The effects of drained and undrained loading on visco-elastic waves in rock-like composites

We outline a micromechanical approach to the acousto-elastic effect in rock-like composites characterized by multiple solid constituents and non-dilute concentrations of interconnected pores and cracks. Estimates of the T-matrix type (known from the theory of stochastic waves) are first transformed from the stiffness to the compliance domain, so that the (small) strain variation within a single (but interacting) inclusion can be related to the (small) applied effective stress (rather than strain) variation, via a non-dilute K-tensor that depends upon the overall properties as well as the (arbitrary) homogeneous reference material and inclusion (particle, cavity) shape/ orientation. In order to deal with large changes in the applied effective stress under dry conditions, one generally has to integrate a system of ordinary differential equations (ODE s) for the evolution of the microstructural variables (crack densities, porosities, mineral concentrations, aspect ratios of inclusions and correlation functions) under loading. Under undrained conditions, the (total fluid mass within a representative volume element is conserved) solution to the single cavity deformation problem can be found from the same system of ODE s (and initial conditions) as in the dry case, provided that one replaces dry with saturated (effective compliances) K-tensors depending on second-rank tensors of pore pressure build-up coefficients that can be found from the boundary conditions, in combination with a higher-order expression for the change in porosity (for each cavity type) and the constitutive relation for the fluid. Under drained conditions, the (fluid pressure is constant) dry system of ODE s can still be used, provided that one replaces (dry with saturated effective compliances) effective with aparent stress variations that depend on the boundary conditions as well as (small) changes in the dry responses during loading. This use of a fluid inclusion-dependent aparent stress in the dry evolution law is possible since the integrated results are independent of the loading-path, in the absence of hysteresis.