A Multiscale Framework for Simulating Multi-phase Flow on Flexible Grids

The geological structure of porous rocks include irregular geometries and heterogeneities on a multiple of scales. Reservoir simulations (e.g. oil recovery, CO2 storage, ground water flow) often involve large spatial scales, where local fine-scale heterogeneities might have an important impact on the global flow. Discretisations of the governing equations of multi-phase flow in general render large systems of non-linear equations. The multiscale nature of the geology is inherited by the discrete mathematical problem, which can be ill-conditioned and thereby hard to solve. In this work, we apply domain decomposition (DD) as a solution strategy for the equations. The splitting of the global system into local subproblems can be done either prior to, or after linearisation of the non-linear system. Comparison between DD as a non-linear and linear solver strategy indicates that splitting into subproblems should be considered for use as preconditioners, also for non-linear systems of equations. When applied after linearisation, DD is best suited as a preconditioner in an iterative solution procedure. In this work, we consider a two-level DD framework for linear systems. The framework is flexible, and can be used both as an upscaling procedure, and as a preconditioner.