# Relating Topological and Electrical Properties of Fractured Porous Media: Insights into the Characterization of Rock Fracturing

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## Abstract

**:**

## 1. Introduction

## 2. Methodological Background

#### 2.1. Electrical Parameters

#### 2.2. Modeling Approach

#### 2.3. Numerical Simulations

## 3. Results for Complex Fractured Porous Media

#### 3.1. Results for a Large Range of Fracture Densities

#### 3.2. Results at the Fracture Percolation Threshold

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Interconnected fracture networks for the domains whose fracture lengths are defined by (

**a**–

**f**) a uniform distribution with ${L}_{ratio}={L}_{max}/L$; and (

**g**–

**l**) a power-law distribution.

**Figure 2.**Electrical formation factor F plotted as a function of porosity $\varphi $ with the fracture lengths being defined from uniform (left column) and power-law (right column) distributions. The fracture networks are generated with the percolation parameter $p=6$, 12 and 18 (black, blue and red symbols, respectively), and we consider 5 simulations for each combination of the parameters {${L}_{max}$,p} and {a,p}. The relationship between F and $\varphi $ is compared to (

**a**,

**b**) values obtained for simple configurations that are described in the text, and (

**c**,

**d**) models based on Archie’s law that are described in Table 1.

**Figure 3.**(

**a**,

**b**) Archie’s exponent m and (

**c**,

**d**) electrical tortuosity $\tau $ plotted as a function of porosity $\varphi $ with the fracture lengths being defined from uniform (left column) and power-law (right column) distributions. These electrical parameters are computed from the results presented in Figure 2.

**Figure 4.**(

**a**)Electrical formation factor F; (

**b**) Archie’s exponent m; and (

**c**) electrical tortuosity $\tau $ plotted as a function of porosity $\varphi $ with the fracture lengths being defined from a uniform distribution and the generated fracture network being at the percolation threshold. We consider 100 simulations for each value of ${L}_{ratio}$ and represent the systems leading to the minimum and maximum values of each electrical parameter. The dashed black lines correspond to the reference values ${F}_{1}^{\ast}$, ${m}_{1}^{\ast}$ and ${\tau}_{1}^{\ast}$.

**Figure 5.**(

**a**) Electrical formation factor F; (

**b**) Archie’s exponent m; and (

**c**) electrical tortuosity $\tau $ plotted as a function of porosity $\varphi $ with the fracture lengths being defined from a power-law distribution and the generated fracture network being at the percolation threshold. We consider 100 simulations for each value of a and represent the systems leading to the minimum and maximum values of each electrical parameter. The dashed black lines correspond to the reference values ${F}_{1}^{\ast}$, ${m}_{1}^{\ast}$ and ${\tau}_{1}^{\ast}$.

**Figure 6.**Averaged (

**a**,

**b**) electrical formation factor F; (

**c**,

**d**) Archie’s exponent m; and (

**e**,

**f**) electrical tortuosity $\tau $ plotted as a function of porosity $\varphi $ from the results presented in Figure 4 and Figure 5 (left and right columns, respectively). In (

**a**,

**b**), the best fit between the models $F={\varphi}^{-{m}^{\prime}}$ (dashed lines), $F={\tau}^{\prime}/\varphi $ (dashdot lines), and $F={(\varphi -{\varphi}_{c})}^{-t}$ (solid lines) is shown for each family of fractures and the corresponding model parameters are presented in Table 2.

**Table 1.**Parameters obtained by fitting the results in Figure 2 with the models $F={\varphi}^{-{m}^{\prime}}$ and $F={\tau}^{\prime}/\varphi $ using a non-linear least-squares method. Bold numbers show the best models for which the coefficient of determination ${R}^{2}$ is larger than 0.8. The corresponding models are plotted in Figure 2c,d.

$\mathit{F}={\mathit{\varphi}}^{-{\mathit{m}}^{\prime}}$ | $\mathit{F}={\mathit{\tau}}^{\prime}/\mathit{\varphi}$ | ||||
---|---|---|---|---|---|

