Percolation parameter and percolation-threshold estimates for 3D random ellipses with widely-scattered distributions of eccentricity and size

Jean-Raynald de Dreuzy (1) , Philippe Davy (1) , Olivier Bour (1)

1 GR - Géosciences Rennes

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 that 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 a new expression of the total excluded volume that remains scale invariant at the percolation threshold and that can thus be used as the proper control parameter, called 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.

Mots clés: Hydrogeology; Percolation; Connectivity

Domaines: Hydrologie

Jean-Raynald de Dreuzy, Philippe Davy, Olivier Bour. Percolation parameter and percolation-threshold estimates for 3D random ellipses with widely-scattered distributions of eccentricity and size. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, 2000, 62, pp.5948-5952. ⟨10.1103/PhysRevE.62.5948⟩. ⟨hal-00135821⟩