GEOTECHNICAL ANALYSIS OF GRAVITY FLOW DURING BLOCK CAVING

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1 GEOTECHNICAL ANALYSIS OF GRAVITY FLOW DURING BLOCK CAVING RAMON VERDUGO Professor of Geotechnical Engineering, University of Chile, Chile. JAVIER UBILLA Graduate Student, Rensselaer Polytechnic Institute, USA. Abstract Block Caving is a mining operation method where taking advantage of the gravity forces, the ore is fragmented into particles and extracted from different draw points. The actual phenomenon, usually called gravity flow that takes place in the mine is complex, being still at the present the operation and design mainly empirical. In this paper the gravity flow is analyzed using the computer code Flac 3D, modeling the fractured ore as a granular material with a Mohr-Coulomb failure criteria and with a modulus of deformation as function of both the minor principal stress and stress path. The computed results are in good agreement with empirical evidences observed in the block caving. The obvious question of how a flow can be modeled with a computer code for a continuum media is also explained in the paper. 1 INTRODUCTION The mining industry is continuously growing and developing new designs, equipment and procedures to efficiently extract the ore from the mother rock. This is a permanent duty since the international prices of the metals are decreasing in the long-term perspective, forcing the mining industry to reduce all the operational costs. In this context, the analysis and better understanding of one of the most economically attractive method of extraction used in underground mines, the block caving, is important for optimizing and improve the scenario of mine production in long-term bases. In all those natural ground conditions where the orebodies are close to the surface, the extraction throughout the open pit technique is usually adopted. Whereas, when the mineralized body is deeper, depending upon the rock quality, shape and dimensions, the so-called Cave Mining procedures usually provide the lowest cost. Cave Mining are called all those mining operations where the orebody caves naturally after undercutting and then it is retrieved through a grill of drawpoints (Kvapil, 1982). Under this denomination is possible to include, block caving, panel caving, inclined-drawpoint caving, and front caving (Laubscher, 1994). This mining operation works due to the fact that the hanging wall of rock caves when the excavation goes forth. In block caving, a thick block of ore is undercut by removing a slice of ore, so the unsupported block is allowed to collapse due to its own weight. Then, the broken ore is drawn off from below as the fractured rock falls driven by the gravity forces. Although in the last decades a tremendous effort has been done in order to understand and predict the phenomenon that control the cave mining, the actual results are still rather insufficient. Consequently, a study oriented to obtain new analytical insights has been undertaken and a part of these results are presented below. 2 MODELING OF THE GRAVITY FLOW In spite of the significant advances in computer hardware, numerical modeling and development of new analytical approaches dealing with gravity flow, the actual designs of block caving are mainly based on both empirical procedures and

2 engineering judgement (Kvapil, 1965; 1982; Douglas, 1984; Mansson, 1995). This fact somehow shows that more studies are still needed to assess the actual behavior of the gravity flow associated to block caving. A rigorous analysis of the phenomenon that takes place during the gravity flow in the mining operation by block caving should consider the following field conditions: The initial intact rock that is caved by the mining operation very seldom is homogeneous and continuos. On the contrary, different geological structures and lithological heterogeneities are normally encountered, which govern the fragmentation, the principal stress directions and stress relaxation. Additionally, the presence of water significantly affects the rock crashability and the resistance of the finer particles that result from the fractured rock. The boundary that separates the intact body and the fragmented rock is a diffuse and movable zone, which affects the dimensions and stress field of the crashed rock. It is recognized that gravity flow associated to isolated draw points corresponds to a continuous flow of particles that is concentrated only in a certain region of the granular mass, the so-called drawzone, while the remainder particles essentially do not move. Therefore, this phenomenon generates a clear discontinuity that encloses the region under flow (drawzone). As the fractured rock is moving toward the draw points, it is possible to expect a continuous secondary crashing of the larger particles due to the high level of pressure involved. Hence, the grain size of the material in the drawzone is modified during the process of flow. Any mathematical model for predicting the block caving that attempts to incorporate all the considerations indicated above definitely will be very complex and in addition, it will have serious problems to obtain the input parameters. Consequently, instead of going into new more refined models, the authors have opted for the appropriate implementation of already well established analytical tools that may permit to clarify fundamental characteristics of the gravity flow. First it is important to realize that after the block caving method of mining breaks the original massive rock, the resulting fractured material resembles a coarse particulate media. Therefore, the use of the mechanical laws of coarse granular materials is totally pertinent. It is postulated that the fundamental feature of gravity flow phenomenon is strictly related to the particulate nature of the body of crashed rock and its mechanical response is capture by its modeling as a granular material. From a practical point of view, the modeling of the fragmented rock as a single granular material imply that all the process of secondary particle crashing that likely occurs in the drawzones is not considered. Additionally, the phenomenon associated to the phase transformation from solid rock to particulate material is not incorporated in the present analysis either. These two simplifications can be considered limitations of the present analysis. On the other hand, it is important to realize that the mining process of extraction of the crashed material that has passed throughout the draw points is rather slow. Typically, the loader equipment can take 6 to 8 ton of fragmented ore each time, retrieving the material and coming back again for the next load. Each of these cycles is ordinarily done in a time interval of approximately 10 minutes. This non-continuous operation of ore extraction together with the rather large period of time involved in each cycle implies that the inertial forces are small enough, so they can be neglected. Thus, the whole phenomenon can be considered essentially static, and therefore, the classical set of differential equations for static equilibrium of a continuum media holds valid. Furthermore, although the operation of extraction is slow, still it generates the flow of the granular material, which creates a discontinuity around the drawzone. The emergence of a discontinuity makes this state to diverge from a continuum media. However, just before the development of the failure of the granular material with the generation of the drawzone, which actually means the initiation of the flow, the granular material can be treated as a continuum. Therefore, just until the starting of the flow, the whole body of fractured rock would satisfy the set of differential equations of a continuous medium. Accordingly, the applicability of the equations of geometric compatibility associated to a continuum

