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analysis was also carried out in recognition that the factors governing slope stability all
exhibit natural variation, especially the values of the rock mass strength.
As was mentioned before, the rock mass strength parameters of joint sets have
been estimated using back analysis based on previous failure cases. This information was
then used in numerical modeling studies to design the ultimate pushback of the West
Wall discussed in Chapter 5. Discrete Element Method (DEM) and Finite Element Meth-
od (FEM) have been applied to evaluate different slope design alternatives. The results of
these two numerical tools were compared to evaluate the suitability of these methods in
reproducing the failure mechanism of the West Wall.
Finally, in Chapter 5, presents the factor of safety was computed by reducing the
rock shear strength in stages until the slope fails. The resulting factor of safety is the ratio
of the actual shear strength of the rock mass to the reduced shear strength at failure. This
method is called the shear reduction technique, SRT. The SRT has a number of ad-
vantages over conventional slope stability analysis based on the method of slices. In this
study, the SRT is discussed and applied to the slope stability analyses of the West Wall.
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CHAPTER II
CHUQUICAMATA OPEN PIT COPPER MINE
2.1 Introduction
The Chuquicamata complex is the largest mineral resource of the five divisions of
the National Copper Corporation of Chile (CODELCO-Chile). It is located in the
province of Loa, II region in Antofagasta, Chile, approximately 1600 km (994 miles)
north of Santiago; the nation’s capital. It is also 240 km (149 miles) northeast of the port
of Antofagasta and 150 km (93 miles) east of the port of Tocopilla, between 2500 and
3000 m (8202 and 9843 ft) above sea level (Figure 2.1). The most important populated
zones in the area are Calama which has a population of 120,000 and Chuquicamata which
has a population of 12,000.
The climate in the region corresponds to marginal high altitude conditions, and is
extremely dry and arid. The exception is during the Bolivian winter, which occasionally
produces heavy rains between December and March. The average annual temperature is
23ºC (50ºF), subject to seasonal and daily variations.
The complex is based upon a porphyry copper deposit, 14 km (9 miles) long from
north to south, with average width of 1 km (0.6 miles) from east to west. In this complex
the areas known as Radomiro Tomic, and Chuquicamata are found, which are the
principal areas under exploitation. Immediately to the south Mina Sur is also being
exploited.
The exploitation of the Chuquicamata Mine is by the open pit method. The
present dimensions of the excavation are 4,500 m (14,764 ft) long, 2,500 m (8,202 ft)
wide, and 750 m (2,461 ft) deep. This copper mine currently produces around 140 Mton
of rock annually, of which around 56 Mton is ore. Future mining plans call for a pit depth
of 1,100 meters (3,609 ft) in the year 2022. Based on this scenario, without precedent in
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the world of open pit mines, extra efforts will have to be made in order to maintain the
safety and economics of the mine.
2.2 Geology
The Chuquicamata porphyry is a porphyritic body of granodiorite vein type 14 km
(8.7 miles) long with a mean width of 1 km (0.6 miles) mineralized, with a nuclei of
greater mineralization in Chuquicamata and Radomiro Tomic. The estimated age of the
porphyry is between 32 and 34 million years.
The deposit is formed by a magnetic porphyry that probably began to develop
during the lower Paleozoic period. During the latest stage of development in the middle
Tertiary, together with a process of potassic alteration, the first mineralizing phase took
place, with the addition of chalcopyrite, bornite, pyrite, and a slight amount of
molibdenite (Lindsay et. al., 1995). After the mineralization phase, the process of
hydrothermal alteration took place. One must distinguish hydrothermal alteration
between 1) the early hydrothermal phase, with limited sericited quartz alteration,
characterized by the addition of maximum of molybdenite, together with chalcopyrite and
pyrite, and 2) the main hydrothermal phase, with a new addition of pyrite, chalcopyrite,
and some enargite, and 3) the later hydrothermal phase that added pyrite, enargite,
sphalerite, galena, and thetrahedrite.
During the Pliocene, large amounts of rainwater formed enriched bodies in each
sector. In the west sector of Chuquicamata, the water was able to penetrate greater depths
(over 1000 m (3,280 ft)) through the principal faults, forming an upper leached zone.
Beneath the leached zone is an important zone rich in chalcocite. Below this zone there is
a zone rich in covelline.
To the east, the rock is less fractured and potassic alteration is present. The copper
in the solutions was neutralized near the surface in the form of mineral oxides and less
important zones rich in chalcocite and covellite were developed below.
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2.2.1 Mine Geology
The following description of the geology of the immediate area of Chuquicamata
is largely based on the work done by Zentilla et al.,1994; Alvarez and Aracena, 1985:
Martin et al.,1993; Maskseav, 1990; Lowell and Guilbert, 1970. The Chuquicamata
deposit is located in the southern portion of the elongated N10E trending Chuquicamata
intrusive complex. The Eocene-Oligocene complex consists of three main lithological
units (1) the Porfido Este: a matrix-poor monzo-granitic porphyry with interstitial
groundmass; (2) the porfido Oeste: a monzogranitic porphyry with an aplitic groundmass;
and (3) the Profido Banco: a matrix-rich monzodioritic porphyry with an aphanitic
groundmass. Eastward, the complex has an obscure relationship with the Cretaceous,
Elena granodiorite, which has intrusive contacts with the Triassic through Cretaceous
metasedimentary and metavolcanic rocks. To the west, a regional and important
cataclastic/gouge fault zone, the West Fault, separates the porphyry complex and the
Chuquicamata deposit from the non-mineralized Fortuna intrusive complex.
The Chuquicamata deposit is concentrated in a zone of pervasive quartz-sericite
alteration immediately east of the West Fault (Figure 2.2 and 2.3). The northern part of
the deposit is controlled by NE trending fault zones and vein arrays. This zone is limited
to the east in the southern portion of the mine by NS trending gouge-bearing faults. These
alteration zones grade eastward into; and variably overprints, a potassium-feldspar-biotite
alteration zone. Potassic alteration gives way to a zone of chlorite-magnetite-specularite
(epidote) alteration in the easternmost part of the deposit.
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2.2.2 Structural Domains
Studies based on 1:100-scale mapping were focused on estimating the structural
system that controls the mineralization of the deposit. Work conducted by Lindsay et al.,
1995 contains a comprehensive explanation regarding the different fault and vein systems
present in the deposit. Within the deposit, at the current level of exposure, the following
structural domains can be defined:
(a) Zaragoza, an early NNE-SSW hydrothermal vein system, possibly related to
a NE-trending shear system.
(b) Estanques Blancos, a NE- shear system with related early and late
hydrothermal veins.
(c) The Nor-Oeste fault system, a post-hydrothermal minaralization NW-SW
system.
(d) Balmaceda, a composite domain consisting of a NE-SW fault system
(Estanques Blancos) related to late hydrothermal activity and the
superimposed components of NW-SE faults (Nor-Oeste).
(e) A weakly defined post-mineralization E-W system (Banco H1).
(f) Mesabi, an early pre-mineralization ductile shear zone, localized in
mesozoic rocks.
(g) West Fault and Americana, a related pair of approximately N-S systems with
regionally important post-mineralization movements.
Additional work was done on the West Wall of the mine by Torres et al., 1997
and two new structural domains were added to the previous work Fortuna Sur and
Fortuna Norte. Figure 2.4 and Table 2.1 present the characteristics of each structural
domain.
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2.2.4 Geotechnical Characterization
An intensive field investigation was carried out from 1996 to 1997 by Torres at al,
1997. This work addressed identification and mapping of joint sets that may affect slope
stability, and engineering geotechnical classification of the rock mass in the pit. The
geotechnical characterization of Chuquicamata Mine is based on Rock Mass Rating
(RMR; Bieniawski, 1976). This description covers the entire mine as can be seen in
Figure 2.5. The Chuquicamata deposit is divided into two main sectors, West Wall and
East Wall, by the main geological fault called the West Fault. The West Wall presents a
RMR of 41-60 corresponding to moderate quality of the rock mass. However, within this
wall, there is a shear zone in which the rock mass was highly sheared and fractured. The
original matrix of the rock is totally destroyed. The clay content in this area is about 25%,
composed mainly of mortmorillonite. According to these conditions, the RMR
description for this rock is less than 25 points corresponding to bad quality rock mass. As
explained later, this shear zone plays a major role in understanding the failure
mechanisms on the West Wall. The RMR on the East Wall varies between 45-80 points,
which qualifies the rock mass as moderate to good quality. It is important to note that the
rock mass rating is largely controlled by porphyries alteration and the existence of
metasedimetary rocks.
2.3 Rock Mass Strength
Taking into account the RMR described above, and the results of laboratory tests
of the intact rock, the rock mass properties were estimated using the Hoek-Brown
methodology (Hoek and Brown, 1990; Hoek, 1997). These properties were then
“adjusted” using engineering judgment and the observed in-situ behavior. This
methodology is described in Chapter III. The “adjusted” properties are summarized in
Table 2.2. The properties of the joints and other structures used in stability analyses are
summarized in Table 2.3.
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E Deformability modulus of the intact rock.
i
RMR Rock Mass Rating (Bieniwaski, 1976).
c Cohesion of the rock mass
Friction angle of the rock mass.
E Deformability modulus of the rock mass.
Poisson’s ratio of the rock mass.
Shear strength properties were estimated mainly from direct shear tests. However,
one of the reasons for this research is that the strength parameters of the joints obtained
from lab test may not represent the true resistance of the joints on a large scale. This
conclusion comes from stability analyses already performed, in the past using the
properties given in table 2.2. The results showed a slope with a factor of safety 1.6. The
observed behavior of the West Wall, however, is that the slope should have a factor of
safety around 1.3, based on the deformations measured for this slope. The conclusion can
then be made that the direct shear test gives an upper bound of resistance for the joints
due to a potential size effect. This is one of the major problems that must be solved in
order to have a reliable slope design.
2.4 Slope Stability of the West Wall
Currently, the West Wall of Chuquicamata Open Pit Mine has an average overall
slope angle of 30 and a depth of 750 m. This slope is experiencing deformations of up to
5 meters a year on average (based on current monitoring in 1999). In an open pit as large
and deep as Chuquicamata, every single degree of the slope inclination has enormous
importance on the economy and safety of the pit operation in an inverse relation: a steep
slope which is favorable for economy can be unfavorable for stability. On the contrary, a
flatter slope is good for stability can be very uneconomical. The case of Chuquicamata,
the segment of each degree represents huge volumes even in mining terms. In
Chuquicamata it is complicated and fascinating to observe the behavior of the West Wall,
which controls the future of the entire open pit operation.
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The West Fault (north-south direction, see Figure 2.6) separates two different
worlds; the East Wall behaves as an intact or a stable slope, while the West Wall is in
continually, non-uniform movement. In addition to a specific structural configuration of
its rock masses, the displacement in the West Wall is enabled by a shear zone, which is a
soft and compressible zone connected with its surface. Both of these rock/soil formations
are in the area of the West Wall toe. The compression of this formation repeats again and
again, because it follows the same rhythm as the pushback of the west slope and
deepening of the pit.
The role and importance of the geotechnical structures in the stability of the West
Wall slope can be summarized as follows:
1. The most important aspect on the failure mechanism is the weak shear zone
present near the bottom of the wall, associated with the West Fault. This shear
zone presents a variable width between 80 to 200 m, in a north-south direction
along the wall (see Figure 2.6).
2. This shear zone has important clay content, especially near the West Fault and
is probably saturated with depth.
3. In addition, two conspicuous and continuous joint sets in the upper part of the
slope play an important role in affecting the large-scale slope behavior. The
first, joint set dips approximately 70 and strikes in a north-south direction
and they are causing a toppling failure mode. The second joint set dips 40
into the pit, also striking in a north-south direction and this set is believed to
control the depth of the rock mass movement.
4. Due to the excavation process, the stress distribution has been overloading the
shear zone, which then squeezes this zone upward.
5. The deformations in the shear zone are observed to be greater near surface
than they are acting in depth.
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6. The upper part of the slope in the shear zone presents notorious displacements
near the surface which decrease with depth.
7. Large displacements allow the formation of an active zone (or block) of rock
in the upper part of the slope. This active zone provokes an active load on the
shear zone, which is quite similar to an active load on a retaining wall (Figure
2.7).
8. An important principal applies regarding inclination of a load behind a
retaining wall. The active load can be reduced proportionally to the angle .
However, the reduction of the angle will have a marginal effect on the
active load if it is reduced to less than or equal to 25o. Therefore, a continuous
reduction of the inclination of the West Wall will be less efficient. This
inefficiency will make it difficult to eliminate deformations in the shear zone.
9. The previously described mechanisms in the shear zone generate movement of
the entire block. The base of the block is defined by a joint system dipping 45o
into the pit center. The thickness of the block is roughly 80 to 100 m,
measured perpendicular to the shear zone.
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stability analysis on the West Wall because they do not only allow an estimation of the
factor of safety, but also give the information about the failure mechanism, the slope
displacement pattern; and the eventual zone of stress concentration.
The design of the West Wall is based on the following principles:
1. The bench-berm system is designed according the volumes of unstable sliding
blocks (e.g., wedges, shear planes, etc.). The inclination of the bench face
depends on the discontinuities and the quality of the blasting. On the West
Wall benches have an inclination between 58 to 65. The minimum berm
width is determined from the width required to contain the material collapsed
with a size that has a 15% probability of exceedance.
2. Having the bench-berm system establish the initial geometrical values for
determining the interramp angle are calculated. The interramp and overall
slopes are checked against the criteria defined for slope design at the mine.
Overall slopes are determined in the same way.
Figures 2.9 and 2.10 show a plan view of the current and final dimensions of the
Chuquicamata Mine. The design of the pit walls is based on the following acceptability
criteria:
1. The factor of safety in an operational condition (with no earthquakes; with the
typical groundwater condition; and using controlled blasting pattern) must be
equal to or larger than 1.30.
2. The factor of safety in extreme conditions (with earthquakes; with a higher
phreatic surface; and with poor blasting pattern), must not reduce to a value
less than 1.10.
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Table 2.4: Current Slope Design of the West Wall at the Chuquicamata Mine.
Bench-berm Interramp Overall
Design h Q b h r H
b b r r r o Comments
Sector (m) (º) (m) (m) (º) (m) (m) (º) (m)
Slope behavior is
West highly affected
26 63 9.0-9.5 15.5 39 162 40 32 750
Wall by the presence
shear zone.
h Bench height Bench face incination Q Back break
b b
h Interramp height Interramp angle b Berm width
r r
H Overall height Overall angle r ramp width
o r
2.6 Summary
Slope design for the West Wall has become a challenge due to the inevitable
geotechnical uncertainties and limitations of the current numerical models in modeling
the behavior of the rock mass. The size of the Chuquicamata Mine and the fact that the
current mine plan predicts reaching a depth of 1100-m, make it necessary to go beyond
the current state of design practice. Applied research needs to be developed to extend the
current concepts of rock mass strength estimation, slope stability analysis, and overall
slope design to designing slopes in rock masses with a poor to fair geotechnical quality.
In assembling the failure mechanism of the West Wall, doubts arose regarding the
properties of the two joint sets present in this wall. These joints are believed to have a
significant influence upon the deformation and failure mechanisms in the wall. Abroad
back-analysis of bench-scales or larger failures can provide improved estimates of
strength of these joints, yielding more realistic results than an adjusted laboratory tests
results.
By the year 2016 the Chuquicamata pit will have reached a depth of 1100 m. At
this scale, it is possible that the West Wall will behave as a continuous slope. Potential
failures could progress from interramp to overall slope. Decoupling the slope by inserting
wide platforms (e.g., 200 m) at strategic intervals down the wall may provide a means of
breaking the slope up into more manageable units.
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CHAPTER III
LITERATURE REVIEW
3.1 Introduction
The purpose of this chapter is to review the concepts and assumptions that are
used in this thesis. The main topics considered being important for slope stability are
grouped into three categories: (1) rock mass strength, (2) slope failure mechanism, and
(3) slope design techniques.
3.2 Rock Mass Strength
The strength of a large-scale rock mass ultimately determines whether slope
failure occurs along a given slope face. It is therefore, of primary importance to be able to
quantify the rock mass strength for design purposes. In order to illustrate the problem,
Hoek (1998) showed that for homogeneous rock mass with a known friction, the required
cohesion for maintaining the stability of the slope is determined by the slope angle and
height. The calculated cohesion values for a rock mass with a friction of = 30º and a
density of 2700 kg/m3 are shown in Figure 3.1, for slope heights of 100 to 1000 meters.
One finds that the cohesion required to maintain the slope stability increases with
increasing slope height and also vary considerably with slope angles. Small changes in
the strength parameters of cohesion and friction angle correspond to relatively large
geometrical changes to the slope geometry, which in turn may have a large impact on the
mines financial viability. Consequently, a great effort should be made in order to estimate
strength properties for the design of large-scale slopes.
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with increasing size. The decrease in rock mass strength with increasing size of the
specimen is attributed the increase in the number of pre-existing discontinuities in the
rock mass, Hoek and Brown (1990) and Pinto da Cunha (1993).
The scale effect implied in Figure 3.2 is prominent in rock materials. However,
there is evidence that the strength approaches to a constant value as volume increases.
From studies on coal pillars, Bieniawski (1968) showed that the strength of the samples
with larger than one cubic meter, practically remain constant. He also showed that the
scatter in strength values decreases as the specimen sizes increases. It is, therefore, quite
likely that the same behavior could be expected for large-scale rock slopes.
