Percolation parameter and percolation-threshold estimates for three-dimensional random ellipses with widely scattered distributions of eccentricity and size

In fractured materials of very low matrix permeability, fracture connectivity is the first-order determinant of the occurrence of flow. For systems having a narrow distribution of object sizes (short-range percolation), a first-order percolation criterion is given by the total excluded volume which is almost constant at threshold. In the case of fractured media, recent observations have demonstrated that the fracture-length distribution is extremely large. Because of this widely scattered fracture-length distribution, the classical expression of the total excluded volume is no longer scale invariant at the percolation threshold and has no finite limit for infinitely large systems. Thus, the classical estimation method of the percolation threshold established in short-range percolation becomes useless for the connectivity determination of fractured media. In this study, we derive an expression for the total excluded volume that remains scale invariant at the percolation threshold and that can thus be used as the proper control parameter, called the parameter of percolation in percolation theory. We show that the scale-invariant expression of the total excluded volume is the geometrical union normalized by the system volume rather than the summation of the mutual excluded volumes normalized by the system volume. The summation of the mutual excluded volume (classical expression) remains linked to the number of intersections between fractures, whereas the normalized geometrical union of the mutual excluded volume (our expression) can be essentially identified with the percolation parameter. Moreover, fluctuations of this percolation parameter at threshold with length and eccentricity distributions remain limited within a range of less than one order of magnitude, giving in turn a rough percolation criterion. We finally show that the scale dependence of the percolation parameter causes the connectivity of fractured media to increase with scale, meaning especially that the hydraulic properties of fractured media can dramatically change with scale.