Nonlinearly preconditioned constraint-preserving algorithms for subsurface three-phase flow with capillarity

The multiphase flow model has been extensively used to describe complicated flow behaviors in subsurface formations, together with sophisticated reservoir models and well-defined fluid property. In this study, the fully implicit method, as one of most promising schemes for subsurface flow modeling, is employed to solve multiphase flow problems. In contrast to the conventional approach where mathematical models often include a pressure equation, the multiphase flow problems are modeled by up to three continuity equations so that mass conservation holds for all present phases. Another challenge that frequently shows up is the computed solution may sit outside its physically meaningful range, thereby leading to inaccurate predictions or even a failure of the simulation process. A simple remedy is to apply a cutting-off operation to the out-of-bound solution but such an action could ruin both local and global mass conservation. Instead, we replace the original model by a variational inequality formulation with box inequality constraints to protect the boundedness requirement on pressure and saturations from being violated. The variational inequality problem is then solved by a well-designed nonlinear solver consisting of the active-set reduced-space method and the nonlinear elimination preconditioning technique. A number of examples are presented to demonstrate that the proposed formulation is bound-preserving and mass-conservative for each of the present phases/components.