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Analytical Solutions for the Characteristic Size Distribution of the Elliptical Model in Fractured Rock Mass

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Abstract

Fracture characteristics have a significant effect on the mechanical and hydraulic properties of rock mass. In this paper, before modeling a discrete fracture network, the fracture is simplified as a non-similar ellipse. First, the multifactor coupling stereological relationship of the sampling trace length distribution is established, which proves that the trace length distribution is independent of the fracture occurrence distribution. Using the Volterra integral equation method, the stereological formula of the trace length is inversely solved to obtain the distribution expressions of the major axis and axial ratio of the elliptical fractures. The analytical solutions of the probability density function (PDF) of the characteristic size of the elliptical fractures are derived for cases in which the trace length follows a uniform distribution, fractal distribution, and polynomial distribution. Second, for cases in which the trace length conforms to a negative exponential distribution, gamma distribution, chi-square distribution, and lognormal distribution, the statistical eigenvalues of the major axis and axial ratio of the elliptical fractures are deduced. Finally, a Monte Carlo statistical simulation is performed using the Rock Mass Joint Network Simulation (RJNS3D) toolkit to verify the applicability of the derivation process and the correctness of fitting a polynomial function to the trace length distribution to solve the PDF of the characteristic size of the elliptical fractures. The proposed method can better predict the distribution of the required parameters, such as the major axis and axial ratio of the elliptical fractures, according to the typical two-dimensional data in the sampling windows. This method can be further applied to reconstruct fracture networks in practical engineering tasks and lay a foundation for the analysis of rock mass strength and deformation, representative elementary volume (REV), seepage, and surrounding rock stability.

Highlights

  • A non-similar elliptical model is developed to simulate a fracture network in rock mass.

  • The multifactor coupling stereological relationship of the sampling trace length distribution is established.

  • The correct analytical solutions of the probability density function of the characteristic size of elliptical fractures are derived.

  • Trace sampling in the sampling window is simulated by the Monte Carlo method.

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Data availability

The data are available from the corresponding author on reasonable request.

Abbreviations

α :

Relative included angle between the elliptical fracture plane and the sampling plane

β :

Rotation angle of the elliptical fracture

H :

Distance from the center point of the elliptical fracture to the sampling plane

a :

Major axis of the elliptical fracture

k :

Axial ratio

l :

Trace length

\(\phi\) :

Strike of the elliptical fracture plane

\(\theta\) :

Dip of the elliptical fracture plane

\(\phi ^{\prime}\) :

Strike of the sampling plane

\(\theta ^{\prime}\) :

Dip of the sampling plane

\(f\left( l \right)\) :

Probability density function of l

\(g\left( a \right)\) :

Probability density function of a

\(u\left( k \right)\) :

Probability density function of k

\(\omega {\kern 1pt} \left( \alpha \right)\) :

Probability density function of α

\(\nu \left( \beta \right)\) :

Probability density function of β

N V :

Volume density of the elliptical fractures

B 0 :

Lower limit of l

B :

Upper limit of l

\(\xi_{0}\) :

Lower limit of a

\(\xi\) :

Upper limit of a

\(\xi_{k}\) :

Upper limit of k

\(u_{l}\) :

Mean value of l

\(u_{a}\) :

Mean value of a

\(u_{k}\) :

Mean value of k

D :

Fractal dimension value of the trace distribution

\(F\left( l \right)\) :

Cumulative distribution function of l

\(G\left( a \right)\) :

Cumulative distribution function of a

\(\sigma_{l}\) :

Standard deviation of l

\(\sigma_{a}\) :

Standard deviation of a

\(f_{{\text{C}}} \left( l \right)\) :

Probability density function of the contained trace length

N C :

Number of contained traces

N D :

Number of dissected traces

\(\gamma\) :

Angle between the trace line and the sampling window

RMSE:

Root-mean-square error

\(R_{{{\text{NL}}}}\) :

Fitting optimization index

\({\text{OC}}\left( u \right)\) :

Operating characteristic function of the mean value

\(\Phi \left( x \right)\) :

Standard normal distribution function

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Acknowledgements

This research was funded by the National Natural Science Foundation of China (No. U1965203 and 52004167), the China Postdoctoral Science Foundation (No. 2021T140485) and the Open Foundation of MOE Key Laboratory of Deep Underground Science and Engineering (No. DESEYU202201).

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Authors and Affiliations

  1. State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resource and Hydropower, Sichuan University, Chengdu, 610065, China

    Kun **ao, Ru Zhang, **g **e, Mingzhong Gao, Zetian Zhang & Chendi Lou

  2. MOE Key Laboratory of Deep Earth Science and Engineering, Sichuan University, Chengdu, 610225, China

    Ru Zhang, Li Ren, Zetian Zhang & Ting Ai

  3. Guangdong Provincial Key Laboratory of Deep Earth Science and Geothermal Energy Exploitation and Utilization, Institute of Deep Earth Science and Green Energy, Shenzhen University, Shenzhen, 518060, China

    Mingzhong Gao

  4. Asian School of the Environment, Nanyang Technological University, 50 Nanyang Avenue, Singapore, 639798, Singapore

    Ersheng Zha

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  1. Kun **ao
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**ao, K., Zhang, R., **e, J. et al. Analytical Solutions for the Characteristic Size Distribution of the Elliptical Model in Fractured Rock Mass. Rock Mech Rock Eng 56, 3927–3948 (2023). https://doi.org/10.1007/s00603-023-03263-w

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