# Dependence Models of Borehole Expansion on Explosive Charge in Spherical Cavity Blasting

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

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

^{3}, VOD 4500 m/s, gas volume 900 l/kg. Therefore, six cases “indicator-explosive” are defined in this way.

- Presenting a new method for determining the value of spherical cavity blasting performance indicators;
- Determining the mathematical model that best describes the dependence of blasting results on the amount of explosive charge.

## 2. Previous Research

## 3. Recent Research

**Locating blasting areas and positioning of the future boreholes**- -
- In order to determine the borehole position coordinates and height, is the determination of the geodetic profile (Figure 6), was performed with the RTK GNSS method, using online transformation parameters via CROPOS (CROatianPOsitioning System).

**Geotechnical field investigations**- -
- For the assessment of the dynamic properties of the soil at the profile depth before and after the blasting, the MASW (multi-channel analysis of surface waves) method for multichannel analysis of surface waves, have been used for measurement of velocity of the shear waves V
_{s}[20,21]. The aim of the geophysical research was to determine the data on the general and mechanical properties of the soil natural layers by depth, and to determine the changes of the soil dynamic properties caused by the activation of explosive charges of different type and masses; - -
- Exploratory drilling of boreholes was carried out with the aim of obtaining disturbed and undisturbed samples of clay, which were taken from characteristic boreholes at certain depth, and sent to geotechnical laboratory for further testing;
- -
- Geomechanical laboratory tests of disturbed and undisturbed samples of the subject clay have been performed in an accredited laboratory of the Faculty of Geotehnical Engineering according to the international standard HRN EN ISO / IEC 17025: 2007. For the purposes of research, the clay moisture is determined, with its undrained shear strength before and after blasting.

**Test blasting and determination of the effective range of masses of two different types of explosives**- -
- It was found that the spherical expansion for the predetermined borehole diameter of 131 mm and the depth from 2.00 to 3.00 m, is possible with the explosive type Permonex V19 and Pakaex, ranging from 0.2 to 1.6 kg. Declared velocity of detonation for Permonex V19 explosive is 4500 m/s, while for the Pakaex explosive is 2950 m/s. The explosive charge activation was performed by the NONEL system started with the instantaneous electric detonators (IED) (Figure 7). During the test blasts, the stem was 0.3 m and 0.5 m, depending on the quantity of explosive in the borehole, not to allow its ejection from it, which would result in partial detonation energy loss. The stone material gradation 0/4 mm was used for stemming. Figure 8a shows the borehole construction. The moment of explosive charge activation of the Permonex V19 is shown in Figure 8b, and the schematic representation of the resulting spherical expansion is shown in Figure 8c [14].

**Determination of spherical expansion after explosive charge activation**- -
- The most important was the determination of the shape and volume of the spherical expansions formed after the explosive charge detonation in a 131 mm in diameter cylindrical borehole.

## 4. Spherical Expansion Volume Measuring Method

## 5. Models

_{rc}); the quantity of explosives–diameter of the expansion (L

_{re}) and the quantity of explosives–depth of expansion (D

_{re}) for both types of explosive. In the scatter plots, regression curves (exponential, logarithmic and power) were implemented, which represent dependence models of increased volume, expansion, and deepening of the borehole on the amount of explosive charge, Table 2.

_{rc0}), diameter (D

_{re0}), and depth of expansion (L

_{re0}) are added to the experimental data. These points in scatter plot have coordinates (0, V

_{rc0}), (0, D

_{re0}), and (0, L

_{re0}) (Figure 10, Figure 11 and Figure 12).

_{α}).

^{2}value is specified for each model (Table 5). However, since the models differ from each other by quantity of data (N) on which, basis of parameters are defined, Akaike ́s Information Criteria (AIC) was used:

_{c}) is used:

_{c}= AIC + (2K(K + 1))/(N-K + 1),

_{c}values for each model are listed in Table 5. The above equation shows that the AIC

_{c}value increases with the increase of N, SS and K, and the best model is considered the one where the AIC

_{c}value is the lowest.