The volume above which the scale-free properties can be obtained is commonly
referred as the Representative Elementary Volume (REV) (Pinto da Cunha, 1993). The
REV is the smallest volume for which there is equivalence between the real rock mass
and ideal continuum material. The REV could be distinct for different rock masses and
properties. Even though much has been written on REV, the focus has been toward
theoretical and laboratory studies of relatively small-scale samples and practical
application of this concept to the large-scale slopes are to be proven.
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Taken together, these findings have important practical significance for open pit
slopes. Caution has to be exercised when extrapolating the strength from laboratory tests
directly to the design of overall pit slope angles.
3.2.1 Strength of Discontinuities
The strength of a rock mass is obviously a function of the strength of both the
discontinuities and the rock bridges (i.e. intact rock) separating discontinuities. The
strength also depends on the stress state in the slope. Tensile strength for a large-scale
rock mass is small and in most cases is assumed to be zero. Pure uniaxial compressive
failure (no confining stress) is relatively uncommon and deserves less attention in slope
applications. The most important shear loading, where the shear resistance of the rock
mass is enhanced by the normal stresses acting within the rock mass. Most failure modes
are believed to involve some shear failure, in particular along discontinuities.
For a planar discontinuity, the shear strength is normally a linear function of the
normal stress acting on the discontinuity. The Coulomb shear strength criteria states that:
= c + tan (3.1)
n
where is the peak shear strength, the effective normal stress, and c and are the
n
cohesion and friction angle of the discontinuity, respectively.
The Coulomb criterion in equation 3.1 is a simplified representation of the
physical processes that take place during the shearing of a discontinuity. According to
Patton (1966) the cohesion only exists for discontinuities with in-fill material. The
frictional resistance is highly dependent on the normal stress. It was found that there is no
strong dependence of friction on rock type or lithology. Instead, the friction is mostly a
function of the surface geometry of the discontinuity.
The factors believed to contribute to the shear strength of a discontinuity are as
follows (Hencher, 1995):
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1. Adhesion bonding.
2. Interlocking of surface asperities and ploughing though asperities.
3. Overriding of surfaces asperities.
4. Shearing of rock bridges and locked asperities.
Chemical bonding is significant for metals but in general less for rocks. Second,
interlocking of minor asperities and damage to these adds to the basic friction angle for
the discontinuity. The third factor, overriding of asperities, occurs on a larger scale along
the discontinuity surface. These asperities, termed first order projection by Patton (1966),
adds significant dilation to the shear behavior. Patton (1966) formulated a shear strength
criterion to account for this effect as:
= c + tan ( + i) (3.2)
f n
where is the peak shear, i is the inclination of the surface asperities, and is the friction
f
angle for a flat surface. This simple extension of the Coulomb slip criterion can explain
several of the effects which have been observed in shear tests of natural rock
discontinuities. The resulting shear stress-normal curve from such tests is often non-
linear, reflecting changes in the failure mechanism as the normal stress increases. With
increasing normal stress, the asperities are sheared off and eventually failure of the intact
rock material occurs. For high normal stress, Patton (1966) suggests the use of the
Coulomb slip criterion with a residual angle of friction and an apparent cohesion. With
these suggestion the Patton criterion become bilinear.
In addition to Patton’s failure criterion several other shear strength criteria have
been formulated. Jeager (1971) proposed a power law criterion, which better agreed with
curved failure envelope observed from shear tests, but this did not explain the mechanism
of the shearing process. Ladanyi and Archambault (1970) proposed a criterion which
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accounted for the dilation rate and the ratio of the actual shear area to the complete
sample area. Barton (1976), Barton and Bandis (1990), and Bandis (1992) developed an
empirical shear failure criterion which included terms for the roughness (asperities) of the
discontinuity surface and the compressive strength of the wall rock. The Barton shear
strength criterion is as follows:
= tan { + JRC log(JCS/
f n b n
(3.3)
where JRC is the Joint Roughness Coefficient, JCS the Joint Wall Compressive Strength
and the basic friction angle for the discontinuity. The basic friction angle corresponds
b
to the friction angle of a flat, unweathered rock surface.
Application of the Barton failure criterion is not always straight forward. In
particular, the determination of the JRC value is difficult. JRC can be back calculated
from a tilt test of the actual discontinuity (Barton and Chouby, 1977). However, if
samples of the actual joints are not available, JRC must be determined by visual
comparison with typical joint profiles. It is notoriously difficult to judge how
representative the standard profiles are. Quantifying the roughness of a discontinuity
surface thus remains a major problem in estimating the shear strength (Lindfords, 1996).
An important aspect to consider of the Barton criterion is the scale effect. An
empirical formula, which adjusted the JRC and JCS values with increasing size, has been
developed (Bandis, 1992). Both JRC and JCS decrease as the physical dimensions
increase, which means that the effects of overriding and failure of the roughness decrease
with increasing scale. The basic friction angle is believed to remain unchanged as the
scale increases. Practical application and verification of these scaling laws is still absent.
3.2.2 Strength of Jointed Rock Masses
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For rock mass, not only the presence of discontinuities but also their orientation in
relation to the loading direction contributes to the overall strength. Depending on the
inclination of the discontinuity relative to the principal loading direction, the rock sample
may exhibit different strength values (Hoek and Brown, 1980). The shear strength of a
rock specimen is much lower for those cases when slip on an existing discontinuity is
possible. Also, as the number of discontinuities increase, the rock specimen tends to
behave more closely to an isotropic material, as preferential direction of weakness
disappear.
The difficulty associated with explicitly describing the rock mass strength based
on the actual mechanism of failure has led to development of strength criteria which treat
the rock mass as an equivalent continuum. A relatively simple empirical failure criterion
for jointed rock masses has been developed by Hoek-Brown (1990), defined as:
= + (m + s 2)1/2 (3.4)
1 3 c 3 c
where and are the major and minor principal stresses at failure, respectively, is
1 3 c
the uniaxial compressive strength of the intact rock, and m and s are parameters which
depend upon the type of rock and the shape and degree of interlocking within the rock
mass. Values for m and s can be determined from rock mass classification, using the
RMR-system (Bieniawski, 1976). The Hoek-Brown failure criterion is widely used in
practical rock mechanics, both for underground and slope applications. However, since
the Hoek-Brown failure criterion represents a curved failure envelope, a transition to the
linear Coulomb criterion (Equation 3.1) is often conducted. This is because a linear
failure envelope is easier to handle in both analytical and numerical design methods.
While using the linear failure envelope one has to be aware that the strength envelope in
reality is curved, which means that the equivalent cohesion and friction angle are
dependent on the normal stress.
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There are practical advantages in using the linear Coulomb failure criterion for
defining the rock mass strength, since only two strength parameters need to be
determined. Common for both Coulomb and the Hoek-Brown criterion is that they do not
provide a true description of the physical processes that occur in the failure of a large
scale rock mass. The cohesion term not only represents the true cohesion due to
fracturing of intact rock bridges but also the effects of crushing the asperities and rotation
and separation of rock blocks. An effective cohesion is thus used to account for several of
the mechanisms that take place during rock mass failure. Such an effective cohesion
could also include the effects of confinement and reinforcement on rock slope. Although
it seems to be advantageous, there are only a few examples available in the literature on
how to choose an “effective” strength parameter (see, e.g. Jennings, 1970) based on
various rock mass and failure characteristics. The assumption of an equivalent continuous
shear failure surface incorporating both discontinuities, intact rock and corresponding
equivalent shear strength properties can also result in an overestimate of the strength
since rock bridges can fail in tension rather than shear (Franklin and Dusseault, 1991).
From the above discussion it follows that determination of the rock mass in
practice can be extremely difficult. There are in principle four different ways to estimate
the strength of a rock mass:
1. Mathematical modeling (described above).
2. Rock mass classification.
3. Large scale testing.
4. Back-analysis of failures (Hoek and Bray, 1981; Krauland, Söder and
Agmalm, 1986).
Rock mass classification is the most common method used to assess the rock
mass strength, in particular in combination with the Hoek and Brown criterion, as
discussed above. This approach has been used also in slope applications. As an example,
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the Chuquicamata Mine uses this methodology to estimate the rock mass strength. Larger
scale discontinuities may be tested in the field using hydraulic jacks. The problem with
this is the same as for a full scale tests of slope strength: high costs. Large scale testing is
therefore seldom feasible economically and practically, and is rarely used in slope
applications.
Remaining is the back-analysis of previous failures in a slope. This is an attractive
method to obtain relevant strength parameters. It requires that the failure mode is well
defined, and information must be collected on the failure geometry, groundwater
conditions and other factors which are believed to have contributed to the failure. Often,
limit equilibrium methods are used to back calculate the strength, assuming that driving
and resisting forces are equal (factor of safety = 1.0).
To summarize, the knowledge of the shear behavior of a single discontinuity in
laboratory scale is fairly well understood, but the transition from small-scale shear
failures along discontinuities, to failures involving the interaction of many discontinuities
and rock bridges in a large scale rock mass, is not well known. Furthermore, failure
criteria which consider some of the actual failure mechanisms, tend to became complex
and difficult to apply in practice. Greatly simplified criteria may be useful when
calibrated against field conditions.
3.2.3 Back-Analysis of Rock Slope Failures
The reliability of any slope stability prediction depends on the accuracy of the
input parameters used in the analysis. In some cases, totally new approaches may have to
be derived to achieve the required accuracy for input data.
Sakurai (1981) defines back-analysis as a technique of finding the governing
parameters of a system by analyzing the system output behavior, Figure 3.3. In back-
analysis of rock structures, strength parameters such us modulus of elasticity, cohesion,
and internal friction angle are determined from displacement, strain and failure measured
during or after construction. Back-analysis which is also referred as “reverse” method
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(Sakurai, 1981), is a method where the force conditions and strength properties are the
input for determining displacement, stress and strain, and stability of a structure. The
opposite approach to back-analysis the “forward” or “ordinary” analysis.
In summary, there are three ways of determining the shear behavior parameters of
a rock mass discontinuity:
a. Through laboratory tests accomplished on representative samples taken from
the field.
b. By means of in-situ shearing test usually located over the critical joints of the
rock mass.
c. Through back-analysis of previous rock slope failures.
In most cases back-analysis is the most realistic and representative way of
obtaining shear strength parameters, especially if the slope failure parameters are
identified reasonably realistically. These parameters are mechanism of failure, slope and
slide geometry, groundwater conditions, acting forces at slope failure, displacement, and
strains. A series of steps are suggested by Denis da Gama (1981) to perform a back-
analysis:
1. Input data
Define slope and slide geometry
Ground water conditions
Acting forces at slope failure
2. Formulation of slope failure model, including its mechanism
3. Stability analysis (limit equilibrium methods, finite element, etc.)
4. Determination of shear strength parameters
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Appropriate geomechanics models can be used to estimate the values of shear
strength parameters on the basis of certain assumptions. These back calculated values
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may then be used for preventive and remedial work for redesigning failed slopes and for
new projects in similar type of rock mass.
Traditionally, the shear strength parameters of a failed slope are calculated
through the following procedures:
1. Assume a value for the internal friction angle, or cohesion, c, to calculate
the other (Figure 3.4a).
2. Utilizing two major slope failures, which have similar geological and
hydrological conditions, establish two equations and then evaluate the values
of c and . This procedure is called a single solution (Figure 3.4b).
3. Utilizing more than two failed slopes, obtain as many as n(n-1)/2 points of
intersections (solutions) for n curves (Figure 3.4c).
4. The set of continuous curves represents the range of back calculation can be
obtained based on engineering judgment, experience and verified with shear
test results if these are available (Figure 3.4d).
The above procedures are normally based on the assumption that the failure is
defined by the linear Mohr-Coulomb failure criterion, which is characterized by c and
values.
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3.3 Slope Failure Mechanism in Rock Slopes
Instability of rock slopes may occur by failure along pre-existing structural
discontinuities failure through intact rock, or failure along a surface formed partly along
discontinuities and partly through intact rock. The mechanism of failure of a particular
slope is governed by the geological conditions of the rock mass and is almost always
unique to a particular site. A clear understanding of the failure mechanism of a slope
requires knowledge of the mechanical and strength properties of the intact rock and
discontinuities that make up the rock mass. As it was discussed in the previous section,
the most important factors that contribute to these properties are the characteristics of
discontinuities and hydrogeological conditions within the slope. The interrelationship of
all the relevant engineering geological parameters, thus must be correctly evaluated in
order to determine the governing failure mechanism(s) in the slope.
Failure mechanisms are easiest to assess if they can be represented in two
dimensions. In these cases, the failure surface is assumed to be subparallel to the dip
direction of the slope so that, the analysis can be performed for a unit width of the slope
(Hoek and Bray, 1981). Failures, which are assessed in this manner generally, require the
presence of upper boundary release surfaces or lateral surfaces of separation which do not
provide any resistance to failure. Two-dimensional failures, which involve a single
discontinuity or a single set of discontinuities, are shear plane, toppling and buckling
failures (Coates 1977). In some cases, a second set of discontinuities in the slope could
result in the tension cracks, or stepped failure surfaces which accordingly modify both the
geometry of the failure surface and failure mechanism.
Wedge failures develop when two intersecting sets of discontinuities form a
tetrahedral failure block which could slide out of the slope along either one of the
discontinuities or along both discontinuities. Solution of these types of failures generally
requires a three-dimensional analysis technique or some method of correctly resolving
forces from the three-dimensional model to a two-dimensional model for analysis
purposes. Analytical solutions for wedge are widely reported. One of the most commonly
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used methods is that reported by Hoek and Bray (1981). Presence of additional
discontinuities can considerably alter the wedge geometry, resulting in a complex shape
for a possible failure surface. Key-block theory published by Goodman and Shi (1981), is
often used for assessing these complex block geometries.
A more complex failure mechanism can be seen in large slopes where numerous
types of geological and hydrological characteristics are present. Analyses of failure
mechanism, in these cases involve one or more discontinuities and, are based more on the
strength of the discontinuities and less on the strength of the intact rock itself. However,
if the intact rock is sufficiently weak and the rock mass is highly jointed, failure could
occur through intact material, or the rock mass itself. In such cases, assessment of
stability must be approached differently since the failure surfaces could develop through
intact material. Unfortunately, this is the most common case in deep open pit mines. In
such situation the failure mechanism is difficult to determine in advance. Numerical tools
need to used to understand the failure mechanism and the parameters that govern the
slope behavior.
In summary, analyses of slope failure problems are generally conducted using
simple limit equilibrium techniques to evaluate sensitivity of possible failure conditions
to slope geometry and rock mass parameters. More detailed limit equilibrium techniques,
numerical modeling, statistical techniques and/or related probability assessment are also
conducted for those cases where failure mechanism, as well as operating parameters, are
sensitive to slope stability.
3.4 Slope Design Techniques
The design of any slope involves some form of analysis in which the disturbing
forces, due to gravity and water pressure, are compared to the available strength of the
rock mass. Traditionally these analyses have been carried out by means of limit
equilibrium models. Recently, numerical models are also used for this purpose.
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Limit equilibrium models fall into two main categories: the models that deal with
structurally controlled planar or wedge slides and the models that deal with circular or
near circular failure surfaces in “homogeneous” materials. Many of these models have
been available for more than 35 years can be considered as reliable slope design tools
(Hoek, 1998). The mathematical expressions required for appropriate calculations are
well known, and can be found in several publications (Bishop and Morgenstern, 1963;
Attewell and Farmer, 1976; Hoek and Bray, 1981).
Numerical modeling of slope deformation behavior is now a routine activity on
many slope stability analyses. Codes such as Finite Element Methods (FEM), Finite
Difference Methods (FDM), and Discrete Element Methods (DEM) are frequently used.
Finite element and finite difference methods are referred to as “domain” methods, in
which the entire problem domain must be discretized into elements. While FEM and
FDM are continuum codes, DEM is a discontinuum code in which discontinuities present
in the rock mass are modeled explicitly.
In continuum models, the displacement field is continuous. The location of the
failure surface can only be judged by the concentration of shear strain in the model. No
actual failure surface discontinuity is formed. The location of the failure surface, judged
from the onset of displacement or shear strain, is in general very similar to that predicted
by limit equilibrium methods.
A common limitation of continuum models of numerical analysis is that they do
not account for pre-existing fractures in the material or combinations of failure through
intact material and along discontinuities. In a discontinuum model, discontinuities are
included into the basic model geometry from the beginning calculation. Among the
discontinuum codes, one can distinguish the distinct element programs UDEC and 3DEC
(Cundall, 1971; Itasca, 1999), and the discontinuous deformation analysis program, DDA
(Goodman and Ke, 1995; Pei and Shi, 1995). Both types of programs require that the
locations of pre-existing discontinuities are known before an analysis has begun. This
often, but not always, requires a rough idea of the governing failure mechanisms. By
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including a large number of discontinuities it is also possible, to some extent, to simulate
the path which failure will take, under the assumption that failure only occurs along
discontinuities. Failure through the intact rock material can be judged in the same manner
as for continuum models, but in these programs, it is not possible to simulate the
formation of a fracture through intact rock.
In this thesis, both continuum and discontinuum model are used for slope
stability. Specifically, SLOPE1 (Griffiths, 1996) and UDEC codes (Itasca, 1999) are the
principal tools for carrying out slope design for the Chuquicamata Open Pit Mine.
In summarizing the advantages and disadvantages of these two main techniques
of slope stability analysis, the following can be concluded:
1. Limit equilibrium methods are simple to use and well adapted for making
rapid first-estimates of slope stability. A number of analytical and graphical
methods are available and have been used successfully for rock slope design
in the past.