3 holds also valid until the gravity flow throughout the draw points is initiated. 3 GEOMECHANICAL MODEL OF THE FRACTURED ROCK At the present a significant number of mathematical models that use different constitutive laws for representing the mechanical response of coarse granular materials are available. It is possible to indicate that the constitutive law provided by the elasto-plastic stress-strain relationship has shown to be the simplest and mathematically fully consistent. Beside, the main features of coarse particulate materials as the dependency of shear strength and stiffness with the level of normal stresses and the increment of stiffness during unloading processes can be easily incorporated in the elasto-plastic stress-strain relationship. The perfect plastic response occurs when the shear strength is achieved, which is a function of the level of normal stresses. The dependence of shear strength with the normal stresses is incorporated by most of the failure criteria, as for example, Mohr-Coulomb, extended Von Mises, Lade-Duncan and Matsuoka-Nakai, among others (Chen and Mizuno, 1990; Matsuoka and Nakai, 1974). Considering that the Mohr-Coulomb failure criterion is well known and it reproduces the ultimate state of stresses to a great extend, it has been adopted in the present analysis. This criteria is expressed as: On the other hand, to take into account the effect of confining pressure on the stiffness of a granular material, the following expression for the deformation modulus as been used (Jambu, 1963): E σ n 3 = E o Pa P (2) a Where, E o and n are material constants, σ 3 is the minor principal stress, and P a represents the atmospheric pressure and its use in this expression is only for having non dimensional values of the constants E o and n. Experimental evidence suggests that n = 0.5 is a reasonable value for coarse materials, so it has been adopted in this study (Duncan et al. 1980). The third important feature of granular materials is related to the sharp change in rigidity that is observed when unloading processes take place. A typical result of a load plate test carried out in a coarse granular material is shown in Fig. 1, where a significant increase of stiffness during unloading can be seen. τ = σ tgφ f n (1) Where, τ f corresponds to the shear strength, σ n represents the normal stress to the plane that contains τ f, and φ is the angle of internal friction of the granular material. The right side of this relationship may have an additional term named cohesion that accounts for those states of a granular material that mobilize strength although the confining pressure is zero. However, the granular materials generated from the blasted rock definitely correspond to cohesionless materials. Only in those situations of fine material (clay and silt sizes) and under a condition of partial saturation, a small level of cohesion may be observed, but these stages are out of the scope of this study. Figure 1: Typical load plate test with loading and unloading performed on coarse soils.

