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2D and 3D Roughness Characterization

Abstract

The quantification of surface roughness for the purpose of linking its effect to mechanical and hydrodynamic behavior has taken many different forms. In this paper, we present a thorough review of commonly used 2D and 3D surface roughness characterization methods, categorized as statistical, fractal, and directional. Statistical methods are further subdivided into parametric and functional methods that yield a single value and function to evaluate roughness, respectively. These statistical roughness metrics are useful as their resultant outputs can be used in estimating shear and flow behavior in fractures. Fractal characterization methods treat rough surfaces and profiles as fractal objects to provide parameters that characterize roughness at different scales. The directional characterization method encompasses an approach more closely linked to shear strength and is more suitable for estimating the influence of fracture roughness on mechanical responses. Overall, roughness characterization methods provide an effective objective measure of surface texture that describe its influence on the mechanics of surfaces without requiring qualitative description.

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Fig. 1
Fig. 2
Fig. 3

(modified from Gadelmawla et al. 2002)

Fig. 4

(modified from Whitehouse 1994)

Fig. 5

(modified from Spragg and Whitehouse 1972)

Fig. 6
Fig. 7

(modified from Fardin et al. 2001)

Fig. 8
Fig. 9

(after Tatone and Grasselli 2009)

Fig. 10
Fig. 11
Fig. 12

(after Candela et al. 2009)

Fig. 13

Abbreviations

\(\overline{{\theta^{*} }}\) :

Modified apparent dip characteristic angle

\(\theta_{ \text{max} }^{ *} /(C + 1)\) :

Directional roughness metric

\(A_{0}\) :

Total area fraction facing queried analysis direction

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

Average asperity inclination

\(R_{\Delta a}\) :

Average slope

\(R_{\Delta a}^{'}\) :

Average curvature

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

Kurtosis

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

Profile roughness coefficient

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

Surface roughness coefficient

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

Skewness

\(R_{\lambda q}\) :

Average wavelength

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

Mean zero-crossing spacing

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

Mean peak spacing

\(Z_{2} , R_{\Delta q}\) :

Textural slope parameters

\(Z_{3} , R_{\Delta q}^{'}\) :

Textural wavelength parameters

\(i_{\text{p}}\) :

Peak dilatancy angle

\(\alpha_{\text{e}}\) :

Effective asperity angle

\(\beta_{\text{roll-off}}\) :

Roll-off parameter

\(\theta^{*}\) :

Apparent dip angle facing queried analysis direction

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

Unconfined compressive strength

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

Normal stress

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

Tensile stress

\(\varphi_{\text{b}}\) :

Basic friction angle

\(\varphi_{\text{r}}\) :

Residual friction angle

\(\varphi_{\text{sr}}\) :

Surface roughness friction angle

\(h\) :

Average joint height

\(A\) :

Fractal amplitude parameter

\({\text{ACF}}\left( \tau \right)\) :

Autocorrelation

\({\text{ACVF}}\left( \tau \right)\) :

Autocovariance function

\(C\) :

Fractal amplitude parameter (spectral characterization) (Sect. 3.2), directional roughness metric fitting parameter (Sect. 4)

\(C'\) :

Modified directional roughness metric fitting parameter

\({\text{CLA, }} R_{a}\) :

Center-line average asperity height

\(D\) :

Fractal dimension

\(G\left( f \right)\) :

Power spectral density (PSD)

H :

Hurst exponent

\({\text{JCC}}\) :

Joint contact state coefficient

\({\text{JCS}}\) :

Joint compressive strength

\({\text{JRC}}\) :

Joint roughness coefficient

\(L\) :

Length of profile

\(P\left( z \right)\) :

Cumulative probability density function

\(R\left( f \right)\) :

Aperture to surface power spectral density ratio

\({\text{RMS}}, Z_{1} , R_{q}\) :

Root-mean-square asperity height

\(S\left( w \right)\) :

Standard deviation of points in roughness-length method

\(b\) :

Crossover length

\(f\) :

Frequency

\(k\) :

Wavenumber

\(n\) :

Modified apparent dip distribution parameter

\(p\left( z \right)\) :

Amplitude probability distribution function

\(w\) :

Window width for roughness-length method

\(x\) :

Profile length axis

\(z\) :

Profile height axis

\(\alpha\) :

Angle between triangle normal vector and queried analysis direction

\(\beta\) :

Spectral exponent

\(\gamma \left( k \right)\) :

Random number sequence weighting function

\(\kappa\) :

Topothesy (Sect. 3.2), displacement-to-joint length ratio (joint contact state coefficient) (Sect. 5.1.1)

\(\lambda\) :

Wavelength

\(\tau\) :

Lag distance (autocovariance) (Sect. 2.2.1), shear strength (Sect. 5.1.1)

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Acknowledgements

This work has been supported through the NSERC Discovery Grants 341275, the NSERC CREATE ReDeveLoP program, and the NSERC/Energi Simulation Research Chair in “Fundamental rock physics and rock mechanics” program. The software developed in this study can be accessed from our data server (http://www.geogroup.utoronto.ca/?ddownload=5130).

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Appendices

Appendix 1

The formulas used and the works that they are derived from are listed below in Table 1.

Table 1 A summary of the roughness parameters and typical symbols seen in literature

Appendix 2

To alleviate the time and effort put into processing roughness, a calculator application was developed as an open-source package by the authors implementing most of the methods discussed in this work (Table 1). The software and source code are open-source and can be obtained at the Geomechanics Group @ University of Toronto website (http://www.geogroup.utoronto.ca/?ddownload=5130). Statistical parameters are immediately calculated and are typically the fastest to process. Functional characterizations are displayed on graphs that can be generated then saved by the user. More complex characterization methods using fractal theory or shear-dependency tend to take a large amount of time depending on the number of points used. Further analysis on the outputs can be done as all raw data is output into text files and input surfaces are rotated and translated to align their mean plane along the x–y plane.

