  Multiphase flow is a critical process in a wide range of applications,
including carbon sequestration, contaminant remediation, and groundwater
management. Typically, this process is modeled by a nonlinear system of partial
differential equations derived by considering the mass conservation of each
phase (e.g., oil, water), along with constitutive laws for the relationship of
phase velocity to phase pressure. In this study, we develop and study efficient
solution algorithms for solving the algebraic systems of equations derived from
a fully coupled and time-implicit treatment of models of multiphase flow. We
explore the performance of several preconditioners based on algebraic multigrid
(AMG) for solving the linearized problem, including "black-box" AMG applied
directly to the system, a new version of constrained pressure residual
multigrid (CPR-AMG) preconditioning, and a new preconditioner derived using an
approximate Schur complement arising from the block factorization of the
Jacobian. We show that the new methods are the most robust with respect to
problem character as determined by varying effects of capillary pressures, and
we show that the block factorization preconditioner is both efficient and
scales optimally with problem size.
