Discontinuity frequency and block volume distribution in rock masses

• Joint spacing density function is found by applying the maximum entropy method.
• The relation between RQD and joint frequency is validated against simulation data.
• The number of fractures per length follow the Weibull distribution.
• The closed-form expression of block volume distribution is found.
• The left-truncated block volume proportion above a certain block volume size is found analytically.

The discontinuity spacing density function is theoretically found by applying the first principles of the Maximum Entropy Theory. It is shown that this function is the negative exponential probability density function. Then, the analytical relation between RQD and discontinuity frequency that may be derived, provided that discontinuity sets follow the negative exponential model, is validated against simulation data. It is also found that if the discontinuity spacings follow the negative exponential distribution, then the number of fractures per length measured along scanlines or drilled cores follow quite well the two-parameter Weibull distribution function. Subsequently, following the methodology proposed originally by Hudson and Priest, the closed-form expression of block volume distribution in a rock mass transected by three mutually orthogonal discontinuity sets is found. Also, the left-truncated block volume proportion above a certain block volume size is found analytically. The theoretical results referring to discontinuity frequency and block volume distributions are finally successfully validated against measurements carried out on drill cores and exposed walls, in a dolomitic marble quarry. The methodology presented herein can be applied to rock engineering applications that necessitate the characterization of rock mass discontinuities and discontinuity spacings are reasonably well represented by the negative exponential probability density function. The proposed method for the prediction of marble block volume distribution was applied to data from a quarry from drill cores and scanlines on exposed quarry walls.