Closed-form and numerical solutions for the probability distribution function of fracture diameters

After summarizing the available methods intended to estimate the fracture size distribution from a given trace length distribution, Santalò closed-form integral solution is used to derive a closed-form expression of the fracture size distribution for each of the most common trace length distributions, i.e. uniform, exponential, gamma, and power law. Numerical integration is used for the lognormal distribution. Expressions are given for the mean fracture diameter as a function of the mean trace length and the minimum fracture diameter. It is shown that, when the trace lengths are uniform, exponential, gamma, lognormal or power law, none of fracture diameter distributions is lognormal, exponential or gamma, as assumed in the literature. Power law trace length distribution yields a power law fracture size. The minimum fracture diameter cannot be equal to zero and plays an important role in determining the fracture diameter distribution. When the trace length distribution is defined on an interval, the diameter lower bound must be contained in the trace length interval, and the upper bound must be equal to the trace length upper bound.