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Sustainability 2024, 16, 4711. https://doi.org/10.3390/su16114711 www.mdpi.com/journal/sustainability
Article
In Situ Stress Paths Applied in Rock Strength Characterisation
Result in a More Correct and Sustainable Design
Andre Vervoort
Department of Civil Engineering, KU Leuven, 3001 Leuven, Belgium; andre.vervoort@kuleuven.be
Abstract: Rock strength is an essential parameter in the design of any underground excavation, and
it has become even more relevant as the focus increasingly shifts to sustainable excavations. The
heterogeneous nature of rock material makes characterising the strength of rocks a dicult and
challenging task. The research results presented in this article compare the impact on the strength
when the classic stress paths in laboratory experiments are applied versus when in situ stress paths
would be applied. In most laboratory experiments, the rock specimens are free of stress at the be-
ginning of the tests, and the load is increased systematically until failure occurs. Opposite paths
occur around an underground excavation; that is, the rock is in equilibrium under a triaxial stress
state and at least one stress component decreases while another component may increase. Based on
discrete element simulations, the research shows that dierent stress paths result in dierent failure
envelopes. The impact of this nding is evaluated in the application of wellbore stability (e.g., the
minimum or maximum mud weight), whereby it is concluded that failure envelopes, based on stress
paths closer to the in situ stress paths, result in a more accurate design. Although the most critical
location along the circumference is not dierent, the required density of the mud is signicantly
dierent if the rock strength criteria are based on the more realistic in situ stress paths. This means
that a change in the way the strength of rocks is characterised improves the sustainable design of all
underground excavations.
Keywords: rock mechanics; rock characterisation; rock strength; failure envelope; numerical
simulations; drilling uid engineering; wellbore stability; drilling mud density
1. Introduction
Due to the specic nature of rock material, the characterisation of its strength is a
dicult and challenging task. Even so, the strength of rocks is an essential parameter in
the design of any underground excavation, and it has become even more relevant as the
focus shifts increasingly to sustainable excavations. Among other things, this means that
failure or poor design must be eliminated at all costs. This applies to all excavations, in-
cluding those aimed at creating a sustainable infrastructure for underground storage,
transport, and clean energy. However, it also applies to the more traditional mine excava-
tions as well as gas and oil wells. For example, the environmental consequences of the
failure of rocks around a gas or oil well are extremely serious, and they can have long-
term disastrous consequences.
The strength of rocks is a non-intrinsic parameter of the material. Apart from the
characteristics of the material, e.g., its composition, heterogeneities, aws, and pore dis-
tribution, the strength of a rock specimen is inuenced by numerous other parameters,
i.e., the geometry of the specimens (e.g., their volumes, shapes, and orientations), envi-
ronmental conditions (e.g., humidity and temperature), the loading rate (e.g., conven-
tional loading rates, dynamic loading, and creep), and the test set-up. The consequence of
this is that one should always know all the conditions of the tests that are conducted when
determining the strength of rocks. This is the reason tests should be conducted following
Citation: Vervoort, A. In Situ Stress
Paths Applied in Rock Strength
Characterisation Result in a More
Correct and Sustainable Design.
Sustainability 2024, 16, 4711.
hps://doi.org/10.3390/su16114711
Academic Editors: Martina
Inmaculada Álvarez Fernández and
Víctor Martínez-Ibáñez
Received: 22 April 2024
Revised: 21 May 2024
Accepted: 30 May 2024
Published: 31 May 2024
Copyright: © 2024 by the authors. Li-
censee MDPI, Basel, Swierland.
This article is an open access article
distributed under the terms and con-
ditions of the Creative Commons At-
tribution (CC BY) license (hps://cre-
ativecommons.org/licenses/by/4.0/).
Sustainability 2024, 16, 4711 2 of 23
the suggested methods by the International Society of Rock Mechanics (e.g., [1]) or stand-
ard methods by various institutes (e.g., [2,3]) that describe the test conditions. When these
recommendations are not followed, this should be clearly stated, including the motivation
for the deviation. In this paper, the question is raised concerning whether the stress paths
presented in the suggested or standard methods are the most appropriate to quantify rock
strength and therefore should be used in the design of in situ excavations.
A recent paper [4] presented the results of a study of the relevance of dierent stress
paths concerning the rock strength and the corresponding failure envelopes. In the paper,
it was shown that dierent stress paths had signicant impacts on the strengths that were
observed. This means that this parameter also is one of the many parameters that make
strength a non-intrinsic parameter. In [4], an overview is given of the research results for
stress paths that are dierent from the ones applied in basic laboratory experiments, i.e.,
following the suggested methods and standards. In most of these cases, multiple loading–
unloading cycles are studied (e.g., [5–7]), but the eect on the entire failure envelope is
not quantied. In a recent published paper [8], a set of laboratory experiments on diorite
specimens was presented with loading and unloading stress paths. These paths were sim-
ilar to the ones studied using numerical simulations in [4], but they were not identical.
(Further, the results of their experiments are discussed in more detail, and they are com-
pared to the results of the simulations in this article).
In most laboratory experiments, specimens are free of stress before the tests start, and
the load is increased systematically until failure occurs. Around an underground excava-
tion, e.g., a tunnel or borehole, the opposite path occurs. Before the excavation, the rock is
in equilibrium under a triaxial stress state. Following the excavation, at least one stress
component decreases, and another component may increase. As a result, the stress paths
in classic laboratory experiments are dierent from the in situ stress paths. The impact of
various stress paths on the failure envelopes was studied in [4]. A distinct element model
was applied, allowing the simulation of the micro-fracturing of the rock. A small repre-
sentative volume element (RVE) was considered, and three basic types of stress paths were
applied to this RVE, i.e., (1) the conventional loading, (2) the unloading of the minor prin-
cipal stress, and (3) the simultaneous loading and unloading of the major and minor prin-
cipal stresses, respectively. The laer stress path is what occurs around a borehole. (See
further for a more detailed discussion). The applied loading and unloading rates in the
simulations are considered to be similar as in classic laboratory experiments and conven-
tional excavations. So, no dynamic loading or unloading is studied, as observed, for ex-
ample, during rock bursts or hydraulic fracturing. Very slow stress variations, as occur-
ring during geological processes, are also not part of the study. The details of the two-
dimensional RVE are provided in [4], including a schematic diagram of the RVE. The
choice of the characteristics of the model was based on the calibrations of our own labor-
atory experiments [9,10]. The merits of the RVE model were established by simulating
published laboratory experiments that were undergoing a complex stress path from an-
other research group [11].
The aim of the current research is to analyse the practical consequences of the earlier
ndings and to determine whether there is reason to change the engineering practice. The
application of these three dierent stress paths showed a clear eect. The main reason for
this is that the micro-fracturing that occurs when loading a rock (from zero or a low stress
state) until failure is dierent from the micro-fracturing that occurs when unloading rocks
(from the in situ stress state) till failure. Thus, the failure envelopes are dierent because
of this dierence in the weakening processes, i.e., the conventional loading results in the
largest strength and uniaxial unloading results in the smallest strength.
The paper focuses on the need to apply more realistic stress paths in the rock charac-
terisation process when studying wellbore stability and, in fact, when designing any ex-
cavation where there is a risk of rock fracturing. First, the eect of the three basic stress
paths on the failure criteria are presented and discussed, including a more detailed
presentation of the RVE model. Second, the wellbore stability for an initial isotropic stress
Sustainability 2024, 16, 4711 3 of 23
state is discussed, and the eect of considering a stress path closer to the in situ stress path
on the design parameters, i.e., the estimation of the maximum and minimum drilling mud
pressure. Third, an initial anisotropic stress state is considered and implemented in the
design of the borehole.
2. Eect of the Three Basic Stress Paths on Failure Envelopes and
Micro-Fracturing Process
Three dierent loading paths are investigated. All three start from an isotropic stress
state, including a zero-stress state. The rst type is characterised by an increase in the
successive steps of the major principal stress until failure (for a constant minor principal
stress). This type of stress path is abbreviated as S1(loa),S3(=). For the second type, the minor
principal stress is decreased and the major principal stress remains constant, i.e., S1(=),S3(unl).
For the third type, the major principal stress is increased, and the minor principal stress is
decreased with the same stress increments, i.e., S1(loa),S3(unl). In comparison to the previous
two types, the change in deviatoric stress is larger for the third type.
The set-up of the research project is to repeat a large number of simulations, starting
from the same model, i.e., a relatively small RVE. This is impossible to apply in real labor-
atory conditions since each specimen is dierent, at least slightly. The focus of the research
is on one single parameter, i.e., the impact of dierent stress paths on the strength. The
RVE model should be considered to represent a black box rock. Before applying a stress
path, one does not know the strength of the RVE model for a specic stress path. The
various input parameters for the RVE model are based on past simulations, whereby the
calibration was conducted based on both the observed fracture paerns and measured
stress–strain curves [9,10]. Crucial for a correct modelling of the behaviour of rock speci-
mens is that the initiation and growth of fractures are possible in the model. Therefore,
the two-dimensional Universal Distinct Element Code (UDEC) is used [12]. The original
main application of this code was the simulation of rock blocks and their deformation and
relative movements, and that is still the case. However, an intact rock can be approximated
by an assembly of individual blocks, whereby these blocks initially are glued together
along all contact lines. These contact lines are possible future fracture paths. A contact line
(i.e., between two adjacent blocks) does not represent a physical crack as long as it is not
activated or has passed the pre-dened failure criterion. The sub-division of the model by
contact lines has the advantage that future fractures are composed of relatively straight
lines. Thus, the contact lines or elements are given strength properties, and hence, they
can fail in shear and/or tension, simulating the occurrence of (micro-)fractures. After acti-
vation, the contact elements can deform, slide, and open. The blocks also can deform, e.g.,
in a linear elastic way. Both the individual blocks and the contact elements within a single
model can have dierent property values. More details on the set-up of the black box RVE
model are given in [4], including a schematic diagram of the RVE, and details on the cali-
bration of the input parameters are available in [9,10].
