Feasibility of Broken Ore Flow Simulation in Block Caving Mining Method Using Attribute Stochastic Medium Theory

Abstract

In this paper, to reveal the flow characteristics of broken mineral-rock of block caving, we present the Attribute Stochastic Medium Theory that combines an attribute block model with stochastic medium theory. An attribute block model was established to fit discrete target regions. A void transfer model and data structure were established based on stochastic medium theory. Our key contribution is that we propose Attribute Stochastic Medium Theory and present the concept of draw-out index and block fragmentation index. It can be used to analyze the flow characteristics of rock block under different drawing heights and different fragmentation conditions. We implement the flow algorithm and simulation with C++ programming. The results show that mineral elemental grade information of the ore-drawing body was obtained, and when the block fragmentation index remains unchanged, with the increase of the draw-out index, the ore-drawing ellipsoid develops gradually, the eccentricity increases gradually, and the average depth of the depression increases gradually. In the isolated extraction zone (IEZ), the length of the long half-axis of the IEZ increases linearly with the increase of the ore-drawing height. But the length of the short half-axis and the eccentricity of the IEZ increase nonlinearly with the increase of the ore-drawing height, showing a general power exponential relationship. Under the condition of constant draw-out index, with an increase of the block index, the eccentricity of the ellipsoid decreases. This method could obtain the properties and flow characteristics of caving ore and rock. Combined with in situ test of ore drawing. This method could provide guidance for the design of ore drawing bottom structure and ore drawing plan.

1. Introduction

1.1. Research Background

Block caving is a high-technology mining method with high capacity, and the application of this mining method can greatly reduce mining costs and produce good economic benefits [1]. The broken ore in this method is directly in contact with the overburden, and its attributes include information on grade and lithology. The broken ore and the overburden have different grade attributes. The former is valuable and is expected to be drawn out for profit, while the latter grade has almost negligible value and is expected to remain in the caving zone after ore drawing. However, in the process of ore drawing, these two components mix with each other actually, resulting in ore loss and dilution. If the design of the bottom structure or parameters is unreasonable, or managements of ore drawing are improper, ore loss and dilution would be worsen, causing a waste of mineral resources and decrease in the economic benefit of mining enterprises. The work of ore drawing is directly related to the success or failure of the application of the caving mining method [2]. Accurately grasping the flow characteristics of broken mineral-rock blocks has important theoretical and practical significance. However, it is difficult to clearly grasp the flow of broken mineral-rock blocks for three reasons. First, the broken ore is covered by the overburden, and personnel and equipment cannot enter the inside of the caving zone. Second, with the progress of ore drawing, the caving ore and rock flow dynamically. Third, the ore-drawing height of the block caving method is generally large, so it is difficult to realize a full-scale drawing test. At present, the research about ore drawing mainly focuses on the physical model test at a certain scale or numerical simulation.

