The effect of stress, pore fluid and pore structure on elastic wave velocities in sandstones

Abstract

A model of elastic wave propagation in fluid-saturated sandstones is developed that takes into account pore fluid properties, and stress. It is based on the assumption that the pore space can be represented by a distribution of crack-like spheroidal pores having a distribution of aspect ratios, and a single family of non-closable spheroidal pores. The Eshelby-Wu formalism is used to find exact expressions for the bulk and shear compliances of the pores. Asymptotic analytical expressions are obtained for the compliances of both dry and fluid-saturated spheroidal pores that are crack-like, needle-like, or nearly spherical. These expressions are incorporated into two commonly used effective medium theories, the Mori-Tanaka and the Differential schemes, to obtain expressions for the effective elastic properties of dry and fluid-saturated rocks containing spheroids of a given aspect ratio, as a function of porosity or crack density. The stress dependence of the elastic velocities is modelled by considering that the elastic moduli vary with stress due to crack closure. This pore structure model is able to explain successfully the pressure dependence of ultrasonic dry velocities on many sets of laboratory data. Predictions of saturated velocities are made using either the Gassmann equation, or using an effective medium theory in conjunction with the aspect ratio distribution found from the dry data. For ultrasonic velocities obtained at high frequencies (MHz), the predictions of effective medium theories are more accurate than the Gassmann predictions. Low-frequency measurements (0.02 Hz) of the bulk modulus were obtained on Fontainebleau sandstone, under pressure, and with different pore fluids. For water and glycerin-saturated samples, both the 4% and 13% porosity rock specimens were more compliant at low frequencies than at high frequencies. Finally, a model is proposed for the frequency dependence of the wave velocities, assuming that at a given frequency, some pores obey the Gassmann equation, and others are isolated, with a critical aspect ratio demarcating the two families that depends on frequency and fluid viscosity.