4 According to the available information in the Institute for Research and Testing of Materials, IDIEM, of University of Chile, the resulting stiffness during unloading is in the order of 4 to 10 times greater than the stiffness developed during loading. This phenomenon is wellrecognized in granular materials and it is associated to the generation of a sort of elastic region by the loading process. Therefore, any change in the state of stresses has to be analyzed in order to identify whether it corresponds to a step of unloading that would need a change in the modulus of deformation. To accomplish this, a simple procedure analyzing the stress path increment in the q-p plane is proposed as shown in Fig. 2. In this stress path plot, p represents the mean stress (σ 1 + σ 2 + σ 3 )/3 and q corresponds to the maximum shear stress ((σ 1 - σ 3 )/2 q Failure envelope Region 2 45º Region 1 p o,q o Region 3 Figure 2: Regions of loading and unloading In Fig. 2, the point (p o, q o ) represents an initial state of stress, which will be modified to a new state. When the stress-path moves toward the failure envelope (EF), it is possible to visualize a condition of loading, which is defined by region 1. On the contrary, when the stress path is moving away from the failure envelope it is possible to recognize a process of unloading. In addition, it is important to bear in mind that the modulus of deformation has been considered explicitly as a function of the minor principal stress. Thus, all those stress paths in the region 3 will automatically increase the soil stiffness because in this region always there is an increase in the minor principal stress. However, in the unloading region 2, the minor principal stress decrease, being necessary to externally modify p and to increase the deformation modulus. When the stress paths go into this region, the deformation modulus has been considered to increase four-time respect to its previous value during loading. This ratio of four between the stiffness in loading and unloading has been adopted considering the minimum ratio observed in coarse materials. 4 DESCRIPTION OF THE NUMERICAL ANALYSIS Considering that the classical equations of a continuum can be applied, the geotechnical model indicated above was implemented in the computer code Flac 3D. To analyze of the gravity flow in a fractured rock mass a medium of 100 m in length, with heights between 50 and 150 m were simulated. A square mesh of nodes of 1 m per 1m was used. To reduce the time of CPU, the analysis was basically done for a plane strain condition. The adopted procedure of loading, that is compatible with the generation of a gravity flow, consisted in the definition of the draw points via a set of nodes where external loads could be applied. Initially, in these nodes an upward vertical load equivalent to the vertical dead weight of the material was applied. Thereafter, these loads were incrementally decreased until the failure of the granular material took place and the deformations become too large to be managed by the computer code that was basically working for the static analysis of a continuum. The previous load step to this collapse was considered the initiation of the gravity flow and its analysis is presented below. 5 ANALYSIS OF THE NUMERICAL RESULTS For a single draw point, the deformation contours that are generated at the initiation of the flow are shown in Fig. 3, where are also indicated the material properties used in the analysis. It can be seen that the typical ellipsoidal shapes of the drawzone of extraction observed in boxes filled with sandy soils are quite well reproduced by the present analysis. Another interesting result is related with the distribution of vertical stresses at the bottom of the granular material that are shown in Fig. 4. The computed results indicate that there is an

5 important concentration of vertical stresses around the draw point, which magnitude and locations are mainly controlled by the friction angle of the material. These results are show in Figs. 5 and 6, where it is observed that the greater the angle of internal friction the greater the peak of the concentrated vertical stress and the closer its location to the opening. These results also show that as the angle of internal friction increases the total zone where exists a stress concentration becomes clearly smaller. Besides, when the friction angle increases the smaller the drawzone created by the flow. From these results, it is also interesting to observe that if the frictional strength is high enough, the peak vertical stress around the draw point can duplicated its value respect to the initial geostatic value. (Eo = 2000 kg/cm 2, Ko= 0.5, Density = 2.3 t/m 3, φ = 35 º) σv / σv Geostatic 2,5 2 1,5 1 0,5 φ =54º φ =48º φ =42º φ = 36º Horizontal distance [m] φ = 30º φ = 24º Figure 5: Distribution of vertical stresses at the bottom, for different friction angles of the materials. Figure 3: Ellipsoidal shape of the drawzone. 2,5 Distance from opening [m] Angle internal friction [º] σv/ σv Geostatic 2 1,5 1 0, Horizontal Distance [m] Figure 4: Distribution of vertical stresses along the bottom of the granular material. Figure 6: Distance from the opening at which occurs the peak vertical stress as a function of the friction angle. On the other hand, from a practical viewpoint it is relevant to estimate both the width of the drawzones, and the critical distance between draw points at which the interaction between drawzones becomes important. To evaluate the maximum width of the drawzone, a single draw point with different opening sizes has been analyzed. The initiation of the gravity flow has been established for this

