed as multiples, β, of the area of engineering interest, Ac = Lc2) and block

contents measured (ABP) (modified from [4]).

The potential of this approach is that 2D mapping surveys are cheaper to perform

than geotechnical exploration drilling programs. However, despite its utility, it may not

be possible to determine the ABP if the heterogeneous ground is not accessible or visible.

In that case, other field measurements must be used, of which the most popular and nec-

essary are 1D borings.

Given the limitations of the studies highlighted above, this paper complements the

statistical approach of Napoli et al. [4] to take into account the uncertainty in estimates of

VBPs yielded by 1D measurements.

After its validation, an update and extension of Medley’s 1999 work was undertaken

in order to provide a new “design-aid” chart, improving that introduced by Medley to

determine the uncertainty in LPB measurements for mélanges and similar block-in-matrix

formations.

2. Uncertainty in VBP Estimates from LBPs

2.1. Validation of Medley’s 1997 Findings

In this study, the statistically rigorous approach proposed by Napoli et al. [4] was

used to review the validity of Medley’s [43] work and address shortcomings in his re-

search findings, related to mélanges and similar block-in-matrix formations.

A Matlab code was created to reproduce Medley’s work as much as possible and so

compare the uncertainties in VBP estimates obtained with those found in 1997. Four vir-

tual bimrock models were computer generated, having the same dimensions (H × B × L =

150 × 100 × 170 mm), block size distribution and VBPs (13%, 32%, 42% and 55%) of Med-

ley’s models. The minimum and maximum block dimensions were determined, according

to the literature, as a function of the characteristic (engineering) dimension, Lc (the ced of

Medley, 1994): the smallest blocks were equal to 0.05√A = 0.05Lc = 6.5 mm and the largest

blocks (dmax) were equal to 0.75√A = 0.75Lc = 98 mm.

Spherical blocks were distributed randomly within the 3D domains according to the

procedure presented in Napoli et al. [4]. The blocks were assumed to have the same (frac-

tal) block-size distribution adopted by Medley [43,50], with a 3D fractal dimension D = 2.3

(i.e., for log–log plots of frequency of blocks vs. sizes of blocks, the slope of the line is 2.3

[51]. In practice, this means that given one large block, there will be about 5-times as many

blocks of half the size; 25 blocks at a quarter the size of the largest block, 125 at one-eighth

the size of the largest blocks, and so on).

The 32% VBP virtual (computer-generated) bimrock model is shown in Figure 5 by

way of example, penetrated by virtual borings. The virtual borings were located on a grid

Figure 4.

Uncertainty in the VBP estimate from 2D measurements, as a function of the total inves-

tigation surface (expressed as multiples,

β

, of the area of engineering interest, A

c

= L

c2

) and block

contents measured (ABP) (modiﬁed from [4]).

Geosciences 2022,12, 405 6 of 15

The potential of this approach is that 2D mapping surveys are cheaper to perform

than geotechnical exploration drilling programs. However, despite its utility, it may not be

possible to determine the ABP if the heterogeneous ground is not accessible or visible. In

that case, other ﬁeld measurements must be used, of which the most popular and necessary

are 1D borings.

Given the limitations of the studies highlighted above, this paper complements the

statistical approach of Napoli et al. [

4

] to take into account the uncertainty in estimates of

VBPs yielded by 1D measurements.

After its validation, an update and extension of Medley’s 1999 work was undertaken

in order to provide a new “design-aid” chart, improving that introduced by Medley

to determine the uncertainty in LPB measurements for mélanges and similar block-in-

matrix formations.

2. Uncertainty in VBP Estimates from LBPs

2.1. Validation of Medley’s 1997 Findings

In this study, the statistically rigorous approach proposed by Napoli et al. [

4

] was

used to review the validity of Medley’s [

43

] work and address shortcomings in his research

ﬁndings, related to mélanges and similar block-in-matrix formations.

A Matlab code was created to reproduce Medley’s work as much as possible and so compare

the uncertainties in VBP estimates obtained with those found in 1997. Four virtual bimrock mod-

els were computer generated, having the same dimensions (

H×B×L = 150 ×100 ×170 mm

),

block size distribution and VBPs (13%, 32%, 42% and 55%) of Medley’s models. The minimum and

maximum block dimensions were determined, according to the literature, as a function of the char-

acteristic (engineering) dimension, L

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 = 0.75Lc= 98 mm.

