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Derivation of the Discontinuity Spacings Distribution from Frequency Data Along Scanlines and Boreholes

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The spacing distribution of natural joints or fractures in rock masses affects the rock mass deformability, strength and transport properties according to their size, distribution in space, deformability and strength. Rock block sizes formed by intersecting joints are also an important engineering parameter influencing the behavior of rock masses around underground openings and surface excavations, extraction of blocks of commercial sizes of decorative stones and in rock fragmentation processes by blasting or mechanical excavation techniques. The rock block shapes and sizes are depicted by the number of joint sets, as well as orientation in space, persistence and spacings of joint sets transecting the rock mass. In this paper, we first elaborate on a new generalized Poisson probability density function of joint frequencies counted along scanlines and boreholes. Based on this probability density function we construct an algorithm for the estimation of joint spacings distribution from joint frequency measurements along scanlines. The validation of the conversion of joint frequencies to joint spacings is performed with the Monte-Carlo simulation technique. Joint count data along with drill cores from a white dolomitic marble quarry transected by three joint sets is used to demonstrate the application of the proposed algorithm. The proposed model and associated algorithm are necessary for the subsequent prediction of rock block volume distribution isolated between mutually intersecting joints. In this context among other applications they may find many applications in Discrete Fracture Network (DFN) modelling of jointed rock masses.

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Cumulative distribution function


Discrete fracture networks


Digital terrain model


Fracture frequency (m1)

\(f_{\lambda }\) :

Number of fractures of the same joint set per fixed length interval along scanline or rock core (m1)


Homogeneous Poisson process


Non-homogeneous Poisson process


Linear elastic fracture mechanics


Probability density function


Random variable

\(s\) :

Apparent joint spacing (m)

\(\hat{s}\) :

True joint spacing (m)

x :

Sampling length (or sample support) (m)

\(\lambda\) :

Linear fracture frequency (or intensity) parameter (m−1)


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The financial support by the Programme “Marble resources estimation based on oriented drill core data (MARBLECORE)” under Contract No. AMΘΡ2-0016310 is kindly acknowledged here.

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Correspondence to M. Stavropoulou.

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Stavropoulou, M., Saratsis, G., Xiroudakis, G. et al. Derivation of the Discontinuity Spacings Distribution from Frequency Data Along Scanlines and Boreholes. Rock Mech Rock Eng 54, 3095–3113 (2021).

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