6 series of different opening sizes considering a medium of 50 m high. The analysis has been performed until the flow changes its mode of failure and the whole material above the opening fails forming a sort of vertical column. When the opening has a size such that the arch effect tends to show up, there is an important stress rotation close to the opening that is reproduced by the computed results presented in Fig. 7. These results also define the mode of failure of the granular material, inducing the shape of the drawzone indicated in Fig. 8 for the simulation of a 50 m high fractured rock mass. On the other hand, it is possible to realize that if the opening size is large enough, no arch effect would emerge and the granular material simply will move down generating a cylindrical drawzone with identical diameter of the opening. It is postulated that the minimum size of the opening that is associated with this type of failure it would coincide with the maximum width of the drawzone where arch effect is manifested. This concept comes from the fact that the column failure is related with those sizes where the arch effect is not developed. Similarly, the maximum width of the drawzone is the critical distance until the arch effect can act. It is important to point out that in this context arch effect is related with a tendency of stress rotation and it does not correspond with the arch that stop the flow. The computed results for the critical opening associated to the column failure is shown Fig. 9, being possible to confirm the hypothesis of the existence of two different failure mechanisms that control the movements of the granular material. Figure 7: Rotation of principal stresses around the opening Figure 8: Typical results of the drawzone shape in a medium of 50 m high and small opening size. Figure 9:

7 Following the explained methodology, the numerical analysis of the drawzone widths was carried out for different values of the angle of internal friction of the simulated crashed rock, considering a medium of 50 m high. The computed results are shown in Fig. 10. It is interesting to observe that these results indicate that the higher the frictional strength the narrower the drawzone, which agrees with what has been observed in experiments with coarse soils of different frictional strength. A normal value for the angle of internal friction of a fragmented rock is around 43º, which according to these results it means a drawzone width of 22 m, what is compatible with the general experience in block caving mining operation. 35 Width of Drawzone [m] Angle of internal friction [º] Figure 10: Variation of the drawzone width with the angle of internal friction of the fractured rock: Furthermore, the critical distance at which there is interaction between draw points was estimated through the computations of a series of two consecutive draw points located at different distance and subjected simultaneously to the loading process that reproduces the gravity flow. The results are shown in Fig. 11, and they suggest that the actual interaction starts at a distance equal to 1.2 times the width of the isolated drawzone. This result is practically similar to the empirical value of 1.5 used in the block caving (kvapil, 1992; Laubscher, 1994). Figure 11: Isolated draw point and level of interaction at different distance of the draw points. 6 CONCLUSIONS The well-established theory for a continuum in conjunction with a simple model for coarse granular materials have been implemented in the

8 computer code Flac 3D. The numerical model is able to work until the initiation of the flow. A simple load procedure to generate the gravity flow has been implemented, so the initiation of the granular flow can be reproduced. The obtained results are in agreements with experimental evidences and field observations. It has been shown that the vertical stresses are concentrated around the draw points, reaching values as high as two times the initial geostatic vertical stress. Laubscher, D. (1994): Cave mining-the state of the art, The Journal of The South African Institute of Mining and Metallurgy. Mansson, A. (1995): Development of body of motion under controlled gravity flow of bulk solids, Licentiate thesis, Lulea University of Technology, Sweden. Matsuoka, H. and Nakai, T. (1974): Stress-deformation and strength characteristics of soils under three different principal stresses, Proceedings Japan Society of Civil Engineers, No. 232, pp The numerical results indicate that the width of the drawzone (zone of flow) is strongly affected by the frictional strength of the fragmented rock and it decreases as the friction strength increases. Beside, the results suggest that the critical distance, at which the interaction between two adjacent draw points is developed, can be approximated to 1.2 times the width of the drawzone generated by a single draw point. ACKNOWLEDGEMENTS The authors want to thank the financial support provided by FONDEF under the grant No that made this research possible. REFERENCES Chen, W. and Mizuno, E. (1990): Nonlinear analysis in soil mechanics. Theory and implementation, Elsevier. Duncan, J., Byrne, P., Wong, K. and Mabry, P. (1980): Strength, stress-strain and bulk modulus parameters for finite element analysis of stresses and movements in soil masses, Report No. UCB/GT / University of California, Berkeley. Douglas, P. (1984): Physical modeling of the draw behavior of broken rock in caving, Colorado School of Mines Quarterly, USA. Vol. 79, No. 1. Jambu, N. (1963): Soil compressibility as determined by oedometer and triaxial tests, European Conference on Soil Mechanics and Foundation Engineering. Wiesbaden, Vol. 1, pp Kvapil, R. (1965): Gravity flow of granular materials in hoppers and bins, Int. J. Rock Mechanics and Mining Science, Parts I and II. Vol. 2. Kvapil, R. (1982): The mechanics and design of sublevel caving system, Underground Mining Methods Handbook, W. Hustrulid ed., SME, New York.