Robust Optimization Using the Mean Model with Bias Correction

Optimization of the expected outcome for subsurface reservoir management when the properties of the subsurface model are uncertain can be costly, especially when the outcomes are predicted using a numerical reservoir flow simulator. The high cost is a consequence of the approximation of the expected outcome by the average of the outcomes from an ensemble of reservoir models, each of which may need to be numerically simulated. Instead of computing the sample average approximation of the objective function, some practitioners have computed the objective function evaluated on the “mean model,” that is, the model whose properties are the means of properties of an ensemble of model realizations. Straightforward use of the mean model without correction for bias is completely justified only when the objective function is a linear function of the uncertain properties. In this paper, we show that by choosing an appropriate transformation of the variables before computing the mean, the mean model can sometimes be used for optimization without bias correction. However, because choosing the appropriate transformation may be difficult, we develop a hierarchical bias correction method that is highly efficient for robust optimization. The bias correction method is coupled with an efficient derivative-free optimization algorithm to reduce the number of function evaluations required for optimization. The new approach is demonstrated on two numerical porous flow optimization problems. In the two-dimensional well location problem with 100 ensemble members, a good approximation of the optimal location is obtained in 10 function evaluations, and a slightly better (nearly optimal) solution using bias correction is obtained using 216 function evaluations.