In this work we develop an a posteriori-based adaptive algorithm for thermal multiphase compositional flows in porous media. The key ingredients are fully computable a posteriori error estimates, bounding the dual norm of the residual supplemented by a nonconformity evaluation term. The theory hinges on assumptions that allow the application to variety of discretization methods. The estimators are then elaborated to estimate separately the space, time, linearization, and algebraic errors. This additional information is used to formulate a fully adaptive algorithm including adaptive stopping criteria for iterative solvers as well as refinement/derefinement criteria for both the time step and the mesh size. Numerical validation is provided on an industrial case study in the context of oil-recovery based on the steam-assisted gravity drainage procedure. Implicit cell-centered finite volumes with phase-upwind and two-point discretization of the diffusive fluxes are considered. It is shown that significant gains in computational cost can be achieved in this example, without hindering the quality of the results as measured by quantities of engineering interest.
Di Pietro, D. A., Vohralík, M., & Yousef, S. (2014). An a posteriori-based, fully adaptive algorithm with adaptive stopping criteria and mesh refinement for thermal multiphase compositional flows in porous media. Computers and Mathematics with Applications, 68(12), 2331–2347. https://doi.org/10.1016/j.camwa.2014.08.008