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

摘要

STABLE METHOD AND APPARATUS FOR SOLVING S-SHAPED NON-LINEAR FUNCTIONS UTILIZING MODIFIED NEWTON-RAPHSON ALGORITHMSAbstract An apparatus and method are provided for solving a non-linear S-shaped function F = f(S) which is representative of a property S in a physical system, such saturation in a reservoir simulation. A Newtoniteration (T) is performed on the function f(S) at Sv to determine a next iterative value Sv+'. It is then determined whether Sv" 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+'is set to S', a modified new estimate. The modified new estimate, S', is preferably set to either the inflection point, Sc, or to an average value between Sv and Sv-', i.e., S1 = 0.5( Sv+ S v"). The above steps are repeated until Sv+i is within the predeterminedconvergence criteria. Also, solution algorithms are described for two-phase and three-phase flow with gravity and capillary pressure.