Inversion of Seismic Velocities to obtain the Crack and Pore Aspect Ratio Distribution

During a hydrostatic experiment, in the elastic regime, P and S elastic wave velocities measured on rock samples generally increase with pressure and reach asymptotic values at high pressures. This increase of seismic velocities with confining pressure is known to be caused by the closure of compliant pores, such as flat “cracks”, and therefore the high-pressure values of the velocities must reflect only the influence of stiff, “equant” pores. If the pores are assumed to be spheroids, the use of an effective medium theory, combined with a crack closure model, gives a model to relate the elastic properties to the microstructure at each pressure. Therefore, the pressure dependence of elastic velocities can be inverted to obtain the pore aspect ratio distribution. This is done more easily using data obtained in dry experiments, since pore fluids have a strong effect on velocities and to some extent mask the effect of the pore geometry. However, thus far most models have used restrictive assumptions, such as assuming that the stiff pores are spherical, or the interactions between inclusions can be neglected (such as Morlier’s method), which is unfortunately not realistic in most cases. Others methods, such as the one developed by Cheng and Toksoz (1979), assume that the rock contains a discrete distribution of crack aspect ratios, and are complicated to implement numerically. Moreover, in most work only the dry data have been inverted, or jointly the dry and wet data, but it seems that few works have tried to look in detail at a consistent pore model, that remains simple and is able to predict the dependence of Vp and Vs under saturated conditions, based on data collected on dry rocks. We assume that the rock contains a distribution of cracks with different aspect ratios, and two families of stiff pores, each with their own finite aspect ratio. We use this model to invert the wavespeeds to obtain aspect ratio distributions of some isotropic sandstones (Berea, Boise, Vosges), using data from the literature (King, 1966; Fortin et al., 2007). Our method, which is based on the algorithm of Zimmerman (1991), fully accounts for interactions between pores, and can make use of any effective medium theory, although we focus here on the results obtained using the differential effective medium theory and the Mori-Tanaka scheme. Finally, we attempt to predict the saturated velocities from the pore aspect ratio distribution model obtained from the dry data.