Streamline based History Matching

Abstract

Streamline based reservoir simulation has been used with great success in the petroleum industry for decades. Fast computational speed together with the fact that sensitivities can be computed analytically along streamlines makes streamline based methods efficient for history matching problems. The key idea for streamline based methods is to approximate 3D fluid flow by a family of 1D transport equations along streamlines. This is done by introducing a time-of-flight coordinate variable. In this thesis a method called generalized travel-time inversion (GTTI) will be discussed for history matching. The main idea behind GTTI is to combine amplitude matching with travel-time matching. The inverse problem associated with the amplitude matching is fully nonlinear and can therefore give unstable and non-unique solutions, while the travel-time inversion has quasi-linear properties. GTTI is able to use all the data available and still preserve the quasi-linear properties of the travel-time inversion. Several authors has studied GTTI previously. In all these articles only the first order sensitivities are used in the history matching. Since the second order sensitivities can be computed along the streamlines at practically no cost, a method that includes information from the second order sensitivities will in theory show better convergence properties. Some synthetic examples shows that this is true for cases where almost no regularization is needed but not for more realistic and illposed problems. To further improve the convergence properties of the minimization a line search and a trust-region strategy is suggested. Both the line search strategy and the trust region method of Levenberg-Marquardt shows good performance. Both in the ability to reduce the objective function and in characterizing the permeability field. A method where the regularization parameters are chosen by the L-curve criteria is also considered and compared to the other methods. An advantage of this method is that the regularization term is reduced as the iterate enters into the neighborhood of the solution and less regularization is needed.