Uncertainties in estimating the roughness coefficient of rock fracture surfaces

Abstract

Joint roughness has a critical role in the deformation behavior of discontinuous rock masses. Several subjective (visual comparison) and quantitative (statistical and fractal) approaches have been proposed for estimating rock joint roughness coefficient (JRC). Using a large collection of 223 published joint profiles, this study investigates variability of the JRC estimates by these approaches. Among the profile parameters, maximum height (R _z), ultimate slope (λ), and fractal dimension (D _h–L, determined using the hypotenuse leg method) show a lower sensitivity to the sampling interval than the root mean square of the first deviation (Z ₂), profile elongation index (δ), fractal dimension (D _c, determined using the compass-walking method), and standard deviation of the angle i (σ _i ). Accordingly, this study proposes two separate sets of equations for quantitatively estimating JRC. The performances of these equations are examined by performing direct shear tests on 23 rock joint samples. The subjective approach is found to underestimate JRC by less than two units because it ignores (1) the main trend of the compared profile and (2) the limited scope of the standard profiles. Following these results, the visual comparison chart is updated by explicitly adding a scale bar for the y-axes of the standard profiles. Several basic rules for visual comparisons are also proposed.

Appendix

The definition and calculation of some of the parameters used in this paper: Z ₂: Root mean square of the first deviation of the profile (Tse and Cruden 1979)

$$ {\text{Z}}_{2} = \left[ {\frac{1}{\text{L}}\mathop \int \limits_{{{\text{x}} = 0}}^{{{\text{x}} = {\text{L}}}} \left( {\frac{\text{dy}}{\text{dx}}} \right)^{2} {\text{dx}}} \right]^{{\frac{1}{2}}} = \left[ {\frac{1}{\text{L}}\mathop \sum \limits_{{{\text{i}} = 1}}^{{{\text{N}} - 1}} \frac{{({\text{y}}_{{{\text{i}} + 1}} - {\text{y}}_{\text{i}} )^{2} }}{{{\text{x}}_{{{\text{i}} + 1}} - {\text{x}}_{\text{i}} }}} \right]^{1/2} $$

where dx is the increment of x of the profile; dy is the increment of y of the profile; N is the number of evenly spaced sampling points; and x _i and y _i are the x- and y-coordinates of sampling point.

σ _i : Standard deviation of the angle i (Yu and Vayssade 1991)

$$ \begin{aligned} \sigma_{i} = { \tan }^{ - 1} \left[ {\frac{1}{L}\mathop \int \limits_{x = 0}^{x = L} \left( {\frac{{{\text{d}}y}}{{{\text{d}}x}} - {\text{tan }}i_{\text{ave}} } \right)^{2} {\text{d}}x} \right]^{{\frac{1}{2}}} \hfill \\ i_{\text{ave}} = \frac{1}{L}\mathop \int \limits_{x = 0}^{x = L} { \tan }^{ - 1} \left( {\frac{{{\text{d}}y}}{{{\text{d}}x}}} \right){\text{d}}x \hfill \\ \end{aligned} $$

R _z: Maximum height of a profile, equals to the vertical distance between the highest peak and the lowest valley.

L: The projected length of the profile

$$ L = \mathop \sum \limits_{i = 1}^{N - 1} \left( {x_{i + 1} - x_{i} } \right). $$

λ: Ultimate slope of the profile, λ = R _z/L.

δ: Profile elongation index, δ = (L _t − L)/L (Maerz et al. 1990)

$$ {\text{L}}_{\text{t}} = \mathop \sum \limits_{{{\text{i}} = 1}}^{{{\text{N}} - 1}} \sqrt {({\text{x}}_{{{\text{i}} + 1}} - {\text{x}}_{\text{i}} )^{2} + \left( {{\text{y}}_{{{\text{i}} + 1}} - {\text{y}}_{\text{i}} } \right)^{2} } $$

D _c: Fractal dimension determined by compass-walking method (Maerz et al. 1990)

$$ D = - \frac{\log (n + f/r)}{\log r}. $$

where n is the number of steps for walking through a joint profile by a divider with a span of r; and f is the remaining length shorter than r after excluding the length of nr.

D _h–L: Fractal dimension determined by the hypotenuse leg (h–L) (method (Li and Huang 2015)

$$ D = \frac{\log 4}{{\log [2(1 + \cos (\tan^{ - 1} \frac{2h}{l}))]}}({\text{where}},\;h = \frac{1}{M}\mathop \sum \limits_{i = 1}^{M} h_{i} \;\;{\text{and}}\;\;l = \frac{1}{M}\mathop \sum \limits_{i = 1}^{M} L_{i} . $$

where l and h are the average base length and the average height of asperities of a joint, respectively; and M is the number of asperities.