A novel inversion method of 2D TD-NMR signals based on realizing unconstrained maximization of objective function

The inversion of time-domain nuclear magnetic resonance (TD-NMR) signals is an ill-posed problem, which presents enormous challenges for the inversion algorithm. We propose a novel inversion method that converts conventional minimum objective function with non-negative constraints into an unconstrained maximization problem in the inversion of TD-NMR signals. Hence, the objective function becomes a differentiable concave function that can be solved more easily. The validity of the proposed method was verified by the uncertainty estimation of NMR inversion spectra with different signal-to-noise ratios (SNR). Through the inversion of simulated 2D D-T₂ and T₁-T₂ signals under different SNR, the proposed method was proved to be less sensitive to noise than the conventional inversion method. We use the proposed method to study the migrations of oil and water in shales, the components change in shale could be identified and quantified according to the 2D T₁-T₂ inversion spectra. The proposed method was also used to analyze the hydration process of cement. The 2D T₁-T₂ inversion spectra could distinctly present the component of tiny volume with short relaxation time, and the migration regularity of capillary water, gel water, and bound water could also be found. In conclusion, the proposed method could be a reliable method to invert TD-NMR signals, especially the identification of the 2D NMR signals with a short relaxation time in low SNR.