The black box rock in this study has the same geometry, the same stiness values,
and the same values for the properties of the individual blocks as in [4]. Only the charac-
teristics of the contact strength have been changed. In this study, the contacts are charac-
terised by a Mohr–Coulomb criterium with a cohesion of 14 MPa, a friction angle of 30°,
and a tensile strength of 7 MPa. In the previous publication, the cohesion was 20 MPa, and
the tensile strength was 10 MPa. These values are for the contacts, and they should not be
confused with the macro-behaviour of an entire rock specimen. As mentioned above, the
failure envelope is unknown prior to the simulations. The contact lines are orientated in
such a way that their orientations are distributed equally over individual classes at 30°
intervals. Therefore, at a macro-scale, the black box rock behaves as an isotropic rock. So
far, the simulations are limited to 2D, and the RVE is a square. The external stresses are
applied directly on the model, without any interaction of platens. The reason for this is
that the focus is on the behaviour of the rock and not on the simulations of laboratory
experiments. No stress–strain curves are recorded. It is assumed that a specimen has failed
Sustainability 2024, 16, 4711 4 of 23
when 50% of all contact elements have been activated. This percentage seems to be the
most reliable. During the variation in the external stress, it is recorded if a contact element
fails in tensile, shear, or mixed mode.
Figure 1 presents the results of these simulations. First, a stress path corresponding
to loading the RVE in one direction (S1(loa),S3(=)) is presented, starting from an isotropic
stress state, i.e., from 0, 5, 10, and 15 MPa (Figure 1: vertical stress paths). The failure load
is determined for each simulation. Figure 1 presents the failure envelope in a major vs.
minor principal stress diagram. The failure envelope of the uniaxial loading is the one that
corresponds the closest to the classic failure envelope of the laboratory experiments. How-
ever, there are dierences, e.g., there are no platens in the model, and stress is applied
instead of displacements. The second set of simulations starts from an isotropic stress
state, which is equal to the failed stress (i.e., the major principal stress) simulated in the
rst set of simulations. In the second set (horizontal stress paths in Figure 1), the minor
principal stress is decreased until failure occurs, and the major principal stress remains
constant (S1(=),S3(unl)). If there is no eect of the stress path, the failure loads in the sec-
ond set of simulations should correspond to the conning stresses applied in the rst set.
The third set of simulations, i.e., S1(loa),S3(unl), also aims to reach the failure loads in the
rst set of simulations (stress paths under an angle of 45° in Figure 1).
Figure 1. Failure envelopes for the three basic types of stress paths (i.e., uniaxial loading
(S1(loa),S3(=)), uniaxial unloading (S1(=),S3(unl)) and simultaneous loading and unloading
(S1(loa),S3(unl))). Start of each stress path (i.e., an isotropic stress state) and failed stress states are
indicated by crosses. Each set of three basic types of stress paths is presented by a dierent colour
(four sets in total).
The three failure envelopes are clearly separated (Figure 1). The one that corresponds
to the uniaxial loading (S1(loa),S3(=)) results in the strongest strength; e.g., for a zero-con-
ning pressure, the failure load is 55 MPa (Table 1a). The failure envelope of the simulta-
neous loading and unloading stress paths (S1(loa),S3(unl)) is situated between the uniax-
ial loading (S1(loa),S3(=)) and uniaxial unloading (S1(=),S3(unl)) envelopes. The strength
reduction along the stress path, when comparing the failure load of the uniaxial unloading
stress path with the uniaxial loading stress path, is situated between 41% and 43% (Table
1b). A comparison between the simultaneous loading and unloading stress paths and the
uniaxial loading shows a strength reduction between 28% and 34% (Table 1c). (Further,
these dierences are compared to the experimental results published in [8]).
Sustainability 2024, 16, 4711 5 of 23
When the stress state in a rock changes, micro-fracturing may occur. This is clearly
illustrated when acoustic emissions are recorded [7,8,13–17]. The type and amount of
micro-fracturing is inuenced by the stress level and by the amount of the change in the
stress. It is the accumulation of all the micro-fracturing that leads to the ultimate macro-
failure of rocks, and this occurs both in laboratory experiments and in situ. The dierence
between the two applications is that, in most experiments, the test is continued until fail-
ure occurs (and sometimes even beyond failure, i.e., further than the maximum stress
level), while in situ, the stress redistribution does not necessarily lead to failure. Dierent
micro-fracturing is induced in dierent stress paths. Hence, micro-fracturing plays an im-
portant role when loading or unloading rock specimens. The dierence in micro-fractur-
ing results in a dierence in the nal macro-strength and macro-fracture paern. The type
of contact activation or failure is recorded by the Universal Distinct Element Code (UDEC)
[12,18]. There are three modes of contact failure, i.e., in tension, in shear, or in a combina-
tion of both. For example, a contact can open (tensile failure), followed, in a later calcula-
tion step, by a closure and/or by a shear displacement. The laer two situations are la-
belled as (1) tensile failure in the past and (2) tensile failure in the past combined with
shear failure. The laer is an example of a mixed failure or activation mode. Figure 2 pre-
sents the variation in the contact failure modes as a function of the stress path. The three
simulations that are directly and indirectly linked to a conning pressure of 15 MPa are
presented, i.e., the three stress paths indicated by the light blue colour in Figure 1. In Fig-
ure 2, the horizontal axis presents a relative scale between 0% and 100% from the initial
load till the failure load. For the rst (S1(loa),S3(=)) and third (S1(loa),S3(unl)) types of
stress paths, the major principal stress is increased systematically until failure occurs. For
the second type (S1(=),S3(unl)), the minor principal stress is decreased systematically until
failure occurs. At the top of each graph, the starting and nal stresses are indicated as
absolute values. (Also, see Table 1). As can be observed, dierent load intervals are needed
to reach 50% of the failed or activated contact elements.
Table 1. Strength values for the three types of stress paths. (a) Uniaxial loading (S1(loa),S3(=)), i.e.,
reference for other stress paths; (b) uniaxial unloading (S1(=),S3(unl)) and reduction along stress
path in comparison to uniaxial loading; (c) simultaneous loading and unloading (S1(loa),S3(unl))
and reduction along stress path in comparison to uniaxial loading.
(a)
Initial Stress State
Stress State at Failure
Isotropic Stresses,
MPa
Minor Principal Stress (S3), MPa
Major Principal Stress
(S1), MPa
0.0
0.0
55.0
5.0
5.0
76.6
10.0
10.0
105.5
15.0
15.0
130.4
(b)
Initial Stress State
Stress State at Failure
Strength Reduction for
S3, MPa (%)
(along Stress Path)
Isotropic Stresses,
MPa
Minor Principal
Stress (S3), MPa
Major Principal
Stress (S1), MPa
55.0
23.2
55.0
23.2 (42%)
76.6
34.3
76.6
29.3 (41%)
105.5
51.4
105.5
41.4 (43%)
130.4
64.8
130.4
49.8 (43%)
(c)
Initial Stress State
Stress State at Failure
Strength Reduction for
S3 and S1, MPa (%)
(along Stress Path)
Isotropic Stresses,
MPa
Minor Principal
Stress (S3), MPa
Major Principal
Stress (S1), MPa
Sustainability 2024, 16, 4711 6 of 23
27.5
7.8
47.2
11.0 (28%)
40.8
15.9
65.7
14.7 (29%)
57.8
26.2
89.3
22.9 (34%)
72.7
33.9
111.5
26.7 (33%)
(a)
(b)
(c)
Figure 2. Variation in the contact failure modes (i.e., tensile-only, shear-only, and mixed modes) as
a function of the stress path, i.e., from the initial isotropic stress state till the load level of failure,
corresponding to 50% failed contacts. The three types of stress paths are presented for the case, with
a conning pressure of 15 MPa during the uniaxial loading (i.e., the three stress paths indicated by
the light-blue colour in Figure 1). The initial and nal failure loads in absolute values are presented
at the top of each graph. (a) Uniaxial loading, S1(loa),S3(=), major principal stress levels are pre-
sented; (b) uniaxial unloading, S1(=),S3(unl), minor principal stress levels are presented; (c) simul-
taneous loading and unloading, S1(loa),S3(unl), major principal stress levels are presented.
