Linear vibrations of porous media

Abstract

In this research, the equations of motion of an isotropic fluid-saturated sedimentary medium are developed from first principles. The equations are limited to the case where linear stress/strain relations are adequate for describing the deformation. The idea of the modeling is to allow for the possibility of relative fluid/solid motion and the possibility of a second (slow) compressional wave. By directly volume averaging the balance laws that hold in the solid and fluid phases while accounting for the boundary conditions on the pore walls separating the fluid and solid, Biot's (1956a,b; 1962a,b) equations are obtained. The definitions of the elastic moduli given by Biot and Willis (1957) are also obtained. However, models for the inertial operator controlling the magnitude of relative flow have been considered that were not allowed for by Biot. In modeling this inertial relative-flow operator, Biot limited himself to the case where the pores are modeled as constant-width flow channels. In this work, three generalizations to Biot's relative flow model are treated. First, pores that possess variation in their width are allowed for with the result that less relative-flow is predicted as compared to Biot's model. Second, bumps on the surface of an otherwise smooth and constant-width flow channel are allowed for with the result that if the bump heights are much smaller than the channel widths, then the relative flow inertial operator remains unaffected by the surface roughness. Third, electrokinetic or "streaming potential" forces are allowed for with the result that if the liquid is a low molarity brine (molarities < 10^-3 ) and if the flow channels have small enough widths (< 1 micron) then the relative flow may be reduced by up to 45% compared to Biot's model. Lastly, expressions for the phase velocities, attenuation coefficients and normal-incidence reflection/transmission coefficients are obtained using a displacement/stress-vector formulation of the equations of motion.