1 INTRODUCTION

Cave mining methods allow for the bulk extraction of large and low grade ore bodies in a cost effective manner. The caving process involves undercutting (blasting a horizon of in situ rock mass) and extraction of the broken rock from drawpoints on a production level lo-cated at depth. When the plan area of the mining footprint reaches a large enough dimen-sion, a self-sustained propagating cave will develop as long as the broken and bulked ore is continued to be withdrawn. A schematic diagram of a typical block cave mine is layout pro-vided in Figure 1a.

Figure 1. (a) Schematic representation of a typical block cave mine (b) Main behavioral regions of a

propagating cave.

A historical review of the development of numerical cave propagation simulations

B.L. Sainsbury Itasca Australia Pty Ltd, Melbourne, Australia

D.P. Sainsbury Itasca Australia Pty Ltd, Melbourne, Australia

M.E. Pierce Itasca Consulting Group, Inc., Minneapolis, MN, USA

ABSTRACT: Cave mining methods allow for the bulk extraction of large, low grade ore bodies in a cost effective manner. The fundamental mechanics of caving involves the self-propagating yield (failure) of an in situ rock mass in response to production draw from a mining horizon lo-cated at depth. Since the inception of cave mining methods in the iron ore mines of northern Michigan, USA, during the early part of the 20th century, researchers have sought to understand and predict the nature of cave propagation through simple one-dimensional volume based rela-tionships and empirical methods. Although these methods have successfully been applied to many cave operations to-date, numerical modeling is considered to be able to provide a more fundamental, rigorous and robust assessment of cave propagation behavior. Over the past 12 years, through funding provided by the International Caving Study (ICS) and Mass Mining Technology (MMT) Projects, significant advancement has been made with respect to the devel-opment and validation of numerical caving methodologies.

2 CAVING TERMINOLOGY

A conceptual model of caving has been developed by Duplancic & Brady (1999). The model includes four main behavioral regions - presented in Figure 1b. The characteristics of each region have been derived from in situ monitoring, underground observations and numerical simulations. They are defined below. − Elastic Region -- The host rock mass around the caving region behaves mainly

elastically and has properties consistent with an “undisturbed” rock mass. − Seismogenic Zone -- Microseismic (and sometimes seismic) activity is concen-

trated in this region primarily due to slip along pre-existing discontinuities and the initiation of new fractures.

− Yielded Zone – The rock mass in this region is fractured and has lost some or all of its cohesive strength and provides minimal support to the overlying rock mass. Rock mass within the yielded zone is subject to significant damage, i.e. open holes are cut-off, TDRs break and cracking is observable in infrastructure. Stress components within this region are typically low in magnitude.

− Mobilized Zone – This zone gives an estimate as to the portion of the orebody that has moved at least 1-2 m in response to the production draw and may be re-coverable.

3 ANALYTICAL AND EMPERICAL METHODS OF CAVEABILIY ANALYSIS

Since the inception of cave mining methods during the early part of the 20th century, re-searchers have sought to understand and predict the nature of cave propagation. Rice (1934) and later Panek (1984) developed simple one-dimensional volume based relationships to characterize cave behavior according to an assumed bulking behavior, as illustrated in Fig-ure 2.

Figure 2. Relationship between thickness of caved ore withdrawn (tc), thickness of in-place ore ex-

tracted (ti) and rise of the caving brow (∆h) (after Panek 1984). Simple volumetric relations are still used today by many researchers to estimate cave

propagation rates (Beck et al. 2006). However, the simplifications assume that (a) cave initi-ation always occurs, and that, (b) cave propagation (and mobilization of material) is always vertical.

Since the inception of cave mining methods, the use of empirical methods to estimate cave behavior have been, and still are, widespread. The most commonly used approach for estimating cavability was developed by Laubscher (Diering & Laubscher 1987, Laubscher 1990, 1994, 2001) and is based on a compilation of rock mass geotechnical characteristics and caving case histories largely derived from low strength kimberlitic deposits in South Af-rica. Laubscher’s chart (Figure 3a) defines, three possible caving states that include; “no caving”, “transitional” whereby the cave initiates, but propagation is minimal and “caving” whereby self-sustained propagation occurs.

