Abstract Shales are highly anisotropic and are generally characterized by very low permeabilities. In this paper we report results of an experimental study of the relationship between elastic wave propagation anisotropy and permeability anisotropy of an illitic shale. Elastic wave velocity was measured as a function of angle relative to the bedding plane orientation. Radial compressional and shear wave velocities were measured around the circumference of a core taken parallel to the bedding of the shale. The shale was highly anisotropic. Permeability measurements were then taken for cores oriented in the directions of the maximum and minimum velocities. The permeability was measured using a novel steady-state-flow permeameter capable of measuring extremely low flow rates of water, brine, or oil. At low effective pressure the shale was highly anisotropic, but the anisotropy reduced to zero at high effective pressure. We compare the shale results with those from measurements on a sandstone and a tuff, where permeability anisotropy is maintained even up to the highest effective pressures used. P. 307