Author(s): 
Dario Grana, Stanford University (US) T. Mukerji, Stanford University (US) L. Dovera, Eni E&P (IT) E. Della Rossa, Eni E&P (IT)

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 discretecontinuous case. This task is achieved by extending the sequential simulations method to the mixed discretecontinuous problem. The sequential approach allows us to generate a Gaussian mixture random field that honors the covariance function of the continuous property, and the available observed data. The traditional inverse theory results, available for the Gaussian case, are first extended to 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. Similarly to the Gaussian case, in Sequential Gaussian Mixture Simulation the means and the covariance matrices of the conditional distribution in 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 nonstationarity. An example of application of the methodology is presented in reservoir modeling domain where realizations of facies distribution and reservoir properties, such as porosity or net togross, are obtained using Sequential Gaussian Mixture Simulation approach. In this example, 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 pointwise probability of the facies. 