Many mines still use Laubscher’s chart to estimate the undercut dimensions required to induce continuous caving. In most cases, good agreement is achieved. However Lorig et al. (1995), van As & Jeffrey (2000), De Nicola Escobar & Fishwick Tapia (2000) have previ-ously reported on instances where significant differences were observed. A detailed review

of these cases by Trueman & Mawdesley (2003) showed that the biggest difference in actual outcome versus prediction was associated with strong (MRMR greater than 50) rock masses and misinterpretation of the application of adjustments in the MRMR rating scheme. As a result of this review, Trueman & Mawdesley proposed an alternate method for the prediction of continuous caving conditions through an extension of the Mathews stope stability chart (Mathews et al. 1980) that included data from non-caving operations. Their extended stabil-ity chart is provided in Figure 3b.

By necessity, empirical methods are limited by the dataset they are developed from. In his review of cave mining practices, Brown (2003) reasons that numerical modeling enables a more fundamental and rigorous assessment of cave initiation and propagation behavior than empirical methods, since it may have advantages in cases for which current experience is lacking.

Figure 3. (a) Laubscher’s stability chart, after Laubscher (1994) (b) Extended Mathews Stability Chart, after Trueman & Mawdesley (2003).

4 NUMERICAL MODELING OF CAVE BEHAVIOR

There are numerous numerical modeling methods (Boundary Element, Finite Element, Fi-nite Difference, Distinct Element, hybrid based) and approaches available for performing stress and deformation analysis in geomechanics. The important aspect of cave modeling is not necessarily the numerical program itself, but the methodology for simulating the caving process, and the estimation of input material models and properties. The following review attempts to report on the development history of numerical caving methodologies from 1970 (the time of the first known numerical caving simulation) to the state-of-the-art routines that are currently being applied in pre-feasibility and feasibility stage geotechnical studies.

4.1 Two-dimensional elastic models Soon after the introduction of the Finite Element Method (FEM) for the numerical analysis of stresses and displacements in continuous structures (Clough 1960), Palma & Agarwal (1973) from Columbia University, New York, developed the first known two-dimensional, elastic, FEM model to study cavability at the El Teniente Mine in Chile. During the plan-ning of an expansion of the existing cave operations into a stronger and more competent ar-ea, researchers Palma & Agarwal (1973) identified the need to consider the nature of the in situ rock mass fracture network and the impact of principal stress direction in relation to the undercut on cavability. The impact of the fracture network was represented in the numerical simulations by assigning zero tensile strength to all zones. Although not many details of the modeling methodology are provided in the literature, it is clear that yielding of the rock mass immediately above the simulated undercut was assumed to propagate when a tensile stress component was identified within a zone. Figure 4 presents results that clearly show the im-pact of cave height based on in situ stress and orientation of the mining footprint. Although these simulations assumed that caving only occurred as a result of tensile failure, they were

fundamental in understanding the importance of the orientation of a rectangular mining block with respect to the in situ principal stress direction.

Figure 4. Impact of principal stress orientation in relation to undercut (after Palma & Agarwal 1973).

4.2 Two-dimensional plasticity models Through the application of two-dimensional FEM simulations at the Grace Mine, located in Pennsylvania, USA, Barla et al. (1980) introduced a softening material model to represent the degradation of the in situ rock to a fully weakened and bulked state during the caving process. The use of such a material model highlighted the understanding, that caving may not only occur due to a tensile (gravity) mechanism, but also a stress (compressional yield-ing) mechanism.

The softening behavior was simulated through a periodic review of the failure states in the zones. If a zone failed via a compressional or tensile mechanism, then the strength, density and stiffness were reduced to a residual value. In addition, production draw was simulated through a force application, determined from querying stresses at the undercut roof. Figure 5 provides a schematic representation of the modeling methodology used to simulate the un-dercutting process and some of the model results (Figure 5d).

The work reported by Barla & Boshkov (1980) provided a greater understanding the changes in strength, modulus and density that accompany caving and the importance in be-ing able to accurately represent the mining process within the numerical model.

