Bayesian rock-physics inversion with Kumaraswamy prior models

Abstract

The prediction of rock and fluid volumetric fractions from elastic attributes often is referred to as petrophysical or rock-physics inversion because it requires rock-physics models to map petrophysical properties into geophysical variables, such as velocities and density. Bayesian approaches are suitable for rock-physics inverse problems because the solution, expressed in the form of a probability distribution, can represent the uncertainty of the model predictions due to the errors in the measured data. Bayesian inverse methods often rely on Gaussian prior distributions for their analytical tractability. However, Gaussian distributions are theoretically not applicable to rock and fluid volumetric fractions because, by definition, they are nonzero on the entire set of real numbers, whereas rock and fluid volumetric fractions are bounded between zero and one. The proposed rock physics inversion is based on a Bayesian approach that assumes Kumaraswamy probability density functions for the prior distribution to model double-bounded nonsymmetric continuous random variables between zero and one. The results of the Bayesian inverse problem are the pointwise probability distributions of the rock and fluid volumetric fractions conditioned on the seismic attributes. In the first application, the method is validated using synthetic well-log data for the soft sand and stiff rock-physics models with comparisons with several prior models. In the second application, the method is applied to a 2D real data set to obtain the posterior distribution, the maximum a posteriori, and the confidence intervals of porosity, mineral volumes, and fluid saturations. The most likely model of rock and fluid properties estimated from the posterior distribution assuming a Kumaraswamy prior model finds higher accuracies compared to the corresponding results obtained with a Gaussian prior model.