Parallel multilevel domain decomposition preconditioners for monolithic solution of non-isothermal flow in reservoir simulation

In reservoir simulation, the non-isothermal flow adds a conservation of the energy equation with the involvement of a temperature variable to the complicated physics dynamics, which brings a higher nonlinearity of the corresponding PDE system and therefore exhibits significantly additional challenges on the linear preconditioner of the simulator. However, the commonly used block preconditioning techniques mostly focus on the isothermal petroleum model and do not have a specific treatment of the energy conservation equation for the non-isothermal flow. In this study, we propose and develop a family of multilevel restricted additive Schwarz (MRAS) preconditioners on parallel computers for the fully implicit solution of the non-isothermal flow in porous media. The proposed parallel reservoir simulator incorporates the restricted additive Schwarz preconditioner into a general multilevel preconditioning framework with the use of the domain decomposition technique and the multigrid method, which allows us to flexibly construct several types of multilevel Schwarz preconditioners by using the additive or multiplicative strategies. In particular, it follows the fully implicit discretization scheme for the monolithic solution, and therefore is unconditionally stable with the relaxation of the time step size by the stability condition. We numerically show that the proposed approach is highly robust and efficient for solving some non-isothermal flow problems with high nonlinearity, and good parallel scalabilities are obtained on a supercomputer with a large number of processors.