For all uniaxial loading simulations, i.e., S1(loa),S3(=), it is typical that the fracturing
starts in tension only, and this occurs at an early stage of the increase in the major principal
stress (Figure 2a). At around 20% of the failure load (relative to the initial isotropic stress
state of 15 MPa), the rst contact element also undergoes a shear displacement (mixed
mode). At the end of the simulation, i.e., when 50% of all contacts have been activated,
about 52% of these activated contacts have failed in shear only and 16% in tension only.
The mixed mode represents 32% of the activated contacts. In comparison to the previous
stress path, the unloading of the minor principal stress, i.e., S1(=),S3(unl) shows that the
rst activation of contacts occurs later, i.e., at 20 to 30% of the entire unloading interval
(Figure 2b). Again, the rst activations are in tension only. When the RVE is assumed to
have failed fully, the largest mode of failure type is the mixed mode with about 45% of all
activated contacts. The other two modes are 32% for the shear-only mode and 23% for the
tension only mode. For the third type of stress path, i.e., S1(loa),S3(unl), characteristics of
the stress paths of uniaxial loading and of uniaxial unloading are observed. The rst acti-
vation takes place at about 20% of the entire stress path, again in the tension-only mode
Sustainability 2024, 16, 4711 7 of 23
(Figure 2c). At failure, the three modes are relatively similar, i.e., 30%, 32%, and 38% for
tension only, mixed mode, and shear only, respectively. All of these observations are sim-
ilar to the observations for the black box rock RVE, published in [4].
As mentioned above, laboratory experiments on diorite specimens were published
recently by Li et al. [8] with loading and unloading stress paths, similar, but not identical,
to the ones studied using numerical simulations here and in [4]. A criticism of the numer-
ical simulations of the black box rock RVE could be that it is not supported directly by
experimental work. However, the choice of the model characteristics is based on calibra-
tions of our own laboratory experiments [9,10], and the merits of the RVE model were
established by simulating published laboratory experiments undergoing a complex stress
path from another research group [11]. Also, one should not forget that the whole set-up
of the RVE is just to eliminate certain shortcomings of experimental work, e.g., problems
linked to repeatability, the eect of size, and the impact of platens. All this does not mean
that a comparison with experimental work is not important. The experiments on the dio-
rite specimens [8] consider three stress paths like the three types of stress paths applied
above. All experiments are on cylindrical specimens, and the starting point is an isotropic
stress state. A rst set is the classic triaxial compression loading, whereby the conning
stress is kept constant. So, this set follows uniaxial loading simulations, i.e., S1(loa),S3(=).
For the two other sets of stress paths, the axial stress is rst increased up to 80% of its
maximum strength for the corresponding conning stress, i.e., the strength determined in
the rst set of experiments. For the second set of stress paths, the conning stress is de-
creased, while the axial stress is kept constant at the 80% level. The third set applies both
a loading of the axial stress from the 80% level and an unloading of the conning stress.
The unloading rate is twice the loading rate. Even though the starting point of the uniaxial
unloading and that of the simultaneous loading and unloading are much closer to the
failure envelope of the rst set of stress paths, a signicant dierence is noted between the
failure envelope of the classic experiments and the ones of the two other sets. The laer
result in a signicantly lower strength. The impact between the second and third set is not
clear. For example, when analysing the data for the uniaxial unloading, a strength reduc-
tion of 42 to 52% is observed when the same calculation is applied as is applied for the
RVE results. For the laer, this reduction is situated between 41% and 43% (Table 1b). The
larger strength reduction in the published experiments [8] is logical since, during the ex-
periments, the axial load rst is increased from the isotropic stress state until it reaches
80% of its strength. During this interval, micro-fractures occur, and the specimen is weak-
ened. This is clearly illustrated in Figure 2a. At the 80% level, about 75% of the activated
contacts at failure already have been activated.
3. Eect of the Stress Path on a Wellbore Stability Analysis
The design of a wellbore is very complex, and numerous aspects play signicant roles
[19–24]. The aim of the paper is not to summarize and discuss all of these aspects. The
focus is on a single aspect of such a design, i.e., the geo-mechanical determination of the
mud weight interval [19,22–24], so that no new fractures are induced around the borehole
wall. The aim of the paper is to evaluate what the impact would be on the mud weight if
the rock strength or the failure envelope were to be determined dierently from how it is
currently carried out, i.e., by applying stress paths that are closer to the in situ stress paths
than the typical stress paths applied in laboratory experiments. Note that the presented
results are a rst evaluation and are not yet a practical engineering design. Some assump-
tions are made for this rst evaluation. At a later stage, more complex models and ap-
proaches could be considered. First, a 2D approach is applied, assuming that the critical
plane is perpendicular to the borehole axis. This assumption is realistic if the axial stress
is the intermediate principal stress component. If this is not the case, a 3D approach is
needed or another 2D section should be considered (e.g., situated along the axial and a
radial orientation). Second, it is assumed that all stresses are positive (i.e., compressive
stresses at the macro-scale), i.e., only the risk of shear macro-fracturing is integrated in the
Sustainability 2024, 16, 4711 8 of 23
model (and thus not tensile failure at the macro-scale). Tensile activation may occur at the
scale of a contact element, as is illustrated in Figure 2. In situ, tensile failure at the macro-
scale may occur around a borehole, but this aspect is not integrated in the current model.
Third, it is assumed that the pore pressure is equal to zero. Although for a real and full
design, these three assumptions are relevant, they do not prevent the crucial question from
being explored, i.e., should one start to characterise the rock strength or the failure enve-
lope dierently?
First, an initial isotropic stress state is studied, which results in a relatively easy stress
redistribution around a circular opening, i.e., the stress state at all points of the borehole
wall is the same. Second, the initial stress state is anisotropic, whereby the axial direction
corresponds to the intermediate principal stress component.
3.1. Initial Isotropic Stress State
The evaluation of the redistribution of the stress is relatively easy if the initial state of
the stress is isotropic. Before drilling the hole, all directions are principal stress directions,
and the circles of Mohr, which represent the stress state, are points. In a major–minor
principal stress diagram, the stress state is situated along the 45° line. After drilling the
hole and away from the boom of the hole, the stress redistribution along radial lines (i.e.,
through the centre of the hole) is the same for all orientations of the radial line, at least if
the rock is homogeneous, continuous, and isotropic. The radial and tangential directions
are the principal stress directions, and the most critical point is situated on the wall of the
borehole. All points along the circumference have the same criticality. Again, this is the
case for a rock that is homogeneous, continuous, and isotropic. If, on the other hand, there
was a spot along the circumference that is a weak spot, the first fracture would occur at this
spot, and the original easy stress redistribution would not necessarily be valid anymore.
As mentioned above, the central question of the research is whether the current prac-
tices of determining the rock strength or the failure envelopes are the right procedures or
if it would be more appropriate to apply a stress path closer to the in situ stress path. To
address this question, some calculations of minimum mud weights are conducted for dif-
ferent values of the initial isotropic stress state.
Three initial stress values are studied, i.e., 25, 50, and 75 MPa. One could assume that
these values correspond to an approximate depth of 1, 2, and 3 km, respectively. These
three depths cover the range of typical borehole depths well. The pore pressure is assumed
to be zero. In the hypothetical case in which there is no well pressure, the stress state along
the circumference is in a 2D approach a radial (principal) stress of 0 MPa and a tangential
(principal) stress of 50, 100, and 150 MPa, respectively (i.e., two times the initial isotropic
stress level). These values are calculated based on the linear elastic theory and the basic
formulas [19,22–24]. The in situ stress path can be assumed to correspond with the third
basic type of stress paths, i.e., the simultaneous loading and unloading with the same in-
crements (the 45° lines in Figure 1). The axial stress remains constant, i.e., the value of the
initial isotropic stress.
Based on the results of the black box rock RVE, the conclusion is that for a depth of 1
km, no shear macro-fracture is induced if the strength is determined by a stress path of
uniaxial loading. The calculated nal tangential stress of 50 MPa (radial stress is equal to
zero) is less than the intersection of 55 MPa for the type 1 failure envelope (S1(loa),S3(=);
black line in Figure 3, Table 1a). However, the intersection of the failure envelope with the
major principal stress axis for the third type of stress path (S1(loa),S3(unl); blue line in
Figure 3), i.e., 30 MPa, is less than the calculated tangential stress of 50 MPa. Hence, the
rock would fail. For the other two depths, the tangential stress is suciently large, so for
these two failure envelopes, shear macro-fracturing occurs. Of course, if the depth is 500
m, the tangential stress is only 25 MPa, i.e., lower than both failure envelopes. Only the
rst and third stress paths are discussed here. The rst is discussed because it corresponds
to the loading in the classic characterisation. The third is discussed because it corresponds
Sustainability 2024, 16, 4711 9 of 23
to the stress variation occurring in situ, i.e., the radial stress is decreased, and the tangen-
tial stress is increased with the same stress increments.
Figure 3. Failure envelopes for the three basic types of stress paths (see also Figure 1) and illustration
of various ways to dene the criticality (α, β, γ). Case of an initial isotropic stress state of 75 MPa.
Red line corresponds to stress path for a linear elastic calculation.
The variation in the contact failure modes as a function of the stress path for the three
cases is similar to the variation presented in Figure 2c. At failure, the three modes of micro-
fracturing are distributed nearly equally (i.e., tension only, mixed mode, and shear only).
