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Colorado School of Mines | CHAPTER 6
CONCLUSIONS AND RECOMMENDATIONS
6.1) Conclusions
The ultimate objective of this research was to establish a set of design protocols
(guidelines) for estimating optimum cavity geometry and orientation for several critical
excavation and geological factors. In the first stage after the literature study, a case study was
used to verify if employing a 2-dimensional modeling software package (Flac2D) had sufficient
accuracy to perform a stress analysis within a borehole mining system as part of a predictive
design protocol. Data derived from an empiric field study was modeled using Flac2D, and the
results were compared to those obtained from a case study that used a three dimensional model
(Flac3D). It was observed that utilizing Flac2D to simulate borehole mining stress analysis will
give a relatively accurate results and the results obtained by Flac2D were in good agreement with
those generated from Flac3D.
Based upon the positive results of the case study and in an effort to address one of the
prevailing technical challenges impeding the commercial applications of borehole mining in
coal, a finite difference method (Flac2D) and dimensional analysis have been applied to do a
parametric sensitivity study to investigate the effect of several parameters like internal pressure,
cavity size, and in-situ stress, on the stability of a cavity developed through excavation of a
mineral resource during borehole mining. In this stage, the research was focused on delineating
the impact of critical factors associated with maintaining cavity stability during the mining
process and how it pertains to the development of a design protocol. The followings are the
observations during parametric study:
162 |
Colorado School of Mines | Increasing internal pressure and cavity length will significantly increase the induced
maximum principal stress around the cavity.
Depth of cover plays an important role, but not as significant as internal pressure and
cavity size.
Applying negative pressure will cause the stress concentration to move from cavity
roof to the side-wall and corner of the cavity.
Increasing the horizontal stress will decrease the induced shear stress at the cavity
roof corner.
Increasing internal pressure inside the cavity will increase the induced shear stress at
the cavity roof corner.
Reaching final optimum cavity size by using horizontal slices extraction is more
stable as compare to using vertical slices.
In cavities with small dimensions, the impact of internal pressures is less than those
with larger geometries.
Based on the observations during the numerical modeling, the followings are part of
proposed design guidelines:
During modelling, no plastic and shear yielding was observed for cavity radii of 2
m or less, even in environments with higher internal pressures.
By keeping internal pressure up to approximately 10 MPa, no plastic and shear
yielding were observed in all the modeled cavities (i.e. radii of 2, 6, 10, and 20 m).
Cavity radius of 10 m was determined to be the maximum size that could be
reached without stability issues, provided the internal pressure was maximum of
163 |
Colorado School of Mines | 30 MPa. It was observed that applying internal pressure of more than 30 MPa
would induce tension failure around the cavity at a radius of 10 m. Applying
internal pressure of 35 MPa will limit the cavity radius of up to 6 m.
Designing a cavity with a radius of more than 10 m without having stability
problems requires applying less internal pressure. In order to reach a cavity radius
of 20 m, the internal pressure shall not exceed 10 MPa. Applying small internal
pressure (extraction and pressurization) may have technical challenges associated
with shorter standoff distance and material removal.
6.2) Future Work and Recommendations
The guidelines presented in this dissertation are achieved from stress analysis on
simplified models. These guidelines are the first stage in order to achieve comprehensive
protocols for estimation of cavity geometry under different geomechanical and operational
conditions. Using separate codes for specific parts of the system is required. Flac is suitable for
stress analysis around the cavity in a continuum model. The followings are the suggestion for the
future works:
The guidelines need to be extended to include the conditions with higher depth of cover.
The numerical models in this dissertation are for intact and homogenous rock. To obtain
more realistic results, structural geologic features and fractures should be implemented
into the models. In addition, different rock types should be applied into these models as
well.
164 |
Colorado School of Mines | A sensitivity analysis comparing the influence of different factors on predicted induced
principal stress around the cavity roof should be conducted. The role of the parameters,
along with their uncertainties, should be applied in the measuring of the sensitivity. The
sensitivity of internal pressure, cavity geometry, horizontal stress, depth of cover and
rock mechanical properties (i.e. cohesion, density, friction angle, Young Module, Poisson
ratio and tensile strength) should be studied in greater detail.
Utilizing a hybrid finite difference-district element numerical tool will give more
accurate results. It can also reveal some other aspects like the direction and extension of
the induced fractures around the cavity.
Economic analyses of utilizing borehole mining for thin seam coal should be conducted
and the results compared with operation parameters of different BHM design protocols.
One of the extraction techniques being considered utilizes superheated fluid and/or steam
as the primary fragmentation mechanism. As such, it is important to understand the
potential impacts that high temperature fluids and gases will have on the stability of the
cavity during excavation process.
A fluid dynamic numerical tool is able to model some other parameters like the effect of
gas absorption and desorption during the mining process. The fluid (waterjet) pressure
can be simulated more accurately by using a fluid dynamic numerical approach.
Applying the recommendations mentioned above will led to the development of numeric
models that possess greater accuracies and capabilities in predicting the ultimate cavity size for
different geomechanical and geological parameters.
165 |
Colorado School of Mines | model mohr group 'User:caprock'
prop density=2833.0 bulk=2.06E10 shear=1.23E10 cohesion=1.158E7 friction=52.0 dilation=0.0
tension=7850000.0 group 'User:caprock'
fix y j 1
fix x i 101 j 2 151
fix x i 1
set flow=off
history 999 unbalanced
solve
initial xdisp 0 ydisp 0
initial xvel 0 yvel 0
free x i 1 4
initial xdisp 0 ydisp 0
initial xvel 0 yvel 0
model null i 1 j 65
group 'null' i 1 j 65
group delete 'null'
model null i 2 j 65
group 'null' i 2 j 65
group delete 'null'
model null i 3 j 65
group 'null' i 3 j 65
group delete 'null'
model null i 4 j 65
group 'null' i 4 j 65
group delete 'null'
model null i 5 j 65
group 'null' i 5 j 65
group delete 'null'
apply pressure 9000000.0 from 1,66 to 1,65
solve
gen circle 0.1389885,62.348698 2.4930167
model null region 1 63
group 'null' region 1 63
group delete 'null'
fix x i 1 j 66 151
fix x i 1 j 3 61
gen circle 1 63 2
null 1 64
model null region 2 64
group 'null' region 2 64
group delete 'null'
solve
174 |
Colorado School of Mines | solve
initial xdisp 0 ydisp 0
initial xvel 0 yvel 0
group delete 'null'
model null i 66 j 66
group 'null' i 66 j 66
group delete 'null'
model null i 66 j 65
group 'null' i 66 j 65
group delete 'null'
model null i 67 80 j 25 65
group 'null' i 67 80 j 25 65
group delete 'null'
model null i 81 85 j 25 64
group 'null' i 81 85 j 25 64
group delete 'null'
model null i 81 85 j 65
group 'null' i 81 85 j 65
group delete 'null'
model null i 66 j 25 64
group 'null' i 66 j 25 64
group delete 'null'
fix x y i 77 j 67 151
fix x y i 76 j 67 151
solve
Chapter 5
In this stage a borehole and cavity with different sizes under different internal pressures were
modeled. The following is the process of expanding a cavity under internal pressure of 22.5 Mpa.
gen 0.0,0.0 0.0,60.0 120.0,60.0 120.0,0.0 i=1,61 j=1,31
model elastic i=1,60 j=1,30
fix x i 61
fix x i 1
fix y j 1
set gravity=9.81
group 'Rock:shale' notnull
model mohr notnull group 'Rock:shale'
prop density=2700.0 bulk=8.81E9 shear=4.3E9 cohesion=3.84E7 friction=14.4 dilation=0.0
tension=1.44E7 notnull group 'Rock:shale'
prop por=0.3 region 2 30
176 |
Colorado School of Mines | group 'null' i 2 j 32
group delete 'null'
model null i 1 j 32
group 'null' i 1 j 32
group delete 'null'
model null i 1 15 j 27 31
group 'null' i 1 15 j 27 31
group delete 'null'
model null i 1 j 26
group 'null' i 1 j 26
group delete 'null'
model null i 2 j 26
group 'null' i 2 j 26
group delete 'null'
model null i 3 j 26
group 'null' i 3 j 26
group delete 'null'
model null i 4 j 26
group 'null' i 4 j 26
group delete 'null'
model null i 5 j 26
group 'null' i 5 j 26
group delete 'null'
model null i 6 j 26
group 'null' i 6 j 26
group delete 'null'
model null i 7 j 26
group 'null' i 7 j 26
group delete 'null'
model null i 8 j 26
group 'null' i 8 j 26
group delete 'null'
model null i 9 j 26
group 'null' i 9 j 26
group delete 'null'
model null i 10 j 25
group 'null' i 10 j 25
group delete 'null'
model null i 9 j 25
group 'null' i 9 j 25
group delete 'null'
model null i 8 j 25
group 'null' i 8 j 25
group delete 'null'
model null i 7 j 25
group 'null' i 7 j 25
182 |
Colorado School of Mines | group delete 'null'
model null i 6 j 25
group 'null' i 6 j 25
group delete 'null'
model null i 5 j 25
group 'null' i 5 j 25
group delete 'null'
model null i 4 j 25
group 'null' i 4 j 25
group delete 'null'
model null i 3 j 25
group 'null' i 3 j 25
group delete 'null'
model null i 2 j 25
group 'null' i 2 j 25
group delete 'null'
model null i 1 j 25
group 'null' i 1 j 25
group delete 'null'
model null i 11 j 32
group 'null' i 11 j 32
group delete 'null'
model null i 12 j 32
group 'null' i 12 j 32
group delete 'null'
model null i 13 j 32
group 'null' i 13 j 32
group delete 'null'
model null i 14 j 32
group 'null' i 14 j 32
group delete 'null'
model null i 16 j 30
group 'null' i 16 j 30
group delete 'null'
model null i 16 j 29
group 'null' i 16 j 29
group delete 'null'
model null i 16 j 28
group 'null' i 16 j 28
group delete 'null'
model null i 16 j 27
group 'null' i 16 j 27
group delete 'null'
model null i 15 j 26
group 'null' i 15 j 26
group delete 'null'
183 |
Colorado School of Mines | ABSTRACT
The formation of explosive gas zones (EGZs) is a critical problem in longwall coal mines.
The investigation of how EGZs might form currently relies on Computational Fluid Dynamics
(CFD) models, which have limitations around long solution times and availability of certain
validation data in a longwall mine. Physical, scaled models are an alternative to investigating
dynamic fluid behavior under complex scenarios that range from aircraft design to airway
investigation. Scaled modeling requires extensive dimensional analysis evaluation to replicate
complex airflow phenomena adequately. This work presents design and manufacturing
considerations to build a 1:40 scaled version of a longwall coal mine to investigate mine
ventilation strategies and the formation of EGZs. The physical model presented is the only
known scaled model of a longwall coal mine built on a modular design and capable of simulating
different ventilation strategies, longwall face advance, and shearer motion. The physical model
has a mine-wide atmospheric monitoring system (AMS) capable of measuring airflow speed and
gas concentrations in the air courses, longwall face, and gob region. Initial experimental data
results prove the model to be consistent with CFD models and published ventilation data of
longwall coal mines. The physical model captured the trends of flow leakage and simulated
methane accumulation across the longwall face compared to published data of actual mines and
CFD models.
iii |
Colorado School of Mines | CHAPTER 1
INTRODUCTION
Longwall working face equipment and mining process
Methane accumulation is an eminent risk in longwall coal mines. Poorly ventilated areas,
such as the gob and air entrapment regions in the longwall face, are known for their methane
ignition hazards. Methane ignition events at the longwall face are significant with a reported total
of 1,637 events between 1983 and 2014 [1]. Methane formation in coal beds happens during the
coalification process, and any disturbances in the coal seam, such as mining, may result in
methane release [2]. As methane mixes with mine air, it can create explosive gas zones (EGZs)
in the mine. Under normal temperature and pressure conditions, a methane-air mixture is
explosive on a range of approximately 4.5 to 14.5% methane [3]. Methane-air mixture hazards in
longwall coal mines have caused numerous mine disasters. The Upper Big Branch (UBB) mine
explosion in 2010, which resulted in 29 fatalities, is the most recent and widely known mine
disaster caused by the ignition of an EGZ in a longwall mine.
Detailed investigation of EGZs in longwall coal mines currently requires computational
fluid dynamics (CFD) models. While CFD models bring an accurate representation of physical
and chemical fluid phenomena in a longwall coal mine, limitations apply. These include long
solution times, lack of realistic validation data in hard-to-reach locations, and possible
oversimplification. These limitations can impact the usability of such models to thoroughly
investigate the formation of EGZs in longwall coal mines. A widely adopted alternative to CFD
modeling is the development of scaled physical models.
The present work uses an optically accessible, 1:40 scaled version of a longwall coal mine
as a research apparatus for investigating mine ventilation and the formation of EGZs in a
longwall coal mine. The apparatus is equipped with a mine-wide atmospheric monitoring system
(AMS) capable of sensing flow velocities, gas concentrations, and pressures. In this scaled
physical model, the coal bed to be mined is designed in a modular way, allowing researchers to
make changes in the gob, strata, and mining sequence, specifically, the advance of the longwall
face. This research apparatus is also equipped with a gas delivery system capable of injecting a
mixture of helium and carbon dioxide as a substitute for methane. The mixture is injected from
the strata, the gob, and the longwall face. The apparatus is built in a modular design capable of
1 |
Colorado School of Mines | simulating different ventilation strategies including bleeder, U-type, and back-return. The scaled
physical model contains 3D-printed elements of a longwall face, including shields and shearer.
The 3D-printed shearer is remote-controlled and can replicate the motion of a real shearer,
including drum rotation, cowl positioning and shearer travel along the face.
The scope of work presented in this thesis includes the criteria used to design and build a
1:40 scaled version of a longwall coal mine. Design elements include mining machinery, mine
layout, ventilation system arrangements, gob, and strata. The thesis covers a range of ventilation
experiments using the scaled apparatus and analyzes experimental data regarding mine
ventilation, methane distribution, and accumulation in a longwall coal mine for various
scenarios. Measurements are compared to published data of full-scale, operating longwall mines
and CFD models at the full and 1:40 reduced scales.
The research project presented in this thesis results from a 5-year long project funded by the
National Institute for Occupational Health and Safety (NIOSH). The project started in 2016.
Prior to my start in the project (2019), the design and fabrication of the physical model were
finished. My contribution to the research focused on deploying the AMS system in the physical
model, the ventilation and gas experiments, and the management of other fellow research
colleagues working on parallel research tasks. The work presented in this thesis includes all
achievements during my involvement with the project and the ones before it.
This research aims to prove the usefulness of a scaled physical model as a valuable resource
for investigating mine ventilation and the formation of EGZs in a longwall coal mine. The scaled
model is expected to guide the deployment of AMS systems capable of preventing mine disasters
related to the ignition of EGZs in longwall mines.
Research objective
This research aims to design, build and test a 1:40, optically accessible, a scaled version of a
longwall mine. This research apparatus will be used to investigate and test ventilation strategies
in longwall coal mines aimed at reducing the formation of explosive gas zones (EGZs) in the
longwall face, deployment of a mine-wide AMS system, and validate corresponding CFD
ventilation models.
2 |
Colorado School of Mines | In the United States, typical longwall panels are 300 to 500 m wide and may be 3 to over 6
km in length. The primary gate entry is the headgate or main gate and serves to supply fresh air,
power, water, and supplies. It also carries the conveyor belt to haul away the coal. The opposite
gate, the tailgate, carries away the exhaust air and serves as a secondary escape route for
emergencies. Longwall mining heights vary from 1.5 m to 3 m [6], [7]. Panel dimensions vary
depending on ground stability and geological and geotechnical features. The mining height is
determined by the coal bed, longwall equipment size, and, in some cases, coal quality [8].
Inside the longwall face (Figure 2.1, inset, and Figure 2.2), the coal seam is mined using a
shearer equipped with two cutting drums on the headgate and tailgate sides. The drum diameter
is typically 0.6 to 0.7 times the seam thickness or cutting height. In the full-size mine with 3 m
seam thickness, the drums are ~2 m in diameter. The drum width or cutting depth is typically
~1 m. The longwall face is supported by hydraulic roof support shields, which are typically 2 m
wide. Depending on its length, a longwall face is supported by 150 to 250 shields [8]. The roof
support system is designed to follow the shearer and the coal cutting process in the longwall
face. Once the coal has been cut in front of a shield, the shield canopy is lowered, and hydraulic
rams advance it forward to close the roof gap between the shield tip and the coal. The roof
immediately behind the advanced shield is left to collapse in a controlled manner, forming the
gob or goaf. Once the shield advance process is completed, the canopy is raised again to support
the roof. The coal cut by the shearer is transported to the headgate by an armored face conveyor
(AFC). The coal is transported to the surface from the headgate by a series of belt conveyors.
5 |
Colorado School of Mines | Sources of methane in a longwall mine
In a longwall coal mine, there are four sources of methane release: the coal-cutting process,
coal transport on conveyors, including the AFC, the longwall face and adjacent ribs, the gob
[12], [13]. The methane contributions from these sources vary from operation to operation, based
on geological factors, ventilation, coal production rate, and environmental conditions such as
barometric pressure [14]. Typically, the gob contributes approximately 80% of the total released
methane, while the other emission sources account for 20% or less [14], [15].
Longwall mine ventilation systems
There are two major ventilation systems used in a longwall mine. In the United States, all
longwall mines operate as retreat mines and are required to maintain a set of bleeder entries at
the rear end of the mined panels unless exempted [16]. Longwall mines outside the United States
use mostly U-type ventilation systems where the gob is progressively isolated using seals and
may also be inertized using nitrogen or Tomlinson boiler exhaust gas. The amount of air required
to ventilate a longwall face is dictated by methane and dust emission rates. The flow velocity in
the face is also a function of the available cross-sectional area across the face [16].
2.3.1 Bleeder ventilation
In a longwall mine that uses a bleeder system, the bleeder entries surround the mined-out
gob. The bleeder ventilation system intends to dilute and carry away methane from the gob.
Figure 2.4 details a bleeder ventilation system and its elements in a longwall mine. The operator
must travel the bleeder entries and examine them for proper function and air quality. Mines must
maintain >19.5% oxygen and <2% methane throughout the traveled bleeder airways and at
specific examination points [17]. Bleeders are commonly ventilated with dedicated exhaust fans.
To maintain acceptable air quality in the bleeder entries, fresh air is provided from both the
headgate and tailgate entries. The air quantity flowing through the bleeders is controlled through
curtains and regulators. In Figure 2.4, fresh air ventilates the face C coming in through the
headgate (HG) entries A. The face air partially leaks into and through the gob. Fresh air is also
coursed into bleeder entries at H. Return air leaves the face and is coursed to the bleeder fan at F
via the tailgate (TG). On the Tailgate side, a small amount of fresh air is run through regulators
D and E. The longwall belt is typically ventilated away from the face (outby) via B. Bleeder
regulators at location G control the amount of air in the headgate entries.
7 |
Colorado School of Mines | Table 2.1: Methane concentration threshold for different parts of the mine and the required
actions listed on 30 CFR §75.323
Methane
Location Actions
concentration
Except for intrinsically safe atmospheric monitoring systems
(AMS), electrically powered equipment in the affected area
shall be de-energized, and other mechanized equipment shall
be shut off
> 1% Changes or adjustments shall be made at once to the
ventilation system to reduce the concentration of methane to
Working
less than 1.0 percent
places and
No other work shall be permitted in the affected area until
intake air
the methane concentration is less than 1.0 percent
courses
Everyone except those persons referred to in §104(c) of
the Federal Mine Safety Act of 1977 shall be withdrawn
from the affected area
> 1.5%
Except for intrinsically safe AMS, electrically powered
equipment in the affected area shall be disconnected at the
power source
Changes should be made to the ventilation system to reduce
> 1 %
the concentration of methane to less than 1%
Return air
Everyone except those persons referred to in § 104(c) of the
split
> 1.5% Federal Mine Safety Act shall be withdrawn from the
affected area
11 |
Colorado School of Mines | Table 2.1 Continued
Methane
Location Actions
concentration
Other than intrinsically safe AMS, equipment in the affected
area shall be de-energized, electric power shall be
disconnected at the power source, and other mechanized
Return air
> 1.5% equipment shall be shut off
split
No other work shall be permitted in the affected area until the
methane concentration in the return air is less than 1.0 percent
Bleeders and > 2%
Changes should be made to the ventilation system to reduce
other return (Provided
the concentration of methane to less than 2%
air courses use of AMS)
CFR §27.24 and §75.342 also require all coal and rock cutting machines, longwall face
equipment, and any other mechanized equipment used to extract or load coal to have a methane
monitoring system. This system shall trigger an alarm when methane concentrations are above
1% and shut the equipment off when concentrations exceed 2%.
