Reservoir history matching by ensemble smoother with principle component and sensitivity analysis for heterogeneous formations

Abstract

Ensemble-based data assimilation methods have been widely investigated and applied for inverse problems of fluid flow in porous media during the past decades. Among these methods, the ensemble Kalman filter and the ensemble smoother are probably the most popular in history-matching applications. For large-scale problems, the ensemble size is limited to the computational resources in running the forward simulation, and is usually much smaller than the total number of gridblocks. In this case, the ensemble forms a reduced-order subspace, and principle component analysis is suggested by retaining the leading eigenvectors after truncation.In this paper, we propose to improve the ensemble smoother by proper orthogonal decomposition with optimal parameter selection. The motivation behind this method is that the estimation would be more accurate if the subspace spanned by the ensemble favors the dominant components. Specifically, the principle component analysis is first applied to obtain the eigenvalues and orthogonal eigenvectors. Then, a number of eigenvectors are selected according to importance from sensitivity analysis, rather than the eigenvalues. Finally, the ensemble smoother is implemented using these eigenvectors.The proposed method is illustrated by a simple mathematical example, and tested in a 2D single-phase and a 3D multiphase flow problem. The numerical results show that the traditional ensemble smoother may provide poor results in some ill-posed inverse problems, because some important features are lost in the randomly generated realizations. The principle component analysis alleviates this issue by sorting the eigenvectors based on energy, but it may still produce unsatisfactory results. The combination of sensitivity analysis and principle component analysis provides more accurate matching and prediction than those obtained from the previous two methods. The local sensitivity analysis is preferred since it is much more efficient than the global sensitivity analysis, and its accuracy is validated in the case studies.