Dynamic permeability and borehole Stoneley waves: A simplified Biot–Rosenbaum model

Abstract

Stoneley waves in permeable boreholes are diagnostic of formation permeability because their propagation is affected by the dynamic fluid flow at the borehole wall. This flow using the concept of dynamic permeability is characterized. The applicability of the dynamic permeability to porous media was examined by applying it to a single fracture case and was found to agree excellently with the fracture conductivity derived from an exact solution. In dealing with the interaction of a Stoneley wave with a porous formation, the problem is decomposed into two parts. The first is the interaction of the Stoneley with an equivalent elastic formation without fluid flow. The second is the interaction with the flow that is governed by the dynamic permeability. In this manner, a simple model for the Stoneley propagation in permeable boreholes was obtained. The Stoneley-wave attenuation and dispersion characteristics from this model were compared with those from the complete model of the Biot–Rosenbaum theory in the case of a hard and a soft formation, respectively. The results from both models were found to agree very well for a hard formation, although they differ at higher frequencies for a soft formation because of the increased solid compressibility. The theoretical predictions from this simple model were also compared with recently published laboratory data of Stoneley-wave measurements, performed at both low- and high-frequency regions of Biot theory. The simple model and experiment are in excellent agreement. Because of the simplicity of the model, it can be easily applied to problems concerning Stoneley propagation in permeable boreholes, especially to an inverse problem to extract formation permeability from Stoneley-wave measurements.