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Scaling of Fracture Patterns in Three Deep Boreholes and Implications for Constraining Fractal Discrete Fracture Network Models

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Abstract

We present a methodology for generating fractal fracture networks in one, two and three dimensions that respects the dual power-law model, in which the scaling characteristics are set by the two independent parameters: (1) the correlation dimension that pertains to separation of fracture centers, and (2) the length exponent that governs the distribution of fracture lengths. Synthetic fracture distributions were generated to evaluate the stereological relationships between the scaling parameters of 2D and 3D networks, and the scaling of fracture intersection points along a scanline through the network. The results showed that it is not possible to estimate the 2D and 3D fractal scaling parameters of the correlation dimension from the 1D correlation dimension of fracture spacing from scanlines through the network, even if the length exponent is known a priori. Synthetic 1D distributions of fracture spacing of known correlation dimension were used as a benchmark to test the consistency of estimates of fractal dimension derived from box-counting, two-point correlation, and power-law fitting. The results showed that the correlation dimension obtained from the two-point correlation method provided the most stable and reliable estimate of the fractal dimension of fractures on 1D scanlines or boreholes. Application of the two-point correlation function to the observed fracture distributions along three deep boreholes in crystalline rock at Basel, Switzerland and Soultz-sous-Forêts, France showed that the distribution was fractal over more than two orders of magnitude in scale, and in all cases the fractal dimensions was in the range 0.86–0.88. Similar results were obtained for fracture sets of common orientation within the wells, although the fractal dimension ranged between 0.65 and 0.75. This constitutes strong evidence that fracturing in rock masses penetrated by the wells follows a fractal organization.

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Abbreviations

D :

Correlation dimension

a :

Fracture length exponent

r :

Distance between fracture centers (m)

l :

Fracture length (m)

C(r):

Correlation function

\({N_{\text{p}}}\left( r \right)\) :

Number of pairs of fractures whose center-to-center distance is less than r

\({N_{\text{t}}}\) :

Total number of fractures

L :

Domain length in 1D, 2D and 3D (m)

m :

Number of equal-sized sub-domains in a Multiplicative Cascade process

\({l_{{\text{min}}}}\) :

Minimum fracture length (m) in DFN generation

\({l_{{\text{ratio}}}}\) :

The ratio of sub-domain side length to the domain side length in Multiplicative Cascade process

P :

Probability of having a fracture in a sub-domain in Multiplicative Cascade process

U :

Fracture assignment vector

CumP:

Cumulative probability density vector

t :

Ruler length in 1D box-counting technique

\({N_{\text{b}}}(t)\) :

Number of rulers of length t containing at least one fracture

D b :

Box-dimension

s :

Fracture spacing

\({N_{\text{s}}}\) :

Number of spacings greater than or equal to a specific spacing s

D s :

Power-law exponent of fracture spacing

κ :

Fischer coefficient

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Acknowledgements

The research leading to these results received funding from the European Community’s Seventh Framework Program under Grant agreement no. 608553 (Project IMAGE). We would like thank Simon Löw for his continuous support and constructive comments during this project. We are also thankful to the anonymous reviewers, associate editor and the editor for their valuable suggestions that led to improvements of the manuscript.

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Authors and Affiliations

  1. Engineering Geology, Department of Earth Sciences, Geological Institute, ETH Zürich, Sonneggstrasse 5, 8092, Zurich, Switzerland

    Mohammad Javad Afshari Moein

  2. Center for Hydrogeology and Geothermics, University of Neuchâtel, Rue Emile-Argand 11, 2000, Neuchâtel, Switzerland

    Benoît Valley

  3. Geothermal Energy and Geofluids, Department of Earth Sciences, Institute of Geophysics, ETH Zürich, Sonneggstrasse 5, 8092, Zurich, Switzerland

    Keith F. Evans

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  1. Mohammad Javad Afshari Moein
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Afshari Moein, M.J., Valley, B. & Evans, K.F. Scaling of Fracture Patterns in Three Deep Boreholes and Implications for Constraining Fractal Discrete Fracture Network Models. Rock Mech Rock Eng 52, 1723–1743 (2019). https://doi.org/10.1007/s00603-019-1739-7

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