An efficient Bayesian approach to history matching and uncertainty assessment

Abstract

Conditioning reservoir models to production data and assessment of uncertainty can be done by Bayesian theorem. This inverse problem can be computationally intensive, generally requiring orders of magnitude more computation time compared to the forward flow simulation. This makes it not practical to assess the uncertainty by multiple realizations of history matching for field applications. We propose a robust adaptation of the Bayesian formulation, which overcomes the current limitations and is suitable for large-scale applications. It is based on a generalized travel time inversion and utilizes a streamline-based analytic approach to compute the sensitivity of the travel time with respect to reservoir parameters. Streamlines are computed from the velocity field that is available from finite-difference simulators. We use an iterative minimization algorithm based on efficient SVD (singular value decomposition) and a numerical ‘stencil’ for calculation of the square root of the inverse of the prior covariance matrix. This approach is computationally efficient. And the linear scaling property of CPU time with increasing model size makes it suitable for large-scale applications. Then it is feasible to assess uncertainty by sampling from the posterior probability distribution using Randomized Maximum Likelihood method, an approximate Markov Chain Monte Carlo algorithms. We apply this approach in a field case from the Goldsmith San Andres Unit (GSAU) in West Texas. In the application, we show the effect of prior modeling on posterior uncertainty by comparing the results from prior modeling by Cloud Transform and by generalized travel time inversion and utilizes a streamline-based analytic approach to compute the sensitivity of the travel time with respect to reservoir parameters. Streamlines are computed from the velocity field that is available from finite-difference simulators. We use an iterative minimization algorithm based on efficient SVD (singular value decomposition) and a numerical Collocated Sequential Gaussian Simulation. Exhausting prior information will reduce the prior uncertainty and posterior uncertainty after dynamic data integration and thus improve the accuracy of prediction of future performance.