Reservoir characterization and inversion uncertainty via a family of Particle Swarm Optimizers and differential evolution. Application to the synthetic Stanford VI reservoir

History matching provides to the reservoir engineers an improved spatial distribution of physical properties to be used in forecasting the reservoir response for field management. The ill-posed character of the history matching problem yields non-uniqueness and numerical instabilities that increase with the reservoir complexity. These features might cause local optimization methods to provide unpredictable results not being able to discriminate among the multiple models that fit the observed data (production history). In this manuscript the ill-conditioned character of this history matching inverse problem is attenuated by reducing the model complexity using the Spatial Principal Component base and by combining as observables flow production measurements and time lapse seismic cross-well tomographic images. Additionally the inverse problem is solved in a stochastic framework. For this purpose we use a family of Particle Swarm Optimizers that have been deduced from a physical analogy of the swarm system. For the synthetic Stanford VI sand-and-shale reservoir we analyze the performance of the different PSO optimizers, both in terms of exploration and convergence rate for two different reservoir models with different complexity and under the presence of different levels of white Gaussian noise. We show that PSO optimizers have a very good convergence rate and provide in addition, approximate measures of uncertainty around the optimum facies model. Uncertainty estimation makes our algorithms more robust in presence of noise which is always the case for real data. This is an important achievement since in cases where the reservoir exhibits small scale features local methods get trapped and clearly fail to find a good solution. Finally we briefly introduce differential evolution and we show some preliminary results of its performance on the Stanford VI reservoir, showing that we are able to achieve similar results. Application of PSO and DE for reservoir characterization and inversion uncertainty