# Comparative Analysis of Basic and Extended Power Models of Boreholes Expansion Dependence on Explosive Charge in Blasting in Clay Soil

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}) and smallest loss of information through Akaike’s Information Criteria (AIC and AIC

_{c}). This paper presents an extended power model of dependence of spherical cavity volume expansion on explosive charge. Extended model is a basic model with an additional parameter to ensure more precise mathematical description and further decrease of error of estimate for all efficiency indicators and for both types of explosive used.

## 1. Introduction

_{rc}) depends on the mass and type of explosive charge (Q) and geotechnical characteristics of the soil [6,7,8,9]. A diagram of relationship between the volume of spherical cavity and the mass of explosive charge has been made based on the results of spherical cavity blasting (Figure 1) [2].

^{2}, internal friction angle φ = 19.8°, and volume weight γ = 18.7 KN/m

^{3}. Since the clay is hydroalumosilicate, which means that it absorbs water and therefore becomes brittle, the exploitation of the clay can only be carried out in dry months and when the air temperature is above 0 °C [13]. Spherical cavity blasting is rare in hard rocks. Namely, the effect of smaller explosive charge for spherical cavities is in principle different than continuously filled borehole in mining and blasting operations. Detonation of smaller explosive charge crushes the material in proximity of activated explosive charge. With the increase of distance from the explosive charge, the released energy is not enough for crushing materials, but instead compacts it [4,11,12,14].

_{rc}), horizontal (L

_{re}), and vertical (D

_{re}) borehole increase, were obtained in field research from 2014 to 2016 [6,15]. Težak D. et al., 2019 [1] introduced the initial values of spherical cavity volume, horizontal, and vertical borehole increase. The inclusion of these values in the database resulted in the reduction of the error of estimation of the dependence of the boreholes expansion on the explosive charge during spherical cavity blasting. Two types of explosive charge were used: ANFO explosives commercially known as Pakaex and the Ammonia Nitrogen Powder Explosive Permonex V19. Technical specifications from the manufacturer for Pakaex: density 0.87 g/cm

^{3}, VOD 2950 m/s, gas volume 984 l/kg, energy 3.7 kJ/kg and for Permonex V19: density 0.95 g/cm

^{3}, VOD 4500 m/s, gas volume 900 l/kg, energy 4.2 kJ/kg [14].

_{rc}, L

_{re}and D

_{re}) and type of explosive (Pakaex and Permonex V19), power model turned out to be the most efficient. Power model gives smallest estimation error values shown as the sum of square residuals (SS), largest values of determination coefficient (R

^{2}) and smallest loss of information shown by Akaike’s Information Criteria (AIC and AIC

_{c}) [1].

- Basic database—basic model
- Extended database—basic model
- Extended database—extended model

## 2. Previous Research

## 3. Extended Power Model

_{rc0}), horizontal (L

_{re0}), and vertical (D

_{re0}) borehole increase, all shown in blue in Table 1 [1].

_{rc}), horizontal (L

_{re}), and vertical (D

_{re}) borehole increase are shown as point data on the ordinate. Basic power model was extended with addition of a new parameter (c):

^{2}) shown in Table 2 and Table 3. However, that value increases with the increase of number of parameters in the model and can only be used to compare the models with the same number of parameters. Since there is a difference between the basic and extended models, it is necessary to compare both on the basis of Akaike’s Information Criteria (AIC) which also takes into account the number of parameters [21,22,23,24]:

^{2}, AIC, and AIC

_{c}are determined and shown for both types of explosive charge seperately (Table 2 and Table 3). Between the two models, the one that gives lesser values of AIC and AIC

_{c}is considered superior.

## 4. Results and Discussion

_{c}was for the extended model. That result justifies the addition of constant to the function argument as a new model parameter.

^{2}), and values of Akaike’s Information Criteria (AIC and AICc).

^{2}, Table 2 and Table 3). This difference was expected, since the value of the coefficient increases with the increase of numbers of parameters in the model. This is the reason behind using AIC and AIC

_{c}for comparison.

_{c}expressions that their values are proportional to the error of estimate value and the number of the parameters. For that reason, it is necessary to establish whether reduction of error of estimated value is sufficient to justify the implementation of the new parameter.

