# Simulation of Electrical and Thermal Properties of Granite under the Application of Electrical Pulses Using Equivalent Circuit Models

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Simulation Methods

#### 2.1. Simulation Model

^{2}), respectively. Capacitance is expressed by the following equation:

#### 2.2. Composition of Granite

#### 2.3. Electrical and Thermal Properties of Minerals

**J**per unit area flowing through a rock is expressed by Ohm’s law:

**E**is the electric field, and ε is the dielectric constant [14]. From Ohm’s law, the current distribution in a rock is affected by the electrical conductivity and dielectric constant. The relationship between temperature and electrical conductivity of minerals (${\sigma}_{s}$) is empirically expressed by the following equation [12]:

^{−5}eV/K). The constants A and B of the granite rocks are shown in Table 2 [12], and the electrical conductivity of the minerals as a function of temperature is shown in Figure 3. The values of conductivity in Figure 3 were calculated for each single mineral using Equation (4). It is worth mentioning that the conductivity values of minerals (i.e., Figure 3) are surely different from the conductivity values of a mix solution (i.e., for NaCl solution of Equation (7)). At low temperature, the electrical conductivity of minerals is mainly affected by the water content in the fractures and pores. Sinmyo and Keppler reported the following empirical equation for NaCl solutions [15]:

#### 2.4. Applied Voltage

#### 2.5. Temperature in the Rock

_{1}and T

_{2}are the temperatures before and after the application of high voltage (i.e., heat generation) to the granite, respectively. The thermal conductivity of each mineral is summarized in Table 5 [18]. Equation (17) was used to calculate the temperature of granite under the HV application to be discussed later.

## 3. Results and Discussion

#### 3.1. Distribution of Minerals in Granite

#### 3.2. Simulation Results

## 4. Conclusions

- (1)
- The electric field of the granite confirmed that the electric field dropped near the plagioclase;
- (2)
- The electrical conductivity of the plagioclase was higher than that of the other minerals due to its higher void volume, which has a significant effect on the electric field distribution;
- (3)
- The temperature change in the granite observed a very high temperature change near the upper/HV electrode, but it was very small as away from that electrode.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Concept of equivalent circuit model. (

**a**) Granite; (

**b**) equivalent circuit model for a small divided cube of granite.

**Figure 5.**Equivalent circuit model for typical dielectric material. (

**a**) Equivalent circuit; (

**b**) vector analysis of the circuit.

**Figure 6.**Simulation result when time changes. The red boxes in (

**b**,

**d**,

**f**) are added for the visual aid. (

**a**) Mineral distribution; (

**b**) cross-section surface viewed from y-z direction; (

**c**) electric field (50% rise of lightning impulse voltage); (

**d**) electric field (100% rise of lightning impulse voltage); (

**e**) heat (50% rise of lightning impulse voltage); (

**f**) heat (100% rise of lightning impulse voltage); (

**g**) variation of temperature (50% rise of lightning impulse voltage); (

**h**) variation of temperature (100% rise of lightning impulse voltage).

**Table 1.**Mineral composition of a typical granite [12].

Mineral | Volume [%] |
---|---|

Quartz | 30 |

Plagioclase | 20 |

K-Feldspar | 45 |

Biotite | 5 |

**Table 2.**Values of the empirical parameters [12].

Mineral | Log(A) [log(s/m)] | B [eV] |
---|---|---|

Quartz | 6.3 | 0.82 |

Plagioclase | 0.041 | 0.85 |

K-Feldspar | 0.11 | 0.85 |

Biotite | −13.8 | 0.00 |

Mineral | Void [%] | Permittivity |
---|---|---|

Quartz | 0.9 | 6.53 |

Plagioclase | 1.8 | 6.91 |

K-Feldspar | 0.9 | 6.2 |

Biotite | 0.9 | 9.28 |

Mineral | Resistance R [Ω] | Capacitance C [F] | tanδ |
---|---|---|---|

Quartz | 1.57 × 10^{7} | 2.90 × 10^{−15} | 175 |

Plagioclase | 5.54 × 10^{6} | 3.08 × 10^{−15} | 467 |

K-Feldspar | 1.57 × 10^{7} | 6.15 × 10^{−15} | 82.5 |

Biotite | 1.57 × 10^{7} | 8.26 × 10^{−15} | 61.4 |

**Table 5.**Heat characteristics of minerals in granite [18].

Mineral | Heat Resistance [K/W] | Heat Conductivity [W/m⋅K] |
---|---|---|

Quartz | 1300 | 7.69 |

Plagioclase | 4673 | 2.14 |

K-Feldspar | 4329 | 2.31 |

Biotite | 4950 | 2.02 |

Voltage | Maximum Value of Electric Field [V/m] | Minimum Value of Electric Field [V/m] | Maximum Value of Heat [W] | Minimum Value of Heat [W] | Maximum Value of Temperature Change [K] | Minimum Value of Temperature Change [K] |
---|---|---|---|---|---|---|

50% rise | $17.7{\times 10}^{6}$ | $106{\times 10}^{3}$ | 0.262 | 0.00 | 1133 | 0.0283 |

100% rise | $29.7{\times 10}^{6}$ | $104{\times 10}^{3}$ | 0.7351 | 0.00 | 3182 | 0.0664 |

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**MDPI and ACS Style**

Fukushima, K.; Kabir, M.; Kanda, K.; Obara, N.; Fukuyama, M.; Otsuki, A.
Simulation of Electrical and Thermal Properties of Granite under the Application of Electrical Pulses Using Equivalent Circuit Models. *Materials* **2022**, *15*, 1039.
https://doi.org/10.3390/ma15031039

**AMA Style**

Fukushima K, Kabir M, Kanda K, Obara N, Fukuyama M, Otsuki A.
Simulation of Electrical and Thermal Properties of Granite under the Application of Electrical Pulses Using Equivalent Circuit Models. *Materials*. 2022; 15(3):1039.
https://doi.org/10.3390/ma15031039

**Chicago/Turabian Style**

Fukushima, Kyosuke, Mahmudul Kabir, Kensuke Kanda, Naoko Obara, Mayuko Fukuyama, and Akira Otsuki.
2022. "Simulation of Electrical and Thermal Properties of Granite under the Application of Electrical Pulses Using Equivalent Circuit Models" *Materials* 15, no. 3: 1039.
https://doi.org/10.3390/ma15031039