Stoneley wave attenuation and dispersion and the dynamic permeability correction

Stoneley waves induce fluid pressure gradients in a permeable formation surrounding the borehole. These gradients are equilibrated through pressure diffusion, that is to say, slow P-waves in the context of Biot’s poroelasticity theory. Because slow P-waves are strongly sensitive to the formation permeability, the Stoneley-slow P-wave interaction can be used to retrieve the formation permeability from the attenuation and dispersion of Stoneley waves. The accuracy of this established technique in high-permeability formations deteriorates when slow P-waves are not pure diffusion waves; hence, the permeability dependence is more complicated. This effect on Stoneley waves is captured by applying the Johnson-Koplik-Dashen dynamic permeability model. Their model depends on a viscous relaxation length. However, in the estimation of formation permeability from Stoneley waves, this parameter is typically not measured but is estimated from an empirical equation, wherein material properties and microstructural descriptors are lumped together. When the so-calculated relaxation length is erroneous, the inverted formation permeability from the Stoneley wave is not correct either. To overcome this limitation and to provide a versatile alternative, the dynamic permeability problem is reformulated within the viscosity-extended Biot framework. Its physical basis is the conversion scattering in random media from slow P- to slow S-waves. The correlation length of this so-called stochastic dynamic permeability model can be derived from pore-scale images, and it also captures the effect of pore interface roughness. This model is then combined with the simplified Biot-Rosenbaum model to predict Stoneley wave attenuation and dispersion. We have applied this hybrid model to interpret laboratory measurements for which the previously suggested choice of the viscous relaxation length does not provide an accurate prediction. The results indicate that the hybrid model can provide another approach to model Stoneley wave attenuation and dispersion across the entire frequency range.