Preconditioner for Reservoir Simulation Jacobian Matrices
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Solution Overview
Problem
Reservoir simulation algorithms, particularly in the oil/gas industry, face significant computational bottlenecks due to the inversion of large Jacobian matrices, which are sparse but have high condition numbers, leading to inefficient convergence in solving linear systems, especially in massively parallel supercomputing environments.
Innovation Solution
A purely algebraic two-level preconditioning method is introduced, where each processor independently determines eigenvalues and eigenvectors for subsets of the Jacobian matrix, constructing a projector Z that reduces the condition number of the preconditioned system, allowing for parallelization without inter-processor communication and enabling efficient hydrocarbon production determination.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If classical iterative methods are used to solve the linear system with the Jacobian matrix, then the solution can be obtained, but the convergence is slow due to the high condition number of the sparse matrix
Solution Approach 1:
The patent introduces a preconditioner M as an intermediary transformation that modifies the original linear system Ax=b into a preconditioned system with better spectral properties. The preconditioner acts as a mediator that transforms the ill-conditioned Jacobian matrix into a well-conditioned system, enabling faster convergence of iterative solvers without changing the underlying physics or solution.
Solution Approach 2:
The patent changes the parameters of the linear system by applying a preconditioning transformation. Specifically, it modifies the matrix A and vector b into M^-1A and M^-1b (or AM^-1 and b), which changes the spectral distribution and condition number of the system, thereby improving the convergence rate of iterative methods while preserving the solution x.
2Measurement precision
If the Jacobian matrix is inverted using sequential methods, then accurate solutions can be obtained, but the computational efficiency is low in parallel environments
Solution Approach 1:
The patent segments the computation of the projector Z into independent tasks that can be distributed across multiple processors. Each processor independently computes a portion of the projector without requiring inter-processor communication, enabling efficient parallelization while maintaining solution accuracy. This segmentation transforms a sequential bottleneck into a parallelizable operation.
3Loss of time
If a two-level preconditioning method with parallel projector determination is used, then computational time is reduced, but the device complexity increases
Solution Approach 1:
The patent performs preliminary computation of the projector Z in parallel before the main iterative solving process. By pre-computing the projector that defines the preconditioning transformation, the method prepares the system in advance to enable faster convergence during the actual solution phase, trading off some initial computational effort for significant time savings during iteration.
Data Source
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AI summary
The present invention relates to a method for determining hydrocarbon production for a reservoir, wherein the method comprises: - determining a projector matrix (Z) based on a Jacobian matrix (201) function of the gridded model, said determining of the projector matrix (Z) comprising: - splitting the Jacobian matrix (201) into subsets of consecutive lines (202), each subset of consecutive lines having a respective order value function of an index of a line in said subset; - for each subset of consecutive lines: - creating a respective square matrix based on said subset; - determining eigenvectors ( ν 1 , ν 2,..., νn ) and respective eigenvalues (λ 1, λ 2,..., λ n ) A 2, associated with the respective square matrix - determining relevant eigenvectors having respective eigenvalues below a predetermined threshold (0), - determining the projector (Z) as a concatenation of the extended eigenvectors ordered according to multiple criteria: - firstly, the respective order value of the subset; - secondly, the respective eigenvalue of the relevant eigenvector.