1.2. Related Works

Because of the importance of ore drawing, some scholars have done theoretical research on this problem [1,3]. Kacmar discussed theoretical aspects of new options of sublevel caving methods and dealt with the proposal of the SMZ exploitation issue [4]. Morner combined both concepts of pseudokarst caves and paleoseismic activity in Sweden into a unified theory on the formation of fractures, fracture caves and angular block heaps [5]. Nezhadshahmohammad presented a mixed-integer linear programming (MILP) model to optimize the extraction sequence of drawpoints over multiple time horizons of block-cave mines [6]. Many other scholars have carried out field tests or experimental research on ore drawing. Power G. made a dent in granular flow in caving mines by large scale physical models and full scale experiments [7]. Brunton and Hodgkinson studied parameters influencing sublevel caving (SLC) material flow behavior from a full scale experimental program, and presented a Self-Organizing Map (SOM) technique. In their study, parameters analyzed included those related to drawpoint location, drill and blast design, geology, drawpoint geometry, and draw control [8]. R. Castro et al. conducted the single funnel drawing test on the physical model at a scale of 1:30 and 1:100, and concluded that the main factors affecting the geometric shape of the drawing body are the drawing amount and drawing height, while the particle size and funnel size have little effect on the geometric shape of the drawing body [9]. R. Trueman studied the multiple draw-zone interaction between multifunnel drawing and adjacent ore bodies through experiments at a model scale of 1:100 [10]. F. Melo studied the kinematic model of particle flow and considered the expansion effect of the shape of the drawing body [11]. In addition, F. Melo and F. Vivanco compared the drawing body of a large-scale physical model and kinematic model, and obtained the morphology and development law of the drawing body and loose bodies [12]. Tao ganqiang and Zhang Xiufeng conducted a 1:25 physical model test of caving mining of a medium-thick inclined ore body. It was concluded that the area affected by the inclined wall showed an increasing trend with the decrease of dip angle or ore width [13]. Jia designed a physical draw model to investigate the gravity flow characterization of granular materials during longitudinal sublevel caving, and found that the gravity flow characteristics of 3D draw experiments under the inclined walls were significantly different from those of the two-dimensional (2D) draw experiments [14]. Sanchez carried out laboratory experiments on a physical model (scale 1:75) using a Load Haul Dump system (LHD) and found that the dimensions, the diameter, and the height of the draw-zone depend on three variables: moisture content, particle size, and accumulated extraction mass [15]. Physical experiments are an effective method to obtain real data for ore drawing, but they are time-consuming and laborious.

If used properly, numerical simulation is an effective method for ore drawing research. Tony Diering developed a program PC-BC for application to the design and evaluation of block cave operations [16]. Huang introduced the software REBOP to conduct three-dimensional simulation and loss-dilution prediction of multiple hopper drawing with regard to hanging wall, footwall, and medium drawing in a nickel mine [17]. Rafiee, R. investigated the effect of seven parameters using a numerical technique named the Synthetic Rock Mass (SRM) and found that in situ stress and hydraulic radius are the most effective parameters in cavability of rock mass in block caving mines [18]. Castro presented the development of FlowSim as an improved model of gravity flow based on the cellular automaton approach, to estimate dilution entry, mixing and ore recovery [19]. Sanchez solve the Brinkman–Darcy equation coupled to the granular kinematic model using the finite elements method in 2D, to describe the entry of water into draw points in the Block Caving mining [20]. Woo presented a sophisticated 3-D numerical modelling method of predicting the extent and magnitudes of caving-induced surface subsidence [21]. In recent years, the rapid development of three-dimensional modeling technology and digital mine technology have made geological modeling and block caving research more convenient. Wang pointed out the key technologies of digital mine and the latest technological progress about mine resource evaluation and modeling, production planning and optimization, digital mine design, production management and safety control [22]. Zhong et al. presented an improved approach to the surface reconstruction of orebody from sets of interpreted cross-sections with geometry constraints [23]. Li proposed a local dynamic updating method of the orebody model based on mesh reconstruction and mesh deformation [24]. Zhong and Wang have done a great deal of successful research on implicit modeling of geological bodies. They presented an approach of orebody implicit modeling from raw drillhole data using the generalized radial basis function interpolant [25]. In addition, they presented a new algorithm for reconstruction of implicit surfaces from a set of cloud points with normals (Hermite data), based on the generalized radial basis functions interpolant with various types of constraints [26]. To construct a discrete model, Gang-hai constructed a spherical particle model based on MFC software [27]. The common software packages PFC2d and PFC3d are based on Newton’s second law of mechanics and often used to simulate particle flow [28,29], but they cannot characterize the attribute information of particles. A block model can represent attributes, and it has been studied and applied by other scholars with good results [30]. Beck proposed a coupled DFE-Newtonian cellular automata to simulate cave initiation, propagation and gravity flow. The coupled simulations incorporate velocity-based instability criteria for cave back instability, assessed by the DFE model allowing direct, explicit forecasting of cave propagation geometry and rates. However, it cannot simulate the attribute information and its change of rock mass during the propagation and gravity flow [31]. In cave mining a large number of rock fragments have to be simulated during flow based on a production plan, and simulations are even more complex when fine rock fragment is involved. Raúl Castro presented a new fine material migration logic based on cellular automata to improve the calculation of this phenomenon. The key variable used is the size ratio between the rock fragments of the broken column. The mean size, d50, is used to represent a characteristic rock size and calculate the coarse and fine fragment ratio. It can be concluded that incorporating the d50 to characterize the granular material in gravity flow logic using cellular automata provides an accurate and quick fine material migration estimation [32]. Statistical analysis, such as grade estimation and lithological analysis, could be carried out on the basis of a three-dimensional geological model and block model, but these analyses are static and cannot reflect the temporal and spatial changes of geological bodies. Stochastic methods have proved to be a good alternative for assessment of dilution degree for different draw strategies to minimize the waste extraction and optimize the ore recovery [33].

