Uncertainty quantification of reservoir performance predictions using a stochastic optimization algorithm

The simultaneous perturbation stochastic approximation (SPSA) algorithm is modified to obtain a stochastic Gaussian search direction (SGSD) algorithm for automatic history matching. The search direction in the SGSD algorithm is obtained by simultaneously perturbing the reservoir model with unconditional realizations from a Gaussian distribution. This search direction has two nice properties: (1) it is always downhill in the vicinity of the current iterate and (2) the expectation of the stochastic search direction is an approximate quasi-Newton direction with a prior covariance matrix used as the approximate inverse Hessian matrix. For Gaussian reservoir models, we argue and demonstrate that the SGSD algorithm may generate more geologically realistic reservoir description than is obtained with the original SPSA algorithm. It is shown that the SGSD algorithm represents an approximation of the gradual deformation method but its search direction has the desirable properties that are lacking in all gradual deformation methods. The SGSD algorithm is successfully applied to the well-known PUNQ-S3 test case to generate a maximum a posteriori estimate and for uncertainty quantification of reservoir performance predictions using the randomized maximum likelihood method.