Parallel algorithms and simulators with good scalabilities are particularly important for large-scale reservoir simulations on modern supercomputers with a large number of processors. In this paper, we introduce and study a family of highly scalable multilevel restricted additive Schwarz (RAS) methods for the fully implicit solution of subsurface flows with Peng-Robinson equation of state in two and three dimensions. With the use of a second-order fully implicit scheme, the proposed simulator is unconditionally stable with the relaxation of the time step size by the stability condition. The investigation then focuses on the development of several types of multilevel overlapping additive Schwarz methods for the preconditioning of the resultant linear system arising from the inexact Newton iteration, and some fast solver technologies are presented for the assurance of the multilevel approach efficiency and scalability. We numerically show that the proposed fully implicit framework is highly efficient for solving both standard benchmarks as well as realistic problems with several hundreds of millions of unknowns and scalable to 8192 processors on the Tianhe-2 supercomputer.

具有良好可扩展性的并行算法和仿真器对于具有大量处理器的现代超级计算机上的大规模油藏仿真尤为重要。在本文中，我们介绍和研究了一系列高度可扩展的多级受限加性Schwarz（RAS）方法，用于用二维和三维状态下的Peng-Robinson状态方程对地下流进行完全隐式求解。通过使用二阶完全隐式方案，所提出的模拟器是无条件地稳定的，并且通过稳定条件来放松时间步长。然后，研究集中在几种类型的多级重叠加性Schwarz方法的开发上，这些方法用于预处理由于不精确的牛顿迭代而产生的线性系统，提出了一些快速求解器技术，以确保多级方法的效率和可伸缩性。我们从数字上显示，所提出的完全隐式框架对于解决标准基准以及具有数亿未知数的现实问题非常有效，并且可在天河2号超级计算机上扩展至8192个处理器。