Borehole NMR forward modeling based on Bloch equation and spatial sensitivity distribution

Abstract Nuclear magnetic resonance (NMR) logging is based on the interaction between electromagnetic fields generated by downhole tools and hydrogen nuclear spins in formation fluids within porous rocks. This study proposes a novel forward simulation framework that integrates the Bloch equation with three-dimensional spatial sensitivity analysis. The methodology comprises three core components: (ⅰ) finite element modeling of the static magnetic field (B0) and the radiofrequency field (B1); (ⅱ) recursive numerical solutions of the Bloch equation to simulate magnetization dynamics, including spin excitation, precession, and relaxation; and (ⅲ) calculation of the spatial sensitivity function, which quantifies the contribution of each volumetric element to the detected signal, with an examination of the influence of temperature-dependent quality factors (Q) and pulse sequence parameters on this function. The results demonstrate that the proposed framework quantitatively reproduces the coupled effects of formation thickness, borehole inclination, and tool motion rate on NMR logging responses. It reveals that high-angle wells effectively enhance thin-layer resolution, while excessive motion rate leads to a shift in T2 spectrum and a reduction in thin-layer resolution. This integrated approach provides a workflow for predicting NMR responses under complex downhole conditions, supporting the optimization of tool design and logging strategies in heterogeneous and unconventional reservoirs.