Effect of inhomogeneous surface relaxivity, pore geometry and internal field gradient on NMR logging:exact and perturbative theories and numerical investigations

Abstract

Nuclear magnetic resonance is widely used as a probe of pore geometry and fluid composition in well logging. One of the critical assumptions often made is that the diffusion of fluid molecules is sufficiently fast to warrant the condition for direct mapping between the surface-enhanced relaxation rate and the pore geometry. In pores satisfying such a condition, but having a significant spatial variation of surface relaxivity (rho), one can show that the one-to-one mapping may break down. The degree to which the NMR logging interpretation is affected has not been systematically studied until now. In this work, we theoretically investigate the interplay between the pore geometry, internal field and the inhomogeneous surface relaxivity. We develop a perturbative framework and compare its results with exact solutions obtained for a class of rho textures in a pore with simple geometry. Its effect is quantified for a wide range of diffusivity and/or rho strength. The result allows us to set the bounds for the change in the final slope of the relaxation curve and may serve as a useful guide for logging applications in real rocks with a wide range of pore sizes and fluid diffusivity. We further employ large scale numerical simulations to perform virtual experiments on more complex situations. Internal field and its gradient distributions were obtained and analyzed for up to 1.5^3 cm^3 based on tomograms of carbonate rocks. We find that the texture of rho based on the internal field gradient induces a small, but observable shift, compared to a random noise generated texture for which no shift is observed.