A Variant of Particle Swarm Optimization for Uncertainty Quantification

In this paper we present an enhanced and new optimization algorithm based on Particle Swarm Optimization (PSO) that can be used in reservoir characterization applications to determine multiple solutions that reproduce measurements within satisfactory accuracy. The method is illustrated on a history-matching problem where significant speed-up is obtained with respect to a standard PSO implementation. PSO is a global-search iterative technique where the solution space is explored using a swarm of particles. In every PSO iteration each particle is individually attracted to: the best point (in terms of objective function) in the swarm during last iteration, the best point visited by that particle in all iterations, and the best point found by any particle in all iterations. In the PSO variant proposed here, rather than single points, collections of points carefully selected are used as attractors. The algorithm ensures by a series of mechanisms that the number of attractors is adequate and that the solutions progressively determined are sufficiently different from one another. A series of experiments were performed with a synthetic model based on a real oil field for which nine years of historical data were generated. Both standard PSO and the PSO variant were tested to obtain several models that reproduced well rates within acceptable accuracy. In order to determine multiple solutions via standard PSO (which usually returns only one solution), a number of independent runs with different initial conditions were considered (i.e., standard PSO was run in multi-start fashion). In our experiments, which had a budget of 3,072 simulations in the two cases, the number of solutions found on average by standard PSO and by our PSO variant was 8 and 837, respectively. That is, multi-start standard PSO may require, on average, around 100 times more simulations than our PSO variant to compute a comparable number of models (possibly) distributed in similar manner in the solution space. These results may be explained by noticing that the search for multiple solutions may be more efficient when all information is considered simultaneously than separately by means of a number of independent runs. Standard PSO, when applied to many uncertainty quantification problems, may provide only one solution because the particles in the swarm generally tend to converge to essentially a single point. The Ensemble Kalman Filter (EnKF), which has lately become a very popular history-matching algorithm, presents a similar problem because the ensemble sometimes collapses into one solution. The PSO variant introduced in this work can be an efficient alternative to these algorithms in reservoir characterization since it has been designed to provide multiple and different solutions in a single run.