| Abstract |
Modern reservoir simulation relies on iterative methods to solve the large sparse linear system. These efficient iterative methods are often built on the projection method and Krylov subspace which ensure the convergence behavior. However, iterative methods, applicable to general large sparse matrices, might encounter problems of instability or inefficiency in various simulation models. In addition to the commonly used preconditioners which are also employed for general sparse systems, such as Incomplete LU (ILU) factorization, specific physics-based preconditioners should also be developed for performance improvement of iterative methods. Constrained Pressure Residual (CPR) is proposed to solve pressure equations first for damping the low-frequency error components before ILU preconditioning, which is proven to be effective in improving the convergence of iterative solver. The combination of CPR-ILU(0) is widely used in the black-oil reservoir simulator. However, this strategy could be less efficient for thermal models, due to the incorporation of energy equation into the linear system. In geothermal reservoir simulation, it is necessary to seek a more efficient physics-based preconditioner for large-scale thermal models. A physics-based multi-stage preconditioner, named as SWIFT, aiming at the large magnitude terms generated by thermal energy equations is the solution to this problem. This preconditioner first scaled the matrix diagonally and applied row-column equilibration in order to transform the linear system into a near diagonally dominant one. Through this transformation, the Crout version of ILU (ILUC) can be used without partial pivoting and become capable of improving the accuracy of factorization by adaptively exploiting the sparsity and modifying fill-in terms. This preconditioner combining CPR, SWIFT and ILUC is observed to significantly improve the convergence performance of iterative methods in thermal models. In this paper, CPR-SWIFT-ILUC multi-stage preconditioner is implemented and applied into geothermal reservoir simulation. The effectiveness of CPR-SWIFT-ILUC is evaluated in several cases. The comparison of results shows that CPR-SWIFT-ILUC can help iterative methods achieve convergence efficiently in difficult thermal models. The overall efficiency with respect to computational time using CPR-SWIFT-ILUC is mainly dependent on problem complexity and should be determined by the simulator based on convergence conditions. |