NMR Integral Transforms for Stable Petrophysical Estimation
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Solution Overview
Problem
The estimation of petrophysical and fluid properties from nuclear magnetic resonance (NMR) measurements is challenging due to the ill-conditioned and non-linear nature of the problem, leading to instability and subjectivity in estimating relaxation time distributions, especially with the presence of noise in the data.
Innovation Solution
The method involves computing integral transforms directly on the measured NMR data in the measurement domain to estimate linear functionals of relaxation times and diffusion coefficients, bypassing the need to first compute the distribution functions, using techniques such as Mellin transforms and convolution-multiplication equivalence to provide more stable and accurate results.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If traditional inversion algorithms are used to estimate relaxation time distributions from NMR magnetization data, then the distributions can be obtained, but the estimation becomes ill-conditioned and non-linear leading to instability and subjectivity especially with noisy data
Solution Approach 1:
The patent extracts only the necessary information (linear functionals such as moments, area under curves, mean, width) directly from the NMR magnetization data through integral transforms, rather than computing the complete relaxation time distribution. This extraction approach bypasses the ill-conditioned inversion problem while obtaining the specific petrophysical properties of interest directly from the measured signal.
Solution Approach 2:
The patent transforms the problem from the relaxation time domain to the frequency domain through integral transforms (Laplace, Mellin, Fourier). This dimensional transformation converts the ill-conditioned inversion problem into a more stable calculation where linear functionals can be computed directly from the magnetization decay signal without requiring distribution estimation.
2Ease of operation
If regularization is applied to minimize the cost function to obtain smooth relaxation distributions, then subjectivity is reduced, but the process remains complex and computationally intensive
Solution Approach 1:
Instead of computing the complete distribution function and then deriving properties from it, the patent directly extracts the required linear functionals (moments, area, mean, width) from the NMR signal using integral transforms. This eliminates the need for regularization and complex inversion algorithms while maintaining objectivity, as the functionals are computed directly through stable mathematical operations on the measured data.
3Productivity
If complete relaxation time distributions are computed first, then petrophysical properties can be derived, but the process involves unnecessary computational steps and propagates uncertainty
Solution Approach 1:
The patent directly extracts the required petrophysical information (linear functionals) from the NMR magnetization signal through integral transforms, bypassing the intermediate step of computing the complete relaxation time distribution. This direct extraction path eliminates unnecessary computational steps and prevents uncertainty propagation, as each property is calculated directly from the measured signal through stable mathematical operations.
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 allows for more accurate and stable estimation of petrophysical properties like rock permeability and hydrocarbon viscosity, reducing uncertainty related to noise and providing a straightforward quantification of uncertainty in the estimated parameters, while being applicable to various NMR pulse sequences and downhole applications.
Implementation Method 1
measuring an indication of a formation using nuclear magnetic resonance measurements
Implementation Method 2
The relaxation time T2 is the characteristic time corresponding to loss of energy by protons in hydrocarbons or water present in pores of a rock or in the bulk fluid
Data Source
AI summary
Apparatus and method of characterizing a subterranean formation including observing a formation using nuclear magnetic resonance measurements, calculating an answer product by computing an integral transform on the indications in measurement-domain, and using answer products to estimate a property of the formation. Apparatus and a method for characterizing a subterranean formation including collecting NMR data of a formation, calculating an answer product comprising the data, wherein the calculating comprises a formulaK(x)≡∫0∞k(t)e-t/xdt.and estimating a property of the formation using the answer product.