Stress fluctuations in fracture networks from theoretical and numerical models
Résumé
We analyze the spatial fluctuations of stress in a simple tridimensional model constituted by a population of disc-shaped fractures embedded in an elastic matrix with uniform and isotropic properties. The fluctuations arise from the classical stress enhancement at fracture tips and stress shadowing around fracture centers that are amplified or decreased by the interactions between close-by fractures. The distribution of local stresses is calculated at the elementary mesh scale with the 3DEC numerical program based on the distinct element method.
As expected, the stress distributions vary with fracture density, the larger is the density, the wider is the distribution. For freely slipping fractures, it is mainly controlled by the percolation parameter $p$ (i.e., the total volume of spheres surrounding fractures). For stresses smaller than the remote deviatoric stress, the distribution depends only on for the range of density that has been studied. For large stresses, the distribution decreases exponentially when increasing stress, with a characteristic stress that increases with entailing a widening of the stress distribution.
We extend the analysis to fractures with plane resistance defined by an elastic shear stiffness $k_s$ and a slip Coulomb threshold. A consequence of the fracture plane resistance is to lower the stress perturbation in the surrounding matrix by a factor that depends on the ratio between $k_s$ and a fracture-matrix stiffness $k_m$ mainly dependent on the ratio between Young modulus and fracture size. $k_m$ is also the ratio between the remote shear stress and the displacement across the fracture plane in the case of freely slipping fractures. A complete analytical derivation of the expressions of the stress perturbations and of the fracture displacements is obtained and checked with numerical simulations. In the limit $k_s >> k_m$, the stress perturbation tends to 0 and the stress state is spatially uniform.
The analysis allows us to quantify the intensity of the stress fluctuations in fractured rocks as a function of both the fracture network characteristics (density and size distribution), and the mechanical properties (fracture shear stiffness vs matrix elastic properties).