Scale Dependence of Waviness and Unevenness of Natural Rock Joints through Fractal Analysis

Abstract

The scale dependence of surface roughness is critical in characterising the hydromechanical properties of field-scale rock joints but is still not well understood, particularly when different orders of roughness are considered. We experimentally reveal the scale dependence of two-order roughness, i.e., waviness and unevenness through fractal parameters using the triangular prism surface area method (TPM). The surfaces of three natural joints of granite with the same dimension of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mn>1000</mml:mn><mml:mtext> </mml:mtext><mml:mtext>mm</mml:mtext><mml:mo>×</mml:mo><mml:mn>1000</mml:mn><mml:mtext> </mml:mtext><mml:mtext>mm</mml:mtext><mml:mtext> </mml:mtext></mml:math>are digitised using a 3D laser scanner at three different measurement resolutions. Waviness and unevenness are quantitatively separated by considering the area variation of joint surface as grid size changes. The corresponding fractal dimensions of waviness and unevenness in sampling window sizes ranging from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mn>100</mml:mn><mml:mtext> </mml:mtext><mml:mtext>mm</mml:mtext><mml:mo>×</mml:mo><mml:mn>100</mml:mn><mml:mtext> </mml:mtext><mml:mtext>mm</mml:mtext></mml:math> to 1<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mn>000</mml:mn><mml:mtext> </mml:mtext><mml:mtext>mm</mml:mtext><mml:mo>×</mml:mo><mml:mn>1000</mml:mn><mml:mtext> </mml:mtext><mml:mtext>mm</mml:mtext></mml:math> at an interval of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mn>100</mml:mn><mml:mtext> </mml:mtext><mml:mtext>mm</mml:mtext><mml:mo>×</mml:mo><mml:mn>100</mml:mn><mml:mtext> </mml:mtext><mml:mtext>mm</mml:mtext></mml:math> are determined. We find that both the fractal dimensions of waviness and unevenness vary as the window size increases. No obvious stationarity threshold has been found for the three rock joint samples, indicating the surface roughness of natural rock joints should be quantified at the scale of the rock mass in the field.