## 6. Results and Discussion

_{rc}(m

^{3}) and diameter, D

_{re}(m) of expansion at the same mass of explosive charge Q (kg), is greater after activation of Permonex V19 than of Pakaex explosive. Conversely, the deepening result, L

_{re}(m) at the same mass of explosive charge Q (kg), is greater when using Pakaex than Permonex V19 explosive (Figure 12).

^{2}) range from 0.69 to 0.89 (Table 5). Based on these results, it can be concluded that all proposed models successfully fit into experimental data. However, because of the difference between the numbers of parameters in the models, the AIC or AIC

_{c}value (Table 5), is relevant for mutual comparison of the models. Namely, in two cases (Pakaex, expansion, logistical model and Pemonex V19, expansion, exponential model) obtained R

^{2}values are higher in the supplemented models, which means that these models are better than the basic ones. However, the obtained AICc values are lower for basic models.

_{c}value gives exactly the opposite conclusion. Namely, if the basic and supplemented models are mutually compared for the same indicator, the same function and the same explosive, 18 pairs of models are obtained. In 77.78% cases (14 out of 18), the AIC

_{c}value is lesser for the supplemented model (Table 5).

_{c}values are obtained for both types of explosive.

_{c}value is obtained for the supplemented model. This fact confirms the justification for adding the constant to the function argument as a new model parameter.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Spherical cavity blasting phases (

**a**) Before the detonation of the explosive charge; (

**b**) After detonation of the explosive charge; (

**c**) Charging the spherical cavity of the borehole [1].

**Figure 4.**Location of the exploitation field Cukavec II [14].

**Figure 5.**Spherical expansion measurement equipment [14].

**Figure 8.**(

**a**) Borehole construction;(

**b**) Moment of explosive charge activation; (

**c**) Formed spherical expansion.

**Figure 10.**The dependence of the resulting volume of expansion V

_{rc}(m

^{3}) on the mass of explosive charge Q (kg), for Permonex V19 and Pakaex.

**Figure 11.**The dependence of diameter of enlargement D

_{re}(m) on the mass of explosive charge Q (kg), for Permonex V19 and Pakaex.

**Figure 12.**The dependence of the resulting deepening L

_{re}(m) on the mass of explosive charge Q (kg), for Permonex V19 and Pakaex.

**Figure 13.**(

**a**) The mathematical model plots of the lowest AIC

_{c}values for the resulting expansion volume V

_{rc}(m

^{3}) for Pakaex. (

**b**) The mathematical model plots of the lowest AIC

_{c}values for the resulting expansion volume V

_{rc}(m

^{3}) for Permonex V19.

**Figure 14.**(

**a**) The mathematical model plots of the lowest AIC

_{c}values for the expansion diameter D

_{re}(m) for Pakaex. (

**b**) The mathematical model plots of the lowest AIC

_{c}values for the expansion diameter D

_{re}(m) for Permonex V19.

**Figure 15.**(

**a**) The mathematical model plots of the lowest AIC

_{c}values for the resulting deepening L

_{re}(m) for Pakaex. (

**b**) The mathematical model plots of the lowest AIC

_{c}values for the resulting deepening L

_{re}(m) for Permonex V19.

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 | |||||

MB37 | 0.20 | 0.1175 | 0.6010 | 0.2500 |

**Table 2.**Dependence models of increased volume, expansion and deepening of the borehole on the amount of explosive charge.

