Multivariate probabilistic rock-physics models using Kumaraswamy distributions

Abstract

Rock-physics models are physical equations that map petrophysical properties into geophysical variables, such as elastic properties and density. These equations are generally used in quantitative log and seismic interpretation to estimate the properties of interest from measured well logs and seismic data. Such models are generally calibrated using core samples and well-log data and result in accurate predictions of the unknown properties. Because the input data are often affected by measurement errors, the model predictions are often uncertain. Instead of applying rock-physics models to deterministic measurements, I have applied the models to the probability density function (PDF) of the measurements. This approach has been previously adopted in the literature using Gaussian distributions, but for petrophysical properties of porous rocks, such as volumetric fractions of solid and fluid components, the standard probabilistic formulation based on Gaussian assumptions is not applicable due to the bounded nature of the properties, the multimodality, and the nonsymmetric behavior. The proposed approach is based on the Kumaraswamy PDF for continuous random variables, which allows modeling double-bounded nonsymmetric distributions and is analytically tractable, unlike beta or Dirichlet distributions. I have developed a probabilistic rock-physics model applied to double-bounded continuous random variables distributed according to a Kumaraswamy distribution and derived the analytical solution of the probability distribution of the rock-physics model predictions. The method is evaluated for three rock-physics models: Raymer’s equation, Dvorkin’s stiff sand model, and Kuster-Toksöz’s inclusion model.