Rock Mechanics and Rock Engineering

, Volume 52, Issue 3, pp 803–825 | Cite as

Minimum Scanline-to-Fracture Angle and Sample Size Required to Produce a Highly Accurate Estimate of the 3-D Fracture Orientation Distribution

  • Lei Huang
  • Huiming TangEmail author
  • Liangqing Wang
  • C. H. Juang
Original Paper


Accurate estimates of the three-dimensional (3-D) rock fracture orientation distribution are crucial for generating a reliable fracture system model. Although an earlier method by Fouché and Diebolt allows one to estimate such distributions from one-dimensional (1-D) samples, this method does not typically produce highly accurate estimates of the 3-D orientation distribution (HAE3DOD). In this study, the minimum scanline-to-fracture angle (minimum θ) and the minimum orientation sample size (minimum n) required to produce HAE3DOD are investigated. Firstly, the factors significantly influencing minimum θ and minimum n are identified, and the influence clarified. For minimum θ, the possible influencing factors include the orientation concentration parameter (к) and n, while for minimum n, the possible influencing factors include к and θ. Fractures from three selected sites in China provide sufficient data for this investigation. The investigation results reveal that minimum θ varies almost linearly with к and n. Moreover, minimum n varies linearly with к and θ. Variations in the minimum θ and minimum n values are strongly associated with discrepancies in sample density resulting from different values of these factors. To ease the estimation of minimum θ and minimum n, empirical relations that take into account the separate factors are proposed for these two variables. A practical example demonstrates that the proposed relations accurately and efficiently estimate minimum θ and minimum n and can provide sampling guidelines (i.e., the intervals of scanline direction and sample size) for producing HAE3DOD. The potential limitations of the proposed relations are also discussed.


Fracture system 3-D orientation distribution Fouché and Diebolt method Scanline mapping Sampling guideline 

List of symbols






Kolmogorov–Smirnov test statistic


Two-tailed asymptotic significance returned by the Kolmogorov–Smirnov test


Error in the 3-D orientation distribution estimate


Highly accurate estimates of the 3-D fracture orientation distribution, defined as the 3-D orientation distribution estimate that results in p ≥ 0.65


Concentration parameter of fracture orientations


Angle of sampling scanline axis to the preferred fracture plane


Sample size in a set of fractures


Size of the weighted sample

minimum θ

The smallest value of θ required to produce HAE3DOD

minimum n

The smallest value of n required to produce HAE3DOD


Sine operator


Cosine operator


Arcsine operator


Arccosine operator


Number of fracture centers per rock volume


Sampling scanline length


Trend of sampling scanline


Plunge of sampling scanline


Fracture radius


Expected value of r


Expected value of r2


Dip direction of the preferred fracture


Dip angle of the preferred fracture

\({\alpha _i}\)

Dip direction of the ith fracture, i = 1, …, n

\({\beta _i}\)

Dip angle of the ith fracture


Dip direction of the fracture as defined at a grid-cell center


Dip angle of the fracture as defined at a grid-cell center


Intersection angle between the scanline and the fracture whose orientation is defined at a grid-cell center

w(φ, n)

Bias compensation factor as a function of φ and n

\(\left\langle {\frac{{n - 1}}{{\sin \varphi }}} \right|\)

The largest integer less than or equal to \(\frac{{n - 1}}{{\sin \varphi }}\)


Observed frequency over a grid cell


Calibrated frequency over a grid cell


Stereographic projection area occupied by fracture poles (unit:°2)


Sample density, defined as the average number of fracture poles per unit stereographic projection area


Partial correlation coefficient returned by partial correlation test


Two-tailed significance returned by partial correlation test


Critical significance level for acceptance/rejection of linear partial correlation hypothesis


Resultant vector of the n orientation unit vectors

\(\left| {{{\mathbf{r}}_{\mathbf{n}}}} \right|\)

Magnitude of rn



This research was funded by Zhejiang Collaborative Innovation Center for Prevention and Control of Mountain Geological Hazards under Grant no. PCMGH-2017-Z03 and the National Key Research and Development Program of China under Grant no. 2017YFC1501305. We are grateful to Dr. Dingjian Wang for assisting in the field sampling of fractures.


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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Engineering Geology and Geotechnical Engineering, Faculty of EngineeringChina University of GeosciencesWuhanChina
  2. 2.Department of Civil EngineeringClemson UniversityClemsonUSA
  3. 3.Department of Civil EngineeringNational Central UniversityTaoyuan CityTaiwan

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