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

摘要

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 Newton iteration (T) is performed on the function f(S) at S<SUP>V</SUP> to determine a next iterative value S<SUP>V+1 </SUP>. It is then determined whether S<SUP>V+1</SUP> is located on the opposite side of the inflection point S<SUP>C</SUP> fromS<SUP>V</SUP>. If S<SUP>V+1 </SUP>is located on the opposite side of the inflection point fromS<SUP>V</SUP> , then S<SUP>V+1</SUP>is set to S<SUP>1</SUP>, a modified new estimate. The modified new estimate, S<SUP>1</SUP>, is preferably set to either the inflection point, S<SUP>C</SUP>, or to an average value between S<SUP>V</SUP> and S <SUP>V+1</SUP> , i.e. , S<SUP>1</SUP> = 0.5( S<SUP>V</SUP>+ S <SUP>V+1</SUP>). The above steps are repeated until S<SUP>V+1</SUP>is within the predetermined convergence criteria. Also, solution algorithms are described for two-phase and three-phase flow with gravity and capillary pressure.