# 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

- Barton, N.; Choubey, V. The shear strength of rock joints in theory and practice. Rock Mech. Rock Eng.
**1977**, 10, 1–54. [Google Scholar] [CrossRef] - Pyrak-Nolte, L.; Nolte, D. Approaching a universal scaling relationship between fracture stiffness and fluid flow. Nat Commun
**2016**, 7, 10663. [Google Scholar] [CrossRef] - Magsipoc, E.; Zhao, Q.; Grasselli, G. 2D and 3D Roughness Characterization. Rock Mech. Rock Eng.
**2020**, 53, 1495–1519. [Google Scholar] [CrossRef] - Sornette, D. Critical Phenomena in Natural Sciences; Springer: Berlin, Germany, 2004. [Google Scholar]
- Alava, M.J.; Nukala, P.K.; Zapperi, S. Statistical models of fracture. Adv. Phys.
**2006**, 55, 349–476. [Google Scholar] [CrossRef] - Gjerden, K.S.; Stormo, A.; Hansen, A. Universality Classes in Constrained Crack Growth. Phys. Rev. Lett.
**2013**, 111, 135502. [Google Scholar] [CrossRef] [PubMed] - Shekhawat, A.; Zapperi, S.; Sethna, J.P. From Damage Percolation to Crack Nucleation Through Finite Size Criticality. Phys. Rev. Lett.
**2013**, 110, 185505. [Google Scholar] [CrossRef] [PubMed] - Aligholi, S. Evaluating Rock Physics–Fracture Mechanics Relationship by Quantifying Fracture Process Zone. Ph.D. Thesis, Monash University, Victoria, Australia, 2022. [Google Scholar] [CrossRef]
- Barenblatt, G.I. The mathematical theory of equilibrium cracks in brittle fracture. Adv. Appl. Mech.
**1962**, 7, 55–129. [Google Scholar] - Aligholi, S.; Ponson, L.; Torabi, A.R.; Zhang, Q.B. A new methodology inspired from the theory of critical distances for determination of inherent tensile strength and fracture toughness of rock materials. Int. J. Rock Mech. Min. Sci.
**2022**, 152, 105073. [Google Scholar] [CrossRef] - Barabasi, A.L.; Bourbonnais, R.; Jensen, M.; Kertész, J.; Vicsek, T.; Zhang, Y.C. Multifractality of growing surfaces. Phys. Rev. A
**1992**, 45, R6951. [Google Scholar] [CrossRef] [PubMed] - Davis, A.; Marshak, A.; Wiscombe, W.; Cahalan, R. Multifractal characterizations of nonstationarity and intermittency in geophysical fields: Observed, retrieved, or simulated. J. Geophys. Res. Atmos.
**1994**, 99, 8055–8072. [Google Scholar] [CrossRef] - Mazeran, P.; Odoni, L.; Loubet, J. Curvature radius analysis for scanning probe microscopy. Surf. Sci.
**2005**, 585, 25–37. [Google Scholar] [CrossRef] - Vernede, S.; Ponson, L.; Bouchaud, J.P. Turbulent fracture surfaces: A footprint of damage percolation? Phys. Rev. Lett.
**2015**, 114, 215501. [Google Scholar] [CrossRef] [PubMed] - Santucci, S.; Maloy, K.J.; Delaplace, A.; Mathiesen, J.; Hansen, A.; Bakke, J.O.H.; Schmittbuhl, J.; Vanel, L.; Ray, P. Statistics of fracture surfaces. Phys. Rev. E
**2007**, 75, 016104. [Google Scholar] [CrossRef] [PubMed] - Barabasi, A.L.; Szepfalusy, P.; Vicsek, T. Multifractal spectra of multi-affine functions. Phys. A
**1991**, 178, 17. [Google Scholar] [CrossRef] - Peng, C.K.; Havlin, S.; Stanley, H.E.; Goldberger, A.L. Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos
**1995**, 5, 82–87. [Google Scholar] [CrossRef] [PubMed] - Bacry, E.; Delour, J.; Muzy, J.F. Multifractal random walk. Phys. Rev. E
**2001**, 64, 026103. [Google Scholar] [CrossRef] [PubMed] - Flandrin, P. On the spectrum of fractional Brownian motions. IEEE Trans. Inf. Theory
**1989**, 35, 197–199. [Google Scholar] [CrossRef] - Martin, W.; Flandrin, P. Wigner-Ville spectral analysis of nonstationary processes. IEEE Trans. Acoust. Speech Signal Process.
**1985**, 33, 1461–1470. [Google Scholar] [CrossRef]

**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