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  • Mining Methods and Applications - Some General Notes

Mining Methods and Applications - Some General Notes


Characterization of Fragmented Rock

The size of fragments and their distribution in the broken material can be classified by the size of the blast, the explosive distribution, the in-situ properties of the rock and the type of the blast (geometry, explosive distribution and confinement). The size of the blast may range from a few tons to thousands of tons while the confinement may vary from a completely confined shot to a surface burst as in secondary blasting.
Rock breakage by explosives involves the detonation of the explosive and the response of the surrounding or adjacent rock when it is subjected to the shock waves and the pressure of the explosion products. The breakage mechanisms occurring in rock fragmentation by explosives depend on the number of free faces, the burden, the hole placement, the explosive properties, the rock properties, the stemming and rock structure. Many theories for rock fragmentation exist. Many of these theories were developed on the basis of high speed movie observations of blasts or from small scale tests. Others have been derived from mathematical concepts while others are the result of pure speculation. Although there is a wide range of consensus. There has been some agreement regarding the rock breakage mechanism by explosives.
The main difference between the different theories proposed is in the amount of fragmentation which is caused by the strain pulse generated by the explosive and the amount of fragmentation attributed to the expansion of the detonation products.
Determining the fragmentation profiles using software products - WipFrag and Split - as well as other software is difficult because of inherent lighting in the underground environment along with the movement of equipment in drifts and drawpoints.  In addition, drawpoints need to have their contents wetted to minimize the dust that is generated.

Measurement of Fragmentation in Underground Mining Operations

Success to date regarding fragmentation measurement at scooptram buckets or drawpoints has been sporadic using optical granulometry. One of the chief reasons is lack of light in the underground environment. Most efforts attempt to use existing light sources that are available – headlamps, flood lamps and scooptram headlights. What is really required is to bring as much light as possible akin to bringing sunlight underground.
AEGIS Analyzer will have a fragmentation prediction element so that ring blasts can have a result metric associated with energy distribution and fragmentation profiling. In order to undertake any measurement in an underground mining operation, a new vision system will be required that deals with the light problem. It turns out that previous work at a large underground mine may have provided the means to develop not only a method for capturing broken ore photos reliably and consistently with sufficient detail that allows optical photographic measurements to be done, but also provides a method for grading ore (sulfides).
The picture below represents the means of trying to come to grips with a methodology for fragmentation measurement. A scooptram bucket is photographed using two lasers set a specific distance apart for scale (two red dots visible in the image). Unfortunately, pictures are usually dark and with the overwhelming quantities of fines in the much, not much detail around fragmentation sizes can be obtained. The scooptram represents is a poor means of sampling ore for fragmentation purposes. 
Laser dots providing scale. The above photograph shows an obvious difficulty using scooptram buckets for fragmentation measurement. The fines component is large normally and even the best image analysis techniques are not able to measure quantities of fines or even percentages of fines at a drawpoint. The intent is to provide the necessary specifications for light sources suitable to performing edge detection with enough detail to image fragment sizes between fines and chunks.
Obviously using buckets as a means of conveying size information is grossly inadequate. Fines are a consequence of borehole pressure, ore strength as well as borehole roughness. There is not a great deal that a mining operation can do Lowering borehole pressure doesn’t seem to be a viable way of reducing fines since the oblique geometry precludes reducing the strength of the detonation reaction. Accurate drilling would probably be a more tangible means of reducing fines with lower energies in the collar region (gassed emulsions or appropriate staggering). A most important result of the breaking process using explosives is to ensure that the ore profile doesn’t exceed the dimensions of the mantle leading into an ore pass. The photographic analysis would then be composed of fragment sizes from fines to the mantle dimension along with a chunk could provided by scooptram operators or beat foremen.
In the photograph shown, using broad spectrum white light (about 4.8 KJ), the ore appears as gold shaded fragments and rock as black chunks. The frequency range of the light (2000W – pulsed) must be carefully chosen to illuminate the back corners of drawpoints. The ore needs to be washed down to minimize dust. Not the red painted rock at the middle right in the picture which is the geologist’s estimation of the ‘rock off’ point. The camera is carefully chosen to match the broad spectrum strobe flash tube.
From the image above it shows very clearly that the rock off position (position beyond which ore could not be recovered without significant dilution) was in fact too lean with valuable ore being left behind. The proper photograph would serve two purposes;
1.     Provide better edge detection due to the increased power of broad spectrum light.
2.     A possible means of grading ore (has been completed for nickel bearing sulfides).
The placement of the vision system camera is critical to ensure a credible sample, either attached over the drawpoint with a video feed to recording equipment, or affixed to the front of a scooptram in an unobstructed view of the drawpoint. The light source is placed on the front of the scooptram. Sample photos are taken before the bucket draws muck providing a continuous stream of data that can be correlated to the previous blasting. This process of collecting data would provide not only an historical view of blasting practice and appraisal but also can be used to fine tune and adjust new blasting operations using the AEGIS Analyzer CMS recovery/dilution.
Its probably superfluous to used buckets as a means of characterizing fragmentation since whatever is in a scoop bucket is the right size for the ore pass. Drawpoints are a built-in sampling mechanism since all muck flows through them. I would be advantageous to do all of the sampling at the drawpoint using the scooptram itself. With the right pulsed light a continuous sample could be obtained since the drawpoint itself would be similar to moving conveyor belts on surface.

