Rapid modeling of borehole measurements of nuclear magnetic resonance via spatial sensitivity functions

Borehole measurements of nuclear magnetic resonance (NMR) are routinely used to estimate in situ rock and fluid properties. Conventional NMR interpretation methods often neglect bed-boundary and layer-thickness effects in the calculation of fluid volumetric concentrations and NMR relaxation-diffusion correlations. Such effects introduce notable spatial averaging of intrinsic rock and fluid properties across thinly bedded formations or in the vicinity of boundaries between layers exhibiting large property contrasts. Forward modeling and inversion methods can mitigate the aforementioned effects and improve the accuracy of true layer properties in the presence of mud-filtrate invasion and borehole environmental effects across spatially complex formations. We have developed a fast and accurate algorithm to simulate borehole NMR measurements using the concept of spatial sensitivity functions (SSFs) that honor NMR physics and incorporate tool, borehole, and formation geometry. Tool sensitivity maps are derived from a 3D multiphysics forward model that couples NMR tool properties, magnetization evolution, and electromagnetic propagation. In addition, a multifluid relaxation model based on Brownstein-Tarr’s equation is introduced to estimate layer NMR porosity decays and relaxation-diffusion correlations from pore-size-dependent rock and fluid properties. The latter model is convolved with the SSFs to reproduce borehole NMR measurements. The results indicate that NMR spatial sensitivity is controlled by porosity, electrical conductivity, excitation pulse duration, and tool geometry. We benchmark and verify the SSF-derived forward approximation against 3D multiphysics simulations for a series of synthetic cases with variable bed thickness and petrophysical properties, and in the presence of mud-filtrate invasion in a vertical well. Results indicate that the approximation can be executed in a few seconds in a central processing unit, by a factor of 1000 times faster than rigorous multiphysics calculations, with maximum root-mean-square errors of 1%.