During the early 1990’s, Rech & Lorig (1992) conducted two-dimensional, finite differ-ence analyses with the numerical code FLAC (Itasca 1991) to reproduce the existing condi-tions at the Henderson Mine, Colorado, USA and predict the expected cave behavior. These are the first simulations that attempt to correlate production schedule tonnes with the simu-lated advance of the cave. The cave zone was initialized within the model through a number of mining increments that corresponded to the historical and planned production schedule. Vertical draw and a bulking factor were assumed based on the volumetric equations outlined in Panek (1984). Stresses were reset to zero within the cave zone and the rock mass proper-ties were reduced to those consistent with a fully-bulked rock. Simulation of a fully-bulked and reduced vertical stress conditions within the caved rock mass ensured that the mining induced stress magnitude and directions were accurately captured. However, by initializing the caved rock mass within the model, and by artificially reducing stresses within the cave, mass and energy were not conserved.

Figure 5. (a) Model mesh (b) section through the mining geometry (c) simulated undercutting pro-

cess (d) contours of resultant mobilized strength – the shaded area represents a fully sof-tened/caved rock mass (after Barla & Boshcov 1980).

During the International Caving Study (ICS), Karzulovic & Flores (2003) considered the

influence of depth, stress, structure, rock mass strength and groundwater on cavability through a sensitivity analysis with the two-dimensional FEM code Phase2 (Rocscience 2002). The caving methodology employed forces vertical cave propagation. It was assumed that rock mass breakage should only occur in a “window” with a width equal to 10% of the undercut width (i.e. if the undercut width is 100 m, then the vertical propagation of the cav-ing requires the breakage of the upper part of the cave along a width of 10 m), as illustrated in Figure 6a.

Based upon stress redistribution around an imposed cave shape, a Cave Propagation Fac-tor (CPF), which is the ratio between the average deviatoric stress acting on the cave back and the maximum deviatoric stress that the rock mass can sustain, was used to determine if caving is problematic, transitional or self-sustained.

Although simple assumptions were used in the representation of geometry, stress redistri-bution (2D), rock mass plasticity and post peak rock mass behavior, assessment of the CPF at the Northparkes Lift 1 Mine provided good correlation with the actual performance of the cave that stalled in 1999 – as shown in Figure 6b.

4.2.1 Axis-symmetric strain softening models In an attempt to include a better representation of the three-dimensional shape of the under-cut and surrounding stress field, during the International Caving Study, Lorig (2000) (also reported in Brown 2003) conducted axis-symmetric simulations of cave propagation using FLAC (Itasca 2000). A cylindrical undercut located at varying depths was considered. The initial state of stress within the models was assumed to be lithostatic and stress boundaries (a support pressure) were imposed at the excavated undercut level to ensure initial stability. To simulate mining, the support pressure was monotonically reduced in the roof of the undercut (similar to the approach of Barla & Boshcov 1980) and the extension of the yielded rock mass (represented by a strain-softening material) was assessed.

Through this approach, even though the true three-dimensional geometry and stress tensor were not accurately represented, Lorig (2000) was able to predict a hydraulic radius (HR) associated with cave initiation and self-sustained instability that compared well to

Laubscher’s empirical cave stability chart. A schematic representation of the modeling methodology used to simulate production draw from these axis-symmetric models is provid-ed in Figure 7.

Figure 6. Cave Propagation Factor at Northparkes Lift 1 Mine (after Karzulovic & Flores 2003)

Figure 7. (a) axis-symmetric model (b) evolution of the undercut pressure and height (c) stepwise

reduction of undercut pressure (d) details of the pressure evolution with a simulated reduction step (after Brown 2003).

