Pressure Preconditioning Using Proper Orthogonal Decomposition

Abstract

Abstract We developed and implemented a new first-stage preconditioning method for large-scale reservoir simulation as an alternative to the popular Algebraic Multi Grid (AMG) method. We used Proper Orthogonal Decomposition (POD) to derive a reduced-order model for the linearized pressure equation in a proprietary reservoir simulator. A small set of pre-computed pressure solutions are used to transform the equation into a lower-order system that can be solved economically yet provides a relatively accurate estimation of the solution of the original high-order system. We present results for a two-phase (oil-water) reservoir model under water flooding conditions. Use of the POD-based preconditioner lead to a significantly faster convergence of the overall simulation compared to AMG, and reduced the linear solver times between approximately 70 % and 80%, i.e. we obtained accelerations with factors between three and five. The highest speed up was achieved for a case in which the flow rates in the eight injection wells were changed frequently during the entire production period, using ten POD basis functions obtained from short training simulations in which the injectors were operated only once each. These first results are very encouraging, especially because there is room to optimize the trade-off between the number of basis functions used in the POD method and the required number of linear iterations. However, further research is required to assess the applicability of the POD-based preconditioner to more complex cases including e.g. strongly compressible flow or compositional effects. The overhead required to pre-compute the POD solutions implies that the new method will be particularly attractive when many solutions of near-similar simulation models are required such as in computer-assisted history matching or flooding optimization.