Velocity-porosity Relationships, 1: Accurate Velocity Model for Clean Consolidated Sandstones

We use numerical simulations to derive the elastic properties of model monomineralic consolidated sandstones. The model morphology is based on overlapping spheres of a mineral phase. We consider model quartzose and feldspathic sands. We generate moduli-porosity relationships for both the dry and water-saturated states. The ability to control pore space structure and mineralogy results in numerical data sets which exhibit much less noise than corresponding experimental data. The numerical data allows us to quantitatively analyze the effects of porosity and the properties of the mineral phase on the elastic properties of porous rocks. The agreement between the numerical results and available experimental data for clean consolidated sandstones is encouraging. We compare our numerical data to commonly used theoretical and empirical moduli-porosity relationships. The self-consistent method gives the best theoretical fit to the numerical data. We find that the empirical relationship of Krief et al. is successful at describing the numerical data for dry shear modulus and that the recent empirical method of Arns et al. gives a good match to the numerical data for Poisson's ratio or Vp/Vs ratio of dry rock. The Raymer equation is the best of the velocity-porosity models for the water-saturated systems. Gassmann's relations are shown to accurately map between the dry and fluid-saturated states. Based on these results, we propose a new empirical method, based solely on a knowledge of the mineral modulus, to estimate the full velocity-porosity relationship for monomineralic consolidated sands under dry and fluid-saturated states. The method uses the equation of Krief et al. for the dry shear modulus and the empirical equation of Arns et al. for the dry Poisson's ratio. Gassmann's relations are applied to obtain the fluid-saturated states. The agreement between the new empirical method, the numerical data and available experimental data for dry and water-saturated states is encouraging