Lasers Used as Scale for Analyzing Fragmentation in Scooptram Buckets

Lasers used as as scale for measuring fragmentation in a scoop bucket.

Fragment Properties


Particle Sizing

Generally the methods used come from the field of mineral processing or dressing. The data from crushing and grinding processes are usually plotted as size-frequency curves on Cartesian coordinates or as cumulative distributions on Cartesian or semilog, or log probability paper.  The curve fitting is done in an attempt to obtain usable mathematical expressions that will represent the distribution curve with some accuracy.
A commonly used equation (Gates-Gaudin-Schuhmann) is the following:
particle size
distribution parameter
maximum particle size
The equation offers a direct correlation for cratering parameters and some crushing and grinding.
Another equation used is the Rosin-Rammler equation, which is expressed as;
is a constant
Both of these equations have the effect of expanding the fine-size region, while the upper limit of the Rosin-Rammler equation is infinite.
For large craters produced by blasting, the Schuhmann equation applies.
The Rosin-Rammler equation has been used by many for blasting analysis in the following form:
fraction of material retained on screen
screen size
constant called characteristic size
constant called uniformity index
The uniformity index typically has values between 0.6 and 2.2. A value of 0.6 means that the muckpile is non-uniform (dust and boulders) while a value of 2.2 means a uniform muckpile with the majority of fragments close to the mean size.
Another equation which is used in blasting is the Kuznetsov equation. This combined with the Rosin-Rammler equation form the Kuznetsov-Rammler model. This is normally called the Kuz-Ram model. The Kuznetsov equation is expressed as:
mean fragment in cm
rock factor
rock volume in m3 broken per blasthole
mass in kg of the TNT equivalent explosive per borehole
The previous parameters will be explained in the following;
It was found that A = 7 for medium rocks, 10 for hard, high fissured rocks and 13 for hard, weakly fissured rocks. Recently A was associated with rock mass description (powdery, friable, jointed or massive), the joint spacing, the rock density, the rock uniaxial compressive strength and Young’s Modulus. V is calculated a burden times spacing time bench height. Q is calculated as;
absolute weight strength of the
explosive (cal/g) and the factor 1260 is
the absolute weight strength of TNT.
From the above equations the following expression can be derived;
powder factor (kg/m3)
For x = xmean, then R = 0.5 and if this is substituted into equation 7.3 the following will result;
Therefore xmean can be determined from the Kuznetsov equation and therefore the critical size xc can be calculated if n is known. If xc and n are known then the distribution is known form the Rosin-Rammler equation. The resulting model is called the Kuz-Ram model.

Relating to Blast Design Parameters    

From all of the mathematics presented above, the usefulness of the equations must be such that the information on blast design parameters must have influence.
Therefore, the following equation has been proposed;
borehole diameter
standard deviation of drilling accuracy
spacing to burden ratio
charge length
hole depth
Of course, all models have their limitations, and this one is no exception. The following potential contributory weaknesses are;
   geologic structure has not been considered

Spacing and Burden and SB Ratio

Burden and spacing should be kept in a staggered configuration. The break overlap range should be within 2% to 12% for best coverage.
Staggered patterns usually provide two working faces
If B < Boptimum strain wave fracturing occurs rapidly and much of the heave energy is lost as air-blast and kinetic energy of the fragmented rock (flymuck).
If B > Boptimum the compressive strain wave is attenuated excessively by the time it reaches the face so that when it is reflected as a tensile wave the strains are too small to extend radial cracks to the face. Excessive burden distances inhibit flexural rupture. Also the pressure of the explosion gas products is not sufficient to unable to move or heave the muck. This will cause excessive vibrations. If the burden is too large, the expansion gases work against the horizontal face resulting in excessive overbreak. Also if structure is highly jointed, large chunks will form as shown in Figure 3.30. In multi-ring blasts the first row must break the burden completely otherwise the subsequent rings will work against one another and excessive burdens will gradually built up. Good blasting results can be obtained with S=B (square pattern). However triangular patterns with S = 1.15 B are more effective in hard rocks because energy is better distributed and there are more flexural forces at work as well as less splitting along the plane containing the blastholes. When S < B premature splitting occurs along the plane of the boreholes with more gases escaping from the collar resulting in some with adverse effects on . In fan drilling rings spacing decreases from the toe region to collar. Therefore the fans are drilled so that the spacing to burden ration in the toe is larger than 1.15.