Methane-air explosion hazards in longwall coal mines
Methane-air mixtures are explosive in a range between 4.5% and 14.5% volumetric
concentration of methane. According to Coward's diagram [3], Figure 2.7, a methane-air mixture
can only become explosive when sufficient oxygen exists.
12 |
Colorado School of Mines | Figure 2.7: Explosibility limits of methane-air mixtures [3] (public domain)
In longwall coal mines, the hazards related to methane-air mixtures are eminent. The diverse
sources of methane, particularly the longwall face and the gob, create explosion hazards that
have long been associated with mine disasters.
Data from GVBs in longwall mines indicate that gob methane concentrations can exceed
80% [20]. As per regulatory matters, the longwall face methane concentration shall be kept
below 1%, with exceptions up to 1.5%. Juganda et al.[21] demonstrated that a fringe region
exists between the face and the center of the gob where the methane concentration is in the
explosive range.
A methane-air explosion in a confined space such as a longwall mine can reach pressures of
several atmospheres [22]. An explosion may propagate as long as fuel and oxygen are present.
For example, in the 2010 Upper Big Branch explosion, the pressure wave from the initial
methane-air explosion stirred up fine coal dust that caused the explosion to propagate through
more than 60 km of mine workings [23].
13 |
Colorado School of Mines | CHAPTER 3
FLUID FLOW AND DIMENSIONAL ANALYSIS
Airflow and head in underground mine ventilation
Air and gases traveling through a mine continuously face changes in pressure and flow
velocity due to friction, shock losses, changes in cross-section area, and changes in elevation.
Flow velocity changes are changes in the kinematic energy of the fluid. Variations in elevation
impact the potential energy of the fluid. The energy retained in the fluid stream is known as flow
work.
Bernoulli's equation (3.1, energy form [J/kg]) represents the fluid flow of air and gases
derived from the principle of energy conservation.
(3.1)
2
𝑝1
2
𝑝2
𝑣1 + 𝜌 +𝑍1𝑔 = 𝑣2 + 𝜌 +𝑍2𝑔+𝐻𝑙
where is the flow speed, p the fluid pressure, the elevation, is gravity, and is the
pressure loss.
𝑣 𝑍 𝑔 𝐻𝑙
Dimensional analysis, scaling, and similarity applied to coal mine ventilation
3.2.1 Dimensional analysis
Fluid flows can be analyzed in scaled models. This process is often done when evaluating
designs for aircrafts and ships. Dimensional analysis is the technique behind scaling the fluid
behavior of these models. The scaling process happens through the representation of
dimensionless parameters, such as Reynolds, Froude, Richardson, Peclet, Mach, etc., with a
similar magnitude between scaled and full-scale models.
Rosen (1989) [24] explains that, in a scaling problem, only some of the derived parameters
require a transformation factor, while others do not. In scaling aerodynamic problems, surface
tension, buoyancy, and the Reynolds number are examples of parameters that must be scaled.
Therefore, the scaling process might introduce errors to the problem, which may considerably
impact the representation of flow phenomena in the scaled version. An example would be the air
compressibility changes when scaling an airway down while maintaining the Reynolds number.
14 |
Colorado School of Mines | For instance, if the hydraulic diameter of an airway is reduced by 40 times, the flow speed would
have to be 40 times higher to maintain the same Reynolds number assuming the same
environmental conditions. If the scaled flow velocities are higher than Mach 0.3 (~103 m/s), the
compressibility properties of air will change.
3.2.2 Similarity, similitude, and scaling
Zohuri (2015) [25] states that similitude is achieved when three types of physical similarity
are achieved: geometric, kinematic, and dynamic similarity. Duncan (1953) [26] defines
similitude as the determination of quantitative conditions in which an event's behavior similarity
is replicated. The concept of reproducing a similar behavior of a physical event is linked to
dimensional analysis. Gibbings (2011) [27] states that similarity can be applied to kinematics and
Newtonian problems by identifying non-dimensional parameters through their physical
representations, such as the Reynolds number and its representation of turbulence in a fluid flow.
Gibbings also exemplifies the limitations of similarity for more complex phenomena, such as
compressible flow. Here, non-dimensional parameters, such as the Reynolds, Mach, and Grashof
numbers represent certain phenomena. However, integral representation cannot be achieved
since no non-dimensional parameter can represent the specific heats.
Zohuri (2015) [25] states that similarity is achieved when equations 3.1 and 3.2 are met.
refers to the full-size model similarity number, the scaled model reference similarity
𝑁nu𝑜m𝑚ber, a non-dimensional variable, a non-dime 𝑁n𝑜s𝑝ional coordinate, and non-dimensional
time.
𝑛 𝜒𝑖 𝜏
(3.1)
𝑁𝑜𝑚 = 𝑁𝑜𝑝 (3.2)
S𝑛c𝑢 a(li𝜒n𝑖 g, 𝜏c)o𝑚 rre=la𝑛te𝑢 s( t𝜒h𝑖 e, 𝜏f)u𝑝 ll-size mine and scaled model variables at their corresponding
locations (Zohuri, 2015 [25]). This implies that shape, velocity, and other variables would be
scaled proportionally to the scaling factor. The scaling factor ( ) relates the dimensional
variables of the scaled and the full-size mine, as described in equation 3.3. The mathematical
𝑘𝑢
definition of scaling (equation 3.3) illustrates why a full-scale model and a scaled model might
have different phenomena representations even though a scaling factor is followed when
considering the geometry.
15 |
Colorado School of Mines | (3.3)
𝑢(𝜒𝑖,𝜏)𝑝
𝑘𝑢 = 𝑢(𝜒𝑖,𝜏)𝑚
3.2.3 Geometric similarity
Geometric similarity refers to the shape and geometrical aspects of the scaled model and the
full-size mine. This similarity exists if two objects look alike except for their overall size. Pallet
(1961) [28] states that if the ratio of any two corresponding dimensions between two systems is
constant, the geometric similarity is achieved. In the scaled model of a longwall mine, this
similarity is achieved when representing the physical elements such as airway and machinery
heights, widths and lengths with a scale factor; in this case, 1:40.
3.2.4 Kinematic similarity
Kinematic similarity considers the ratio of motion velocities between a full-scale mine and a
scaled model. Kinematic similarity can be observed in similar streamlines that can be plotted
based on CFD computations [29]. Kinematic similarity must also be considered for the rotation
of the shearer drums. In a 1:40 scaled version of the drums, to maintain kinematic similarity, the
scaled model drums would have to rotate 40 times faster than those of a full-size shearer.
3.2.5 Dynamic similarity
Dynamic similarity implies that the fluid and boundary forces acting in a scaled model
should scale when compared to the full-size mine. This implies that the viscous and inertial
forces are scaled such that their ratio remains constant [25]. If the ratios of the forces are
maintained, the dimensionless variables that represent physical phenomena in the fluid flow,
such as the Reynolds number and turbulence, shall also be maintained to ensure dynamic
similarity.
3.2.6 The Buckingham π theorem
An essential concept in dimensional analysis is the Buckingham π theorem. The theorem
states that physical relationships can be expressed using independent dimensionless variables
composed of relevant parameters [30]. In dimensional analysis, a natural phenomenon is
described mathematically using equation 3.4, in which all terms represent m physical variables
[25].
16 |
Colorado School of Mines | (3.4)
A𝑓n( 𝑄e1x,a𝑄m2,p…le ,t𝑄o𝑚 i)llu=st0rate equation 3.4 is the pressure drop (Δp) in a horizontal pipe with
turbulent flow. This problem is mathematically expressed in equation 3.5 as a function of the
diameter (D), length (L), and roughness (η) of the pipe, and fluid viscosity (µ) and density (ρ).
(3.5)
W𝑓(h𝛥e𝑝n ,a𝐷p,p𝐿ly,𝜂in,g𝜌 ,thµe) B=u0ck ingham π theorem, equation 3.4 transforms into equation 3.6. All π
variables in equation 3.5 are non-dimensional products of some or all variables in equation 3.4.
The theorem transforms elements of an equation describing a problem into non-dimensional
elements while maintaining a functional relation [25]. Buckingham referred to the dimensionless
parameters of variables such as mass, length, and time in a problem as π-groups. The (m-n)
independent dimensionless parameters are different combinations of the many dimensional
variables out of the independent variables that define a problem. refers to the number of
fundamental dimensions (mass, temperature, time, length, etc.) in the variables.
𝑚 𝑛
𝑚 (3.6)
T𝑓o( πso1,lvπe2 ,e…qu,aπt𝑚io−n𝑛s )th=ro0ugh the Buckingham π theorem, two requirements must be satisfied.
The first is that each fundamental dimension (such as distance, time, and mass.) must appear in at
least one of the variables representing a π-group. The second is that all π-groups must be
independent, and no group must be linear or exponentially dependent of another group.
3.2.7 Important dimensionless parameters in coal mine ventilation scaling
The airflow in the working airways of longwall coal mines is generally turbulent. These
airways include intakes, returns, and the longwall face. In the gob, there may exist zones of
laminar flow where the ventilation airflow velocity is low and almost non-existent [7], [31], [32].
Sales and Hinsley (1951) state that the Reynolds numbers in the working airways of an
underground mine ventilation system typically range from 50,000 to 20,000,000 [33]. Jones and
Hinsley (1958) [34], while studying the applications of scaled models to investigate underground
mine ventilation, noted that maintaining a Reynolds number above 30,000 was sufficient to
maintain similarity and replicate the turbulent effects of the flow.
17 |
Colorado School of Mines | The Reynolds number ( ) (equation 3.7) is a non-dimensional number representing the
ratio of inertial forces to viscous forces in a fluid. The inertial forces are a product of the fluid
𝑅𝑒
density ( ), flow velocity ( ), and characteristic linear dimension ( ), while the viscous forces
are represented by the dynamic viscosity ( ). A laminar flow ( < 2,000) indicates that viscous
𝜌 𝑢 𝐿
forces dominate and characterize a smooth fluid motion. A turbulent flow ( > 4,000) is
𝜇 𝑅𝑒
characterized by dominant inertial forces, creating a chaotic motion in which flow instabilities
𝑅𝑒
are present. Reynolds numbers between 2,000 and 4,000 represent a transitional flow regime
between laminar and turbulent characteristics.
𝜌𝑢𝐿
𝑅𝑒 = 𝜇 (3.7)
Besides the Reynolds number, Gangrade et al. (2019) [35] and Jones et al. (1995) [36] also
noted the importance of the Richardson number in scaled models where flow layer dispersion is
evident, such as in methane-air mixtures in longwall coal mines. The Richardson number ( )
(equation 3.8) is a non-dimensional parameter that represents the ratio of buoyancy to flow shear,
𝑅𝑖
a function of gravity ( ), fluid density ( ), flow speed ( ), and depth ( ). The greater the
Richardson number, the greater is the effect of buoyancy on the fluid dynamic behavior.
𝑔 𝜌 𝑢 𝑧
(3.8)
𝑔 𝜕𝜌/𝜕𝑧
2
𝑅𝑖 = 𝜌(𝜕𝑢/𝜕𝑧)
Scaled modeling for coal mine ventilation
In 1952, Sales & Kinsley [33] developed a physical model to investigate Atkinson friction
factors in full-size airways. The model had airways with a cross-sectional area of 58 cm2with
variable cross-section dimensions and lengths. Entries were supported by round and square
timbers. The model proved to replicate real mining conditions, observing that the friction factors
might vary depending on the Reynolds number. Sales and Kinsley noticed only minor changes in
the friction factor over a range of 50,000 < Re < 300,000. Sales and Kinsley determined that the
spacing, size, and shape of the timbers had the most significant impact on the friction factor.
Stein et al. (1975) [37] built a 1:10 scale model of a low room-and-pillar coal mine section.
Entry dimensions were 0.9 m high by 7.9 m wide. The section used a line brattice exhaust
ventilation arrangement. The model was used to investigate the impact of distance from the end
18 |
Colorado School of Mines | of the brattice to the mining face. Stein et al. concluded that keeping the brattice within 3 m of
the face was more effective to control dust than keeping the curtain further back and increasing
the ventilation quantity. Additionally, the authors correlated the auger rotational speed to the
amount of dust generated, demonstrating that a scaled model effectively represented the
ventilation system and contaminant concentrations in a mine.
Aitken et al. (1988) [38] tested scaled aerodynamic models to study local build-up of
methane and frictional ignition risks in coal mines. Aitken et al. (1988) identified bulk mine
airflow and contaminant generation rates as critical parameters relevant to the physical model.
The experiments demonstrated that scaled models were useful to simulate methane behavior in
coal mines. Aitken et al. developed meaningful scaling relationships based on non-dimensional
airflow parameters and Reynolds number similarity.
Jones and Lowrie (1993, 1997) [39], [40] developed a scaled model of a longwall face to
investigate the accumulation of methane and airborne pollutants. The model represented a 1:70
scale of a 200 m long and 3 m high coal face. The authors evaluated flow scaling criteria by
comparing inertial dispersion, dispersion by molecular diffusion, and pressure gradients.
Geometric scaling was applied to the mine layout, while dynamic scaling was used for the gob
and flow. The main factor guiding flow scaling was to maintain the Reynolds number to values
comparable in actual mines. While modeling the gob, the authors kept the permeability and
Reynolds number similar to full-scale mines. Table 3.1 details the chosen scaling criteria for the
1:70 scaled model when simulating a full-scale flow rate of 10 m3/s in the longwall face.
Table 3.1: Scaling criteria used by Jones and Lowrie (1997) in a 1:70 scaled version of a
longwall coal mine [40]
Property Scaling/substitute
Reynolds number Constant
Peclet number (gob) × 3
Nominal grain size × 0.48
Flow speed × 0.21
Gob permeability × 0.23
Geometry × 1/70
Transit time (face) ×0.0068
19 |
Colorado School of Mines | CHAPTER 4
DESIGN AND MANUFACTURING OF THE PHYSICAL MODEL
Dimensions and layout of the 1:40 scaled model
The objective is to develop a 1:40 scaled version of a longwall coal mine capable of
providing an accurate representation of the mine with high-fidelity flow behavior and
comprehensive flow and gas sensing abilities. This model is used to investigate the dynamic
behavior of airflow mixtures in a longwall coal mine and the formation of EGZs.
The model was designed to have a footprint of 7.0 m by 6.1 m (280 and 244 m, respectively,
at full scale). The mine layout (Figure 4.1) incorporates a portion of a single longwall panel,
main entries, tailgate, headgate, and bleeder entries. The longwall face is designed to be 5.5 m
wide (220 m), and the face can advance 3.89 m (155 m) in increments of 0.61 m (24.4 m). The
coal seam thickness and mining height are assumed to be 7.62 cm (3 m). The pillar width in the
gate and bleeder was adjusted to reduce space usage within the physical model as it has a
negligible impact on ventilation. Airway shape and layout maintain a high degree of geometric
similarity with the modeled longwall coal mine. Air courses range from 14 cm to 15.5 cm (5.6 to
6.2 m) in width and are as tall as the mining height. The variability in width is due to the cutting
tolerances of the plexiglass sheets and pillar material.
Modularity was an important design criterion for the physical model. The scaled model can
be adjusted to simulate a variety of bleeder, U-type, and back-return ventilation scenarios.
Researchers can represent several longwall face advance positions, different shearer locations,
and both cutting directions.
22 |
Colorado School of Mines | Figure 4.7: Flow visualization with glycerin smoke. This picture represents how the flow
progresses as the fog density increases. The flow interactions with the pillars were successfully
captured. The smoke shows eddies forming between the pillars
4.2.3 Pillars and airway design
The pillars in the physical model are made of MDF. Since this scaled physical model aims to
investigate the dynamic behavior of airflow mixtures in longwall coal mines, the pillar layout
was designed to provide an accurate representation of the airways when compared to a real
longwall mine.
The airways (Figure 4.8 [44]) in the model maintain a realistic aspect ratio, with the height
fixed to 7.6 cm (3 m, in full-scale) cm and the width ranging from 14 to 15.5 cm (5.6 to 6.2 m, in
full-scale) depending on location in the physical model (Figure 4.1).
The pillars are fixed in the physical model using double-sided tape. Pillar bodies are hollow
to permit the installation of flow and gas sensor electronics inside (Figure 4.9 [44]). Figure 4.10
[44] shows the fully assembled physical model.
27 |
Colorado School of Mines | 4.2.4 Gob and strata modules
As part of the modular design, the mining and gob regions can be changed to simulate the
advance of the longwall face. Gob modules can be filled with a variety of materials to simulate
changes to the porosity and permeability of the gob and strata.
Figure 4.11 shows a gob module filled with orange plastic balls and shredded plastic
sheeting representing parts of the gob strata. Gob modules are 50.8 cm wide and are 60.1 or 63.5
cm long, depending on the modules' location. The differences in length for the gob modules
serve to simulate cut and uncut portions of the longwall. The longer gob modules are used where
the shearer has already cut the coal. The module base is a 12.7 mm thick plexiglass panel
mounted on casters that run in a guide rail in a guide rail (Figure 4.12). Above the plexiglass
panel sits a cage made from expanded steel that is filled with materials representing the gob
strata. Filling with different shapes and materials allows adjustment of gob and strata
permeabilities. In scaling the permeability of the gob, researchers focused on an accurate
representation of the gob airflow through dynamic scaling, detailed in Chapter 5
Figure 4.11: A) CAD design of a gob and face module. B) Gob module attached to a face
module. The gob is simulated as 58-mm-diameter spheres and the strata as shredded plastic
sheeting
30 |
Colorado School of Mines | Figure 4.12: Guide rail and track system that allows the face and gob modules to slide back and
forth, simulating different longwall face advance scenarios
Changing the gob material adapts the physical model to different simulation scenarios. This
includes reproducing the gradient change in permeability of the gob (Figure 4.13), which results
from differences in compaction between the fringes and the central region of the gob. The
representation of variable resistance in the gob is important to capture subsidence and caving of
the gob subject to the properties of the host rock in the immediate roof, which will impact the
overall resistance and permeability values of the gob [45].
The face advance simulation happens through the insertion of additional rows of gob
modules into the physical model (Figure 4.14). Every row accommodates 11 modules for the
5.5 m (220 m) face length. The model can accommodate up to 6 rows, simulating face advances
from 0.9 m (36 m) to 4.6 m (184 m) in steps of 0.62 m (25 m). The top plexiglass panels are
hinged to facilitate the insertion of additional gob modules when advancing the longwall face.
31 |
Colorado School of Mines | 4.2.5 Longwall face
The longwall face is equipped with a total of 110 ground support shields, a shearer, and a
track to advance the shearer (Figure 4.15). The shields are based on a Caterpillar design and were
3D printed using acrylonitrile butadiene styrene (ABS) plastic. The shields are 5 cm (2 m) wide
and are as tall as the mining height, 7.5 cm (3 m). They are held in place in the longwall face
through a push ram attached to the cable tray.
Figure 4.15: Side view of a face module detailing the 3D printed longwall face machinery and
dimensions
The shearer (Figure 4.16) design is based on the Caterpillar EL2000 Longwall Shearer, with
the cutting drums based on the Globoid® Drums with the design provided by German
manufacturer Krummenauer.
Figure 4.16: Longwall shearer inside a face module
Besides matching the geometry of an actual longwall shearer, the 3D printed shearer can
reproduce the following actions: rotation of the drums, movement of the ranging arms and cowls,
33 |
Colorado School of Mines | and translation along the face via a rack-and-pinion system. All movements are controlled via a
microcontroller unit (MCU) and servo controllers (Figure 4.17). The servo controllers provide
the ranging arm and cowl actuation, while a separate servo motor powers the rotation of the
drums for both the tailgate and headgate sides of the shearer.
Figure 4.17: Illustration of the shearer automation process
A Bluetooth module allows the remote operation of the shearer from the data acquisition
computer. Shearer movement is controlled by a color sensor that stops the shearer when it sees a
red tag on the track. Red tags along the track control how far the shearer will move for the next
experimental stage. A lithium-ion battery system (Figure 4.18) runs the shearer for
approximately 1 hour per charge.
Figure 4.18: Shearer internal circuits and power system
To allow the shearer to run for more extended periods of time, it is also equipped with a
wireless charging system. Through power transmission coils embedded in the shearer tracks
(Figure 4.19), it is possible to recharge the shearer and to keep running continuously for longer
experiments. The wireless control system eliminates wires, avoiding airflow disturbances in the
physical model.
