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 connectivi ty is the first-order determinant of the occurrence of flow. For systems ha ving a narrow distribution of object sizes (short-range percolation), a fir st-order percolation criterion is given by the total excluded Volume which is almost constant at threshold. In the case of fractured media, recent obs ervations have demonstrated that the fracture-length distribution is extrem ely large. Because of this widely scattered fracture-length distribution, t he classical expression of the total excluded volume is no longer scale inv ariant at the percolation threshold and has no finite limit for infinitely large systems. Thus, the classical estimation method of the percolation thr eshold established in short-range percolation becomes useless for the conne ctivity determination of fractured media. In this study, we derive an expre ssion for the total excluded volume that remains scale invariant at the per colation threshold and that can thus be used as the proper control paramete r, called the parameter of percolation in percolation theory. We show that the scale-invariant expression of the total excluded volume is the geometri cal union normalized by the system volume rather than the summation of the mutual excluded volumes normalized by the system volume. The summation of t he mutual excluded volume (classical expression) remains linked to the numb er of intersections between fractures, whereas the normalized geometrical u nion of the mutual excluded volume (our expression) can be essentially iden tified with the percolation parameter. Moreover, fluctuations of this perco lation parameter at threshold with length and eccentricity distributions re main limited within a range of less than one order of magnitude, giving in rum a rough percolation criterion. We finally show that the scale dependenc e of the percolation parameter causes the connectivity of fractured media t o increase with scale, meaning especially that the hydraulic properties of fractured media fan dramatically change with scale.