Acoustoelastic Mori-Tanaka model for third-order elastic constants of fractured rocks

Abstract

Insights into the stress-dependent elastic moduli of fractured rocks are important in monitoring geopressure and tectonic stress. This can be addressed by the theory of acoustoelasticity that describes elastic nonlinearity due to the cubic strain-energy function through third-order elastic constants (TOECs). The acoustoelastic TOECs are strictly valid for an isotropic homogeneous medium. The extension to fractured rocks remains largely unaddressed. A group of fractures (ellipses) is embedded into a homogeneous background medium subjected to a uniform confining pressure. We formulate an acoustoelastic Mori-Tanaka (MT) model for the effective TOECs of fractured rocks which can be defined as the weighted average of individual TOECs between the background and fractures in terms of respective volume fractions and stiffnesses. For homogeneously distributed fractures, the stress-induced strains are isotropic in the background and in the fractures. For aligned fractures, the resulting anisotropic strains over the background and fractures make the rock a vertical transverse isotropic medium. The effective elastic moduli and TOECs of fractured rocks depend on background elastic moduli, fracture contents, aspect ratios, and fracture orientations. The prestressed aligned fractures induce nonlinear effects on the effective TOECs that increase with increasing fracture contents and decreasing aspect ratios. We validate the acoustoelastic MT model by experimental data and investigate its limitation by comparing it with finite-element simulations.