This work describes the combination of adjoint methods for derivative calculation with a trust-region Quasi-Newton optimization algorithm for solving history matching problems. The techniques employed allowed for the resolution of problems with a larger number of parameters, when compared to other approaches, based on derivative free optimization or the forward method for derivative calculation.
Applications to both synthetic and real cases will be shown. A substantial improvement of the match was obtained in all examples. Problems with more than 250 parameters were solved efficiently. The ability to handle problems with a large number of parameters is exploited to discuss the possibility of accessing the uncertainty in the history matching process through exausting the space of possible models.
Assisted history matching tools are becoming more common in the last few years in the oil industry.[1,2] Common to most of these methodologies is the use of a least-squares objective function to quantify the misfit between simulated and observed data. This objective function can be expressed as
where Nobs is the number of observed data, di/sim and di/obs represent the i-th simulated and observed data, respectively, and wi is a weight. After defining a set of reservoir parameters to be determined, an optimization algorithm is employed to obtain the parameter values for which this objective function is the least possible. Each step of the optimization algorithm requires one or more simulations, so, in general, the process becomes quite expensive.
Although considerable progress has been made in this area, the existing tools are still limited to problems with a small number of parameters and they suffer from a lack of integration with the geological modeling. Consequently, in addition to a negative effect on the productivity of reservoir studies, the history matching process does not assess the uncertainties inherent in the problem, since typically a single model is obtained.
This work shows the results of the development of an assisted history matching tool for a full-featured black-oil simulator. The techniques employed, based on derivatives calculated internally by the simulator, are briefly described and applications to synthetic and real cases, including problems with more than 250 parameters, are presented. In all examples, a substantial improvement in the match over the initial model was reached.
The results indicate that the tool is able to solve, in a practical manner, problems with a large number of parameters. In two of the examples, the computational efficiency is explored to generate a set of models matched to the production data, in order to assess the uncertainty in the history matching problem through the exaustive sampling of the space of possible models.