STABLE METHOD AND APPARATUS FOR SOLVING S-SHAPED NON-LINEAR FUNCTIONS UTILIZING MODIFIED NEWTON-RAPHSON ALGORITHMS

摘要

This invention relates to an apparatus and method are provided for solving a non-linear S-shaped function F = t'(S) which is representative of a property S in a physical system, such saturation in a reservoir simulation. A Newton iteration (T) is performed on the function f(S) at Sv to determine a next iterative value SV+1 . It is then determined whether SV+1 is located on the opposite side of the inflection point Sc fromSv. If Sv+1 is located on the opposite side of the inflection point fromSv , then Sv+l is set to S1, a modified new estimate. The modified new estimate, S1. is preferably set to cither the inflection point. S1, or to an average value between Sv and S v+l , i.e. . S1 ;-- 0.5( Sv+ S V+1). The above steps are repeated until SV+1 is within the predetermined convergence criteria. Also, solution algorithms are described for two-phase and three-phase flow with gravity and capillary pressure.