Initiation Sequence

For optimum  the initiation sequence should be such that;
As stated previously, staggered patterns are considered preferable since two free faces are created for most of the charges. Initiation should start at the point which gives best progressive relief for the maximum number of holes.

Borehole Diameter

The blasthole or borehole diameter is influenced by
When blasthole diameter is small, the costs of drilling and blasting can be high, time consuming and labour intensive. Small diameter holes are prone to deviation over the length of hole drilled. In underground blasting, small diameter blastholes are usually confined to drifting, and very narrow ore veins. In narrow vein mining, distribution of explosive and thus powder factor/energy is important when near ore/waste rock contacts.
If the blasthole diameter is large with respect to the width of ore, with poor explosives distribution, with the large pattern producing  will result in coarse  as well as dilution. It is important to ensure that the blasthole diameter is several diameters removed from the minimum diameter of an explosive and more so with respect to the critical diameter. In large diameters of charge, commercial explosives exhibit behavior close to the ideal with maximum energy yield. However if  is to remain constant an increase in borehole diameter often means an increase of the powder factor/energy.
In heavily jointed ore that exhibits widely-spaced open joints, fewer larger diameter blastholes intersect a smaller percentage of effective blocks. When these joints are parallel to the blastholes the strain waves produced from the detonation of the charges will be reflected and fragment the rock between the borehole and the joint quite well. However the rock beyond the joint will be poorly fragmented. Therefore the spacing in this case can be adjusted to offset the effect of reflected joint surfaces.
In highly jointed rocks and in bad ground, the  is very dependent upon the orebody's structural characteristics. The energy of the gases of the detonation is more important in this case while the explosive generated strain waves are attenuated drastically. In these cases the borehole diameter can be increased.
Increased diameter also means increased stemming lengths. For this purpose a charge might have to be located in the stemming area in surface blasting operations so that the collar rock is fragmented.  very much depends on the ability of the first row or ring to break and displace the burden.

Blasthole Alignment

Alignment errors in drilling are any one or in worst cases all of the following;
The above drawing shows the various types of drilling errors responsible for borehole deviation.
Concentration of explosives in the collar region of rings usually manifests itself with easily broken burdens at the start but as powder factors decreases up the ring plane due to every increasing borehole spacings benching may result. Blasthole pressure at the toe is reduced preventing adequate breakage.

Blasthole Length

Due to blasthole deviation excessive blasthole lengths result in burdens and spacings different than the intended pattern design. This reflects on the  which can be excessive or inadequate. The deviation can occur within the plane of the ring or along the axis of the ring. In either case the break radius must be sufficient to overcome the limitations of burden and spacing.
From measurements taking using a variety of borehole deviation measuring instruments, the drilling accuracy figure can be anywhere from 1% to 10% of the total borehole depth. Some of the new automated drills claim accuracies of 1%, which means for a 30 m hole depth the radial accuracy is .3 m.
A graphical representation of an idealized ring plan along with one in which there is at least 5% deviation.

Highly Stressed Ground

When ground that is stressed from both gravity and tectonic stress, blasts should be loaded and detonated as quickly as possible. Some in-ground stresses are high enough to actually close the holes that have been drilled. Classic signs of stressed holes include elliptical closure with cracks at the apex of the elliptical shape as well as scabbing of the sides of the borehole.
Drill hole in highly stressed ground - note the cracking formed as well as scabbing on the borehole wall.

The Role of the Free Face

Generally the free face within a critical distance of a detonating explosive serves two purposes as follows:
As mentioned previously, when a compressive shock wave reaches a free face it is reflected and the rock may fail in tension (spalling). The reflected wave interacts with the growing outbound cracks formed by hoop stress, placing additional stress at the crack tips assisting in further growth and migration . Since longer cracks propagate more easily they develop preferentially. Therefore the reflected wave determines both which fractures develop and in which direction they travel.
Good  break is difficult to achieve for the cases in which:
 Slots should be designed so that all future rings are drilled parallel to the free face generated by the slot blast. Burdens must be constant between first ring to detonation and the free face. Too light a burden may cause a situation where the next ring to detonate is simply burdened so heavily that there is no opportunity for cracks to be developed into the next burden distance. Rings must be burdened enough so that there is some degree of back cracking into the next ring to be detonated. Each ring should be initiated with a timing design that allows cracks to be established ring burden to ring burden.
Choke blasting represents an important concern (blasting next to broken material). Sometimes there is a requirement to build up broken muck as inventory when mucking is tight. Although it is much more advantageous from the standpoint to blast to a void space, in underground mining it is recognized that this is not always possible and that blasting to broken ore is not only necessary but often desirable especially from a ground control situation.
In tunneling and drifting, blast designs are such that the major face is normal to the axes of the boreholes. Satisfactory advance depends upon development of an effective face that is parallel to the boreholes. Relief holes or empty holes are drilled parallel to the cut holes such that there is at least 25% void space in the cut that is developed. To prevent these empty holes from becoming choked, longer delays must be used.

