Automatic history matching of production data for large-scale problems

Within the context of Bayesian statistics, realizations of rock property fields can be generated by automatic history matching of production data using a prior model to provide regularization. Automatic history matching requires the minimization of an objective function which includes the sum of squared production data mismatch as well as a regularization term arising from the prior geostatistical model. For large scale problems, the computational efficiency and robustness of the optimization algorithms used for minimization are of paramount importance.; We consider a variety of optimization algorithms for history matching production data. For history matching problems where tens of thousands of parameters are estimated, preconditioned conjugate gradient methods and quasi-Newton methods appear to be the only viable gradient based methods. Based on several examples considered in this work, a particular implementation of the limited memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm is more robust and computationally efficient for large scale problems than the preconditioned conjugate gradient methods that we have tried. It is shown that computational efficiency of the limited memory BFGS can he improved by a proper choice of the scaling factor and the initial approximation of the inverse Hessian. To the best of our knowledge, the particular implementations of these algorithms presented here are new to the petroleum engineering literature.; An iterative linear solver based on orthomin theory was implemented in this work. For large problems, the iterative solver is orders of magnitude faster than the direct solver which is based on the LU decomposition. The iterative solver was used to solve the adjoint equation system which is a linear system. The solution obtained by the iterative solver is in excellent agreement with the solution obtained by a sparse matrix technique.; The computational algorithms for history matching are applied to condition rock property fields generated from a prior geostatistical model to production data. The procedure allows one to consider the errors in prior means as model parameters.