_{c}are listed in Table 2 and Table 3 in separate columns. Lower AIC values were obtained for extended models for spherical volume increase for both explosive charges and horizontal borehole increase for Pakaex. However, AIC

_{c}lower values for basic models were obtained for all cases.

## 5. Conclusions

_{c}were obtained for the basic models, making the basic model superior and the introduction of the new parameter not completely justifiable.

_{c}values.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Diagram of relationship between the volume of spherical cavity V

_{rc}(m

^{3}) and the mass of explosive charge Q (kg) [2].

**Figure 2.**Schematic of basic and extended power model for spherical cavity volume increase V

_{rc}(m

^{3}) in dependence on the mass of explosive charge Q (kg).

**Figure 3.**Basic and extended power model for Pakaex: (

**a**) Spherical volume increase; (

**b**) Borehole horizontal increase; (

**c**) Borehole vertical increase.

**Figure 4.**Basic and extended power model for Permonex V19: (

**a**) Spherical volume increase; (

**b**) Borehole horizontal increase; (

**c**) Borehole vertical increase.

**Table 1.**The results of spherical cavity blasting for explosives Pakaex and Permonex V19 [1].

Pakaex | Permonex V19 | ||||||||
---|---|---|---|---|---|---|---|---|---|

Borehole | Explosive Charge Mass | Volume of the Resulting Cavity | Resulting Expansion of the Borehole | Deepening of the Resulting Expansion | Borehole | Explosive Charge Mass | Volume of the Resulting Cavity | Resulting Expansion of the Borehole | Deepening of the Resulting Expansion |

Q (kg) | V_{rc}(m ^{3}) | L_{re}(m) | D_{re}(m) | Q (kg) | V_{rc}(m ^{3}) | L_{re}(m) | D_{re}(m) | ||

MB20 | 1.00 | 0.7100 | 1.1570 | 0.5200 | MB24 | 0.80 | 0.6184 | 1.1900 | 0.3100 |

MB41 | 1.00 | 0.8095 | 1.1110 | 0.6000 | MB26 | 0.80 | 0.5690 | 1.1310 | 0.3600 |

MB34 | 0.80 | 0.3935 | 0.9530 | 0.3300 | MB45 | 0.80 | 0.7405 | 1.0700 | 0.4000 |

MB18 | 0.80 | 0.3440 | 0.8770 | 0.4600 | PMB5 | 0.80 | 0.7227 | 1.0710 | 0.4200 |

MB19 | 0.80 | 0.3626 | 0.8750 | 0.4800 | MB23 | 0.60 | 0.5276 | 1.1040 | 0.3500 |

MB40 | 0.80 | 0.5190 | 1.0600 | 0.4000 | MB25 | 0.60 | 0.6330 | 1.0850 | 0.2900 |

MB35 | 0.60 | 0.2555 | 0.7830 | 0.2500 | PMB6 | 0.60 | 0.6151 | 1.1520 | 0.3500 |

MB17 | 0.60 | 0.6160 | 1.0430 | 0.3400 | MB36 | 0.40 | 0.1135 | 0.6930 | 0.2300 |

MB39 | 0.60 | 0.3785 | 1.0880 | 0.4000 | MB21 | 0.40 | 0.2925 | 0.9360 | 0.2600 |

MB15 | 0.40 | 0.2445 | 0.6980 | 0.3100 | MB27 | 0.40 | 0.2160 | 0.5850 | 0.3200 |

MB16 | 0.40 | 0.1945 | 0.7870 | 0.3000 | MB43 | 0.40 | 0.2815 | 0.8660 | 0.3000 |

MB38 | 0.40 | 0.2980 | 0.8480 | 0.4000 | MB22 | 0.20 | 0.0825 | 0.5570 | 0.2600 |

MB13 | 0.20 | 0.1005 | 0.5760 | 0.1800 | MB28 | 0.20 | 0.0700 | 0.5050 | 0.2200 |

MB14 | 0.20 | 0.0645 | 0.5770 | 0.2200 | MB42 | 0.20 | 0.1480 | 0.6620 | 0.2000 |

MB29 | 0.20 | 0.0980 | 0.6870 | 0.2400 | - | 0.00 | 0.0005 | 0.1310 | 0.0390 |

MB37 | 0.20 | 0.1175 | 0.6010 | 0.2500 | |||||

- | 0.00 | 0.0007 | 0.1310 | 0.0390 |

Pakaex (N = 17) | a | b | c | K | SS | R^{2} | AIC | AIC_{c} | |
---|---|---|---|---|---|---|---|---|---|

