两相储层流问题的多尺度摄动方法
The Multiscale Perturbation Method for Two-Phase Reservoir Flow Problems
AI总结：
本文提出两相流多尺度摄动方法（MPM-2P），结合区域分解与多尺度Robin耦合方法，通过重用基函数避免完整更新，在保证精度的同时将两相流模拟计算成本降低近90%。
摘要：
在这项工作中，我们提出并测试了一种新方法，即用于两相流的多尺度摄动方法（MPM-2P），用于快速、准确且天然可并行地数值求解由算子分裂方法逼近的多孔介质中两相、不可压缩、不混溶驱替问题。该方法基于区域分解，将多尺度摄动方法（MPM）与多尺度Robin耦合方法（MRCM）相结合。当算子分裂算法要求更新速度场时，MPM-2P可以根据一个无量纲算法参数的大小，通过重用先前计算的多尺度基函数，为速度场提供准确且计算成本低的近似。从而避免了MRCM构建新速度场所需的全部多尺度基函数的完整更新。MPM-2P的构建包含两个主要步骤：首先，对每个子域求解一个带有平凡Robin边界条件的局部边值问题（而非MRCM所需的一整套多尺度基函数）；然后，通过求解一个低成本的界面问题，获得速度场在区域分解骨架上的解。最终的速度场近似通过降尺度得到。我们考虑了具有挑战性的两相流问题，包括高对比度渗透率场和均质介质中的水油指进增长。数值实验表明，使用MPM-2P为两相流模拟带来了显著的加速——计算成本降低了近90%。数百次MRCM求解可由低成本的MPM-2P求解替代，且水突破现象仅需极少的MRCM多尺度基函数更新即可完成模拟。
英文摘要：
In this work we formulate and test a new procedure, the Multiscale Perturbation Method for Two-Phase Flows (MPM-2P), for the fast, accurate and naturally parallelizable numerical solution of two-phase, incompressible, immiscible displacement in porous media approximated by an operator splitting method. The proposed procedure is based on domain decomposition and combines the Multiscale Perturbation Method (MPM) with the Multiscale Robin Coupled Method (MRCM). When an update of the velocity field is called by the operator splitting algorithm, the MPM-2P may provide, depending on the magnitude of a dimensionless algorithmic parameter, an accurate and computationally inexpensive approximation for the velocity field by reusing previously computed multiscale basis functions. Thus, a full update of all multiscale basis functions required by the MRCM for the construction of a new velocity field is avoided. There are two main steps in the formulation of the MPM-2P. Initially, for each subdomain one local boundary value problem with trivial Robin boundary conditions is solved (instead of a full set of multiscale basis functions, that would be required by the MRCM). Then, the solution of an inexpensive interface problem provides the velocity field on the skeleton of the decomposition of the domain. The resulting approximation for the velocity field is obtained by downscaling. We consider challenging two-phase flow problems, with high-contrast permeability fields and water-oil finger growth in homogeneous media. Our numerical experiments show that the use of the MPM-2P gives exceptional speed-up - almost 90% of reduction in computational cost - of two-phase flow simulations. Hundreds of MRCM solutions can be replaced by inexpensive MPM-2P solutions, and water breakthrough can be simulated with very few updates of the MRCM set of multiscale basis functions.
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