For example, at the moment of failure for the initial isotropic stress state of 75 MPa, the
percentages of failed contacts are 28%, 38%, and 34% for the tension-only mode, the mixed
mode, and the shear-only mode, respectively. The rst activation takes place at about 20%
of the entire stress path and is in the tension-only mode.
The conclusion of this rst set of calculations is that, for certain stress levels, the cho-
sen stress path for the characterisation of rock strength can make the dierence between
predicting failure correctly and non-failure incorrectly. The degree of criticality or of
safety is dierent for all stress levels. There are various ways to dene this criticality, i.e.,
the amount above the failure envelopes. The simplest one is just looking at the dierence
between the tangential stress calculated by the linear elastic theory and the intersection of
the failure envelope with the vertical axis (intervals α in Figure 3). For the three depths,
these dierences for S1(loa),S3(=) are -5 MPa (no failure), 45 MPa, and 95 MPa, respectively
(Table 2). For S1(loa),S3(unl), these dierences are larger, i.e., 20 MPa, 70 MPa, and 120
MPa, respectively. A second way is to look at the interval between the stress state calcu-
lated in the linear elastic domain and the failure envelope along the stress path, i.e., the
45° line (intervals β in Figure 3). These values also can be expressed as a percentage of the
total interval of possible stress change, i.e., for a linear elastic calculation (Table 2). The
intervals for the stress paths (S1(loa),S3(unl)) are about two to three times larger than the
intervals for the stress paths (S1(loa),S3(=)). For example, for an initial isotropic stress state
of 50 MPa, the values β1 and β3 are equal to 11.2 MPa and 29.7 MPa, respectively. So, this
means that macro-failure occurs about 8.5 MPa earlier when the stress path is equal to the
in situ stress path in comparison to the conventional strength characterisation. The third
way applied is the most visual and easily interpretable method, i.e., by calculating the
Sustainability 2024, 16, 4711 10 of 23
minimum wellbore pressure needed to eliminate the risk of a shear macro-fracture being
induced. By applying this pressure, the radial stress on the circumference is increased by
its value and the tangential stress is decreased by it. This is indicated by the intervals γ in
Figure 3 and Table 2. By using the depths indicated above, this minimum wellbore pres-
sure is translated into a minimum mud weight. Looking blindly at the gures, it means
that the mud density when characterising a rock by the third type of stress path
(S1(loa),S3(unl)) should be two to three times larger than the density when the rock is
characterised by the rst type of stress path (S1(loa),S3(=)). For example, for the deepest
section (i.e., 3000 m), the minimum density is 0.546 kg/dm3 and 1.171 kg/dm3, respectively.
From a practical perspective and for this particular black box rock, one must point out that
there is no risk of shear macro-failure in four of the six combinations if the density is 1 kg/dm3.
For a weaker material, the latter is not necessarily still true. However, the conclusion remains
that the impact of rock characterisation by one of the two stress paths is significant.
Table 2. Stress redistribution for an initial isotropic stress state and parameters describing the criti-
cality of failure at the borehole wall (see Figure 3), including the minimum mud weight.
Initial isotropic stress state
Approx. depth, m
1000
2000
3000
Stress, MPa
25
50
75
Pore pressure, MPa
0
0
0
Well pressure, MPa
0
0
0
Stress redistribution (lin. elastic)
Radial stress, MPa
0
0
0
Tangential stress, MPa
50
100
150
Criticality parameters
α1, MPa
(no failure)
45
95
α3, MPa
20
70
120
β1, MPa
(no failure)
11.2 (15.8%) *
22.9 (21.0%) *
β3, MPa
9.4 (26.5%) *
29.7 (42.0%) *
50.0 (47.1%) *
γ1, MPa
(no failure)
7.8
16.4
γ3, MPa
5.9
20.5
35.1
Minimum mud weight, kg/dm3
Cf. γ1
-
0.388
0.546
Cf. γ3
0.585
1.024
1.171
* Relative to the entire stress change interval for a linear elastic model.
3.2. Initial Anisotropic Stress State
Even for the relatively simple case of a circular excavation, the stress state in each
location along the circumference is dierent if the initial stress state is anisotropic (Figure
4). As one of the conclusions of the previous study was that failure envelopes are dierent
for each dierent stress path, it means that the study of a circular excavation becomes very
complex. Here, the focus is on the quantication of this complexity. So far, the simulations
of an RVE have shown that the various stress paths result in failure envelopes that are
situated between the one whereby the major principal stress is increased (type 1,
S1(loa),S3(=), or a stress path comparable to the loading in classic laboratory experiments)
and the one whereby the minor principal stress is decreased (type 2, S1(=),S3(unl)). Hence,
by conducting experiments for these two types of stress paths, one gets a relatively good
idea about the possible eect of a stress path on the failure envelope. If the second type of
stress path (S1(=),S3(unl)) is applied, but starting from an anisotropic stress state, the fail-
ure envelope for a larger anisotropy moves closer to the one of the conventional loading
paths. This is studied for the case in which the initial minor principal stress is decreased.
Sustainability 2024, 16, 4711 11 of 23
(a)
(b)
Figure 4. Stress redistribution along borehole wall for the initial anisotropic stress state. (a) 50 and
30 MPa; (b) 50 and 40 MPa. 0° corresponds to orientation of initial major principal stress (Figure 5).
(a)
(b)
Figure 5. Conceptual drawing of orientation of principal stresses (each in a dierent colour) along
borehole wall. (a) Before drilling; (b) after drilling.
In comparison to an initial isotropic stress state, the stress redistribution around a
borehole for the initial anisotropic stress state is more complex. The tangential stress is
dierent in each point of the circumference (Figure 4) (to be precise, per quarter of the
circumference). The two extreme values are reached on the two radial lines that corre-
spond to the orientation of the initial major and minor principal stresses, i.e., for the angles
of 0° and 90°, respectively. For these two angles, the orientation of the principal stresses
does not change, i.e., it is always in the radial and tangential direction in comparison to
the borehole (Figure 5). Along any other radial line, the principal stress orientations before
drilling are dierent from the principal stress orientations on the circumference of the
borehole wall. The laer are always radial and tangential. For the radial line that corre-
sponds to the orientation of the minor principal stresses (i.e., 90° in Figures 4 and 5), this
orientation (i.e., radial orientation) remains the orientation of the minor principal stress.
However, for the radial line that corresponds to the orientation of the major principal
stresses (i.e., 0°), this orientation (i.e., the radial orientation) evolves from the major to the
minor principal stress orientation. All of these aspects should be considered when analys-
ing the RVE behaviour.
In earlier studies, we specically analysed the eect of the principal stress orienta-
tions. Although the aim of the research and the experimental set-up were dierent from
the current research, the importance of the change in stress orientation was illustrated
clearly. Some experiments focused on cyclic Brazilian tensile tests, whereby the disks were
rotated before reaching the nal macro-failure [25]. These tests aimed to evaluate the
Sustainability 2024, 16, 4711 12 of 23
sensitivity of the Kaiser eect towards the principal stress orientation. The experiments
and the discrete simulations showed that the eect gradually became less pronounced as
the rotation angle increased. In another study, the damage around a circular opening was
studied [14] and two congurations were investigated. In the rst set of experiments, the
damage was induced mainly by shear stresses (only macro-compressive stresses were pre-
sent). In the second set, the tangential stresses were tensile stresses at the macro-scale. For
specimens rst damaged by a tensile macro-stress, followed by macro-compressive
stresses, more damage was observed than in the other sequence.
3.2.1. No Change in the Orientation of the Principal Stresses
The two points situated at the angles of 0° and 90° along the circumference are dis-
cussed rst. As illustrated in Figure 5, the radial and tangential orientations are principal
stress orientations before and after drilling the borehole. Elastic calculations clearly show
that the point at 90° is more critical than the point at 0°, i.e., for shear failure at a macro-
scopic scale. An unknown factor for all analyses presented in this paper is how the stress
state exactly evolves from before drilling the hole until after the hole has been drilled. For
all simulations, it is assumed that the stress paths follow linear variations in the major-
minor principal stress diagrams. This assumption seems to be realistic, at least if one con-
siders a 2D approach. For a 3D model and taking the stress variation around the boom
of the hole into account, the real stress paths are most likely more complex [21,23,24,26].
For the point at 90°, the situation is comparable to the results presented earlier. The
tangential and radial orientations correspond to the major and minor principal stress ori-
entations over the entire stress path, respectively. Four dierent initial anisotropic stress
states are studied, i.e., (30, 50), (40, 50), (45, 75), and (60, 75) MPa (Figure 6 and Table 3).
The same method used above is applied here. For each initial stress state, the nal stress
state is calculated for the borehole wall, based on a linear elastic calculation. The RVE
model is loaded in the vertical direction (i.e., corresponding to the tangential orientation)
and unloaded in the horizontal direction (i.e., corresponding to the radial orientation)
aiming at the nal linear elastic stress state (Figure 6a). However, the laer is, in most
cases, never reached because failure occurs somewhere along the stress path. In other
words, the maximum strength is calculated for each stress path (Figure 6a; Table 3a). These
four strength values correspond to four dierent failure envelopes, i.e., for each of the four
stress paths. The four simulated strength values are situated between the failure envelopes
of uniaxial loading (type 1; black line in Figure 6a) and of the simultaneous loading and
unloading with the same increments (type 3; blue line). The larger the initial anisotropy,
the closer the strength for the in situ stress paths is situated to the failure envelopes of
uniaxial loading. Most likely, the behaviour for the various stress paths is a combination
of the mentioned basic types of stress paths (black and blue lines). For a larger anisotropy,
the in situ stress path is oriented more vertically and thus more towards the type of uni-
axial loading (and farther away from the type of stress path under an angle of 45°).