To grasp the flow characteristics and attributes of broken mineral-rock blocks, an Attribute Stochastic Medium Theory that combines an attribute block model with stochastic medium theory was proposed, and the concept of draw-out index and block fragmentation index were presented. Computer simulation was carried out on a software platform with a digital mine. First, the target region of ore drawing was discretized by a block model, and then the model was combined with an attribute model to establish an “attribute block model”. Stochastic medium theory was applied to the attribute block model to establish a void transfer model and data structure for programming. Then, a simulation process and flow subsystem were developed. The flow of broken mineral-rock blocks under several representational numerical examples considered conditions with changes in such parameters as ore-drawing height, block index, and draw-out index.

2. Methods

The main research contents and key points of this method include block discretization, attribute assignment, Attribute Stochastic Medium Theory and block flow algorithm.

2.1. Block Discreteness and Attribute Modeling

In block caving mining method, the object of caving is the orebody above the under cut level. For simplicity, the target orebody is regarded as a large cuboid region that can be defined by a starting point and three extensions, while the starting point and three extended axes constitute a Cartesian coordinate system. Alternatively, the target region can be defined by a three-dimensional surface model. After determining the target region, the next step in simulation of broken rock flow is to discretize the target region as block elements, often tetrahedrons or hexahedrons. The commonly used octree block model construction method uses hexahedron elements. Ore bodies have grade attributes, so discrete blocks, which are called attribute blocks, should also have corresponding attributes.

In this study, the elements used are attribute hexahedrons. We define the attribute blocks model as attribute hexahedrons. A center point is added to each geometric hexahedron in the three-dimensional space to construct an attribute hexahedron. As the bearing point of block attribute information, the center point is used to record block attribute information. The discretized attribute blocks model is shown in Figure 1. The attribute blocks have the attributes information defined in Section 2.3.1 The attribute value can be assigned a constant value or can be obtained with an attribute difference method by the sample values, such as geostatistical method or distance power inverse ratio method. The attribute value stores in the structure object struc_Particle of the corresponding point.

2.2. Attribute Stochastic Medium Theory

2.2.1. Modeling Technique of Combining Stochastic Media Theory and Attribute Blocks

The research object is regarded as a random medium with statistical characteristics consistent with stochastic medium theory, which was put forward by Polish scholar Litwiniszyn to study strata movement and surface subsidence [34]. YU studied top-coal drawing law of LTCC mining based on stochastic medium theory [35]. Liu Bo presented a prediction model based on stochastic medium theory for ground surface settlement induced by non-uniform tunnel deformation [36]. Liu developed stochastic medium theory into probability integral method [37]. The classical stochastic medium theory takes the motion characteristics of discrete bodies into consideration and is applicable to the study of surface subsidence and ore drawing [38], but it fails to consider attributes of discrete bodies.

On the basis of classical stochastic medium theory, we combined the attribute block model with stochastic medium theory to attempt to establish attribute stochastic medium and form Attribute Stochastic Medium Theory to study the flow of broken mineral-rock of block caving mining. Zhu made a preliminary study about ore and rock flow under multifunnel and optimization of ore drawing scheme of block caving [39,40]; that study is a companion to this paper. In this theory, the flow of caving mineral-rock is regarded as the flow of attribute blocks, which is to say, the flow objects are attribute blocks. Attribute blocks are used to represent discrete body attributes. There are two important points in this theory. First, the caving block is flowing in the process of ore drawing, which should be described by some probability models; second, the broken blocks have attributes, which can be lithology, mineral grade information and so on. Attribute Stochastic Medium Theory adds the consideration of attribute information on the basis of the classical stochastic medium theory. The latter mainly describes the spatial position change of the flow object in the flow process, without considering the attribute information of the flow object. While Attribute Stochastic Medium Theory could take both spatial position change and attributes change into consideration in the process of flowing, and it could be used in the fields of ore drawing research, debris flow research, and attribute particle flow research.