These simulations showed that cave propagation is dependent on the post-peak brittleness

of the rock mass, i.e. as the material was made more brittle, the cave height increased. Trueman & Mawdesley (2003) suggest that continuum methods that use a strain softening model are not robust as they are highly sensitive to mesh size and the post peak material properties which impose mesh dependency within the numerical results. However, Lorig (2000) showed that, provided the mesh-dependency was considered by introducing a stand-ard length scale to the numerical mesh and the critical plastic strain softening input parame-ters, that the cave simulation results are repeatable with different sized meshes. This scaling approach, also termed the Standard Regularization Method has also been used by other re-searchers to account for mesh dependency in the numerical simulation of other geomechani-cal processes (Crook et al. 2003). Table 1 provides a summary of Lorig’s results.

In addition, an analysis of the Northparkes Lift 1 cave was completed, as illustrated in Figure 8. For each expanding undercut increment, the resulting cave height was assessed. A hydraulic radius of 42.5 m was required to reproduce the observed in situ cave height. This is consistent with the stalled undercut geometry. Although the cave height was reproduced, it is clear the shape of the cave volume (flat-back) is not necessarily realistic.

Table 1. Cave height as a function of brittleness (after Brown 2003). _____________________________________________________________________ Grid Critical Strain (εs

crit) Cavern Height (m) _____________________________________________________________________ Coarse 0.1 160 Fine 0.02 150 Coarse 0.005 200 Fine 0.01 205 Coarse 0.0025 225 Fine 0.005 250 _____________________________________________________________________

Figure 8. Cave geometry resulting from a numerical simulation, hydraulic radius = 42.5m (a) dis-

placement vectors (b) softened zone (after Lorig 2000).

4.3 Distinct element models Two dimensional, distinct element models were developed by Lorig et al. (1995) with the PFC code to provide a richer understanding of the fracturing of the in situ rock mass and an improved predicted shape of the cave region. Conceptual models of a cave in a high initial stress state were developed and two fundamental failure mechanisms associated with cave propagation were identified- intact rock block failure and slip along pre-existing joints. The model results are presented in Figure 9a and Figure 9b. Brown (2003) reports on the exten-sion of the two-dimensional particle models to three-dimensional axis-symmetric models during the International Caving Study – results of which are provided in Figure 9c.

During the Mass Mining Technology Project (MMT I) Synthetic Rock Mass Modeling (SRM) was developed (Pierce et al. 2006a, Mas Ivars et al. 2008) to allow for the detailed consideration of the rock mass joint fabric in the determination of rock mass response at large scales – i.e. 10 to 100 m. The SRM methodology uses PFC3D (Itasca 2007) to explicit-ly represent a discrete fracture network (DFN) embedded within an intact rock matrix. The resulting SRM can explicitly account for the presence of intact rock bridges between termi-nating fractures – similar to in-situ rock mass conditions. Through simulated testing of these synthetic materials, it is possible to derive rock mass properties such as modulus, strength and brittleness. At present, it is not practical to simulate large-scale mining/geological pro-cesses within PFC3D due to the computational intensity of the numerical technique. For this reason, a Ubiquitous Joint Rock Mass model was developed (Sainsbury et al. 2008) to repre-sent PFC-based behavior within large (mine-scale) simulations.

Additional Smooth Particle Hydrodynamics (SPH) codes have also been used to represent a cave scale model in two-dimensions (Karakel et al. 2010); however, due to their limited validation and application, they are considered research tools at the current time.

Figure 9. (a) Particle clusters early in the caving process with superimposed contact force chains

(after Lorig et al. 1995). (b) Particle clusters after significant cave propagation showing internal fractures of blocks in the caving zone chains (after Lorig et al. 1995). (c) Forces arching around the unstable rock mass(after Brown 2003).

4.4 Three-dimensional methods Pierce & Lorig (1998) describe an improved methodology developed during the Internation-al Caving Study (compared to the axis-symmetric approach reported by Brown (2003)), whereby sequential undercuts of constant width were simulated to reproduce the increasing undercut hydraulic radius during cave start-up. Production was simulated in the model by monotonically reducing stresses in the undercut level using the same methodology presented in Figure 7. For each undercut increment, the model was stepped to equilibrium before sub-sequent undercut increments were taken. In addition to the dynamic nature of the undercut expansion, Pierce & Lorig implemented a user-defined function that modified material prop-erties and stresses based on plasticity state and strain. Through this approach the point at which self-sustained instability (the critical HR) could be determined based on actual three-dimensional stresses distributing around the undercut. A diagram that represents the model-ing methodology is provided in Figure 10a along with typical model results (Figure 9b). It can be seen that the modeling methodology generally results in a flat cave back (Figure 10B-7), which is not considered realistic based on the conceptual model of a cave.