Timing and Sequencing

Electronic detonator timing was derived from measured P wave velocities for these specific rock masses including S wave and estimated crack velocities under typical powder factors/energies representative of underground mining operations. Table 1 lists the properties used along with travel times;
Table 1. Delay Calculations Based on an Equation for Dynamic Motion
Rock/Ore Property
Transit Time
P wave Velocity
6010 mps,      19718 fps
0.166 х 10-3 spm ,     0.051 х 10-3 spf
S wave Velocity
3400 mps,      11155 fps
0.294 х 10-3 spm ,     0.089 х 10-3 spf
Crack Velocity
2160 mps,        7085 fps
0.463 х 10-3 spm ,     0.141 х 10-3 spf
Burden Movement Velocity at 30% detached
55 mps,          180 fps
18.182 х 10-3 spm ,     5.556 х 10-3 spf
*spm – seconds per meter.  *spf – seconds per foot
Delays were adjusted between holes and rows according to the timing constraints presented in table 2 below. From high speed cameras used in open pit face movement monitoring, burden movement velocities range from 15.2 mps (50 fps) to 30.5 mps (100 fps) roughly at 30% detachment. Little work has been done in underground blasting regarding the measurement of face movement with the exception of development. Since powder factors for underground mining are typically double those for surface mining, an assumption was made that the burden movement velocities would also double. The delay times for the inclusion of dynamic events was determined from the following equation (for single holes detonated). Between rings timing was 3 times this value or roughly 65 ms;
Note that the above equation takes into account the explosive column height, in this case, Tuc , which corresponds to the length of a unit charge. Substitute any value for the longest column height in the ring and use the calculated value as the delay between holes.
Table 2. Parameters Used to Calculate Delays Between Holes.
Input Values and Units
Length of unit charge
2.38 m
7.80 ft
Detonation time of unit charge
0.42 ms
0.42 ms
Length of explosive column
30 m
98.42 ft
Pattern Spacing
2.8 m
9.19 ft
Pattern Burden
2.5 m
8.20 ft
P wave velocity
6010 mps
19717 fps
S wave velocity
3400 mps
11155 fps
Detonation velocity
 5723 mps
18776 fps
Crack velocity
2160 mps
7087 fps
Gas expansion velocity
55 mps
Burden Percent Detached
Time delay for this hole
21.53 ms
21.53 ms
Normally, long holes are found near the center of the ring so the delay time for the longest hole would be used throughout the ring with 65 ms used between rings.

The Role of Flaws in the Rock Mass

Propagation of stress waves
It has been found by high speed photography that inherent flaws in the rock mass influence the rock fragmentation significantly. New fractures can developed at pre-existing ones. Old fractures reactivate or are intersected and reactivated along part of their length by new fractures. They serve as initiation sites for new fractures because they also provide compressive-tension reflective surfaces.
Obviously blasting results will be affected by the joints and their orientation. The joints represent surfaces such that stress waves will be reflected or suffer attenuation. Fragmentation will thus be affected by the presence of joints. The cracks which are aided by the rebounding tensile stress wave develop the most, extending the point of propagation along the crack tips.