Synthetic surface roughness is also implemented based on the power spectrum methodology described in Sect. 5.1.3 (Brown 1995; Candela et al. 2009). A computer generated pseudo-random seed is used to generate a 3D square surface complying with user-specified Hurst exponents along the x- and y-axes based on the code by Candela et al. (2009). Afterwards, the surface elevation is scaled down to match a target \(R_{q}\) value and the spacing between points scaled to match a user-requested length. The 3D surface can be saved and triangulated for input using the Delaunay triangulation algorithm. For 2D profiles, a slice can be selected from the surface along either the x- or y- axis and placed in the import queue for processing (Fig. 14).

Fig. 14
figure14

Synthetic profile generated in the program using the algorithm presented by Candela et al. (2009)

Before attempting to use 3D characterization methods, the surface mesh should be inspected for any holes, gaps, and non-useful features. Cropping the surface may be necessary to ensure that the mesh is contiguous. It is recommended to use a third-party STL viewer such as GOM Inspect (GOM 2018) or MeshLab (Cignoni et al. 2008) prior to processing the data to ensure data quality. Understanding 3D characterization methods is important to ensure that the results given are not of poor quality as the results would have no indication of such. Surface roughness fractal characterization using the roughness length method and the directional roughness characterization is computationally demanding. The implementations for both methods are briefly described below.

2.1 Fractal Characterization: Roughness Length Method

The roughness length method requires a range of specified “window sizes” on which the surface is analyzed. These windows are divisions of the surface as discussed in Sect. 3.1. Although window sizes can be specified, the actual windows themselves are sized to capture the minimum and maximum vertices along the x- and y-axes. This is done by dividing the surface with a set number of windows along the shortest side of the surface bounding box (defined by the minimum and maximum x- and y-axis coordinates) then dividing the surface accordingly. The windows are aligned based on the reference system given and along the bounding box of the surface.

2.2 Directional Roughness Characterization

In evaluating the directional roughness of a surface, the potentially contacting facets are determined by the difference between the triangle facet normal and the direction of shear (Eq. 25). Some differences to the mathematical description of the apparent dip angle used for calculation of potential contact areas are necessary. While Eq. (25) can be applied to each triangular facet of the 3D surface, reliance on trigonometric functions should be reduced especially with calculations sensitive to rotation. To provide a more consistent approach to calculating the apparent dip angle, the relationship

$$\cos \left( {\theta^{ *} + 90^\circ } \right) = \frac{{\left[ {\varvec{n} - {\text{proj}}_{{\varvec{t}_{\text{norm}} }} \left( \varvec{n} \right)} \right]}}{\lVert{\left[ {\varvec{n} - {\text{proj}}_{{\varvec{t}_{\text{norm}} }} \left( \varvec{n} \right)} \right]}\rVert}\cdot \varvec{t}$$
(39)

is used, where \(\varvec{n}\) is the triangle facet normal vector, \(\varvec{t}\) is the analysis direction vector and \(\varvec{t}_{\text{norm}}\) is the vector normal to the analysis direction plane, and \({\text{proj}}_{{\varvec{t}_{\text{norm}} }}\) is the projection function against \(\varvec{t}_{\text{norm}}\). This equation provides mathematically equivalent results to Eq. (25) but with more robust computation than trigonometric functions.

After determining the apparent dip angle of each triangular facet, the triangle areas are binned by the apparent dip angle to obtain a cumulative distribution on which Eq. 26 can be fitted. A non-linear fitting method is required since logarithmic transformation of the equation yields a non-linear relationship (Tatone and Grasselli 2009). As such, the Gauss–Newton fitting algorithm (Björck 1996) was directly implemented for the cumulative distribution fitting curve. The fitting process is iterative and is stopped once the change in the calculated value is less than a user-defined threshold.

2.3 Graphical Output

In addition to providing the results of roughness data using the various methods, the program can provide graphical imaging of data. Graphical imaging for 2D profiles is more developed due to the simplicity of 2D characterization. These graphs provide a view of the fitting performed to provide some quality assurance of the resultant parameters. Amplitude density characterization is immediately comparable with the provided profile and can be used to visually judge the validity of the obtained statistical parameters (Fig. 2). The apparent angle distribution and fitting for the directional roughness method is also provided to ensure the quality of the fitting (Fig. 15). Fractal roughness characterization with the roughness-length method is plotted along with the regression function. The autocovariance function, autocorrelation function, and structure function are also provided graphically. Finally, the PSD can be viewed either with respect to wavelength (Fig. 6) or wavenumber.

Fig. 15
figure15

Directional roughness distribution for a 2D profile along the positive direction

Graphical presentation in 3D does not have the same variety of graphs that the 2D version produces, but its usefulness is best shown with directional characterization. Radial plots can be generated to see the directional change in roughness and the roughness metric’s components (Fig. 16a) and a 3D graphic of a surface can be produced given that the computer’s rendering capabilities are sufficient (Fig. 16b).

Fig. 16
figure16

a Directional roughness is plotted along with the \(C\) parameter in the middle of the plot. b The surface analyzed can be imaged in 3D with basic functionality

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Magsipoc, E., Zhao, Q. & Grasselli, G. 2D and 3D Roughness Characterization. Rock Mech Rock Eng 53, 1495–1519 (2020). https://doi.org/10.1007/s00603-019-01977-4

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Keywords

  • Roughness characterization
  • Rock joint roughness
  • Aperture
  • Joint shear strength
  • Fractal roughness
  • Synthetic roughness