34 |
Colorado School of Mines | Figure 4.19: Power transmission coil embedded to the shearer track
4.2.6 Gas delivery and distribution system
A gas delivery system (Figure 4.21 through Figure 4.23) allows the physical model to inject
gas to the longwall face and the gob, simulating the most critical methane source locations of an
actual mine. The gas delivery system has a total of 26 discrete injection points (Figure 4.20). The
main lines in the system are 12.7 mm diameter stainless steel, while the lines supplying
individual injection points are 6.3 mm in diameter.
Of the 26 injection points, 11 are dedicated to the longwall face (one injection point per face
module), and the remaining 15 are dedicated to the gob. The gob injection points are distributed
across three rows of gas supply lines, used according to how far the longwall face and the gob
region have advanced. The system allows the flow quantities directed to the gob and longwall
face to be adjusted independently. Needle valves individually adjust the flow rate to each
injection point in the longwall face and gob.
Figure 4.20: Schematic of the lines and fittings in the gas delivery system in the physical model
35 |
Colorado School of Mines | There are currently two different manifolds used to simulate the gas injection to the longwall
face. The first manifold (Figure 4.24, a) simulates a distributed gas source. This manifold has 10
rows of 3 mm diameter holes spaced 6 mm (center-to-center), each row having 72 holes. The
perforated manifold allows a distributed gas injection panel if the line is attached to the central
slot. It can also simulate two discrete injection points by connecting the gas supply lines to the
two lateral gas slots. Figure 4.25 details how the gas line attaches to the perforated manifold. The
second manifold (Figure 4.24, b) has two discrete injection points and allows a clear view of the
longwall face, which is helpful during flow visualization experiments.
Figure 4.24: Two different manifolds used to inject gas into the longwall face. A) Laser-cut
manifold with 3 mm perforations. B) Discrete injection point manifold
Figure 4.25: Gas supply line to the perforated manifold presented in Figure 4.24
The system is sized based on an airflow rate that reproduces Reynolds numbers in the
physical model above 50,000 and a maximum 4% (by volume) concentration of gas at the
longwall face, requiring a gas flow of approximately 580 SLPM. These numbers are beyond the
38 |
Colorado School of Mines | typical values the physical model will operate. Based on an inlet pressure of 690 kPa, the
pressure drop in the lines is estimated to be approximately 100 kPa.
An Alicat M-Series, 500SLPM mass flow controller (MFC, Figure 4.26) regulates the
overall flow rate of gas injected into the physical model. The calibration of the MFC is adjusted
for gases with the same density as methane (0.668kg/m³, at normal temperature and pressure
conditions). The MFC can handle a maximum flow rate of 500 SLPM and a maximum gas
pressure of 1 MPa.
Figure 4.26: Alicat M-500SLPM mass flow controller installed into the physical model
4.2.7 Ventilation system
A Fantech FKD 10XL axial-vane fan (Figure 4.27) serves as the main fan for the physical
model. When coupled to the inlet of the physical model, the fan produces a maximum airflow
quantity of 0.2 m³/s at ~240 Pa static pressure. Figure 4.28 shows the characteristic curve for this
fan. The inlet and outlet of the fan are 25.4 cm in diameter. The fan outlet is connected to a 25.4
cm diameter flexible duct that discharges at the airflow to the mains of the physical model
through a register.
39 |
Colorado School of Mines | Figure 4.34: Schematic showing a front view of a face module and the location of the sensor slot
A total of 49 IST flow sensors are distributed in strategic locations of the physical model
(Figure 4.35) to observe the behavior and distribution of the airflow throughout the physical
model. 37 sensors are installed in the mine airways and 12 in the gob. The sensor locations are
selectedto capture airflow changes in all airways, monitor flow quantities across the longwall
face, and investigate flow behavior in the gob.
The installation of the flow sensors in the gob differs from the overall installation on the
airways. While a flow visualization test can guide determining the airflow direction in the
airways, smoke travel cannot be observed in the gob. Therefore, gob flow sensors are installed in
pairs mounted on an L-bracket with an 90º offset orientation (Figure 4.36, b) that provides
separate vectors in two directions, X and Y. These velocity vectors define the direction and
magnitude of the flow in the gob.
Figure 4.35: Location of the IST FS7 flow sensors in the physical model.
44 |
Colorado School of Mines | 4.3.2 Methane substitute gas sensors
For safety reasons, the physical scaled model does not use methane gas. A surrogate gas
mixture of 30% CO and 70% He (by volume) is used to replace it instead. Chapter 5 details the
2
selection process and properties of this surrogate gas mixture.
One benefit of using a CO mixture in the physical model is the wide availability of suitable,
2
compact sensors. Since the surrogate gas used is premixed, tracking the concentration of CO in
2
the physical model suffices to calculate an equivalent methane concentration.
The sensors used in the physical model are the ExplorIR®-W CO sensors (Figure 4.37). To
2
increase the output resolution, the measuring ranges of these sensors vary by location. The gob
and tailgate region uses sensors with ranges of 0 to 20% and 0 to 60% of CO . The sensor closer
2
to the headgate in the longwall face and the bleeder entries use sensors with a 0 to 5%
CO range.
2
Figure 4.37: ExplorIR®-W CO sensor and its dimensions
2
46 |
Colorado School of Mines | The response time for the CO sensors depend on the diffusion process. Using a 15% CO -
2 2
85% N gas mixture in the calibration vessel, the response time for a 0-60% range CO the
2 2
sensor was tested (Figure 4.40). For this specific experiment, the sensor reached its peak
concentration within approximately 30 seconds. Once the sensor was removed from the vessel,
the response took approximately 30 seconds to read less than 1% of CO . This experiment
2
guided the running time of a gas experiment in the physical model.
Figure 4.40: Response time for a CO sensor
2
The physical scaled model has a total of 12 CO sensors installed in the longwall face, gob,
2
and bleeder entries, as shown in Figure 4.41. Three sensors are installed along the longwall face
(Figure 4.42). In the gob, the sensors are located in the same meshed enclosure that
accommodates the bidirectional flow sensors. The gob CO sensors are placed approximately
2
15 cm above the bottom of the gob module. This height allows measuring the simulated methane
when buoyancy is present. The sensor height will be adjusted in future experiments to investigate
the impact of buoyancy in the measurements. The sensors in the bleeder entries are installed
directly in the pillars. A flexible tube attaches to the infrared sensing element, allowing the gas
mixtures to reach the sensors (Figure 4.43).
48 |
Colorado School of Mines | 4.3.3 Data acquisition and processing
The physical scaled model uses a National Instruments (NI) cDAQ-9179 chassis to
accommodate up to 14 data acquisition modules. There are currently three NI-9209 data
acquisition modules connected to the chassis. These modules have 32 analog input channels,
adjusted to a ± 10 V input. Each module can sample at 500 samples/second. Since the flow
sensors (7.5 V) run on a different voltage than the CO sensors (3.3 V), two modules are
2
dedicated to the flow sensors, and one is dedicated to the CO sensors. Each sensor uses an
2
analog, single-ended channel with a common ground connected to the module.
A LabVIEW application (Figure 4.45 and Figure 4.46) configures the channel settings for
each data acquisition module, sampling rate, voltage range, terminal configuration, and the path
to export the acquired data. The acquired data is exported as a Technical Data Management
System (TDMS) file, which is more efficient in terms of memory and storage considerations
when compared to an Excel file format.
Figure 4.45: Settings tab of the LabVIEW code used for data acquisition. This tab serves to
configure the data acquisition modules, sampling settings, and file export location
51 |
Colorado School of Mines | CHAPTER 5
AIRFLOW SCALING
Dynamic scaling of the turbulent airflow in the physical scaled model
During earlier stages of this research, both Reynolds and Richardson numbers were potential
dimensionless parameters to guide the flow similarity design of the physical scaled model. The
Richardson number is useful in regions of the mine where flow speeds are lower and the buoyancy
forces are more predominant, such as the gob. In regions where the flow is turbulent, such as the
longwall face and air courses, the Reynolds more accurately represents flows dominated by inertia
forces and turbulent mixing.
Comparing the initial CFD models of the physical model with full-scale CFD models of a
mine is necessary to understand how the flow behaves following different scaling criteria.
Assuming a 500,000 Reynolds number in a full-scale longwall face, the inertial forces overcome
buoyancy forces. Although the velocity profile of the scaling criteria using Reynolds and
Richardson numbers might be similar, the methane-air mixture behavior for these criteria might
differ substantially.
Juganda [48] developed more comprehensive CFD models of the scaled model. These
models (Figure 5.1 through Figure 5.4) investigated the impacts of varying the Reynolds number
in turbulent regions of the physical scaled model. Figure 5.1 and Figure 5.2 demonstrates that
Reynolds numbers greater than 36,000 are sufficient to reproduce the dynamic behavior of the
turbulent airflow in the longwall face. When the Reynolds number is reduced to 7,000, the shape
of the streamlines is still reproduced, but the magnitude of the normalized velocity differs from
Reynolds numbers greater than 35,000.
54 |
Colorado School of Mines | The CFD models indicated that the 30% CO – 70% He mixtures successfully replicated the
2
same diffusion and buoyancy effects of methane in the air, and this mixture was selected as the
surrogate gas. An additional benefit of using a gas mixture with CO is the wide availability of
2
gas supplies and also CO sensors.
2
Gob resistance
The gob region of the scaled physical model does not have the intention to scale
geometrically to an actual mine. The focus of the materials used as the gob is to have an accurate
representation of the airflow and gaseous mixtures' dynamic behavior compared to an actual
mine. This happens through matching resistance and permeability values of an actual gob region.
Currently, the gob in the physical model is simulated as packed spherical objects (Figure 5.9).
The modularity allows the objects to be easily replaced, so permeability and resistance values
can be adjusted to desired levels.
Figure 5.9: Currently available spheres for simulating the gob
The permeability of the packed spheres is calculated using Darcy’s law (Equation 5.1), in
which is the airflow quantity (m³/s), is the absolute permeability (m²), is the pressure
drop across the channel (Pa), is the dynamic viscosity of the air (Pa.s), and is the length of
𝑄 𝑘 ∆𝑃
the packing/porous medium. Packings with the three available sphere diameters presented in
𝜇 𝑥
Figure 5.9 were tested.
(5.1)
−𝑘𝐴∆𝑃
𝑄 = 𝜇𝑥
Figure 5.10 presents the calculated permeability values for three different packing lengths
(15.2, 38.1, and 76.2 cm) and sphere sizes in Figure 5.9. The calculated permeability values for
the different sphere sizes are approximate to actual gob permeability values available in the
60 |
Colorado School of Mines | Figure 6.7: Illustration of the probing depth and height for the data presented in Figure 6.6
Simulated methane investigation and gob flow
For the flow experiments presented in Figure 6.3, simulated methane (30% CO and 70%
2
He) was injected into the longwall face and gob. The flow rate of methane was constant at 180
SLPM. 20% of the total gas volume (36 SLPM) was directed to the longwall face, injected at
each of the 11 face modules, while the gob had 5 injection points, injecting the remaining 80%
(144 SLPM) of the simulated methane. The 180 SLPM represents 2% of the flow quantity into
the mains for the fan setting that provides approximately 9000 LPM to the mains. Figure 6.8
details the location of the injection points in the physical model for these experiments.
As expected, the flow quantities in the longwall face of the physical model (Figure 6.3)
affect the methane concentrations in the longwall face. For the shearer in the headgate scenarios,
the methane concentration increased as the flow speed decreased. The equivalent concentrations
of methane are in the explosive range for the two experiments in which the shearer is at the
headgate. The EGZ for this scenario expands from tailgate to mid-face. Since the simulated
methane is a premixed gas mixture with 30% CO , the equivalent concentration of methane is
2
obtained by multiplying the measured CO 2concen tration by 3.33 (100/30).
66 |
Colorado School of Mines | CHAPTER 7
CONCLUSIONS AND RECOMMENDATIONS
Conclusions
This work demonstrates that a 1:40 scaled physical model of a longwall coal mine can be
used to investigate mine ventilation strategy and the formation of EGZ. When developing such a
model, an important part of the work is how to accurately maintain the geometric, kinematic, and
dynamic similarity with a full-scale longwall mine. It is important to note that different areas of
the mine require different approaches to properly represent flow similarity. Therefore, the gob
region does not match geometric similarity but instead focuses on representing the dynamic
similarity of the airflow and gaseous mixtures.
Additive manufacturing techniques, such as 3D printing, can assist with ensuring geometric
similarity. It is also important to understand how the motion and actuation of machinery in a
mine might impact the dynamic behavior of the airflow. The dynamic similarity of turbulent
zones can be achieved by maintaining a Reynolds number above 35,000. Although a lower
Reynolds number might still simulate similar flow streamlines, it impacts the overall
representation of gaseous mixtures in air. On lower flow speeds, buoyant forces prevail over
momentum and shear forces, affecting the dynamic behavior of airflow and gaseous mixtures.
Too high Reynolds numbers might reproduce flow speeds that could change the compressibility
of air, no longer maintaining a similar flow. This happens because the flow speed would have to
increase to compensate for the reduced hydraulic diameter.
The gob region can be scaled by using similar permeability values of actual gobs. An
indication that the flow in the gob is behaving as expected in the scaled model is the airflow
leakage across the longwall face. For two rows of gob modules, the physical scaled model has
presented a flow leakage in the upper limits of what is observed in a longwall mine. Once the
longwall face advances and the gob region expands, the overall permeability of the gob will
decrease, and so will the leakage.
For a bleeder ventilation scenario, gas experiments proved that EGZs might exist in the gob
region as the air travels towards the bleeder fan. In the longwall face, an EGZ is more likely to
73 |
Colorado School of Mines | exist closer to the tailgate since there will be less ventilation due to airflow leakage to the gob.
Both these findings are consistent with in-mine observations as well as CFD modeling. The
reduced ventilation towards the tailgate might be insufficient to dilute methane from the coal
face.
The shearer location also impacts the airflow in the longwall mine. The shearer acts as an
obstacle, creating flow separation in the longwall face. This flow separation affects the overall
ventilation at the cutting location. The scaled model verified that EGZs in the longwall face are
likely to exist at the airflow boundary layer with the coal face, confirming the methane ignition
risks during coal cutting. To prevent ignition hazards in these circumstances, it is recommended
to maintain airflow quantites at the longwall face on sufficient levels to adequately dilute
methane. Ignition control measures, such as water sprays at the cutting drums, are recommended
to prevent methane ignition during coal cutting.
Future work
On the instrumentation side, a recommendation for future work is to continue the expansion
of gas sensors in the physical model and flow sensors in the gob region. The complexity of
understanding flow direction will increase as the gob region expands, requiring more sensors to
capture the flow with more accuracy. A setup that includes at least an additional flow sensor per
location in the gob can make it easier to understand the flow behavior under increased
complexity.
Expanding the gas sensing capabilities of the physical model in the bleeder entries, headgate
portions of the gob, and the airways at the tailgate are recommended. This expansion will allow
the physical model to measure the simulated methane in currently impossible locations. Gas
sensing capabilities at the tailgate airways will be necessary when simulating a U-type
ventilation system.
On the experimental side, it is recommended to continue the experiments with bleeder
ventilation with a longer face advance. This will be important when evaluating how a more
extended gob region can affect the flow leakage across the longwall face. Also, researchers
should test the U-type ventilation scenario with and without back return and compare the
experimental data to CFD models and in-mine measurements. Researchers can provide insights
74 |
Colorado School of Mines | ABSTRACT
Open-pit mines must be designed to develop the Earth’s natural resources in the most
responsible, sustainable, and economic way. Traditional mine planning optimization methods do
not consider operational constraints; such as minimum mining width or minimum pushback width
constraints, and often do not generate realistic, actionable designs. This dissertation develops
techniques to incorporate operational constraints into open-pit mine planning which allows for
engineers to more accurately convert mineral resources into mineral reserves and better evaluate
the economic viability of open-pit mining projects. A major practical challenge is that the
resulting mathematical models are very large, with potentially hundreds of millions of variables
and constraints. Addressing this challenge and delivering tools which are usable on real-world 3D
datasets requires a theoretically motivated and computationally grounded approach.
The first contribution of this dissertation is an efficient implementation of the pseudoflow
algorithm for the well known ultimate pit problem. Modest theoretical improvements and
practical computational improvements combine to create a fast and efficient open source ultimate
pit optimizer, called “MineFlow,” which is more performant than all evaluated commercial
alternatives. A model with sixteen million blocks which takes over three minutes to solve with a
commercial ultimate pit optimizer is solved in nine seconds with this implementation.
The second contribution is a formulation and methodology for the ultimate pit problem with
minimum mining width constraints. These operational constraints restrict the shape of the
ultimate pit in order to provide suitably large operating areas which can accommodate the large
machinery used in open-pit mining. This problem is shown to be -complete and several
NP
optimization approaches are developed in order to compute high quality results for large block
models in a reasonable amount of time. The two most effective approaches use Lagrangian
relaxation and the Bienstock-Zuckerberg algorithm which are modified for this problem.
Moreover, the formulation is extended to open pit direct block scheduling problems with
operational constraints and solved using a newly developed method based on the
Bienstock-Zuckerberg algorithm. This approach is applicable to large, realistic, open-pit planning
problems that span multiple time periods and multiple possible destinations for each block.
iii |
Colorado School of Mines | Figure 2.14 Cleaning an ultimate pit with mathematical morphology as with the Maptek
Vulcan automated pit designer. The initial planar section on the left is
moderately cleaned (middle), and aggressively cleaned (right). . . . . . . . . . . 48
Figure 3.1 Left: a network which is a tree with associated terminology. Right: a network
which is a directed acyclic graph. Figure adapted with permission from
Deutsch, Da˘gdelen, and Johnson 2022 . . . . . . . . . . . . . . . . . . . . . . . 63
Figure 3.2 An example network flow model. Source S and sink T nodes are labeled.
Numbers on arcs indicate ‘flow’ / ‘capacity’. The bolded arcs show a possible
augmenting path. Figure adapted with permission from Deutsch, Da˘gdelen,
and Johnson 2022 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Figure 3.3 The four different possible s-t cuts for the network in Figure 3.2. Numbers on
arcs are the arc’s capacity. The cut-set arcs are bolded and the total cut
capacity is shown below each cut. Figure adapted with permission from
Deutsch, Da˘gdelen, and Johnson 2022 . . . . . . . . . . . . . . . . . . . . . . . 64
Figure 3.4 Left: The maintained pseudoflow on the flow network is notated with numbers
on arcs for the flow, and numbers within nodes for excesses or deficits. Right:
Thick arcs are a part of the normalized tree, and dotted arcs are not. Gray
nodes are strong, white nodes are weak. Figure adapted with permission from
Deutsch, Da˘gdelen, and Johnson 2022 . . . . . . . . . . . . . . . . . . . . . . . 67
Figure 3.5 Left: The input ultimate pit problem. Right: The initial normalized tree,
letters near nodes are the node names and not a part of the notation. . . . . . . 68
Figure 3.6 Left: The starting network, circled numbers on nodes indicate the order of
merging arcs in this example. Right: The result of the first merge between
nodes b and e. Figure adapted with permission from Deutsch, Da˘gdelen, and
Johnson 2022 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Figure 3.7 Left: The result of the second merge between nodes b and f. Right: The
result of the third merge between nodes a and c. Figure adapted with
permission from Deutsch, Da˘gdelen, and Johnson 2022 . . . . . . . . . . . . . 72
Figure 3.8 Left: The result of the fourth merge between nodes a and d. Right: The result
of the fifth merge between nodes a and e which requires splitting on the arc
between b and e. Figure adapted with permission from Deutsch, Da˘gdelen,
and Johnson 2022 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Figure 3.9 Left: The modified example with a different sequence of merges as numbers
on arcs. Right: The network after three merges. Labels are given as numbers
next to the node names. Figure adapted with permission from Deutsch,
Da˘gdelen, and Johnson 2022 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Figure 3.10 An example slope definition with six azimuth slope pairs. Linear and cubic
interpolation for unspecified directions is shown.. . . . . . . . . . . . . . . . . . 78
xi |
Colorado School of Mines | Figure 3.11 An example minimum search pattern for 45° slopes and a maximum vertical
offset of 9 blocks. Numbers in cells are the z offset. . . . . . . . . . . . . . . . . 79
Figure 3.12 Left: The ‘one-five’ precedence pattern extended 30 blocks vertically. Right:
the true set of antecedent blocks for a single base block . . . . . . . . . . . . . . 80
Figure 3.13 The slope accuracy of several minimum search patterns when used for block
models with the indicated number of benches. . . . . . . . . . . . . . . . . . . . 82
Figure 4.1 Manually contouring a block section. Shaded blocks are included in the
ultimate pit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Figure 4.2 Example minimum mining width constraint templates . . . . . . . . . . . . . . 94
Figure 4.3 Left: the original ultimate pit. Right: the ultimate pit calculated with
minimum mining width constraints. . . . . . . . . . . . . . . . . . . . . . . . . . 95
Figure 4.4 Inset from Figure 4.3. Left: the original ultimate pit. Right: the ultimate pit
calculated with minimum mining width constraints. Most changes required by
the minimum mining width constraints are at the bottom of the ultimate pit. . 95
Figure 4.5 Small example mining width configuration with three variables and three
minimum mining width constraints consisting of two variables each . . . . . . . 98
Figure 4.6 Left: An inoperable configuration of blocks permitted by the original
formulation. Right: Another inoperable configuration . . . . . . . . . . . . . . . 100
Figure 4.7 A graphical depiction of the two-dimensional ultimate pit problem with
minimum mining width constraints. The input economic block value section
(top left) is shown with three different solutions with mining widths of size
one (the original ultimate pit problem, top right), two (bottom left), and
three (bottom right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Figure 4.8 Example input transformation from an economic block value model (left) to
the cumulative value model (right). Note the extra row of ‘air’ blocks and the
change of indices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Figure 4.9 A very small example ultimate pit problem with minimum mining width
constraints. Left: the seven block variables and two auxiliary variables. Right:
The block values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Figure 4.10 The small example’s corresponding flow problems. Left: With a dual
multiplier of 0. Right: With a dual multiplier of 15.. . . . . . . . . . . . . . . . 117
Figure 5.1 Synthetic 2D scheduling dataset. Left: the economic value of each block,
darker is higher. Right: A block schedule consisting of three phases . . . . . . . 140
xii |
Colorado School of Mines | Figure 5.2 Example block schedule for the synthetic 2D scheduling dataset that satisfies
a maximum sink rate operational constraint . . . . . . . . . . . . . . . . . . . . 140
Figure 5.3 Example block schedule for the synthetic 2D scheduling dataset that satisfies
a minimum mining width constraint of six blocks . . . . . . . . . . . . . . . . . 141
Figure 5.4 Example block schedule for the synthetic 2D scheduling dataset that satisfies
a minimum pushback width constraint of six blocks . . . . . . . . . . . . . . . . 142
Figure 5.5 A small example block model used to illustrate the BZ subproblem . . . . . . . 148
Figure 5.6 The base precedence constraints and block nodes in the BZ subproblem.