2. The main disadvantage of the limit equilibrium method is the assumption that
the failure of rigid body; that is deformations within the sliding body are
completely ignored. Judging from observed failure modes in large-scale
slopes, this is an over-simplification. Furthermore, the failure surface must to
some extent be known previously. Also, block flow failures and crushing
failures at the slope toe cannot be analyzed using limit equilibrium methods.
3. Despite their relative simplicity, the necessary assumption of failure mode and
rigid body movements somewhat restricts the applicability of limit
equilibrium methods. However, if the shape of the failure surface is known
and the strength properties can be assessed, experience has shown that
equilibrium methods are sufficiently accurate for practical slope design.
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4. Numerical analysis can be used to calculate both stress and deformation in a
slope, and different materials and constitutive laws can be relatively easily
incorporated. Numerical models can, with the correct input data, be used also
for making predictions of slope behavior.
5. To simulate the current failure mechanism using numerical methods, an
assumption of the failure surface is generally not necessary. Existing
continuum codes can simulate the location and shape of the failure surface
developing in a slope. Discontinuum codes can simulate failure along existing
discontinuities.
6. More development is necessary before general method is available which can
simulate both fracturing through intact rock, slip, and separation of pre-
existing discontinuities.
3.4.1 Slope Stability Analysis by Incorporating Uncertainty in Critical Parameters
The objective of this section is to review and compare the deterministic and
probabilistic methods of slope stability analysis. It is only possible to use deterministic
approach by means of a sensitivity analysis. A sensitivity analysis can produce a good
qualitative understanding of the factors that are most important for a specific rock slope,
but cannot quantify the actual chance of failure. The basis for probabilistic design
methods is the recognition that the factors that govern slope stability all exhibit some
natural variation (Harr, 1987; Pentz, 1981; Call, 1983, Hoek, 1995). In a probabilistic
method, the stochastic nature of the input parameters is included and the resulting chance,
or probability, of failure is estimated. In the following, a short description of the
deterministic and probabilistic design methods is given.
3.4.2 Deterministic Methods
The most common methods of slope analysis are deterministic. For any slope
failure mechanism, slope geometry, failure surface, groundwater pressure distribution,
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material unit weight and shear strength, a factor of safety against the assumed mode of
failure is calculated using a chosen method of analysis. This calculated factor of safety is
then compared with an “allowable” value, often taken as around 1.3 for temporary slopes
and 1.5 for permanent slopes. On the basis this comparison, the trial slope design is either
accepted or rejected.
When using this approach, however, there exists the danger of attaching undue
importance to the single calculated value of factor of safety. The uncertainty involved in
the sampling and testing of the rock mass, the method of analysis and the selection of the
admissible values of the factor of safety should be taken into consideration.
It is clear that the deterministic approach ignores the variability of the rock
properties. The rock strength and weight parameters that enter the factor of safety
expression are treated as single-valued quantities. However, rock and soil are natural
materials with acknowledge spatial variability; they are seldom homogenous and
isotropic, and samples obtained from the ground are always disturbed to some degree.
Rock samples are often not truly representative of the rock mass especially if the rock is
jointed or damaged by drilling.
In a conventional deterministic analysis, it is difficult to allow for the
uncertainties with regard to material parameters. A more logical approach is to use
statistical characteristics, such as the mean, standard deviation, coefficient of variation,
and concepts from the theory of probability to draw broader conclusion from the
available data.
3.4.3 Probabilistic Methods
Probabilistic methods have long been used in engineering disciplines with
significant success. Examples of this can be found in civil engineering where
probabilistic design methods are used almost routinely in building structures.
To illustrate the methodology, assume that the load and the strength of a structure
element, for example, a slope, can be described by two probability density functions,
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respectively, as shown in Figure 3.5a. The strength, or resistance, of the element is
termed R and the load is denoted S. The respective mean and standard deviations of each
distribution are denoted m and s for the resistance, and m and s for the load. From
r r s s
Figure 3.5a one can see that the two curves overlap, meaning that there exist values of
resistance which are lower than the load, thus implying that failure is possible. In a purely
deterministic approach when using only the mean strength and load, the resulting factor
of safety would be significantly larger than unity, implying stable conditions.
In estimating the probability that the load exceeds the strength of the construction
element it is common to define a safety margin, SM (Figure 5b):
SM = R – S (3.5)
The safety margin is one type of performance function which is used to determine
the probability of failure. The performance function is often denoted G(x), Harr (1987),
consequently:
G(x) = R(x) – S(x) (3.6)
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where x is the collection of random input parameters, which make up the resistance and
the load distribution. An alternative formulation of the performance function, which is
often used in rock mechanics, involves the factor of safety, FS. Failure occurs when FS is
less than unity; thus the performance function is defined as:
G(x) = FS – 1 (3.7)
The probability density function for the safety margin is illustrated in Figure 3.4b.
In this case, failure occurs when the safety margin is less that zero. The probability of
failure, P(failure), is the area under the density function curve for values less than zero, as
shown in Figure 3.5b. On the other hand, the reliability of a structure is defined as the
probability that the construction will not fail.
Assuming that the performance function can be expressed according to either
Equation 3.6 or 3.7, and that the resistance and load distribution can be defined, how can
the failure probability be calculated? For this, three levels of analysis can be identified. A
level 1 analysis is basically a deterministic analysis, i.e., only one parameter value is used
for every variable. In a level 2 analysis, each stochastic variable is characterized by two
parameters, namely the mean and the standard deviation. A level 3 analysis is the most
complete method of estimating the probability since exact statistical characteristics of all
variables are taken into account and the joint probability density functions are calculated.
However, level 3 analysis is fairly uncommon, in particular in rock mechanics
applications, because it is very difficult to describe and quantify the joint probability
function (Mostyn and Li, 1993).
For practical design, level 2 analysis is used. In this approach, the probability of
failure is evaluated using a reliability index, , defined as:
= ( – 1.0) / (3.8)
FS FS
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In this equation, FS, is the estimated value of the factor of safety (usually the value
calculated from the best estimates of the parameters), and is the standard deviation of
f
the factor of safety. As the name implies, reliability theory describes analytically the
reliability of a facility or structure (Christian et al., 1997). Based on the best estimates of
the geometry and soil properties, Christian et al. (1997) calculated the factors of safety
against failure for two different cases as shown in Figure 3.6. For one slope, the factor of
safety is 1.2 and the other it is 1.5. However, the standard deviation of the factor of safety
for the first slope is 0.1. In the other case the best estimate of the standard deviation is
estimated to be 0.5. Figure 3.6 shows the probability density functions for a reasonable
assumption that the factors of safety are normally distributed. The probability of failure in
each case is the area under the probability density function to the left on the vertical line
at FS = 1.0. It is clear that the area for the second case is much larger. In other words, the
probability of failure is greater than for the case with the larger factor of safety. Here, it
can be seen that the factor of safety by itself does not provide enough information to
evaluate the safety factor of a slope (see e.g. Duncan, 2000). The criteria for a
satisfactory value depend on the degree of confidence that the engineer has in the
information that went into the calculation. Hence, the reliability index provides a measure
of how much confidence one can have in the computed value of the factor of safety and
leads to an estimate of the probability of failure. Now, the question is how to compute the
reliability index.
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describing load and resistance, it is often non-linear. This condition prohibits
exact or direct calculation.
2. First Order Second Moment (FOSM) methods - These techniques are based
on using the first terms of a Taylor series approximation for the variance.
They give a direct approximation for and at the same time provide direct
f
insight into the relative contributions of various parameters. The basic method
is easy to use. Its major disadvantage is that it gives an approximate solution
that may be quite inaccurate if the variance is hieght.
3. The Hasofer-Lind method - Hasofer and Lind (1974) developed an
improvement to the FOSM method to account for the fact that the derivatives
used in the FOSM should be evaluated at a point on the failure surface, that is,
when FS = 1.0. When the underlying assumptions of the FOSM method are
satisfied, this technique improves the accuracy of the results. However, it does
require an iterative procedure that is not intuitively obvious, and, when the
FOSM assumptions are not reasonable, the technique can actually give results
that are less accurate than other techniques. Ang and Tang (1990) give a very
clear description of the computational procedures.
4. The Point Estimate method (PEM) – Rosenblueth (1975) has described this
approximate technique, which has been widely adopted in geotechnical
engineering. It involves computing the factor of safety using values of each
variable that are one standard deviation above or below the expected value
and combining the results to estimate the expected value and standard
deviation of the factor of safety. For n variables this requires 2n computations.
The method is simple but approximate. When the variables are not normally
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or log-normally distributed, it may not be intuitively obvious how to choose
the pairs of values to be used.
5. Random field modeling in combination with Finite Elements. This method
includes the full statistical distribution of shear strength properties and
accounts for spatial correlation effects (Griffiths and Fenton, 2000).
All of the above methods are analytical means of determining the reliability index
from a number of stochastic variables, which make up the performance function. In cases
where the performance function is complex and contains a large number of variables, a
simulation technique can be used. The most common simulation technique is the Monte
Carlo method. In this method, the distribution function of each stochastic variable must
be known. For each distribution, a parameter value is allowed to vary randomly
according to its prescribed probability solution. This technique is simple and can be
applied to almost any problem (see Griffiths and Fenton, 2000) and there is practically no
restriction to the type of distribution for the input variables. The drawback is that it can
require running a very large number of cases to get robust results. To overcome this,
more efficient sampling techniques have been developed, among them the Latin
Hypercube sampling technique. In this method, stratified sampling is used to ensure that
the samples are obtained from the entire distribution of each input variable. This results
in fewer samples to produce the distribution of the performance function.
It is clear that a probabilistic approach is very attractive since it remedies some of
the limitations of a purely deterministic approach. The limitations for working with
probabilities in geotechnical engineering are mainly the assumption of a distribution
function for stochastic variables.
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3.4.4 Slope Stability Applications
In civil engineering, probabilistic design has advanced to the stage that virtually
all building regulations are based on a probabilistic approach. An example of this is the
use of partial safety factors, which have been calibrated using more sophisticated
methods of analysis. Probabilistic design has not yet reached this point in the field of
geomechanics. One of the reasons for this is the difficulty associated with describing a
rock mass quantitatively and defining a model which describes both the load and the
strength acting on a specific construction element. In the previous section the assumption
was made that both the load and strength could be described explicitly. This requires,
similarly to deterministic design methods, knowledge of the failure mechanism and a
model which describes how failure occurs. In slope design, probabilistic methods have
therefore primarily been based on the same failure models as those employed in limit
equilibrium methods but with statistical distributions assigned to all input parameters
(Franklin and Dusseault, 1991). Using this approach one must trust completely in the
failure mechanism, but several other uncertainties, such as the distribution of
discontinuity parameters can be taken into account.
The well-known works in this area are those by McMahon (1971, 1975), Call
(1985), Call, Savely and Nicholas (1977), Call et al. (1977), and Call and Savely (1990).
These works emphasize the importance of geological structure on slope behavior. The
commonly assumed failure modes are plane shear, step path, and wedge failures. The
failure mechanism is, in most of the cases, determined by the distributions of the
discontinuity parameters such as dip, dip direction, length and spacing. Once the
kinematically possible failure mode is determined, random samples are taken from the
distributions of joint strength, and the factor of safety is calculated. The probability of
failure is then calculated as the ratio between the number of iterations, which yielded a
safety factor less than unity, and the total number of Monte Carlo simulations.
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At this point, one must discuss what failure probabilities are acceptable for a
slope. This was brought up by Priest and Brown (1983) and Pine (1992), who defined
acceptance criteria according to Table 3.1.
Table 3.1: Acceptance Criteria for Rock Slopes (after Priest and Brown, 1983; Pine,
1992)
Category and
Failure probability
consequence of Example Reliability index,
P (FS<1)
failure
Non-critical
1. Not serious 1.4 0.1
benches
2. Moderately Semi-permanent
2.3 0.01-0.02
serious slopes
High/permanent
3. Very serious 3.2 0.003
slopes
There are several potential merits for adopting a probabilistic approach to slope
design but also significant drawbacks and limitations. These may be summarized as
follows:
1. Probabilistic methods have the ability to include the inherent variation
exhibited by almost all parameters that influence slope stability. Also, a
probabilistic approach emphasizes the fact that a slope collapse cannot be
completely neglected for any choice of slope angle, although the likelihood
for failure can be very small.
2. A probabilistic approach to design is more easily integrated with mine design
than a deterministic approach, as it can serve as input to risk analysis for an
open pit. Cost-benefit analyses of failures are very attractive but suffer from
difficulty associated with estimating all costs involved.
3. These methods can be applied to almost any type of problem but a major
drawback is the assumption of failure model, commonly based on existing
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deterministic limit equilibrium methods. Probabilistic methods share this
problem with almost all other design methods.
4. Probabilistic methods require extensive input data. Statistical uncertainty,
sampling bias and spatial variation must be accounted for, which is not easily
done. Specifically, probabilistic methods are less suited to deal with variations
in the load acting on the slope.
5. It can be concluded that application of probabilistic design methods appear to
be the best way to include the uncertainties of the parameters controlling the
slope stability. These methods seem workable when one has enough input data
and the mechanism of failure is known. This is especially true when one is
trying to design a large-scale slope. In these cases, the failure mechanism is
sometimes too complex that it cannot be represented in a formulation that can
be analyzed by probabilistic approaches.
3.5 Summary
The following conclusion can be made from this review of rock mass strength,
slope failure mechanism, and slope design techniques:
1. The knowledge of the shear behavior of single discontinuities is fairly good but
the transfer from small scale shear failures along discontinuities to failures
involving complex interaction of many discontinuities and rock bridges in a large
scale rock mass, is not well known. Furthermore, failure criteria which consider
some of the actual failure mechanisms tend to become complex and difficult to
apply in practice. More simplified criteria greatly oversimplified the actual
behavior but can be useful when calibrated against field conditions. Back-analysis
of similar cases can provide valuable knowledge on some aspects of the large-
scale strength, but requires a very precise description of the prevailing conditions
at failure.
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2. Slope failure mechanism along one or two discontinuities is well understood. As
well as when a rock mass behaves as a homogenous media. However, large-scale
slope behavior is more complex and less understood due to interaction of
discontinuities, intact rock and different types of geological conditions. To
overcome this problem, numerical analysis can be used to increase the
understanding of the fundamental failure mechanisms.
3. Limit equilibrium methods are simple to use and thus are very popular. However,
the problem with this method is the assumption that the rock behaves as a rigid
material, and the shear strength being mobilized at the same time along the entire
failure surface. On the other hand, numerical modeling appears to be more
versatile and can simulate progressive failure behavior and deforming materials.
The current numerical codes do not allow fracture propagation through intact
material, and new development in this area has not yet reached required maturity
for practical applications in slope stability.
4. Probabilistic methods can be used to assess the risk for failures in a better
quantitative way than the use of the factor of safety, and account for inherent
uncertainties of rock mass parameters. They require a minimum amount of input
data and assumptions regarding the distributions function. In addition, cost
integration into design can, to some extent, be accomplished using probabilistic
methods. However, the vast amount of input data required has rendered these
cost-benefit-methods difficult to use in practical applications.
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CHAPTER IV
SHEAR STRENGTH OF DISCONTINUITIES FROM BACK-CALCULATION
4.1 Introduction
The primary objective of this section is to establish a methodology of back-
analysis in order to estimate the shear strength of the discontinuities present in
Chuquicamata Open Pit Mine. This can then be used with reasonable confidence for
slope design.
As was mentioned in Chapter 2, the West Wall presents two conspicuous joint
sets that may be controlling the amount of deformation in the wall. For this reason, it is
important to know, with a high level of confidence, the strength properties of these joint
sets. As previously concluded in Chapter 3, due to scale effect, it is difficult to have
reliable strength values from laboratory tests. One alternative that can be used to
overcome this problem is to estimate the strength using back-analysis of past failures.
This technique has been used in the past in rock engineering for verifying rock mass
strength estimates and provided valuable information.
In this chapter, the statistical technique known as maximum likelihood is
described and used to estimate the shear strength properties of the two major joint sets in
the West Wall of the Chuquicamata Mine. The approach is based on the back-analysis
method used by Salamon and Munro (1967) in calculating strength of coal pillars.
4.2 Field Data
For the back-analysis studies, data from failure cases were collected from
Chuquicamata Open Pit Mine. The first field survey was carried out during November
1999, for the purpose of this research. The development of a statistical analysis depends
on the availability of an adequately large and tolerably reliable database of examples of
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failed and unfailed cases. This task was completed in Chuquicamata Mine after a second
field reconnaissance performed during August 2000.
The first task was to identify the most common failure mechanism of sliding
present on pit walls that involved the two joint sets present in the West Wall. The field
reconnaissance was thus concentrated in three structural domains, which featured these
two joint systems. The structural domains mapped were Fortuna Norte, Fortuna Sur, and
Noroeste. In each structural domain, several slope failures were observed. Geometric,
geomechanic and hydrologic conditions were obtained and processed to further analyses.
Addition data on unfailed cases were also gathered.
The attention focused solely on plane and wedge failure cases. This was done
since these are the most common failure mechanism controlled by the two discontinuity
sets. In addition, these failure mechanisms allow the computation of the load, which is a
fundamental prerequisite to the determination of discontinuity strengths.
A number of criteria that had to be specified before a particular case are included
in the database:
1. The failure mechanism can be determined by means of field observation (e.g.,
wedge and shear plane failure).
2. The plane or planes upon which sliding has occurred are clearly defined and
exposed.
3. The dip and dip directions of these planes can be measured with a high degree
of accuracy.
4. The discontinuities involving the wedge failure must have the same geologic
and geotechnical characteristics (e.g., the same infill material, which means
one can assume that they have the same resistance).