Table 3. Stress redistribution for the initial anisotropic stress state and parameters describing the
criticality of failure at the borehole wall (see Figure 3), including the minimum mud weight. (a) For
point corresponding to 90° along the circumference; (b) for point corresponding to 0° along the cir-
cumference.
(a)
Initial anisotropic stress state
Approx. depth, m
2000
2000
3000
3000
Major principal stress, MPa
50
50
75
75
Minor principal stress, MPa
30
40
45
60
Pore pressure, MPa
0
0
0
0
Well pressure, MPa
0
0
0
0
Stress redistribution (lin. elastic)
Sustainability 2024, 16, 4711 13 of 23
Radial stress, MPa
0
0
0
0
Tangential stress, MPa
120
110
180
165
Strength
Major principal stress, MPa
87.3
84.2
132
124.5
Minor principal stress, MPa
14
17.2
20.6
27
Criticality parameters
α1, MPa
65
55
125
110
αreal, MPa
78
73
132
125
α3, MPa
90
80
150
135
β1, MPa
23.1 (30.4%) *
15.7 (21.8%) *
44.5 (38.9%) *
31.5 (29.1%) *
βreal, MPa
35.5 (46.7%) *
31.0 (43.0%) *
52.2 (45.7%) *
48.7 (45.0%) *
β3, MPa
48.1 (63.2%) *
36.8 (51.1%) *
80.1 (70.1%) *
62.1 (57.4%) *
γ1, MPa
11.2
9.5
21.6
19
γreal, MPa
18.4
19.5
26
30.3
γ3, MPa
26.3
23.4
43.9
39.5
Minimum mud weight, kg/dm3
Cf. γ1
0.56
0.474
0.718
0.632
Cf. γreal
0.921
0.975
0.867
1.009
Cf. γ3
1.317
1.171
1.463
1.317
(b)
Initial anisotropic stress state
Approx. depth, m
2000
2000
3000
3000
Major principal stress, MPa
50
50
75
75
Minor principal stress, MPa
30
40
45
60
Pore pressure, MPa
0
0
0
0
Well pressure, MPa
0
0
0
0
Stress redistribution (lin. elastic)
Switch between major-minor principal
stress, MPa
33.3
43.8
50
65.6
Final radial stress, MPa
0
0
0
0
Final tangential stress, MPa
40
70
60
105
Strength
Major principal stress, MPa
(no failure)
68.5
(no failure)
99.8
Minor principal stress, MPa
2.5
8.6
Criticality parameters
α1, MPa
(-)
15
5
50
αreal, MPa
(-)
12
2
47
α3, MPa
10
40
30
75
β1, MPa
(-)
3.2 (6.3%) *
(-)
10.8 (14.1%) *
βreal, MPa
(-)
2.9 (5.7%) *
(-)
10.1 (13.1%) *
β3, MPa
4.4 (13.2%) *
29.7 (42.0%) *
11.8 (23.1%) *
29.2 (38.1%) *
γ1, MPa
(-)
2.6
(-)
8.6
γreal, MPa
(-)
2
(-)
8
γ3, MPa
2.9
11.7
8.8
22
Minimum mud weight, kg/dm3
Cf. γ1
(-)
0.129
(-)
0.287
Cf. γreal
(-)
0.102
(-)
0.267
Cf. γ3
0.146
0.585
0.293
0.732
* Relative to the entire stress change interval for a linear elastic model.
Sustainability 2024, 16, 4711 14 of 23
(a)
(b)
Figure 6. Stress paths between the initial anisotropic stress state and the linear elastic calculation of
the stress state on the borehole wall. The starting points and the stress states at failure are indicated
by squares and circles. All information is superimposed on the failure envelopes for the three basic
types of stress paths (see Figure 1). Four dierent anisotropic stress states are studied (each with a
dierent colour), i.e., (30, 50), (40, 50), (45, 75), and (60, 75) MPa. (a) 90° angle (see Figure 5); (b) 0°
angle.
It is important to stress again that, in the approach put forward in this study, the
strength values in Table 3a are considered to be the “correct” ones (indicated by the label
“real”; see also the squares and circles in Figure 6a). In comparison with these values, the
values determined by stress path type 1 (S1(loa),S3(=)) are an overestimation of the “real”
strength, and the ones for the type 3 stress path (S1(loa),S3(unl)) are an underestimation
of the “real” strength. The type 2 stress path ((S1(=),S3(unl), brown line) results in a larger
underestimation and is not integrated in Table 3. In Table 3a, the values for the various
parameters of criticality are presented for the type 1 stress path, the in situ or real stress
path, and the type 3 stress path. For detailed values, the reader is referred to Table 3a, but
the conclusion is that the values for the in situ (or real) stress paths are signicantly dif-
ferent from the two basic types of stress paths.
This conclusion also is valid for the calculated mud weight (Table 3a). The mud
weight only can be calculated if a depth is known. Here, it is assumed that the major prin-
cipal stress corresponds to the depth. So, the value of 50 MPa corresponds to an approxi-
mate depth of 2000 m, and the value of 75 MPa corresponds to 3000 m. So, the lay-out
could be a horizontal borehole at these depths or a vertical borehole whereby the major
horizontal principal stress is equal to the vertical (axial) stress. For example, for an initial
stress state of (60, 75) MPa, the minimum mud weight for the real stress path is 1.009
kg/dm3. When one determines the strength by applying stress paths similar to the conven-
tional testing (type 1), the minimum mud weight is only 0.632 kg/dm3. This is a signicant
dierence, and it cannot be neglected. When comparing the various values of the mini-
mum mud weights (Table 3a), there is a peculiar observation. For both basic types of stress
paths, the minimum mud weight is greater when the initial anisotropy is larger (i.e., com-
paring the rst column to the second column and comparing the third column to the
fourth column). It is just the opposite for the in situ or real stress paths. The reason for this
is that, in the case of the real stress paths, both the linear elastic calculated stress values
Sustainability 2024, 16, 4711 15 of 23
and the strength values are dierent as a function of the initial anisotropy. For the cases
with an initial major principal stress of 75 MPa, the linear elastic calculated stress values
are 180 MPa (third column) and 165 MPa (fourth column) for the large and small initial
anisotropy, respectively. As can be seen in Figure 6a, the two real stress paths cross. This
means that, the value of 180 MPa is farther away from the basic types of failure envelopes
than the value of 165 MPa. So, a larger minimum mud weight is needed. When consider-
ing the real stress path, the linear elastic value of 180 MPa is situated closer to the corre-
sponding calculated strength than it is to the value of 165 MPa and its corresponding
strength value. Note that the entire four failure envelopes that correspond to the four real
or in situ stress paths are not drawn in Figure 6a; only one point of each failure envelope
is represented. In conclusion, the calculations of the critical mud weights illustrate well
the importance of applying a stress path close to the in situ stress path when testing rocks.
The calculations also highlight the complexity of the problem.
As pointed out in Figure 4 above, the point that corresponds to an angle of 90° along
the circumference is more critical than the 0° point when studying macroscopic shear frac-
turing. In Figure 6b and Table 3b, the same exercise is repeated for an angle of 0°. A sig-
nicant dierence between the two points is that for 0°, the orientation of the major and
minor principal stresses switches along the stress path (Figure 5). Initially, the radial stress
component is the major principal stress. Along the stress path, this component decreases,
and at a certain moment, it becomes equal to the tangential stress, which increases sys-
tematically starting from the initial minor principal stress (Figure 6b). Afterwards, the tan-
gential stress increases further and the radial stress decreases. Table 3b gives the stress
value for which both components are equal. The result of switching between the principal
stress orientations is that the global stress state rst evolves away from the failure enve-
lopes (Figure 6b). At the moment of this switch, the circle of Mohr is represented by a
point. During that part of the entire stress variation, no micro-fracturing is induced (Fig-
ure 7b). For the two cases with the largest anisotropy, no failure occurs (Figure 6b). For
the other two cases, the strength following the real stress path is similar to the strength
that corresponds to the type 1 stress path (S1(loa),S3(=)).
(a)
(b)
Figure 7. Variation in the contact failure modes (i.e., tensile-only, shear-only, and mixed modes) as
a function of the stress path, i.e., from the initial stress state of (60, 75) MPa till the load level of
failure, corresponding to 50% failed contacts. (a) 90° angle (see Figure 5); (b) 0° angle.
