A deconvolution-based method for nuclear magnetic resonance T₁-T₂ inversion

Nuclear magnetic resonance (NMR) relaxation data are typically analyzed by different methods from simple multi-exponential nonlinear fitting to ill-posed linear inversion. However, linear sampling with a constant echo spacing captures signals from short T₂ components only in the first few echoes when measuring samples containing very short T₂ components such as organic matter or bitumen. Consequently, fast decay of short relaxation signals leads to incompleteness of short T₂ components in two-dimensional Laplace inversion. Based on the conversion of multi-exponential nonlinear fitting to a linear inversion problem, this study further converts the linear inversion problem into a deconvolution problem. A uniform logarithmic fitting and regularized deconvolution (LFD) algorithm is proposed, effectively addressing the incomplete and inaccurate recovery of short T₂ components in traditional T₁-T₂ inversion algorithms. Specifically, the uniform logarithmic fitting method is employed to reconstruct both echo data and the inversion kernel function, which mitigates the ill-conditioned nature of the conventional inversion and improves computational efficiency. Numerical simulations and core experiments indicate that the proposed method significantly improves the accuracy of short T₂ component inversion in T₁-T₂ maps: for the shale model example, the relative error for kerogen saturation decreases from 20.20% to 2.60%, for bitumen from 14.97% to 1.90%, and for clay bound water saturation from 5.85% to 0.90%. Applicable conditions are also discussed.