The management of the field can be formulated as an optimization problem in which the rates in the producer and injector wells are to be obtained fulfilling specific constraints. For this particular problem a commonly used objective function is the net present value (NPV) of the field. The design variables are the allocated rates at producer and injector wells so each particular design will require a complete reservoir simulation.

History matching in reservoir simulation is an inverse problem whose aim is to understand the reservoir and to predict future field performance. The history matching problem consists of adjusting a set of parameters, reservoir properties, in order to match the data considered for the numerical experiments to the actual production data of the reservoir. Automatic history match can be conducted using optimization techniques in which the objective function is some error measure that is related to the difference between the calculated and observed field responses. The design variables are the reservoir rock and rock-fluid interaction properties in the reservoir.

Both optimization problems described above commonly involve several calls of the numerical simulator, which may turn the optimization task into a very time consuming process. In order to ameliorate such a drawback, a Kriging data fitting scheme is employed to build surrogate models to be used in substitution for the numerical reservoir simulations.

The optimization algorithm of choice is the sequential quadratic programming. This is embedded here in an interactive procedure, named sequential approximate optimization. A trust region based method is used to update the design variable space for each local (subproblem) optimization solution (SAO iteration). The required numerical reservoir simulations to build the data fitting models were performed using the IMEX commercial software from CMG.

Some applications illustrate the use of the presented methodology to conduct the management and history matching (HM) of the reservoirs. For a more realistic permeability field representation the HM application considers the Karhunen-Loève expansion procedure.

For the examples studied here, the proposed methodology performed well with the optimum solutions found. As both WF management and HM are typically multisolution problems, global or hybrid (global + local) optimization strategies will be appropriate in further studies.

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