${\mathit{m}}^{\prime}$ | ${\mathit{R}}^{\mathbf{2}}$ | ${\mathit{\tau}}^{\prime}$ | ${\mathit{R}}^{\mathbf{2}}$ | ||

p (Figure 2c) | 6 | $1.65$ | $0.06$ | $27.79$ | $0.18$ |

12 | $\mathbf{1.43}$ | $\mathbf{0.95}$ | $7.98$ | $0.85$ | |

18 | $\mathbf{1.37}$ | $\mathbf{0.96}$ | $5.55$ | $0.89$ | |

p (Figure 2d) | 6 | $1.66$ | $-0.06$ | $30.47$ | 0 |

12 | $\mathbf{1.5}$ | $\mathbf{0.83}$ | $12.4$ | $0.81$ | |

18 | $\mathbf{1.42}$ | $\mathbf{0.97}$ | $7.56$ | $0.94$ |

**Table 2.**Parameters obtained by fitting the results in Figure 6a,b with the models $F={\varphi}^{-{m}^{\prime}}$, $F={\tau}^{\prime}/\varphi $ and $F={(\varphi -{\varphi}_{c})}^{-t}$ using a non-linear least-squares method. Bold numbers show the best models for which the coefficient of determination ${R}^{2}$ is larger than 0.8. The best model of each configuration is plotted in Figure 6a,b.

$\mathit{F}={\mathit{\varphi}}^{-{\mathit{m}}^{\prime}}$ | $\mathit{F}={\mathit{\tau}}^{\prime}/\mathit{\varphi}$ | $\mathit{F}={(\mathit{\varphi}-{\mathit{\varphi}}_{\mathit{c}})}^{-\mathit{t}}$ | ||||||
---|---|---|---|---|---|---|---|---|

${\mathit{m}}^{\prime}$ | ${\mathit{R}}^{\mathbf{2}}$ | ${\mathit{\tau}}^{\prime}$ | ${\mathit{R}}^{\mathbf{2}}$ | ${\mathit{\varphi}}_{\mathit{c}}$ | $\mathit{t}$ | ${\mathit{R}}^{\mathbf{2}}$ | ||

${L}_{ratio}$ (Figure 6a) | 1 | $1.68$ | $0.56$ | 33 | $0.36$ | $\mathbf{0.004}$ | $\mathbf{1.33}$ | $\mathbf{0.98}$ |

$0.5$ | $1.7$ | $0.69$ | 38 | $0.46$ | $\mathbf{0.003}$ | $\mathbf{1.44}$ | $\mathbf{0.94}$ | |

$0.2$ | $1.72$ | $0.88$ | 38 | $0.62$ | $\mathbf{0.002}$ | $\mathbf{1.57}$ | $\mathbf{0.96}$ | |

$0.1$ | 1.74 | $0.94$ | 40 | $0.75$ | $\mathbf{0.0006}$ | $\mathbf{1.7}$ | $\mathbf{0.95}$ | |

$0.05$ | 1.77 | 0.94 | 42 | $0.79$ | $\mathbf{2}\times {\mathbf{10}}^{-\mathbf{11}}$ | 1.77 | 0.94 | |

a (Figure 6b) | $1.5$ | $1.7$ | $0.33$ | 37 | $0.22$ | $0.003$ | $1.44$ | $0.47$ |

2 | $1.69$ | $0.64$ | 35 | $0.43$ | $\mathbf{0.003}$ | $\mathbf{1.46}$ | $\mathbf{0.83}$ | |

$2.5$ | $1.7$ | $0.78$ | 37 | $0.57$ | $\mathbf{0.002}$ | $\mathbf{1.59}$ | $\mathbf{0.83}$ | |

3 | $\mathbf{1.73}$ | $\mathbf{0.84}$ | 39 | $0.82$ | ${\mathbf{10}}^{-\mathbf{11}}$ | $\mathbf{1.73}$ | $\mathbf{0.84}$ | |

$3.5$ | $1.76$ | $0.85$ | 42 | $\mathbf{0.92}$ | $5\times {10}^{-12}$ | $1.76$ | $0.85$ |

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**MDPI and ACS Style**

Roubinet, D.; Irving, J.; Pezard, P.A.
Relating Topological and Electrical Properties of Fractured Porous Media: Insights into the Characterization of Rock Fracturing. *Minerals* **2018**, *8*, 14.
https://doi.org/10.3390/min8010014

**AMA Style**

Roubinet D, Irving J, Pezard PA.
Relating Topological and Electrical Properties of Fractured Porous Media: Insights into the Characterization of Rock Fracturing. *Minerals*. 2018; 8(1):14.
https://doi.org/10.3390/min8010014

**Chicago/Turabian Style**

Roubinet, Delphine, James Irving, and Philippe A. Pezard.
2018. "Relating Topological and Electrical Properties of Fractured Porous Media: Insights into the Characterization of Rock Fracturing" *Minerals* 8, no. 1: 14.
https://doi.org/10.3390/min8010014