In Figure 7, the variation in the three contact failure modes is presented as a function
of the full in situ stress path for the case of an initial stress state of (60, 75) MPa and for an
angle of 90° and 0°. As mentioned earlier, during the stress relaxation for an angle of 0°,
no contacts are activated (Figure 7b). At the moment of failure (i.e., 50% of all contacts
have been activated), the main activation mode is shear (55% of all activated contacts in
the shear-only mode and 35% in the mixed mode). The activation in the tensile-only mode
is small (10% of all activated contacts). The overall shape of the curves for a 0° angle is
Sustainability 2024, 16, 4711 16 of 23
comparable to the type 1 stress path (Figures 2a and 7b), although the decrease in the
tensile-only mode is more pronounced in Figure 7b. Part of the explanation could be that
the strength following the real stress path is situated close to the type 1 failure envelope
(S1(loa),S3(=)). For an angle of 90° (Figure 7a), the activation in shear only is less dominant
than for 0° (Figure 7b). For 90°, the percentages of all activated contacts are 20% tensile-
only mode, 37% mixed mode and 43% shear-only mode. The situation at failure is similar
to the type 1 stress path (Figure 2a).
3.2.2. Change in the Orientation of the Principal Stresses
Apart from the two points discussed above, the orientation of the principal stresses
for the other points is dierent between the initial stress state and the nal stress state
along the borehole wall. This means that both the sizes and the orientations of the princi-
pal stresses change. For these variations, it is assumed that the size of the principal stresses
follows a linear variation in successive steps and that the orientation changes simultane-
ously. This is illustrated in Figure 8 for the point on the circumference at an angle of 45°.
For this angle, the nal normal stresses on the planes of 0° and 90° (i.e., the original prin-
cipal stress orientations) are equal. (See the red triangle for the nal (i.e., largest) Mohr
circle). Note that only a limited number of successive steps are presented in Figure 8.
Along the entire stress path, rst, the normal stress increases, and this is followed by a
decrease for one of these two planes (illustrated by the right set of triangles). For the other
plane, rst, there is a decrease in the normal stress, and this is followed by an increase (the
left set of triangles). Along the entire stress path, there is a systematic increase in the shear
stresses on these planes of 0° and 90°. For the orientations closer to 0° than 45°, the princi-
pal stress paths also cross, similar to the case of 0° (Figure 6b).
Figure 8. Evolution of the stress state starting from the initial anisotropic stress of (60, 75) MPa till
the stress state of (0, 135) MPa, based on linear elastic calculations. Red triangles represent the com-
bination of normal and shear stresses for the successive Mohr circles, acting on the planes with an
orientation of 0° and 90°.
For the initial four principal stress combinations, the strength values following the
real stress paths for the point at an angle of 45° along the circumference are situated rela-
tively close to the type 1 failure envelope (S1(loa),S3(=); Figure 9). Note that the point of
45° along the circumference in most cases is situated outside the danger zone for a mac-
roscopic shear fracture [20]. The stress path applied here, i.e., with a systematic change in
principal stress orientation, is globally an unknown territory in rock characterisation, and
the translation of it in laboratory experiments is far from evident. In the experiments by
Lavrov et al. [25], a systematic change in principal stress orientations was applied, but it
was for Brazilian tensile tests and for cyclic testing, which is dierent from the stress var-
iation studied here. The black box RVE simulation indicates that the amount of the tensile-
only mode of activation is very limited over the entire stress path (Figure 10). When 50%
of all contacts are activated, the percentage for the three activation modes of all activated
contacts are 5% tensile-only, 38% mixed, and 57% shear-only, respectively. The rst con-
tact that is activated is in shear (at about 35% of the entire stress path). For all of the pre-
vious stress paths that were studied, the rst contact activation was in tension.
Sustainability 2024, 16, 4711 17 of 23
Figure 9. Stress paths between the initial anisotropic stress state and the linear elastic calculation of
the stress state on the borehole wall for an angle of 45°. The starting points and the stress states at
failure are indicated by squares and circles. All information is superimposed on the failure enve-
lopes for the three basic types of stress paths (see Figure 1). Four dierent anisotropic stress states
are studied (each with a dierent colour), i.e., (30, 50), (40, 50), (45, 75), and (60, 75) MPa.
Figure 10. Variation in the contact failure modes (i.e., tensile-only, shear-only, and mixed modes) as
a function of the stress path, i.e., from the initial stress state of (60, 75) MPa till the load level of
failure, corresponding to 50% failed contacts. Information is for an angle of 45°.
For the initial anisotropic stress state of (60, 75) MPa, the exercise is repeated for all
angles along the circumference at 11.25° intervals (Figure 11 and Table 4). The three angles
that already have been presented (i.e., 0°, 45°, and 90°) also are included. The stress path
for each angle is dierent because the linear elastic stresses at the circumference are dif-
ferent. This means that the real strength in the approach taken in this study is presented
by dierent points in the major–minor principal stress diagram. Figure 11b presents the
stress ranges of interest. In this diagram, the failure envelopes also are drawn for each
individual angle. The stress paths starting from the initial anisotropic stress state are not
drawn because doing so would overload the graph. For the interval between 45° and 90°,
i.e., the most critical zone, the real failure envelopes move away from the type 1 failure
envelope (S1(loa),S3(=)) and move towards the type 3 failure envelope (S1(loa),S3(unl)).
For an angle of 90°, the “real” strength is closest to the laer for all angles between 0° and
90°. The strength for the angles of 0° and 11.25° are situated close to the type 1 failure
Sustainability 2024, 16, 4711 18 of 23
envelope. For practical reasons, in the further analysis, it is assumed that the failure enve-
lopes of these two angles are the same as the type 1 failure envelope. In Figure 12, critical-
ity parameter α (Figure 3) is presented as a function of the angle along the circumference.
The parameter is presented for the three types of failure envelopes (i.e., type 1, type 3, and
the series of the real envelope for each angle). Figure 12 conrms the observations formu-
lated for Figure 11. The curve that corresponds to the real stress paths is closer to the fail-
ure envelope of the type 3 stress path for larger angles and further away for smaller angles
(and the opposite is true when referring to the type 1 stress path).
(a)
(b)
Figure 11. Strength values following the real stress paths starting from an anisotropic stress state of
(60, 75) MPa for all angles along the circumference between 0° and 90° with 11.25° intervals. All
information is superimposed on the failure envelopes for the three basic types of stress paths (see
Figure 1). (a) Full view; (b) zoomed-in view and all individual failure envelopes added (each with
a dierent colour).
Figure 12. Variation in criticality parameter α (Figure 3) along circumference of borehole wall for
three dierent stress paths starting from the initial anisotropic stress state of (60, 75) MPa: the real
stress path (green), the type 1 stress path (S1(loa),S3(=); black), and the type 3 stress path
(S1(loa),S3(unl); blue).
Sustainability 2024, 16, 4711 19 of 23
Table 4. Stress redistribution for the initial anisotropic stress state and parameters describing the criticality of failure along the circumference of the borehole wall,
including the minimum mud weight.
Angle Along Circumference (°)
0
11.25
22.5
33.75
45
56.25
67.5
78.75
90
Stress redistribution (lin.elastic)
Radial stress, MPa
0
0
0
0
0
0
0
0
0
Tangential stress, MPa
105.0
107.3
113.8
123.5
135.0
146.5
156.2
162.7
165
Strength
Major principal stress, MPa
99.8
101.7
103.5
109.3
116.3
119.0
122.0
124.0
124.5
Minor principal stress, MPa
8.6
9.5
15.0
17.1
18.7
23.1
25.3
26.5
27.0
Criticality parameters
α1, MPa
50
52.3
58.8
68.5
80
91.5
101.2
107.7
110
αreal, MPa
47
51.3
67.8
77.5
89
102.5
114.2
121.7
125
α3, MPa
75
77.3
83.8
93.5
105
116.5
126.2
132.7
135
γ1, MPa
8.6
9.0
10.1
11.8
13.8
15.8
17.4
18.6
19.0
γreal, MPa
8.0
8.8
14.0
16.5
18.2
24.1
27.4
29.4
30.3
γ3, MPa
22.0
22.6
24.5
27.4
30.7
34.1
36.9
38.8
39.5
Minimum mud weight, kg/dm3
Depth 3000 m
Cf. γ1
0.287
0.300
0.338
0.394
0.460
0.526
0.582
0.619
0.632
Cf. γreal
0.267
0.292
0.468
0.550
0.607
0.804
0.914
0.981
1.009
Cf. γ3
0.732
0.754
0.817
0.912
1.024
1.136
1.231
1.295
1.317
Depth 2400 m
Cf. γ1
0.359
0.376
0.422
0.492
0.575
0.657
0.727
0.774
0.790
Cf. γreal
0.333
0.365
0.584
0.688
0.759
1.004
1.143
1.227
1.261
Cf. γ3
0.915
0.942
1.022
1.140
1.280
1.420
1.539
1.619
1.646
Sustainability 2024, 16, 4711 20 of 23
In Table 4, the minimum mud weight is calculated as a function of the angle. Two
cases are considered for the initial stress state of (60, 75) MPa. The rst case is the case in
which the initial major principal stress of 75 MPa corresponds to an approximate depth of
3000 m (Figure 13a). As explained above, this could be a horizontal borehole or a vertical
borehole at that depth, whereby the major horizontal principal stress is equal to the verti-
cal (axial) stress. Second is the case in which the initial minor principal stress of 60 MPa
corresponds to an approximate depth of 2400 m (Figure 13b). The main reason to consider
both depth values is to obtain a good idea about the impact on the critical mud weight.
