  Multiphase flow is a critical process in a wide range of applications,
including oil and gas recovery, carbon sequestration, and contaminant
remediation. Numerical simulation of multiphase flow requires solving of a
large, sparse linear system resulting from the discretization of the partial
differential equations modeling the flow. In the case of multiphase
multicomponent flow with miscible effect, this is a very challenging task. The
problem becomes even more difficult if phase transitions are taken into
account. A new approach to handle phase transitions is to formulate the system
as a nonlinear complementarity problem (NCP). Unlike in the primary variable
switching technique, the set of primary variables in this approach is fixed
even when there is phase transition. Not only does this improve the robustness
of the nonlinear solver, it opens up the possibility to use multigrid methods
to solve the resulting linear system. The disadvantage of the complementarity
approach, however, is that when a phase disappears, the linear system has the
structure of a saddle point problem and becomes indefinite, and current
algebraic multigrid (AMG) algorithms cannot be applied directly. In this study,
we explore the effectiveness of a new multilevel strategy, based on the
multigrid reduction technique, to deal with problems of this type. We
demonstrate the effectiveness of the method through numerical results for the
case of two-phase, two-component flow with phase appearance/disappearance. We
also show that the strategy is efficient and scales optimally with problem
size.
