Characterizing Joint Distribution of Uncertainty Parameters and Production Forecasts Using Gaussian Mixture Model and a Two-Loop Expectation-Maximization Algorithm
Authors: Gao Guohua , Lu Hao , Blom Carl 
Published: September 2024

Abstract
Uncertainty quantification of reservoirs with multiple geological concepts and robust optimization are key technologies for oil/gas field development planning, which require properly characterizing joint distribution of model parameters and/or production forecasts after conditioning to historical production data. In this work, an ensemble of conditional realizations is generated by a multi-realization history-matching (MHM) 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.
This paper first discusses major limitations of the traditional EM algorithm--not robust and converging to suboptimal solutions. We develop a two-loop EM algorithm (EM-EVD-TL) using the compact form of eigenvalue decomposition (EVD) and propose new strategies to overcome these limitations: (1) Reduce the dimension of a Gaussian component if its covariance matrix becomes singular; (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.
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 MCMC method or 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 performs robustly. In contrast, the traditional EM-CD algorithm without regularization fails to converge for most testing cases. The EM-EVD-TL algorithm converges faster to better solutions than the EM-CD algorithm.
The proposed two-loop EM-EVD-TL algorithm has many potential applications and thus helps make better decisions: (1) Close gaps between theoretical formulations of history matching and real applications; (2) characterize posterior distribution of reservoir models having multiple geological concepts or categories; (3) select high-quality P10-P50-P90 representative models; (4) reparametrize gridblock-based properties; and (5) conduct robust well-location and well-control optimization (WLO/WCO) under uncertainty, e.g., through seamless integration of EM-GMM with our advanced multi-objective optimization techniques.