Source and sink arcs are omitted. . . . . . . . . . . . . . . . . . . . . . . . . . . 148
Figure 5.7 BZ Subproblem with collapsed destination nodes and two minimum mining
width constraints. Source and sink arcs are omitted. . . . . . . . . . . . . . . . 149
Figure 5.8 Planar sections through three schedules computed with Da˘gdelen’s data. Left:
No operational constraints, Middle: 2 2 minimum mining width constraints,
×
Right: 3 3 minimum mining width constraints. Top: Bench five, Middle:
×
Bench three, Bottom: Bench one . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Figure 5.9 A 3D overview of the McLaughlin area of interest. Left: Original topography.
Right: Na¨ıve ultimate pit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Figure 5.10 Cross sections through the McLaughlin block schedules, lighter blocks are
mined earlier. Top: No operational constraints, Bottom: 5x5 minimum mining
width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Figure A.1 Example simplified operational ultimate pit decision problem in the plane.
Numbers in blocks are the EBV. If the requested lower bound is less than 28
the set of shaded blocks on the right is a valid selection corresponding to a
‘yes’ answer to the decision problem. . . . . . . . . . . . . . . . . . . . . . . . . 177
Figure A.2 The wire concept in the -completeness proof. Each of the two solutions
NP
(left and right) have the same value and correspond to the two possible
assignments (true and false). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Figure A.3 The wire in Figure A.2 as a schematic instead of explicit block values. Bolded
arcs correspond to mined mining width sets in the equivalent solutions. . . . . . 179
Figure A.4 A negation ‘wiggle’ incorporated into a wire. The top wire has exactly two
parities (only one is shown) that appears the same on both sides. The bottom
wire also has exactly two parities (again only one is shown), but the parities
appear different on either side of the wiggle. . . . . . . . . . . . . . . . . . . . . 180
xiii |
Colorado School of Mines | Figure A.5 A configuration demonstrating how to cross wires over other wires. The extra
negative valued blocks next to the circled crossover points ensure that the
parity of each wire is retained. Only two of the four possible solutions are
shown alongside the relevant schematic. . . . . . . . . . . . . . . . . . . . . . . 181
Figure A.6 An example clause point consisting of three small wires brought close together
around the circled clause point. The shown solution is suboptimal, and does
not capture the one unit of value available in the circled clause point. . . . . . . 182
Figure A.7 The seven optimal solutions for the simple clause point example
corresponding to the three cases where one variable is true (top row), three
cases where exactly two variables are true (middle row), and when all
variables are true (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Figure A.8 An overview of the construction used to transform a 3-SAT problem into an
ultimate pit problem with minimum mining width constraints.. . . . . . . . . . 184
Figure B.1 An example pit by pit graph with the selected, evenly spaced, pushbacks
indicated with darker bars.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
xiv |
Colorado School of Mines | ACKNOWLEDGMENTS
I have led a very privileged and fortunate life which has allowed me to pursue this
undertaking. This good fortune has extended to a great many wonderful people in my life which I
would like to acknowledge and thank here.
Special thanks are dedicated to my advisors: Dr Kadri Da˘gdelen and Dr Thys Johnson who
both provided guidance and assistance throughout the process of preparing this dissertation. I
also greatly appreciate the contributions and considerations from Dr Alexandra Newman, Dr
Dinesh Mehta, and Dr Rennie Kaunda. All of these individuals contributed not only ideas and
advice relating to the work itself, but also important life lessons.
I would also like to thank Dr Marcelo Godoy, Larry Allen, Dr Conor Meagher, and Dr Ryan
Goodfellow at the Newmont Mining Corporation who provided many useful discussions and the
majority of the funding for the work contained herein. Bill Blattner, Bruce Ramsay, Dr Keith
Ramsay, and Rob Hardman at Maptek also deserve special thanks for their advice and for
generously allowing me to split my time between developing software and pursuing this effort.
My good fortune has also extended to my family and friends. Thank you to Pauline, Clayton,
Jared, and Kelsey for your encouragement, advice, and support. Thank you to my friends for the
much needed distractions and emotional support.
However my greatest thanks are owed to my wife Amanda and my children Philip, Isaac, and
Lilah. They had to put up with the most stress and trouble to make this whole thing a reality
and despite that they maintained a positive attitude and provided never-ending love and support.
Especially Amanda whose tireless efforts and self sacrifice throughout this endeavor can not be
overstated. Thank you.
xvii |
Colorado School of Mines | CHAPTER 1
INTRODUCTION
1.1 Problem Setting and Background
Our society is built on natural resources, many of which are extracted from the Earth via large
scale mining operations. It is of the utmost importance that these limited resources are developed
in a responsible and sustainable way such that future generations will be able to enjoy a quality of
life that is the same, or better, than our own. The mining companies which explore for, plan, and
ultimately develop these natural mineral deposits require suitable mathematical tools and
techniques in order to make the best decisions possible. Better, and more informed, decisions in
the planning process will lead to better plans that yield an improved return on investment and
also provide more benefit for local communities and accommodate our growing global needs.
In recent decades the global trends in mining have been towards mining larger and lower grade
deposits. The demand for metal is increasing and improvements in all aspects of the mining and
mineral processing chain have made these previously uneconomic deposits viable. Underground
mining, especially through large scale bulk mining methods such as block caving, are increasingly
being chosen for green and brownfield projects alike. However, in many situations surface mining
is still preferred due to lower capital and operating costs, among other considerations.
The field of operations research is a vital part of mine planning. The techniques provided by
operations research allow practitioners to form mathematical models of the complex engineering
problems that they face and analyze them in order to gain necessary insight. The solutions
provided by operations research guides mining engineers in navigating the complex operational,
geological, financial, environmental, and social challenges in mining. In this dissertation these
tools are developed further in order to provide additional insight, tackle previously
under-represented operational constraints, and provide necessary improvements in the field of
long range open-pit mine planning which are applicable to operating and future surface mines.
A mathematical model is only useful in so far as it represents the real underlying decision
problem, and how well the resulting analysis can be used. A poor model can often do more harm
than good, and unrealistic or inappropriate assumptions in the modeling process can mislead
1 |
Colorado School of Mines | engineers and practitioners, sometimes drastically. For this work, the emphasis is on more
accurately representing underlying decision problems and more closely matching the mathematical
models with the realities of open-pit mining. Specifically operational constraints, or constraints
that must be imposed on the plan because large equipment is used, are considered herein.
The unfortunate reality, however, is that increased accuracy generally leads to increased
complexity and can limit the applicability to smaller and less useful models because the
computational process of solving for the correct answer takes an inordinate amount of time.
Therefore, special emphasis has been placed on developing techniques that are computationally
efficient and are directly usable with full sized, realistic, 3D datasets from real mining operations.
1.2 Goals and Outline
The goal of this dissertation is to develop high quality and computationally efficient models
which addresses the most fundamental operational constraints in open-pit long range mine
planning. The models must be correct, flexible, and applicable to a wide range of real world
deposits. To achieve this goal the dissertation is structured as follows.
Chapter 2 reviews relevant literature and establishes the necessary prerequisites in the fields of
mine planning and operations research. The focus is on long range mine planning problems,
specifically the ultimate pit limit problem and the block scheduling problem. Current approaches
to incorporating operational constraints are discussed.
Chapter 3 documents a series of improvements to the foundational ultimate pit problem. The
general network flow based pseudoflow algorithm is dissected and reimagined solely for the
ultimate pit problem. Small optimizations are presented with a dramatic impact on the
computational performance of the algorithm. This provides the ability to solve hundreds or even
thousands of ultimate pit problems extremely rapidly. A novel notation is developed for the
pseudoflow algorithm with moderate pedagogical value, and a novel means by which the accuracy
of a given precedence pattern can be computed is presented. Finally, Chapter 3, concludes with a
computational comparison between this new, and permissive open-source, implementation of the
pseudoflow algorithm and long standing commercial alternatives.
Chapter 4 extends the ultimate pit problem to consider minimum mining width constraints
which are conventionally addressed late in the mine design process by hand. Incorporating these
2 |
Colorado School of Mines | CHAPTER 2
OPEN-PIT MINE PLANNING: REVIEW AND ALGORITHMS
Mining is responsible for providing many of the raw materials necessary to build and maintain
our civilization. The design, planning, and scheduling of mining operations is critically important
to maintaining our way of life and providing the building blocks of our society. The mine design
process helps determine the economic viability of mining projects around the globe and provides
plans which guide the operation through development, production, and reclamation. This is an
extremely difficult and complex problem because it is both very large with many different
concerns including geological, financial, environmental, and social; and contains many different
interdependent systems and interactions.
To contend with this complexity mining engineers and technologists in the mining industry
use a wide range of techniques from operations research and other fields in order to make the best
decisions possible. The concepts and prior work described in this chapter provide the necessary
background and the basis for the new methods introduced in the subsequent chapters.
Section 2.1 reviews the broader field of open-pit mine planning including its main objectives,
modeling, and the modern approach to open-pit mine planning. The circular nature of mine
planning is introduced and several of the subproblems are discussed. The subproblems most
relevant to this dissertation are then described in the following sections.
Section 2.2 discusses the foundational ultimate pit problem which helped launch the field of
computational open-pit mine planning. The ultimate pit problem has a well defined mathematical
structure which has afforded multiple different exact solution techniques and heuristics which are
described herein. The max-flow / min-cut approach to identifying the ultimate pit limit is
described in detail as it is used extensively when considering operational constraints and is
further developed in Chapter 3.
Where the ultimate pit problem is concerned only with what is mined throughout the life of
the mine, the block scheduling problem concerns itself with when to mine it and how to process
the mined material. Section 2.3 discusses the block scheduling problem where the ultimate pit is
divided into a set of smaller nested pits which are extracted in sequence. These nested pits are
4 |
Colorado School of Mines | also referred to as pushbacks, phases, or cutbacks and are chosen to maximize the net present
value (NPV) of the mine while adhering to additional constraints such as minimum and
maximum mill grades, blending, stockpiling and more.
Section 2.4 presents prior work relating to extending the ultimate pit problem and the block
scheduling problem to consider operational constraints. Special attention is given to extensions
which consider minimum mining width constraints as many are implemented and considered in
Chapter 4.
Section 2.5 presents the necessary background around mathematical programming and
optimization. The Lagrangian relaxation approach to dualizing certain constraints is given special
attention as it is used later in the dissertation to accommodate operational constraints.
2.1 Open-Pit Mine Planning
Open-pit mines are characterized by mining deposits which are near-surface using horizontal
benches and dumping waste material outside of the pit limits. Deeper deposits are generally more
well suited to underground methods which can leave much of the waste undisturbed by using a
variety of methods to support overlying rock and extract material through shafts or declines.
Underground mines generally have higher development costs, longer start-up times, and are less
preferable to open-pit mines all else being equal [1]. Quarries are distinct from open-pit mining in
that a quarry extracts aggregate material or dimensioned stone instead of selectively extracting
ore, and strip-mines depart from open-pit mining by operating with larger, shallower footprints
and depositing the waste in previously mined panels [2].
Production rates in excess of 100 MT/year are achievable with open-pit mines, where even the
largest block caving underground mines rarely exceed 30 MT/year [3]. This high production is
possible in part by using larger equipment and many concurrent working areas which are more
difficult in confined underground workings which also have to contend with complex ventilation
requirements. Open-pit mines are well suited for extracting industrial metals including iron ore
and copper, along with precious metals such as gold so long as the ore bodies are near enough to
the surface to maintain an operable stripping ratio. The high production rates and lower mining
costs allows even low-grade porphyry copper deposits to be economical with open-pit techniques.
5 |
Colorado School of Mines | An operating open-pit mine has several main features which are in place to accommodate the
large equipment and mining operations. The pits themselves are large holes in the ground which
are divided into many horizontal benches. These benches typically range in height from 10 to 20
meters which depends on geotechnical considerations, the equipment being used, and the deposit
itself. Some benches, which are currently being excavated, are called working benches and must
be of a sufficient size to host the necessary equipment. The pit also maintains at least one large
haul road and several smaller access roads. These roads provide necessary access to the working
benches and a means by which material is transported to the relevant processing facilities or
waste dumps using haul trucks.
Open-pit mines generally extract much more overburden and waste than ore which has to be
put somewhere. It is important to handle this waste material as little as possible and great care is
generally taken to avoid having to re-handle it. The waste, by definition, is not processed for
revenue but may require processing to remove deleterious elements or avoid issues associated with
acid rock drainage or other concerns. Waste is typically stored in large waste dumps which, at
least in the early stages of an open-pit mine, are located outside of the pit limits.
Different ores have different processing requirements that may involve many unit operations
including comminution, classification, and concentration. The facilities for these operations may
be on the mine site, or accessed by transporting the raw bulk material or intermediate products
by rail, large conveyor systems, cargo ships, or other bulk transportation methods. If the facilities
are on site their by-products may also have to be stored on site or processed to avoid untenable
environmental impacts. Many mineral processing plants generate tailings; small particles which
have been separated from the elements of interest and are mixed with water and other processing
reagents. These tailings are stored within large tailings storage facilities on site which have to be
carefully engineered and located, as moving them later is essentially impossible.
Historically open-pit mine planning was completed by hand, with hand drawn sections and
maps. This approach relied on trial and error, tedious hand calculations, and manual smoothing
between cross sections [4]. Many decisions during this era of mine planning were based on visual
criteria that emphasized operational concerns or rules of thumb. Alternative mine designs were
expensive and time consuming to consider so only a very small number of trials or alternatives
could be generated.
6 |
Colorado School of Mines | Following the development of computers and the acceptance of computerized methods for
mine planning many of the traditional limitations were lifted. Computers are capable of
evaluating many different designs and solving different subproblems that are infeasible to solve by
hand. In general, operations research has proven to be one of the most important tools for the
modern open-pit mine planning engineer and a great deal of research has been completed in this
area [5]. The following sections describe much of the modern, computerized, approach to open-pit
mine planning including its objectives, the mine planning process, mathematical modeling, and
the most relevant subproblems.
2.1.1 Objectives of Open-Pit Mining Operations
The objective of an open-pit mining operation is to extract and process relevant material in
the most cost effective and economic way possible. Generally the most commonly accepted
objective is to maximize the net present value (NPV) of the project which accounts for the time
value of money and rewards generating profit as early as possible typically by reducing early
capital costs and mining high value material quickly. However the NPV of a project is not
all-encompassing and modern mining projects are increasingly designed to be developed in a
sustainable manner that responsibly considers the social and environmental impact of the
operation both locally and globally. Some mines may be developed or operated for reasons other
than purely economic; such as for social reasons or as a matter of national security. Even in these
circumstances decisions are generally still made based on economic criteria [6].
The net present value of a project is calculated as follows:
n
(cid:88) R t
NPV = (2.1)
(1+i)t
t=1
R is the net cash flow at time t, i is the discount rate, and n is the number of time periods.
t
The discount rate is generally a difficult parameter to define and is often dictated by corporate
policy and other concerns. There are valid criticisms of the NPV approach to mine design -
especially when the project may have long-term benefits which can be hard to quantify [7].
Additionally it is often common in the mining industry to use a relatively high discount rate such
that the denominator in each term of Equation 2.1 grows rapidly and the cash flows in later years
7 |
Colorado School of Mines | become much less meaningful, which may not be desired.
It is imperative for the open-pit mine plan to operate in line with all relevant legal,
environmental, and social requirements. These objectives generally take the form of constraints
which necessarily reduce the net present value. Indeed, all secondary objectives can only lead to a
decrease in NPV which should always be quantified and understood [8].
Mine planning is a field which is rife with uncertainty. The typical open-pit mine plan samples
a minuscule fraction of the deposit prior to mining which can lead to large unknowns with respect
to the underlying deposit geometry and grades. The mine planner does not know with a high
degree of precision how much ore there is, or its exact makeup. Additionally, the fundamental
economic quantities such as commodity price and external costs are uncertain. The value of these
parameters are dictated by external markets and may change drastically during the decades that
the mine is operating. Some mines are located in countries where political upheaval represents a
substantial risk. Addressing uncertainty explicitly, instead of simply being conservative, is
preferred [8]. This may involve incorporating risk into the objective, such as using a
Risk-adjusted NPV or incorporating a maximum allowable level of risk as a constraint.
2.1.2 Mine Planning Process
The design of open-pit mines is characterized by a cyclical process of assuming, planning,
evaluating, refining, and then revisiting the original assumptions. It is not feasible to design an
entire open-pit mine plan straight through because there is no way to make a single starting
decision that does not depend on later ones.
Consider, for example, trying to define the ultimate extents of the open-pit. These extents are
very useful for quantifying the contained resource, and thereby the total potential revenue from
the mining operation. Additionally, the extents help inform where to place any supporting
infrastructure such as dumps, processing facilities, and tailings ponds. The ultimate extents also
give a planner a means to assess the total mined tonnage and thereby the mine life, provided that
the mining rate is known. However, it is not possible to determine the optimal ultimate open-pit
extents without knowing the mining cost which directly depends on things such as the
infrastructure locations, chosen equipment, and so on. Therefore the overall open-pit mine
planning process is a circular process which relies on slowly converging on a good mine plan
8 |
Colorado School of Mines | through repeatedly revisiting earlier decisions and assumptions [9–11].
In general an open-pit mine is planned and designed at three main levels of detail:
• Long Range Plans are typically updated yearly or at longer intervals and provide a basis for
reporting contained reserves and contain the actionable life-of-mine plan. A long range plan
gives a good handle on the mineral inventory and decisions which have a lasting impact,
such as where infrastructure is placed and any major operational changes that may require
large capital expenses. Long-term price forecasts are considered and a higher degree of
uncertainty is accepted. In a long range plan the optimization problems are concerned with
much larger volumes at a lower level of precision.
• Medium Range Plans are updated at a higher frequency than long range plans and generally
dictate the operation on a monthly or quarterly basis. A medium term plan would only be
concerned with short-term price forecasts and decisions that impact the more immediate
operation such as blending or stockpiling concerns. Medium range plans may look at
extraction sequences on a monthly or quarterly level with consideration for equipment
allocation or shorter term haul roads.
