  Simulating compositional multiphase flow in porous media is a challenging
task, especially when phase transition is taken into account. The main problem
with phase transition stems from the inconsistency of the primary variables
such as phase pressure and phase saturation, i.e. they become ill-defined when
a phase appears or disappears. Recently, a new approach for handling phase
transition has been developed, whereby the system is formulated as a nonlinear
complementarity problem (NCP). Unlike the widely used primary variable
switching (PVS) method which requires a drastic reduction of the time step size
when a phase appears or disappears, this approach is more robust and allows for
larger time steps. One way to solve an NCP system is to reformulate the
inequality constraints as a non-smooth equation using a complementary function
(C-function). Because of the non-smoothness of the constraint equations, a
semi-smooth Newton method needs to be developed. In this work, we consider two
methods for solving NCP systems used to model multiphase flow: (1) a
semi-smooth Newton method for two C-functions: the minimum and the
Fischer-Burmeister functions, and (2) a new inexact Newton method based on the
Jacobian smoothing method for a smooth version of the Fischer-Burmeister
function. We show that the new method is robust and efficient for standard
benchmark problems as well as for realistic examples with highly heterogeneous
media such as the SPE10 benchmark.