The rst observation is that the largest minimum mud weight is needed for the point
along the circumference at an angle of 90°. The second observation is that the mud weight
for all angles, taking into account the real or in situ stress paths, is situated between the
values if one applies the type 1 loading or the type 3 loading and unloading of both prin-
cipal stresses. The third observation is a conrmation that the dierence in mud weights
for small angles is small between the real stress paths and the type 1 loading. The fourth
observation is linked to the ratio between the minimum mud weight for an angle of 90°
and the one for an angle of 0°. For the type 1 loading, this ratio is equal to 2.2 for both
depths. However, for the real stress paths, this ratio is about 3 (i.e., 2.9 for an assumed
depth of 3000 m and 3.5 for 2400 m). The laer value is equal to 1.261 kg/dm3 divided by
0.333 kg/dm3 (Table 4). Although one could say that the only relevant value is the overall
largest minimum weight of mud, which is applied along the entire circumference, the var-
iation in the individual calculated values along the circumference gives a good idea about
the critical arc. This leads to the fth observation, when, for example, for a depth of 2400
m, one looks at the arc where a minimum mud weight of 1 kg/dm3 is needed. For the real
stress path, this arc corresponds to angles between about 56° and 90°. However, if the rock
is characterised by classic laboratory experiments, all values are below a minimum mud
weight of 1 kg/dm3.
(a)
(b)
Figure 13. Variation in the minimum mud density along circumference of borehole wall for three
dierent stress paths starting from the initial anisotropic stress state of (60, 75) MPa: the real stress
path (green), the type 1 stress path (S1(loa),S3(=); black) and the type 3 stress path (S1(loa),S3(unl);
blue). (a) For a depth of 3000 m; (b) for a depth of 2400 m.
4. Discussion and Conclusions
Since halfway through the 20th century, the rock mechanical community has been
testing intact rock specimens by starting from a zero-stress state and systematically in-
creasing the load. On the one hand, the main goals of these experiments are the determi-
nation of the strength of rocks and possibly the entire failure envelope. On the other hand,
there is the goal to obtain a beer understanding of the fracturing process that leads to the
failure of the specimens. However, it is important to admit that we still do not fully un-
derstand either the entire process of rock failure at the lab scale or the link with the in situ
behaviour of a rock mass. This statement clearly is supported by the numerous research
projects that are still being conducted that focus on the basic type of loading applied in
Sustainability 2024, 16, 4711 21 of 23
laboratory experiments. These projects often focus on a specic aspect, for example, the
impact of heterogeneities (e.g., [10,27,28]), the orientations of aws or weak planes (e.g.,
[21,29–31]), and the characterisation of the (micro-)fracturing process by the analysis of
acoustic emission signals (e.g., [15–17,32,33]). These are only a few of the many examples
that use classic testing procedures. The correct characterisation of the rock strength and a
good understanding of the failure process are essential for all types of design, but their
importance even has increased because, more and more, the design should integrate the
concepts of long-term behaviour and sustainability.
Some researchers have looked at alternative loading, e.g., dynamic loading (e.g.,
[6,34–37]), cyclic loading (e.g., [5–7]), or even more complex loading paths (e.g., [8,11]),
but the focus of these projects is on developing a beer understanding of the phenomenon
rather than on characterising the strength of a rock and the failure envelope. In this paper,
as was the case in the previous publication, the question is whether one would benet
from having a more accurate and more reliable design when the testing procedure of the
specimens are changed signicantly. As mentioned above, the current laboratory practice
starts from a zero-stress state, and this is followed by the loading of the specimen. In situ,
the starting point is the in situ stress state and, due to the excavation, this is followed by
an unloading or a combination of unloading one stress component and the loading of
another. The paper addresses two questions, i.e., (1) has the application of dierent stress
paths during rock characterisation a signicant impact on rock strength and the failure
envelope; (2) if so, is the impact on the design parameters relevant? In this paper, the laer
is the minimum mud weight that is needed to eliminate the fracturing of the rock around
a circular borehole.
The answer to the rst question clearly is positive, and it also was the conclusion of
an earlier publication [4]. The main advantage of numerical simulations and working with
the same black box rock RVE is that one can test the same model as often as needed. This
is clearly not the case when testing real specimens. Apart from the fact that the amount of
available rock material is almost always limited, rock material still is heterogeneous to a
certain degree, which means that no two specimens are exactly the same. Also, there is the
restrictions of time and cost when conducting experiments. The simulations show that
large reductions in the strength of a rock can occur. For example, the reduction in the
strength for the uniaxial unloading stress path in comparison to the uniaxial loading stress
path is about 40 to 45%. A comparison between the simultaneous loading and unloading
stress paths and the uniaxial loading shows a strength reduction of about 30%. These val-
ues are obtained when the initial stress state is isotropic. The reduction in the strength is
less when the initial stress state is anisotropic.
The answer to the second question also is positive. For an initial isotropic stress state,
the minimum density of the mud when characterising the rock by stress paths that are
close to the in situ stress variation can be two to three times larger than when the rock is
characterised by stress paths that are close to the classic experimental procedures. For the
initial anisotropic stress state, the answer remains positive, but the analysis becomes very
complex because each point along the circumference undergoes a dierent stress path.
The impact is discussed in detail above, but the main conclusions are as follows: (i) the
most critical point along the circumference remains the point with the largest tangential
stress; (ii) depending on the input parameters, the minimum mud weight can be up to
60% larger when the stress path is close to the in situ stress path instead of the one close to
the classic experiments; (iii) and the arc along the circumference where a minimum mud
weight of 1 kg/dm3, for example, is required is larger when the in situ stress path is considered.
The overall conclusion of the analysis presented in this paper is that the impact of
dierent stress paths needs further aention. The characterisation of the strength or the
characterisation of the entire failure envelope based on a stress path closer to the in situ
stress paths leads to more correct values than if one characterises the rock by loading rock
specimens in the classic experiments. Such a signicant change in the procedure of rock
characterisation will not be realised from one day to the next. It also is logical that further
Sustainability 2024, 16, 4711 22 of 23
research is needed, e.g., a 3D analysis, a more detailed RVE model, typical in situ stress
paths for other applications, and a verication by laboratory experiments. It is only at that
moment that one can start thinking about implementing it in the engineering design.
However, some of the more complex stress paths, i.e., with a change in principal stress
orientations, cannot be implemented easily in a laboratory. After learning more about the
impact of dierent stress paths, one must probably make some choices, one of which is
whether such implementations will be possible or whether they would be impractical.
However, the combination of laboratory experiments and the simulation of RVE models
could become a worthwhile combination. The analysis also highlighted again that discrete
models really are needed to study the behaviour of rock material, as the micro-fracturing
during nearly the entire variation in the stress state plays an important role. A good next
step in the discrete modelling of rock material is to calibrate the models by also using
experiments whereby in situ stress paths are applied.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Data are contained within the article.
Conicts of Interest: The author declares no conicts of interest.
References
1. Bieniawski, Z.T.; Bernede, M.J. Suggested Methods for Determining the Uniaxial Compressive Strength and Deformability of
Rock Materials. Int. J. Rock Mech. Min. Sci. 1979, 16, 138–140. hps://doi.org/10.1016/0148-9062(79)91451-7.
2. European Standard EN1997-2. Eurocode 7, Geotechnical design, Part 2, Ground Investigation and Testing, Chapter 5.14,
Strength Testing of Rock Material, 2010. Available online: hps://eurocodes.jrc.ec.europa.eu/EN-Eurocodes/eurocode-7-ge-
otechnical-design (accessed on 21 April 2024).
3. ASTM D7012-14e1. Standard Test Methods for Compressive Strength and Elastic Moduli of Intact Rock Core Specimens under
Varying States of Stress and Temperatures, 2023. Available online: hps://www.astm.org/d7012-14e01.html (accessed on 21
April 2024).
4. Vervoort, A. Dierent Stress Paths Lead to Dierent Failure Envelopes: Impact on Rock Characterisation and Design. Appl. Sci.
2023, 13, 11301. hps://doi.org/10.3390/app132011301.
5. Ding, Z.W.; Jia, J.D.; Tang, Q.B.; Li, X.F. Mechanical Properties and Energy Damage Evolution Characteristics of Coal Under
Cyclic Loading and Unloading. Rock Mech. Rock Eng. 2022, 55, 4765–4781. hps://doi.org/10.1007/s00603-022-02884-x.
6. Ning, Z.; Xue, Y.; Li, Z.; Su, M.; Kong, F.; Bai, C. Damage Characteristics of Granite Under Hydraulic and Cyclic Loading–
Unloading Coupling Condition. Rock Mech. Rock Eng. 2022, 55, 1393–1410. hps://doi.org/10.1007/s00603-021-02698-3.
7. Zhang, A.; Xie, H.; Zhang, R.; Gao, M.; Xie, J.; Jia, Z.; Ren, L.; Zhang, Z. Mechanical properties and energy characteristics of coal
at dierent depths under cyclic triaxial loading and unloading. Int. J. Rock Mech. Min. Sci. 2023, 161, 105271.
hps://doi.org/10.1016/j.ijrmms.2022.105271.