Influence of Controllable Blast Parameters on Fragmentation

When an explosive charge detonates it creates a radially expanding strain wave in the surrounding rock. The wave is responsible for the fragmentation of the rock in the vicinity of the charge. The energy of the expanding gases is responsible for extending the cracks and for displacing the rock. However, evening impacted the boundary of the free face, the tensile component reflects backwards into the burden again opening up cracks and discontinuities. Optimum fragmentation is the aim of blasting since there are obvious economic advantages associated with it. Oversized fragments necessitate secondary blasting.
Effect of Fragmentation on drill, blast and muck costs
On the other hand, fines represent a waste of the drilling and blasting expenditure and have some real detriment effects not only in material handling but in underground mining may represent a substantial portion of oxidized muck which is difficult to reprocess and may get diverted to tailings as part of the waste feed to the mill. When blasthole diameters are small, the costs of drilling and blasting are high, time consuming, and labour intensive. If the diameter is too small these disadvantages outweigh the benefit of the distribution of explosive and thus powder factor. If the diameter is large with poor explosives distribution, the large pattern will result in coarser fragmentation.
With larger diameters, commercial explosives exhibit behavior close to the ideal with maximum energy yield. However if fragmentation is to remain constant an increase in borehole diameter often means an increase of the powder factor.
In strata which exhibits widely-spaced open joints, fewer larger diameter blastholes intersect a smaller percentage of effective blocks. Where these joints are parallel to the blastholes the strain waves produced from the detonation of the charges will be reflected and fragment the rock between the borehole and the joint quite well. However the rock beyond the joint will be poorly fragmented. Therefore the spacing in this case should be reduced.
In highly jointed rocks and in bad ground, the fragmentation is totally dependent upon the materials structural characteristics. The energy of the gases of the explosion is more important in this case while the explosive generated strain waves are attenuated drastically. In these cases the borehole diameter can be increased.
Increased diameter also means increased stemming lengths. For this purpose a charge might have to be located in the stemming area in surface blasting operations so that the collar rock is fragmented. Fragmentation very much depends on the ability of the first row or ring to break and displace the burden. UnitCharge

A New Model For Predicting Fragment Size Due to Blasting - Jonathan Zimmerling for iRing AEGIS Break Analyzer, November 30, 2011

1. Introduction

The current aim is to develop the most complete model possible for fragmentation due to blasting in underground mines, while still maintaining relatively short computation time in order to facilitate optimization. We are currently using the Single Ring model put forth by Onederra [1], and although up till now it was, to the best of our knowledge, the best underground model, it is far from comprehensive and may be overly simplistic. This is by design, as the Single Ring model is developed as an engineering tool to provide rough guidelines, not as a rigorous description of physical processes. A full description of Onederra's model will not be given here, for a summary of our implementation see Alkins [5]. A key issue with the model (although perhaps not the most glaring) is the application of peak particle velocity in order to determine a break radius around a blasthole. In Onederra's model, a linear interpolation of a set of points is used to create a peak particle velocity field using the Holmberg- Persson model of peak particle velocity [4]. Since Onederra's model is an engineering tool, the need to calculate the full field is unnecessary. It can also be argued that peak particle velocity gives an acceptable approximation of stress for this model, since it is just an engineering tool. However, it is known that, in the case of a cylindrical charge, the stress will not be proportional to the peak particle velocity. Further, the Holmberg-Perrson model of peak particle velocity (which is what is used in Onederra's model) is physically inaccurate, and makes some unreasonable simplifying assumptions.
The Holmberg-Persson model does not give a model of particle motion, rather it seeks to give an estimate of what the highest velocity of each particle will be. The values generated by the Holmberg- Persson method are empirically fit to observed data within Onederra's model. It behooved us to pursue other avenues of estimation to attempt to achieve a more realistic (and thus more statistically significant) prediction of a break radius. Work began by investigating the Scaled Heelan Model [2] as a potential replacement for the Holmberg-Persson approximation as a piece of Onederra's model.
Although the results show that is would be a reasonable step to take (along with re-calibrating the tuning coefficients to take into account the new predictions), research has led to the development of an entirely new model of fragmentation. This new model uses the physical processes to give a fragmentation distribution, and is based on material and explosive properties. It currently has restrictions and shortcomings, but is not merely contained to either above or underground blasting – it addresses both. It is clear that more work needs to be done before this model can be applied to a full set of rings, but it shows a lot of promise.

