Acoustoelastic DZ-MT model for stress-dependent elastic moduli of fractured rocks

Understanding the stress-dependent elastic moduli of fractured rocks is essential for monitoring geopressure and tectonic stress. The elastic nonlinearity with finite strains can be described by the theory of acoustoelasticity through the third-order elastic constants (3oECs) that are strictly valid for an isotropic homogeneous medium. The extension to fractured rocks becomes particularly interesting because of the sensitivity of fractures to prestress, but remains largely unaddressed. We address this issue through theoretical modeling with confirmed validity by considering a group of homogeneously distributed fractures (ellipses) embedded into a homogeneous background medium subject to a uniform confining pressure. We incorporate both the David-Zimmerman (DZ) and Mori-Tanaka (MT) models into the theory of acoustoelasticity. The DZ model accounts for the closure influence of cracks with different aspect ratios, whereas the MT model is used to estimate the combining effect of the background acoustoelasticity and the elastic nonlinearity due to the closure of cracks. We validate the acoustoelastic DZ-MT model of fractured rocks by experiment data with Fontainebleau and Vosges sandstones. We show that the effective elastic moduli of fractured rocks mainly depend on the background acoustoelasticity and nonlinear elastic deformations induced by closing cracks. The inclusions inside fractures have too low 3oECs that the associated acoustoelastic effect could be neglected for both interconnected and isolated cracks. The effective elastic moduli of rocks change significantly during the closing period of cracks with increasing pressure under low pressure. The influence of cracks can be reduced only when cracks are closed completely, where, unlike the theoretical prediction by the DZ model, the effective elastic moduli of rocks still increase along with increasing pressure because of the acoustoelastic effect of the background.