Semi-analytical solution to the problem of frequency dependent anisotropy of porous media with an aligned set of slit cracks

Abstract

Fractures control fluid flow in geological formations because their presence can greatly affect effective permeability. Existence of cracks in a porous rock can induce seismic dispersion attenuation and frequency-dependent anisotropy, due to wave-induced fluid flow (WIFF) between the fractures and pores. Thus, dispersion and attenuation can be valuable seismic signatures for fracture characterization. In previous studies these effects were modeled by considering cracks as low aspect-ratio inclusions embedded in a porous background medium modeled by Biot's poroelasticity equations. However, previous works are limited to normal wave incidence. In this research, we study the P-wave dispersion and attenuation for oblique incidence in a saturated porous medium with aligned slit (two-dimensional) fractures, through solving a mixed boundary value problem for Biot's dynamic poroelasticity equations. The semi-analytical solution of this problem gives dispersion and attenuation as functions of the incidence angle. The strongest dispersion and attenuation occur when the P-wave propagates along the fracture normal and decrease with increasing incident angle (as measured from the fracture normal). For the frequency-dependent anisotropy at low frequencies, the sample behaves as nearly elliptical, and at high frequencies the medium exhibits a strong anellipticity. The attenuation anisotropy parameters reach their maximum when the fluid diffusion length is comparable to the crack radius. Comparison of the heuristic solution, which is s based on an interpolation between low- and high frequencies using a relaxation function, to the semi-analytical solution of the rigorous equations shows excellent agreement, demonstrating the validity of the heuristic approach.