• Short Range Plans consider smaller volumes of material with higher precision and greater
frequency. Short range plans may be developed weekly or to dictate a single day’s
operation. They consider the current state of the mine, the plant, and immediately available
blast hole data to decide where material should be routed to maximize profit and maintain
appropriate blending. The decisions made at this timescale are irreversible and have
immediate impacts on the operation.
The information effect is an important aspect of open-pit mine planning. The long range plans
are developed with different data availability than short range plans, as new data is constantly
being collected and incorporated throughout the mine life. This new data allows short range
plans to operate with less uncertainty, but flexibility is greatly reduced as it is not possible to
revisit earlier decisions at this stage.
9 |
Colorado School of Mines | 2.1.3 Open-Pit Mine Modeling
An open-pit mine model is a mathematical representation of a real-world mine developed to
aid in decision making and provide insight. These models depart from the real world in many
ways to make them more manageable and more useful. Models are designed in conjunction with
geologists, geostatisticians, geotechnical engineers, and others. These models are then used for a
wide range of tasks described in Section 2.1.4.
In this section we describe the general character of open-pit mine models, how they are
constructed, the typical variables present in a workable model and the relevant global parameters.
The most fundamental set of geometric constraints to open-pit mine modeling, the precedence
constraints, are also described.
2.1.3.1 Model Framework
The primary type of model used in modern open-pit mine planning is the block model. A
block model divides the area of interest into a set of non-overlapping volumes, called blocks,
which are then populated with a wide range of different attributes. Regular block models are
constructed from blocks all of the same shape and size arranged in a regular rectangular grid,
however irregular block models are also used which may have blocks of different shapes or sizes.
The regular block model is the most common and consists of a few main parameters.
• The origin, which is a point in space defined relative to some datum or at some site specific
location. Commonly set to the lowest, leftmost, frontmost point of the model but may be at
any of the other 8 corners of the model. The origin is either at the centroid of the block or
its outside corner. In Figure 2.1 the origin is denoted with the dot and notated o ,o ,o .
x y z
• The block size or, equivalently, the block spacing. This parameter defines the shape of each
individual block and is chosen to balance a few concerns. The vertical size of the blocks are
typically linked to the bench height, on the range of 10 to 20 meters [12], and the horizontal
dimensions are generally similar to maintain a roughly cubical aspect ratio. In Figure 2.1
the block sizes are notated s ,s ,s .
x y z
• The number of blocks along the 3 coordinate axes. This, along with the block size and
origin, controls the extent of the block model. In Figure 2.1 the number of blocks is notated
10 |
Colorado School of Mines | n ,n ,n .
x y z
• The block model orientation, often a single azimuth angle, allows for the model to rotate
around the z axis and not be aligned with the underlying coordinate system in order to
more closely follow the local geology. Block models are very rarely rotated again around the
x or y axes.
n
z
o ,o ,o
x y z s
z
z n
y
y
s
x n x s x y
Figure 2.1 The structure and parameters of a regular 3D block model
Each block in a regular block model is often considered a selective mining unit or SMU, which
corresponds to “The smallest volume of material that can be selectively extracted as ore or
waste” [13]. However, in Section 2.4 we see evidence that open-pit mine planners do not typically
allow for mine plans to treat single blocks, or even a handful, as extractable. Instead most mine
plans require a small contiguous group of blocks to be extracted all as one.
The primary reason that most mines do not scale the blocks themselves larger to avoid this
issue is that you lose out on modeling precision, you may have difficulty recreating the precedence
constraints discussed in Section 2.1.3.4, and there may be costly aliasing effects. Larger blocks
include additional dilution which may significantly impact the valuation of a deposit and mine
plan. If that dilution is in excess of the bare minimum dilution required due to a mine’s operating
parameters, it could lead to under-estimating the value of a deposit and making sub-optimal
decisions or incurring substantial opportunity costs.
11 |
Colorado School of Mines | On the other hand, if a modeler elects to reduce the block size in order to more precisely
represent the underlying geology they face two major challenges. The first is that it is much
harder to accurately estimate smaller block sizes and the uncertainty inherent in these estimates
may not be properly understood or accounted for in downstream applications. The second is
more practical; as you reduce the size of each block you must also increase the number of blocks
in order to retain coverage over the area of interest which can quickly overwhelm downstream
algorithms and procedures. Reducing the size of blocks in half along each axis leads to an eight
times increase in the number of blocks required.
Regular block models generally lead to computationally efficient algorithms and are well
suited to the procedures in the following sections. For example, regular block models do not need
to store every block’s individual coordinates as they can be calculated rapidly from the block’s
indices. If the model is organized such that 0th block is the lowest, frontmost, leftmost block and
the blocks are numbered sequentially first along the x direction, then y, and finally z the block’s
one dimensional index i is given as:
i = i n n +i n +i (2.2)
z y x y x x
· · ·
Where the number of blocks in the x, y, and z directions are denoted n , n , n respectively.
x y z
The indices along the x, y, and z directions are similarly denoted i , i , i . The individual indices
x y z
can be recovered from the one dimensional index i using the following three equations using
integer division (truncating the fractional part) and where % is the modulo operator which yields
the remainder following division.
i = i % n (2.3)
x x
i
i = % n (2.4)
y y
n
x
i
i = (2.5)
z
n n
x y
·
Regular block models, with their efficient and consistent structure, are well suited for
accurately representing precedence constraints (Section 2.1.3.4), and minimum mining widths
(Section 2.4). However, there are downsides to regular block models. They generally include a lot
of superfluous information especially around the edges and the lower benches which wastes space
12 |
Colorado School of Mines | and slows processing. Additionally there may be certain deposit types for which a one size fits all
approach is not warranted.
Irregular block models are much more varied and flexible than regular block models and
therefore more complex. Sub-blocked models retain rectangular prism shaped blocks but vary the
sizes of individual blocks to align with the underlying geometry. For example, large waste regions
may be modeled with just a few blocks that take up a lot of space. Then along a geologic contact
the blocks are made much smaller to capture the boundary. A vertical cross section through a sub
blocked model is included on the left in Figure 2.2. This form of block model is more typically
used with underground mines, but still has its place in open-pit mine modeling.
z
x
Figure 2.2 Vertical sections through two irregular block model types (left) a sub-blocked model
(right) a stratigraphic block model
A second form of irregular block model, the stratigraphic block model, is shown on the right in
Figure 2.2. This type of block model is used for stratigraphic ore bodies where there are generally
laterally expansive layers where the thickness of each layer is important to capture accurately.
There are many other different types of irregular block models including ones which deviate from
rectangular prisms and those which cover irregular non rectilinear areas. However, by far the
most common model framework is the regular 3D block model.
2.1.3.2 Model Estimation and Simulation
Once the framework, typically a regular 3D block model, is defined geologists and
geostatisticians use a wide range of data and techniques to fill that model with the relevant
information required for mine planning. Rossi and Deutsch [13], describe the four main
13 |
Colorado School of Mines | components of the mineral resource estimation process as:
• Data collection and management, which includes concerns involving drilling and sampling
theory. The truism ‘Garbage in Garbage out’ is especially relevant during model estimation
and simulation as any errors at this stage propagate through all the following stages and
lead to erroneous information which can lead to costly mistakes and opportunity costs.
• Geologic interpretation and modeling, is the process by which geologic data and
mathematical techniques are combined to create realistic geologic domains for downstream
steps. The domain model assigns rock types, or similar classifications, to areas within the
model and can be used to evaluate contained tonnage and provides the groupings within
which the downstream grade estimation and simulation techniques are applied.
• Grades assignment, involves identifying the variables of interest and filling them in around
and between the drill holes. These are generally the concentrations of different elements,
called ‘grades’, or other continuous properties. However, geometallurgical variables, which
may be nonlinear, might also be considered; which necessitates more complicated
procedures. The grade assignment process can use many different approaches that have
several practical and theoretical challenges which need to be overcome to create a accurate
and precise model free of bias or other types of error.
• Assessing and managing geologic and grade uncertainty. As previously discussed, there are
many sources of uncertainty in the mining process and some of these uncertainties can be
evaluated during model estimation. There is uncertainty in the data, the geologic modeling,
and within the grade assignment process which must be quantified and considered carefully.
Note that there may be several different models used for different purposes in mine planning
all following the above four steps. This is in part due to the information effect discussed in Section
2.1.2. For example, the new data obtained from blast holes may permit a higher resolution model
over a smaller area, such as a single blast or bench, that can be used in short term planning.
The data used for model estimation and simulation must be of good quality and free of
systematic bias. It must be representative; such that it is spatially spread out and not overly
concentrated in one area or biased to specific types of material. This can be complicated by both
14 |
Colorado School of Mines | the expense of collecting data and the desire of project managers to drill high value holes that
provide better sounding press than statistical merit. Mining operations sample a minute fraction
of the mineral deposit through different means including; trenches, drill holes; including both
reverse circulation and diamond drill holes, and test pits. The sampled material is then processed
through a variety of tests and assay procedures to quantify the relevant mineral concentrations.
Site specific data may also be inferred through geophysical methods such as magnetic surveys,
electromagnetic methods, or reflection seismology [14]. Geophysical data has different
considerations than directly sampled data, but the principles of ensuring quality control,
understanding sampling variance, and avoiding errors are the same. In all situations the data must
be correctly managed, safely stored, and verified through regular audits as all of the subsequent
steps use that data to develop realistic models which are then used for mine planning and design.
The geologic model is responsible for defining estimation domains and dictating how to pool
data together into relevant stationary zones. A mining operation is primarily concerned with
making decisions to most economically develop the deposit so the geologic model is designed to
provide the necessary information for this task. Domains identified in a geologic model may be
based on both alteration and structural geologic variables depending on the mineral deposit.
Stationarity refers to only grouping variables of like statistical and geologic properties together.
Combining data from different statistical populations leads to poorer quality estimates.
There are several techniques used to develop geologic models including; manual interpretation
on sections, indicator Kriging methods, implicit modeling, machine learning approaches and
others [15]. Each technique has different advantages, disadvantages, and areas of applicability. In
all cases the mixing of different mineralization types should be strictly minimized as this leads to
poor quality estimates that may skew resulting tonnage and minerals concentration estimates. It
is important to consider uncertainty within the geologic model as the boundaries are never
sufficiently sampled to give an absolutely precise delineation between zones. Some methods can
consider uncertainty directly; as is the case with indicator simulation or using the so-called
‘uncertainty parameter’ within implicit modeling, however others may require additional effort.
This information has to be accurately communicated to the downstream mine planning engineer
and other model users.
15 |
Colorado School of Mines | Once a domain model has been developed, audited, and verified it is necessary to estimate
mineral concentrations and other variables of interest within the identified domains. The variables
in mineral deposits exhibit spatial dependence because they are formed through natural processes
following well defined, if not completely understood, rules [16]. It is valuable to describe this
spatial dependence, typically through a variogram or similar measure, in order to inform the
estimation process. Earlier, less sophisticated, estimation techniques, such as nearest neighbor or
inverse distance, do not incorporate a variogram directly.
Kriging is a method for estimating continuous variables using a direct model of spatial
variability that minimizes the expected error variance [13]. There are several types of Kriging
with varied levels of applicability to different models including simple Kriging, ordinary Kriging,
Kriging with a trend or external drift model. Each approach is selected based on the geologic and
statistical nature of the domain, data availability and the variable of interest [17]. Regardless of
the method selected, modelers must justify their choice and perform various checks including
cross validation, and visual and statistical verification. Additionally models should be calibrated
to other data sources such as production data if possible.
Of extreme importance to the entire modeling process is the understanding that the models
are subsequently used to make specific mine planning decisions. The technical nature of the
model and the different considerations must be accurately transmitted to the downstream
consumers of the models, including considerations such as how dilution has been handled and how
to use each variable in the model.
As introduced in Section 2.1.1, uncertainty is inherent in the mine planning and geologic
modeling process. A typical mining operation only samples a minuscule fraction of the mineral
deposit before development. The estimation techniques introduced above provide a single,
hopefully high quality, estimate for the unsampled locations but this averaging process does not
explicitly consider the geologic uncertainty. Simulation is a process by which multiple geologic
realizations can be generated from the input data and, when taken as a whole, can aid in
understanding. One major advantage of simulation is demonstrated in Figure 2.3, where we see
how multiple realizations can be used to create multiple plans which, when taken as an ensemble,
provide information on multiple possible outcomes.
16 |
Colorado School of Mines | Single Estimate Single Plan Single Response
Multiple Realizations Multiple Plans Multiple Responses
Figure 2.3 Applying mine planning to estimated and simulated models. A single response
provides no information on uncertainty. Multiple realizations can provide more information
A growing challenge in mine planning is to properly account for the geologic uncertainty.
However, the workflow implied by Figure 2.3 is not the ultimate ideal. Instead of determining the
best mine plan for a specific realization or even the best plans for multiple realizations it is
preferred to create the best mine plan in the presence of uncertainty.
2.1.3.3 Typical Model Variables and Global Parameters
An open-pit mining operation is planned and developed in response to a wide range of model
variables and global parameters. A model variable is a metric that varies by location throughout
the mining area and is typically stored within the block model described earlier. Parameters do
not vary by location however they may vary by time. All inputs including model variables, global
parameters, and other information are uncertain and considered as either estimates or as a part of
some potentially unknown distribution.
At minimum a geologic model for mine planning consists of a usable rock type model, density,
relevant mineral concentrations, and a resource classification. More sophisticated models may
include additional variables such as geometallurgical indices, or additional information that can
help guide the mine planning process.
The most impactful global parameter on the open-pit mine planning process is the commodity
price [8]. This parameter can singlehandedly make or break a mining operation and
17 |
Colorado School of Mines | fundamentally alters the way that the mine operates including dictating the optimal production
rate and the final pit limits. Unfortunately the commodity price is also one of the most uncertain
parameters and depends on many factors outside of the mining operation’s control. To combat
this uncertainty mines may negotiate longer term contracts or use a variety of other economic and
financial means to try and avoid disruptive price drops and still be able to take advantage of price
increases.
The planning engineer typically designs multiple plans and schedules based around at least
three different price forecasts: a middling price forecast which aligns with the general expectation,
a low price forecast which is pessimistic, and a high price optimistic forecast. In certain
circumstances many more price forecasts can be considered which are generated using a
simulation process or a method akin to random walks as shown in Figure 2.4.
2,800
2,600
2,400
2,200
2,000
1,800
1,600
1,400
Historical Price Price Forecasts
1,200
0 100 200 300 400 500 600 700 800 900 1,000 1,100 1,200
Time (Days)
Figure 2.4 Example price forecasts for mine planning generated by using random walks
These variables and global parameters are drivers for some the most important design
parameters of an open-pit mine planning operation [8] including:
• The total product produced which is an upper bound on any potential revenue from
processing a block and directly informs material routing and valuation. For example in a
gold mining operation the gold grade (typically measured in grams per tonne) along with
18
)$(
ecirP |
Colorado School of Mines | density and the volume of the block gives an upper bound on the potential product
available for recovery. No current processing methodology can achieve 100% recovery of the
minerals of interest, and different element interactions can be quite complicated.
• The mining costs include costs associated with drilling, blasting, loading, and hauling the
material. Mining costs are applied on a dollar per ton basis and varies by depth, and
distance to the relevant processing facility or waste dump.
• The processing costs are applied to material which is processed in, for example, a processing
plant or a leach pad. Processing costs have to include the cost of any reagents used, energy
consumed, and tailings rehabilitation.
• The material destination, or material classification, is the selected process stream for the
mined material. A given mine may have many different possible material classifications
including different streams for waste and ore of different grades and properties.
• The economic block value, or EBV, is a combination of the above design parameters which
indicates the, generally undiscounted, value of a block if it were mined and sent to the
appropriate material destination. This parameter may be provided on a per destination
basis, or assuming that the material can be processed via the most economic process.
• The pit slopes dictate the precedence constraints described in the following section and vary
based on rock type, and direction. Large scale structural geology features, such as faults
and joint sets, have a strong impact on the allowable pit slopes.
The economic block value is of utmost importance in the following chapters. At its core
economic block value is defined as revenues less costs. A simple economic block value formula
may have the following form:
v = T g r P T PC T MC (2.6)
b o o
· · · − · − ·
Where v is the economic block value for block b. T is the ore tonnage, T is the total tonnage,
b o
g is the grade, r is the recovery as a proportion of recovered material between 0 and 1. P is the
commodity price, PC is the processing cost in dollars per ton of ore, and MC is the mining cost.
19 |
Colorado School of Mines | In practice a much more complicated procedure is often used to calculate economic block value
based on a wide range of additional local and global variables and parameters [18, 19].
In Chapters 3 and 4 the economic block value variable and pit slopes are used. Chapter 5
requires additional variables including the discount rate, material classification, and others.
2.1.3.4 Precedence Constraints
Unlike underground mines, which use a variety of techniques to support the material above
the deposit, open-pit mines progress downwards, removing successive benches of material in a
cone-like shape. The slope of the cone is based on the strength of the material being mined, its
geotechnical properties, and other structural geologic factors. This slope generally varies by both
location and direction throughout the mining area as the composition of the material changes and
certain geologic features are more prone to different failure modes in different directions.
Geotechnical engineers are responsible for using field data and sophisticated modeling techniques
to determine safe pit slopes that protect individuals, equipment, and prevent slides that could
lead to injury, expensive re-handling, and other difficulties.
The geotechnical understanding of the area is used to create precedence constraints, which are
used in combination with the variables described in Section 2.1.3.3 as inputs for the techniques in
Section 2.1.4. Precedence constraints encode the physical relationships between blocks and
indicate that ‘this block cannot be mined, until all of these other blocks are mined.’ They are
generally modeled as pairwise relationships between blocks and denoted with a directed arc from
the lower, base block to the higher block. Precedence relationships are common across any
optimization or scheduling technique that operates on an open-pit mine and must be constructed
correctly and efficiently to avoid unsafe deviations from the geotechnical model.
Lerchs and Grossmann proposed two precedence schemes in their seminal 1965 paper on the
ultimate pit problem [20]. The first scheme uses an irregular model where the odd layers are offset
from the even layers by half of the block size; which is an uncommon block model configuration.
The second precedence scheme, commonly now called the 1:5 pattern, connects each block to the
five blocks immediately above it in a cross pattern which is locally precise but globally inaccurate.
When the 1:5 pattern is applied with larger block models and over several benches it creates
diamond shaped pits that deviate substantially from 45° in the off axis directions.
20 |
Colorado School of Mines | To account for the deviations from 45° in the off axis directions, Gilbert in 1966 developed the
1:5:9 pattern which varies the precedence scheme by level [21]. The 1:5:9 pattern alternates
between connecting every block to the five above it in a cross pattern and connecting every block
to the nine above it in a square. Lipkewich and Borgman introduced the “Knight’s move” pattern
in 1969 which connects each base block to the five immediately above it in a cross and eight more
blocks two benches above that are offset by a knight’s move as in chess [22]. Note that both of
these patterns only approximates 45° when the block model has isometric block sizes. These small
precedence patterns are shown in Figure 2.5.
A) B)
z
y C) D)
x
Figure 2.5 Small precedence schemes. (A, B) from Lerchs and Grossmann, (C) the “1:5:9” scheme
from Gilbert, (D) The “Knight’s Move” scheme from Lipkewich and Borgman [20–22]
Chen, in 1976, apply variable slope angles and varying cone templates across the deposit to
more faithfully recreate the geometrical and operational constraints [23]. Pit slopes that vary by
direction require some form of interpolation between the specified directions. Researchers have
used splines, ellipsoids, and inverse distance interpolation for this purpose [24–26].
Precedence schemes must be generated so that they realistically reflect the geometrical
constraints and such that they are computationally tractable and efficient. The techniques in
subsequent chapters are sensitive to the problem size, so generating unnecessary precedence
constraints should be avoided. The “Minimum search patterns” from Caccetta and Giannini are
designed to accurately recreate arbitrary slope requirements and consist of the fewest constraints
possible [24]. Their approach is included in Algorithm 1, and is discussed further in Section 3.2.2
alongside an efficient and open-source implementation.
21 |
Colorado School of Mines | Algorithm 1: Algorithm to generate minimum search patterns, adapted from Caccetta
and Giannini 1988[24]
Input : A list of azimuth slope pairs A: as a list of pairs defining the corresponding
azimuth and pit slope.
Input : The maximum number of benches in the output pattern N . Note: this is
z
generally less than the number of benches in the input model.
Input : The block dimensions: s ,s ,s
x y z
Output: The minimum search pattern S: as a list of 3-tuple offsets in the x,y, and z
directions.
