Estimation of Distribution Algorithms for History Matching

In previous history matching studies it has been shown that there may be multiple local optima to the response surface, even when the inverse problem is well defined in a mathematical sense. Practical algorithms that allow the identification of these local optima, such as Genetic Algorithms, have been demonstrated to work. Whilst it is useful to know where in parameter space multiple solutions exist, it is not every thing we would wish to know. At each local optimum we would like to know the range of uncertainty for each parameter and how the parameters are correlated. This will allow us to make more useful predictions, including better estimates of the uncertainty in those predictions.<br><br>In this paper we demonstrate the use of a simple Estimation of Distribution Algorithm, Probability Based Incremental Learning, on a simple reservoir cross sectional model with three parameters which is known to have multiple high quality local optima. The probability distribution function, for each parameter, can be approximated by a histogram which is adjusted using the results of the search. The sampling of the parameter space is guided by the current pdfs. We show that this algorithm can evolve steady-state pdfs which would allow us to sample the parameter space more efficiently when estimating uncertainty.<br><br>We have also introduced a modified version of the sum of square objective function. This allows a better treatment of water break through as part of the objective function. The result is that some of the optima are wider and this allows optimisers to find them more easily.<br>