Spherical blocks were distributed randomly within the 3D domains according to

the procedure presented in Napoli et al. [

4

]. The blocks were assumed to have the same

(fractal) block-size distribution adopted by Medley [

43

,

50

], with a 3D fractal dimension

D = 2.3 (i.e., for

log–log plots of frequency of blocks vs. sizes of blocks, the slope of the line

is 2.3 [

51

]. In practice, this means that given one large block, there will be about 5-times

as many blocks of half the size; 25 blocks at a quarter the size of the largest block, 125 at

one-eighth the size of the largest blocks, and so on).

The 32% VBP virtual (computer-generated) bimrock model is shown in Figure 5by

way of example, penetrated by virtual borings. The virtual borings were located on a grid

like that of Medley’s [

43

]. As in Medley, block/boring intercept lengths were measured

and LBPs calculated for each boring.

Geosciences 2022, 12, x FOR PEER REVIEW 7 of 16

like that of Medley’s [43]. As in Medley, block/boring intercept lengths were measured

and LBPs calculated for each boring.

Figure 5. 32%VBP virtual bimrock model with example virtual borings (dimensions in millimeters).

Statistical analyses were performed in the same fashion as Medley [43] to assess the

error in estimates of VBPs, based on the assumption that they are equivalent to the meas-

ured LBPs, that is: the full dataset of 100 LBPs per model (ϑ = 100); the same subsets of

randomly selected and combined scanlines (β = 2, 4, 6, 8, 10, 15 and 20); and the same

number of randomizations (λ = 40).

The results of these analyses are presented in Figure 6, where the uncertainty factors

(UFs) found by Medley are also shown by way of comparison.

The UF values are uncertainty factors defined as standard deviation/mean (or CV,

Coefficient of Variation, since for sufficient sampling, the VBP is very close to the mean

value). In Figure 6, the UFs are associated with the corresponding values of N, which is

equal to the Ndmax 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 (dmax).

It is apparent from the graphs in Figure 6 that the UFs obtained from this research

are consistent with Medley’s 1997 findings, although a slight difference can be observed.

A deviation between Medley’s and new findings was expected and is not surprising be-

cause of the unavoidable differences between the virtual and physical bimrock models

(e.g., location of the combined scanlines; block distributions; ellipsoid/spherical shapes of

the blocks; etc.). That there was a difference between Medley’s [43] ellipsoidal blocks and

our spherical blocks may not have greatly influenced our findings, which is fortuitous

given the difficulties involved in modelling populations of oriented ellipsoids or irregu-

lar-shaped blocks in virtual bimrocks [52,53]. Further, Lu et al. [45] and Ramos-Cañón et

al. [6] discovered that the influence on VBPs of block shapes is apparently minor.

Figure 5.

32%VBP virtual bimrock model with example virtual borings (dimensions in millimeters).

Geosciences 2022,12, 405 7 of 15

Statistical analyses were performed in the same fashion as Medley [

43

] to assess

the error in estimates of VBPs, based on the assumption that they are equivalent to the

measured LBPs, that is: the full dataset of 100 LBPs per model (

ϑ

= 100); the same subsets

of randomly selected and combined scanlines (

β

= 2, 4, 6, 8, 10, 15 and 20); and the same

number of randomizations (λ= 40).

The results of these analyses are presented in Figure 6, where the uncertainty factors

(UFs) found by Medley are also shown by way of comparison.

Geosciences 2022, 12, x FOR PEER REVIEW 8 of 16

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.

Furthermore, the points on the N-UF plane obtained by the numerical analyses are

not aligned vertically with those of Medley (1997). The reasons for this slight dislocation

are that the Medley physical (hand-made) models had four different dmax 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 con-

stant heights, scanline lengths and dmax were considered in the present study.

Despite the differences between the results obtained from analyzing the physical and

virtual models, the virtual exploration campaign yielded additional data that “infilled”

between Medley’s 1997 data.

It is also apparent in Figure 6 that there is good correlation between the trends of our

data and Medley’s 1997 [43] results and, thus, the approach taken of using virtual bim-

rocks was clearly justified. Additionally, at this point of the research, it was evident that

geopractitioners who used Medley’s 1997 “design charts” in Figure 2 could be assured

that any estimates of VBP they have made were reasonable.

2.2. Extension of Medley’s 1997 Findings

To extend the statistical validity of Medley’s limited approach, the computer-derived

bimrock models were further explored by using the same scanlines and scanline subsets

and a greatly increased maximum number of randomizations (λ) to 1000 instead of the 40

measured by Medley [43]. The maximum number of randomizations was set so as to fall

within the calculation and storage capacity of the workstation used.

Figure 7 compares the trend lines of the results of this expanded set of analyses for λ

= 1000, with those obtained using λ = 40 (based on Medley’s 1997 approach). The trend

lines shown in this graph were obtained using the fitting equations found on an N interval

between N = 2 and N = 30 and were then extrapolated into the region of N > ~30.