5. The geometry involved in the failure must be kinematically admissible. This
point is especially important for the unfailed cases.
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6. If the failure occur immediately after blasting it cannot be included in the
database.
The justification of these requirements is mainly done for two reasons. First, there
must be no doubt of the failure mechanism in order to apply the limit equilibrium
solutions to estimate the factor of safety. It is not uncommon to observe plane or wedge
failure where the discontinuity planes do not daylight on the slope surface. These failures
occur if blasting breaks the bottom part of the block. Second, the method proposed for
back-calculation assumes that the discontinuities for wedge failure have the same
strength characteristic as the plane failure cases. This situation is not a limitation of the
method because the definition of the structural same domain requires that the structures
belonging to the domain have the same geological characteristics.
The database contains a total of 178 cases, of which 94 cases correspond to shear
plane failures. Among them, there are 48 failed cases and 46 unfailed cases. In addition to
these data, there are 84 wedge failure cases, where 43 failed cases and 41 unfailed cases
were identified. This collection is subdivided into three separate sets of data as shown in
Table 4.1 and 4.2.
Table 4.1: The Range of Variables for Plane Failures.
Variable Failed Unfailed
Dip of slip plane 1 (º) 37-66 32-69
Dip-direction of the slip plane (º) 075-342 092-322
Height of the sliding block (m) 10.8-32.0 7.54-25.60
Dip of the slope face (º) 57-85 58-77
Weight of the sliding block (tonnes) 39.0-1100.3 11.5-376.4
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Table 4.2: Range of Variables for Wedge Failures.
Variable Failed Unfailed
Dip of slip plane 1 (º) 45-85 42-80
Dip-direction of slip plane 1 (º) 052-312 010-325
Dip of slip plane 2 (º) 52-85 42-85
Dip-direction of slip plane 2 (º) 004-360 005-360
Dip of slope face (º) 63-85 58-86
Dip-direction of slope face (º) 045-360 085-340
Height of sliding block (m) 11.5-52.0 13.0-30.6
Weight of sliding block (tonnes) 2.54-121.5 2.31-28.23
4.3 Methods of Back-Analysis
Prior to back-calculation, it is necessary to define basic principles. They are:
1. In carrying out a back-analysis of sliding on a plane or planes failure surface,
the equations defining the condition of limiting equilibrium is solved by
assuming that the shear strength, , of the failure surface or surfaces is
represented by a simple Mohr-Coulomb criterion, given as:
= c + tan (4.1)
n
where, c, is cohesion strength, , the normal stress and, , internal friction
n
angle.
2. The normal stress acting across on a potential sliding surface is given by
(Figure 4.1):
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4.3.1 Back-Analysis at Chuquicamata Mine
Chuquicamata staff follows a traditional methodology of estimating the strength
of discontinuities by means of laboratory tests and back-analyses. This procedure
involves carrying out laboratory tests in a shear box to estimate the friction angle of the
discontinuities. Back stability analyses are then performed to estimate cohesion values for
discontinuities. The following steps are completed to estimate the strength parameters:
1. Samples with rough natural joints are taken out from the benches on the pit.
These samples are tested in a in-house shear box that can accommodate
samples with maximum size of 20 cm x 20 cm.
2. Failures are identified in the pit and complete configurations of the failed
slopes are recorded (e.g., geometry of the slope, joint or fault geometry).
3. Limiting equilibrium equations are set up and solved for a cohesion at FS = 1.
4. From a plot of cohesion against friction angle, the cohesion and friction angle
are determined for using in design.
An estimation of cohesion and friction angle is performed the in-situ data
collected for this research. Twenty “undisturbed” samples are considered. This examples
were taken from the West Wall. Table 4.4 shows the results obtained from the laboratory
test. A linear regression analysis estimated the curve shown in Figure 4.4. An estimation
of confidence intervals is also incorporated, assuming that the strength parameters are
normally distributed. The mean and standard deviation of the variables are listed in Table
4.3.
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4.3.2 Maximum Likelihood Method
The method of maximum likelihood was first introduced by R. A. Fisher in the
1920s. The method finds the estimate of a parameter, which maximizes the probability of
observing the data for a specific model for that data. The likelihood function is defined
as:
n
L() f(x ;) (4.7)
i
i1
The likelihood function is simply the joint probability of observing the data. The
large means “product”, x is the random variable and is the parameter to be estimated.
i
The likelihood function is obtained by multiplying the probability function for each
outcome. It is a function of the unknown parameters and the estimation proceeds under
the assumption that the data observed were the most likely data to occur. For
computational convenience the log likelihood is generally used instead of (4.7):
n
logL(;x) log f(x ;) (4.8)
i
i1
Taking the first derivative of (4.8) with respect to and set it equal to zero, the
value of can be solved from the resulting equations. This is frequently difficult
analytically, and an iterative numerical method may be required. Alternatively, it may be
convenient to numerically maximize the likelihood function directly.
This technique is well known as a procedure of parameter estimation and is well
adapted to solve the present problem. The main advantage of this approach is that it
permits the use of the data referring to both failed and unfailed cases.
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The approach taken here is similar to that used by Salamon and Munro (1967). It
is assumed that a frequency distribution of factors of safety results if the calculation is
repeated for all failed cases. As seen in Figure 4.6, the central value and the spread of this
distribution are measures of the precision that can be achieved with the prediction ability
of the statistical method. The closer the central value is to unity and less it is scattered,
the better is the prediction method.
A practical solution procedure takes advantages of the knowledge that the safety
factors form a frequency distribution. The form of this distribution depends mainly on the
values of the strength parameters used in the safety factor formula. In this research, a
distribution is adequate if the estimated values are centered around unity and closely
grouped.
S 4
F
,ro
tc
a
f
y te 2
fa
S
0
0 1 2
Frequency distribution
Figure 4.6 Efficiency of the Statistical Approach of Estimating Factors of Safety for
Different Standard Deviation, = 0.1(); = 0.2 (); = 0.3 (---) (after
Salamon and Munro, 1967).
An important factor to consider in this analysis is to include data concerning
unfailed cases. This information contributes to improving the reliability of strength
parameters obtained from the analysis. The “full” maximum likelihood method is also
used to incorporate the unfailed cases into the analysis.
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To develop this method for this specific application, it is necessary to assume a
particular statistical distribution for the “critical safety factor” (FS) values. In this study,
it is assumed that a natural logarithm of the critical safety factor is normally distributed
with zero mean and a standard deviation . The frequency distribution of the logarithms
of the critical safety factors then becomes:
1 1 lnFS
f lnFS exp lnFS2 Z (4.9)
2 22
where the function Z(.) is the standard Gaussian normal distribution. This distribution is
symmetric in the logarithmic scale and mean is at FS = 1 (which is to say lnFS = 0).
The frequency distribution of FS on the logarithm scale is then defined by the
following function:
1lnFS2
1
f(FS) e 2 (4.10)
2FS
The mean, , and standard deviation, s, of the distribution of FS values are given
by the following expressions (Devore, 1995):
2
e 2 s(e2 1)12 (4.11)
Having expressed the required concepts of this statistical method, let us
concentrate on the problem of estimating of the unknown strength parameters. First,
consider the failed cases. As mentioned above, the objective is to concentrate the failed
cases to as closely as to FS =1. The condition that the failed cases should concentrate
around the median value is conveniently expressed in terms of the frequency distribution
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in the logarithmic scale (Salamon and Munro, 1967). This distribution is the greatest
when the factor of safety is one and decreases as the factor moves away in either
direction. Therefore, having all factor of safety values concentrated around one, is
achieved by the multiplication of the frequency distribution corresponding to each failed
case collected. The function to be maximized is presented by:
m
M f lnFS c, (4.12)
1
i1
where FS(.) represents the factor of safety for m failed cases in the database. This
function by itself can be used to perform the back-analysis and estimate the strength
parameters. The solution is obtained by maximizing the function M with respect to
1
cohesion and friction angle (unknown parameters), and the standard deviation, s. This
technique is called the limited maximum likelihood method.
The unfailed cases make contributions to the likelihood of the unknown
parameters, in others word, the estimation is more precise concentrating more values
around the unity. This concept can be introduced by having the cumulative frequency
distribution of the unfailed cases close to the unity. This also can be expressed as the
probability function between zero and FS, defined by:
FS lnFS/ lnFS
F(FS) f(x) Z()d P (4.13)
0
the function F(.) is the standard cumulative normal distribution. The same distribution
was applied in equation (4.9) to the standard frequency distribution of the ln(FS).
The objective is no failure occurs. To achieve this, the probability of failure not to
occur should be as high as possible. This is done by maximizing the cumulative
distribution of the factor of safety for the unfailed cases, such that:
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n
M F(FS) (4.14)
2
im1
the total number of cases in the database is n, therefore, the number of unfailed cases is n-
m.
The most effective estimation is achieved by including both groups of cases into
the estimation process. The optimum solution is obtained by maximizing function M,
where:
M M M (4.15)
1 2
with respect to standard deviation and the strength parameters. It is necessary to point out
that the function M is known as the likelihood function and the process of parameter
estimation for the maximum likelihood method. The function M can be solved easier by
maximizing the natural logarithm, as follows:
m 1 m n lnFS
lnM mln 2mlnlnFS (lnFS )2 ln P i (4.16)
i 22 i
i1 i1 im1
The maximum of this function can be found when the first derivatives are taken with
respect to and the unknown parameters (cohesion and friction angle of the
discontinuities) are set to zero. A numerical solution was performed to evaluate this
function using the computational software Mathcad 8 (Mathsoft, 1998).
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4.3.2.1 Plane Failure Analysis
Back-analysis was performed using the technique detailed above. First, the
limited maximum likelihood was used to study the failed cases of plane failures. Figure
4.7 shows the results obtained from 46 of plane failures. The values of safety factors are
concentrated around the unity with a median of 1.005. Additionally, a plot of shear stress
acting on the plane against the plane’s shear strength is presented in Figure 4.8. The
straight line in this plot shows the location of the point where shear stress and shear
strength are equal (FS=1). Most of the values of factor of safety lies on this line, so that it
can be concluded that the method produce results that fit the observation reasonably well.
Second, utilizing the full maximum likelihood method derived from equation
(4.10) and applying equation (4.16) has been tested in order to evaluate its efficacy in
evaluating the unknown parameters including both failed and unfailed cases. The
histogram in Figure 4.9 shows that the safety factors are fairly close to unity with a
median of 1.02. Also was plotted the shear stress acting on the plane against the plane’s
shear strength. In this case, it is expected that the all crosses (failed cases) scatter around
the FS = 1 line and all circles (unfailed cases) are located above this line, Figure 4.10.
25
18.75
se
ic
n
e u 12.5
q
e
rF
6.25
0
0 0.5 1 1.5 2
Safety Factor
Figure 4.7: Results Obtained Using Plane Failures and Limited Maximum Likelihood.
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4.4 Summary of Results
The relevance of the results obtained from the maximum likelihood back-
calculation to the design of slopes at the Chuquicamata Mine is as follows:
1. In order to perform the conventional back-calculation method, numerous samples
from the field must be gathered and tested for various parameters. From
laboratory tests, the mean value for friction angle and cohesion are 36.2º and
181.5 kN/m2, respectively. According to research by Barton, 1981, the friction
angle for coarse-grained granite was 31º to 35º. It can be assumed that the results
of the laboratory test were reasonable for this parameter based on Barton’s results.
Hoek and Bray, 1981, concluded that cohesion value for coarse-grained granite
was estimated to be 242 kN/m2. It can be assumed that results of the laboratory
tests were reasonable with respect to the result from Hoek and Bray research.
2. Based on research conducted by Barton, 1981; McMahon, 1975, the basic friction
angle remains relatively constant for the rock joint characteristics over the length
of the discontinuity. The cohesion, however, must be scaled with respect to the
length of the discontinuity. In order to have a reasonable value for cohesion, back-
analysis was performed using the friction angle from the laboratory tests to derive
the in-situ cohesion. Based on Barton’s results for a 30-m discontinuity, it is
suggested that a reduction factor of 3 may be appropriate for estimating the in-situ
cohesion, which in this case is 56 kN/m2. The mean value from back-calculation
for the in-situ cohesion is 19 kN/m2, for an average discontinuity length of 30-m.
It can be assumed that the results from conventional back-analysis were
reasonable with respect to Barton’s suggestions.
3. The conventional back-calculation used currently at Chuquicamata Mine yields
reasonable results for strength parameters of the discontinuities present in the
West Wall. However, the results of the strength parameters are based on an
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iterative process using friction angle as an initial known variable. If this value
were inaccurate, the entire back-calculation (estimation) would be inaccurate.
4. In the maximum likelihood method the data needed are the geometry of the
failure, such as bench height, bench orientation, discontinuity angle, etc, and the
specific gravity of the rock mass. The results from this method are not based on a
single, initial value. Furthermore, no values for strength need to be assumed for
the analysis, as the data is gathered from a variety of failed geometries.
5. The values of cohesion and friction angle as estimated from the maximum
likelihood method of the discontinuities in the West Wall at Chuquicamata Mine
are the given in Tables 4.6 and 4.7:
Table 4.6: Estimated Values of Strength Parameters from Plane Failures.
Cohesion Friction angle
Plane failures
(kN/m2) (º)
Limited maximum
19.91 35.40
likelihood
Full maximum
20.67 35.27
likelihood
Table 4.7: Estimated Values of Strength Parameters from Wedge Failures.
Cohesion Friction angle
Wedge failures
(kN/m2) (º)
Limited maximum
13.62 35.33
likelihood
Full maximum
15.22 35.12
likelihood
These values are comparable with the values obtained from conventional back-
calculation analysis. The friction angles between limited and full maximum
likelihood are consistent with one another for plane and wedge failures. The
cohesion values, however, are slightly different between plane and wedge failure.
This may be attributed to the normal stress component of the load acting on the
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sliding block. In the case of plane failure, the normal stress is distributed over the
plane, yielding a certain value for the strength of the discontinuity. In the case of
wedge geometry the load is distributed over two planes, thus, reducing overall
strength for the discontinuities.
6. The maximum likelihood statistical method yielded consistent results when
compared to the results from conventional back-calculation and previous research
completed by Barton, 1981; McMahon, 1985; Hoek and Bray, 1981.
7. The statistical parameters obtained from limited maximum likelihood of the factor
of safety are clustered around the unity, with a median of 1.005. For the full
maximum likelihood, the distribution of the factor of safety values is clustered
around the unity, with a median value of 1.016, which shows that the model is in
concordance with the observations. This statement is valid for plane and wedge
failures.
8. The mean and standard deviation of the factor of safety values, corresponding to
the failed cases are given in Table 4.7 below. Note that the standard deviation, ,
is a direct output of the estimation and S is obtained from with the help of the
definition in equation (4.11). The mean and the standard deviation Se are
statistical properties of the factor of safety obtained from the strength parameters
estimated from maximum likelihood technique.
Table 4.7: Summary of Statistical Results of Back-Analysis Using Maximum Likelihood.
Range of
Method of Estimation S Se
FS
Limited Max. Likelihood
1.005 0.101 0.102 0.097 0.74-1.22
for plane failures
Full Max. Likelihood for
1.007 0.099 0.099 0.099 0.75-1.23
plane failures
Limited Max. Likelihood
1.017 0.181 0.186 0.241 0.73-1.53
for wedge failures
Full Max. Likelihood for
1.016 0.179 0.184 0.254 0.75-1.66
wedge failures
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CHAPTER V
NUMERICAL MODELING AND SLOPE DESIGN
5.1 Introduction
The main purpose of the numerical modeling studies is to examine the effects of
changes in the slope geometry on the magnitude and distribution of stresses on the slope.
The stress distribution is especially important in the shear zone, as this is the area of
weakness. The complex failure mechanism governing the stability of the West Wall
requires evaluating different alternatives for designing the final push back of the wall. In
this section three alternative designs are evaluated using two different numerical
approaches. These alternatives are current continuous slope, 200-m wide platform and
250-m wide platform over the shear zone. A Finite Element Method (FEM) model and a
Discrete Element Method (DEM) model were used for evaluating three proposed slope
designs. The probabilistic analysis is also carried out to further analyses of the safety
factors obtained from numerical simulations.
5.2 Shear Strength Reduction Technique
The factor of safety, FS, for slopes, is often defined as the ratio of the actual shear
strength to the minimum shear strength required preventing failure. Therefore, a logical
way to compute the factor of safety with a finite element or finite difference program is to
reduce the shear strength until failure occurs (e.g., Griffiths and Lane, 1999; Cundall and
Dawson, 1996). Similarly, the factor of safety is the ratio of the soil or rock’s actual
strength versus the reduced shear strength at failure. This shear strength reduction
technique was first used with finite element models by Zienkiewicz et al (1975) to
compute the safety factor of a slope composed of multiple materials.
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To perform a slope stability analysis with the shear strength reduction technique,
simulations are run for a series of increasing trial factors of safety, FStrial. Actual shear
strength properties such as cohesion, c, and friction angle, , are reduced for each trial
according to the equations:
1
ctrial c (5.1)
FStrial
1
trial arctan tan (5.2)
FStrial
If multiple materials and/or discontinuities are present, the reduction is made
simultaneously for all materials. The trial factor of safety is gradually increased until the
slope fails. The safety factor at failure is the trial safety factor. Zienkiewicz et al (1975)
has shown that the shear strength reduction factors of safety are generally within a few
percent of limited equilibrium solutions.