Model | Basic | Linearized |
---|---|---|

Exponential | y = a*e^(b*x) | ln(y) = b*x + ln(a) |

Logarithmic | y = a*ln(x) + b | e^((y - b)/a) = x |

Modified logarithmic | y = a*ln(x + b) | e^((y/a)) = x + b |

Power | y = a*x^b | (y/a)^(1/b) = x |

Volume | Expansion | Deepening | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

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

Basic | Bs.+ nul.v. | Basic | Bs.+ nul.v. | Basic | Bs.+ nul.v. | Basic | Bs.+ nul.v. | Basic | Bs.+ nul.v. | Basic | Bs.+ nul.v. | ||

Exp. | a | 0.09 | 0.09 | 0.10 | 0.10 | 0.57 | 0.51 | 0.52 | 0.46 | 0.19 | 0.17 | 0.20 | 0.17 |

b | 2.03 | 2.10 | 2.38 | 2.47 | 0.68 | 0.84 | 1.02 | 1.20 | 1.04 | 1.16 | 0.79 | 1.03 | |

Log. | a | 0.31 | 0.93 | 0.44 | 1.24 | 0.29 | 1.37 | 0.43 | 1.77 | 0.16 | 0.66 | 0.11 | 0.54 |

b | 0.57 | 0.92 | 0.75 | 0.90 | 1.07 | 1.31 | 1.23 | 1.15 | 0.47 | 1.16 | 0.39 | 1.24 | |

Pow. | a | 0.66 | 0.66 | 0.94 | 0.94 | 1.09 | 1.09 | 1.30 | 1.30 | 0.50 | 0.50 | 0.40 | 0.40 |

b | 1.20 | 1.20 | 0.21 | 1.32 | 0.35 | 0.35 | 0.51 | 0.51 | 0.53 | 0.53 | 0.37 | 0.37 |

Linearized Models | PAKAEX | PERMONEX V19 | |||||||
---|---|---|---|---|---|---|---|---|---|

Basic | Supplemented | Basic | Supplemented | ||||||

r | t | r | t | r | t | r | t | ||

Volume | Exp. | 0.9054 | 7.979 | 0.7614 | 4.549 | 0.9125 | 7.727 | 0.9125 | 8.043 |

Log. | 0.8458 | 5.932 | 0.8543 | 6.366 | 0.9299 | 8.758 | 0.9418 | 10.101 | |

Pow. | 0.8893 | 7.276 | 0.9088 | 8.436 | 0.9369 | 9.284 | 0.9436 | 10.276 | |

Expansion | Exp. | 0.8754 | 6.776 | 0.7522 | 4.421 | 0.8746 | 6.249 | 0.8269 | 5.302 |

Log. | 0.8602 | 6.311 | 0.8866 | 7.424 | 0.8935 | 6.893 | 0.9225 | 8.536 | |

Pow. | 0.7308 | 5.816 | 0.8624 | 6.598 | 0.8944 | 6.927 | 0.9182 | 8.358 | |

Deepening | Exp. | 0.863 | 6.394 | 0.787 | 4.934 | 0.913 | 7.727 | 0.936 | 9.604 |

Log. | 0.848 | 5.984 | 0.890 | 7.572 | 0.916 | 7.883 | 0.935 | 9.466 | |

Pow. | 0.846 | 5.937 | 0.872 | 6.889 | 0.893 | 6.889 | 0.896 | 7.275 | |

df | 14 | 15 | 12 | 13 | |||||

t(α)
| 1.761 | 1.753 | 1.782 | 1.771 |

**Table 5.**The values of the comparison measures. Blue is the lowest AIC

_{c}value for the measured data. Red is the lowest AIC

_{c}value for the measured data + V

_{rc0}, D

_{re0}and L

_{re0}

Volume | Expansion | Deepening | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

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

Measured | Me.+ V_{rc0} | Measured | Me.+ V_{rc0} | Measured | Me.+ D_{re0} | Measured | Me.+ D_{re0} | Measured | Me.+L_{re0} | Measured | Me.+ L_{re0} | ||

