Acoustic Dispersion Curve Identification via Reciprocal Condition Number
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Solution Overview
Problem
Solving the challenge of determining accurate dispersion curves for acoustic waves in wellbores, which is hindered by the computational instability and complexity of large matrices in existing methods, particularly when modeling multiple thin layers.
Innovation Solution
Employing the reciprocal condition number (RCN) method to measure the singularity of matrices, generating more accurate and detailed dispersion curves by identifying local minima in the RCN, rather than relying on the determinant method, which is unstable for large systems.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If the determinant method is used to solve large matrices for dispersion curves, then the method can handle complex multi-layer systems, but the computational stability deteriorates and results become unreliable
Solution Approach 1:
The patent transforms the mathematical approach by changing from determinant-based singularity detection to reciprocal condition number-based detection. This parameter change in the mathematical method enables stable computation for large matrices representing multi-layer systems while maintaining the ability to identify dispersion curves accurately.
Solution Approach 2:
The patent replaces the traditional determinant method (mechanical approach) with a reciprocal condition number method. This substitution introduces a more numerically stable mathematical approach that handles large matrices from multi-layer acoustic models without suffering from computational instability.
2Device complexity
If traditional matrix methods are used for dispersion curve identification, then the approach is computationally simpler, but the measurement precision and accuracy of dispersion curves deteriorate
Solution Approach 1:
The patent introduces the reciprocal condition number as an intermediary metric between the matrix representation and dispersion curve identification. This intermediary provides a more precise and stable measure of matrix singularity, enabling accurate dispersion curve detection even in complex multi-layer acoustic systems.
3Ease of manufacture
If the determinant method is applied to identify dispersion curves in large matrices, then the process can be implemented with standard algorithms, but the loss of information increases due to numerical instability
Solution Approach 1:
The patent changes the mathematical parameter used for singularity detection from determinant to reciprocal condition number. This parameter change reduces information loss by providing a more numerically stable measure that accurately captures matrix singularity conditions without suffering from the numerical instability that plagues determinant-based methods.
Data Source
AI summary
To generate dispersion curves for acoustic waves in a radially layered system, a matrix M containing solutions to the wave equation subject to the boundary conditions of the system is constructed. The reciprocal condition number (RCN) of the matrix M is determined as a function of acoustic wave frequency and slowness. The local minima of the RCN in the frequency-slowness plane produces the dispersion curves corresponding to allowable acoustic modes in the system. A sensitivity analysis which identifies the dispersion curves dependent on a selected parameter. The dispersion curves independent of the perturbed parameters are eliminated by perturbing the modeling parameters and generating the RCN of the perturbed matrix M and then subtracting the RCN values of the unperturbed matrix M, leaving the dispersion curves that exhibit dependence on the selected parameter.