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A Practical Method for Estimating the Volumetric Intensity of Non-persistent Discontinuities on Rock Exposures


The volumetric discontinuity intensity (\(P_{32}\)) is defined as the total discontinuity area by rock mass volume. \(P_{32}\) has been considered the most suitable parameter to quantify discontinuities within rock masses because it takes into account density and size and does not depend on orientation. Currently, determining the \(P_{32}\) from discontinuity mapping of rock exposures requires a series of discrete fracture network (DFN) simulations. This method is time-consuming and requires advanced software; thus, it is generally not used in practical rock engineering applications. This paper proposes a new method to obtain the \(P_{32}\) of rock masses from 2D rock exposures, based on well-known weighted joint density (\({\text{wJd}}\)) and mean trace length (\(\mu_{\text{l}}\)) estimates. The method was developed and validated using DFN modeling and compared with \(P_{32}\) estimates obtained by computational simulations in a real case study (Monte Seco tunnel). Finally, the synthetic rock mass (SRM) modeling technique was used to investigate the effects of \(P_{32}\) on the mechanical behavior of a hypothetical rock mass, highlighting the individual contributions of \({\text{wJd}}\) and \(\mu_{\text{l}}\). The results demonstrate that the proposed method is reliable and can estimate the \(P_{32}\) of rock masses without requiring computation simulations. Moreover, the SRM analysis showed that discontinuity density and size have a similar impact on the mechanical behavior of rock masses with non-persistent discontinuity sets.


  • A new method for obtaining the volumetric intensity of rock masses from discontinuity mapping of 2D rock exposures.

  • The method was developed and validated using parametric discrete fracture network analysis and applied in a tunneling case study.

  • A synthetic rock mass case demonstrated the volumetric intensity and discontinuity size's contributions to the mechanical behavior of rock masses.

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\(P_{32}\) :

Volumetric discontinuity intensity (m1)

\(P_{32}^{{{\text{max}}}}\) :

Maximum volumetric discontinuity intensity (m1)

\(P_{20}\) :

Areal discontinuity density (m2)

\(P_{10}\) :

Linear discontinuity intensity (m1)

\({\text{wJd}}\) :

Weighted joint density (m1)

\({\text{wJd}}^{{\text{u}}}\) :

Unbiased weighted joint density (m1)

\(\delta_{\text{i}}\) :

Acute angle between the discontinuity dip and the mean rock face (°)

\(A\) :

Area of the square-shape sampling window (m2)

\(L\) :

Length of the square-shape sampling window’s edge (m)

\(r\) :

Radius of the circular window (m)

\(m\) :

Number of trace endpoints in the window

\(n\) :

Number of trace endpoints cut by the window boundary

\(f_{\text{i}}\) :

Correction factors for \(\delta_{\text{i}}\)

\(\mu_{{\text{l}}}\) :

Unbiased mean trace length (m)

\(\sigma_{{\text{l}}}\) :

Unbiased standard deviation of trace lengths (m)

\(\mu_{{\text{D}}}\) :

Mean discontinuity diameter (m)

\(\sigma_{{\text{D}}}\) :

Standard deviation of discontinuity diameter (m)

\({\text{CoV}}\) :

Coefficient of variation of discontinuity diameters

\(C_{31}\) :

Constant factors for \(P_{10}\)

\(C_{32}\) :

Constant factors for \(P_{20}\) (m)

\(S_{{\text{f}}}\) :

Biased size factor

\(S_{{\text{f}}}^{{\text{u}}}\) :

Unbiased size factor

\(V_{{\text{f}}}\) :

Size variation factor

\({\text{UCS}}_{{\text{i}}}\) :

Uniaxial compressive strength of the intact rock (MPa)

\({\text{UCS}}_{{\text{m}}}\) :

Uniaxial compressive strength of the synthetic rock mass (MPa)


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The authors would like to thank the Brazilian National Council for Scientific and Technological Development (CNPq) logistical and financial support.

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Correspondence to Pedro Pazzoto Cacciari.

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Cacciari, P.P., Futai, M.M. A Practical Method for Estimating the Volumetric Intensity of Non-persistent Discontinuities on Rock Exposures. Rock Mech Rock Eng 55, 6063–6078 (2022).

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