N | 16 | 17 | 14 | 15 | 16 | 17 | 14 | 15 | 16 | 17 | 14 | 15 | |

Exponential | R^{2} | 0.78 | 0.80 | 0.82 | 0.83 | 0.74 | 0.70 | 0.75 | 0.76 | 0.77 | 0.77 | 0.71 | 0.70 |

R^{2}_{adj} | 0.75 | 0.77 | 0.78 | 0.81 | 0.70 | 0.66 | 0.71 | 0.72 | 0.73 | 0.74 | 0.66 | 0.65 | |

SS | 0.16 | 0.17 | 0.15 | 0.16 | 0.15 | 0.32 | 0.20 | 0.33 | 0.05 | 0.07 | 0.02 | 0.04 | |

AIC | −69.35 | −74.12 | −59.37 | −63.95 | −70.45 | −63.56 | −55.42 | −53.34 | −88.58 | −89.59 | −90.24 | −85.83 | |

AIC_{C} | −68.55 | −73.37 | −58.44 | −63.09 | −69.65 | −62.81 | −54.49 | −52.14 | −87.78 | −88.84 | −89.32 | −84.97 | |

Logarithmic | R^{2} | 0.72 | 0.78 | 0.84 | 0.87 | 0.77 | 0.80 | 0.81 | 0.87 | 0.69 | 0.80 | 0.70 | 0.79 |

R^{2}_{adj} | 0.67 | 0.74 | 0.81 | 0.84 | 0.74 | 0.77 | 0.77 | 0.84 | 0.64 | 0.77 | 0.64 | 0.76 | |

SS | 0.21 | 0.19 | 0.13 | 0.13 | 0.13 | 0.22 | 0.16 | 0.18 | 0.07 | 0.06 | 0.02 | 0.03 | |

AIC | −69.05 | −72.10 | −61.22 | −66.98 | −72.78 | −70.13 | −58.93 | −62.01 | −83.95 | −91.55 | −89.58 | −91.42 | |

AIC_{C} | −68.25 | −71.35 | −60.30 | −66.13 | −71.98 | −69.38 | −58.01 | −61.16 | −83.15 | −90.80 | −88.66 | −90.57 | |

Power | R^{2} | 0.77 | 0.80 | 0.87 | 0.89 | 0.78 | 0.86 | 0.81 | 0.87 | 0.72 | 0.80 | 0.71 | 0.85 |

R^{2}_{adj} | 0.74 | 0.77 | 0.85 | 0.87 | 0.74 | 0.84 | 0.77 | 0.85 | 0.68 | 0.78 | 0.66 | 0.83 | |

SS | 0.17 | 0.17 | 0.11 | 0.11 | 0.13 | 0.15 | 0.16 | 0.17 | 0.06 | 0.06 | 0.02 | 0.02 | |

AIC | −68.70 | −74.28 | −64.26 | −70.17 | −72.92 | −76.67 | −59.01 | −62.97 | −85.92 | −92.13 | −90.32 | −96.80 | |

AIC_{C} | −67.90 | −73.53 | −63.34 | −69.32 | 72.12 | −75.92 | −58.09 | −62.12 | −85.12 | −91.38 | −89.41 | −95.94 |

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**MDPI and ACS Style**

Težak, D.; Stanković, S.; Kovač, I.
Dependence Models of Borehole Expansion on Explosive Charge in Spherical Cavity Blasting. *Geosciences* **2019**, *9*, 383.
https://doi.org/10.3390/geosciences9090383

**AMA Style**

Težak D, Stanković S, Kovač I.
Dependence Models of Borehole Expansion on Explosive Charge in Spherical Cavity Blasting. *Geosciences*. 2019; 9(9):383.
https://doi.org/10.3390/geosciences9090383

**Chicago/Turabian Style**

Težak, Denis, Siniša Stanković, and Ivan Kovač.
2019. "Dependence Models of Borehole Expansion on Explosive Charge in Spherical Cavity Blasting" *Geosciences* 9, no. 9: 383.
https://doi.org/10.3390/geosciences9090383