Using this approach, the amount of material extracted within the model could be calculat-ed based on changes in mesh volume, stress and bulk modulus. However, by reducing the stresses uniformly across the undercut, material was drawn preferentially from the higher stress areas in the undercut. This methodology highlighted the need for better control on production draw simulation and during the Mass Mining Technology (MMT I) project, Pierce et al. (2006b) simulated production draw based on the pseudo-static application of small downward-oriented velocities in the back of the undercut within the model. By doing so, the amount of material being removed from an area could be controlled. Details of this methodology have previously been documented by Sainsbury et al. (2008) and are summa-rized in Figure 11.

Through a velocity controlled production draw algorithm, Pierce et al. (2006b) were able to simulate the evolving cave behavior at Northparkes Lift 2 that was consistent with the conditions observed onsite. The numerical results presented in Figure 12 have been validat-ed against measurements of open hole blockages, TDR breakages and the progression of the seismogenic zone with production draw.

Figure 10. (a) Logic sequence to simulate caving (b) typical simulation results (after Pierce & Lorig

1998).

Figure 11. Simulation of production draw from FLAC3D model based on velocities (after Sainsbury et

al, 2008a).

Figure 12. Progression of predicted mobilized zone limit (white iso-surface) and overlying yield zone

limit (blue iso-surface) versus TDR breakage locations (blue spheres) and open hole blockage lo-cations (red squares).

4.5 Hybrid techniques Most of the methodologies described so far have represented the rock mass as an isotropic material (i.e. the same strength in all directions). However, it is known that the cave propa-gation behavior of a jointed rock mass is strongly governed by the unique nature of joints and discontinuities together with the intact strength of rock-bridges that make up a rock mass.

Vyazmensky et al. (2007) used the combined finite element-discrete element ELFEN code to insert physical fractures into a continuum finite element mesh that is gradually de-graded into discrete blocks through systemic sampling of the tensile strength and principal stress magnitudes and directions. Results of recent simulations conducted with this ap-proach are presented in Figure 13.

The model results are considered qualitative as the caving methodology, has yet to be val-idated against actual observations and measurements of seismicity, TDR breakages, break-through timing or subsidence limits. Due to the complexity of model setup and the signifi-cant computational time required to run mine scale subsidence problems this technology is currently limited to two-dimensions.

Figure 13. Simulation of cave development using a hybrid approach (after Vyazmensky et al. 2007).

5 KEY COMPONENTS OF THE CURRENT NUMERICAL CAVING ALGORITHM

As discussed, a numerical model of caving has been researched within the industry funded International Caving Study (ICS) and MMT I (Mass Mining Technology) projects. The cur-rent (MMTII) caving algorithm is implemented within, but not limited to FLAC3D (Itasca 2009) and 3DEC (Itasca 2007). The model is able to represent the true three-dimensional mining geometry, structural model and tonnes-based production schedule in addition to a strain-softening rock mass response that can account for a discrete fracture network – through a Ubiquitous Joint Rock Mass approach (Sainsbury et al. 2008).

The methodology allows the seismogenic, yielded and mobilized zones to evolve as a re-sult of a specified production schedule and rock mass behavior. The cave volume is not in-troduced manually into the model; rather it is allowed to develop based on the specified draw strategy, evolving induced stress conditions and the simulated constitutive behavior of the rock mass. In doing so, hang-ups, over-breaks and rapid advance rates can all be predict-ed. Key components of the caving algorithm are summarized below.

