Characterizing Joint Distribution of Uncertainty Parameters and Production Forecasts Using Gaussian Mixture Model and a Two-Loop Expectation-Maximization Algorithm

Summary Uncertainty quantification of reservoirs with multiple geological concepts and robust optimization are key technologies for oil/gasfield development planning, which require properly characterizing joint distribution of model parameters and/or production forecasts after conditioning to historical production data. In this work, we generate an ensemble of conditional realizations by a multirealization history-matching workflow. The posterior probability density function (PDF) of model parameters and/or production forecasts is non-Gaussian, and we approximate it by a Gaussian mixture model (GMM) using an expectation-maximization (EM) algorithm. We first discuss major limitations of the traditional EM algorithm—not robust and converging to suboptimal solutions. We develop a two-loop (TL) EM algorithm [EM-eigenvalue decomposition (EVD)-TL] using the compact form of EVD and propose new strategies to overcome these limitations: (1) Reduce the dimension of a Gaussian component if its covariance matrix becomes singular and (2) introduce an inner EM loop in which only the diagonal matrix in EVD of the covariance matrix is updated. The first strategy improves the stability and convergence of the EM algorithm in dealing with degeneration of Gaussian components. The second strategy reduces the computational cost and further improves the convergence rate and accuracy of solutions. The proposed EM-EVD-TL algorithm was validated on an analytical testing example, and its performance is compared against the single-loop, traditional EM algorithms, which use either Cholesky decomposition (EM-CD) or EVD (EM-EVD). An ensemble of conditional realizations is generated from sampling the actual PDF using the Markov Chain Monte Carlo (MCMC) approach. For the analytical example, the GMMs approximated by three EM algorithms are very close to the actual distribution with negligible difference. Finally, we applied the proposed EM-EVD-TL algorithm to realistic history-matching problems with different number of uncertainty parameters and production forecasts. We first generate an ensemble of conditional realizations using either the MCMC method or the distributed Gauss-Newton (DGN) optimization method. Then, we construct GMMs using different EM algorithms by fitting the conditional realizations, starting from different initial configurations and settings. Our numerical results confirm that the proposed EM-EVD and EM-EVD-TL algorithms perform robustly. In contrast, the traditional EM-CD algorithm fails to converge for most testing cases. The EM-EVD-TL algorithm converges faster to better solutions than the EM-CD algorithm.