Automatic History Matching in a Bayesian Framework, Example Applications

Abstract

Summary The Bayesian framework allows one to integrate production and static data into an a posteriori probability density function (pdf) for reservoir variables (model parameters). The problem of generating realizations of the reservoir variables for the assessment of uncertainty in reservoir description or predicted reservoir performance then becomes a problem of sampling this a posteriori pdf to obtain a suite of realizations. Generation of a realization by the randomized-maximum-likelihood method requires the minimization of an objective function that includes production-data misfit terms and a model misfit term that arises from a prior model constructed from static data. Minimization of this objective function with an optimization algorithm is equivalent to the automatic history matching of production data, with a prior model constructed from static data providing regularization. Because of the computational cost of computing sensitivity coefficients and the need to solve matrix problems involving the covariance matrix for the prior model, this approach has not been applied to problems in which the number of data and the number of reservoir-model parameters are both large and the forward problem is solved by a conventional finite-difference simulator. In this work, we illustrate that computational efficiency problems can be overcome by using a scaled limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithm to minimize the objective function and by using approximate computational stencils to approximate the multiplication of a vector by the prior covariance matrix or its inverse. Implementation of the LBFGS method requires only the gradient of the objective function, which can be obtained from a single solution of the adjoint problem; individual sensitivity coefficients are not needed. We apply the overall process to two examples. The first is a true field example in which a realization of log permeabilities at 26,019 gridblocks is generated by the automatic history matching of pressure data, and the second is a pseudofield example that provides a very rough approximation to a North Sea reservoir in which a realization of log permeabilities at 9,750 gridblocks is computed by the automatic history matching of gas/oil ratio (GOR) and pressure data.