Elastic properties of rocks with fracture-induced anisotropy1
Aligned sets of fractures cause elastic anisotropy. The change in the stiffness coefficients (and the subsequent velocity modifications) caused by fracturing in a background medium, is analyzed using effective medium theory. A general non-linear expression for the stiffness modifications due to the fracturing has been derived. For media with arbitrary symmetry a linearized expression provides a simple way to calculate analytically the first order influence of the fractures on the stiffness matrix. Non-linearized closed-form solutions are obtained for vertical orthorhombic fracturing in a monoclinic medium, and for three perpendicular sets of orthorhombic fracturing in an orthorhombic background medium. For an orthorhombic background mediumwith one set of orthorhombic fractures, re-parameterization of the fracture compliances makes the resulting stiffness parameters depend linearly on the new fracture parameters.
We then consider the velocity modifications as a function of direction caused by transversely isotropic (TI) fracturing in an isotropic background medium. Exact analytical formulae are derived and subsequently simplified under the assumption of weak fracturing influence. The phase velocity modifications due to the fracture parameters are discussed for three principal directions of wave propagation. An increase in the normal fracture compliance gives a velocity decrease for the P-wave with largest effect perpendicular to the fracturing, and a velocity decrease for the S V-wave at 45 degrees to the fracture orientation. An increase in the tangential fracture compliance results in a small velocity decrease for the P-wave with no effect perpendicular to the fracturing, a velocity decrease for the SV-wave, with the least effect being at 45 degrees to the fracturing, and a velocity decrease for the SH-wave with no effect parallel to the fracturing. Two examples are presented, where fracturing in a weakly anisotropic background medium results in similar velocity modifications as for fracturing in an isotropic background medium.