// Maintain a list of tagged blocks, as offsets
T ;
← ∅
for l 1 to N do
z
←
for all untagged blocks on this level which violate the slope constraints do
S S this block ;
← ∪{ }
T T this block ;
← ∪{ }
for all tagged blocks do
Apply current search pattern to this block;
2.1.4 Open-Pit Planning Subproblems
An open-pit mine plan consists of several different studies at different levels of detail all with
different individual subproblems. The scope of a particular study may allow for large strategic
changes to the mine’s development or instead operate within a pre-defined strategy which was
chosen based on an earlier study. The subproblems present in the open-pit planning process are
applied as necessary to answer specific questions and inform subsequent decision making [27, 28]
Some of the most important subproblems in the open-pit mine planning process are listed below.
• The ultimate pit problem was originally described in 1965 by Lerchs and Grossmann [20]
and is concerned with determining the final limits of an open-pit mining operation such that
mining any more material would require removing so much waste that it would be
uneconomic. The majority of this dissertation is devoted to this subproblem and it is
discussed in Section 2.2 in detail and further in Chapters 3 and 4.
• Pit parameterization is the process by which many ultimate pits are constructed for further
analysis originally found in Lerchs and Grossmann’s seminal paper and further described by
Matheron in 1975 [29]. There are several techniques for pit parameterization including using
a revenue factor, cost factor, or a constant decrement on block values among others
22 |
Colorado School of Mines | infrastructure placement. The following sections discuss two of the fundamental subproblems in
more detail which are the focus of this dissertation: The ultimate pit problem, and the block
scheduling problem.
Mill Reclassified
Leach
MREV $23.88 Dump
LREV $30.08
DREV -$3.00
MREV $13.92
LREV -$3.22
MREV $165.78
DREV -$3.00
LREV $132.66
DREV -$3.00
y
x
Figure 2.6 The grade control polygon optimization problem takes a bench classification (left) and
creates an operable set of polygons (right). The operable polygons satisfy minimum mining width
constraints and maximize undiscounted cash flow
2.2 The Ultimate Pit Problem
The ultimate pit problem, [20], is an important subproblem in the open-pit mine planning
process. Its narrow focus allows it to be applied to large models and give initial results that can
be used to inform infrastructure placement, assist in equipment selection, perform parametric
analysis, and guide many other aspects of designing and scheduling a surface mining operation
[39–42]. The ultimate pit problem possesses a unique mathematical structure which makes it
amenable to several different approaches including very efficient methods from network
optimization. Specifically the problem can be cast as a network flow problem which is the current
best approach for large models with dozens or hundreds of millions of blocks.
Graphically the ultimate pit problem is shown in Figure 2.7 for a single vertical section
through a synthetic ore body. The economic block values and the precedence constraints are used
to identify the blocks which together maximize the undiscounted value and satisfy the precedence
constraints. This subset of blocks is called the ultimate pit and is not only used in the context of
24 |
Colorado School of Mines | determining when a mining operation should cease production but also in informing many
engineering decisions in the mine planning process, generating outer bounds for scheduling, and
several other contexts.
-1 -1 -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 -1 2 -1 -1 -1 -1 -1 -1 -1 2 -1 2 -1
-2 -2 -2 1 5 4 -2 -2 -2 -2 -2 -2 -2 1 5 4 -2 -2 -2 -2
z -2 -2 -2 2 3 2 -2 -2 -2 -2 -2 -2 -2 2 3 2 -2 -2 -2 -2
x
Figure 2.7 The ultimate pit problem requires an economic block value model, (left) and
precedence constraints, as directed arcs between blocks, (middle) in order to identify the ultimate
pit limits (right)
In practice the ultimate pit can be calculated rapidly for very large 3D models. The example
3D ultimate pit shown in Figure 2.8 is for a relatively small model with 374,400 input blocks and
7,116,016 precedence constraints however even models with hundreds of millions of blocks and
billions of precedence constraints are attainable. This is not unrealistic; as many modern open-pit
mining complexes use very large models with smaller blocks for a variety of reasons [43].
The remainder of this section describes three formulations for the ultimate pit problem
including the original network based formulation, the linear programming formulation, and the
max-flow / min-cut formulation. Then four methods for solving the ultimate pit problem are
described: The Lerchs & Grossmann algorithm, the heuristic floating / moving cone algorithm,
the pseudoflow algorithm, and the special case 2D dynamic programming algorithm. Finally we
present a brief survey of alternative methods which are less commonly used.
2.2.1 Original Network Formulation
In 1965, Lerchs and Grossmann originally described the ultimate pit problem in the context of
a custom network algorithm for determining the smallest maximum valued closure of a weighted
directed network [20]. A network, also called a graph, is a structure for modeling pairwise
relationships between objects. Networks are extremely useful structures for all kinds of real world
problems and are commonly found in many different areas in open-pit mine optimization and
planning.
25 |
Colorado School of Mines | z
y
x
Figure 2.8 An example 3D ultimate pit calculated from a 374,400 block model
The nodes of the network represent some object and are connected in pairwise relationships
via directed or undirected arcs. A directed arc is used when there is some order to the nodes, or a
specific orientation inherent in the relationship. The beginning node is called the tail, and the
ending node is called the head. Undirected arcs do not have any order but are merely used to
indicate that the relationship exists. A weighted network may have weights associated with nodes,
arcs, or both. The weights are some scalar value that could have many different meanings
depending on the context.
In Lerchs and Grossmann’s description of the ultimate pit problem blocks are represented by
weighted nodes with the weight being the economic block value. Blocks are then connected by
directed arcs from lower blocks to upper blocks following the precedence constraints. The
resulting network has no cycles by construction and therefore is classified as a directed acyclic
network. A closure of a directed acyclic network is a set of nodes such that there are no arcs with
their tail inside the closure and their head outside. All closures of this network are valid pits and
do not violate any precedence constraints. The ultimate pit is the smallest maximum valued
closure and Lerchs and Grossmann developed an algorithm for identifying this closure, described
in Section 2.2.4.
Figure 2.9 shows an example network that corresponds to a very small example ultimate pit
problem. This model consists of only 12 blocks which are not arranged in the typical rectilinear
grid, instead they each only have two precedence constraints indicated by the directed arcs. The
26 |
Colorado School of Mines | marked closure is the maximum valued closure and corresponds to the ultimate pit with a total
contained value of 2 units.
closure
node
-1 -1 -1 -1 -1
head -2 -2 1 2
directed
arc
tail -3 5 -3
Figure 2.9 Example weighted directed acyclic network with labeled maximum valued closure
A few more definitions are required to properly describe the Lerchs and Grossmann algorithm
and network flow approaches. Paths are sequences of arcs such that each subsequent arc shares a
common node. In many situations paths are constrained such that they can only go through
nodes and arcs at most once and they may be restricted to only travel along directed arcs from
tail to head. Paths in networks with undirected arcs do not have this restriction. In directed
networks where the sequence of arcs does not have any orientation restrictions this is sometimes
called a chain. A network such that any two nodes are connected by exactly one path is called a
tree. The tree may contain a special node designated a root from which there could be many
subtrees or branches, which are themselves trees when ‘detached’ from the root.
2.2.2 Linear Programming Formulation
The linear programming formulation for the ultimate pit problem is useful theoretically and as
a means of communicating the problem clearly. Ultimate pit solvers do not typically use this
formulation directly with the simplex algorithm or another general purpose linear programming
solver; but it is still useful and is strongly related to the subsequent max-flow / min-cut
formulation in Section 2.2.3. Linear programming is discussed in more detail in Section 2.5.2.
The ultimate pit problem as a linear program is defined as follows [44–46]:
27 |
Colorado School of Mines | Sets:
• b , the set of all blocks. Commonly based on a regular block model. See Section 2.1.3.1.
∈ B
• ˆb ˆ , the set of antecedent blocks that must be mined if block b is to be mined in order to
b
∈ B
honor pit slope requirements. These sets are derived from the precedence constraints
described in Section 2.1.3.4.
Parameters:
• v , the economic block value of block b. See Section 2.1.3.3.
b
Variables:
• X , the proportion of block b which is mined in the ultimate pit
b
The Ultimate Pit Problem:
(cid:88)
maximize v X (2.7)
b b
b∈B
s.t. X X 0 b ,ˆb ˆ (2.8)
b
−
ˆb
≤ ∀ ∈ B ∈
Bb
0 X 1 b (2.9)
b
≤ ≤ ∀ ∈ B
Equation 2.7 maximizes the total contained value of ore and waste blocks mined within the
ultimate pit. Equation 2.8 enforces precedence constraints. Note that for any block b to attain a
value of 1 all of its antecedent blocks ˆb must already have a value of 1 or else this constraint
would be violated. Equation 2.9 enforces bounds on the main variable to disallow mining a
negative proportion or more material than is present.
Importantly it is not necessary to restrict the variable X to only integral values. This is
b
inherently satisfied because the precedence constraints form a totally unimodular system [47].
2.2.3 Min Cut Formulation
In 1968 Johnson showed the ultimate pit problem can be formulated as a max-flow / min-cut
network problem [44]. This reformulation was also demonstrated by Picard in 1976 alongside
further mathematical justifications [48]. The construction is based on taking the dual of the linear
28 |
Colorado School of Mines | programming formulation in the previous section and performing some appropriate manipulations.
Johnson recommended Ford and Fulkerson’s labeling algorithm to solve the max-flow / min-cut
formulation of the ultimate pit problem. This formulation gives rise to the currently fastest known
methods of solving the ultimate pit problem; specifically Hochbaum’s pseudoflow algorithm [46].
A flow network differs from other networks in a few respects. First two nodes are identified as
the source, denoted S, and the sink, denoted T. Each directed arc is then capable of carrying
some non negative integer amount of flow from its tail to its head up to some specified capacity.
Each node, other than the source and the sink, are required to satisfy a flow balance constraint
such that the inflow is equal to the outflow. Flow networks are used for many purposes such as
modeling traffic on roads, fluids flowing through pipes, power flowing along electrical grids, and
others [49]. Ford and Fulkerson developed much of the initial theory on flow networks including
the important max-flow / min-cut theorem [50].
Each well-formed flow network generally has many possible valid flows, which are assignments
of flow values to each arc in the network. As previously mentioned these flows are constrained
such that the flow on an arc cannot exceed the arc’s capacity, and also such that for every node
(except the source and sink) the total inflow into the node is equal to the total outflow from the
node. Each flow network has a max flow which is an assignment of flow such that the outflow
from the sink (and thereby the inflow into the source) is maximized. There are many procedures
for determining the max flow including augmenting paths [50, 51], the push-relabel algorithm
[52, 53], and the pseudoflow algorithm [46, 54, 55].
Also in any given flow network there are many ways to cut the network into two pieces. This
cut is accomplished by removing a set of arcs, called the cut-set. If, once the cut-set arcs are
removed, the flow network is divided neatly into two pieces with the source on one side, the sink
on the other, and such that there are no paths along directed arcs from the source to the sink this
cut is called an s-t cut. In an s-t cut it is only required to remove arcs which are directed from
the source side to the sink side and any arcs that are directed from the sink side to the source side
are not a part of the s-t cut. The capacity of the s-t cut is the sum of the arc’s capacity in its
cut-set. The s-t cut which corresponds to the minimum capacity is called the minimum cut of the
flow network.
29 |
Colorado School of Mines | Ford and Fulkerson’s max-flow / min-cut theorem states that the maximum flow of a flow
network is equal to the minimum cut of the same flow network. Intuitively this is because the arcs
which are a part of the minimum cut’s cut-set form the ‘bottleneck’ of the flow network. There is
no way to sneak more flow through the cut-set without increasing its capacity. The formal proof
of this theorem proves that if the maximum flow did not equal the minimum cut there would be a
contradiction [50].
For our purposes, in the mining industry, this is a very useful result because using Johnson’s
construction our desired ultimate pit directly corresponds to the source side of the minimum cut.
If one is able to find the maximum flow through some means then the minimum cut can be
extracted, and thereby the ultimate pit. Or, if an algorithm gives the minimum cut directly this
also immediately reveals the ultimate pit.
Practically a given ultimate pit problem is transformed by representing each block as a node
in the network, however unlike Lerchs and Grossmann’s network formulation the nodes do not
have any associated weight. The source and sink nodes are added, and each node corresponding
to a positively valued block is connected by a directed arc from the source with a capacity equal
to the economic value. Nodes with negative values are connected with a directed arc towards the
sink with a capacity equal to the absolute value of their economic value. Nodes that correspond
to blocks with zero economic value do not need to be connected to the source or the sink. Finally,
each precedence constraint is incorporated by including a directed arc between the lower and
higher block with infinite capacity. An example transformation of an ultimate pit problem is
shown in Figure 2.10.
In Figure 2.10 the capacity of each arc in the flow network is the denominator of the fraction
shown on each arc. The flow, which in this case corresponds to the maximum flow, is the
numerator. The minimum cut is represented by the dashed line which goes through the four
bolded arcs, the sum of the capacities of the arcs in this set is 9 which is equal to the maximum
flow. The difference between this value and the sum of the capacities of the source adjacent arcs
is the ultimate pit value. Note that the nodes on the source side of the minimum cut, which are
colored darker, are exactly the ultimate pit.
30 |
Colorado School of Mines | 2/
∞
2/ 2/2
∞
6/7
-2 -2 -2 -4 2/ 2/2
∞
S T
0/ 2/2
7 3 ∞
z 3/3
0/ 3/4
x ∞
3/
∞
Figure 2.10 Example transformation of an ultimate pit problem (left) into source-sink form for
solving with a max-flow or min-cut algorithm. Figure adapted with permission from Deutsch,
Da˘gdelen, and Johnson 2022 [56]
2.2.4 The Lerchs & Grossmann Algorithm
In their seminal 1965 paper, Lerchs and Grossmann proposed two approaches to solving the
ultimate pit problem [20]. The simpler approach based on dynamic programming which is only
applicable to two dimensional cross sections is described in Section 2.2.7. The more general
approach, based on network theory, is reviewed here.
The Lerchs and Grossmann algorithm is an iterative algorithm which identifies the maximum
closure of a directed network. It begins with a infeasible solution and iterates through primal
infeasible solutions until it identifies the first primal feasible solution which is the optimal
solution. The solution at each stage is the current collection of strong nodes, which begins as the
set of all positive blocks and ends as the set corresponding to the ultimate pit.
Instead of operating with the original directed network which contains all of the precedence
constraints, the Lerchs and Grossmann algorithm uses an augmented network which contains a
special artificial root node which is originally connected to every node with a directed arc
beginning at the artificial root. The augmented network is a tree at initialization and remains a
tree throughout the entire process. Nodes within the augmented network are denoted as strong or
weak depending on the sum of all the associated block values within their branch. Initially
positive nodes are classified as strong, and non-positive nodes are classified as weak. This
initialization step is shown in Figure 2.11, the example ultimate pit problem (left) is transformed
into the augmented network (right) with each node connected to the artificial root. The strong
branches (colored darker) form the initial solution.
31 |
Colorado School of Mines | -2 -2 -2 -2 -2
-2 -2 -2 -2 -2
7 6 -2
7 6 -2
4
4
z
x
x
o
Figure 2.11 The initialization step for the Lerchs and Grossmann algorithm.
In addition to classifying nodes based on strength, arcs within the augmented network are
classified based on their direction and strength. The direction classification of an arc depends on
its direction relative to the chain originating at the artificial root and passing through the arc. If
the arc is pointing toward the root it is considered a m-arc or minus-arc and if it is pointing away
from the root it is a p-arc or a plus-arc. The strength of an arc depends on the net value of the
nodes which are supported by the arc, this is called the mass of the branch. If the mass supported
by an arc is positive it is a strong arc, and if non-positive it is a weak arc. The augmented
network is said to be normalized if and only if all strong arcs are adjacent to the artificial root.
The two main steps of the Lerchs and Grossmann algorithm are to first ‘move toward
feasibility’ and then ‘normalize’ the resulting tree. These two steps are repeated until the move
toward feasibility is no longer possible at which point the ultimate pit is identified. In the linear
programming context the move toward feasibility corresponds to a change of basis on the path
toward primal feasibility. The normalization step is to fix the current iteration when it strays
from dual feasibility [57].
In practice the Lerchs and Grossmann algorithm continuously scans over all of the precedence
arcs and once an arc is identified that has a ‘strong’ tail and a ‘weak’ head it is selected for the
move toward feasibility. In the move toward feasibility the offending precedence arc is added to
the augmented network, which creates a cycle which has to be broken so that the augmented
network can remain a tree. The Lerchs and Grossmann algorithm breaks the cycle by removing
the arc which is adjacent to the artificial root. If, following the introduction of the offending
32 |
Colorado School of Mines | precedence arc, there are any arcs which are strong p-arcs that are not adjacent to the artificial
root the normalize procedure must be used to prepare the tree for the next move toward
feasibility.
A strong p-arc in the non-normalized tree is replaced with the artificial arc between the
artificial root and the head of the p-arc. A strong m-arc is replaced with the arc between the
artificial root and the tail of the m-arc. In effect this is to guard against inappropriate allocations
of values between the branches of the tree. For example, a strong p-arc implies that a node
farther along the branch is using its value to pay for blocks that are not within its cone of
extraction. This is avoided by the normalization step as shown in Figure 2.12.
-2 -2 -2 -2 -2 -2 -2 -2 -2 -2
7 6 -2 7 6 -2
4 4
x x
o o
Figure 2.12 The normalization step in the Lerchs and Grossmann algorithm transforms the non
normalized tree (left) to the normalized tree (right) by replacing any non root adjacent strong
arcs.
The Lerchs and Grossmann algorithm terminates when there are no possible moves toward
feasibility at which point the remaining strong arcs are the smallest maximum valued closure of
the directed network: the ultimate pit. Practically this may require many complete scans over the
entire precedence network, along with many moves toward feasibility and normalization steps.
The performance of the Lerchs and Grossmann algorithm is not very good when the number of
nodes and precedence constraints is large. Heuristic methods, primarily the floating cone method
in the following section, were preferred for many years following 1965 until a strong commercial
implementation of the Lerchs and Grossmann algorithm was developed by Whittle [34]. In recent
years the Pseudoflow algorithm is the fastest approach to the ultimate pit problem.
33 |
Colorado School of Mines | 2.2.5 Floating Cone Methods
The floating cone method was originally described by Pana in 1965 in the context of
simulating the mining process in order to determine both the ultimate pit limits and the optimal
mining sequence [58, 59]. Pana, and members of the Systems and Data Processing Division at
Kennecott Copper Corporation, developed a system whereby estimated geologic attributes were
transformed into economic block values, coded on punch cards, and the mining sequence was
determined by evaluating many hundreds or thousands of ‘frustums,’ or cones, of material. An
open-pit mine can be thought of as a set of intersecting cones which together define the ultimate
pit limits. Pana described an approach which later became known as the ‘floating cone’ because
of the way one can visualize an inverted cone floating from block to block and either mining that
entire cone or not.
Floating cone algorithms are iterative in nature and work to define the ultimate pit limits by
successively visiting different possible cone bottom locations and deciding to extract that cone
based on the net contained value. That is, if a cone contains blocks that together have a positive
economic value then it is extracted and if non-positive it is left in place. The mining engineer
defines the cones in the same manner as precedence constraints, with the added flexibility that
they can very easily define minimum bottom widths by simply requiring the cone to consist of
multiple blocks at the bottom. Floating cone algorithms terminate once there are no remaining
cones containing a positive value.
The fundamental issue with floating cone methods is that they do not correctly consider the
contribution of multiple cones at one time. This can lead to both overmining: wherein a larger
cone than necessary is mined that includes a subset of material with net non-positive economic
value that does not need to be mined, and undermining: wherein the floating cone method is
unable to identify a situation whereby two or more cones could ‘share’ the cost of extracting
negative valued material and end up net positive [27, 60–62].
The example in Figure 2.13 demonstrates both overmining and undermining, on the left and
right respectively. The true ultimate pit in both models (hatched section) has a value of 2 and
relies on sharing the top middle waste block between the two ore blocks. However when the
floating cone algorithm is applied to the model on the left it creates the too large pit that
34 |
Colorado School of Mines | incorrectly mines two extra blocks, and when the floating cone algorithm is applied to the model
on the right it is unable to find any economic cone to extract and terminates with the
‘mine-nothing’ solution.
-4 -4 -4 -4 -4 -4 -4 -4 -4 -4
11 -4 11 11 -4 11
3 -4
z
x
Figure 2.13 Two issues with the conventional floating cone algorithm. The pit on the left is too
large, and no pit is identified on the right which is too small compared to the optimum ultimate
pit (hatched blocks).
