NMR T2 Distribution Estimation via Integral Transforms
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Solution Overview
Problem
The estimation of petrophysical and fluid properties from nuclear magnetic resonance (NMR) data is challenged by the ill-conditioned and non-linear nature of the T2 distribution estimation, leading to non-unique solutions and susceptibility to noise in the data.
Innovation Solution
The use of integral transforms allows for direct computation of linear functionals of the T2 distribution without inverting the T2 distribution function, employing methods such as analytical forms, numerical approaches, and convolution analysis to estimate parameters like porosity and permeability, thereby reducing uncertainty and instability.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If the conventional inverse Laplace transform method is used to estimate the T2 distribution, then the estimation process can be performed, but the solution is ill-conditioned and non-unique, leading to high sensitivity to noise and instability
Solution Approach 1:
The patent applies preliminary action by using integral transforms to compute linear functionals of the T2 distribution directly from the NMR data before any inversion process. This preliminary computation of moments and other linear functionals provides stable, noise-resistant estimates that constrain the subsequent inversion process, preventing the ill-conditioned nature of direct inversion while still obtaining the T2 distribution.
Solution Approach 2:
The patent introduces integral transforms as an intermediary between the raw NMR data and the T2 distribution estimation. Instead of directly inverting the Laplace transform, the method uses integral transforms to compute intermediate linear functionals (moments, tapered areas) that serve as stable constraints, thereby mediating the inversion process and reducing sensitivity to noise.
2Reliability
If regularization is applied to obtain a smooth T2 distribution, then a unique solution can be obtained, but additional assumptions and parameters must be introduced
Solution Approach 1:
The patent performs preliminary computation of linear functionals using integral transforms before the inversion process. These pre-computed moments and tapered areas serve as constraints that guide the inversion toward a unique solution without requiring complex regularization schemes. The preliminary action reduces the solution space in a physically meaningful way.
Solution Approach 2:
The patent changes the approach from regularizing the T2 distribution directly to regularizing the linear functionals computed via integral transforms. By transforming the problem into the moment space and back, the method achieves uniqueness through different mathematical parameters (the integral transform weights) rather than through traditional regularization of the distribution itself.
3Loss of information
If direct inversion of the T2 distribution is performed, then the full distribution can be obtained, but the process is computationally intensive and sensitive to measurement noise
Solution Approach 1:
The patent extracts the essential linear functionals (moments, tapered areas) from the full T2 distribution using integral transforms. By taking out these key characteristics directly from the NMR data through stable integral computations, the method obtains the most important distribution properties without performing the full, noise-sensitive inversion process.
Solution Approach 2:
The patent creates a simplified copy of the T2 distribution information through its linear functionals (moments and tapered areas). Instead of working with the complete, noisy distribution, the method uses these copied statistical properties that can be computed stably and then used to reconstruct or constrain the full distribution, reducing noise impact while preserving essential information.
Applied Scientific Principles
This section explains which scientific principles are used to turn an abstract innovation direction into a practical engineering solution.
Function Achieved in This Case
This approach provides a more straightforward and stable method for estimating petrophysical properties, reducing the impact of noise and improving the accuracy of parameters like porosity and permeability, and can be applied to various NMR data types, including fully and imperfectly polarized data.
Implementation Method 1
Measured nuclear magnetic resonance (NMR) data resulting from a multi-component sample can be denoted by G(t) which represents a multi-exponential decay, with time constants T2
Implementation Method 2
The conventional approach to estimating ƒ(T2) utilizes an inverse Laplace transform (ILT)
Data Source
AI summary
A nuclear magnetic resonance (NMR) related distribution is estimated that is consistent with NMR measurements and uses linear functionals directly estimated from the measurement indications by integral transforms as constraints in a cost function. The cost function includes indications of the measurement data, Laplace transform elements and the constraints, and a distribution estimation is made by minimizing the cost function. The distribution estimation may be used to find parameters of the sample. Where the sample is a rock or a formation, the parameters may include parameters such as rock permeability and/or hydrocarbon viscosity, bound and free fluid volumes, among others. The parameters may be used in models, equations, or otherwise to act on the sample, such as in recovering hydrocarbons from the formation.