The shear strength reduction technique has two advantages over the slope stability
analyses with limited equilibrium. First, the critical failure surface is found automatically
making it unnecessary to specify the shape of the failure in advance (Griffiths and Lane,
1999). In general the failure mode of rock slopes is more complex than simple circles or
segmented surfaces. Second, numerical methods automatically satisfy transitional and
rotational equilibrium.
This technique has been used in this study to estimate the factor of safety for
different geometries of the final pit slope of the Chuquicamata Mine. Program SLOPE1,
finite element method, has an in-built subroutine to estimate the safety factor (Griffiths,
1996). Program UDEC is a discrete element method (Itasca, 1999) that utilizes a
subroutine called P3_FS.FIS to perform the safety factor calculations.
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5.3 Rock Mass Properties
The most important factors to consider in the numerical analysis reported in this
section are the rock mass and discontinuities strengths and stiffness. The Figure 5.1
shows a cross-section along the coordinate 4200 N, located roughly at the center of the
West Wall. In this cross-section the current slope geometry is shown together with
geotechnical units, and major and minor discontinuities.
The numerical models, Slope1 and UDEC, were run with a perfectly plastic,
constitutive model. To estimate the rock mass strength and stiffness, the approach
suggested by Hoek and Brown (1990) was used, with a few important adjustments. Most
importantly, parameters m and s in the Hoek-Brown criterion were calculated assuming
disturbed rock mass conditions, since recent work, at the Chuquicamata Mine had shown
that this category is more appropriate for large-scale slopes. For each geotechnical unit, a
RMR (Rock Mass Rating) and estimated values of the uniaxial compressive strengths of
the intact rock, , were used.
c
Other minor modifications were also made to make the Hoek-Brown technique to
make it more applicable to this case. The curved Hoek-Brown failure envelope was
“translated” to a linear Mohr-Coulomb envelope, for use as input into the numerical
models. Cohesion and friction angle for the Mohr-Coulomb model were determined using
linear regression over a representative stress range of the Hoek-Brown envelope.
Furthermore, the intercept with the vertical axis, was fixed at the value of the uniaxial
1,
compressive strength of the rock mass, as calculated from the Hoek-Brown criterion.
cm,
The regression stress range was determined from elastic stress analysis. Young’s modulus
of the rock mass, E , was calculated from the RMR using the empirical relation of
m
Serafin and Pereira (1983).
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))22
22
00
TT 00 00
TT
IIPP
EE TT
22
RR
AA
EE YY
IIPP CC EE00 DD22 -- AA MM IITT
LL UU
((SS TT IIMM
IILL
EE
00
00
55
44 SS
DD
NN
SS TT LL UU AA
FF
TT LL UU AA FF
YY
EE 00 00
EE GG EE
LL
LL AA PP IICC
NN IIRR
RR AA DD NN
UU CC EE
00 PP SS
44
XX EE LL PP EE TT IIRR
MM DD OO
tt lluu aa FF ttss ee EE 00 00 55 33 SS
OO
CC EE VV IISS UU RR TT NN II AA NN UU TT
EE
RR AA EE HH SS EE TT AA RR EE DD OO MM DD EE RR AA EE HH SS YY LL HH GG IIHH
IIDD
OO NN AA RR GG EE NN UU TT RR OO FF 0 0 2 4 e.N -
WW EE EE
00 00 00 00 50 50 23 23
TT II NN
UU LL AA CC II NN HH
CC EE TT OO XX EE LL PP MM
OORR OO FF
TT NN EE MM IIDD EE SS AA TT EE MM
CC IITT SS AA LL CC
EE TT IIRR OO IIDD OO NN AA RR
GG AA NN EE LL EE
EE TT IIRR OO IIDD OO NN AA
RR GG TT SS AA EE
t a n
i d r o o C e h t
g n o l A
EE
CC
AA n
GG TT AA LL o
EE 00 00 00 22 MM AA CC IIUU QQ UU HH CC KK CC OO RR CC IITT IICC IIRR
EE
SS
CC IITT IICC IIRR EE SS YY RR YY
HH
PP
RR
MM UU IISS SS AA TT OO PP YY RR YY
HH
PP
RR
AA CC IITT IIRR OO LL HH CC YY RR YY
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PP
RR
s s o r C
:
1i t c e S -
LL EE 00 00 00 33 LL EE 00 00 55 22 LL EE 00 00 00 22
ZZ
TT RR AA UU QQ
OO
PP TT SS AA EE
OO
PP TT SS AA EE
OO
PP TT SS AA EE
.5
e r u g
i
F
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Previous stress measurements have been used to estimate the horizontal-to-
vertical stress ratio K. Based on the results of these measurements, an estimation of K =
1.2 was used for numerical modeling. Table 5.1 shows the values of rock mass properties
for each geotechnical unit used for the analysis.
Table 5.1: Strength and Deformability Parameters for the Rock Mass Used in Numerical
Modeling
Quartz-
Fortuna East
Parameter Shear Zone Sericitic
Granodiorite Porphyry
rock
Cohesion c (kN/m2) 675 125 2,254 365
Friction angle (º) 43 25 46 43
Dilation angle (º) 0 0 0 0
Young’s modulus E
9,000,000 700,000 4,900,000 9,000,000
(kN/m2)
Poisson’ ratio 0.25 0.35 0.25 0.25
Unit weight (kN/m3) 26 25 26 26
The cohesion and friction angle values for the discontinuities modeled explicitly
in the UDEC model are 20 kN/m2 and 35º respectively.
5.4 Design of the West Wall
In order to evaluate the stability of the final geometry of the West Wall, two steps
have to be accomplished. First step is there-designing the interramp geometry using the
new strength parameters estimated from back-analysis. The second step includes re-
designing the overall slope geometry for the West Wall with the new interramp geometry.
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2. Shear strength obtained from back-calculation in a previous chapter are
applicable.
3. According to safety criteria at the Chuquicamata Mine, the maximum height
for an interramp must be equal to or less than 160-m. Also, the minimum
factor of safety for the interramp must be equal to or greater than 1.3.
4. Factor of safety for a dry slope with no tension cracks is defined in Equation
(4.8) and has been used for this analysis. This equation can be rearranged in
the following form:
2ccosec
H p (5.3)
(cot cos tan)(sin FS cos tan)
p p p p
Solving this equation for a range of slope angles, assuming = 26 kN/m3, c = 20
kN/m2, = 35º, = 44º, and = 30º to 44º gives the results shown in Figure 5.3.
p f
According to these results, the slope height of 160 m, the maximum interramp
slope angle is 44º. The final design proposed for the interramp in the West Wall is
presented in Figure 5.4. It is important to note that this analysis deals with the stability of
the interramp slope and not with the possible failures of individual benches. In a large pit
such as the Chuquicamata pit, it would be uneconomic to attempt to analyze the stability
of each bench. Normally the bench design is been based in operational consideration as
described in Section 2.5.
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5.4.2 Overall Slope Design
According to the description of the failure mechanism of the West Wall discussed
in Chapter II, the final slope design should be modified if the mining is to progress
dipper. The new slope design for this slope need to be planned in such a way that the
probability of failure, based on the mechanism described in Chapter II, is minimized.
This requirement can be accomplished by reducing the active load on the shear zone. A
potential solution is to break up the continuous slope into two or more segments, using
wide platforms located at strategic intervals in the slope. However, for economical
reasons, a new solution should at least maintain the current overall slope angle. This is
critical for keeping the actual stripping ratio considered in the mine plan.
Taking into account these considerations, three alternatives of slope geometry are
proposed. Both proposals involve an increasing the interramp slope angle from 39º
(current design) to 44º, based on the findings given in Section 5.4.1. A wide platform
located on top of the shear zone will allow the slope to be broke up into two sections, thus
reducing the active load on the shear zone. The current design and the two alternative
geometries are presented in Figures 5.5, 5.6, and 5.7
Due to the complexity of failure mechanism of the West Wall, the analyses had to
be carried out using numerical models. Two numerical codes were tested for their
applicability to the problem. A Finite Element Method (FEM) and Discrete Element
Method (DEM) were chosen to evaluate the stability of the final design of the West Wall.
In the next section, an explanation how these methods were applied and the results
obtained are given.
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5.5 Numerical Modeling
The numerical methods can provide information regarding the internal conditions
of the rock mass slope in term of stress and displacement distribution, such that the stable
zones can be differentiated from unstable zones. This is especially important for this
study due to the complex nature of the geology of the West Wall. Since numerical
methods allow development of the failure surface naturally as function of the
combination of material properties and geometry, no previous assumption of the failure
mechanism surface needs to be made.
Pit slope design using numerical methods requires the following information:
1. Final pit wall geometry, including height and pit slope angles.
2. General geology, location of the lithologic units, major and minor
discontinuities, and alteration variations within the rock types.
3. Rock mass properties and shear strength of the discontinuities and elastic
properties of the rock mass.
4. Pore water pressure in the pit walls.
5.5.1 Finite Element Model
The code used for this analysis is a modified version of the Finite Element
program, SLOPE1 (Smith and Griffiths, 1998). The program is a two-dimensional plane
strain code and incorporates the Mohr-Coulomb failure criterion. Each quadrilateral
element has eight-nodes with four Gauss points per element. The model assumes that the
material is initially elastic and calculates normal and shear stresses at all Gauss points
within the mesh. After that, the stresses are compared to the Mohr-Coulomb strength. If
the stresses at specific Gauss point rest within the Mohr-Coulomb failure envelope, then
that Gauss point is assumed to remain elastic. On the contrary, if the stresses rest on or
outside the failure envelope, then that Gauss point is considered yielding. The yielding
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stresses are redistributed through the mesh utilizing the visco-plastic algorithm. Overall
failure occurs when a sufficient number of Gauss points have yielded to allow a failure
mechanism to progress.
The program defines the failure to occur when the algorithm cannot converge
within a user-specified maximum number of iterations. The meaning of this is that no
stress distribution can be found that is simultaneously able to satisfy both the Mohr-
Coulomb criterion and the global equilibrium (Griffiths, 1999). When the algorithm is
unable to satisfy these criteria, the slope failure has occurred.
The model allows the introduction of six parameters to define the mechanical and
elastic characteristics of the rock mass. The properties used in this model were presented
in Table 5.1 and the slope geometry in Figures 5.5 through 5.7.
The results obtained for three of the slope geometries analyzed as part of the study
are shown in Figures 5.8, 5.9 and 5.10. The factors of safety for the three options
analyzed are 1.30, 1.6 and 1.6 respectively. The results clearly show that a geometry with
a continuous slope in the West Wall lead to less stability in this slope. The model,
however, does not show any significant difference in platform lengths of 200 m and 250
m. However, indicating that the improvement in stability after adding 50 m to the
platform width is small.
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realistic values of factor of safety, it is necessary to include the two discontinuity sets
present in the West Wall.
Although, the FEM model is not completely appropriate to be used for this study,
it provided a good understanding of the failure mechanism of the West Wall. The FEM
model can estimate the location of the critical failure surface with no a priori assumption
of its position. As such the model can be used for sensitivity analyses for testing different
slope geometries in relatively short run time.
5.5.2 Discrete Element Method
In the Discrete Element Method, the rock mass is represented as an assemblage of
discrete blocks. Joints are viewed as interfaces between distinct bodies. The discontinuity
is treated as a boundary condition rather than a special element in the model. The contact
forces and displacement at the interfaces of a stressed assembly of blocks are found
through a series of calculations which trace the movements of the blocks. Movements
result from the propagation through the block system of a disturbance applied at the
boundary (Cundall, 1971). This is a dynamic process in which the speed of propagation is
a function of the physical properties of the discrete system. The dynamic behavior is
described numerically by using a timestepping algorithm, in which the size of the
timestep is selected such that velocities and accelerations can be assumed constant within
the timestep. The Distinct Element Method is based on the concept that the timesteps are
sufficiently small such that, during a single step, disturbances cannot propagate between
one discrete element to its immediate neighbors. This solution scheme is identical to the
one used by the explicit finite difference method for continuum numerical analysis (Hart,
1988). The calculation performed in the Distinct Element Method alternates between
applications of a force-displacement law at the contacts and Newton’s second law of
motion at the blocks. This numerical formulation satisfies momentum and energy
conservation laws by satisfying Newton’s laws of motion.
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A rock joint is represented numerically as a contact surface formed between two
block edges. The joint is assumed to extend between two contacts and divided in half,
with each half-length supporting its own contact stress. Incremental normal and shear
displacements are calculated for each point contact and associated length.
The basic model used in the code captures several features of the physical
response of joints. In the joint-normal direction, the stress-displacement relation is
assumed to be linear and governed by the stiffness, k . Similarly, in shear, the response is
n
controlled by a constant shear stiffness, k . The shear stress is limited by a combination of
s
cohesive strength and friction angle. This model is known as the Coulomb Slip Model.
The Coulomb model can appropriate a displacement-weakening response, which is often
observed in physical joints. This is accomplished by setting both, the tensile strength and
cohesion to zero.
5.5.2.1 UDEC Modeling
In UDEC, each block is automatically discretized into triangular constant strain
elements. These elements may follow an arbitrary, nonlinear constitutive law (e.g., Mohr-
Coulomb criterion). Other nonlinear, plasticity models included in UDEC are ubiquitous
joint and strain softening models. The complexity of deformation of the blocks depends
on the number of elements used to discretize the blocks (Itasca, 1994).
The UDEC code was used to examine the effect of discontinuities found in the
west wall. Since UDEC represents joints explicitly, it is possible to study the relative
shear and opening behavior of the joints. The UDEC model for this study was
constructed directly from coordinates taken from geotechnical units including both the
joint set of 65º with 20 m spacing and the joint set 44º with 10 m spacing in the Fortuna
Granodiorite. The blocks in the model were subdivided into finite difference zones to
ensure accurate that plasticity solutions could be obtained.
The results of simulations led to the following conclusions:
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1. The factor of safety for the continuous slope (current final slope design) is
1.20. This value was reached after modeling 7 trial factor of safety. Figure
5.10b shows the plot indicating that when FS=1.20, there is a sudden increase
in the velocity, which in turn shows that the model is unable to reach
equilibrium. The Figures 5.11a and 5.11b gives the horizontal displacement
and deformed block-mesh corresponding to the “no convergence” situation at
FS=1.20. As seen in these, figures the failure propagates through the shear
zone and causes the entire slope to move downward. Also seen is the toppling
mechanism, induced by the shear displacements on the joint sets. This is
consistent with the observed behavior of the West Wall.
2. The factor of safety for the final slope design with a 200-m platform on top of
the shear zone is 1.40. Figures 5.12a and 5.12b shows a plot of factor of safety
against velocity and horizontal displacements and deformed block-mesh
respectively. It can be noticed that displacements in the shear zone are less
than those in continuous slope simulations. This condition can be attributed to
shear zone being less exposed and thus more confined in the flat slope case.
3. The factor of safety for the final slope design with a 250-m platform on top of
the shear zone is 1.45. Figures 5.13a and 5.13b show the plot of factor of
safety against velocity, horizontal displacements and deformed block-mesh
respectively. Here, it can be seen that the displacements in the shear zone are
only slightly less than the case of 200-m platform. This option may be more
favorable than the 200-m platform, due to uncertainties regarding boundaries
of the shear zone.
As a conclusion the change in the geometry of the final pushback of the West
Wall by including a platform over the shear zone improves the stability of the slope. The
factors of safety for the two platform options analyzed are significantly greater than that
for the current design, however, the difference between thems is small. The minimum
requirement of factor of safety has to be equal to or greater than 1.3 is satisfied only for
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5.6 Probabilistic Analysis of the Resulting Factor of Safety
Conventional slope stability methods compute the factor of safety of a rock mass
slope normally for a fixed set of conditions or a given data set which includes shear
strength parameters, pore-water pressure and slope geometry. Uncertainties of the input
parameters are not taken into consideration. Rock mass properties of the West Wall
estimated using RMR (Section 5.3), exhibit a distribution about a mean value under ideal
conditions. This variability can have a significant impact upon the design calculations.
The values of cohesion c, friction angle , the uniaxial compressive strength of
the rock mass , and the deformation modulus E of the continuum rock mass were
m
calculated by the procedure described by Hoek and Brown (1990). In order to assess the
impact of the variation on output parameters, a calculation of the factor of safety for each
alternative was carried out. The distribution of the factor of safety values were
determined by Rosenbleuth’s Point Estimate Method (1981) in which two values are
chosen at one standard deviation on either side of the mean for each variable. The factor
of safety is then calculated for every possible combination of point estimates, producing
2n solutions, where n is the number of variables considered. The mean and standard
deviation of the factor of safety are then calculated from these 2n solutions. The two
variables that have the largest influence in the slope stability are cohesion and friction
angle.
The mean and standard deviation of cohesion and friction angle of the rock mass
were estimated using the approach suggested by Hoek, 1997. The analyses were carried
out for each geotechnical units that are present in the West Wall. The mean and standard
deviation obtained for all the geotechnical units are presented in Table 5.2.
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In deciding the acceptability of the level of risk associated with a given slope, use
is made of the quantity P(FS<1.0), which is a value between 0 and 1.0, giving the
probability that the factor of safety of a given slope will be less than 1.0 (McMahon,
1975). The value P(FS<1.0) is assumed be equal to the proportion of factor of safety
values less than 1.0, and this can be read from the cumulative distribution of safety
factors. The minimum acceptable values of the mean FS and the maximum acceptable
values of P(FS<1.0) adopted in the present analysis are described in Chapter 2.
