SWIFT Preconditioner for Reservoir Simulation Linear Systems
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Solution Overview
Problem
Current methods for solving linear systems of equations in reservoir simulation, such as those used in the petroleum industry, face inefficiencies and slow convergence rates, especially when dealing with large-scale, highly non-symmetric fully implicit or adaptive implicit matrices, due to the lack of effective exploitation of sparsity and adaptive modification of factorization patterns.
Innovation Solution
The implementation of a SWIFT (Strong-Weak Incomplete Factorization with Thresholds) preconditioner, which applies block diagonal scaling and threshold incomplete lower-upper approximate factorization to obtain a preconditioning matrix, allowing for iterative solution of linear systems without pivoting and improving convergence rates.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If direct methods (Gaussian elimination) are used to solve linear systems, then solution accuracy is improved, but computational cost becomes prohibitively expensive for large sparse matrices
Solution Approach 1:
The patent extracts only the essential diagonal elements and immediate off-diagonal elements to form an incomplete factorization, discarding the need to compute and store all matrix elements. This extraction creates a sparse approximate factorization that maintains solution accuracy while dramatically reducing computational cost and memory requirements for large reservoir simulation problems.
Solution Approach 2:
The patent uses a disposable incomplete factorization that is computationally inexpensive to construct and store. Rather than investing in expensive full LU decomposition, the method creates a simplified factorization structure that is sufficient for iterative solution and can be easily regenerated if needed, optimizing the trade-off between accuracy and computational resources.
2Productivity
If iterative methods with standard incomplete factorization are used, then computational cost is reduced, but convergence becomes very slow for large systems
Solution Approach 1:
The patent introduces adaptive implicit time-stepping that dynamically adjusts the factorization pattern based on problem characteristics. The incomplete factorization adapts its structure during iteration, allowing the solver to converge faster by modifying the factorization pattern to better match the current state of the reservoir simulation, thus reducing convergence time while maintaining computational efficiency.
Solution Approach 2:
The patent changes the parameters of the incomplete factorization by allowing variable block-size and adaptive infill criteria. By adjusting these parameters dynamically during the solution process, the method optimizes the balance between factorization accuracy and sparsity, leading to faster convergence without significantly increasing computational cost.
3Measurement precision
If full implicit time discretization is used, then solution accuracy is improved, but matrix size and computational complexity increase
Solution Approach 1:
The patent extracts only the necessary diagonal and immediate off-diagonal elements from the full implicit matrix to create an incomplete factorization. This extraction preserves the essential physics and solution accuracy while removing the need to handle the full complex matrix structure, thereby reducing computational complexity for large-scale reservoir simulations.
Solution Approach 2:
The patent segments the matrix into variable-sized blocks corresponding to different grid cells and their neighbors. This segmentation allows the incomplete factorization to handle each block independently with appropriate computational resources, reducing overall complexity while maintaining the accuracy required for full implicit time discretization.
4Measurement precision
If adaptive implicit time-stepping is used, then flexibility and accuracy are improved, but matrix structure becomes variable block-size increasing solver complexity
Solution Approach 1:
The patent uses dynamic adaptive implicit time-stepping where the factorization pattern automatically adjusts to match the variable block-size matrix structure. This dynamic adaptation allows the solver to handle the complexity of adaptive time-stepping without requiring separate treatment for each matrix type, maintaining solver flexibility while managing complexity through automated pattern adjustment.
Data Source
AI summary
A computer-implemented system, method, and software are provided for solving linear systems of equations arising in reservoir simulation. A linear system of fluid flow equations, including a plurality of unknown variables, is provided that represents fluid flow properties in a geological formation of a subterranean reservoir. Block diagonal scaling is applied to the linear system of fluid flow equations. Threshold incomplete lower-upper approximate factorization is applied to obtain a preconditioning matrix, which can be used to iteratively solve unknown variables associated with the system of fluid flow equations. Preconditioning can be used directly as part of a preconditioned Krylov subspace method or alternatively as a 2nd stage preconditioner in a CPR algorithm.