Modified Newton-Raphson Algorithm for Stable S-Shape Convergence
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Solution Overview
Problem
Implicit solution algorithms for solving non-linear transport problems with S-shaped flux functions in reservoir simulations are not unconditionally stable, leading to convergence issues when large time steps are used, resulting in redundant computation or time step cuts.
Innovation Solution
A modified Newton-Raphson scheme that checks if the iterative value is on the opposite side of the inflection point and applies a modified estimate, either setting it to the inflection point or using selective under-relaxation, to ensure unconditional stability and convergence.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If the Newton-Raphson method is used to solve S-shaped non-linear functions in reservoir simulations, then the solution convergence is achieved for convex, concave, or linear flux functions, but the method becomes unstable and fails to converge when large time steps are used with S-shaped flux functions
Solution Approach 1:
The patent modifies the Newton-Raphson iteration formula by introducing a parameter α that adapts based on the convexity/concavity of the flux function. When the flux function is detected to be S-shaped (changing convexity), the parameter α adjusts the iteration step to prevent overshooting the inflection point, thereby maintaining numerical stability while allowing larger time steps compared to the conventional method.
Solution Approach 2:
The patent implements a feedback mechanism that monitors the convexity/concavity of the flux function during the iteration process. By detecting changes in the second derivative of the flux function, the algorithm dynamically adjusts the iteration parameter α to ensure the solution remains on the correct side of the inflection point, preventing divergence and maintaining convergence reliability.
2Reliability
If empirical time step control techniques are used to overcome convergence issues with S-shaped flux functions, then convergence is achieved, but the time step size becomes conservative or too large, resulting in redundant computation or time step cuts
Solution Approach 1:
The patent makes the iteration parameter α dynamic rather than fixed, adjusting it based on the local convexity/concavity characteristics of the flux function. This dynamic adaptation allows the algorithm to take larger steps when appropriate (improving productivity) while automatically reducing steps when approaching inflection points (maintaining reliability), eliminating the need for conservative empirical time step control.
3Device complexity
If the conventional Newton-Raphson method is applied without modification to S-shaped functions, then the algorithm remains simple and straightforward, but unconditional stability is lost when time steps are too large
Solution Approach 1:
The patent introduces a single adaptive parameter α into the Newton-Raphson formula, which changes based on the convexity/concavity detection of the flux function. This minimal modification maintains the algorithm's simplicity and straightforward implementation while restoring unconditional stability, avoiding the need for complex alternative methods.
Data Source
AI summary
An apparatus and method are provided for solving a non-linear S-shaped function F=ƒ(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 ƒ(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 from Sv. If Sv+1 is located on the opposite side of the inflection point from Sv, then Sv+1 is set to Sl, a modified new estimate. The modified new estimate, Sl, is preferably set to either the inflection point, Sc, or to an average value between Sv and Sv+1, i.e., Sl=0.5(Sv+Sv+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.