The acceptable values of mean and P(FS<1) for acceptable design of the West
Wall are FS equal to or greater than 1.3 and P(FS<1) <= 5%. According to these criteria
and guidelines, both option 2 and 3 are stable and acceptable as designed. These slope
geometries involve a small risk of failure. On the other hand, the current design for the
West Wall does not satisfy both criteria and is considered to be unstable and represents
an unsatisfactory design.
5.7 Summary of Results
According to results obtained above the following main conclusion can be
mentioned:
1. For increasing the interramp angle in the West Wall, it is necessary to take
measures for reducing loading on the shear zone, which can be accomplished
allowing a platform as discusses in Section 5.4. Having a continuous slope in
the West Wall will increase loading on the shear zone thus increasing the
possibility of slope failure, Calderon at al. (1997).
2. The rock mass strength data required for the input for numerical analysis was
obtained using Hoek-Brown empirical strength criterion. This methodology
offers an alternative means of estimating rock mass strength from examination
of borehole core and surface exposures, coupled with simple laboratory
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testing (Hoek and Brown, 1997). The data obtained from this procedure are
reasonable for representing the strength of the rock mass in the West Wall.
3. The factors of safety obtained from Finite Element Method were consistently
greater than those obtained by Discrete Element Method. This situation may
be attributed to the fact that the Finite Element Method used does not take into
account the two set of discontinuities present in the slope. This condition is
especially important in representing the failure mechanism of the slope as
these discontinuities represent a plane of weakness (anisotropy) in the rock
mass.
4. As mentioned above, one of the potential limitations of Finite Element
Method for this analysis is that the method does not account for directional
weakness in the rock mass. The Discrete Element Method allows explicit
modeling of the discontinuities and thus provides more realistic results.
5. According to the factor of safety values obtained using UDEC, both options,
with platforms on the shear zone, are feasible to develop for the final push
back of the West Wall.
6. Analyzing the probability of failure of each case, it is clear that the current
final design for the slope does not satisfy maximum probability of failure of
5%, established for this particular open pit mine. Therefore, a continuous
slope design for the West Wall is not feasible for the depth studied.
7. It is clear that the alternative geometries proposed for the slope give an
improvement both in the safety of the slope and with a small probability of
failure. Even though the differences in factor of safety and probability of
failures of these options are small, it was decided that the safest option for
design the final push back of the West Wall is 250-m wide platform which
gives the maximum factor of safety 1.45 with a minimum probability of
failure among the three options studied.
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CHAPTER VI
CONCLUSIONS AND RECOMMENDATIONS
6.1 Conclusions
The main conclusions emerged from this study are as follows:
1. The study achieved its main objective in proposing a justified slope design for the
large pushback at the Chuquicamata Copper Mine. The proposed design is to
leave 250 m platform over the shear zone and increase the interramp angle from
39º to 44º. It is important point out that this increment can be implemented only if
the overall slope design may decrease the deformation of the shear zone. It is
shown that this prerequisite can be achieved by placing a platform over the shear
zone.
2. The Maximum Likelihood statistical technique has proven successful in back
calculating strength parameters of the discontinuities. This technique can be used
routinely at the open pit mines for discontinuity strength estimation.
3. A statistical method of back-analysis eliminates two major disadvantages of
laboratory or even in-situ testing. Size effect is not a problem since the strength
values are derived from information of real cases. Similarly, shape effect is not a
problem because the data represent real cases where no assumption of the
geometry is made. In addition to this, coupling both failed and unfailed cases in
the analysis helps to improve the accuracy of the estimation.
4. An associated limitation with the statistical technique used is the choice of the
distribution function for the factor of safety. In this study a lognormal distribution
was tested. It is necessary to check the method with other distribution functions.
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5. It is important to emphasize that, the data used to estimate the strength of
discontinuities came from a specific mine site. All the data were collected at
Chuquicamata Mine. Therefore, the strengths represent a particular type of
discontinuities belong a unique rock mass and cannot be used in different
geologic environment. This is especially important at Chuquicamata Mine, where
the slopes are excavated in different geological unit in which the discontinuities
present different characteristic, therefore, different strengths.
6. Rock mass properties were obtained from a combination of rock mass
classification and laboratory tests. To estimate the rock mass strength and
stiffness, the approach suggested by Hoek and Brown, 1990, was used..
Parameters m and s in the Hoek-Brown criterion were calculated assuming
disturbed rock mass conditions, since recent work has shown this category to be
more appropriate for large-scale rock slopes. The curved Hoek-Brown failure
envelope was “translated” to a linear Mohr-Coulomb envelope, for use as an input
into numerical models. This approach seems to be satisfactory for estimating rock
mass strength. However, care should be taken using this approach due to the
cohesion values are sensible to the RMR rating assigned to the rock mass.
7. Due to the complex failure mechanism governing the West Wall at Chuquicamata
Mine was required numerical modeling to study the slope stability of the wall. To
achieve this goal, two numerical methods were used. Finite Element Method
(FEM) and Discrete Element Method (DEM) were formulated to represent the
geometry and strength properties of the problem. In general sense, both numerical
methods captured the failure mechanism of the wall which is mainly controlled by
the shear zone. However, FEM model is not capable to include explicitly or
implicitly discontinuities in the formulation. This situation has an important
impact on the result of the model due to the strength of the rock mass is over
estimated. On the contrary, DEM model allows including explicitly
discontinuities that are important on the failure mechanism of the West Wall.
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8. This study confirms that the current slope geometry generates low factor of safety
and higher probability of failure, i. e., 1.220 and 8.2% respectively. These results
were expected according observation recorded at the mine. The influence of the
shear zone established by the numerical modeling is similar to the current
behavior of the slope. The results obtained for the proposed slope geometries with
platform over the shear zone show that the factors of safety are 1.40 and 1.45 and
the probability of failure of 0.38% and 0.25%, respectively, can be achieved.
These results suggest that the proposed slope geometries have a potential to
improve the stability of the final pushback of the West Wall.
6.2 Recommendations
The research reported on this thesis, leads to the following recommendations:
1. The back-analysis technique presented in this study can be used at Chuquicamata
Mine as a direct method to estimate strength parameters of the discontinuities.
Extending this methodology to the different sector of the mine, and other mines
should be explored as this would further evaluate the validity of the model
2. In order to improve the accuracy of the statistical method utilized in this research,
additional investigation needs to be done to evaluate other statistical distribution
function for the factor of safety.
3. Refinement of the rock mass property estimation methods should be done. The
method employed in this study is sensible to the rock mass rating assigned to each
geological unit. A small change in this variable can lead to high change in the
strength of the rock mass. Exploring different methodologies is required to
improve the reliability of strength of the rock mass.
4. The numerical modeling has shown to be a powerful tool for evaluating the
stability of the West Wall. Therefore, it is recommended to continue applying this
tools for analyzing the slope stability problems.
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Data concern to the plane failures.
Block
Dip Dip-dir Dip Dip-dir Water
No Code height Type Condition
(º) (º) (º) (º) Condition
(m)
1 SPV-2 56 105 64 110 23.00 JS dry Failed
2 SPV-3 43 93 66 105 17.41 JS dry Failed
3 SPV-4 44 85 63 102 22.50 JS dry Failed
4 SPV-5 44 95 66 104 15.61 JS dry Failed
5 SPV-6 39 95 65 106 19.44 JS dry Failed
6 SPV-7 44 92 61 110 20.00 JS dry Failed
7 SPV-8 47 87 65 105 18.36 JS dry Failed
8 SPV-9 48 97 61 105 18.99 JS dry Failed
9 SPV-10 47 89 63 110 16.54 JS dry Failed
10 SPV-11 42 95 59 105 15.31 JS dry Failed
11 SPV-12 45 95 62 97 25.20 JS dry Failed
12 SPV-13 42 102 65 102 22.07 JS dry Failed
13 SPV-14 45 111 61 102 19.68 JS dry Failed
14 SPV-15 45 109 63 108 25.30 JS dry Failed
15 SPV-16 37 112 65 105 19.00 JS dry Failed
16 RV-1 57 250 70 269 14.00 JS dry Failed
17 RV-3 63 261 73 269 15.89 JS dry Failed
18 RV-5 61 265 68 266 22.36 JS dry Failed
19 RV-6 55 259 69 262 16.33 JS dry Failed
20 RV-12 45 282 70 288 12.39 JS dry Failed
21 RV-15 60 284 72 290 12.31 JS dry Failed
22 RV-17 66 271 75 288 17.54 JS dry Failed
23 RV-19 57 291 73 294 10.77 JS dry Failed
24 RV-22 59 268 68 275 13.66 JS dry Failed
25 RV-23 56 272 68 292 13.78 JS dry Failed
26 RV-24 65 342 75 295 12.21 JS dry Failed
27 RV-26 54 297 71 285 16.05 JS dry Failed
28 RV-27 60 286 70 292 18.31 JS dry Failed
29 RV-29 53 285 71 285 11.98 JS dry Failed
30 RV-30 64 284 72 293 18.16 JS dry Failed
31 GF-W1 48 98 60 93 22.00 JS dry Failed
32 GF-W2 55 108 62 110 26.00 JS dry Failed
33 GF-W3 42 118 62 85 22.80 JS dry Failed
34 GF-W4 47 115 65 95 15.66 JS dry Failed
35 GF-W5 48 95 62 101 22.00 JS dry Failed
36 GF-W6 56 110 65 122 20.30 JS dry Failed
37 GF-W7 47 112 57 112 22.40 JS dry Failed
38 GF-W8 55 115 64 105 25.00 JS dry Failed
39 GF-W9 45 98 61 102 26.00 JS dry Failed
40 GF-W10 46 100 63 108 24.00 JS dry Failed
41 GF-W11 50 125 64 97 23.45 JS dry Failed
42 GF-W12 50 75 68 113 24.38 JS dry Failed
43 GF-W13 47 105 85 98 20.70 JS dry Failed
44 GF-W14 52 276 70 100 17.00 JS dry Failed
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Block
Dip Dip-dir Dip Dip-dir Water
No Code height Type Condition
(º) (º) (º) (º) Condition
(m)
45 GF-W15 40 266 69 103 32.00 JS dry Failed
46 GF-W16 55 285 73 96 23.00 JS dry Failed
47 GF-W17 58 262 70 88 23.20 JS dry Failed
48 GF-W18 60 304 68 101 24.70 JS dry Failed
49 SPP-1 42 95 63 110 11.71 JS dry Unfailed
50 SPP-2 39 114 60 105 10.97 JS dry Unfailed
51 SPP-3 32 122 58 105 8.97 JS dry Unfailed
52 SPP-4 40 115 64 102 13.93 JS dry Unfailed
53 SPP-5 38 110 69 94 13.53 JS dry Unfailed
54 SPP-6 40 121 66 105 14.09 JS dry Unfailed
55 SPP-7 37 112 65 97 12.90 JS dry Unfailed
56 SPP-8 38 110 67 95 14.89 JS dry Unfailed
57 SPP-9 33 99 67 95 15.97 JS dry Unfailed
58 SPP-10 45 102 67 88 12.17 JS dry Unfailed
59 SPP-11 38 110 69 100 9.86 JS dry Unfailed
60 SPP-12 42 100 62 98 10.19 JS dry Unfailed
61 RP-2 65 255 70 265 17.20 JS dry Unfailed
62 RP-4 57 254 65 265 15.50 JS dry Unfailed
63 RP-7 53 264 63 265 9.33 JS dry Unfailed
64 RP-8 68 233 77 268 9.58 JS dry Unfailed
65 RP-9 57 254 67 280 10.84 JS dry Unfailed
66 RP-10 65 268 76 285 11.78 JS dry Unfailed
67 RP-11 66 254 76 295 8.15 JS dry Unfailed
68 RP-13 58 273 65 286 15.39 JS dry Unfailed
69 RP-14 66 289 76 295 9.09 JS dry Unfailed
70 RP-16 62 264 77 290 7.54 JS dry Unfailed
71 RP-18 64 297 74 296 9.55 JS dry Unfailed
72 RP-20 55 322 76 295 10.95 JS dry Unfailed
73 RP-21 64 255 70 295 13.18 JS dry Unfailed
74 RP-25 68 232 75 286 16.65 JS dry Unfailed
75 RP-28 58 266 71 300 15.90 JS dry Unfailed
76 GP-W1 52 94 65 94 14.00 JS dry Unfailed
77 GP-W2 45 108 64 108 18.00 JS dry Unfailed
78 GP-W3 42 92 63 87 19.00 JS dry Unfailed
79 GP-W4 55 118 65 96 12.90 JS dry Unfailed
80 GP-W5 57 120 66 101 15.00 JS dry Unfailed
81 GP-W6 60 105 64 122 21.10 JS dry Unfailed
82 GP-W7 60 120 64 112 25.60 JS dry Unfailed
83 GP-W8 55 112 63 105 10.50 JS dry Unfailed
84 GP-W9 50 110 62 102 12.60 JS dry Unfailed
85 GP-W10 48 100 61 108 15.35 JS dry Unfailed
86 GP-W11 50 129 68 98 9.35 JS dry Unfailed
87 GP-W12 55 275 65 114 12.00 JS dry Unfailed
88 GP-W13 45 275 62 96 12.54 JS dry Unfailed
89 GP-W14 40 235 63 101 18.00 JS dry Unfailed
90 GP-W15 38 276 64 102 18.00 JS dry Unfailed
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A short solution for a wedge with a horizontal slope crest and with no tension
crack. Each plane may have a different cohesion and friction angle. Also water pressure
can be setup on each discontinuity plane. This solution of the wedge problem is presented
by Hoek and Bray, 1981.
The geometry of the problem is illustrated in Figure 4.6 in chapter 2. The discon-
tinuities are denoted by 1 and 2, the upper part of the slope by 3 and the slope face by 4.
The input data required for the solution of the wedge problem are:
1. Unit weight of the rock .
2. Height of the block wedge H.
3. The dip and dip direction of each discontinuity plane.
4. The cohesion c and the friction angle for both planes.
Other terms used in the calculation are:
FS: = factor of safety against the sliding wedge.
A = area of a face of the wedge.
W = weight of the wedge.
N = effective normal reaction on a plane.
S = Actuating shear force on a plane.
x,y,z = coordinate axes with origin at 0. The z-axis is directed vertically upwards,
the y-axis is in the dip direction of plane 2.
a = unit vector in the direction of the normal to plane 1 with components
(a ,a ,a ).
x y z
b = unit vector in the direction of the normal to plane 2 with components
(b ,b ,b ).
x y z
f = unit vector in the direction of the normal to plane 4 with components
(f ,f ,f ).
x y z
g = vector in the direction of the line of intersection of planes 1 and 4 with com-
ponents (g ,g ,g ).
x y z
i = vector in the direction of the line of intersection of planes 1 and 4 with com-
ponents (i ,i ,i ).
x y z
i = -i
z
q = component of g in the direction of b.
r = component of a in the direction of b.
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k = /i/2 = i 2 + i 2 + i 2.
x y z
l = W/A
2
p = A /A
1 2
n = N /A
1 1 2
n = N /A
2 2 2
li k SA
2
m = N /A
1 1 2
denominator of FS = S /A contact on plane 1 only.
1 2
m = N /A
2 2 2
denominator of FS = S2/A2 contact on plane 2 only.
Sequence of calculations
The factor of safety of a wedge against sliding along a line of intersection can be
calculated as follows:
1. a ,a ,a (sin.sin( ),sin cos( ),cos)
x y z 1 1 2 1 1 2 1
2. f , f , f (sin.sin( ),sin cos( ),cos )
x y z 4 4 2 4 4 2 4
3. b sin
y 2
4. b cos
z 2
5. i a b
x y
6. g f a f a
z x y y xz
7. q b (f a f a )b g
y z x x z z z
8. If q/i > 0, or if (f – q/i) tan > 1 f 2 and (1)2, no
z 3 z 3 4
wedge is formed the calculations should be terminated.
9. r a b a b
y y z z
10. k 1r2
11. l (Hq) (3g )
z
12. p b f g
y x z
13. n (l/k)(a rb ) pu p/ p
1 z z 1
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14. n
(l/k)(b ra )u
2 z z 2
15. m (la ru pu )p/ p
1 z 2 1
16. m (lb rpu u )
2 z 1 2
17. a) if n 0 and n 0, there is contact on both planes and
1 2
FS (n tan n tan pc c ) k /li
1 1 2 2 1 2
b) if n 0 and m 0, there is a contact on plane 1 only and
1 1
m tan pc
FS 1 1 1
l2(1a2)ku2 2(ra b )lu
12
z 2 z z 2
c) if n 0 and m 0, there is a contact on plane 2 only and
1 2
m tan c
FS 2 2 2
l2b2 kp2u2 2(rb a )plu
12
y 1 z z 1
d) if m 0 and m 0, contact is lost on both planes and the wedge floats as
1 2
a result of water pressure acting on planes 1 and 2. In this case, the factor of safety falls
to zero.
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Mathcad 8 spreadsheet, it is showing the application of maximum likelihood ap-
plication for plane and wedge failures. Based on the solution of factor of safety for plane
and wedge solution presented in Chapter 4, the maximum likelihood is set.
Application of maximum likelihood to plane failures:
Input data:
Read input file from c:\input\planeinput.xls and define data array as:
data = planeinput.xls
Define the following variables:
p = data(3) array that contains all dips of discontinuities
A = data(11) array that contains all areas of discontinuities
W = data(12) array that contains all weights of the sliding block
m = 47 number of failed cases
n = 93 number of total cases failed and unfailed
Define an array that contains all the factor of safety as:
cA W cos(p )tan()
FS(c,,i) i i i
W sin(p )
i i
Define the limited maximum likelihood function as follows:
m
M1(c,,) ln(dlnorm(FS(c,,i),0,)
i0
then maximize the function M1 given:
= 0.3 c = 30 = 30
utilize maximize function from Mathcad 8 as follows:
P = Maximize(M, c, , )
Outputs from P will be the estimate of c, , .
|
Colorado School of Mines
|
139
The input and output files from SLOPE1 computer code are presented in this Ap-
pendix
Input file for option 1: continuous slope.
