Many methods for calculating the volume of rock blocks have been developed in the last decades. The first attempts to estimate such crucial quantity produced analytical equations to calculate the mean and variance of volume, considering blocks created by three discontinuity sets with a certain spacing probability distribution. From then, the research community followed three kinds of approaches for calculating block volume: the fully analytical one (e.g., Palmstrøm’s formula), the fully probabilistic one (e.g., Discrete Fracture Network generators), and the mixed one (e.g., In Situ Block Size Distribution). In this paper, a comparison among the different methods is presented, supported by numerical examples, highlighting their strengths and disadvantages in terms of reliability and repeatability.
In rock masses, blocks are delimited by discontinuity planes and can have various shapes depending on the number and orientation of the discontinuities which form them. Therefore, the calculation of rock block volume is not trivial, and many attempts have been made to propose simple analytical methods.
One of the most commonly used is Palmstrøm’s formula [1] for calculating the volume of a block created by the intersection of three discontinuity sets, based on the spacing values and the angles between pairs of sets.
Lopes and Lana [2] proposed an analytical solution developed for tabular, prismatic, and tetrahedral blocks. The solution is based on linear algebra and vectorial analysis concepts. It depends on discontinuity orientations, spacing, and block shape. A discontinuity plane with dip ϕn, dip direction θn, and spacing En can be represented by its director vector (equation), where (equation) and
(equation)
They demonstrate that the volume V of a block created by three discontinuity sets can be calculated as:
(equation)
where
(equation)
The triple product gives the volume of a six-face solid. Still, spatially these direction vectors may also define a solid with a different number of faces; therefore, the result could need to be multiplied by a constant.