Several authors have worked to address these issues to varying degrees of success. Wright, in
1999, introduced the floating cone II algorithm which was able to outperform the original
approach by varying the float sequence and adjusting the extraction criteria in certain situations
[63]. This was later modified and further developed by Khalokakaie in 2006, Kakaie in 2012,
among many others as recently as 2022 [64–68]. Floating cone methods are inherently heuristic
methods and do not guarantee the optimal ultimate pit limits in all scenarios, however their
strength comes from the ease with which minimum mining width constraints can be incorporated
and their relative ease of implementation and use. Interestingly with the advent of flow based
techniques, and efficient implementations of exact ultimate pit optimization algorithms, floating
cone methods are often slower than their optimal counterparts when used without any
operational extensions.
2.2.6 The Pseudoflow Algorithm
The pseudoflow algorithm is very similar to the Lerchs and Grossmann algorithm, in that
both algorithms start with a primal infeasible solution and move towards feasibility by
incorporating violated precedence constraints at each iteration [46, 54, 55]. In Chapter 3 a
variation of the pseudoflow algorithm that is customized specifically for the ultimate pit problem
along with relevant implementation details is described. Therefore, this section focuses on some of
35 |
Colorado School of Mines | the practical differences between the pseudoflow algorithm and the Lerchs and Grossmann
algorithm instead of the details of the pseudoflow algorithm which are discussed later.
The biggest practical difference between the pseudoflow algorithm and the Lerchs and
Grossmann algorithm is that the pseudoflow algorithm is much faster. A computational
comparison conducted by Hochbaum and Chen in 2000 showed the same model being solved by
the push-relabel algorithm in forty minutes whereas the Lerchs and Grossmann algorithm took
around five and a half hours [69]. Hochbaum indicated in the same paper that the pseudoflow
algorithm, which was in part inspired by this comparison, ran faster than all known
implementations of the push-relabel algorithm. Muir, in 2007, showed how the pseudoflow
algorithm was superior to the Lerchs and Grossmann algorithm in practice [70]. Chandran and
Hochbaum also showed that the pseudoflow algorithm was superior to known implementations of
the push-relabel algorithm [71].
In 2015, a comparison between all three algorithms; the Lerchs and Grossmann algorithm, the
push-relabel algorithm, and the pseudoflow algorithm was conducted [40]. Deutsch et al. showed
for a model with 16 million blocks that pseudoflow could compute identical results in four
seconds, to those by push-relabel in nine seconds, and Lerchs and Grossmann in forty five
minutes. With one particularly egregious dataset, which consisted of a steeply dipping vertical ore
body, the Lerchs and Grossmann algorithm took fifteen hours compared to pseudoflow requiring a
mere nine seconds.
Another practical difference is that the pseudoflow algorithm, which is based on routing units
of flow around a network, is limited to operating with integral economic block values, unlike the
Lerchs and Grossmann algorithm which can use floating point numbers [72]. This is not a
detriment for mining engineers, as fractional economic block values can be multiplied by a large
constant and then rounded to the nearest integer while retaining the same solution. In certain
circumstances there may be a concern for integer overflow within the solution process, but this
implementation detail should be handled by any serious ultimate pit solver.
There are several different variants of the pseudoflow algorithm which control the order in
which nodes are processed and the means by which nodes are labeled. Labeling, which is another
implementation detail responsible for much of the performance improvements relative to the
conventional Lerchs and Grossmann algorithm, is discussed further in the subsequent chapter.
36 |
Colorado School of Mines | However the general notion is to assign a monotonically increasing label to nodes that prevents
the algorithm from reprocessing certain nodes until other nodes have been processed. This has the
effect of avoiding certain sequences of merging operations that would require additional iterations
and is one of the key differentiators between the pseudoflow algorithm and the Lerchs and
Grossmann algorithm. Of the lowest-label and highest-label strategies described by Hochbaum in
2001, the highest-label variant generally performs the best on ultimate pit problems [46, 70].
2.2.7 2D Dynamic Programming Algorithm
The two-dimensional ultimate pit algorithm from Lerchs and Grossmann is a dynamic
programming based technique to solve a special case of the ultimate pit problem [20]. The
precedence constraints are restricted to the simple case of requiring each mined block to mine the
block immediately above it and the two blocks above and to the left and right. A dynamic
programming method is characterized with a straightforward data preparation step, a iteration
through the section, and a traceback step. In Section 4.3, a novel extension of this algorithm
which accounts for minimum mining width constraints is developed as a part of this dissertation.
The data preparation step is to construct an initial tableau which converts the economic block
values into a cumulative value model which corresponds to extracting the entire column of
material above each block. This avoids having to recompute these cumulative values over and over
again throughout the process. An example of this data preparation step is shown in Figure 4.8.
The iteration step proceeds down each column individually from left to right keeping the top
row zero, and filling in each cell of a new tableau as the maximum of three previously computed
values corresponding to mining an additional bench downwards, mining straight across and
reducing the depth of the pit by one bench. The entries of this new tableau represent the
maximum possible contributions of the columns to the left and thereby construct the ultimate pit
one column at a time from the left of the section. Traceback information is recorded on a per
block basis indicating which of the three options were the best.
Lerchs and Grossmann describe keeping the top row as zero throughout the entire process and
following the traceback information from the maximum valued block in the first row [20]. This
can present difficulties when there are multiple disparate pits within the cross section, because
there is no means by which value can be transferred between the pits without mining the
37 |
Colorado School of Mines | uppermost bench, which generally consists of overburden, and reduces the value. It is therefore
better to only set the leftmost value of the upper air row to zero and use the iteration step mostly
unchanged, allowing the values of the ‘air blocks’ to increase if necessary. Finally, the traceback
step should be initiated from the rightmost block in the air row instead of the maximum valued
block in the first bench. This minor issue in Lerchs and Grossmann’s orignal description was
addressed early on [22, 73], and can also be seen corrected in Section 4.3.
2.2.8 Alternate methods
Zhao and Kim proposed another network theoretic algorithm in 1991 that was purported to
have better performance than the Lerchs and Grossmann algorithm [74]. Their algorithm is of a
similar character and operates with a similar direct network representation of the problem,
however Zhao indicates that the normalization step is avoided by enforcing a different set of
invariants between operations [75].
The original dynamic programming approach from Lerchs and Grossmann was restricted to
two dimensional cross sections of an open-pit mine. Johnson and Sharp, in 1971 showed a possible
means to apply this approach to a three dimensional mine which involved running the algorithm
repeatedly on sections and then smoothing between those sections [73]. This approach found use
as it is substantially less computationally intense than exact approaches. As computers have
become more powerful and faster algorithms are more available a heuristic is no longer applicable
when the optimal answer is known.
Koenigsberg, in 1982, present a three dimensional application of dynamic programming to the
ultimate pit problem [76]. This approach has faced valid criticism for not identifying the optimal
ultimate pit contours due to how it handles precedence constraints and how in certain cases it
creates additional constraints which preclude the true optimal result [77–79]. A proper three
dimensional dynamic programming approach to the ultimate pit problem which flexibly handles
precedence constraints has not been developed.
As previously mentioned in Section 2.2.3 any max-flow/min-cut algorithm would be usable
with the ultimate pit problem. Such as the push-relabel algorithm [52, 53], the Ford-Fulkerson
algorithm [50], the Edmonds-Karp algorithm [80], Dinic’s algorithm [81], and several others
[82–85].
38 |
Colorado School of Mines | It is not expected that all max flow techniques will always be faster than the Lerchs and
Grossmann algorithm. In Lerchs and Grossmann’s original paper they specifically mention that
their direct approach is preferred for “obvious reasons,” although they neglect to say what those
reasons are. It is possible that the max flow algorithms available at the time, which have quite
steep memory requirements, were not well suited to the existing computer hardware. Yegulalp
and Aries applied the excess-scaling max flow algorithm from Ahuja and Orlin to the ultimate pit
problem but were unable to solve problems as quickly as the Lerchs and Grossmann
implementation from Whittle [86, 87].
Recently, in 2022, Chen et al. have described an algorithm for determining the maximum flow
in near-linear time which could be the fastest approach yet described [88]. This approach appears
to be quite involved, relying on identifying certain minimum cycles and using custom data
structures. An implementation of Chen’s algorithm does not yet exist however this could be a very
valuable result for large models if the algorithm translates well to existing computer hardware.
2.3 The Block Scheduling Problem
Where the ultimate pit problem is concerned with determining which blocks should be mined
at all, the block scheduling problem is concerned with when those blocks should be mined, and
where they should be routed. With these additional concerns, the dimensionality of the problem
vastly increases which allows practitioners to define additional constraints which more closely
approximate the true open-pit mining process. However, this increased flexibility and accuracy
also increases the problem’s complexity and the required solution time. Block scheduling
problems are more complicated, more specialized, and less tractable than ultimate pit problems.
Overcoming these challenges has received a great deal of academic and commercial effort over
the years. Researchers have evaluated a variety of approaches. In Section 2.3.1 the pushback
approach to block scheduling is discussed. This approach begins with the ultimate pit, and then
calculates a sequence of nested pits through some means, which approximate an extraction
sequence wherein parts of the smallest nested pit are extracted first, followed by the next largest
pit next, and so on until the ultimate pit. Nested pits only consider the time component of the
block scheduling problem, and only indirectly. Integer programming is often employed to properly
handle routing, blending, stockpiling, and more relevant concerns which is discussed in Section
39 |
Colorado School of Mines | 2.3.2. Finally Section 2.3.3 discussed several other approaches to the block scheduling problem
that do not rely on Integer programming, and instead use heuristics to inform long range open-pit
mine planning.
2.3.1 The Pushback Approach to Block Scheduling
Pushbacks, or phases, are nested pits calculated via the ultimate pit problem and some pit
parameterization method introduced in Section 2.1.4. The central idea is that these pushbacks
approximate an optimal extraction sequence where benches from the smallest pit are generally
extracted before those in subsequent pushbacks.
Using pushbacks to approximate the block scheduling problem first appeared in Lerchs and
Grossmann’s paper which introduced the ultimate pit problem [20]. Lerchs and Grossmann
proposed reducing the block values by a constant, and resolving for the ultimate pit with the
modified block values. This pit will be the same or smaller than the original ultimate pit, and if
smaller it will be smaller such that it tends to prefer the higher valued blocks. By repeating this
process, reducing by even larger constants, an entire series of nested pits can be determined which
Lerchs and Grossmann claimed to maximize the integral of the cash flow curve. One useful way of
(cid:80)
thinking about this process is by dualizing a constraint of the form X T into the objective
x x ≤
where T is some tonnage constraint whose value isn’t important. The dual on this constraint is
exactly the constant that Lerchs and Grossmann use to decrease all of the block values.
Matheron later expanded on the idea of pit parameterization alongside Vallet in the late 70s
[29, 30]. Da˘gdelen and Francois-Bongarcon used more finely grained variations on the commodity
prices and mining / processing costs [31]. Whittle also developed several approaches to pit
parameterization based on price and cost factors [11]. Meagher, Dimitrakopoulos and Avis review
several of these approaches, and others in a recent review paper [89].
A key feature of pit parameterization is that the pits are nested within one another and do not
overlap. This makes each pit potentially a good candidate to use for downstream pushback design
and can be used to determine rough production schedules or used in more sophisticated block
scheduling algorithms. A disadvantage of pit parameterization is that in some circumstances the
ore body does not lend itself to creating operable pushbacks. For example, a steeply dipping
vertical ore body often generates nested pits that are concentric cones which do not share a
40 |
Colorado School of Mines | Risk constraints are a common addition to mine schedules following Johnson’s seminal work.
Dimitrakopoulos et al. incorporate grade uncertainty and risk into open-pit design by
incorporating additional constraints and considering geostatistical simulation [90–92]. Godoy
presented a multi-stage approach for profitable risk management [93]. Van Dunem incorporated a
form of risk constraints based on limiting the number of blocks mined that are classified as
measured, indicated, or inferred in a given year [94].
Another example of an additional constraint is in mining complexes with both a surface and
underground component. King et al. developed a model which incorporated the
surface-to-underground transition, leaving space for a suitable crown pillar [95]. However, the
model was still very difficult to solve and a linear relaxation approach was used alongside a
rounding heuristic to achieve an integer solution.
Several researchers have worked to relax certain constraints or change the variable types in
order to solve larger models. Gershon allowed for continuous variables in certain parts of the
formulation to allow certain blocks to be mined partially [9].
One method of improving the tractability of the block scheduling problem is to reduce the
number of variables. Several approaches rely on aggregated blocks together into larger groups of
blocks that are considered as one unit [96–98]. These aggregation approaches yield smaller
problems with fewer variables that are easier to solve however the larger aggregated units may
not provide enough granularity to identify the optimal mine design for the original problem.
A useful approach to solving large block scheduling problems is via Lagrangian relaxation,
which is discussed further in Section 2.5.3 and in Chapter 4. This approach was originally
introduced by Da˘gdelen in the mid 1980s [27]. The value of this approach is that the sub problem
remains a network model which has a useful mathematical structure which can be exploited for
very fast solutions. Several researchers have expanded on Da˘gdelen’s approach included Tachefine
and Sumois who use an alternative method to obtain the necessary Lagrange multipliers and
Akaike who worked to incorporate stockpiles and a more involved cutoff grade strategy [99, 100].
The Bienstock Zuckerberg algorithm is a very useful column generation approach used for the
block scheduling problem which takes advantage of the large precedence constraint structure and
the handful of knapsack constraints [101, 102]. This algorithm is discussed further in Section
2.5.4. Aras, in 2018, showed in detail how the BZ algorithm could be used for direct block
42 |
Colorado School of Mines | scheduling for complex datasets with three destinations: a mill, a leach pad, and a waste dump
[103]. Additionally, Aras incorporated uncertainty and risk into the scheduling process by limiting
the number of blocks that are measured, indicated, or inferred on a per period basis. A novel
integerization method was also developed which achieves a small gap between the integer solution
and the linear relaxation in practice.
2.3.3 Heuristic methods
The number of variables and constraints in the open-pit mine planning process is often too
large for exact methods. The techniques described in the previous section to aggregate variables
and solve relaxed models help to alleviate some of the difficulty, although there has also been
efforts towards using heuristics and other inexact methods that sacrifice an optimality guarantee
to be faster. Several approaches have been developed by Chicoisne et al., Lamghari et al.,
Lambert et al., and others [104–106]. Many of these heuristic methods are discussed further in
Fathollahzadeh et al [42]. Several of these methods use meta-heuristics, of which several are
described in Section 2.5.5.
One distinct advantage of a heuristic approach to open-pit mine design is that it is much
easier to incorporate nonlinear or complicated constraints that cannot be modeled directly with
integer programming, or other exact methods that are more prescriptive. Additionally, heuristics
can be used not only to generate feasible solutions but also to improve existing ones. For
example, the local search heuristic developed by Amaya et al. can take an initial integer feasible
solution and evaluate nearby solutions in order to find a local optimum [107].
2.4 Open-Pit Mine Planning with Operational Constraints
Both the ultimate pit problem and the block scheduling problem do not, in their classical
descriptions, consider operational constraints such as minimum mining width constraints or
minimum pushback width constraints. The ultimate pit problem considers only the barest
minimum of constraints. Straightforward bounds on the variable preclude impossible values and
precedence constraints govern the shape of the ultimate pit to ensure geotechnical stability.
Outside of these considerations there is no prerogative other than to simply maximize the
undiscounted economic value. The block scheduling problem has many constraints but the most
43 |
Colorado School of Mines | common formulations are generally concerned with managing blending, plant capacities,
sequencing, and certain economical considerations instead of operational ones.
Many of the researchers that have developed extensions to the ultimate pit problem and block
scheduling problem to handle operational constraints, including minimum mining width
constraints, are discussed in this section.
The very first floating cone methods from Pana and Carlson were able to handle minimum
mining width constraints by restricting the volumes of extraction to consist of several blocks
[58, 59]. Whittle, in 1990, had this to say about floating cone methods:
Apart from being easy to understand and program, the one advantage that the
floating cone method has over other methods is that, if instead of using just one block
the program uses a disk of blocks as its starting point, then this can ensure a
particular minimum mining width at the bottom of the pit [108].
Of course extending the floating cone algorithm to address minimum mining width constraints
does not address the fundamental concerns with floating cone methods described in Section 2.2.5.
Overmining and undermining errors will still occur and may even be exacerbated.
Wharton, in 1997, describe a series of geometric operators on nested pits used to create more
operable designs and assess the impact of minimum mining width on the NPV of long-term
schedules [109]. The original nested pits used as input are calculated with conventional
parametric analysis and the Lerchs Grossmann algorithm using, for example, a mining cost
adjustment factor. The user then specifies a rectangular mining width template and a iterative
procedure is carried out to remove small contiguous blocks, remove protrusions along the outer
pit wall, handle inaccessible blocks, remove small holes, and overall perform some geometric
cleaning. This cleaning is performed based only on the shape of the nested pits and does not
consider block values. Run times and typical block model sizes are not reported, but the
description implies a constant number of linear passes over the model and is likely to be very
quick if programmed efficiently. However, the impact on NPV is substantial. In their case study
NPV decreases between 3 and 22% depending on how aggressive the cleaning is.
An early optimization based approach that was not solely geometric was described by
Dimitrakopoulos in 2004. Dimitrakopoulos et al. develop a “risk-based production-scheduling
44 |
Colorado School of Mines | formulation for complex, multielement deposits”, and incorporate specific equipment constraints
and mining sequence feasibility [91] . Their model provides a benefit for mining blocks in the same
period as blocks within their immediate neighborhood, or equivalently a penalty when adjacent
blocks are mined in different periods. This benefit is realized by applying cost coefficients, which
are determined through trial and error, to relevant terms in the objective function and not as
hard constraints. Using cost coefficients allows the user to specify how important mining width
constraints are to them in actual dollar terms, however this is typically not straightforward to
determine a priori and requires iterating on those parameters considering the final results. This
formulation is applied to a model with 2030 blocks and the elapsed time is not reported.
Stone et al. in 2007, describe an in-house optimization tool developed at BHP Billiton called
‘Blasor’ which generates optimized mine schedules [110]. Blasor internally uses CPLEX, a well
known MILP solver developed by IBM, in order to solve specially constructed formulations
accounting for mining, transport, comminution, and market constraints. Minimum mining width
constraints and pushback width constraints are specifically handled by aggregating blocks
together using a ‘proprietary fuzzy clustering algorithm’ and scheduling based on those
operational units. Additionally Blasor provides a graphical tool to allow practitioners to make
manual modifications to incorporate other operational constraints which are difficult to encode
algorithmically. The precise details are unavailable.
In 2008 Zhang introduced a heuristic approach to incorporate minimum mining width
constraints in nested pits similar to Wharton in 1997 [109, 111]. The initial heuristic operator
developed in 2008 applies three geometric operators in an unspecified sequence:
• Removing drop cuts: small contiguous groups of blocks which are smaller than some
constant and are surrounded by later phases or blocks outside of the pit are removed.
• Small wall removal: similar to the drop cut removal step, this operator removes small
contiguous groups that contact blocks from earlier phases.
• Interior minimum mining width enforcement: reassign blocks which do not satisfy a
rectangular mining width template to nearby common phases.
45 |
Colorado School of Mines | These operators do not consider economic value and instead aim to make the pit operational
by changing relatively few blocks. It is generally true that changing fewer blocks leads to a
smaller reduction in economic value with most datasets. However, it is possible to construct
counterexamples and a true optimizing approach must not rely on this. The following year Zhang
extended their heuristic cleaning technique with a meta-heuristic that would use the cleaning as a
sub procedure [112]. The working solution is repeatedly modified in a stochastic manner then
made feasible by the cleaning procedure. This new solution would be accepted as the working
solution always if the NPV improves, but also if the NPV reduces with some decreasing
probability. Accepting a ‘worse’ design with some non-zero probability is one of the defining
features of simulated annealing and works to prevent the algorithm from getting stuck in local
optima. Zhang applied their method to two synthetic models of 16,000 and 120,000 blocks
respectively but do not indicate runtime.
Another two optimization formulations for the block scheduling problem with minimum
mining width constraints are given by Pourrahimian in 2009 [113]. Their first formulation
restricts blocks such that they can only be extracted in a given period if a certain number of
nearby blocks are also extracted in that period. This does not explicitly disallow inoperable
configurations of blocks, but it does preclude many common issues (such as single block pit
bottoms). Crucially, because this extra constraint is appended to a full optimization method it
does consider economic block values. The second works by aggregating blocks prior to
optimization which had the added benefit of making the model much smaller and easier to
optimize while handling mining width considerations by design. The authors apply both of their
methods to a single bench of 415 blocks and do not report the total run time.
