Keywords: oil reservoir engineering, waterflooding, surrogate based optimization.

In oil reservoir engineering applications one problem of great interest is the dynamic optimization of production scheduling, considering constraints on the platform's total rate. In this context, we will tackle here the waterflooding problem which is by far the most commonly used method to improve oil recovery after primary depletion. In general, the numerical simulation of above application tends to have a high computational cost in a single simulation process. Therefore any procedure which requires multiple numerical simulations for its optimization is prohibitively expensive. Approximation strategies are identified in the literature as a powerful tool to overcome the above mentioned problem [1]. In this work we will consider a kriging data fitting approximation approach. The central idea of kriging is that the sample response values exhibit spatial correlation with response values modelled by a Gaussian process around each sample location. The main advantages pointed for such scheme are: the ability to accommodate irregularly spaced data, the ability to model functions with many peaks and valleys. From a proper choice of a design of experiments (DOE) scheme [1], followed by the evaluations of the true function at the samplings, a kriging predictor is built in order to evaluate the functions at untried points during the optimization algorithm iterations. The surrogate model built will be used here in the context of both global and local optimization algorithms.

The local optimization algorithm of choice is the sequential quadratic programming (SQP). This will be embedded here in an interactive procedure, named the sequential approximate optimization (SAO). A trust region based method is used to update the design variable space for each local (subproblem) optimization solution (SAO iteration). The global algorithm considered is the efficient global optimization (EGO) proposed by Jones et al. [1], in which the metamodel is used to guide the search for promising designs which are then added to update the model until a suitable termination criterion is fulfilled. The selection of designs which are adaptively added to the sample is the infill sampling criterion (ISC). The ISC balances the need for improving the value of the objective function with that of improving the quality of the prediction so that one does not get trapped in a local minimum. In the EGO algorithm this balance is achieved through the usage of the expected improvement merit function. The DIRECT algorithm [2] is employed to maximize the expected improvement. The present work will compare the solutions obtained when using the above mentioned different optimization strategies in the dynamic optimization of production scheduling. Constraints on the platform's total rates for different situations in which the constraints on the water cut and/or bottom hole pressure at the wells are activated or not, are considered.

1

D.R. Jones, M., Schonlau, W.J. Welch, "Efficient Global Optimization of Expensive Black-Box Functions", Journal of Global Optimization, 13(4), 455-492, 1998. doi:10.1023/A:1008306431147

2

D.E. Finkel, "DIRECT optimization algorithm users guide", Center for Research and Scientific Computation, CRSC-TR03-11, North Carolina State University, Raleigh, 2003.

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