"Chuquicamata Final Slope: option 1 continuous slope"
"Width of top of embankment (w1)"
1000
"Width of sloping portion of embankment (s1)"
1537
"Distance foundation extends to right of embankment toe (w2)"
500
"Height of embankment (h1)"
1000
"Thickness of foundation layer (h2)"
343
"Number of x-elements in embankment (nx1)"
50
"Number of x-elements to right of embankment toe (nx2)"
10
"Number of y-elements in embankment (ny1)"
50
"Number of y-elements in foundation (ny2)"
10
"Number of different property groups (np_types)"
4
"Material properties (phi,c,psi,gamma,e,v) for each group"
35. 675.0 0. 26. 7e7 0.25
25. 125.0 0. 25. 7e7 0.25
35. 675.0 0. 26. 7e7 0.25
35. 675.0 0. 26. 7e7 0.25
"Property group assigned to each element (data not needed if np_types=1)"
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
|
Colorado School of Mines
|
140
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4
"Convergence tolerance (tol)"
0.0001
|
Colorado School of Mines
|
143
"Number of different property groups (np_types)"
4
"Material properties (phi,c,psi,gamma,e,v) for each group"
43. 675.0 0. 26. 7.e7 0.25
25. 125.0 0. 25. 7.e7 0.25
46. 2450.0 0. 26. 7.e7 0.25
43. 365.0 0. 26. 7.e7 0.25
"Property group assigned to each element (data not needed if np_types=1)"
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
|
Colorado School of Mines
|
144
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 3 3 3 3 3 3 3 3 3
2 2 3 3 3 3 3 3 3 3 3 3
2 3 3 3 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4
"Convergence tolerance (tol)"
0.0001
"Iteration ceiling (limit)"
1000
"Number of trial factors of safety (nfos)"
8
"Trial factors of safety"
0.8 1.0 1.2 1.3 1.4 1.5 1.6 1.7
Output file for option 2: 200-m platform.
“Chuquicamata final slope: option 2 200-m platform”
w1= 1000.00
s1= 1032.00
w2= 250.00
s2= 223.00
w3= 500.00
h1= 842.00
h2= 162.00
h3= 343.00
nx1= 50
nx2= 8
nx3= 10
ny1= 16
ny2= 4
ny3= 8
|
Colorado School of Mines
|
146
"Number of x-elements in embankment (nx1)"
50
"Number of x-elementsin berm (nx2)"
8
"Number of x-elements to the right of the berm (nx3)"
10
"Number of y-elements in embankment (ny1)"
16
"Number of y-elements in berm (ny2)"
4
"Number of y-elements in foundation (ny3)"
8
"Number of different property groups (np_types)"
4
"Material properties (phi,c,psi,gamma,e,v) for each group"
43. 675.0 0. 26. 7.e7 0.25
25. 125.0 0. 25. 7.e7 0.25
46. 2450.0 0. 26. 7.e7 0.25
43. 365.0 0. 26. 7.e7 0.25
"Property group assigned to each element (data not needed if np_types=1)"
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
|
Colorado School of Mines
|
147
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 3 3 3 3 3 3 3 3 3
2 2 3 3 3 3 3 3 3 3 3 3
2 3 3 3 3 3 3 3 3 3 3 3
3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4
4 4 4 4 4 4 4 4
"Convergence tolerance (tol)"
0.0001
"Iteration ceiling (limit)"
1000
"Number of trial factors of safety (nfos)"
8
"Trial factors of safety"
0.8 1.0 1.2 1.3 1.4 1.5 1.6 1.7
|
Colorado School of Mines
|
ABSTRACT
The mining and milling activities associated with extraction of metals directly
generates waste in the form of mine tailings. This material is one of the largest sources of
heavy metal contamination via water, air, flora, and fauna in the world. The re-use of this
waste as an input to a construction material such as concrete could lead to a preventive
method of reducing the environmental impact. This method of encapsulation of heavy
metals has been applied to paste backfill; however, the compressive strength
requirements are much lower compared to the ASTM standards for structural concrete.
The objectives of this study were: (a) to examine the feasibility of maintaining the
structural integrity of concrete, with compressive strength of 4,000 psi or greater with a
slump of 3-4 inches, when using mine tailings as a fine aggregate, (b) investigate the
ability of this material to encapsulate heavy metals, sulfates, and acid.
The waste material, collected from the Pride of the West mill in Silverton, CO, was
first physically and chemically characterized. After performing batch leach extraction
tests, the raw mine tailing leachate contained heavy metal concentrations above
conservative regulatory limits. Then, the optimal tailing to fine aggregate ratio was
investigated. It was found that the compressive strength was comparable to control
samples made with aggregate and the concentration of heavy metals found in the leachate
were consistently low when the ratio varied below 50%. Therefore, the ASTM standard
for the minimum allowable fineness modulus was used to obtain in maximum amount of
mine tailings allotted in the concrete mixture.
To examine whether metals could be leached from the concrete-tails mix, three
extraction fluids varying in pH were used to accelerate the weathering process. The
metals of concern were shown to have been thoroughly encapsulated in the concrete
matrix, with a 2-4 log encapsulation capacity when compared to the metals leached from
the raw tailings. Finally, a strength development experiment was conducted to observed
changes over time. It was found that the specimens that contained mine tailings
maintained comparable compressive strengths as the controls cylinders, above the
minimum compressive strength requirements for structural concrete.
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ACKNOWLEDGEMENTS
First, I would like to thank my advisor, Dr. Ron Cohen, for our meetings were always
uplifting and intriguing. Even his dutiful supervision during field work was time well
spent. I am also grateful the words of wisdom from my committee members, Dr. Susan
Reynolds for helping shed light on the application of this research. Her senior design
class also catalyzed the brainstorming of good ideas. I am grateful for Dr. Alexandra
Wayllace’s patience when dealing with my incessant need for laboratory equipment,
without your help I would have not gotten this project off the ground. And, of course I
am grateful for Dr. Panos Kiousis, for when I am lost and unsure in my experimentation,
you were there to help guide me along the right path.
Second, I would like to thank the research team who helped me collect concrete
materials and run experiments, Rachel Schoen for her dedicated energy and warm
presence. Andriena Barendt could not have been a better lab partner when working long
hours in the lab. And last but not least Shannon Campos, who has been a part of the
research team from the very beginning.
Next, I would like to thank folks who lent their time to help me organize lab
equipment, space, and materials: Dr. Chris Higgins, Dr. Josh Sharp, Dr. Tom Wildeman,
Dr. Guterriez, Dr. Kathryn Lowe, Luke Frash, John Hood, Simon Farrell, Tom Cuifo,
John Jezek, and Steve Coleman. The generous donations of sand and aggregate from
Metro Mix, LLC, and Portland cement from Quikrete could not have been done without
the help from Eric Tyrrell and Nathan Lawing. This project was possible because John
Ferguson was kind enough to let me go dig up some mine tailings in Silverton, CO.
I would also like to thank the funding entity, the USEPA. The conference calls were
always a pleasure to exchange and spark new ideas. Many thanks goes out to: Tim
Rehder, Kendra Morrison, Kate Fenlon, Patti Tyler, and Souhail Al-Abed.
Finally, I would like to thank my family for helping me reach the point in my life
where I conduct research.
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CHAPTER 1
INTRODUCTION
The manufacturing of concrete consumes a large amount of energy and generates
significant CO emissions. When inspecting the individual ingredients of concrete,
2
cement, water, aggregate and supplementary additives, the environmental footprint of
aggregate production is a significant portion of the total concrete supply chain. According
to a life cycle assessment report, (Nisbet, Marceau and VanGeem 2000), 8% of energy
consumption, 39% of particulate emissions, and 7 lbs. of CO per cubic yard of concrete
2
produced comes from aggregate production.
An industrial waste- or by-product substitute for the aggregates in concrete could
reduce the amount of energy consumed and help preserve the atmosphere by utilizing a
material that has already been processed. Solid waste streams such as industrial processes
and mining operations already exist and could be a source of aggregates. The use of
wastes in concrete is not a new concept. Slag, fly-ash, and cement kiln dust are
alternative cementitious material that not only minimize the environmental impacts but
also serve as additives that enhance the quality of concrete. For example, the re-use of
slag has structural benefits, increasing the long-term compressive strength, (Bouzoubaa
and Foo 2004).
One source of a waste material, derived from mine activity, that has not been fully
utilized are mine tailings. Substitution of tailings for the fine aggregates in concrete not
only may result in an effective structural concrete, it also acts as a remediation strategy
for a potential source of environmental contamination.
1.1 Problem Statement
Mine tailings represent a potential impact to the environment. Tailings storage
facilities also have a very large footprint, utilizing huge areas of the earth’s surface.
When mine tailings that contain sulfide minerals, particularly pyrite, come in contact with
water, air, and microbes, mining impacted water (MIW) is formed. There are 20,000 –
50,000 abandoned mines which have an estimated volume of 3 million cubic meters of
mine tailings per mine that affect approximately 23,000 kilometers of water ways in the
United States (Matlock, Howerton and Atwood 2002). The sources of MIW are often
found in the Northeast USA due to coal mining, where 10% of the Northern Appalachian
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regional streams are affected by MIW (Herlihy, et al. 1990). If these mine tailings could
be used as a component of concrete for building materials, safely and effectively, this
could reduce the potential environmental impact of those tailings.
There are several important issues that need to be addressed when considering the
use of a potentially hazardous material as a component of a value added product such as a
construction material. First, how does the waste modify the structural integrity of the
construction material, in this case, concrete? Second, how effectively are the tailings
encapsulated in the concrete such that the potential of leaching MIW is reduced to
insignificance? And, finally, if the presence of the tails does change the process needs of
making structural concrete, what are the most efficacious substitution ratios of tailings to
aggregates and how are setting times changed?
1.2 Significance of Study
One main focus of this paper is to examine the feasibility of maintaining the
structural integrity of concrete when using mine tailings as a fine aggregate. Another
focus is minimizing the leachability of the heavy metals, sulfates and acid from pyritic
materials once the tailings are encapsulated in structural concrete. If one of these
objectives are not met then the feasibility of this technology is compromised. The
disposal of mine tailings is thought to be the largest source of environmental impact from
mining processes (Vick 1990). Therefore, the successful use of tailings as fine aggregate
in structural concrete could have multiple environmental benefits: reducing the quantity
of mine tailings, disposal in the environment and decreasing the amount of CO
2
emissions attributed to the extraction of new aggregate. The social implications of mining
activity could be improved while creating a revenue steam for mining companies, thus
increasing economic prosperity.
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CHAPTER 2
BACKGROUND AND LITERATURE REVIEW
Before developing a methodology to test the compressive strength and leachability
of concrete containing mine tailings, research of the important factors that pertain to this
waste stream was conducted. First, the processing of ore bodies and the typical disposal
pathways are investigated. Second, background information in relation to the
environmental impacts such as the formation of MIW is reviewed. Next, a brief
description of the alternative strategies, including encapsulation, is described. Finally, the
hydration process, leaching, and regulatory issues are discussed.
2.1 Mill Processing
The processes of taking ore bodies and producing concentrated metals will dictate
the characteristic of the mine tailing waste. For example the reduced particle size depends
on the operations of ball mills, while the chemical composition depends on the
mineralogy, percent removal, and chemical additions. The chemical and physical make-
up of mine tailings is fundamental to the production of MIW. The steps from when the
ore body enters the milling facility to when the mine tailings are disposed of are
important to understand because of the changes in the chemical and physical make-up.
The Pride of the West is outside of Silverton, CO and the primary metals of interest
are gold, silver, lead, copper, and zinc. The processing facility currently has the capacity
to refine 700 tons of ore per day. There are hundreds of mines within a 10 mile radius
that contain an estimated 20 million tons of ore.
The mill processing starts with filling the coarse ore bin, which is connected to a
jaw crusher with a conveyer belt. Before entering the jaw crusher, the ore flows
uniformly over a vibratory feeder, which separates the finer particles. The coarse particles
are allowed to enter the top of the jaw crusher to reduce the size of the ore. The finer
particles and the effluent from the jaw crusher are sent to another vibratory separator,
where material that is fine enough is sent to the grinding processes. The cone crusher is
used to further change the physical characteristics between ⅜” to ½” in size. If the ore
passes through certain mesh size, the ore is transferred to the fine ore bin. If the ore does
not pass the mesh, then the ore is sent back to the second vibratory separator.
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The fine ore is ground in a rod mill and is reduced to the size of sand particles. The
ground ore passes through a mineral jig where the ore is initially concentrated by the
stratification of particles due to the different densities of the ground minerals. This is
completed by sending fluid, typically water, through a bed of ground ore at alternating
velocities. This operation acts like a sophisticated gold pan where minerals containing
gold and silver are concentrated while the rest of the minerals are sent to the flotation
cells. The gold and silver concentrate is then sent to a clarifier that uses gravity to
separate the fine and coarse minerals. The fine gold and silver particles are sent to the
melting furnace, where particles that are too large to undergo the formation of doré, are
sent to a ball mill. The steel balls in the ball mill are 2” to 3¾” in size. A mineral jig also
is used after the ball mill. The gold and silver doré is created by heating the concentrate
with the addition of flux. Flux is a lime, aluminosilicate mixture. When heated with the
gold and silver concentrate, a relatively less dense slag is created with the doré settling to
the bottom of the melting furnace. The slag is separated from the doré, where the doré is
poured in a conical mold. The doré is sent to a refinery, where further processing occurs.
The rest of the ground minerals from the mineral jig are sent to four flotation cells.
The minerals are suspended by pumping air into the cell and xanthate salts are used as a
flotation agent and surfactants are used to create a flocculation of minerals of interest. It
is hypothesized that 1 mg/L of xanthates could be toxic to aquatic biota, however, the
rapid degradation rates of these salts reduce the risk for aquatic biota (Xu, Lay and Korte
1988).
The lead, copper, and zinc concentrate is then sent to a thickener where the
minerals settle to the bottom of the tank. The thickened lead, copper, and zinc concentrate
is pumped through a filter, where the bottom of the filter is operated under a vacuum and
the top half is at atmospheric pressure conditions. When the minerals of interest pass the
section of the filter with air, they fall into the vacuum portion of the filter and drop out
into a container that is shipped to a smelter. The waste material is the mine tailings that
are found in the tailing storage facility (TSF).
2.2 Tailing Storage Facilities
Disposing of mine tailings has social implications and possibly create economic
prosperity. For example, mining communities often enjoy the visual aesthetics of small
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piles of yellow/orange mine waste seen on the mountain sides, whereas large scale
destruction of forested land is seen as unsightly. Mine tailings can often be re-milled after
new processing technology is available. The potential value of disposed mine tailings
needs to be taken into account when considering the outcome of this waste stream. The
usefulness of mine tailings can be measured with respect to the alternative remediation
technologies. The alternative remediation technologies can be beneficial so that local
communities, mining companies, government, and other interested stakeholders
agreeably reduce the environmental impacts while creating economic prosperity.
The storage of mining tailings has changed over the course of the mining industry’s
history. Small abandoned mines sites, consisting of waste rock and finer mine tailings,
are easily discernible in mountainous regions. These slopes and valleys are essentially
receptacles, with no to very little preventive methods used to limit the transport of MIW
in surface or groundwater. Modern tailing storage facilities utilize the construction of
impoundments, liners, and dust suppression equipment to reduce the environmental
issues that arise due to the large amount of waste material that is generated at milling
processing facilities.
2.2.1 Design of Impoundments
The EPA identifies the following four types of impoundment designs that can
contain tailings: valley, ring-dike, in-pit, and specially dug pit impoundments, (USEPA,
Design and evaluation of tailings dams 1994). Often times large earthen dam structures
are used to contain the waste material.
Earthen dams are typical structures when disposing of large volumes of mine
tailings. An earthen dam structure is shown in Figure 2.1. An impervious core
constructed out of clay or even asphalt, is used to reduce seepage. The filter and under
drains help keep the overlying material as dry as possible. Internal erosion occurs if the
filter is clogged and seepage channels out of the intended drainage zone. The overlying
material could be the dry tailings themselves, which could be a cost effective design
method. To do so, the tailings must be dried to certain moisture content. A common way
of increasing the percent solids is to use a thickener. With these structural and technical
components the risk of the failure mechanisms can be minimized.
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Figure 2.1: The important components of a tailings dam, (Vick 1990)
The embankments are commonly constructed in three different factions, upstream,
downstream, and centerline, which is shown in Figure 2.2. Tailing dams are usually built
in successive parts because of the economic benefits of spreading the capital cost across
long periods of time. The upstream method is considered a more efficient design because
it uses less material, but it is used rarely in new dam construction because of liquefaction
risks. The downstream method uses more building material and more land area; however,
it can generally withstand more seismic stresses that occur during an earthquake. The
third method, the centerline class, is a compromise between the upstream and
downstream methods; it uses less material than the downstream method and is safer than
the upstream method.
Understanding the deposition of the tailings is also important. The settling rate is just
one component that will determine the fate of tailings in the impoundment. The type of
tailings will also affect the permeability in relation to the distance from where the tailings
are discharged at a point source, or spigot. Some deposited tailings, such as heavy-laden
lead-zinc mixtures, will reduce in permeability very quickly. This is due to the fact of the
settling rates, where the finer particles take longer to deposit, creating slimes downstream
of the spigot. Knowing the variation in permeability is important so one can determine
the flow path of potentially toxic water. The use of cyclones helps with separating and
depositing tailings by size, gaining insight to where these permeability regimes exist.
