# Intermittency of Rock Fractured Surfaces: A Power Law

^{*}

## Abstract

**:**

_{i}is related to the crossover length of correlation function ξ, and how this critical length scale can be objectively identified.

## 1. Introduction

## 2. Materials and Methods

## 3. Results and Discussions

#### 3.1. Roughness Correlation

#### 3.2. Multifractality of Roughness

#### 3.3. Determining the Cut-Off Length

#### 3.4. Limitations and Future Work

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The 3D X-ray computed tomography images of the fractured area and topographic images of fractured surfaces of sandstone (

**a**), marble (

**b**) fine-grained granite (

**c**), and coarse grained granite (

**d**). The x-axis and z-axis correspond to the crack propagation direction and the crack front direction, respectively. The real length of reconstructed CT images is around 18 mm in the propagation direction.

**Figure 2.**The outline of the adopted methodology for modelling the multiscaling spectrum of FPZ of studied rocks.

**Figure 3.**Spatial correlations of ${\mathsf{\omega}}_{\u03f5}$ for sandstone (

**a**), marble (

**b**) fine-grained granite (

**c**), and coarse grained granite (

**d**). The correlations are represented for ${\mathsf{\omega}}_{\u03f5}$ computed at different scales $\u03f5$. The true cutoff length $\xi $ is represented for each case. Dashed red lines are showing the initial guess for estimating ${\lambda}_{c}$; green solid lines are passing through the point of convergence (red points) and the true $\xi $ with the slope ${\lambda}_{c}={\lambda}_{i}/c$; and black dashed lines are showing different iterations that are converging to the green lines.

**Figure 4.**Multiscaling spectra of the rock fractured surfaces: sandstone (

**a**), marble (

**b**) fine-grained granite (

**c**), and coarse grained granite (

**d**). The spectra are computed both below (blue curve) and above (red curve) $\xi $. Both regimes are predicted with power laws; the best fit lines and their equations are represented with the same colour. Intermittency of multifractal regimes ($\delta r<\xi $ ) are showing perfect power laws with ${R}^{2}\approx 1$ and some exponents between 0.25 and 0.5. Monofractal regimes ($\delta r>\xi $), however, show insignificant intermittency whose exponents are less than 0.05. Coefficients of presented equations represent $H\left(1\right)={\zeta}_{1}/1$, which is sometimes considered as a Hurst exponent. Its values range from 0.5 to 0.6 for monofractal regimes, and from 0.65 to 0.85 for multifractal regimes.

**Figure 5.**Experimental and predicted multiscaling spectra of the rock fractured surfaces: sandstone (

**a**), marble (

**b**) fine-grained granite (

**c**), and coarse grained granite (

**d**). Predicted spectra by the proposed model are very close to experimental ones.

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**MDPI and ACS Style**

Aligholi, S.; Khandelwal, M.
Intermittency of Rock Fractured Surfaces: A Power Law. *Water* **2022**, *14*, 3662.
https://doi.org/10.3390/w14223662

**AMA Style**

Aligholi S, Khandelwal M.
Intermittency of Rock Fractured Surfaces: A Power Law. *Water*. 2022; 14(22):3662.
https://doi.org/10.3390/w14223662

**Chicago/Turabian Style**

Aligholi, Saeed, and Manoj Khandelwal.
2022. "Intermittency of Rock Fractured Surfaces: A Power Law" *Water* 14, no. 22: 3662.
https://doi.org/10.3390/w14223662