Unified Theory of Elastic Nonlinearity for Stress-Dependent Wave Propagation in Porous and Fractured Rocks With Weakly Cemented Contacts

Abstract

Mechanical deformations of porous and fractured rocks with weak intergranular cementation involve significantly different varieties of nonlinear stress–strain behaviors due to the presence of compliant microstructures such as cracks and grain contacts, generally including nonlinear elastic (due to crack closure and intergranular compaction), hyperelastic (due to stress accumulation), and inelastic (due to crack growth) deformations prior to mechanical failure. Various piecewise modeling approaches have been proposed to describe stress-dependent wave propagation by focusing on certain elastic behavior. However, these highly differentiated mechanical deformations are not exclusive mutually but coexist with different levels of contributions in different stress segments during the progressive deformation process. We address this issue by integrating these diverse-source elastic nonlinearities into a coupled framework where the total energy function consists of hyperelastic strains in the background (grains and stiff pores) and nonlinear strains by intergranular compaction and crack closure. By assuming intergranular compaction to be the category of nonlinear elasticity, we propose a penny-shaped, cement-filled crack to approximate the mechanical behavior of intergranular contact structures, facilitating the construction of strain energy functions for intergranular compaction. We investigate the effects of stiff and compliant pores, contact structures area, and coordination numbers on the effective elastic moduli. Applications to experimental data with Fontainebleau (porosity 4%), Vosges (porosity 25%), and Bleurswiller (porosity 25%) sandstones show that predicted wave velocities agree well with ultrasonic measurements at different effective stresses.