8. Li, Y.; Fu, J.; Hao, N.; Song, W.; Yu, L. Experimental study on unloading failure characteristics and damage evolution rules of
deep diorite based on triaxial acoustic emission tests. Geosci. J. 2023, 27, 629–646. hps://doi.org/10.1007/s12303-023-0020-y.
9. Debecker, B.; Vervoort, A. Two-dimensional discrete element simulations of the fracture behavior of slate. Int. J. Rock Mech. Min.
Sci. 2013, 61, 161–170. hps://doi.org/10.1016/j.ijrmms.2013.02.004.
10. Van Lysebeen, G.; Vervoort, A.; Maertens, J.; Huybrechts, N. Discrete element modeling for the study of the eect of soft
inclusions on the behavior of soil mix material. Comput. Geotech. 2014, 55, 342–351.
hps://doi.org/10.1016/j.compgeo.2013.09.023.
11. Song, Z.; Zhang, J.; Wang, S.; Dong, X.; Zhang, Y. Energy Evolution Characteristics and Weak Structure—“Energy Flow” Impact
Damaged Mechanism of Deep-Bedded Sandstone. Rock Mech. Rock Eng. 2023, 56, 2017–2047. hps://doi.org/10.1007/s00603-022-
03150-w.
12. Itasca, 2004. UDEC v4.0 manual. Itasca Consulting Group, Inc., 2004. Available online: hps://www.itascacg.com/software/udec
(accessed on 21 April 2024).
13. Dong, L.J.; Chen, Y.C.; Sun, D.Y.; Zhang, Y.H.; Deng, S.J. Implications for identication of principal stress directions from acous-
tic emission characteristics of granite under biaxial compression experiments. J. Rock Mech. Geotech. Eng. 2023, 15, 852–863.
hps://doi.org/10.1016/j.jrmge.2022.06.003.
14. Ganne, P.; Vervoort, A.; Wevers, M. Quantication of pre-peak brile damage: Correlation between acoustic emission and ob-
served micro-fracturing. Int. J. Rock Mech. Min. Sci. 2007, 44, 720–729. hps://doi.org/10.1016/j.ijrmms.2006.11.003.
Sustainability 2024, 16, 4711 23 of 23
15. Chang, S.H.; Lee, C.I. Estimation of cracking and damage mechanisms in rock under triaxial compression by moment tensor
analysis of acoustic emission. Int. J. Rock Mech. Min. Sci. 2004, 41, 1069–1086. hps://doi.org/10.1016/j.ijrmms.2004.04.006.
16. Du, K.; Li, X.; Tao, M.; Wang, S. Experimental study on acoustic emission (AE) characteristics and crack classication during
rock fracture in several basic lab tests. Int. J. Rock Mech. Min. Sci. 2020, 133, 104411. hps://doi.org/10.1016/j.ijrmms.2020.104411.
17. Liu, X.; Zhang, H.; Wang, X.; Zhang, C.; Xie, H.; Yang, S.; Lu, W. Acoustic Emission Characteristics of Graded Loading Intact
and Holey Rock Samples during the Damage and Failure Process. Appl. Sci. 2019, 9, 1595. hps://doi.org/10.3390/app9081595.
18. Cundall, P.A. A computer model for simulating progressive large-scale movements in block rock systems. In Proceedings of
the ISRM Symposium, Nancy, France, 4–6 September 1971, Paper II-8.
19. Fjær, E.; Holt, R.M.; Horsrud, P.; Raaen, A.M.; Risnes, R. (Eds.) Petroleum Related Rock Mechanics. In Developments in Petroleum
Science, 2nd ed.; Elsevier: Amsterdam, The Netherlands, 2008; Volume 53, 491p. ISBN: 978-0-444-50260-5
hps://doi.org/10.1016/S0376-7361(07)53003-7.
20. Xiang, Z.; Moon, T.; Oh, J.; Si, G.; Canbulat, I. Analytical investigations of in situ stress inversion from borehole breakout geom-
etries. J. Rock Mech. Geotech. Eng. 2023, in press. hps://doi.org/10.1016/j.jrmge.2023.08.018.
21. Zhang, Y.; Long, A.; Zhao, Y.; Zang, A.; Wang, C. Mutual impact of true triaxial stress, borehole orientation and bedding incli-
nation on laboratory hydraulic fracturing of Lushan shale. J. Rock Mech. Geotech. Eng. 2023, 15, 3131–3147.
hps://doi.org/10.1016/j.jrmge.2023.02.015.
22. Wang, S.; Liao, G.; Zhang, Z.; Wang, X. Study on Wellbore Stability Evaluation Method of New Drilled Well in Old Reservoir.
Processes 2022, 10, 1334. hps://doi.org/10.3390/pr10071334.
23. Deangeli, C.; Omwanghe, O.O. Prediction of Mud Pressures for the Stability of Wellbores Drilled in Transversely Isotropic
Rocks. Energies 2018, 11, 1944. hps://doi.org/10.3390/en11081944.
24. Shi, Y.; Ma, T.; Zhang, D.; Chen, Y; Liu, Y.; Deng, C. Analytical model of wellbore stability analysis of inclined well based on the
advantageous synergy among the ve strength criteria. Geouids 2023, 2023, 2201870. hps://doi.org/10.1155/2023/2201870.
25. Lavrov, A.; Vervoort, A.; Wevers, M.; Napier, J.A.L. Experimental and numerical study of the Kaiser eect in cyclic Brazilian
tests with disk rotation. Int. J. Rock Mech. Min. Sci. 2002, 39, 287–302. hps://doi.org/10.1016/S1365-1609(02)00038-2.
26. Eberhardt, E. Numerical Modelling of Three-Dimension Stress Rotation Ahead of an Advancing Tunnel Face. Int. J. Rock Mech.
Min. Sci. 2001, 38, 499–518. hps://doi.org/10.1016/S1365-1609(01)00017-X.
27. Malan, D.F.; Napier, J.A.L. Computer modelling of granular material microfracturing. Tectonophysics 1995, 248, 21–37.
hps://doi.org/10.1016/0040-1951(95)00019-J.
28. Van de Steen, B.; Vervoort, A.; Sahin, K. Inuence of internal structure of crinoidal limestone on fracture paths. Int. J. Eng. Geol.
2002, 67, 109–125. hps://doi.org/10.1016/S0013-7952(02)00148-5.
29. Wang, T.; Ye, W.; Liu, L.; Li, A.; Jiang, N.; Zhang, L.; Zhu, S. Impact of Crack Inclination Angle on the Spliing Failure and
Energy Analysis of Fine-Grained Sandstone. Appl. Sci. 2023, 13, 7834. hps://doi.org/10.3390/app13137834.
30. Chang, X.; Zhang, X.; Dang, F.; Zhang, B.; Chang, F. Failure Behavior of Sandstone Specimens Containing a Single Flaw Under
True Triaxial Compression. Rock Mech. Rock Eng. 2022, 55, 2111–2127. hps://doi.org/10.1007/s00603-021-02761-z.
31. Van de Steen, B.; Vervoort, A.; Napier, J.A.L.; Durrheim, R.J. Implementation of a aw model to the fracturing around a vertical
shaft. Rock Mech. Rock Eng. 2003, 36, 143–161. hps://doi.org/10.1007/s00603-002-0040-2.
32. Dong, L.; Zhang, Y.; Bi, S.; Ma, J.; Yan, Y.; Cao, H. Uncertainty investigation for the classication of rock micro-fracture types
using acoustic emission parameters. Int. J. Rock Mech. Min. Sci. 2023, 162, 105292. hps://doi.org/10.1016/j.ijrmms.2022.105292.
33. Xie, N.; Tang, H.M.; Yang, J.B.; Jiang, Q.H. Damage Evolution in Dry and Saturated Brile Sandstone Revealed by Acoustic
Characterisation Under Uniaxial Compression. Rock Mech. Rock Eng. 2022, 55, 1303–1324. hps://doi.org/10.1007/s00603-021-
02716-4.
34. Hu, L.; Yu, L.; Ju, M.; Li, X.; Tang, C. Eects of intermediate stress on deep rock strainbursts under true triaxial stresses. J. Rock
Mech. Geotech. Eng. 2023, 15, 659–682. hps://doi.org/10.1016/j.jrmge.2022.06.008.
35. Du, K.; Tao, M.; Li, X.B.; Zhou, J. Experimental study of slabbing and rockburst induced by true-triaxial unloading and local
dynamic disturbance. Rock Mech. Rock Eng. 2016, 49, 3437–3453. hps://doi.org/10.1007/s00603-016-0990-4.
36. Gong, F.Q.; Wu, C.; Luo, S.; Yan, J.Y. Load–unload response ratio characteristics of rock materials and their application in pre-
diction of rockburst proneness. Bull. Eng. Geol. Environ. 2019, 78, 5445–5466. hps://doi.org/10.1007/s10064-019-01474-6.
37. Liu, D.; Sun, J.; Li, R.; He, M.; Cao, B.; Zhang, C.; Meng, W. Experimental study on the eect of unloading rate on gneiss rock-
burst. J. Rock Mech. Geotech. Eng. 2023, in press. hps://doi.org/10.1016/j.jrmge.2023.09.028.
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