2. The Scaled Heelan Model

Blair and Minchinton [2], [7] suggest using equations based on Heelan's work on particle displacement caused by a cylindrical element of charge [3]. This gives a full description of displacement over time, which can be used to determine velocity, strain and stress. The development of these equations is given in a previous paper [6], but is repeated verbatim here for completeness.
Heelan equations describe the displacement of particles over time in an isotropic, homogenous continuum due to cylindrical waves emanating from a single charge element. Of course an ore body is not an isotropic, homogenous medium but these simplifying assumptions are necessary for any physical model of wave behavior. Moreover, a charge in a blasthole cannot be represented as a single charge element, as the detonation begins at a priming point and occurs over a time period. However, we can
divide the charge into M smaller charge elements, with each one having a time of detonation that can be expressed as a function of detonation velocity and distance from the priming point. So for any time t and point P(r, z) we can sum the contribution to displacement of every element of charge to find the total displacement (Figure 2 shows the contribution of elements of charge towards velocity). Thus we have a physically accurate model of particle displacement that incorporates priming point and timing.
Using these displacements we can determine velocity, strain and stress, which, in conjunction with yield criteria found in the literature (as will be discussed in sections 3 and 4), will give a more realistic approximation of rock failure than the Holmberg-Persson model.
Figure 1:  Contribution of individual elements of charge to the velocity of a point.
The blasthole source pressure-time function is given by;
......... 1
Figure 2:  Pressure Time Plot of Source Function
where b and n are constants, is the Heaviside step function (0  when , 1  otherwise) and is a normalizing constant (given in GPa/tn). Blair makes the substitution;
Assume that the first charge element detonates at t = 0. Given any ordering of the charge elements (it makes sense to order them in terms of their detonation order) then a dimensionless time associated with the detonation of the mth element is given by;
.......... 2
where is the length of each individual charge element, and ve is the velocity of detonation.
The Heelan equations describe the motion  , ) of a particle due to a blast from a cylindrical source at time Ƭ as;
.......... 3
•     z is the vertical axis of the cylinder (m)
•     r is the axis perpendicular to the z-axis (m)
•     Rm is the distance from the center of the mth charge to the point of interest (defined in equation 6) in meters
•     is the angle between the z-axis and (as illustrated in Figure 1)
•     is the P-wave contribution of the mth element of charge at time (defined in equation 4) in meters
•     is the S-wave contribution of the mth element of charge at time (defined in equation 4) also in meters
.......... 4
•  Vm is the volume of the mth charge (m3)
•  ρe is the density of the explosive (kg/m3)
•  μ is the rock shear modulus (Pa)
•  vs is the S-wave velocity (m/s)
•   vp is the P-wave velocity (m/s)
.......... 5
Figure 3: Derivative of the source function.
Now, the distance between the center of the mth charge element and  P (r, z) is
.......... 6
where  zm is the distance between the z-axis component of the m th charge element () and point P(r, z) given by;
.......... 7
Furthermore, the sine and cosine of the angles can be written as;
.......... 8
Now define;
   .......... 9
.......... 10
Using these equations we can then compute the strain tensor, given by;
.......... 11
Again, these equations can be broken up into four principle parts, then partial derivatives can be taken with respect to r and z. Equation 12 shows the partial derivatives with respect to r, the partial derivatives with respect to z are similar.
.......... 12
•       and  are given in Equation 13
•     Pp    and Ps are the derivative of the source function with respect to Ƭ , shifted by the P-wave and S-wave velocities, respectively and shown in Equation 14
 .......... 13
.......... 14
In order to calculate  , , ,   we must first solve the derivative of P' with respect to Ƭ , which is;
..........  15
Now, we will make a simple substitution to make notation more elegant (and to reduce the number of computations required). Below, let;
...........  16
and recall
..........  17
which gives
 .......... 18
From the strain tensor given in Equation 11 we can calculate the stress tensor, which is;
........... 19
This tensor can be reformulated with respect to the principal axes of stress, which are the eigenvalues of the matrix given in Equation 19. These principle stresses are;
..........  20
Note that these stresses are not ordered by magnitude.
Without intimate knowledge of the vibrational load of an explosive at the blasthole walls it is impossible to accurately determine . Unfortunately, little is currently known, and the model must be scaled in another way in order to achieve accurate absolute values of strain and stress. In order to do this we use the familiar charge weight scaling laws (where K, A, and B are site constants, ɣn is a constant based on scaling the maximum of the waveform to 1, and  wc is the mass of each charge element). The assumption that is 0 is made throughout the remainder of the paper. This is known as the Scaled Heelan Model.
Our equations for displacement become;
..........  21
with partial derivatives (shown only with respect to r for brevity)
.......... 22
 ..........  23
With partial derivatives
 .......... 24
The “recipes” given in equations 10, 11 and 12 remain the same, and from these equations the values of the stress tensor may be calculated.
The strain rate tensors are given by
.......... 25
where vr is the point's velocity with respect to the r axis, and is the derivative of displacement (ur) with respect to time. Now
..........  26
 ... by Clairaut's Theorem.
Taking the derivative with respect to time of the partial derivatives given in equation 22 yields
  ..........  27
Since     and    are invariant over time, it follows that
  ..........  28
  ..........  29
..........  30
This leads to
 .......... 31
and since
  ..........  32
we have a full characterization of the component of the strain rate tensor. The other components may be computed similarly.
From here, the volumetric strain, Δ s , and the volumetric strain rate,  Δ' can be found using;
.......... 33
This model can take timing, priming, site/explosion specific attenuation and local rock properties into account. To perform a full field calculation it is rather computationally expensive, but interpolation techniques can be applied. We also currently have no method to incorporate free face effects, but work is currently being done in this direction.

