A comparison of Bayesian versus deterministic formulation for dynamic data integration into reservoir models

Abstract

The integration of dynamic data into reservoir models is known as automatic history matching, and it requires the solution of an inverse problem through the minimization of an objective function. The objective function to minimize is defined as the misfit between the observed production data and the computed production response given a set of parameters. The objective function as described above suffers from ill-posedness. In order to alleviate the problem, we can resort to one of the two available techniques viz. the deterministic and Bayesian formulation. For purposes of automatic history matching, the Bayesian approach has been used extensively, whereas the deterministic formulation has been introduced only recently. This study compares their performance and explores the relative advantages and disadvantages. The Bayesian approach requires knowledge of prior information in the form of a variogram model, which is never well known. For this reason, first we made a sensitivity study to evaluate how critical the knowledge of variogram parameters is. It was found that low sills and high ranges in the variogram model lead to poor results in the inversion. The LSQR algorithm is used to minimize the deterministic formulation; and the Levenberg-Marquardt (L-M) algorithm, and a modified Gauss-Newton (G-N) routine are used to solve the Bayesian formulation. In the second study, we show how the L-M algorithm improves the convergence of the objective function compared to G-N for cases when the solution gets trapped in a local minimum. The algorithms mentioned above require the computation of sensitivities of dynamic data with respect to reservoir parameters. In the third study, we introduce the idea of logarithm sensitivity as presented by Kabala. The results show a lot of improvement in the inversion when using logarithm sensitivity for the deterministic approach, whereas for the Bayesian approach the improvement was minimal. Finally, the fourth study makes a full comparison of both techniques. The deterministic approach was more efficient; the CPU time required for the inversion was very small; the convergence of the objective function was well behaved; and the parameter estimate closely reproduced the main features of the reference model.