V_{rc} | 1 | 0.66 | 0.66 | - | 2 | 0.170 | 0.770 | −68.700 | −67.900 |

2 | 0.66 | 1.20 | - | 2 | 0.170 | 0.800 | −74.278 | −73.528 | |

3 | 0.64 | 1.32 | 0.02 | 3 | 0.169 | 0.804 | −74.358 | −72.758 | |

L_{re} | 1 | 1.09 | 0.35 | - | 2 | 0.130 | 0.780 | −72.920 | −72.120 |

2 | 1.09 | 0.35 | - | 2 | 0.148 | 0.863 | −76.667 | −75.917 | |

3 | 0.96 | 0.42 | 0.13 | 3 | 0.131 | 0.878 | −76.718 | −75.118 | |

D_{re} | 1 | 0.50 | 0.53 | - | 2 | 0.060 | 0.720 | −85.920 | −85.120 |

2 | 0.50 | 0.53 | - | 2 | 0.060 | 0.804 | −92.130 | −91.380 | |

3 | 0.45 | 0.64 | 0.05 | 3 | 0.057 | 0.812 | −90.894 | −89.294 |

_{rc}, made spherical volume; L

_{re}, horizontal borehole increase; D

_{re}, vertical borehole increase.

**Table 3.**Parameter values and results of the power model analysis for explosive charge Permonex V19.

Permonex V19 (N = 15) | a | b | c | K | SS | R^{2} | AIC | AIC_{c} | |
---|---|---|---|---|---|---|---|---|---|

V_{rc} | 1 | 0.94 | 0.21 | - | 2 | 0.110 | 0.870 | −64.260 | −63.340 |

2 | 0.94 | 1.32 | - | 2 | 0.110 | 0.891 | −70.173 | −69.316 | |

3 | 0.96 | 1.22 | −0.03 | 3 | 0.105 | 0.893 | −70.399 | −68.553 | |

L_{re} | 1 | 1.30 | 0.51 | - | 2 | 0.160 | 0.810 | −59.010 | −58.090 |

2 | 1.30 | 0.51 | - | 2 | 0.170 | 0.870 | −62.972 | −62.115 | |

3 | 1.18 | 0.59 | 0.12 | 3 | 0.157 | 0.885 | −62.384 | −60.538 | |

D_{re} | 1 | 0.40 | 0.37 | - | 2 | 0.020 | 0.710 | −90.320 | −89.410 |

2 | 0.40 | 0.37 | - | 2 | 0.018 | 0.853 | −96.798 | −95.941 | |

3 | 0.36 | 0.43 | 0.04 | 3 | 0.017 | 0.866 | −96.187 | −94.472 |

_{rc}, made spherical volume; L

_{re}, horizontal borehole increase; D

_{re}, vertical borehole increase.

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**MDPI and ACS Style**

Kovač, I.; Težak, D.; Mesec, J.; Markovinović, I. Comparative Analysis of Basic and Extended Power Models of Boreholes Expansion Dependence on Explosive Charge in Blasting in Clay Soil. *Geosciences* **2020**, *10*, 151.
https://doi.org/10.3390/geosciences10040151

**AMA Style**

Kovač I, Težak D, Mesec J, Markovinović I. Comparative Analysis of Basic and Extended Power Models of Boreholes Expansion Dependence on Explosive Charge in Blasting in Clay Soil. *Geosciences*. 2020; 10(4):151.
https://doi.org/10.3390/geosciences10040151

**Chicago/Turabian Style**

Kovač, Ivan, Denis Težak, Josip Mesec, and Ivica Markovinović. 2020. "Comparative Analysis of Basic and Extended Power Models of Boreholes Expansion Dependence on Explosive Charge in Blasting in Clay Soil" *Geosciences* 10, no. 4: 151.
https://doi.org/10.3390/geosciences10040151