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.

The UF values are uncertainty factors deﬁned as standard deviation/mean (or CV,

Coefﬁcient of Variation, since for sufﬁcient sampling, the VBP is very close to the mean

value). In Figure 6, the UFs are associated with the corresponding values of N, which is

equal to the Nd

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 (dmax).

It is apparent from the graphs in Figure 6that the UFs obtained from this research

are consistent with Medley’s 1997 ﬁndings, although a slight difference can be observed.

A deviation between Medley’s and new ﬁndings was expected and is not surprising

because of the unavoidable differences between the virtual and physical bimrock models

(e.g., location of the combined scanlines; block distributions; ellipsoid/spherical shapes

of the blocks; etc.). That there was a difference between Medley’s [

43

] ellipsoidal blocks

and our spherical blocks may not have greatly inﬂuenced our ﬁndings, which is fortuitous

given the difﬁculties involved in modelling populations of oriented ellipsoids or irregular-

shaped blocks in virtual bimrocks [

52

,

53

]. Further, Lu et al. [

45

] and Ramos-Cañón et al. [

6

]

discovered that the inﬂuence on VBPs of block shapes is apparently minor.

Furthermore, the points on the N-UF plane obtained by the numerical analyses are not

aligned vertically with those of Medley (1997). The reasons for this slight dislocation are

that the Medley physical (hand-made) models had four different d

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

Geosciences 2022,12, 405 8 of 15

resultant N values. However, as indicated above, virtual bimrock models with constant

heights, scanline lengths and dmax were considered in the present study.

Despite the differences between the results obtained from analyzing the physical and

virtual models, the virtual exploration campaign yielded additional data that “inﬁlled”

between Medley’s 1997 data.

It is also apparent in Figure 6that there is good correlation between the trends of

our data and Medley’s 1997 [

43

] results and, thus, the approach taken of using virtual

bimrocks was clearly justiﬁed. Additionally, at this point of the research, it was evident

that geopractitioners who used Medley’s 1997 “design charts” in Figure 2could be assured

that any estimates of VBP they have made were reasonable.

2.2. Extension of Medley’s 1997 Findings

To extend the statistical validity of Medley’s limited approach, the computer-derived

bimrock models were further explored by using the same scanlines and scanline subsets

and a greatly increased maximum number of randomizations (

λ

) to 1000 instead of the

40 measured by Medley [

43

]. The maximum number of randomizations was set so as to

fall within the calculation and storage capacity of the workstation used.

Figure 7compares the trend lines of the results of this expanded set of analyses for

λ= 1000

, with those obtained using

λ

= 40 (based on Medley’s 1997 approach). The trend

lines shown in this graph were obtained using the ﬁtting equations found on an N interval

between N = 2 and N = 30 and were then extrapolated into the region of N > ~30.

Geosciences 2022, 12, x FOR PEER REVIEW 9 of 16

Figure 7. Influence of λ (maximum number of randomizations) on the uncertainty in the VBP esti-

mate 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 7 shows that the increase in the maximum number of randomizations λ, from

40 to 1000, generated marked variations in outcomes, as evident by the separation be-

tween the “λ = 40” and “λ = 1000” continuous lines, for each VBP model. The differences

are ascribed to the increase in the total quantity of LBPs generated for the higher number

of scanline combinations analyzed for each β value (numbers of sub-sets of scanlines).

This resulted in statistically more representative datasets and in a consequent increase in

the standard deviations, i.e., in the UFs, particularly for low N values.

Further, it was observed that for VBPs equal to 13%, 32% and 42%, there were slight

“rotations” of the paired “λ = 40” and “λ = 1000” straight lines about intersection points

for the pairs that occur between N of 10 and 25. We interpret these “rotations” as revealing

that a small number of realizations (e.g., λ = 40) entails a relatively minor underestimation

of the UF for small values of N (say 5) compared to an overestimation of the UF for large

values of N (say 25). Further, N~10 seems to be an optimum value of sampling, for which

there is little over- or under-estimation. In other words, as pointed out by Medley [7,49],

drilling or performing scanlines with total lengths of about 10-times the length of the larg-

est block (dmax = 0.75Lc) 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.

For VBP = 55%, the results of the analyses with different values of λ were almost

identical, indicating that the influence of the increase in sampling due to the much higher

λ parameter is negligible.

