A mathematical model of NMR transverse relaxation for pore size distribution estimation in porous media

Nuclear Magnetic Resonance (NMR) is a widely useful technique for studying porous media. Of particular interest are transverse relaxation times ($T_2$), which are often associated with pore size when surface relaxation is the dominant mechanism. Under specific physical assumptions, a distribution of $T_2$ can be used to infer the pore size distribution (PSD). However, in real porous rocks, a combination of diffusion and relaxation mechanisms complicates this interpretation. Despite recent advancements in industrial applications, conventional models frequently rely on simplifying assumptions, particularly when pore size is considered in the fast diffusion regime. This results in the neglect of transverse bulk relaxation ($T_{2B}$) effects, leading to underestimations of pore sizes. To address this, numerical methods, particularly the Finite Element Method (FEM), offer flexibility in modeling symmetric geometries while significantly reducing computational complexity. This paper presents a mathematical NMR model and numerical implementation based on FEM to simulate transverse magnetization signals for a PSD, validated with semi-analytical solutions and applied to synthetic and real samples, such as Berea sandstone. Additionally, a change of variable in the Inverse Laplace Transform (ILT) model is introduced for the direct PSD estimation, demonstrating a strong agreement between experimental and simulated data.