Large-scale history matching with quadratic interpolation models

Abstract

Due to the limited availability of adjoint code in commercial reservoir simulators for gradient calculations, there is a need to explore the applicability of derivative-free optimization algorithms for large-scale history matching. This paper tests the utility of three derivative-free optimization algorithms (stochastic Gaussian search direction (SGSD), new unconstrained optimization algorithm (NEWUOA), and quadratic interpolation model-based algorithm guided by approximate gradient (QIM-AG)) for history matching. The SGSD method uses a negative stochastic gradient which is obtained by simultaneously perturbing all the model parameters using a Gaussian random vector. For a continuous objective function and a sufficiently small perturbation size, the stochastic gradient is always uphill and the expectation of the stochastic gradient converges to the true gradient as the perturbation size goes to zero. NEWUOA is a quadratic interpolation model-based optimization algorithm. At each iteration, the objective function is first approximated by a quadratic interpolation model. The quadratic model is then minimized to obtain an updated reservoir description for the next iteration. The number of interpolation points (reservoir simulation runs) required by NEWUOA must be larger than the dimension of reservoir model parameter space in order to construct the initial quadratic model. QIM-AG reduces the required number of interpolation points by replacing the first-order coefficients that appear in the quadratic model by an approximate gradient. The approximate gradient used in this study is an average of several stochastic gradients from SGSD. To reduce the dimension of the optimization problem, a simple parameterization method based on the prior covariance matrix is applied. The prior covariance matrix is approximated using an ensemble of unconditional realizations. The parameterization avoids the calculation of the inverse of the prior covariance matrix during optimization and may further regularize the ill-posed inverse problem.