Analysis of Dynamic Fracture Compliance Based on Poroelastic Theory. Part I: Model Formulation and Analytical Expressions

The presence of bedding-parallel fractures at any scale in a rock will considerably add to its compliance and elastic anisotropy. Those properties will be more significantly affected when there is a relatively high degree of connectivity between the fractures and the corresponding interconnected pores. This contribution uses linear poroelasticity to reveal the characteristics of the full frequency-dependent compliance of an infinitely extended fracture model assuming the periodicity of the fractured structures. The fracture compliance tensor is complex-valued due to the wave-induced fluid flow between fractures and pores. The interaction between the adjacent fractures is considered under fluid mass conservation throughout the whole pore space. The quantitative effects of fracture (volume) density (the ratio between fracture thickness and spacing) and host rock porosity are analyzed by the diffusion equation for a relatively low-frequency band. The model in this paper is equivalent to the classical dry linear slip model when the bulk modulus of fluid in the fractures tends to zero. For the liquid-filled case, the model becomes the anisotropic Gassmann’s model and sealed saturated linear slip model at the low-frequency and high-frequency limits, respectively. Using the dynamic compliance definition, we can effectively distinguish the saturating fluids in the fractures with the same order magnitude of bulk modulus (e.g., water and oil) using the compliance ratio method. Additionally, the modified dynamic model can be simplified as acceptable empirical formulas if the strain on the fractures induced by the incoming waves is small enough.