Reservoir Pressure Solver for Indefinite Matrices
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Solution Overview
Problem
Current reservoir simulation methods face challenges in achieving convergence when dealing with indefinite coefficient matrices, leading to costly time step reduction and divergence of iterative methods, especially in large-scale reservoir simulations with heterogeneous rock formations.
Innovation Solution
A computer-implemented method and system that determines pressure distribution using an approximate analytical preconditioner and Krylov vectors, allowing the simulation to proceed without reducing the time step size, even with indefinite coefficient matrices, by forming an initial computer matrix and vector of reservoir attributes and applying an approximate analytical preconditioner to generate Krylov vectors for determining fluid flow rates.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If iterative methods are used to solve pressure distribution in reservoir simulation, then convergence can be achieved for diagonally dominant matrices, but the method diverges when encountering indefinite coefficient matrices
Solution Approach 1:
The patent introduces an indefinite matrix detector as an intermediary component that identifies when the coefficient matrix becomes indefinite. This detector acts as a mediator between the iterative solver and the time step controller, triggering appropriate responses (such as reducing time step size) only when needed, rather than applying conservative measures continuously. This resolves the contradiction by enabling the iterative method to converge reliably for diagonally dominant matrices while adapting its behavior when indefinite matrices are detected.
Solution Approach 2:
The patent implements dynamic adjustment of simulation parameters based on the properties of the coefficient matrix. The time step size is adjusted dynamically - maintained at larger values when the matrix is diagonally dominant (ensuring fast convergence) and reduced when indefinite matrices are detected (preventing divergence). This dynamic adaptation resolves the contradiction between maintaining fast convergence and handling indefinite matrices robustly.
2Reliability
If time step size is reduced to handle indefinite matrices, then convergence can be achieved, but computational cost increases significantly
Solution Approach 1:
The indefinite matrix detector serves as an intermediary that identifies the specific condition (indefinite matrix) causing convergence failure. By detecting this condition precisely, the system can apply time step reduction only when necessary, rather than continuously. This resolves the contradiction by maintaining fast simulation speed for the majority of time steps where the matrix is diagonally dominant, while only reducing time step size when indefinite matrices are actually encountered.
Solution Approach 2:
The patent changes the time step parameter dynamically based on the matrix properties. When the coefficient matrix is diagonally dominant, larger time steps are used for efficient simulation. When indefinite matrices are detected, the time step size is reduced to ensure convergence. This parameter adaptation resolves the contradiction between computational efficiency and convergence reliability.
3Reliability
If conservative time step reduction is applied continuously, then convergence is ensured for all cases, but productivity of reservoir simulation decreases
Solution Approach 1:
The indefinite matrix detector acts as a selective trigger that distinguishes between cases requiring conservative time step reduction and cases where larger time steps can be used. By introducing this detection mechanism, the system avoids continuous conservative reduction and instead applies it only when indefinite matrices are detected. This resolves the contradiction by maintaining high productivity for diagonally dominant matrices while ensuring convergence when needed.
Solution Approach 2:
The patent applies time step reduction partially - only when indefinite matrices are detected - rather than excessively applying it continuously. This partial action is sufficient to ensure convergence for indefinite matrices while avoiding the productivity loss that would result from continuous conservative reduction. This resolves the contradiction between convergence guarantee and simulation throughput.
Data Source
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AI summary
Performance of computers is improved during determination of pressure distribution among grid cells of a reservoir model during reservoir simulation by computer processing. Convergence can prove difficult to obtain when conditions cause a coefficient matrix involved in the processing to become indefinite. An indefinite coefficient matrix can occur either due to physical conditions in the reservoir related to vapor liquid equilibria, or due to nonphysical conditions created numerically due to improper derivatives. The conventional previously taken corrective action of time step cutting is avoided when convergence becomes difficult during reservoir simulation. Time step cutting has proven to be very costly, in terms of computer usage and time, for very large reservoirs having models involving millions, billions or more number of unknown parameter values.