A model for wave propagation in a composite solid matrix saturated by a single-phase fluid

This paper presents a theory to describe wave propagation in a porous medium composed of two solids saturated by a single-phase fluid for spatially variable porosity. This problem has been previously solved for constant porosity when one of the solids is ice or clay, but that model is not useful for most realistic situations. The equations for variable porosity are derived from the virtual work principle, where the generalized coordinates are identified as the displacements of the two solid phases and a new variable associated with the relative fluid flow, whose divergence is the change in fluid content. The generalized forces are the fluid pressure and combinations of the stress tensor of each solid phase and the fluid pressure. The Lagrangian equations of motion are derived for the isotropic case and a theorem on the existence and uniqueness of their solution is given. The plane wave analysis reveals the existence of three compressional and two shear waves. The theory is applied to wave propagation in shaley sandstones showing that phase velocities of the faster P and S waves agree very well with experimental data for varying porosity and clay content. A simulation through a plane interface separating two frozen sandstones of different ice contents is presented.