2.2.2. Probability Model of Void Block Transfer in Stochastic Media Theory

After block discretization, the key to the flow simulation of broken mineral-rock blocks is to determine the stochastic flow model of the discretized target region. That is to determine the flow probability model. In ore drawing or geotechnical engineering, different probability models correspond to different boundary conditions. In ore drawing simulation, the common probability models are four-block model, hexagonal model, seven-block model, and nine-block model, which is called n-block model for short, as shown in Figure 2. For example, the commonly used Jolley simulation model is a nine-block model [41]. These probability models satisfy the following Formula (1):

$${\displaystyle \sum }_{i=1}^n{p}_i=1$$

(1)

where n defines numbers of block in the model, p_i defines the flow probability of corresponding block. The flow probability model is determined by discretization and the flow probability of the target region. The discrete regional block flow probability model is represented by the void block transfer model. In this study, nine-block, six-block, and four-block models are selected to consider the void block transfer model under different boundary conditions, as shown in Figure 2. The nine-block model is suitable for the case of no boundary restrictions in the target region; the six-block model is suitable for the end-limited boundary condition; and the four-block model corresponds to the “corner” boundary condition. Each block movement corresponds to a stochastic process, and its probability is determined by the Monte Carlo Method or by test. Taking these three models into account, the flow in a discrete target region can be simulated completely.

2.3. Flow Algorithms of Caved Ore-Rock Blocks Based on Attribute Stochastic Medium Theory

To study the flow of caving ore-rock with Attribute Stochastic Medium Theory, it is necessary to assign attributes to the block and then simulate the flow process. To simplify the problem, the flow object is replaced by the center point of the attribute blocks, and the flow of the attribute blocks is simplified as a flow of particles, which can reduce the data storage space without distorting the important information. Therefore, it is necessary to study the particles flow, including particle data structure, flow model and the influence of block size on flow characteristics, etc.

2.3.1. Data Structure Design and Simulation Process

Design of Flow Particle Structure

A flow particle structure is designed to simulate the flow of attribute blocks. According to the C + + program syntax, this data structure is designed as follows:

struct struc_particle

{

dmDPoint origin_location;

int ijkID[3];

int curijkID[3];

double grade;

bool drawn;

dmDPoint cur_location;

int oreFlag;

int index;

bool used;

……

};

In this data structure, truc_particle is a structure type, and this data structure includes such variables as origin_location, ijkID[3], curijkID[3], grade, drawn, cur_location, oreFlag, index, used. These variables have different types, such as int, double, bool and dmDPoint, where dmDPoint is a data type that is used to store the three-dimensional coordinates of a point in floating-point numbers in digital mine system. Variables origin_location and cur_location are used to record the original position and current position of the particles, respectively, and they are actually three-dimensional coordinates. Array ijkID and curijkID are used to assist in locating the ID of particles in three directions in Cartesian coordinate system. Variables oreFlag and grade represent mineral element type and element grade, respectively. Variables drawn indicates whether the particle was drawn out. Variables used and index represent the access status and index records, respectively.

Flow Model of Particle and Simulation Process

In the process of flow simulation, the flow objects are particles, and the discrete attribute blocks are used to locate and assign attributes to particles. The particle flow model is shown in Figure 3.

