Regularization in History Matching Using Multi-Objective Genetic Algorithm and Bayesian Framework

Abstract

Abstract The history matching procedure can be divided into three sections: decision variables definition, objective function formulation and optimization. The most widespread approach regarding objective function formulation is the Bayesian framework. A Bayesian framework allows the incorporation of prior knowledge into the objective function which acts as a regularization method. In this approach, objective function consists of two terms; likelihood and prior knowledge functions. In order to maximize posterior probability function, usually a summation of prior and likelihood functions is minimized in which the prior and observed data covariance matrixes relate these two functions. Inappropriate covariance matrixes can lead to an incorrect domination of one of the functions over the other one and accordingly result in a false optimum point. In this study, to decrease the chance of convergence into a false optimum point, due to inaccurate covariance matrixes, an application of multi-objective optimization in history matching is introduced while likelihood and prior functions are the two objective functions. By making use of Pareto optimization (multi-objective optimization), a set of solutions named the Pareto front is provided which consists of nondominated solutions. Hence, an inaccuracy in the covariance matrixes cannot allow one objective function to dominate over the other one. After providing the set of solutions, usually a number of solutions are taken out from the set based on postoptimzation trade-offs for uncertainty analysis purposes. For this study, a synthetic case is constructed and history matching is carried out with two different approaches; the conventional and the proposed approach. In order to compare the approaches, it is assumed that covariance matrix of the observed data is not exactly known. Then, history matching is carried out, using a single objective genetic algorithm with different covariance matrixes and also, using a multi-objective genetic algorithm. A comparison between the outcomes of the conventional approach and the proposed approach demonstrates that decisions can be made with more confidence using the proposed approach.