Effect of asperities on stress dependency of elastic properties of cracked rocks

Rocks are complex heterogeneous materials consisting of solid minerals and fluid-filled pore space. Sedimentary rocks often undergo significant continuous stiffening under stress. The main mechanism of this stiffening is an increase of the number of contacts between adjacent grains and asperities on opposite surfaces of cracks. We propose an analytical model for effective compliance of finite cracks with contacting surfaces. To this end we utilize known exact solutions for an annular crack and a contact of the welded-area type. A set of numerical tests verifies applicability of the model to realistic rock microstructures. A micromechanical theory of changes of the elastic moduli with external loading is based on the crack stiffening by contacting asperities. The proposed theory is then used to analyze results of ultrasonic experiments on a sandstone sample. Parameters of the rock sample recovered from the pressure dependencies of seismic waves are consistent with each other and physically reasonable. Demonstrated applications of the model are based on several simplified assumptions but the theory may incorporate further generalizations, namely: interaction of contacts, inclusion size distribution and more complicated modeling of the microstructure evolution with confining stress.