The discrete particle point has the flow particle attribute structure defined in Section 2.3.1, which is stored in the particle structure array m_vecParticle Struc. The outlet location array is determined by the Monte Carlo method to start the flow simulation of the attribute particle. The simulation process is shown in Figure 4. The isolated extraction zone (IEZ), that is, the drawn body, remnants, and contact surface, were obtained by simulation, and its shape and properties were analyzed. The IEZ refers to a virtual three-dimensional body formed in the original space before granular flow, after which the grains were finally drawn out from the drawbell [9]. Remnants refer to a real three-dimensional body formed in the caving heap space after granular flow. The contact surface is the interface between the ore blocks and rock blocks.

2.3.2. Block Flow Characteristics under Different Block Size Conditions

Particle flow simulation relies on the following variables and functions: nDrawOut, integer variable, represents the number of drawn out particles. κ, the draw-out index, a floating point number variable, is the ratio of the number of drawn out particles to the total number of particles, and $k\in \left(0,1\right)$. curNullIJK and preNullIJK are arrays used to locate void blocks. TmpLocation, TmpLocation2, and preLocation are DmDPoint type variables used to indicate temporary locations of void blocks. The function pointInFun() is used to determine the particles in the funnel area during the flow process, and the location array of particles satisfying the conditions is stored in the pointInFun array. For the pointInFun array with capacity n, the location array of the i-th particle is pointInFun[i]. The function IJK, 0 < i < n; FlowMode() is used to determine the flow model based on Attribute Stochastic Medium Theory. Finally, the function FromIJKtoVecScript() is used to obtain subscripts from location arrays.

The fragmentation of broken mineral-rock blocks is one of the important research points of the block caving method, which includes the size and gradation of loose mineral-rock blocks. Rock fragmentation has a great influence on the movement performance of mineral-rock blocks, IEZ, ore loss, and dilution. It determines the shape of the IEZ and the size of the drawbell, and it affects the design of ore-drawing structure parameters, the selection of ore-drawing equipment, and production capacity.

In the process of ore drawing, the size, shape and composition of mineral-rock blocks have an important influence on the drawing body. In the actual production process, the fragmentation changes dynamically because of the flow, mutual friction, and collisions of mineral-rock blocks. Based on stochastic medium theory, the ore-rock heap was discretized first, and then the flow model was adopted to compete the following simulation. If the shape and size composition of mineral-rock blocks are considered, it will greatly increase the difficulty of ore drawing simulation. Therefore, only uniform mineral-rock block sizes and regular shape were considered to simplify the model in this study. However, to be more representative, different sizes of such regular blocks were considered.

The particle flow model considering the fragmentation of mineral-rock blocks is consistent with the particle flow model of Section 2.3.1, as shown in Figure 4. The consideration of fragmentation in Figure 4 is embodied by a parameter called the fragmentation index or block index, which is the ratio of the size of a mineral-rock block to the size of the ore-drawing drawbell. In the simulation of mineral-rock flow, block index β is defined as

β = dbell/dbloc

(2)

where dbell represents the minimum size of the drawbell and dbloc represents the maximum or average size of the broken rock mass. In the simulation of mineral-rock flow, the outlet location array is determined by the Monte Carlo method. Different values of β determine the ore-drawing location of ore rock from a certain drawbell, and also determine the outlet location array, which plays an important role in the simulation program.

2.4. System Development

Taking a digital mine software system as a platform, the secondary development is carried out by means of platform + plug-in. The platform development framework includes a data layer, operation layer, algorithm layer, and display layer, and it has good practicability and scalability [42]. The data layer uses a self-defined data structure, the operation layer uses the interactive mechanism provided by the platform to complete the definition of the plug-in, dialog box, and data interaction, and the algorithm layer encapsulates the function of the attribute block flow as BlockFlowOperator () class to complete the flow model of attribute blocks. The display layer calls the display interface of the platform. The development model is shown in Figure 5. We developed a plug-in subsystem for attribute block flow using Microsoft Studio 2010 (C++ language).

3. Results

Based on the algorithm proposed in Section 2, we implemented this simulation method of the attribute particles flow and tested it. We carried out three kinds of simulation, including ① mineral-rock flow in fixed target drawing heap, ② mineral-rock flow at different drawing heights, ③ mineral-rock flow under different fragmentation conditions. The first simulation is used to reveal the development and change law of drawing body and remnant in the fixed drawing heap. The second one is used to reveal the development of drawing body at different drawing heights, and the third one is used to reveal the influence of block fragmentation on mineral-rock flow.

