Nearwell Local Space and Time Refinement for Multi-phase Porous Media Flows
W. Kheriji, R. Masson and A. Moncorgé
Event name: ECMOR XIV - 14th European Conference on the Mathematics of Oil Recovery
Session: Nonlinear Solvers
Publication date: 08 September 2014
Info: Extended abstract, PDF ( 1.68Mb )
Price: € 20
Nearwell regions in reservoir simulations usually require fine space and time scales due to several physical processes such as higher Darcy velocities, the coupling of the stationary well model with the transient reservoir model, high non linearities due to phase appearance (typically gas), complex physics such as formation damage models. If Local Grid Refinement is commonly used in reservoir simulations in the nearwell regions, current commercial simulators still make use of a single time stepping on the whole reservoir domain. It results that the time step is globally constrained both by the nearwell small refined cells and by the high Darcy velocities and high non linearities in the nearwell region. A Local Time Stepping with a small time step in the nearwell regions and a larger time step in the reservoir region is clearly a promising field of investigation in order to save CPU time. It is a difficult topic in the context of reservoir simulation due to the implicit time integration, and to the coupling between a mainly elliptic or parabolic unknown, the pressure, and mainly hyperbolic unknowns, the saturations and compositions. Our proposed approach is based on a Schwarz Domain Decomposition (DDM) Robin-Neumann algorithm using a full overlap at the coarse level to speed up the convergence of the iterative process. The matching conditions at the nearwell reservoir interfaces use optimized Robin conditions for the pressure and Dirichlet conditions for the saturations and compositions. At the well interfaces, a Neumann condition is imposed for the pressure (assuming to fix ideas that the well condition is a fixed pressure) and input Dirichlet conditions are imposed for saturations and compositions. The optimization of the Robin coefficients can be done on a pressure equation only using existing theory for elliptic/parabolic equations while the algorithm is applied on fully implicit discretization of multi-phase Darcy flows. Numerical experiments on 3D test cases including gas injection and gas-condensate reservoir exhibit the efficiency of the method both in terms of improved accuracy compared with the classical sequential windowing algorithm, and in terms of convergence of the DDM algorithm using Robin coefficients optimized once and for all on the single phase flow equation only.