  Inexact Newton Methods are widely used to solve systems of nonlinear
equations. The convergence of these methods is controlled by the relative
linear tolerance, $\eta_\nu$, that is also called the forcing term. A very
small $\eta_\nu$ may lead to oversolving the Newton equation. Practical
reservoir simulation uses inexact Newton methods with fixed forcing term,
usually in the order of $10^{-3}$ or $10^{-4}$. Alternatively, variable forcing
terms for a given inexact Newton step have proved to be quite successful in
reducing the degree of oversolving in various practical applications. The
cumulative number of linear iterations is usually reduced, but the number of
nonlinear iterations is usually increased. We first present a review of
existing inexact Newton methods with various forcing term estimates and then we
propose improved estimates for $\eta_\nu$. These improved estimates try to
avoid as much as possible oversolving when the iterate is far from the solution
and try to enforce quadratic convergence in the neighbourhood of the solution.
Our estimates reduce the total linear iterations while only resulting in few
extra Newton iterations. We show successful applications to fully-coupled
three-phase and multi-component multiphase models in isothermal and thermal
steam reservoir simulation as well as a real deep offshore west-African field
with gas re-injection using the reference CPR-AMG iterative linear solver.