3.1. Mineral-Rock Flow in Fixed Target Drawing Heap

The size of selected mineral-rock heap is 20 × 20 × 30 m³, the size of the discrete attribute block model is the unit size, the recording attribute is the grade of Cu, and the size of the drawbell outlet is 3.5 × 3.5 × 3.2 m³. After building the block model, the block grade is obtained by attribute difference technology and stored in the particle structure corresponding to the center-point of the attribute block. The discrete particle model of the selected mineral-rock heap is shown in Figure 6.

Based on the algorithm proposed in Section 2, we implemented this simulation method of the attribute particles flow and tested it. The test was divided into five groups, corresponding to draw-out index κ values of 3%, 5%, 10%, 15% and the cutoff value of κ. The selected κ is not very large because some particles cannot be drawn out due to caving heap ridge remnants, so there is a maximum κ value (cutoff κ). In addition, for the flow simulation of the fixed target mineral-rock heap, a too large draw-out index κ will also lead to obvious drawing truncated ellipsoid, which is not conducive to describing the shape of the IEZ. The simulation results are shown in Figure 7 and Figure 8 and

Table 1. Result of particle flow simulation under different draw-out index values.

Draw-Out Index κ (%)	3	5	10	15	16.4
Cu grade of IEZ (%)	0.51	0.49	0.49	0.53	0.51
Long Half Axis of IEZ (m)	7.5	9.56	13.2	15.5	16.1
Short Half Axis of IEZ (m)	4.8	5.55	6.7	7.5	8.0
Eccentricity	0.76	0.81	0.86	0.87	0.87
Remnants volume (m3)	11,646	11,407	10,820	10,212	10,032
Average Depth of Depression Pit (m)	0.75	1.48	2.95	4.47	5.91

.

As can be seen from Figure 7, the shape of the IEZ is approximately ellipsoid. It can be seen from Table 1 that Cu grade information of the ore-drawing body was obtained by this method, and under the conditions that the block index remains unchanged, with the increase of κ, the ore-drawing ellipsoid develops gradually and the eccentricity of drawing ellipsoid gradually increases and approaches 1, while the volume of drawing remnants decreases gradually, and the depth of the depression increases gradually. However, as the κ value approaches 16.4%, the particles in this fixed drawing heap are no longer drawn out. The reasons for this phenomenon will be discussed in Section 4.

3.2. Mineral-Rock Flow at Different Drawing Heights

Based on Attribute Stochastic Medium Theory, the flow subsystem was used to simulate the shape development of the ore-drawing body under the condition that the drawbell outlet size is 3.5 × 4.0 m and the ore-drawing height H = 20 m, 40 m, 60 m, 80 m, 100 m, and 150 m, respectively. The simulation can provide a reference and basis for the design of ore-drawing structural parameters. The results of the shape development of the IEZ are shown in Figure 9.

According to the specific shape of the IEZ at different drawing heights in Figure 8, the lengths of the corresponding long and short half-axes of the IEZ were measured, and the development law of the IEZ shape was analyzed, as shown in

Table 2. Parameters of the IEZ.

Drawing Height (m)	20	40	60	80	100	150	200	300
Length of Long Half Axis (m)	10.56	21.34	31.75	42.05	52.78	74.68	106.6	157.5
Length of Short Half Axis (m)	6.3	8.7	11.7	12.6	14.1	16.4	19.5	20.4
Eccentricity	0.80	0.91	0.93	0.95	0.96	0.98	0.98	0.99
Cu (%)	0.51	0.54	0.55	0.52	0.51	0.49	0.50	0.47

.

Based on Table 2, the relationships between ore-drawing height and the length of long half-axes of the IEZ, the length of short half-axes of the IEZ, and the IEZ eccentricity were obtained with the least-squares method of the curve using Formulas (3)–(5), respectively.