3.  Liu and Katsabanis Model of Rock Damage

Liu and Katsabanis [8] propose a model of continuum damage based on continuum mechanics and statistical fracture mechanics. Damage is a function of rock failure probability, which is a function of strain and time. Again, the details of this model are given in an earlier paper [9], but are repeated here with slight modifications.
It is intuitive that damage is a function of both strain and time. Liu and Katsabanis suggest that fragmentation due to tensile strain is caused by nucleation of pre-existing microcracks as a process of time. When tensile strain exceeds the value of the critical tensile strain of a material (which is proportional to the static tensile strength) crack growth occurs, otherwise there is no effect. Over time, the tensile strain will extend these microcracks until a full crack is formed, fragmenting the rock. In fact, it is suggested that gas pressure does not play an important role in determining fragment size, but rather expands existing cracks and pushes the already fractured rock masses apart.
Liu and Katsabanis posit that for a mass point contained in a volume V, assuming a constant volumetric strain Δ, the crack density within a local area of the mass point at time under tensile strain t can be modeled by
......... 34
where  are material constants, and  is the critical volumetric strain. can either be analytically determined using Young's modulus, Poisson's ratio, fracture stress at a given strain rate and the static strength or it can be empirically determined using the results from a calibrated blast.  is typically taken to be 2.
In a blasting process, volumetric strain is also a function of time. Equation 34 can be reformulated in differential form as
.......... 35
Integrating over time yields
.......... 36
.......... 37
where v is Poisson's ratio, E is Young's modulus and Sc is static tensile strength. This is simply a conversion from stress to strain. Thus, for any point, if we know the volumetric strain and the volumetric strain rate as functions of time we can calculate the crack density at any point in time.
Damage, D, is modeled throughout the ore body using the probability of failure for a material point with volume V and crack density Cd, given by
.......... 38
This is based off the work of Lundborg [10], who formulated a statistical theory of failure of brittle rock based off of Weibull's statistical approach [11]. Given a fixed crack density, it is more probable for a large volume to fail than a small volume. Moreover, if the volume of the fragment containing a material point decreases, the probability of failure will also decrease, and given a constant strain will reach equilibrium. Although strain due to blasting is not constant, eventually the strain and strain rate become zero, and no further change in the crack density will occur. As the rock is damaged, the physical properties change. The degradation of Young's modulus is modeled by
..........  39
where E is the Young's modulus of the undamaged rock. It is assumed that Poisson's ratio remains constant in order to preserve the relationships between elastic parameters. For known strain and strain rate, this gives a model of the damage of rock and rock properties at any material point for any time under strain. Observable damage is likely to occur when  , when there is a full crack in the volume of interest.

4.  Grady Model of Fracture

Grady [12] posits a model of fragmentation based on the conservation of energy. Suppose a material point is undergoing expansion. The kinetic energy associated with the expansion provides the fracturing force, while the increase in surface tension caused by new fractures resists the fracture process. Once fragmentation has occurred, momentum will be conserved (i.e. the new fragments will continue to travel according to the direction of expansion). Intuitively this means that only a portion of the available kinetic energy will be used during the fracture process.
Suppose we have a sphere of mass, with radius r and density ρ .  Suppose there is some strong impulse acting on this sphere (i.e. a nearby blasthole detonation). In particular, suppose that the sphere has been compressed and is currently in a state of rapid uniform expansion (at a density rate ρ' ). When the volumetric strain exceeds some critical value associated with the material the sphere will fragment. The particles traveling from the center of mass will consume part of the energy, and the rest will be used by the creation of new fracture surfaces. Let T' be this local kinetic energy, given by
..........  40
Dividing by the volume and expressing in terms of the fragment surface area to volume ratio (A=3/r) yields
.......... 41
where T is the local kinetic energy density in terms of the surface area created by fragmentation.
The surface energy density of the new fragment ( Γ ) is given by
..........  42
where Ɣ is the specific surface energy calculated as
..........  43
and where is the fracture toughness and c is the speed of sound in the medium (given by , where E is Young's modulus). So the total energy density of the system is given by
.......... 44
Minimizing the energy with respect to the surface area will give the size of the surface area at equilibrium.
..........  45
Now,    can be expressed in terms of the volumetric strain rate    as follows
..........  46
Intuitively this makes sense, because when volumetric strain is changing in the positive direction, density will be decreasing and vice versa. This gives;
..........  47
We are most interested in passing size (diameter, assuming a sphere), which is given by;
..........  48
So, given a known strain rate, density, fracture toughness and Young's modulus we are able to predict the resulting fragment size.

