A model forP-wave attenuation and dispersion in a porous medium permeated by aligned fractures

Abstract

SUMMARY Fractures in a porous rock can be modelled as very thin and highly porous layers in a porous background. First, a dispersion equation for a P wave propagating in periodically layered poroelastic medium is obtained using propagator matrix approach applied to Biot equations of poroelasticity with periodic coefficients. Then in the limit of low stiffness and thickness this dispersion equation yields an expression for the effective P-wave modulus of the fractured porous material. When both pores and fractures are dry, this material is equivalent to a transversely isotropic elastic porous material with linear–slip interfaces. When saturated with a liquid this material exhibits significant attenuation and velocity dispersion due to wave-induced fluid flow between pores and fractures. In the low-frequency limit the material properties are equal to those obtained by anisotropic Gassmann (or Brown–Korringa) theory applied to a porous material with linear-slip interfaces. At low frequencies inverse quality factor scales with the first power of frequency ω. At high frequencies the effective elastic properties are equal to those for isolated fluid-filled fractures in a solid (non-porous) background, and inverse quality factor scales with ω−1/2. The magnitude of both attenuation and dispersion strongly depends on both the degree of fracturing and background porosity of the medium. The characteristic frequency of the attenuation and dispersion depends on the background permeability, fluid viscosity, as well as fracture density and spacing.