5.1 Cohesion and tension weakening There is significant uncertainty regarding post-peak rock mass behavior during the failure process and the manner in which the rock mass transitions from an intact to a caved materi-al. In this complex process, creation of a propagating cave results in a) deformation and stress redistribution of the rock mass above the undercut, b) failure of the rock mass in ad-vance of the cave, with associated progressive reduction in strength from peak to residual levels, and, c) dilation, bulking, fragmentation and mobilization of the caved material. This

overall response is often termed a “strain-softening” process, and is the result of strain-dependent material properties. In the caving model the rock mass material is described by a Mohr-Coulomb strain-softening, failure criterion in which the behavior is constrained by the results of SRM simulations.

Calibration of a SRM material response in FLAC3D (as described by Sainsbury et al. 2008) has been termed a Ubiquitous Joint Rock Mass (UJRM). UJRM samples are calibrated to SRM test results under three different stress paths (UCS, triaxial and direct tension), in three different loading directions and at three different samples sizes. In each of these environ-ments, not only is the peak and post-peak response matched, but the failure mechanisms, de-termined from SRM testing, are also honored.

5.2 Density (porosity) changes It is known that the density of broken rock varies greatly within a cave and that it is related to the volumetric changes that accompany dilation under shear failure and or expansion un-der tensile failure. As a result of this, the density tends to be lowest in actively flowing re-gions and higher in areas that have been stagnant for some time and may have been subject to re-compaction. In the caving algorithm, volumetric expansion can occur via both mecha-nisms (shear dilation and tension). The resulting changes in porosity (n) within each zone are tracked and the density updated according to Equation (1).To prevent bulking of the rock mass to unrealistic levels, a maximum porosity is set within the model that cannot be ex-ceeded.

( )( )( )bulk insitu / 1 / 1ρ = ρ + η − η (1)

5.3 Modulus softening During caving, a rock mass will increase in volume (or bulk) as intact rock blocks fracture, separate and rotate during the yielding and mobilization process. Along with this bulking, a reduction in modulus is expected to occur. The rate at which a rock masses modulus softens within the MMTII-based numerical caving algorithm is based on a non-linear relation de-scribed by Hoek &Diederichs (2006).

5.4 Production simulation Evolution of the mining footprint shape and hydraulic radius is simulated within the MMT II-based caving algorithm through the constant updating of the active production. Production draw is simulated by the same methodology developed during MMT I, by applying a small downward-oriented velocity to grid points in the model that correspond to draw point loca-tions. Production draw is controlled by mass balance calculations that are performed at regu-lar intervals during model stepping.

Contrary to the limited experience and views of Vyazmensky et al. (2009), Sainsbury et al. (2008) have previously shown how a continuum can be calibrated to an anisotropic re-sponse (derived from SRM testing) and used in large (mine-scale) three-dimensional caving analyses that span a production history of 40 plus years.

5.5 Validation of caving algorithm Although Vyazmensky et al. (2009) report that available modeling tools are yet to reach the computational efficiency required to allow detailed and realistic three-dimensional analysis of block caving subsidence, validation of the MMT Caving Algorithm has most recently been completed on a historical case study of the Grace Mine located in Pennsylvania, USA (Sainsbury et al. 2010). Evolution of the cave (both mobilized and yield zones) after simula-tion of the historical mining schedule are presented Figure 14 along with photographs of the observed surface subsidence at the time.

Figure 14. Predicted mobilized (yellow iso-surface) and yield zones (blue iso-surface).

6 SUMMARY

Since the development of numerical methods in the 1970’s, the numerical assessment of cave propagation has been continuously researched and improved to help minimize the ge-otechnical risks associated with cave mining methods. Numerical methodologies have evolved from being able to accurately assess the primary risk of whether a cave will stall and develop an airgap, to now being able to assess detailed cave behavior. The current state-of-the-art caving algorithm, that has been developed within the industry funded MMT2 project provides a robust assessment of the evolving mobilized, yield and seismogenic zones in re-sponse to the actual production draw at the undercut/extraction level. The evolving cave shapes, propagation rates, abutment stresses and subsidence limits can be readily assessed with the one numerical model.

ACKNOWLEDGEMENTS

The authors wish to acknowledge the members of the ICS and MMT I and MMT II projects for sponsoring the development of the numerical caving algorithm. In addition, the authors would also like to thank Itasca and The University of New South Wales for their continued support.

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