The Elastic Stiffness C13 in Transversely Isotropic Shales

Shales are often anisotropic due to fine-scale layering and preferred alignment of clay particles with bedding. They may be approximated as transversely isotropic (TI) media with symmetry axis (x3) perpendicular to bedding. TI media have five independent elastic stiffnesses C11, C33, C55, C66 and C13, with C12 = C11−2C66. While C11, C33, C55 and C66 may be estimated from velocities measured parallel and perpendicular to x3, C13 requires measurements at oblique angles, resulting in greater uncertainty. Despite this, C13 is important in seismic velocity analysis, imaging, amplitude variation with offset (AVO), and geomechanical applications including wellbore stability, stress estimation, hydraulic fracture design, and reservoir compaction. In isotropic media C13 = C33 − 2C55 and C13 = C12 , but shale microstructure may break these equalities due to preferred orientation of clay particles, fine-scale layering, and bedding-parallel microcracks. For a TI orientation distribution (ODF) of clay particles, the elastic properties may be characterized by two coefficients W200 and W400 in an expansion of the ODF in generalized Legendre functions. C13 may be greater than C12 if W200 is significantly larger than W400, whereas C13 may be lower than C33 − 2C55 if W200 is significantly smaller than W400. Fine-scale layering also influences C13. Depending on layer properties, sequences with C12 less than C13 and C13 less than C33 − 2C55 may occur. Bedding-parallel microcracks typically increase C12 − C13 and C13 −C33 +2C55 , but C13 −C33 +2C55 may decrease if the ratio of normal-to-shear crack compliance is small, as expected for isolated fluid-filled cracks. These observations provide insight into the physical controls on C13 and improve understanding of its variability in shale formations.