Exploring Vp-Vs relations: Approach from effective medium theories.

Abstract

Several empirical relations between Vp and Vs have been proposed for the lithologies of oil and gas reservoirs. Although recent technology provides us an improved sonic log tool to measure Vs as well as Vp, often the Vs measurement is distorted due to the very sensitive nature of S-waves to borehole conditions. In addition, the difficulty of picking the first breaks of S-waves often makes the quality of the Vs log insufficient for seismic analysis. On the other hand, the demand for Vs information is expanding since the Vp/Vs ratio (or Poisson's ratio calculated from the Vp/Vs ratio) is a key to characterizing subsurface lithology and/or estimating the formation fluids in the reservoir, by analyzing the AVO response of the seismic data. In this paper, I present a theoretical approach for understanding rock Vp-Vs relations using the Hashin-Shtrikman bounds and effective-medium theories. To find the bounds of the Vp/Vs ratio, I adopt the upper and lower Hashin-Shtrikman bounds of the mixture of a rock and brine. For any mixture, the upper Hashin-Shtrikman bound corresponds to the lowest Vp/Vs bound and the lower Hashin-Shtrikman bound is equivalent to the highest Vp/Vs bound. Then, I compare the Vp-Vs relations calculated by a couple of mathematical models. I explore the Vp-Vs relations of the computed results in the crossplot domain among Vp, Vs and porosity. The Vp-Vs trend of sandstone is represented better by the model with tubular pores. I discuss that the location in the crossplots is an indicator of the pore shape. I show that (1) the upper Hashin-Shtrikman bound resembles Castagna's trend in carbonates and (2) Mavko's randomly-oriented tube model mimics the Castagna's regression of brine saturated sandstone.