# An Integrated Cellular Automata Model Improves the Accuracy of Secondary Fragmentation Prediction
## 2 Materials and Methods
### Fragmentation Model  
In block caving mining, the size of the rock fragments is affected by the fragmentation that occurs during gravity flow in the ore column while material is being extracted from the drawpoints [4]. In addition, smaller particles or fragments can migrate between coarser fragments during this process [29; 30; 31; 32]. The dynamics of these two phenomena, rock fragmentation and particle percolation, present challenges for large-scale simulation in the context of block caving. To meet this challenge, in a study carried out by [20], a gravity flow simulator using the dynamics of cellular automata with a void diffusion rule was used to create a fragmentation model [29] (Figure 4). The fragmentation model uses rock strength, vertical stresses, and distance traveled as input variables to estimate rock fragmentation on a caving scale and was calibrated using experimental and mining operations data.  
Figure 3: (**a**) Profile view of mass transmission in a six-cell arrangement with stagnant and motion zones, modified from [22]. (**b**) Stagnant zone and flow zone in a physical experiment [26].  
The fragmentation model applied a reduction to the mean size (d\({}_{50}\)) associated with each cell to replicate the fragmentation that the material undergoes when it flows due to gravity. By applying a fragmentation model that modifies the d\({}_{50}\) (Equation (5)), the fragmentation mechanisms are integrated. In this case, the size reduction ratio, R\({}_{50}\), is defined as follows [21]:  
\[R_{50}=\frac{d_{50,i}-d_{50,f}}{d}. \tag{5}\]  
In the equation above, R\({}_{50}\) is the reduction in size suffered by the average size of the fragments (d\({}_{50}\), cm) of a cell that travels a distance \"\(d\)\" (m). d\({}_{50,i}\) and d\({}_{50,f}\) are the average initial and final sizes, respectively (in cm), measured when a distance \(d\) is traveled. Then, knowing the reduction ratio R\({}_{50}\) for a cell, it is possible to determine the fragmentation that a block will undergo, quantified through the decrease in its d\({}_{50}\) when traveling a distance \(d\) from its original position. Additionally, the d\({}_{50}\) is used in the flow simulator to incorporate migration during extraction [33]; for this reason, the d\({}_{50}\) was selected in the fragmentation model.  
There is an experimental correlation (Figure 5) between the proposed R\({}_{50}\) and the UCS/\(\sigma_{\rm v}\) relation. Thus, Equation 6 was proposed based on the observed results. Data from previous comminution experiments performed under confined conditions and during extraction were used to analyze the behavior of R\({}_{50}\), presented in Figure 5.  
Figure 4: Void diffusion rule applied to the flow simulator [21]. (**a**) State prior to the ascent of a void. (**b**) State after exchanging cell “\(c\)” with the void.  
Figure 5: Fragmentation results from physical experiments under confined flow [12].  
From an adjustment to these data, the fragmentation equation was defined as follows:  
\[R_{50}=\alpha\Big{(}\frac{\sigma_{v}}{UCS}\Big{)}^{\beta}. \tag{6}\]  
In Equation (6), \(R_{50}\) is the reduction ratio of the mean size (cm/m), \(\sigma_{v}\) is the mean vertical stress (MPa), UCS is the uniaxial compressive strength (MPa), and \(\alpha\) and \(\beta\) are fitting parameters. In a case evaluated, the estimation errors were 9% and 7% [21]; those percentages were considered low error rates due to the wide variability of the phenomena involved. It should be noted that the R\({}_{50}\) can be replicated for any characteristic size \(i\), such as R\(i\), if the respective fitting parameters (\(\alpha\), \(\beta\)) for each size \(i\) are known. Then, Equation 6 can be rewritten as Equation (7):  
\[R_{i}=\alpha\Big{(}\frac{\sigma_{v}}{UCS}\Big{)}^{\beta}. \tag{7}\]  
Equations (6) and (7) show that stress is one of the variables used by the fragmentation model to estimate size reduction. For this reason, it is essential that stress be correctly calculated and integrated as detailed below.