A sliding time window heuristic method to a variant of the general open-pit block sequencing
problem handling multiple periods and typical resource constraints is given by Cullenbine in 2011
[114]. They additionally incorporated a small operational consideration. Each block in Cullenbine
et al’s model is required to extract the five blocks above in a conventional cross sign configuration
but also required to extract at least one of the neighboring blocks on the same level. This
additional constraint precludes single block pit bottoms, and other locations where a single block
is mined in a period by itself with no nearby support. The largest example considered by
Cullenbine et al. contains 25,620 blocks and achieves an optimality gap of 4.3% in just under
46 |
Colorado School of Mines | three hours. However the impact of the operational constraint itself is not the object of this work
and is not isolated.
Pourrahimian and Cullenbine both incorporate the idea of restricting blocks to be mined if
and only if a minimum number of additional blocks within the original block’s neighborhood are
also mined. As the neighborhood grows the minimum number of blocks must also grow, but you
cannot, in general, rule out inoperable configurations with only these kind of constraints.
Additionally, if the minimum number of blocks grows too large relative to the size of the
neighborhood then the constraint is too restrictive, and may lead to poor results.
Instead of incorporating constraints directly into the optimization model Tabesh, in 2014,
suggests clustering blocks together into operable shapes and scheduling on those results [115].
This form of clustering can be guided based on both the perceived performance of the clusters in
downstream scheduling steps and such that they are big enough for operation. Additionally this
form of clustering precludes minimum pushback width violations.
Version 10 of Maptek Vulcan included a tool called the “Automated Pit Designer” which took
pit numbers, from conventional nested pit analysis, and created polygonal designs - without
ramps - satisfying some operational parameters [116]. The help documentation indicates three
operational preprocesses which may be applied to modify the input pit numbers to satisfy mining
width constraints. Two are based on mathematical morphology operators [117], and the third is a
custom geometric operator to ‘snap walls’ together between nested pits. Mathematical
morphology is a useful tool for operational constraints relating to minimum mining width and is
discussed in more detail later, however it is a geometric method and does not consider block
values. These routines operate on a bench by bench basis and do not consider precedence
constraints, as the pit slopes are handled later explicitly in the polygonization process, this allows
them to operate very quickly and have been applied to models with tens of millions of blocks in
seconds. In full disclosure, the author of this dissertation developed this version of the automated
pit designer in version 10 of Maptek Vulcan.
In Figure 2.14 there are three planar sections through an example ultimate pit model to
demonstrate some of the mathematical morphology procedures used in Maptek Vulcan’s
Automated Pit Designer. The first section on the left exhibits several examples of an inoperable
ultimate pit model. There are both missing blocks, which would realistically be mined, and
47 |
Colorado School of Mines | isolated blocks which would realistically be either not mined or mined along with their neighbors.
The middle section shows the result following a cleaning operation consisting of a closing of four
blocks and an opening of three blocks, and the right section shows a more extensive cleaning
operation consisting of a closing of seven blocks and opening of seven blocks. In both cases the
actual economic values of the blocks are not considered and although the resulting sections are
now operational, the sacrificed value may be much greater than necessary.
Mined
Changedtomined
Changedtonotmined
Notmined
y
x
Figure 2.14 Cleaning an ultimate pit with mathematical morphology as with the Maptek Vulcan
automated pit designer. The initial planar section on the left is moderately cleaned (middle), and
aggressively cleaned (right).
Juarez et al. in 2014, introduce a technique whereby operational constraints are considered
within a broader tree search style heuristic approach to open-pit mine scheduling and phase
design [118]. Their algorithm, and associated implementation, consider minimum mining widths
and minimum pushback widths by only generating designs which satisfy operational constraints in
the tree search. Typical block model sizes and times are not reported.
Bai et al. in 2018, describe a custom mathematical morphology approach to handling
minimum mining width and minimum push-back width constraints that does not explicitly
consider the value of the design changes [119]. Their method goes beyond the basic application of
the standard mathematical morphology operators by considering connected components, carefully
48 |
Colorado School of Mines | considering cycling, and accounting for precedence constraints. They report that their methods
are suitable on reasonably small block models, an ultimate pit with 550,000 blocks took 2 hours
to incorporate mining width constraints.
An additional clustering approach is presented by Farmer et al. in 2018 [120]. This approach
is similar to Pourrahimian’s, described earlier in that the blocks are clustered together into larger
operable groups prior to schedule optimization. Specifically Farmer et al. amalgamate the
scheduled blocks using breadth first search, and divide the aggregations into mineable and
non-mineable groups before proceeding. Their specification for minimum mining width is based
on the number of connected blocks in each spatial dimension which corresponds to a rectangular
mining width. They also describe a heuristic post process for smoothing phase designs and
avoiding both minimum mining width and minimum pushback width violations. The method is
applied to two orebodies however the model sizes and achieved runtimes are not reported.
Deutsch, in 2019, introduced a formulation for the ultimate pit problem with a minimum
mining width based on auxiliary variables [121]. These auxiliary variables follow arbitrary mining
width sets such that before any block is mined at least one of its corresponding operable mining
width sets must be completely mined. This formulation is presented as a maximum satisfiability
problem and was only applied to very small 2D examples, on the order of a few thousand blocks.
The two sets of mining width specific constraints in this formulation are appended to a full
optimization approach and guarantee optimal economic results satisfying operational constraints
if ran to completition. The general maximum satisfiability solvers tested did not scale to full size
models however the formulation has merit and is developed further in Chapter 4.
Muir, in 2020, presents a practical method to incorporating minimum mining width
constraints into ultimate pit models and successfully apply it to large models with over 21 million
blocks [122]. Their method involves solving for the ultimate pit, modifying it, and resolving
multiple times - the aforementioned 21 million block model took 10 iterations and achieved a
usable 2x2 mining width satisfying pit in 2 hours and twelve minutes. Muir’s method encodes
known mining width violating patterns along with an ‘appropriate’ action which are then used to
correct pit designs. The process is applicable to large, practical models, and is quite involved.
The main drawbacks of this method are that only 2x2 mining width templates are currently
supported, and there is no guarantee of optimal results.
49 |
Colorado School of Mines | A formulation for the geometrically constrained ultimate pit problem also using arbitrary
mining width sets and auxiliary variables was introduced by Nancel in 2021, however they also
consider additional operational constraints [123]. They incorporate specific constraints to disallow
thin connections between adjoining operable zones and to avoid ‘cavities’ or small collections of
unmined blocks contained within the ultimate pit. Nancel et al. 2021 describe approaches to
preprocess the input and show that on a moderately sized block model, on the order of half a
million blocks, preprocessing can reduce the total solution time for the geometrically constrained
ultimate pit problem from just under 20 minutes to just over a minute. Nancel et al. 2021 use
off-the-shelf optimization software, Gurobi in this instance, to solve their ultimate pit and block
scheduling problems.
Yarmuch et al. 2021 consider operational constraints including mining width, block
connectivity, and ramp access by introducing a so called ‘compactness factor’ to the objective
[124]. The compactness factor preferentially guides the optimization model to select blocks which
are close to the designed ramp when optimizing a single pushback. This method is applied to
either very small models, or models which have been made small by aggregating blocks together.
Manual intervention is required to decide on an appropriate compactness factor. They also
consider a ‘closeness factor’ whereby the output of the optimization model is biased to align with
a predefined manual, operable, schedule.
Another paper from Yarmuch et al. solve an open-pit pushback design problem considering
mining width and connectivity [125]. They use rectangular mining width elements and restrict
blocks from being assigned to specific pushbacks similar to [121] and [123]. If a block is assigned
to a pushback, then at least one of the rectangular mining width templates must be assigned to
that pushback and all of the blocks within that pushback must be extracted. Extensive
preprocessing and sliding window approximations are required to reduce the size of the problem.
Their method is applied to small models, on the order of a few tens of thousands of blocks, and
take many hours to achieve results with an optimality gap of around 6%.
2.5 Optimization
The three main components of an optimization problem are the decision variables which
encapsulate the different choices available, the constraints which limit those choices, and the
50 |
Colorado School of Mines | objective function which ranks the different possible outcomes. In open-pit mine planning there
are many subproblems which use these components to define optimization models, or
mathematical programs, which can be analyzed to guide decision making in the planning and
operating phases of an open-pit mining project. Fundamentally optimization is applicable to
many different problem domains within engineering, management, and more [126, 127].
This section discusses the necessary background regarding optimization within the field of
operations research which is used in this dissertation to develop tools for open-pit mine design
with operational constraints. Section 2.5.1 describes the processes for developing and applying
optimization models to real world problems. The main limitations of this approach are discussed.
Section 2.5.2 introduces the linear programming paradigm for mathematical optimization which is
an extremely useful approach for problems that are inherently linear, or can be approximated as
such. A relevant approach to solving large linear programming problems, Lagrangian relaxation,
is described in Section 2.5.3 because it is used in the following chapters alongside necessary
modifications and extensions. Finally, Section 2.5.5 describes relevant heuristic approaches which
are useful when exact approaches are too inflexible or too slow.
2.5.1 Solving Problems with Optimization
Within this chapter several decision problems have already been presented such as; which
blocks should be mined and how should those blocks be routed. Or when should different areas of
the deposit be developed in order to maximize net present value while satisfying relevant
environmental and operational constraints. Within these problems there is an element of choice.
There must be some flexibility in what can be done, or how a desired outcome can be achieved,
for optimization to be relevant. The flexibility within the system is always bounded by relevant
constraints which capture the real-world nature of the problem. Additionally, there must be some
means of evaluating different outcomes or decisions.
The process of applying optimization begins with taking the problem and modeling it in some
capacity. The variables, constraints, and objective function are all defined in such a manner to
capture the essence of the problem while considering the tractability and validity of the model.
Tractability is the degree to which the model can practically be solved and the extent to which it
admits necessary analysis. An extremely large model with billions of variables and interrelated
51 |
Colorado School of Mines | constraints might be desired to accurately represent the original problem, but if there is no
current technology that can generate a solution then that model is not very tractable, and
therefore not very useful. On the other hand, the validity of a model describes the extent to
which the resulting inferences and conclusions are applicable to the original real world problem.
Simplifications in the modeling process which improve the tractability of a model generally have a
negative effect on that model’s validity, and the trade-off between these two concerns is a
omnipresent dilemma within the modeling process.
The model must then be analyzed in order to draw necessary conclusions. Various
technologies, including some discussed in the following sections, and mathematical analyses are
used in order to extract relevant information from the model, such as the optimal decision policy.
The conclusions are derived from the model and not from the original problem, so they may need
to be modified and certainly considered within the context of any concessions taken during the
modeling process before making any final decisions. Additionally, the problem and modeling
process may need to be revisited once the conclusions are reviewed.
It is extremely important to understand the disconnect between the real-world problem and
the mathematical model, which is a fundamental limitation of optimization. For a wide variety of
reasons models can never fully capture every possible outcome and consideration within the real
world. But that does not preclude optimization as a valuable technique in real world scenarios,
because without optimization one would find themselves adrift in an endless sea of possibilities
and concerns with only ‘rules of thumb’ and their ‘best judgment’ to guide them. Even with the
numerous concessions and approximations required, open-pit mine planning benefits from the
judicious application of optimization techniques to guide decision making.
There are many approaches that may be considered to take a mathematical model through to
its conclusions. These approaches can generally be divided into two groups, exact methods that
provide not only an optimal solution but also a certificate which guarantees that it is as good as
possible, and in-exact or heuristic methods which generally provide a good solution but cannot
guarantee optimality. Of the exact methods there are many approaches such as na¨ıve
enumeration, dynamic programming, network methods, linear programming, maximum
satisfiability, and others. Heuristic approaches include a wide range of ad-hoc methods, simulated
annealing, genetic algorithms, tabu search, and others. In the following section several of these
52 |
Colorado School of Mines | approaches are considered in detail as they are used in the following chapters to develop high
quality solutions to relevant problems in open-pit mine planning.
2.5.2 Linear Programming
Linear programming is a technique whereby a linear system is analyzed to find a vector which
maximizes (or minimizes) some linear objective function subject to linear equality and inequality
constraints [128–130].
A straightforward linear program expressed in standard form with vector notation is given in
statements 2.10 to 2.12.
maximize cX (2.10)
s.t. AX = b (2.11)
X 0 (2.12)
≥
In this notation c are the objective function coefficients, X are the decision variables, A is the
i j matrix, where i is the number of rows (or constraints), and j is the number of columns (or
×
variables), and b are the righthand side of the constraints. It is possible to switch the sense of the
objective from maximize to minimize, or to change the equalities of constraints from to or =
≤ ≥
and still remain a linear program. But this format is preferred because it is straightforward to
transform any other formats to this one. For mine planning many problems can be expressed as
linear programs, which is very useful because linear programs are generally quite easy to solve
with the simplex algorithm or interior point methods [126, 129].
Every linear program has an associated dual, which is a closely related linear programming
problem which uses the same parameters. For a linear program in standard form the dual is given
in Equations 2.13 to 2.14.
minimize bV (2.13)
s.t. ATv c (2.14)
≥
AT is the transpose of the original A matrix and v are the new dual variables, which are
unrestricted when the original constraints are equality constraints. We have already seen the
53 |
Colorado School of Mines | application of duality to optimization problems in open-pit mine planning. In Section 2.2.3 the
dual of the ultimate pit problem was used to create an equivalent network flow model.
In many real world applications of linear programming it is desirable to restrict the decision
variables to integer values. This greatly increases the difficulty of the problem because duality is
no longer applicable, and many of the most useful theoretical developments are disrupted. Integer
linear programs (ILPs) are typically solved through a combination of analyzing their linear
relaxations and branch and bound, which is a exponential technique which enumerates many
integer solutions in order to find the best [126].
2.5.3 Lagrangian Relaxation
Lagrangian relaxation is a strategy that can be applied to integer programming problems to
help compute the linear-programming relaxation of very large models more quickly. In some cases
the Lagrangian relaxation can even be used to give a tighter bound than the linear relaxation,
although this is not guaranteed. The results from a Lagrangian relaxation model can also be
rounded to provide an heuristic ILP solution, or used for other purposes.
Lagrangian relaxation relaxes specific constraints from the input model but does not remove
them entirely. Instead, these relaxed constraints are dualized into the objective function and their
violation is penalized by using an appropriate multiplier - called a Lagrange multiplier denoted
with λ. For a particular constraint the new objective will have the new term in Equation 2.15.
(cid:88)
...+λ ib
i
a i,jX j+... (2.15)
−
j
Where λ is the Lagrange multiplier for this constraint, b is the right hand side of the
i i
constraint, X are the variables, and a are the constraint coefficients for each row i and each
j i,j
column j. The sign of λ is carefully controlled based on the sense of the objective and the
i
direction of the inequality in the constraint. A constraint of the form σ a X b requires a
i i,j j i
≤
non-negative (λ 0) multiplier when maximizing, and non-positive when minimizing. A
i
≥
constraint of the form σ a X b requires a non-positive (λ 0) multiplier when maximizing,
i i,j j i i
≥ ≤
and non-negative when minimizing. Equality constraints have unrestricted multipliers regardless
of the objective sense.
54 |
Colorado School of Mines | There are two important aspects of a Lagrangian relaxation that make it a valid operation.
• Every feasible solution of the original model is feasible within the relaxed model. This is
straightforward to see, because removing constraints will never exclude additional solutions.
• And, the objective value in the relaxed model for every feasible solution must be equal to or
better than the objective function in the full model. This follows because of the sign rules
on the λ multipliers, and how the new terms in the objective are constructed. A solution
i
which satisfies a constraint will have a term in the objective of the necessary sign. For
(cid:80)
example a constraint of the form a X b when maximizing, will have a term that is
(cid:32) (cid:33) j i,j j ≤ i
(cid:80)
of the form λ b a X where lambda is non negative and, because the constraint
i i i,j j i
−
j
(cid:32) (cid:33)
(cid:80)
is satisfied, b a X will be non-negative.
i i,j j
−
j
The primary goal when using Lagrangian relaxation is to determine the best possible bound
on the solution to the original ILP. However, if an optimal solution to the Lagrangian relaxation
is found such that it is feasible for the full model and either all multipliers are zero or the
constraint is satisfied at equality - the solution satisfies complementary slackness - it is optimal in
the full model.
In some cases, the solution to the Lagrangian relaxation will be a tighter bound on the integer
solution than the straightforward linear relaxation. However this is not guaranteed. If the
constraints which were chosen to dualize in the Lagrangian relaxation admit too easy of a model,
that is, one that can be solved by linear programming alone, then the bound will not be improved
[126]. The Lagrangian relaxation guided solver in Section 4.4.6 must contend with this fact.
Additionally there is a practical challenge which arises when using Lagrangian relaxation. The
values of the multipliers must be determined through some means, which can be difficult. It is
often possible to ascertain how to improve the multipliers after a solution is determined, and
potentially when improvement is unlikely which can inform when to stop. A popular method of
determining Lagrangian multipliers is by subgradient search. In this method the full collection of
Lagrange multipliers is updated at each iteration by using a step size and the subgradient which
is computed by looking at the previous solution’s relaxed constraints. That is, the new multipliers
are given in Equation 2.16.
55 |
Colorado School of Mines | λt+1 λt+s δλ (2.16)
i ← i t
Where λt is the multiplier for constraint i for iteration t, s is the step size for iteration t, and
i t
δλ follows in Equation 2.17. The step size, s , is a matter for some flexibility as well. But it is
t
known that a step size which converges to zero, when the sum of all step sizes does not, will
converge [126].
(b AXt)
δλ − (2.17)
← b AXt
|| − ||
Where b is the vector of right hand sides to the dualized constraints, A is the constraint
coefficients, and Xt are the variable values as calculated for the current iterations solution. All λ
values may need to be projected in order to satisfy the necessary sign conventions as discussed
earlier. If an optimal solution is found that satisfies complementary slackness we can terminate
with the optimal answer. However if this fortunate scenario does not occur, then the best solution
and the current bound can be reported once the step size has reached a very small number or
whenever further computation does not seem justified.
2.5.4 The Bienstock-Zuckerberg algorithm
The Bienstock-Zuckerberg algorithm (BZ) was first discussed in Section 2.3.2, where it has, in
recent years, seen use for solving specific instances of the open-pit block scheduling problem. At
its core BZ is an extension of the column generation approach to solving linear programming
problems [101, 102]. Bienstock has colloquially referred to the approach as “Column generation
on steroids”.
Column generation is an approach where the large scale linear program is decomposed into a
master problem and sub-problem. Column generation is primarily useful where the optimization
needs to address combinatorially many decision options that can be re-expressed as columns
(variables representing full solutions) in a partial master problem [126]. This partial master
problem considers the few columns currently available in order to inform a column generating sub
problem about which additional columns may be necessary in order to improve the objective. The
duals from the solution to the partial master are used to inform the column generating procedure,
56 |
Colorado School of Mines | which in the presence of complicating side constraints may be a heuristic. If there is no way to
construct attractive new columns then the model terminates with an optimal, or near-optimal,
solution to the original problem.
The BZ algorithm deviates from the conventional column generation procedure in two ways.
The first is to require the columns to always contain the optimal solution of the master problem
in the previous iteration, which is used at times during the algorithm to prevent the number of
columns from getting too large. The second is to construct the columns as orthogonal 0-1 vectors.
Orthogonal means that for each variable in the full master problem it obtains the value of one in
exactly one column and no others.
The BZ algorithm is most applicable to problems where there is a large submatrix consisting
of X X constraints which can be solved with a network flow procedure, and fewer knapsack
i j
≤
(cid:80)
constraints which are of the form: X y. The precedence constraints are placed in the
i i ≤
column generating subproblem, and both the precedence constraints and knapsacks are retained
in the master.
2.5.5 Heuristic Optimization
Linear programming, and other similar techniques, can admit exact solutions to a given
optimization model which are provably as good, or better, than any other possible solution in
terms of objective function value. A heuristic is an approach that admits a feasible solution, in
that it satisfies all necessary constraints, and generally tries to achieve as good a solution as
possible but it is not guaranteed to obtain the exact optimum. In general exact approaches are
much more satisfying and are preferred. If one is comfortable with the model they have developed
and any assumptions therein, the exact optimal result is going to give the best feasible solution
alongside a certificate that no other solution is going to be better. However this may not be
possible for large models which cannot easily be made smaller without sacrificing model validity.
In these instances a heuristic approach to optimization, either ad-hoc or following an
established meta-heuristic methodology may be appropriate. Additionally, often the losses from
settling for a heuristic instead of the exact optimal result will not exceed the losses and variations
from the concessions taken in the modeling processes or variations and uncertainties present in
the input data or parameters [126]. An exact optimal solution to a shaky model with uncertain
57 |
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