Sequential Simulations of Mixed Discrete-Continuous Properties: Sequential Gaussian Mixture Simulation

Abstract

We present here a method for generating realizations of the posterior probability density function of a Gaussian Mixture linear inverse problem in the combined discrete-continuous case. This task is achieved by extending the sequential simulations method to the mixed discrete-continuous problem. The sequential approach allows us to generate a Gaussian Mixture random field that honors the covariance functions of the continuous property and the available observed data. The traditional inverse theory results, well known for the Gaussian case, are first summarized for Gaussian Mixture models: in particular the analytical expression for means, covariance matrices, and weights of the conditional probability density function are derived. However, the computation of the weights of the conditional distribution requires the evaluation of the probability density function values of a multivariate Gaussian distribution, at each conditioning point. As an alternative solution of the Bayesian inverse Gaussian Mixture problem, we then introduce the sequential approach to inverse problems and extend it to the Gaussian Mixture case. The Sequential Gaussian Mixture Simulation (SGMixSim) approach is presented as a particular case of the linear inverse Gaussian Mixture problem, where the linear operator is the identity. Similar to the Gaussian case, in Sequential Gaussian Mixture Simulation the means and the covariance matrices of the conditional distribution at a given point correspond to the kriging estimate, component by component, of the mixture. Furthermore, Sequential Gaussian Mixture Simulation can be conditioned by secondary information to account for non-stationarity. Examples of applications with synthetic and real data, are presented in the reservoir modeling domain where realizations of facies distribution and reservoir properties, such as porosity or net-to-gross, are obtained using Sequential Gaussian Mixture Simulation approach. In these examples, reservoir properties are assumed to be distributed as a Gaussian Mixture model. In particular, reservoir properties are Gaussian within each facies, and the weights of the mixture are identified with the point-wise probability of the facies.