Bimrocks tend to be problematic at VBPs greater than 10% to 15% (for excavations)

or 25% (for strength). Hence, it is apparent that Figure 7 needs to be infilled with findings

from investigations covering LBP ranges between 13% and 70% (when blocks start to

touching and the materials are no longer considered bimrocks but blocky rock masses

with weak infillings). Such expansion of LBP ranges is considered further in the paper.

Medley’s (1999) chart in Figure 2b is compared with our data in Figure 7 by means

of the dashed lines. Observation reveals significant differences in the trends of the UF lines

between our data and Medley’s, for almost all the VBP values. This is not surprising, con-

sidering that the linearization of Medley’s data was achieved by best-estimate manual

sketching of trend lines through and beyond the data shown in Figure 2a. Despite the

slight mismatches in plots, the general patterns are similar between our findings and those

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 7shows that the increase in the maximum number of randomizations

λ

, from

40 to 1000, generated marked variations in outcomes, as evident by the separation between

the “

λ

= 40” and “

λ

= 1000” continuous lines, for each VBP model. The differences are

ascribed to the increase in the total quantity of LBPs generated for the higher number of

scanline combinations analyzed for each

β

value (numbers of sub-sets of scanlines). This

resulted in statistically more representative datasets and in a consequent increase in the

standard deviations, i.e., in the UFs, particularly for low N values.

Further, it was observed that for VBPs equal to 13%, 32% and 42%, there were slight

“rotations” of the paired “

λ

= 40” and “

λ

= 1000” straight lines about intersection points for

the pairs that occur between N of 10 and 25. We interpret these “rotations” as revealing

that a small number of realizations (e.g.,

λ

= 40) entails a relatively minor underestimation

of the UF for small values of N (say 5) compared to an overestimation of the UF for large

Geosciences 2022,12, 405 9 of 15

values of N (say 25). Further, N~10 seems to be an optimum value of sampling, for which

there is little over- or under-estimation. In other words, as pointed out by Medley [

7

,

49

],

drilling or performing scanlines with total lengths of about 10-times the length of the largest

block (d

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.

For VBP = 55%, the results of the analyses with different values of

λ

were almost

identical, indicating that the inﬂuence of the increase in sampling due to the much higher

λ

parameter is negligible.

Bimrocks tend to be problematic at VBPs greater than 10% to 15% (for excavations) or

25% (for strength). Hence, it is apparent that Figure 7needs to be inﬁlled with ﬁndings from

investigations covering LBP ranges between 13% and 70% (when blocks start to touching

and the materials are no longer considered bimrocks but blocky rock masses with weak

inﬁllings). Such expansion of LBP ranges is considered further in the paper.

Medley’s (1999) chart in Figure 2b is compared with our data in Figure 7by means

of the dashed lines. Observation reveals signiﬁcant differences in the trends of the UF

lines between our data and Medley’s, for almost all the VBP values. This is not surprising,

considering that the linearization of Medley’s data was achieved by best-estimate manual

sketching of trend lines through and beyond the data shown in Figure 2a. Despite the slight

mismatches in plots, the general patterns are similar between our ﬁndings and those of

Medley [

7

,

49

]. Our predicted uncertainties are slightly more conservative than Medley’s

but not enough to cause discomfort for geopractitioners who have used Medley’s original

1997 and 1999 charts.

2.3. Effects of Increasing the 3D Domain Size

The virtual modelling validates Medley’s simple [

43

] results and proves the great

value of virtual investigations of computer-generated bimrocks, as suggested by Medley for

several years. Indeed, it is the experience of many workers in the bimrock/bimsoil ﬁeld that

fabrication of physical models is tedious and often inaccurate compared to the construction

of virtual bimrock/bimsoil models. Given the relative ease of constructing virtual models,

the research sought to extend beyond the geometry of physical and computer models

considered so far.

The Matlab routine was modiﬁed to investigate if the small domain size (volume)

of the bimrock models Medley [

43

] analyzed affected the results obtained. Larger 3D

domains were created, which included blocks with the same shape and characteristics

used previously (i.e., spherical, smallest and largest block dimensions, VBPs and block-

size distribution). Compared to the bimrock models analyzed previously, the new 3D

models could correspond to site scales for an equivalent longer underground excavation

with the same diameter or a larger landslide area with the same failure surface depth.

Indeed, correspondence between small and large domains is valid when working with

many bimrocks (particularly mélanges) because of the scale independence of block-size

distributions over many orders of magnitude.

The choice of investigating the effects of larger three-dimensional domains on the

results was driven by three main considerations:

1.

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;

2.

With higher

ϑ

values (total number of scanlines analyzed), it was possible to con-

sider more varied borehole location distributions by adopting a greater number of

randomizations (λ) for each sub-set of combined scanlines (β);

3.

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