5.  The Proposed Model

The proposed model requires some mean measure of sample block size (used to calculate the volume of the block containing the material point of interest), Young's modulus and Poisson's ratio (or P- and S- wave velocities), rock density, ɑ,β,  static tensile strength of the rock, fracture toughness, charge length, borehole diameter, explosive density, VOD, and charge weight scaling law coefficients. From the Scaled Heelan model we can compute volumetric strain and the volumetric strain rate for a point at any given timestep i. This can be used to determine time under volumetric strain greater than the critical volumetric strain, which, along with volumetric strain and the volumetric strain rate, can be used to determine crack density. Crack density, in conjunction with volume, will determine the probability of failure of the rock.
If  then equations 47 and 48 will be used to predict the passing size of the new fragment, assuming a spherical particle. The degraded Young's modulus from equation 39 will be used, and the density will be found iteratively using;
..........  49
The diameter of the new particle will be used to compute the volume used in the next time step. So as the volume decreases, the probability of continued fracture will also decrease until the equilibrium is reached, or volumetric strain decreases below the critical volumetric strain. This will lead to a fragmentation being dependent on the in situ block size, the rock properties, and the explosive properties, all of which are measurable.
It should be noted that this only models damage caused in brittle solids by positive volumetric strain (i.e. tensile strain). To some extent this is acceptable, as the majority of rock broken in a blast is broken by tensile strain, since tensile strength tends to be much lower than compressive strength. Given that compressive failure around the borehole is widely thought to be the main contributor to fines from blasting (d < 1 mm), it is important to in some way model or approximate the volume of these fines. If rock is to fail compressively, it makes sense that it must first pass its threshold of tensile failure, and deform in some way. In may be possible to model compressive failure via iterative tensile failure (i.e. the rock will fracture until fines are created. In fact, the combined model does lead to fines around the borehole (and elsewhere throughout the break envelope).
Another method is given by Onederra [1]. Onederra proposes an equation to calculate a radius of crushing, wherein everything is reduced to fines. This approximation is given by
.........  50
where rc is the crushed radius, r0 is the borehole radius, Pb is borehole pressure, Ed is dynamic Young's modulus, vd is dynamic poisson's ratio and σc is the uniaxial compressive strength. Borehole pressure is calculated using an equation given by Liu and Katsabanis [13], which is
..........  51
Below is a comparison of the two methods, holding all properties equal except for in situ block size.

Common Elements

Explosive Properties
VOD (m/s)
Length of Charge (m)
Borehole Radius (m)
Density (tonnes/m3)
Rock/Ore Properties
Youngs Modulus
Poissons Ratio
Static Tensile Strength (MPa)
Static Compressive Strength (MPa)
Fracture Toughness
Charge Weight Scaling Law Coefficients
In Situ Block Size-0.5m
Combined Model
Onederra's Borehole Crushing in Conjunction with the Combined Model
Total Mass of Rock
Fractured (t)
5% Passing (m)
95% Passing (m)
% of fines (< 1 mm)
Total Mass of Fines (tonnes)
In Situ Block Size-1m
Combined Model
Onederra's Borehole Crushing in Conjunction with the Combined Model
Total Mass of Rock
Fractured (t)
5% Passing (m)
95% Passing (m)
% of fines (< 1 mm)
Total Mass of Fines (tonnes)
When the average block size is 0.5, both methods return identical results. The method of using tensile failure of the rock to model fines approaches Onederra's method. However, as the average block size increases, we see the methods diverge. The tensile model returns fewer fines, and in fact fails to model break near the crush zone in certain areas. This leads to a lower total mass of fracture, which is intuitively incorrect (everything within the break volume should undergo some measure of fracture, and things sufficiently close to the borehole should be severely damaged). This is illustrated in figures 4 and 5.
Figure 4:  Fragment Size Distribution Predicted Using the Combined Model
Figure 5:  Fragment Size Distribution Predicted Using the Combined Model in Conjunction with Onederra's Crush Zone
These results suggest that incorporating Onederra's crush zone model of compressive failure will give a more realistic fragmentation profile than using just tensile failure. From these results we also see that the fragmentation profile and the break envelope also both rely heavily on the choice of starting average block size. Larger blocks are more likely to fail, creating a larger break volume, and the new fragments formed by lower strains will be larger than those closer to the column of charge. This seems reasonable, as the joints and seams in the ore body should play a major role in fragmentation.

6.  Blair and Minchinton Pressure-time Function vs. Liu and Katsabanis Peak Pressure Function

It should be noted that equations [1] and [51] model the same phenomenon: borehole pressure. Equation [1] gives borehole pressure as a function of time (a large pulse followed by a decrease to zero) whereas [51] gives a peak value. It is important to make sure that these approximations are in agreement with each other.
Beginning at equation 21 we make the substitution;
..........  52
which gives;
..........  52
Using experimental methods, we can find the true pressure at various distances, and from here can test the validity of our borehole pressure-time function, as well as estimate peak borehole pressure using different methods.