A = 0.3727 H + 19.25,

(3)

$$B=20.5-18.87e^{-\frac{H}{82.8}},$$

(4)

$$\epsilon =1-0.65e^{-\frac{H}{25}},$$

(5)

where A is the length of the long half-axis of the IEZ in meters, H is the drawing height in meters, B is the short half-axis length of the IEZ in meters, and ε is the IEZ eccentricity.

According to Table 2 and Formulas (3)–(5), it can be concluded that the length of the long half-axis of the IEZ has an approximately linear relationship with the ore-drawing height. With an increase in the ore-drawing height, the length of the long half-axis of the IEZ increases linearly. With an increase in the ore-drawing height, the length of the short half-axis and the IEZ eccentricity increase gradually, but the growth rate tends to decrease gradually, showing a general power exponential relationship. Formulas (3)–(5) can be used to define the IEZ parameters in subsequent software calculations, and also can provide the basis and reference for the selection of ore-drawing structural parameters.

3.3. Mineral-Rock Flow under Different Fragmentation Conditions

The same drawing region as in Section 3.1 was used to carry out mineral-rock flow simulations with different block sizes using the method described in Section 3. The simulation results are shown in Figure 10 and

Table 3. Result of particle flow simulation under different block indexes (κ = 10%).

Block Index β	Cu Grade of IEZ (%)	Length of Short Half Axis (m)	Eccentricity of IEZ	Average Depth of Depression Pit (m)
1	0.50	6.3	0.84	2.93
2	0.49	6.5	0.83	2.95
3	0.51	7.5	0.8	2.94

. From Table 3, it can be concluded that the eccentricity of the ore-drawing ellipsoid decreases with increasing block index when the draw-out index κ remains unchanged.

4. Discussion and Conclusions

In this paper, we proposed the Attribute Stochastic Medium Theory, and presented the concept of draw-out index and block fragmentation index. It can be used to analyze the flow characteristics of rock block under different drawing heights and different fragmentation conditions. We designed the algorithm for mineral-rock flow based on the Attribute Stochastic Medium Theory and implemented it with C++ language.

Through research, the main conclusions are presented below:

The grade distribution of a flowing IEZ was obtained with our method. With an increase in drawing height, the length of the long half-axis of the IEZ increased linearly, and the length of the short half-axis and the eccentricity of the IEZ increased gradually, but the growth rate tended to decrease gradually, showing a general power exponential relationship. Under the condition that the block index remained unchanged, with an increase of the draw-out index the ore-drawing ellipsoid developed gradually, the eccentricity increased gradually, and the average depth of the depression increased gradually. With an increase of the block index, the eccentricity of the ore-drawing ellipsoid decreased gradually. However, as the κ value approached 16.4%, the particles in this fixed drawing heap were no longer drawn out. This phenomenon was caused by the flow characteristics of the loose mineral-rock, which is to say, the loose mass of the ore heap had a certain angle of repose. When drawing with a single funnel in the fixed ore heap, as the drawing index approached a certain value (cut-off draw-out index), the remnants automatically entered a stable state, so that the remnants will no longer be drawn out. The cut-off value of k depends on the material properties of the loose mass and the ratio of the size of the funnel to the size of the ore heap. For a fixed ore heap, the larger the funnel size is, the larger the cut-off value of κ is.

The above research preliminarily reveals the flow characteristics of caving ore and rock, which is the basis of drawing bottom structure design and production plan, especially the basis for determining the spacing of drawing funnel. The complete bottom structure design needs more in-depth research, including multifunnel drawing research. A good design of bottom structure can reduce ridge residue and reduce ore loss and dilution during ore drawing.

The research results show that the proposed method and the algorithm are feasible and practical. But there are still some limitations to our work that need to be improved. One of them is that the discreted block models are uniform. This is a simplification of the problem. Because of the complexity of uneven blocks flow, no better method has been called to our mind at present. Another limitation to this approach is that we only consider a single funnel drawing. In order to be more in line with the actual mine production, we need to consider multifunnel drawing. Although the research on multifunnel drawing is more complex, it is very meaningful and is the focus of follow-up work.