The size of tailings range from 0.01 to 0.1 mm in diameter, (Mainali 2006). Since,
there is a large portion of fine grain material, settling of the tailings takes time in a
viscous slime and sludge. A typical copper slime, which is a mixture of the finer tailings
and the effluent water from the beneficiation process, the sedimentation rate is roughly
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between 0.14 to 0.31 ft/hr. However, if zinc is found in the slime, the settling rate
increases to 0.38 to 0.54 ft/hr, (Vick 1990).
Figure 2.2: (a) Upstream, (b) Downstream, (c) Centerline embankments (Vick 1990)
Designing a sound tailings dam is difficult due to failure mechanisms such as
liquefaction, slope stability, erosion, and seeping. Most often the failure of a TSF is due
to natural events, such as a heavy rain storm, erosion, and earthquakes, (Rico, et al.
2008). For example, a heavy rain event can cause seepage and piping through the
impoundment’s embankments. Earthquakes are sources of destruction due to liquefaction
and wind and water causes erosion over time. Creating water balances and understanding
the hydraulic parameters of these earthen structures will help shed light onto where water
is traveling and is seepage issues may arise.
2.2.2 Pride of the West TSF
The TSF from which the material used for this project originates from a relatively
small impoundment. The Pride of the West TSF is 100 yards from the processing
equipment. There are two inactive storage sections that are partitioned by one active
storage area. The active storage liner, which can be seen in Figure 2.3, acts a barrier. The
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use of a liner is a conventional way of disposing of mine tailings. The liner minimizes the
exchange of MIW to the subsurface.
Inactive
Active
Inactive
Figure 2.3: Silverton's Pride of the West TSF
When observing the local lands for acid-generating and acid-buffering minerals,
this area is abundant in acid generating material. The USGS has found an innovative way
of scanning the terrain with an Airborne Visible and InfraRed Imaging Spectrometer
(AVIRIS) to map where acid-generating vs. neutralizing material consist along the
Animas River, (Dalton, et al. 2000). The results show that the Pride of the West TSF is
comprised of jarosite and nanocrystalline hematite, however, no neutralizing material is
found close to the TSF.
2.3 Formation of Mining Impacted Water
The removal of gold, silver, lead, copper, and zinc leaves the rest of the sulfur
bearing minerals available for bacteria to oxidize the sulfides, release metal ions, and
produce high concentrations of hydrogen ions. The principle sulfur bearing mineral that
is responsible for the formation of MIW is pyrite. To generate MIW, air (oxygen) and
water are required for the chemical and microbiological reactions for this exothermic
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reaction to occur. Therefore, there are both chemical and biological processes that
contribute to the overall reaction.
2.3.1 Pyrite to MIW and Ferrous Iron Production
The formation of free metals in water is a multi-step process. When considering
pyrite, an overall reaction representing the formation of free metals can be written as the
following chemical equation, (Herlihy, et al. 1990):
( ) → ( ) ( ) (2.1)
There are many individual steps involved in the formation of MIW such as: the
oxidation of pyrite by oxygen to yield ferrous iron, the dissociation of ferrous iron from
pyrite, oxidation of ferrous iron to yield ferric iron, and the formation of iron(III)
hydroxides. The two pathways for the production of ferrous iron from pyrite will be
analyzed. First, the oxidation of pyrite by oxygen,
→ (2.2)
is important because of heat generation and the consumption of oxygen. The standard
enthalpies of reaction can be calculated to validate the exothermic property. An enthalpy
of reaction value of ∆H = -5645 KJ/mol, which represents an exothermic reaction, was
calculated from the standard enthalpies of formation values, (Brezonik and Arnold 2011).
The second pathway for the production of ferrous iron, the dissociation of ferrous
iron, consumes ferric iron. MIW is difficult to remediate because of the cyclical nature of
this reaction coupled with the oxidation of ferrous iron. Once ferrous iron is in its
oxidized form, then ferric iron can precipitate in solution. The chemical reaction written
below is a more complete form of the dissociation of ferrous iron, (Singer and Stumm
1970):
→ (2.3)
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Since, the oxidation of ferrous iron compliments this reaction because of the
production of ferric, it is important to analyze the driving forces.
2.3.2 Rate of Reaction and the Biological Cyclical Reaction
The rate determining step is of interest and the rates of reaction and/or the half-
lives must be estimated for a deeper understanding of the overall reaction. The oxidation
of pyrite by oxygen is not as important of a pathway for producing ferrous iron when
compared to the dissociation of ferrous iron, (Lefebvre, et al. 2001) . This is true because
the oxidation of pyrite by oxygen is a relatively slow reaction with a half-life of
1000 days under acidic conditions and a reaction rate k = 10-7 atm-1 min-1 below a pH
value of 3.5, (Singer and Stumm 1970). While, the dissociation of ferrous iron is much
quicker due to the half-life of ferric iron, for this reaction, was calculated to be
50 minutes at a pH of 1.0, (Singer and Stumm 1970). Therefore, the rate determining step
is the oxidation of pyrite by oxygen.
To produce ferric iron, ferrous iron must be oxidized. When there is no microbial
activity this reaction takes time, but can be accelerated by a factor of a million, (Singer
and Stumm 1970), when microbes are present. A common microbe studied under low pH
conditions is thiobacillus ferrooxidan, (Fortin, Davis and Beveridge 1996). The chemical
equation for ferrous iron oxidation is as follows:
→ (2.4)
As more microbes convert ferrous iron to ferric iron, increasing the concentration
gradient in the dissociation stated above. Therefore, increasing amounts of pyrite will be
consumed and increasing amounts of ferric iron will be produced.
The dissociation of iron to form iron(III) hydroxides is what creates MIW and the
following chemical equation represents only one of the iron(III) hydroxides in solution:
→ ( ) (2.5)
The acidity of the solution will increase due to the release of protons. The cyclical
process which produces MIW will go on until reducing conditions are implemented,
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water is not present in the system, pH is driven upward, microbes are limited due to food
or removed by the use of surfactants, immobilization of the sulfide bearing minerals, or
the mineral containing metals are fully consumed.
In the context of the Pride of the West, the formation of MIW occurs in the TSF
where the waste is disposed in a controlled manner with the use of synthetic liners and
the use of holding ponds. However, mine waste is not always disposed in a manner that
reduces the risk of MIW to contaminate ground and surface water.
2.4 Alternatives for Treating MIW
When considering ways to treat mine waste and the impact of MIW, the
alternatives are often comparatively examined in feasibility studies. For example, if the
TSF has a low probability for producing MIW, then no action may be taken. This would
be the cheapest remediation method. However, there are other alternatives such as:
monitoring, containment technologies, excavation, collection, diversion, solid treatment
technologies, water management technologies, water treatment technologies, and
demolition/treatment activities.
2.4.1 Collection, Containment, Diversion, Excavation, and Monitoring Strategies
At the Pride of the West mill site collecting and containing the tailings was the
strategy from minimizing the environmental impact of MIW. After the useful life of the
TSF it would be advantageous to use a cover to prevent emitting particulate matter into
the atmosphere. Organic chemicals can be added to the deposited tailings to also inhibit
the emission of air borne particles. A cap can be installed using a porous material, such as
pea gravel, silty sand, or synthetic material. This can limit the flux of water due to the
capillary barrier effect (Stormont and Anderson 1999), creating a closed system. While
this method of a barrier system is a common way of minimizing MIW; reusing a portion
of the reactive mine tailings and using it as a fine grained material for the barrier is a
containment option (Doye and Duchesne 2003).
Reusing is advantageous, however, reactive mine tailings must be neutralized so it
can be used and/or immobilized safely. Therefore, water treatment technologies are often
needed if the tailings not only generate enough acid to neutralize all CaCO equivalent
3
material but also dissolve heavy metals.
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The discussion concerning containment strategies is more in-depth than just using
clay liners and impermeable caps. The application of shotcrete or construction of
concrete barriers are often used to immobilize contaminates in portals, drifts, shafts, and
other mine working exposed to sulfide bearing minerals. But, sometimes the mobilization
of MIW is controlled by the use of flow-through bulkhead and outlet piping. This
diversion strategy can help with collecting or moving MIW to a more desirable area. If
the mobilization of contaminants is traveling towards an undesirable area, the direct
removal of the source is an option. To help enhance the use of the waste, soil can be
mixed with the waste to dilute the containments. Also, neutralizing material can be added
to help immobilize the waste.
Preventive and on-site monitoring strategies can be implemented in conjunction
with other treatment technologies. Having knowledge of the environment in which a
contaminant be transported is important. The hydraulic parameters of surface water and
ground water can be estimated, enhancing the decision making.
In the end, the remediation of heavy metals has to be addressed. Therefore,
treatment technologies are either planned from the start of the mines inception or used
when MIW is found after the improper disposal of mine waste.
2.4.2 Solid Waste Treatment Technologies
This category of treatment technologies incorporates encapsulation by the use of
cementitious material. One stabilization method in use is paste backfill, where it has been
estimated that a TSF can be reduced by 60% in size, (Benzaazoua, et al. 2004). This
technology has been used for refilling the open space for such mining excavation
strategies such as room and pillar. This non-Newtonian fluid hardens and prevents the
potential for subsidence, reduces labor, dewatering, and impoundment requirements. The
hardened direct re-use material usually has compressive strength in the ranges of 1.5 to
3.5 MPa (217 to 507 psi), (Brackebusch 1995). The fluid must be economically feasible
to pump long distances, often up to 0.6 miles from the TSF.
These technical and economic issues with delivering the cementitous mixture to
locations under nether the ground can be subdued with technology such as concentrated
foam. The petroleum industry has used this controllable viscous non-Newtonian fluid for
air/gas drilling and in the removal of liquids from gas reservoirs. This technology has
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been used to pump mine tailings back into the ground. This same technology has been
proposed as a method of transporting remediation “medicines” for targeting MIW,
(Gusek, Masloff and Fodor 2012). This could be a viable option when mixing tailings and
ordinary Portland cement (OPC) with foam because of the reduced cost for pumping the
mixture.
There are many different types of fixation processes available in the commercial
treatment technologies industry. Some methods use ordinary cements, while other
methods use silicate, lime, or pozzolanic based materials. Chelating additives are also in
use, such as EDTA, (Jackman and Powell 1991). With the intent of enhancing the
physical mechanism of encapsulation, the chelates ligands could limit the reactions of
heavy metals during hydration process when the material is hardening.
The alternative that is examined in this project is the use of mine tailings in
construction materials, specifically concrete. The stabilization and solidification of
contaminates of concern, mainly heavy metals, is one benefit of hydration chemistry. The
alkaline conditions reduce the amount of acid generating potential.
The disadvantage of using mine waste as an ingredient for concrete is the potential for
reduced quality of the concrete, particularly the strength and durability. It is known that
the addition of sulfide bearing minerals can reduce the compressive strength of concrete
due to the formation of sulfuric acid and degrading the cement paste, (Neville 1996).
However, if the mine tailings can be added to concrete mixes so that the structural
integrity and the environmental impacts can be minimized to acceptable levels, then there
might be an untapped resource that can be utilized for the benefit of local communities,
mining companies, consulting firms, government entities, and other interested stake
holders.
In this case of this project, mine tailings are substituted for the sand in the concrete
mix; therefore, mine tailings should mimic physical and chemical properties of sand. One
project that has used mine tailings directly in concrete mixes was conducted by Freeport
McMoRan in Indonesia, where have built bridges and buildings. However, there is a
large carbon footprint that is associated with OPC.
There has been a push in the academic community to focus on the cementitous
material, namely the use organic polymerization on inorganic substrates (Komnitsas and
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Zaharaki 2007). However, the substitution for the aggregates with a waste stream has
only seen little research and development progress.
Research has been conducted on forming unit sized blocks using mine tailings and
other cementitious material, such as geopolymers. Geopolymers are derived from
polymerization technology that is utilized for organic polymers, (J. Davidovits 1976).
These inorganic polymers are created from mixing caustic, such as sodium hydroxide and
potassium hydroxide, water, and natural source of alumino-silicates while being fired at
low temperatures. The rate of polymerization is increased when thermally triggered.
Temperatures upward of 100 °C can be applied to create the geopolymer.
The synthetic structure is a mixture of amorphous, semi-crystalline, and crystalline
alumino-silicates. This is due to the reaction kinetics. The faster the geopolymerization
occurs; the structure becomes more amorphous and less crystalline. If the reaction
occurred in a closed system with dilute solutions of alkali activators, then the final
product will be more of a crystalline zeolite mineral and less amorphous. Volcanic
zeolites created under low temperature and pressures from sedimentary rocks are created
in closed systems, while laboratory conditions maintain an open system.
The structure is dependent of the type of cations that are introduced to the mix.
Common activators come from the dissociation of NaOH, Na CO , Na SO , KOH,
2 3 2 4
K CO , K SO , and Na SiO . The different types of structures that can be formed from
2 3 2 4 2 3
these two cations are shown in Figure 2.4, (Khale and Chaudhary 2007).
Figure 2.4: Conceptual models of geopolymer structures
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This technology is advantageous because of the carbon footprint reduction and the
high strength capabilities of this cementitious material. To do so, sustainable concrete
material (SCM) such as slag, fly-ash, and metakaolin can be used as precursors for
geopolymers. From an environmental stand point, Figure 2.5 shows the amount of SCM’s
that are needed to off-set the current CO emissions associated with OPC (Mehta 2007).
2
This has a direct effect for the mining, construction and the waste disposal industries. If
business as usual occurs, then the CO emissions will not decrease in the fashion shown
2
below, but will create a burden on the cement industry.
Figure 2.5: Shows the amount of concrete made in 1990 and 2005. Predictions are
made for future years and the amount of CO emissions are estimated by red dots
2
This is the first major benefit for using geopolymers instead of OPC. To produce
1 ton of cement, the cement industry contributes roughly 1 ton of CO to the
2
environment, (J. Davidovits 1994). Since geopolymers are fired at low temperatures and
come from used waste, there is a reduction in CO emissions, up to 80%.
2
The second advantage when using geopolymers instead of OPC is the cost
reduction. For example geopolymer concrete from fly-ash is 10-30% cheaper than
concrete made from OPC. However, there are disadvantages such as the non-uniformity
of fly-ash geopolymeric material.
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The third benefit of using geopolymerization is the higher acid resistance properties
of a cured concrete specimen. One research team used tungsten tailings from a mine in
Portugal as a precursor for the binder, while OPC was used as a control, (Torgal, Gomes
and Jalali 2007). The agitated specimens in sulfuric, nitric, and chloric acid showed that
OPC lost more of its mass compared to the geopolymerized mine tailings. The acid
resistant capabilities of geopolymers may exceed that of common OPC, however, what
leaches off is more important. Leaching tests have been performed on geopolymer
specimens, however, the metals that are introduced into the experiment often come in the
form of salts, not in their natural state. Figure 2.6 shows the current encapsulation
capabilities of geopolymers (Khale and Chaudhary 2007).
Figure 2.6: The amount of heavy metals that can be encapsulated in geopolymers is
shown
Some of the properties of geopolymers that can enhance material applications are
fast/slow setting times, low shrinkage, fire resistance, and low thermal conductivity are
also benefits that can be utilized using this technology. One application would be in an
emergency/military landing scenario for an airplane. Concrete can be set in just 6-hours
gaining 70% of the final compressive strength, thus, being able to withstand the weight of
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an airplane during landing. Other applications can be found in foam, resin, paint, and
high-tech fiber reinforcement.
2.4.3 Water Management Technologies
One approach when considering treatment options is directly remediating the MIW.
Conventional water treatment can be used to physically remove contaminants, such as
sedimentation/clarification, solvent extraction, and sand bed filtration. Where advanced
water treatment entails carbon absorption and nanofiltration (NF). These are viable
options; however, they are labor intensive and require high operation costs versus more
appealing passive technologies. Evaporation is a potential strategy, but it requires large
surface area and adequate climate conditions. These physical type technologies are all
considered unit operations. Where chemical technologies are considered unit processes.
Chemical water treatment technologies involve precipitation, oxidation, and ion
exchange. They are also labor intensive, however, there are ways to engineer systems so
they operate in a more passive manner. For example, limestone can be placed in drainage
pathways, which the MIW flows through.
The neutralizing process can be an effective way of limiting the formation MIW
with the use of alkali materials. The reaction chemistry is applicable to the hydration
process during the curing of concrete due to the components found in cement. Lime is a
common alkaline material in use because of the chemical and biological responses such
as raising the pH, precipitating metals out of solution and reducing the activity of
microbes. With respect to iron, the chemical response can be explained by the chart in
Figure 2.7, where the speciation of iron hydroxides can be mapped on pe vs. pH diagram
(Doye and Duchesne 2003). The spectrum of reductive and oxidative states is shown by
the electric potential on the vertical axis, where the concentration of hydrogen ions varies
on the horizontal axis. It is important to note as the pH increases the more Fe(OH) (s)
3
that precipitates out of solution. Also, the pH at which optimal precipitation of free
metals occurs is higher than the acidic conditions of MIW. This optimal pH changes for
different kind of metals, therefore, this example for iron cannot be used for more
potentially dangerous heavy metals such as cadmium and lead.
As mentioned before the source of MIW is pyritic material, water, oxygen, and
microorganisms. In this treatment case with alkali material, more ferric iron is found in
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