For large-scale history matching problems, optimization algorithms which require only the gradient of the objective function and avoid explicit computation of the Hessian appear to be the best approach. Unfortunately, such algorithms have not been extensively used in practice because computation of the gradient of the objective function by the adjoint method requires explicit knowledge of the simulator numerics and expertise in simulation development. Here we apply the simultaneous perturbation stochastic approximation (SPSA) method to history match multiphase flow production data. SPSA, which has recently attracted considerable international attention in a variety of disciplines, can be easily combined with any reservoir simulator to do automatic history matching. The SPSA method uses stochastic simultaneous perturbation of all parameters to generate a down hill search direction at each iteration. The theoretical basis for this probabilistic perturbation is that the expectation of the search direction generated is the steepest descent direction.
We present modifications for improvement in the convergence behavior of the SPSA algorithm for history matching problems and compare its performance to the steepest descent, gradual deformation and LBFGS algorithm for history matching examples. Although the convergence properties of the SPSA algorithm are not nearly as good as our most recent implementation of a quasiNewton method (LBFGS), the SPSA algorithm is not simulator specific and it requires only a few hours of work to combine SPSA with any commercial reservoir simulator to do automatic history matching.
To the best of our knowledge, this is the first introduction of SPSA into the history matching literature. Thus, we make considerable effort to put it in a proper context.